Research ArticleOptimal Manufacturing-Remanufacturing ProductionPolicy for a Closed-Loop Supply Chain under Fill Rate andBudget Constraint in Bifuzzy Environments

Soumita Kundu1 Tripti Chakrabarti1 and Dipak Kumar Jana2

1 Department of Applied Mathematics University of Calcutta 92 APC Road Kolkata West Bengal 700009 India2Department of Applied Science Haldia Institute of Technology Purba Midnapur West Bengal 721657 India

Correspondence should be addressed to Dipak Kumar Jana dipakjanagmailcom

Received 31 December 2013 Accepted 12 May 2014 Published 24 June 2014

Academic Editor Tamer Eren

Copyright copy 2014 Soumita Kundu et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study a closed-loop supply chain involving a manufacturing facility and a remanufacturing facility The manufacturer satisfiesstochastic market demand by remanufacturing the used product into ldquoas-newrdquo one and producing new products from rawmaterialin the remanufacturing facility and the manufacturing facility respectively The remanufacturing cost depends on the quality ofused product The problem is maximizing the manufacturerrsquos expected profit by jointly determining the collected quantity of usedproduct and the ordered quantity of raw material Following that we analyze the model with a fill rate constraint and a budgetconstraint separately and then with both the constraints Next to handle the imprecise nature of some parameters of the model wedevelop themodel with both constraints in bifuzzy environment Finally numerical examples are presented to illustrate the modelsThe sensitivity analysis is also conducted to generate managerial insight

1 Introduction

In recent years more and more attention has been paidto recycling and remanufacturing of used products dueto the increased environmental concerns reduced wasteand awareness of natural resources limitation worldwideRemanufacturing the used product and then sending themback to market ldquoas-newrdquo product are considered a part ofclosed-loop supply chain (CLSC) along with operations likeacquisitioncollection testing repairingmanufacturing anddistribution Many product categories from car batteries toprinter cartridge and computers can bemade new in thisway

With the integration of a remanufacturing facility ina manufacturing system the complexity is increasing andtherefore also the production planning is getting more chal-lenging to themanufacturer van der Laan et al [1] and Krikke[2] study the production planning and inventory controlproblem for a closed-loop system where manufacturing andremanufacturing operations occur simultaneously All theproducts produced by the manufacturing process and the

remanufacturing process can be used to fulfil customerdemands Two control strategies are analyzed the PUSHstrategy where all returned products are remanufacturedas early as possible the PULL strategy where all returnedproducts are remanufactured as late as it is convenient Inder-furth [3] analyzes the optimal policies to control a hybridmanufacturing-remanufacturing system in which the twooperations are not directly interconnected if remanufactureditems are downgraded andhave to be sold inmarkets differentfrom those for new products But in case of a shortage ofremanufactured products brand-new products can be usedto substitute the remanufactured ones Dobos and Richter[4] discussed a similar system in which the disposal optionof the returned items is allowed and all the recycling batchesfollow the production batches Choi et al [5] present a jointEOQ and EPQ model for an inventory control problem in aclosed-loop system in which a stationary demand is satisfiedby recovered products and newly purchased products Rubioand Corominas [6] investigate a reverse logistics systemwhen it is operated in a lean production environment They

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 690435 14 pageshttpdxdoiorg1011552014690435

2 International Journal of Mathematics and Mathematical Sciences

analyze the coordination of capacity between manufacturingand remanufacturing and develop the optimal productionpolicies of the system

One of the core management issues in remanufacturingindustry is to effectivelymatch demand and supply by dealingwith the uncertainty of the quality of the returned productsand of the market demand since the returned products arenot presorted in many cases and the information about theirquality is usually limited to firms The ldquoyield raterdquo which isthe remanufacturable portion of the used item is randomSome comprehensive reviews of the problem can be foundin Yano and Leersquos work [7] Hsu and Bassok [8] obtain theoptimum production quantity by solving a single-periodmultiproduct downward substitution model with randomyields and demand Bollapragada and Morton [9] presentheuristics for the random yield problem of periodicallyreviewed inventory Kazaz [10] studies production planningwith random yield and demand with a particular focus onolive oil production

Competitive pressure in todayrsquos global market is forcingcompanies to offer superior service to customers Customersatisfaction or the ability to effectively respond to customerdemand can be gauged by measuring service level (Steven[11]) Service level is defined in many ways the simplestdefinition is the fraction of orders that are filled on or beforetheir delivery due date (Steven [11]) There are two typesof service level measures The first one is type 1 servicelevel which measures the probability of no stock-out over aplanning period The second one is fill rate which measuresthe fraction of demand that is satisfied immediately from on-hand inventory Zipkin [12] considered a single-stage modelwith a compound Poisson demand process He obtainedexact and approximate fill rate expressions and presentedmethods for minimizing inventory cost subject to a fill rateconstraint Axsater [13] considered the problem of finding anoptimal (119877 119876) policy under a fill rate constraint and normallydistributed lead-time demand and he concluded that thesavings are large for low service levels but small for highservice levels

All the research works discussed above consideredparameters of inventory model as constant or as functionof time or as random variable with known probabilitydistribution that is crisp in nature But in real life themost part of the information about inventory parameters areavailable in imperfect form So it becomes impossible tomakeprecise statement about the different inventory parametersFuzzy set theory by Zadeh [14] is very appropriate tool forhandling these situations Based on these theories if theinventory parameters are treated as fuzzy parameters suchmodel becomes more realistic During last two decades alot of work related inventory problems have been done infuzzy environments (cf Jana et al [15ndash17]) But when wedig into the uncertainty of a fuzzy set there are two casesthe membership is also fuzzy and the element is also fuzzySo there exist a level 2 fuzzy set and type 2 fuzzy set Themathematical properties of fuzzy set of type 2 are investigatedby Zadeh [18ndash20] and Mendel et al [21] The concept of level2 fuzzy set was introduced by Zadeh [22] and was moreelaborated by Gottwald [23] Based on level 2 fuzzy set Xu

and Zhou [24] defined bifuzzy variable and they adopt thethree models (i) bifuzzy EVM (ii) bifuzzy CCM and (iii)bifuzzy DCM to deal with multiobjective decision makingmodels under bifuzzy environment [25]

In this paper we investigate a closed-loop system involv-ing a manufacturing facility and a remanufacturing facilityThe manufacturer satisfies stochastic market demand byremanufacturing the used product into ldquoas-newrdquo one andproducing new products from raw material in the remanu-facturing facility and the manufacturing facility respectivelyThe remanufacturing cost depends on the quality of usedproduct The problem is maximizing the manufacturerrsquosexpected profit by jointly determining the collected quantityof used product and the ordered quantity of raw materialFollowing that we analyze themodel with a fill rate constraintand a budget constraint separately and then with both theconstraints Next to handle the imprecise nature of someparameters of the model we develop the model with bothconstraints in bifuzzy environment Finally numerical exam-ples are presented to illustrate the models The sensitivityanalysis is also conducted to generate managerial insight

2 Bifuzzy Preliminaries

21 Bifuzzy Set A fuzzy set over universal set 119880 is a set ofordered pair

= (119909 120583

119881

(119909)) | forall119909 isin 119880 120583

119881

(119909) gt 0 (1)

where membership function 120583

119881

is of the form

120583

119881

119880 997888rarr [0 1] (2)

Let the fuzzy set as defined above be called ordinaryfuzzy set Ordinary fuzzy set can successfully handle imper-fect information of one single source which is impreciseuncertain or vague But the modelling facilities of fuzzysets are limited to handle imperfect information of two ormore sources of imperfection Various generalizations of theconcept ldquofuzzy setrdquo have been discussed by researchers whichare ldquotype 2 fuzzy setrdquo and ldquolevel 2 fuzzy setrdquo Type 2 fuzzysets are fuzzy sets whose membership grades are themselvesordinary fuzzy sets The membership function of a type 2fuzzy set has the following form

120583

119881

119880 997888rarr weierp ([0 1]) (3)

where weierp([0 1]) denotes the fuzzy power set of [0 1] A level2 fuzzy set 119881 is a fuzzy set whose elements are ordinary fuzzysets The membership function of a level 2 fuzzy set has thefollowing form

120583

119881

weierp (119880) 997888rarr [0 1] (4)

where weierp(119880) denotes the fuzzy power set of the universal set119880A formal definition of level 2 fuzzy set proposed by Gottwald[23] is given as follows

International Journal of Mathematics and Mathematical Sciences 3

Definition 1 A level 2 fuzzy set 119881 defined over a universal set119880 is defined by

119881 = ( 120583

119881

()) | forall isin weierp (119880) 120583

119881

() gt 0 (5)

where each ordinary fuzzy set is defined by

= (119909 120583

119881

(119909)) | forall119909 isin 119880 120583

119881

(119909) gt 0 (6)

For convenience the membership grades 120583

119881

() of the fuzzysets isin weierp(119880) are called ldquoouter-layerrdquo membership gradeswhereas the membership grades 120583

119881

(119909) of the element 119909 isin

119880 are called ldquoinner-layerrdquo membership grades Normallyspeaking a bifuzzy variable is a fuzzy variable with fuzzyparameters

Example 2 120585 = (120585 1205721

1205731

)119871119877

with 120585 = (120585 1205722

1205732

)119871119877

iscalled 119871119877 bifuzzy variable if the outer-layer and inner-layermembership function as follows

120583

120585

(119905) =

119871(120585 minus 119905

1205721

) 120585 minus 1205721

le 119905 le 120585 1205721

gt 0

119877(119905 minus 120585

1205731

) 120585 le 119905 le 120585 + 1205731

1205731

gt 0

120583

120585

(119905) =

119871(120585 minus 119905

1205722

) 120585 minus 1205722

le 119905 le 120585 1205722

gt 0

119877(119905 minus 120585

1205732

) 120585 le 119905 le 120585 + 1205732

1205732

gt 0

(7)

where 120585 is center of 120585 which is also a fuzzy number1205721

1205722

1205731

1205732

are the left and right spread of 120585 and 120585 the basisfunctions119871(119909)119877(119909) are continuous nonincreasing functionsand 119871 119877 [0 1] rarr [0 1] satisfies that 119871(1) = 119877(1) = 0119871(0) = 119877(0) = 1

120572 cut 120585120572

of 119871119877 bifuzzy variable 120585 is 120585120572

= (120585119871120572

120585119877120572

) = [120585 minus

119871minus1(120572)1205721

120585 + 119877minus1(120572)1205731

]

22 Possibility Measure Let Θ be a nonempty set and 119875(Θ)the power set ofΘ For each119860 sube 119875(Θ) there is a nonnegativenumber Pos119860 called its possibility such that

(1) Pos120601 = 0 and PosΘ = 1(2) Poscup

119896

119860119896

= sup119896

Pos119860119896

for any arbitrary collec-tion 119860

119896

in 119875(Θ)

The triplet Θ 119875(Θ)Pos is called possibility space and thefunction Pos is referred to as a possibility measure

Lemma 3 Let 1205851

and 1205852

be two fuzzy variables then

119875119900119904 1205851

ge 1205852

= sup 120583

120585

1

(119906) and 120583

120585

2

(V) | 119906 gt V

119875119900119904 1205851

gt 1205852

= sup 120583

120585

1

(119906) and infV1 minus 120583

120585

2

(V) | 119906 le V (8)

By using the 120572-level of fuzzy variables 1205851

and 1205852

[119898119871120572

119898119877120572

] and[119899119871120572

119899119877120572

] Lemma 3 can be written as

119875119900119904 1205851

ge 1205852

ge 120572 lArrrArr 119898119877

120572

ge 119899119871

120572

119875119900119904 1205851

gt 1205852

ge 120572 lArrrArr 119898119877

120572

ge 119899119877

1minus120572

(9)

221 Chance Operator of Bifuzzy Variables

Definition 4 (Liu [26]) Let 120585 be a bifuzzy variable and 119861 aborel set of 119877 Then the primitive chance of bifuzzy event 120585 isin119861 is a function from (0 1] to [0 1] defined as

Ch 120585 isin 119861 (120572) = supMe119860ge120572

inf120579isin119860

Me 120585 (120579) isin 119861 (10)

Usually we use Pos or Nec to measure the chance of bifuzzyevents

Definition 5 (Xu and Zhou [24]) Let 120585 = 1205851

1205852

120585119899

be abifuzzy vector defined on Θ 119875(Θ)Pos and 119891 119877119899 rarr 119877 isa real valued continuous functionThen the primitive chanceof a bifuzzy event characterized by119891(120585) le 0 is a function from(01] to [0 1] defined as follows

(1) Pos-Pos chance isCh 119891 (120585) le 0 (120572)

= sup120572isin[01]

120579 | Pos 120579 isin Θ | Pos 119891 (120585 (120579)) le 0 ge 120573 ge 120572

(11)

(2) Nec-Nec chance isCh 119891 (120585) le 0 (120572)

= sup120572isin[01]

120579 | Nec 120579 isin Θ | Nec 119891 (120585 (120579)) le 0 ge 120573 ge 120572

(12)

where 120572 120573 isin [0 1] are predetermined confidence level

23 Model for Bifuzzy CCM Based on Pos Measure Let usconsider the following single objective bifuzzy model

Max ℎ (119909 120585)

st 119892119903

(119909 120585) le 0 119903 = 1 2 119901

119909 isin 119883

(13)

where 120585 = (1205851

1205852

120585119899

) is a bifuzzy vector and 119909 =

(1199091

1199092

119909119899

) is a decision vector then the objective func-tion ℎ(119909 120585) and constraint functions 119892

119903

(119909 120585) become bifuzzyvariables 119903 = 1 2 119901

In order to obtain optimistic and pessimistic equivalent ofbifuzzy model (13) we use the chance operator to transformthe fuzzy uncertain model into the crisp model which wecalled bifuzzy chance constrained model (bifuzzy CCM)

The general bifuzzy CCM is as follows

Max 119908

st Ch ℎ (119909 120585) ge 119908 (120577) ge 120575Ch 119892

119903

(119909 120585) le 0 (120578119903

) ge 120579119903

119903 = 1 2 119901

119909 isin 119883

(14)

4 International Journal of Mathematics and Mathematical Sciences

where 120577 120575 120578 120579 are the predetermined confidence level and 119908is the decision variable

We adopt Pos to measure the fuzzy event then thespectrum of chance constrainedmodel based on Posmeasureis as follow

Max 119908

st Pos 120579 | Pos ℎ (119909 120585 (120579)) ge 119908 ge 120575 ge 120577

Pos 120579 | Pos 119892119903

(119909120585 (120579)) le 0 ge 120579

119903

ge 120578119903

119903 = 1 2 119901

119909 isin 119883

(15)

24 Linear Bifuzzy Model For linear bifuzzy single objectivefunction model it is assumed that the combination of fuzzyvariables is linear but not the decisionmaking variable119909 thenthe objective function and constraints are written as linear inbifuzzy variables We consider the linear programming withbifuzzy parameters 119888

119894

119890119903119895

119887119903

Consider

Max 119888119879

ℎ (119909)

st 119890119879

119903

119892119903

(119909) le 0 119903 = 1 2 119901

119909 ge 0

(16)

where ℎ(119909) = (ℎ1

(119909) ℎ2

(119909) ℎ119896

(119909)) and 119892(119909) = (1198921199031

(119909)

1198921199032

(119909) 119892119903119897

(119909)) are 119896 dimensional and 119897 dimensional vec-tors respectively

241 Equivalent Crisp Model Next using the bifuzzy CCMbased on Pos measure with the above model we can get thefollowing model

Max 119908

st Pos 120579 | Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 ge 120577

Pos 120579 | Pos 119890119903

(120579)119879

119892119903

(119909) le119887119903

(120579) ge 120579119903

ge 120578119903

119903 = 1 2 119901

119909 ge 0

(17)

In order to solve model (17) we apply the following twotheorems to transform the chance constrained model into itscrisp model based on Pos-Pos measure

Theorem 6 Assume that 119888(120579) = (1198881

(120579) 1198882

(120579) 119888119896

(120579))119879 is

bifuzzy vector and 119888119894

(120579) is 119871119877 bifuzzy variable denoted by119888119894

(120579) = (119888119894

(120579) 1205721198881198941

1205731198881198941

)119871119877

with fuzzy 119888119894

(120579) = (119888119894

1205721198881198942

1205731198881198942

)119871119877

forany 120579 isin Θ and ℎ

119894

(119909) ge 0 Then 119875119900119904120579 | 119875119900119904119888(120579)119879ℎ(119909) ge 119908 ge120575 ge 120577 is equivalent to

119888119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909) + 119877minus1

(120577) 120573119888119879

2

ℎ (119909) ge 119908 (18)

where 120575 120577 isin [0 1] are predetermined confidence level

Proof For certain 120579 isin Θ 119888119894

(120579) are fuzzy number and itsmember ship function is 120583

119894(120579)

(119905) By extension principle themembership function of fuzzy number 119888(120579)119879ℎ(119909) is

120583

(120579)

119879ℎ(119909)

(119903) =

119871(119888(120579)119879

ℎ (119909) minus 119903

1205721198881198791

ℎ (119909)) 119903 le 119888(120579)

119879

ℎ (119909)

119877(119903 minus 119888(120579)

119879

ℎ (119909)

1205731198881198791

ℎ (119909)) 119903 ge 119888(120579)

119879

ℎ (119909)

(19)

For convenience denote that 119888(120579)119879ℎ(119909) = (119888(120579)119879

ℎ(119909)

1205721198881198791

ℎ(119909) 1205731198881198791

ℎ(119909))119871119877

Since 119888(120579) is also a 119871119877 fuzzy vectorso 119888(120579)119879ℎ(119909) = (119888119879ℎ(119909) 120572119888119879

2

ℎ(119909) 1205731198881198792

ℎ(119909))119871119877

According toLemma 3 we can get

Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 lArrrArr 119888(120579)119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909)

ge 119908

(20)

So for predetermined level 120575 120577 isin [0 1]

Pos 120579 | Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 ge 120577

lArrrArr Pos 120579 | 119888(120579)119879ℎ (119909) ge 119908 minus 119877minus1 (120575) 1205731198881198791

ℎ (119909) ge 120577

lArrrArr 119888119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909) + 119877minus1

(120577) 120573119888119879

2

ℎ (119909) ge 119908

(21)

Theorem 7 Assume that 119890119903119895

(120579) and 119887119903

(120579) are 119871119877 bifuzzyvariables denoted by 119890

119903119895

(120579) = (119890119903119895

(120579) 1205721198901199031198951

1205731198901199031198951

)119871119877

with fuzzy

119890119903119895

(120579) = (119890119903119895

1205721198901199031198952

1205731198901199031198952

)119871119877

and 119887119903

(120579) = (119887119903

(120579) 1205721198871199031

1205731198871199031

)119871119877

withfuzzy 119887

119903

(120579) = (119887119903

1205721198871199032

1205731198871199032

)119871119877

for 119903 = 1 2 119901 119895 = 1 2 119897and 119892

119903119895

(119909) ge 0 Then 119875119900119904120579 | 119875119900119904119890119903

(120579)119879

119892119903

(119909) ge119887119903

(120579) ge

120579119903

ge 120578119903

is equivalent to

119877minus1

(120579119903

) 120573119887

1199031

+ 119871minus1

(120579119903

) 120572119890119879

1199031

119892119903

(119909) minus 119890119879

119903

119892119903

(119909) + 119887119903

+ 119871minus1

(120578119903

) (120572119890119879

1199032

119892119903

(119909) + 120573119887

1199032

) ge 0

(22)

Proof The proof is similar to Theorem 6

3 Notations and Assumptions

The Notations and Assumptions of the proposed models aregiven below

31 Assumption

(i) The brand-new product and remanufactured productare sold at same selling price

(ii) The return rate of used product is infinite

International Journal of Mathematics and Mathematical Sciences 5

SupplierRejection

Inspection

Y gt 120574

Raw material flowUsed product flow

Returned stock

Y lt 1205740

CustomerRemanufacturing

Manufacturing

Remanufacturableinventories

Finished product flow

1205740 le Y lt 120574

Serviceable stock

Figure 1 The frame work of the closed-loop system

(iii) Remanufacturing cost depends on the quality of usedproduct

(iv) Storage capacities of raw material used productmanufactured product and remanufactured productare infinite

(v) Lead time is zero

4 Model Description

In this paper we study a closed-loop system involving amanufacturing facility and a remanufacturing facility Themanufacturer satisfies stochastic market demand by remanu-facturing the used product into ldquoas-newrdquo one and producingnew products from raw material The manufacturer collectsused items by offering low price to customer In reality thecollected used items are of different qualities In the reman-ufacturing facility the used products are inspected carefullyand sorted with respect to the quality of the used productThe products with good quality require less remanufacturingeffort than the products the quality level of which is below 120574but above 120574

0

and the remaining used products are rejectedAfter sorting process the used product is remanufacturedinto as-new product and stocked at serviceable inventorythat is used to satisfy market demand Most of the timethe remanufactured products can not satisfy all the demandThe manufacturer purchases raw material at high price Newproducts are produced at the manufacturing facility Ourobjective is to maximize the manufacturerrsquos expected profitby jointly determining the collected quantity of used productand the ordered quantity of raw material The frame work ofthe system is presented in Figure 1

5 Models Formulation in Crisp Environment

To formulate the problem we first present the unconstrainedmodel to evaluate the optimal order quantities Following thatwe analyze the model with a fill rate constraint and a budgetconstraint separately and then with both the constraints

51 Unconstrained Model (Model-1) Initially when collectedlot of used items confirms good quality standard that isquality of the used items satisfies GQL (with probability 119866

1

)the associate expected profit per period TP

1

(119876 1198760

) is

TP1

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

1119876+119891

0119876

0

[119901 (1198911

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198911

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

1119876+119891

0119876

0

0

[119901119909 + 119903 (1198911

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199011

119876 minus 1199010

1198760

(23)

In the above expected profit function the first term representsthe acquisition price of used product The second termis purchasing cost of raw material The third term is theexpected revenue minus the shortage cost when the demandis higher than production quantity that is 119909 gt 119891

1

119876 + 1198910

1198760

The fourth term is the expected revenue plus the salvage costwhen the demand is lower than production quantity that is119909 lt 1198911

119876+1198910

1198760

the surplus stock can either be offered with adiscount or sold to a secondary market at a unit salvage cost

6 International Journal of Mathematics and Mathematical Sciences

119903 with 119903 lt 119901 The fifth term is remanufacturing cost of useditems The sixth term is manufacturing cost of raw material

When the quality of collected lot of used items is belowGQL but above RQL (with probability 119866

2

) the associateexpected profit per period TP

2

(119876 1198760

) is

TP2

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

2119876+119891

0119876

0

[119901 (1198912

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198912

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

2119876+119891

0119876

0

0

[119901119909 + 119903 (1198912

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199012

119876 minus 1199010

1198760

(24)

Similarly when the quality of collected lot of used itemsis below RQL (with probability 119866

3

) the associate expectedprofit per period TP

3

(119876 1198760

) is

TP3

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

0119876

0

[1199011198910

1198760

minus 119904 (119909 minus 1198910

1198760

)] 119891 (119909) 119889119909

+ int119891

0119876

0

0

[119901119909 + 119903 (1198910

1198760

minus 119909)] 119891 (119909) 119889119909 minus 1199010

1198760

(25)

Therefore combining all possible qualities of collected useditems the total weighted expected profit per period becomes

TP (119876 1198760

) = 1198661

TP1

(119876 1198760

) + 1198662

TP2

(119876 1198760

)

+ 1198663

TP3

(119876 1198760

) (26)

The value of 1198661

1198662

and 1198663

can be estimated from previousquality history Our problem is

Max TP (119876 1198760

) (27)

Proposition 8 (a) The maximum value of 119879119875(1198761198760

) for theproblem (27) is attained for119876lowast and119876lowast

0

by solving the followingsystem of equations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

(28)

(b)The total expected profit TP(119876 1198760

) is concave in119876 and1198760

Proof (a) The first order partial derivatives of (26) withrespect to 119876 and 119876

0

are the following

120597TP (119876 1198760

)

120597119876= 0

120597TP (119876 1198760

)

1205971198760

= 0 (29)

which give (28)(b) The second order partial derivatives of (26) with

respect to 119876 and 1198760

are given below

1205972TP (119876 1198760

)

1205971198762

= minus (119901 + 119904 minus 119903)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

(1198912

119876 + 1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

12059711987620

= minus (119901 + 119904 minus 119903) 1198912

0

times [1198661

119891 (1198911

119876 + 1198910

1198760

) + 1198662

119891 (1198912

119876 + 1198910

1198760

) + 1198663

119891 (1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

1205971198761205971198760

= minus (119901 + 119904 minus 119903) 1198910

times [1198661

1198911

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119891 (1198912

119876 + 1198910

1198760

)]

le 0

(30)

Next the first (|1198671

|) and second (|1198672

|) order determinants ofHessian matrix are

100381610038161003816100381611986711003816100381610038161003816 =

1205972TP (119876 1198760

)

1205971198762lt 0

100381610038161003816100381611986721003816100381610038161003816

=1205972TP (119876 119876

0

)

1205971198762

1205972TP (119876 1198760

)

12059711987620

minus [1205972TP (119876 119876

0

)

1205971198761205971198760

]

2

= 1198912

0

(119901 + 119904 minus 119903)2

times [(1198911

minus 1198912

)2

1198661

1198662

119891 (1198911

119876 + 1198910

1198760

) 119891 (1198912

119876 + 1198910

1198760

)

+ 1198663

119891 (1198910

1198760

)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]]

gt 0

(31)

Therefore TP(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

International Journal of Mathematics and Mathematical Sciences 7

52 Model with Fill Rate Constraint (Model-2) The marketdemand and quality of returned products are uncertain inthe above model thus the manufacturer has to face twotypes of overstocking and understocking risks Under thesecircumstances we analyze the problem of maximizing theexpected overall profit of the hybrid system subject to a fillrate-type customer-service level Fill rate 120573measures the partof stochastic demand that is met from finished new brandproduct Consider

120573 = 1 minusExpected number of stockout unit

mean demand

= 1 minus119878 (119876119876

0

)

120583

(32)

where

119878 (119876 1198760

) = 1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(33)

Hence the resulting optimization model which representsthe maximization of total weighted expected profit subjectedto a fill rate constraint is

Max TP (119876 1198760

)

subject to 120573 ge 1205730

(34)

Proposition 9 (a) The maximum value for problem (34) isattained for 119876lowast 119876lowast

0

and 120582lowast1

by solving the following system ofequations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

(35)

(b)The problem (34) is a convex programming problem in119876 and 119876

0

Proof (a) The Lagrangian relaxation of the problem (1198751

) isthe following

Max 1198711

(119876 1198760

1205821

) (36)

where 1198711

(119876 1198760

1205821

) = TP(119876 1198760

) + 1205821

1198651

(119876 1198760

) and 1205821

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

1205971198711

120597119876= 0

1205971198711

1205971198760

= 0 1205821

1205971198711

1205971205821

= 0 (37)

gives (35)(b) The model (119875

1

) will be a convex programming prob-lem in 119876 and 119876

0

if the objective function to be maximizedshould be concave while the fill rate constraint (119865

1

= 120573 minus

1205730

) should be concave The objective function TP(119876 1198760

) isconcave in 119876 and 119876

0

The first (|119863

1

|) and second (|1198632

|) order determinants ofHessian matrix nabla2119865

1

(119876 1198760

) are

100381610038161003816100381611986311003816100381610038161003816 =

12059721198651

(119876 1198760

)

1205971198762

= minus1

120583

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]

lt 0

100381610038161003816100381611986321003816100381610038161003816 =

1205972

1198651

(119876 1198760

)

1205971198762

1205972

1198651

(119876 1198760

)

12059711987620

minus [1205972

1198651

(119876 1198760

)

1205971198761205971198760

]

2

=1

1205832[1198663

1198912

0

119891 (1198910

1198760

) (1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

)

+ 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

))

+ 1198661

1198662

1198912

0

119891 (1198911

119876 + 1198910

1198760

)

times 119891 (1198912

119876 + 1198910

1198760

) (1198911

minus 1198912

)2

] gt 0

(38)

As a result 1198651

(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

therefore the model 1198751

is a convex programmingproblem

53 Model with Budget Constraint (Model-3) In this sectionwe maximize the manufacturerrsquos expected profit by jointlydetermining the collected quantity of used product andthe ordered quantity of raw material subject to a budgetconstraint If 119861 is the available budget amount for purchasingused product and raw material and converting them to newproduct then problem is reduced to

Max TP (119876 1198760

)

subject to 119888119876 + 119894119876 + 1198880

1198760

+ 1198661

1199011

119876 + 1198662

1199012

119876 + 1199010

1198760

le 119861

(39)

8 International Journal of Mathematics and Mathematical Sciences

Proposition 10 The maximum value for problem (39) isattained for 119876lowast 119876lowast

0

and 120582lowast2

by solving the following system ofequations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205822

(119861 minus 119894119876 minus 119888119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(40)

Proof The Lagrangian relaxation of the problem (39) is thefollowing

Max 1198712

(119876 1198760

1205822

) (41)

where 1198712

(119876 1198760

1205822

) = TP(119876 1198760

) + 1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus

1198661

1199011

119876minus1198662

1199012

119876minus1199010

1198760

) and 1205822

is the Lagrangian multiplierMaximizing the above problem using Kuhn-Tucker condi-tions

1205971198712

120597119876= 0

1205971198712

1205971198760

= 0 1205822

1205971198712

1205971205822

= 0 (42)

gives (40)

54 Model with Both Fill Rate and Budget Constraint (Model-4) In this section we maximize the manufacturerrsquos expectedprofit subject to a fill rate constraint and a budget constraintand the problem reduces to

Max TP (119876 1198760

)

subject to 120573 minus 1205730

ge 0

119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876

minus1198662

1199012

119876 minus 1199010

1198760

ge 0

(43)

Proposition 11 (a) The maximum value for problem (43) isattained for119876lowast119876lowast

0

1205821

and 120582lowast2

by solving the following systemof equations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(44)

Proof The Lagrangian relaxation of the problem (43) is thefollowing

Max 11987112

(119876 1198760

1205821

1205822

) (45)

where 11987112

(119876 1198760

1205821

1205822

) = TP(119876 1198760

) + 1205821

(120573 minus 1205730

) + 1205822

(119861 minus

119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) and 1205822

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

12059711987112

120597119876= 0

12059711987112

1205971198760

= 0 1205821

12059711987112

1205971205821

1205822

12059711987112

1205971205822

= 0

(46)

gives (44)

6 Models in Bifuzzy Environments

To handle the imprecise nature of the parameters 120574 1205740

1205730

and 119861 of the above models we have developed the models inbifuzzy environment

And the bifuzzy variables are triangular 119871119877 bifuzzynumbers and denoted by

120574 (120579) = (120574 (120579) 12057211

12057311

)119871119877

with 120574 (120579) = (120574 12057212

12057312

)

1205740

(120579) = (1205740

(120579) 12057221

12057321

)119871119877

with 1205740

(120579) = (1205740

12057222

12057322

)

1205730

(120579) = (1205730

(120579) 12057231

12057331

)119871119877

with 1205730

(120579) = (1205730

12057232

12057332

)

119861 (120579) = (119861 (120579) 120572

41

12057341

)119871119877

with 119861 (120579) = (119861 12057242

12057342

)

(47)

61 Unconstraint Model (Model-5) In this section 120574 and 1205740

are assumed to be bifuzzy variablesThen the constraints and

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Mathematics and Mathematical Sciences

analyze the coordination of capacity between manufacturingand remanufacturing and develop the optimal productionpolicies of the system

One of the core management issues in remanufacturingindustry is to effectivelymatch demand and supply by dealingwith the uncertainty of the quality of the returned productsand of the market demand since the returned products arenot presorted in many cases and the information about theirquality is usually limited to firms The ldquoyield raterdquo which isthe remanufacturable portion of the used item is randomSome comprehensive reviews of the problem can be foundin Yano and Leersquos work [7] Hsu and Bassok [8] obtain theoptimum production quantity by solving a single-periodmultiproduct downward substitution model with randomyields and demand Bollapragada and Morton [9] presentheuristics for the random yield problem of periodicallyreviewed inventory Kazaz [10] studies production planningwith random yield and demand with a particular focus onolive oil production

Competitive pressure in todayrsquos global market is forcingcompanies to offer superior service to customers Customersatisfaction or the ability to effectively respond to customerdemand can be gauged by measuring service level (Steven[11]) Service level is defined in many ways the simplestdefinition is the fraction of orders that are filled on or beforetheir delivery due date (Steven [11]) There are two typesof service level measures The first one is type 1 servicelevel which measures the probability of no stock-out over aplanning period The second one is fill rate which measuresthe fraction of demand that is satisfied immediately from on-hand inventory Zipkin [12] considered a single-stage modelwith a compound Poisson demand process He obtainedexact and approximate fill rate expressions and presentedmethods for minimizing inventory cost subject to a fill rateconstraint Axsater [13] considered the problem of finding anoptimal (119877 119876) policy under a fill rate constraint and normallydistributed lead-time demand and he concluded that thesavings are large for low service levels but small for highservice levels

All the research works discussed above consideredparameters of inventory model as constant or as functionof time or as random variable with known probabilitydistribution that is crisp in nature But in real life themost part of the information about inventory parameters areavailable in imperfect form So it becomes impossible tomakeprecise statement about the different inventory parametersFuzzy set theory by Zadeh [14] is very appropriate tool forhandling these situations Based on these theories if theinventory parameters are treated as fuzzy parameters suchmodel becomes more realistic During last two decades alot of work related inventory problems have been done infuzzy environments (cf Jana et al [15ndash17]) But when wedig into the uncertainty of a fuzzy set there are two casesthe membership is also fuzzy and the element is also fuzzySo there exist a level 2 fuzzy set and type 2 fuzzy set Themathematical properties of fuzzy set of type 2 are investigatedby Zadeh [18ndash20] and Mendel et al [21] The concept of level2 fuzzy set was introduced by Zadeh [22] and was moreelaborated by Gottwald [23] Based on level 2 fuzzy set Xu

and Zhou [24] defined bifuzzy variable and they adopt thethree models (i) bifuzzy EVM (ii) bifuzzy CCM and (iii)bifuzzy DCM to deal with multiobjective decision makingmodels under bifuzzy environment [25]

In this paper we investigate a closed-loop system involv-ing a manufacturing facility and a remanufacturing facilityThe manufacturer satisfies stochastic market demand byremanufacturing the used product into ldquoas-newrdquo one andproducing new products from raw material in the remanu-facturing facility and the manufacturing facility respectivelyThe remanufacturing cost depends on the quality of usedproduct The problem is maximizing the manufacturerrsquosexpected profit by jointly determining the collected quantityof used product and the ordered quantity of raw materialFollowing that we analyze themodel with a fill rate constraintand a budget constraint separately and then with both theconstraints Next to handle the imprecise nature of someparameters of the model we develop the model with bothconstraints in bifuzzy environment Finally numerical exam-ples are presented to illustrate the models The sensitivityanalysis is also conducted to generate managerial insight

2 Bifuzzy Preliminaries

21 Bifuzzy Set A fuzzy set over universal set 119880 is a set ofordered pair

= (119909 120583

119881

(119909)) | forall119909 isin 119880 120583

119881

(119909) gt 0 (1)

where membership function 120583

119881

is of the form

120583

119881

119880 997888rarr [0 1] (2)

Let the fuzzy set as defined above be called ordinaryfuzzy set Ordinary fuzzy set can successfully handle imper-fect information of one single source which is impreciseuncertain or vague But the modelling facilities of fuzzysets are limited to handle imperfect information of two ormore sources of imperfection Various generalizations of theconcept ldquofuzzy setrdquo have been discussed by researchers whichare ldquotype 2 fuzzy setrdquo and ldquolevel 2 fuzzy setrdquo Type 2 fuzzysets are fuzzy sets whose membership grades are themselvesordinary fuzzy sets The membership function of a type 2fuzzy set has the following form

120583

119881

119880 997888rarr weierp ([0 1]) (3)

where weierp([0 1]) denotes the fuzzy power set of [0 1] A level2 fuzzy set 119881 is a fuzzy set whose elements are ordinary fuzzysets The membership function of a level 2 fuzzy set has thefollowing form

120583

119881

weierp (119880) 997888rarr [0 1] (4)

where weierp(119880) denotes the fuzzy power set of the universal set119880A formal definition of level 2 fuzzy set proposed by Gottwald[23] is given as follows

International Journal of Mathematics and Mathematical Sciences 3

Definition 1 A level 2 fuzzy set 119881 defined over a universal set119880 is defined by

119881 = ( 120583

119881

()) | forall isin weierp (119880) 120583

119881

() gt 0 (5)

where each ordinary fuzzy set is defined by

= (119909 120583

119881

(119909)) | forall119909 isin 119880 120583

119881

(119909) gt 0 (6)

For convenience the membership grades 120583

119881

() of the fuzzysets isin weierp(119880) are called ldquoouter-layerrdquo membership gradeswhereas the membership grades 120583

119881

(119909) of the element 119909 isin

119880 are called ldquoinner-layerrdquo membership grades Normallyspeaking a bifuzzy variable is a fuzzy variable with fuzzyparameters

Example 2 120585 = (120585 1205721

1205731

)119871119877

with 120585 = (120585 1205722

1205732

)119871119877

iscalled 119871119877 bifuzzy variable if the outer-layer and inner-layermembership function as follows

120583

120585

(119905) =

119871(120585 minus 119905

1205721

) 120585 minus 1205721

le 119905 le 120585 1205721

gt 0

119877(119905 minus 120585

1205731

) 120585 le 119905 le 120585 + 1205731

1205731

gt 0

120583

120585

(119905) =

119871(120585 minus 119905

1205722

) 120585 minus 1205722

le 119905 le 120585 1205722

gt 0

119877(119905 minus 120585

1205732

) 120585 le 119905 le 120585 + 1205732

1205732

gt 0

(7)

where 120585 is center of 120585 which is also a fuzzy number1205721

1205722

1205731

1205732

are the left and right spread of 120585 and 120585 the basisfunctions119871(119909)119877(119909) are continuous nonincreasing functionsand 119871 119877 [0 1] rarr [0 1] satisfies that 119871(1) = 119877(1) = 0119871(0) = 119877(0) = 1

120572 cut 120585120572

of 119871119877 bifuzzy variable 120585 is 120585120572

= (120585119871120572

120585119877120572

) = [120585 minus

119871minus1(120572)1205721

120585 + 119877minus1(120572)1205731

]

22 Possibility Measure Let Θ be a nonempty set and 119875(Θ)the power set ofΘ For each119860 sube 119875(Θ) there is a nonnegativenumber Pos119860 called its possibility such that

(1) Pos120601 = 0 and PosΘ = 1(2) Poscup

119896

119860119896

= sup119896

Pos119860119896

for any arbitrary collec-tion 119860

119896

in 119875(Θ)

The triplet Θ 119875(Θ)Pos is called possibility space and thefunction Pos is referred to as a possibility measure

Lemma 3 Let 1205851

and 1205852

be two fuzzy variables then

119875119900119904 1205851

ge 1205852

= sup 120583

120585

1

(119906) and 120583

120585

2

(V) | 119906 gt V

119875119900119904 1205851

gt 1205852

= sup 120583

120585

1

(119906) and infV1 minus 120583

120585

2

(V) | 119906 le V (8)

By using the 120572-level of fuzzy variables 1205851

and 1205852

[119898119871120572

119898119877120572

] and[119899119871120572

119899119877120572

] Lemma 3 can be written as

119875119900119904 1205851

ge 1205852

ge 120572 lArrrArr 119898119877

120572

ge 119899119871

120572

119875119900119904 1205851

gt 1205852

ge 120572 lArrrArr 119898119877

120572

ge 119899119877

1minus120572

(9)

221 Chance Operator of Bifuzzy Variables

Definition 4 (Liu [26]) Let 120585 be a bifuzzy variable and 119861 aborel set of 119877 Then the primitive chance of bifuzzy event 120585 isin119861 is a function from (0 1] to [0 1] defined as

Ch 120585 isin 119861 (120572) = supMe119860ge120572

inf120579isin119860

Me 120585 (120579) isin 119861 (10)

Usually we use Pos or Nec to measure the chance of bifuzzyevents

Definition 5 (Xu and Zhou [24]) Let 120585 = 1205851

1205852

120585119899

be abifuzzy vector defined on Θ 119875(Θ)Pos and 119891 119877119899 rarr 119877 isa real valued continuous functionThen the primitive chanceof a bifuzzy event characterized by119891(120585) le 0 is a function from(01] to [0 1] defined as follows

(1) Pos-Pos chance isCh 119891 (120585) le 0 (120572)

= sup120572isin[01]

120579 | Pos 120579 isin Θ | Pos 119891 (120585 (120579)) le 0 ge 120573 ge 120572

(11)

(2) Nec-Nec chance isCh 119891 (120585) le 0 (120572)

= sup120572isin[01]

120579 | Nec 120579 isin Θ | Nec 119891 (120585 (120579)) le 0 ge 120573 ge 120572

(12)

where 120572 120573 isin [0 1] are predetermined confidence level

23 Model for Bifuzzy CCM Based on Pos Measure Let usconsider the following single objective bifuzzy model

Max ℎ (119909 120585)

st 119892119903

(119909 120585) le 0 119903 = 1 2 119901

119909 isin 119883

(13)

where 120585 = (1205851

1205852

120585119899

) is a bifuzzy vector and 119909 =

(1199091

1199092

119909119899

) is a decision vector then the objective func-tion ℎ(119909 120585) and constraint functions 119892

119903

(119909 120585) become bifuzzyvariables 119903 = 1 2 119901

In order to obtain optimistic and pessimistic equivalent ofbifuzzy model (13) we use the chance operator to transformthe fuzzy uncertain model into the crisp model which wecalled bifuzzy chance constrained model (bifuzzy CCM)

The general bifuzzy CCM is as follows

Max 119908

st Ch ℎ (119909 120585) ge 119908 (120577) ge 120575Ch 119892

119903

(119909 120585) le 0 (120578119903

) ge 120579119903

119903 = 1 2 119901

119909 isin 119883

(14)

4 International Journal of Mathematics and Mathematical Sciences

where 120577 120575 120578 120579 are the predetermined confidence level and 119908is the decision variable

We adopt Pos to measure the fuzzy event then thespectrum of chance constrainedmodel based on Posmeasureis as follow

Max 119908

st Pos 120579 | Pos ℎ (119909 120585 (120579)) ge 119908 ge 120575 ge 120577

Pos 120579 | Pos 119892119903

(119909120585 (120579)) le 0 ge 120579

119903

ge 120578119903

119903 = 1 2 119901

119909 isin 119883

(15)

24 Linear Bifuzzy Model For linear bifuzzy single objectivefunction model it is assumed that the combination of fuzzyvariables is linear but not the decisionmaking variable119909 thenthe objective function and constraints are written as linear inbifuzzy variables We consider the linear programming withbifuzzy parameters 119888

119894

119890119903119895

119887119903

Consider

Max 119888119879

ℎ (119909)

st 119890119879

119903

119892119903

(119909) le 0 119903 = 1 2 119901

119909 ge 0

(16)

where ℎ(119909) = (ℎ1

(119909) ℎ2

(119909) ℎ119896

(119909)) and 119892(119909) = (1198921199031

(119909)

1198921199032

(119909) 119892119903119897

(119909)) are 119896 dimensional and 119897 dimensional vec-tors respectively

241 Equivalent Crisp Model Next using the bifuzzy CCMbased on Pos measure with the above model we can get thefollowing model

Max 119908

st Pos 120579 | Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 ge 120577

Pos 120579 | Pos 119890119903

(120579)119879

119892119903

(119909) le119887119903

(120579) ge 120579119903

ge 120578119903

119903 = 1 2 119901

119909 ge 0

(17)

In order to solve model (17) we apply the following twotheorems to transform the chance constrained model into itscrisp model based on Pos-Pos measure

Theorem 6 Assume that 119888(120579) = (1198881

(120579) 1198882

(120579) 119888119896

(120579))119879 is

bifuzzy vector and 119888119894

(120579) is 119871119877 bifuzzy variable denoted by119888119894

(120579) = (119888119894

(120579) 1205721198881198941

1205731198881198941

)119871119877

with fuzzy 119888119894

(120579) = (119888119894

1205721198881198942

1205731198881198942

)119871119877

forany 120579 isin Θ and ℎ

119894

(119909) ge 0 Then 119875119900119904120579 | 119875119900119904119888(120579)119879ℎ(119909) ge 119908 ge120575 ge 120577 is equivalent to

119888119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909) + 119877minus1

(120577) 120573119888119879

2

ℎ (119909) ge 119908 (18)

where 120575 120577 isin [0 1] are predetermined confidence level

Proof For certain 120579 isin Θ 119888119894

(120579) are fuzzy number and itsmember ship function is 120583

119894(120579)

(119905) By extension principle themembership function of fuzzy number 119888(120579)119879ℎ(119909) is

120583

(120579)

119879ℎ(119909)

(119903) =

119871(119888(120579)119879

ℎ (119909) minus 119903

1205721198881198791

ℎ (119909)) 119903 le 119888(120579)

119879

ℎ (119909)

119877(119903 minus 119888(120579)

119879

ℎ (119909)

1205731198881198791

ℎ (119909)) 119903 ge 119888(120579)

119879

ℎ (119909)

(19)

For convenience denote that 119888(120579)119879ℎ(119909) = (119888(120579)119879

ℎ(119909)

1205721198881198791

ℎ(119909) 1205731198881198791

ℎ(119909))119871119877

Since 119888(120579) is also a 119871119877 fuzzy vectorso 119888(120579)119879ℎ(119909) = (119888119879ℎ(119909) 120572119888119879

2

ℎ(119909) 1205731198881198792

ℎ(119909))119871119877

According toLemma 3 we can get

Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 lArrrArr 119888(120579)119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909)

ge 119908

(20)

So for predetermined level 120575 120577 isin [0 1]

Pos 120579 | Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 ge 120577

lArrrArr Pos 120579 | 119888(120579)119879ℎ (119909) ge 119908 minus 119877minus1 (120575) 1205731198881198791

ℎ (119909) ge 120577

lArrrArr 119888119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909) + 119877minus1

(120577) 120573119888119879

2

ℎ (119909) ge 119908

(21)

Theorem 7 Assume that 119890119903119895

(120579) and 119887119903

(120579) are 119871119877 bifuzzyvariables denoted by 119890

119903119895

(120579) = (119890119903119895

(120579) 1205721198901199031198951

1205731198901199031198951

)119871119877

with fuzzy

119890119903119895

(120579) = (119890119903119895

1205721198901199031198952

1205731198901199031198952

)119871119877

and 119887119903

(120579) = (119887119903

(120579) 1205721198871199031

1205731198871199031

)119871119877

withfuzzy 119887

119903

(120579) = (119887119903

1205721198871199032

1205731198871199032

)119871119877

for 119903 = 1 2 119901 119895 = 1 2 119897and 119892

119903119895

(119909) ge 0 Then 119875119900119904120579 | 119875119900119904119890119903

(120579)119879

119892119903

(119909) ge119887119903

(120579) ge

120579119903

ge 120578119903

is equivalent to

119877minus1

(120579119903

) 120573119887

1199031

+ 119871minus1

(120579119903

) 120572119890119879

1199031

119892119903

(119909) minus 119890119879

119903

119892119903

(119909) + 119887119903

+ 119871minus1

(120578119903

) (120572119890119879

1199032

119892119903

(119909) + 120573119887

1199032

) ge 0

(22)

Proof The proof is similar to Theorem 6

3 Notations and Assumptions

The Notations and Assumptions of the proposed models aregiven below

31 Assumption

(i) The brand-new product and remanufactured productare sold at same selling price

(ii) The return rate of used product is infinite

International Journal of Mathematics and Mathematical Sciences 5

SupplierRejection

Inspection

Y gt 120574

Raw material flowUsed product flow

Returned stock

Y lt 1205740

CustomerRemanufacturing

Manufacturing

Remanufacturableinventories

Finished product flow

1205740 le Y lt 120574

Serviceable stock

Figure 1 The frame work of the closed-loop system

(iii) Remanufacturing cost depends on the quality of usedproduct

(iv) Storage capacities of raw material used productmanufactured product and remanufactured productare infinite

(v) Lead time is zero

4 Model Description

In this paper we study a closed-loop system involving amanufacturing facility and a remanufacturing facility Themanufacturer satisfies stochastic market demand by remanu-facturing the used product into ldquoas-newrdquo one and producingnew products from raw material The manufacturer collectsused items by offering low price to customer In reality thecollected used items are of different qualities In the reman-ufacturing facility the used products are inspected carefullyand sorted with respect to the quality of the used productThe products with good quality require less remanufacturingeffort than the products the quality level of which is below 120574but above 120574

0

and the remaining used products are rejectedAfter sorting process the used product is remanufacturedinto as-new product and stocked at serviceable inventorythat is used to satisfy market demand Most of the timethe remanufactured products can not satisfy all the demandThe manufacturer purchases raw material at high price Newproducts are produced at the manufacturing facility Ourobjective is to maximize the manufacturerrsquos expected profitby jointly determining the collected quantity of used productand the ordered quantity of raw material The frame work ofthe system is presented in Figure 1

5 Models Formulation in Crisp Environment

To formulate the problem we first present the unconstrainedmodel to evaluate the optimal order quantities Following thatwe analyze the model with a fill rate constraint and a budgetconstraint separately and then with both the constraints

51 Unconstrained Model (Model-1) Initially when collectedlot of used items confirms good quality standard that isquality of the used items satisfies GQL (with probability 119866

1

)the associate expected profit per period TP

1

(119876 1198760

) is

TP1

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

1119876+119891

0119876

0

[119901 (1198911

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198911

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

1119876+119891

0119876

0

0

[119901119909 + 119903 (1198911

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199011

119876 minus 1199010

1198760

(23)

In the above expected profit function the first term representsthe acquisition price of used product The second termis purchasing cost of raw material The third term is theexpected revenue minus the shortage cost when the demandis higher than production quantity that is 119909 gt 119891

1

119876 + 1198910

1198760

The fourth term is the expected revenue plus the salvage costwhen the demand is lower than production quantity that is119909 lt 1198911

119876+1198910

1198760

the surplus stock can either be offered with adiscount or sold to a secondary market at a unit salvage cost

6 International Journal of Mathematics and Mathematical Sciences

119903 with 119903 lt 119901 The fifth term is remanufacturing cost of useditems The sixth term is manufacturing cost of raw material

When the quality of collected lot of used items is belowGQL but above RQL (with probability 119866

2

) the associateexpected profit per period TP

2

(119876 1198760

) is

TP2

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

2119876+119891

0119876

0

[119901 (1198912

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198912

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

2119876+119891

0119876

0

0

[119901119909 + 119903 (1198912

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199012

119876 minus 1199010

1198760

(24)

Similarly when the quality of collected lot of used itemsis below RQL (with probability 119866

3

) the associate expectedprofit per period TP

3

(119876 1198760

) is

TP3

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

0119876

0

[1199011198910

1198760

minus 119904 (119909 minus 1198910

1198760

)] 119891 (119909) 119889119909

+ int119891

0119876

0

0

[119901119909 + 119903 (1198910

1198760

minus 119909)] 119891 (119909) 119889119909 minus 1199010

1198760

(25)

Therefore combining all possible qualities of collected useditems the total weighted expected profit per period becomes

TP (119876 1198760

) = 1198661

TP1

(119876 1198760

) + 1198662

TP2

(119876 1198760

)

+ 1198663

TP3

(119876 1198760

) (26)

The value of 1198661

1198662

and 1198663

can be estimated from previousquality history Our problem is

Max TP (119876 1198760

) (27)

Proposition 8 (a) The maximum value of 119879119875(1198761198760

) for theproblem (27) is attained for119876lowast and119876lowast

0

by solving the followingsystem of equations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

(28)

(b)The total expected profit TP(119876 1198760

) is concave in119876 and1198760

Proof (a) The first order partial derivatives of (26) withrespect to 119876 and 119876

0

are the following

120597TP (119876 1198760

)

120597119876= 0

120597TP (119876 1198760

)

1205971198760

= 0 (29)

which give (28)(b) The second order partial derivatives of (26) with

respect to 119876 and 1198760

are given below

1205972TP (119876 1198760

)

1205971198762

= minus (119901 + 119904 minus 119903)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

(1198912

119876 + 1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

12059711987620

= minus (119901 + 119904 minus 119903) 1198912

0

times [1198661

119891 (1198911

119876 + 1198910

1198760

) + 1198662

119891 (1198912

119876 + 1198910

1198760

) + 1198663

119891 (1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

1205971198761205971198760

= minus (119901 + 119904 minus 119903) 1198910

times [1198661

1198911

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119891 (1198912

119876 + 1198910

1198760

)]

le 0

(30)

Next the first (|1198671

|) and second (|1198672

|) order determinants ofHessian matrix are

100381610038161003816100381611986711003816100381610038161003816 =

1205972TP (119876 1198760

)

1205971198762lt 0

100381610038161003816100381611986721003816100381610038161003816

=1205972TP (119876 119876

0

)

1205971198762

1205972TP (119876 1198760

)

12059711987620

minus [1205972TP (119876 119876

0

)

1205971198761205971198760

]

2

= 1198912

0

(119901 + 119904 minus 119903)2

times [(1198911

minus 1198912

)2

1198661

1198662

119891 (1198911

119876 + 1198910

1198760

) 119891 (1198912

119876 + 1198910

1198760

)

+ 1198663

119891 (1198910

1198760

)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]]

gt 0

(31)

Therefore TP(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

International Journal of Mathematics and Mathematical Sciences 7

52 Model with Fill Rate Constraint (Model-2) The marketdemand and quality of returned products are uncertain inthe above model thus the manufacturer has to face twotypes of overstocking and understocking risks Under thesecircumstances we analyze the problem of maximizing theexpected overall profit of the hybrid system subject to a fillrate-type customer-service level Fill rate 120573measures the partof stochastic demand that is met from finished new brandproduct Consider

120573 = 1 minusExpected number of stockout unit

mean demand

= 1 minus119878 (119876119876

0

)

120583

(32)

where

119878 (119876 1198760

) = 1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(33)

Hence the resulting optimization model which representsthe maximization of total weighted expected profit subjectedto a fill rate constraint is

Max TP (119876 1198760

)

subject to 120573 ge 1205730

(34)

Proposition 9 (a) The maximum value for problem (34) isattained for 119876lowast 119876lowast

0

and 120582lowast1

by solving the following system ofequations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

(35)

(b)The problem (34) is a convex programming problem in119876 and 119876

0

Proof (a) The Lagrangian relaxation of the problem (1198751

) isthe following

Max 1198711

(119876 1198760

1205821

) (36)

where 1198711

(119876 1198760

1205821

) = TP(119876 1198760

) + 1205821

1198651

(119876 1198760

) and 1205821

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

1205971198711

120597119876= 0

1205971198711

1205971198760

= 0 1205821

1205971198711

1205971205821

= 0 (37)

gives (35)(b) The model (119875

1

) will be a convex programming prob-lem in 119876 and 119876

0

if the objective function to be maximizedshould be concave while the fill rate constraint (119865

1

= 120573 minus

1205730

) should be concave The objective function TP(119876 1198760

) isconcave in 119876 and 119876

0

The first (|119863

1

|) and second (|1198632

|) order determinants ofHessian matrix nabla2119865

1

(119876 1198760

) are

100381610038161003816100381611986311003816100381610038161003816 =

12059721198651

(119876 1198760

)

1205971198762

= minus1

120583

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]

lt 0

100381610038161003816100381611986321003816100381610038161003816 =

1205972

1198651

(119876 1198760

)

1205971198762

1205972

1198651

(119876 1198760

)

12059711987620

minus [1205972

1198651

(119876 1198760

)

1205971198761205971198760

]

2

=1

1205832[1198663

1198912

0

119891 (1198910

1198760

) (1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

)

+ 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

))

+ 1198661

1198662

1198912

0

119891 (1198911

119876 + 1198910

1198760

)

times 119891 (1198912

119876 + 1198910

1198760

) (1198911

minus 1198912

)2

] gt 0

(38)

As a result 1198651

(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

therefore the model 1198751

is a convex programmingproblem

53 Model with Budget Constraint (Model-3) In this sectionwe maximize the manufacturerrsquos expected profit by jointlydetermining the collected quantity of used product andthe ordered quantity of raw material subject to a budgetconstraint If 119861 is the available budget amount for purchasingused product and raw material and converting them to newproduct then problem is reduced to

Max TP (119876 1198760

)

subject to 119888119876 + 119894119876 + 1198880

1198760

+ 1198661

1199011

119876 + 1198662

1199012

119876 + 1199010

1198760

le 119861

(39)

8 International Journal of Mathematics and Mathematical Sciences

Proposition 10 The maximum value for problem (39) isattained for 119876lowast 119876lowast

0

and 120582lowast2

by solving the following system ofequations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205822

(119861 minus 119894119876 minus 119888119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(40)

Proof The Lagrangian relaxation of the problem (39) is thefollowing

Max 1198712

(119876 1198760

1205822

) (41)

where 1198712

(119876 1198760

1205822

) = TP(119876 1198760

) + 1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus

1198661

1199011

119876minus1198662

1199012

119876minus1199010

1198760

) and 1205822

is the Lagrangian multiplierMaximizing the above problem using Kuhn-Tucker condi-tions

1205971198712

120597119876= 0

1205971198712

1205971198760

= 0 1205822

1205971198712

1205971205822

= 0 (42)

gives (40)

54 Model with Both Fill Rate and Budget Constraint (Model-4) In this section we maximize the manufacturerrsquos expectedprofit subject to a fill rate constraint and a budget constraintand the problem reduces to

Max TP (119876 1198760

)

subject to 120573 minus 1205730

ge 0

119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876

minus1198662

1199012

119876 minus 1199010

1198760

ge 0

(43)

Proposition 11 (a) The maximum value for problem (43) isattained for119876lowast119876lowast

0

1205821

and 120582lowast2

by solving the following systemof equations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(44)

Proof The Lagrangian relaxation of the problem (43) is thefollowing

Max 11987112

(119876 1198760

1205821

1205822

) (45)

where 11987112

(119876 1198760

1205821

1205822

) = TP(119876 1198760

) + 1205821

(120573 minus 1205730

) + 1205822

(119861 minus

119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) and 1205822

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

12059711987112

120597119876= 0

12059711987112

1205971198760

= 0 1205821

12059711987112

1205971205821

1205822

12059711987112

1205971205822

= 0

(46)

gives (44)

6 Models in Bifuzzy Environments

To handle the imprecise nature of the parameters 120574 1205740

1205730

and 119861 of the above models we have developed the models inbifuzzy environment

And the bifuzzy variables are triangular 119871119877 bifuzzynumbers and denoted by

120574 (120579) = (120574 (120579) 12057211

12057311

)119871119877

with 120574 (120579) = (120574 12057212

12057312

)

1205740

(120579) = (1205740

(120579) 12057221

12057321

)119871119877

with 1205740

(120579) = (1205740

12057222

12057322

)

1205730

(120579) = (1205730

(120579) 12057231

12057331

)119871119877

with 1205730

(120579) = (1205730

12057232

12057332

)

119861 (120579) = (119861 (120579) 120572

41

12057341

)119871119877

with 119861 (120579) = (119861 12057242

12057342

)

(47)

61 Unconstraint Model (Model-5) In this section 120574 and 1205740

are assumed to be bifuzzy variablesThen the constraints and

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

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Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 3

Definition 1 A level 2 fuzzy set 119881 defined over a universal set119880 is defined by

119881 = ( 120583

119881

()) | forall isin weierp (119880) 120583

119881

() gt 0 (5)

where each ordinary fuzzy set is defined by

= (119909 120583

119881

(119909)) | forall119909 isin 119880 120583

119881

(119909) gt 0 (6)

For convenience the membership grades 120583

119881

() of the fuzzysets isin weierp(119880) are called ldquoouter-layerrdquo membership gradeswhereas the membership grades 120583

119881

(119909) of the element 119909 isin

119880 are called ldquoinner-layerrdquo membership grades Normallyspeaking a bifuzzy variable is a fuzzy variable with fuzzyparameters

Example 2 120585 = (120585 1205721

1205731

)119871119877

with 120585 = (120585 1205722

1205732

)119871119877

iscalled 119871119877 bifuzzy variable if the outer-layer and inner-layermembership function as follows

120583

120585

(119905) =

119871(120585 minus 119905

1205721

) 120585 minus 1205721

le 119905 le 120585 1205721

gt 0

119877(119905 minus 120585

1205731

) 120585 le 119905 le 120585 + 1205731

1205731

gt 0

120583

120585

(119905) =

119871(120585 minus 119905

1205722

) 120585 minus 1205722

le 119905 le 120585 1205722

gt 0

119877(119905 minus 120585

1205732

) 120585 le 119905 le 120585 + 1205732

1205732

gt 0

(7)

where 120585 is center of 120585 which is also a fuzzy number1205721

1205722

1205731

1205732

are the left and right spread of 120585 and 120585 the basisfunctions119871(119909)119877(119909) are continuous nonincreasing functionsand 119871 119877 [0 1] rarr [0 1] satisfies that 119871(1) = 119877(1) = 0119871(0) = 119877(0) = 1

120572 cut 120585120572

of 119871119877 bifuzzy variable 120585 is 120585120572

= (120585119871120572

120585119877120572

) = [120585 minus

119871minus1(120572)1205721

120585 + 119877minus1(120572)1205731

]

22 Possibility Measure Let Θ be a nonempty set and 119875(Θ)the power set ofΘ For each119860 sube 119875(Θ) there is a nonnegativenumber Pos119860 called its possibility such that

(1) Pos120601 = 0 and PosΘ = 1(2) Poscup

119896

119860119896

= sup119896

Pos119860119896

for any arbitrary collec-tion 119860

119896

in 119875(Θ)

The triplet Θ 119875(Θ)Pos is called possibility space and thefunction Pos is referred to as a possibility measure

Lemma 3 Let 1205851

and 1205852

be two fuzzy variables then

119875119900119904 1205851

ge 1205852

= sup 120583

120585

1

(119906) and 120583

120585

2

(V) | 119906 gt V

119875119900119904 1205851

gt 1205852

= sup 120583

120585

1

(119906) and infV1 minus 120583

120585

2

(V) | 119906 le V (8)

By using the 120572-level of fuzzy variables 1205851

and 1205852

[119898119871120572

119898119877120572

] and[119899119871120572

119899119877120572

] Lemma 3 can be written as

119875119900119904 1205851

ge 1205852

ge 120572 lArrrArr 119898119877

120572

ge 119899119871

120572

119875119900119904 1205851

gt 1205852

ge 120572 lArrrArr 119898119877

120572

ge 119899119877

1minus120572

(9)

221 Chance Operator of Bifuzzy Variables

Definition 4 (Liu [26]) Let 120585 be a bifuzzy variable and 119861 aborel set of 119877 Then the primitive chance of bifuzzy event 120585 isin119861 is a function from (0 1] to [0 1] defined as

Ch 120585 isin 119861 (120572) = supMe119860ge120572

inf120579isin119860

Me 120585 (120579) isin 119861 (10)

Usually we use Pos or Nec to measure the chance of bifuzzyevents

Definition 5 (Xu and Zhou [24]) Let 120585 = 1205851

1205852

120585119899

be abifuzzy vector defined on Θ 119875(Θ)Pos and 119891 119877119899 rarr 119877 isa real valued continuous functionThen the primitive chanceof a bifuzzy event characterized by119891(120585) le 0 is a function from(01] to [0 1] defined as follows

(1) Pos-Pos chance isCh 119891 (120585) le 0 (120572)

= sup120572isin[01]

120579 | Pos 120579 isin Θ | Pos 119891 (120585 (120579)) le 0 ge 120573 ge 120572

(11)

(2) Nec-Nec chance isCh 119891 (120585) le 0 (120572)

= sup120572isin[01]

120579 | Nec 120579 isin Θ | Nec 119891 (120585 (120579)) le 0 ge 120573 ge 120572

(12)

where 120572 120573 isin [0 1] are predetermined confidence level

23 Model for Bifuzzy CCM Based on Pos Measure Let usconsider the following single objective bifuzzy model

Max ℎ (119909 120585)

st 119892119903

(119909 120585) le 0 119903 = 1 2 119901

119909 isin 119883

(13)

where 120585 = (1205851

1205852

120585119899

) is a bifuzzy vector and 119909 =

(1199091

1199092

119909119899

) is a decision vector then the objective func-tion ℎ(119909 120585) and constraint functions 119892

119903

(119909 120585) become bifuzzyvariables 119903 = 1 2 119901

In order to obtain optimistic and pessimistic equivalent ofbifuzzy model (13) we use the chance operator to transformthe fuzzy uncertain model into the crisp model which wecalled bifuzzy chance constrained model (bifuzzy CCM)

The general bifuzzy CCM is as follows

Max 119908

st Ch ℎ (119909 120585) ge 119908 (120577) ge 120575Ch 119892

119903

(119909 120585) le 0 (120578119903

) ge 120579119903

119903 = 1 2 119901

119909 isin 119883

(14)

4 International Journal of Mathematics and Mathematical Sciences

where 120577 120575 120578 120579 are the predetermined confidence level and 119908is the decision variable

We adopt Pos to measure the fuzzy event then thespectrum of chance constrainedmodel based on Posmeasureis as follow

Max 119908

st Pos 120579 | Pos ℎ (119909 120585 (120579)) ge 119908 ge 120575 ge 120577

Pos 120579 | Pos 119892119903

(119909120585 (120579)) le 0 ge 120579

119903

ge 120578119903

119903 = 1 2 119901

119909 isin 119883

(15)

24 Linear Bifuzzy Model For linear bifuzzy single objectivefunction model it is assumed that the combination of fuzzyvariables is linear but not the decisionmaking variable119909 thenthe objective function and constraints are written as linear inbifuzzy variables We consider the linear programming withbifuzzy parameters 119888

119894

119890119903119895

119887119903

Consider

Max 119888119879

ℎ (119909)

st 119890119879

119903

119892119903

(119909) le 0 119903 = 1 2 119901

119909 ge 0

(16)

where ℎ(119909) = (ℎ1

(119909) ℎ2

(119909) ℎ119896

(119909)) and 119892(119909) = (1198921199031

(119909)

1198921199032

(119909) 119892119903119897

(119909)) are 119896 dimensional and 119897 dimensional vec-tors respectively

241 Equivalent Crisp Model Next using the bifuzzy CCMbased on Pos measure with the above model we can get thefollowing model

Max 119908

st Pos 120579 | Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 ge 120577

Pos 120579 | Pos 119890119903

(120579)119879

119892119903

(119909) le119887119903

(120579) ge 120579119903

ge 120578119903

119903 = 1 2 119901

119909 ge 0

(17)

In order to solve model (17) we apply the following twotheorems to transform the chance constrained model into itscrisp model based on Pos-Pos measure

Theorem 6 Assume that 119888(120579) = (1198881

(120579) 1198882

(120579) 119888119896

(120579))119879 is

bifuzzy vector and 119888119894

(120579) is 119871119877 bifuzzy variable denoted by119888119894

(120579) = (119888119894

(120579) 1205721198881198941

1205731198881198941

)119871119877

with fuzzy 119888119894

(120579) = (119888119894

1205721198881198942

1205731198881198942

)119871119877

forany 120579 isin Θ and ℎ

119894

(119909) ge 0 Then 119875119900119904120579 | 119875119900119904119888(120579)119879ℎ(119909) ge 119908 ge120575 ge 120577 is equivalent to

119888119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909) + 119877minus1

(120577) 120573119888119879

2

ℎ (119909) ge 119908 (18)

where 120575 120577 isin [0 1] are predetermined confidence level

Proof For certain 120579 isin Θ 119888119894

(120579) are fuzzy number and itsmember ship function is 120583

119894(120579)

(119905) By extension principle themembership function of fuzzy number 119888(120579)119879ℎ(119909) is

120583

(120579)

119879ℎ(119909)

(119903) =

119871(119888(120579)119879

ℎ (119909) minus 119903

1205721198881198791

ℎ (119909)) 119903 le 119888(120579)

119879

ℎ (119909)

119877(119903 minus 119888(120579)

119879

ℎ (119909)

1205731198881198791

ℎ (119909)) 119903 ge 119888(120579)

119879

ℎ (119909)

(19)

For convenience denote that 119888(120579)119879ℎ(119909) = (119888(120579)119879

ℎ(119909)

1205721198881198791

ℎ(119909) 1205731198881198791

ℎ(119909))119871119877

Since 119888(120579) is also a 119871119877 fuzzy vectorso 119888(120579)119879ℎ(119909) = (119888119879ℎ(119909) 120572119888119879

2

ℎ(119909) 1205731198881198792

ℎ(119909))119871119877

According toLemma 3 we can get

Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 lArrrArr 119888(120579)119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909)

ge 119908

(20)

So for predetermined level 120575 120577 isin [0 1]

Pos 120579 | Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 ge 120577

lArrrArr Pos 120579 | 119888(120579)119879ℎ (119909) ge 119908 minus 119877minus1 (120575) 1205731198881198791

ℎ (119909) ge 120577

lArrrArr 119888119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909) + 119877minus1

(120577) 120573119888119879

2

ℎ (119909) ge 119908

(21)

Theorem 7 Assume that 119890119903119895

(120579) and 119887119903

(120579) are 119871119877 bifuzzyvariables denoted by 119890

119903119895

(120579) = (119890119903119895

(120579) 1205721198901199031198951

1205731198901199031198951

)119871119877

with fuzzy

119890119903119895

(120579) = (119890119903119895

1205721198901199031198952

1205731198901199031198952

)119871119877

and 119887119903

(120579) = (119887119903

(120579) 1205721198871199031

1205731198871199031

)119871119877

withfuzzy 119887

119903

(120579) = (119887119903

1205721198871199032

1205731198871199032

)119871119877

for 119903 = 1 2 119901 119895 = 1 2 119897and 119892

119903119895

(119909) ge 0 Then 119875119900119904120579 | 119875119900119904119890119903

(120579)119879

119892119903

(119909) ge119887119903

(120579) ge

120579119903

ge 120578119903

is equivalent to

119877minus1

(120579119903

) 120573119887

1199031

+ 119871minus1

(120579119903

) 120572119890119879

1199031

119892119903

(119909) minus 119890119879

119903

119892119903

(119909) + 119887119903

+ 119871minus1

(120578119903

) (120572119890119879

1199032

119892119903

(119909) + 120573119887

1199032

) ge 0

(22)

Proof The proof is similar to Theorem 6

3 Notations and Assumptions

The Notations and Assumptions of the proposed models aregiven below

31 Assumption

(i) The brand-new product and remanufactured productare sold at same selling price

(ii) The return rate of used product is infinite

International Journal of Mathematics and Mathematical Sciences 5

SupplierRejection

Inspection

Y gt 120574

Raw material flowUsed product flow

Returned stock

Y lt 1205740

CustomerRemanufacturing

Manufacturing

Remanufacturableinventories

Finished product flow

1205740 le Y lt 120574

Serviceable stock

Figure 1 The frame work of the closed-loop system

(iii) Remanufacturing cost depends on the quality of usedproduct

(iv) Storage capacities of raw material used productmanufactured product and remanufactured productare infinite

(v) Lead time is zero

4 Model Description

In this paper we study a closed-loop system involving amanufacturing facility and a remanufacturing facility Themanufacturer satisfies stochastic market demand by remanu-facturing the used product into ldquoas-newrdquo one and producingnew products from raw material The manufacturer collectsused items by offering low price to customer In reality thecollected used items are of different qualities In the reman-ufacturing facility the used products are inspected carefullyand sorted with respect to the quality of the used productThe products with good quality require less remanufacturingeffort than the products the quality level of which is below 120574but above 120574

0

and the remaining used products are rejectedAfter sorting process the used product is remanufacturedinto as-new product and stocked at serviceable inventorythat is used to satisfy market demand Most of the timethe remanufactured products can not satisfy all the demandThe manufacturer purchases raw material at high price Newproducts are produced at the manufacturing facility Ourobjective is to maximize the manufacturerrsquos expected profitby jointly determining the collected quantity of used productand the ordered quantity of raw material The frame work ofthe system is presented in Figure 1

5 Models Formulation in Crisp Environment

To formulate the problem we first present the unconstrainedmodel to evaluate the optimal order quantities Following thatwe analyze the model with a fill rate constraint and a budgetconstraint separately and then with both the constraints

51 Unconstrained Model (Model-1) Initially when collectedlot of used items confirms good quality standard that isquality of the used items satisfies GQL (with probability 119866

1

)the associate expected profit per period TP

1

(119876 1198760

) is

TP1

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

1119876+119891

0119876

0

[119901 (1198911

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198911

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

1119876+119891

0119876

0

0

[119901119909 + 119903 (1198911

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199011

119876 minus 1199010

1198760

(23)

In the above expected profit function the first term representsthe acquisition price of used product The second termis purchasing cost of raw material The third term is theexpected revenue minus the shortage cost when the demandis higher than production quantity that is 119909 gt 119891

1

119876 + 1198910

1198760

The fourth term is the expected revenue plus the salvage costwhen the demand is lower than production quantity that is119909 lt 1198911

119876+1198910

1198760

the surplus stock can either be offered with adiscount or sold to a secondary market at a unit salvage cost

6 International Journal of Mathematics and Mathematical Sciences

119903 with 119903 lt 119901 The fifth term is remanufacturing cost of useditems The sixth term is manufacturing cost of raw material

When the quality of collected lot of used items is belowGQL but above RQL (with probability 119866

2

) the associateexpected profit per period TP

2

(119876 1198760

) is

TP2

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

2119876+119891

0119876

0

[119901 (1198912

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198912

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

2119876+119891

0119876

0

0

[119901119909 + 119903 (1198912

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199012

119876 minus 1199010

1198760

(24)

Similarly when the quality of collected lot of used itemsis below RQL (with probability 119866

3

) the associate expectedprofit per period TP

3

(119876 1198760

) is

TP3

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

0119876

0

[1199011198910

1198760

minus 119904 (119909 minus 1198910

1198760

)] 119891 (119909) 119889119909

+ int119891

0119876

0

0

[119901119909 + 119903 (1198910

1198760

minus 119909)] 119891 (119909) 119889119909 minus 1199010

1198760

(25)

Therefore combining all possible qualities of collected useditems the total weighted expected profit per period becomes

TP (119876 1198760

) = 1198661

TP1

(119876 1198760

) + 1198662

TP2

(119876 1198760

)

+ 1198663

TP3

(119876 1198760

) (26)

The value of 1198661

1198662

and 1198663

can be estimated from previousquality history Our problem is

Max TP (119876 1198760

) (27)

Proposition 8 (a) The maximum value of 119879119875(1198761198760

) for theproblem (27) is attained for119876lowast and119876lowast

0

by solving the followingsystem of equations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

(28)

(b)The total expected profit TP(119876 1198760

) is concave in119876 and1198760

Proof (a) The first order partial derivatives of (26) withrespect to 119876 and 119876

0

are the following

120597TP (119876 1198760

)

120597119876= 0

120597TP (119876 1198760

)

1205971198760

= 0 (29)

which give (28)(b) The second order partial derivatives of (26) with

respect to 119876 and 1198760

are given below

1205972TP (119876 1198760

)

1205971198762

= minus (119901 + 119904 minus 119903)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

(1198912

119876 + 1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

12059711987620

= minus (119901 + 119904 minus 119903) 1198912

0

times [1198661

119891 (1198911

119876 + 1198910

1198760

) + 1198662

119891 (1198912

119876 + 1198910

1198760

) + 1198663

119891 (1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

1205971198761205971198760

= minus (119901 + 119904 minus 119903) 1198910

times [1198661

1198911

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119891 (1198912

119876 + 1198910

1198760

)]

le 0

(30)

Next the first (|1198671

|) and second (|1198672

|) order determinants ofHessian matrix are

100381610038161003816100381611986711003816100381610038161003816 =

1205972TP (119876 1198760

)

1205971198762lt 0

100381610038161003816100381611986721003816100381610038161003816

=1205972TP (119876 119876

0

)

1205971198762

1205972TP (119876 1198760

)

12059711987620

minus [1205972TP (119876 119876

0

)

1205971198761205971198760

]

2

= 1198912

0

(119901 + 119904 minus 119903)2

times [(1198911

minus 1198912

)2

1198661

1198662

119891 (1198911

119876 + 1198910

1198760

) 119891 (1198912

119876 + 1198910

1198760

)

+ 1198663

119891 (1198910

1198760

)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]]

gt 0

(31)

Therefore TP(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

International Journal of Mathematics and Mathematical Sciences 7

52 Model with Fill Rate Constraint (Model-2) The marketdemand and quality of returned products are uncertain inthe above model thus the manufacturer has to face twotypes of overstocking and understocking risks Under thesecircumstances we analyze the problem of maximizing theexpected overall profit of the hybrid system subject to a fillrate-type customer-service level Fill rate 120573measures the partof stochastic demand that is met from finished new brandproduct Consider

120573 = 1 minusExpected number of stockout unit

mean demand

= 1 minus119878 (119876119876

0

)

120583

(32)

where

119878 (119876 1198760

) = 1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(33)

Hence the resulting optimization model which representsthe maximization of total weighted expected profit subjectedto a fill rate constraint is

Max TP (119876 1198760

)

subject to 120573 ge 1205730

(34)

Proposition 9 (a) The maximum value for problem (34) isattained for 119876lowast 119876lowast

0

and 120582lowast1

by solving the following system ofequations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

(35)

(b)The problem (34) is a convex programming problem in119876 and 119876

0

Proof (a) The Lagrangian relaxation of the problem (1198751

) isthe following

Max 1198711

(119876 1198760

1205821

) (36)

where 1198711

(119876 1198760

1205821

) = TP(119876 1198760

) + 1205821

1198651

(119876 1198760

) and 1205821

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

1205971198711

120597119876= 0

1205971198711

1205971198760

= 0 1205821

1205971198711

1205971205821

= 0 (37)

gives (35)(b) The model (119875

1

) will be a convex programming prob-lem in 119876 and 119876

0

if the objective function to be maximizedshould be concave while the fill rate constraint (119865

1

= 120573 minus

1205730

) should be concave The objective function TP(119876 1198760

) isconcave in 119876 and 119876

0

The first (|119863

1

|) and second (|1198632

|) order determinants ofHessian matrix nabla2119865

1

(119876 1198760

) are

100381610038161003816100381611986311003816100381610038161003816 =

12059721198651

(119876 1198760

)

1205971198762

= minus1

120583

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]

lt 0

100381610038161003816100381611986321003816100381610038161003816 =

1205972

1198651

(119876 1198760

)

1205971198762

1205972

1198651

(119876 1198760

)

12059711987620

minus [1205972

1198651

(119876 1198760

)

1205971198761205971198760

]

2

=1

1205832[1198663

1198912

0

119891 (1198910

1198760

) (1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

)

+ 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

))

+ 1198661

1198662

1198912

0

119891 (1198911

119876 + 1198910

1198760

)

times 119891 (1198912

119876 + 1198910

1198760

) (1198911

minus 1198912

)2

] gt 0

(38)

As a result 1198651

(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

therefore the model 1198751

is a convex programmingproblem

53 Model with Budget Constraint (Model-3) In this sectionwe maximize the manufacturerrsquos expected profit by jointlydetermining the collected quantity of used product andthe ordered quantity of raw material subject to a budgetconstraint If 119861 is the available budget amount for purchasingused product and raw material and converting them to newproduct then problem is reduced to

Max TP (119876 1198760

)

subject to 119888119876 + 119894119876 + 1198880

1198760

+ 1198661

1199011

119876 + 1198662

1199012

119876 + 1199010

1198760

le 119861

(39)

8 International Journal of Mathematics and Mathematical Sciences

Proposition 10 The maximum value for problem (39) isattained for 119876lowast 119876lowast

0

and 120582lowast2

by solving the following system ofequations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205822

(119861 minus 119894119876 minus 119888119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(40)

Proof The Lagrangian relaxation of the problem (39) is thefollowing

Max 1198712

(119876 1198760

1205822

) (41)

where 1198712

(119876 1198760

1205822

) = TP(119876 1198760

) + 1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus

1198661

1199011

119876minus1198662

1199012

119876minus1199010

1198760

) and 1205822

is the Lagrangian multiplierMaximizing the above problem using Kuhn-Tucker condi-tions

1205971198712

120597119876= 0

1205971198712

1205971198760

= 0 1205822

1205971198712

1205971205822

= 0 (42)

gives (40)

54 Model with Both Fill Rate and Budget Constraint (Model-4) In this section we maximize the manufacturerrsquos expectedprofit subject to a fill rate constraint and a budget constraintand the problem reduces to

Max TP (119876 1198760

)

subject to 120573 minus 1205730

ge 0

119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876

minus1198662

1199012

119876 minus 1199010

1198760

ge 0

(43)

Proposition 11 (a) The maximum value for problem (43) isattained for119876lowast119876lowast

0

1205821

and 120582lowast2

by solving the following systemof equations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(44)

Proof The Lagrangian relaxation of the problem (43) is thefollowing

Max 11987112

(119876 1198760

1205821

1205822

) (45)

where 11987112

(119876 1198760

1205821

1205822

) = TP(119876 1198760

) + 1205821

(120573 minus 1205730

) + 1205822

(119861 minus

119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) and 1205822

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

12059711987112

120597119876= 0

12059711987112

1205971198760

= 0 1205821

12059711987112

1205971205821

1205822

12059711987112

1205971205822

= 0

(46)

gives (44)

6 Models in Bifuzzy Environments

To handle the imprecise nature of the parameters 120574 1205740

1205730

and 119861 of the above models we have developed the models inbifuzzy environment

And the bifuzzy variables are triangular 119871119877 bifuzzynumbers and denoted by

120574 (120579) = (120574 (120579) 12057211

12057311

)119871119877

with 120574 (120579) = (120574 12057212

12057312

)

1205740

(120579) = (1205740

(120579) 12057221

12057321

)119871119877

with 1205740

(120579) = (1205740

12057222

12057322

)

1205730

(120579) = (1205730

(120579) 12057231

12057331

)119871119877

with 1205730

(120579) = (1205730

12057232

12057332

)

119861 (120579) = (119861 (120579) 120572

41

12057341

)119871119877

with 119861 (120579) = (119861 12057242

12057342

)

(47)

61 Unconstraint Model (Model-5) In this section 120574 and 1205740

are assumed to be bifuzzy variablesThen the constraints and

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Mathematics and Mathematical Sciences

where 120577 120575 120578 120579 are the predetermined confidence level and 119908is the decision variable

We adopt Pos to measure the fuzzy event then thespectrum of chance constrainedmodel based on Posmeasureis as follow

Max 119908

st Pos 120579 | Pos ℎ (119909 120585 (120579)) ge 119908 ge 120575 ge 120577

Pos 120579 | Pos 119892119903

(119909120585 (120579)) le 0 ge 120579

119903

ge 120578119903

119903 = 1 2 119901

119909 isin 119883

(15)

24 Linear Bifuzzy Model For linear bifuzzy single objectivefunction model it is assumed that the combination of fuzzyvariables is linear but not the decisionmaking variable119909 thenthe objective function and constraints are written as linear inbifuzzy variables We consider the linear programming withbifuzzy parameters 119888

119894

119890119903119895

119887119903

Consider

Max 119888119879

ℎ (119909)

st 119890119879

119903

119892119903

(119909) le 0 119903 = 1 2 119901

119909 ge 0

(16)

where ℎ(119909) = (ℎ1

(119909) ℎ2

(119909) ℎ119896

(119909)) and 119892(119909) = (1198921199031

(119909)

1198921199032

(119909) 119892119903119897

(119909)) are 119896 dimensional and 119897 dimensional vec-tors respectively

241 Equivalent Crisp Model Next using the bifuzzy CCMbased on Pos measure with the above model we can get thefollowing model

Max 119908

st Pos 120579 | Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 ge 120577

Pos 120579 | Pos 119890119903

(120579)119879

119892119903

(119909) le119887119903

(120579) ge 120579119903

ge 120578119903

119903 = 1 2 119901

119909 ge 0

(17)

In order to solve model (17) we apply the following twotheorems to transform the chance constrained model into itscrisp model based on Pos-Pos measure

Theorem 6 Assume that 119888(120579) = (1198881

(120579) 1198882

(120579) 119888119896

(120579))119879 is

bifuzzy vector and 119888119894

(120579) is 119871119877 bifuzzy variable denoted by119888119894

(120579) = (119888119894

(120579) 1205721198881198941

1205731198881198941

)119871119877

with fuzzy 119888119894

(120579) = (119888119894

1205721198881198942

1205731198881198942

)119871119877

forany 120579 isin Θ and ℎ

119894

(119909) ge 0 Then 119875119900119904120579 | 119875119900119904119888(120579)119879ℎ(119909) ge 119908 ge120575 ge 120577 is equivalent to

119888119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909) + 119877minus1

(120577) 120573119888119879

2

ℎ (119909) ge 119908 (18)

where 120575 120577 isin [0 1] are predetermined confidence level

Proof For certain 120579 isin Θ 119888119894

(120579) are fuzzy number and itsmember ship function is 120583

119894(120579)

(119905) By extension principle themembership function of fuzzy number 119888(120579)119879ℎ(119909) is

120583

(120579)

119879ℎ(119909)

(119903) =

119871(119888(120579)119879

ℎ (119909) minus 119903

1205721198881198791

ℎ (119909)) 119903 le 119888(120579)

119879

ℎ (119909)

119877(119903 minus 119888(120579)

119879

ℎ (119909)

1205731198881198791

ℎ (119909)) 119903 ge 119888(120579)

119879

ℎ (119909)

(19)

For convenience denote that 119888(120579)119879ℎ(119909) = (119888(120579)119879

ℎ(119909)

1205721198881198791

ℎ(119909) 1205731198881198791

ℎ(119909))119871119877

Since 119888(120579) is also a 119871119877 fuzzy vectorso 119888(120579)119879ℎ(119909) = (119888119879ℎ(119909) 120572119888119879

2

ℎ(119909) 1205731198881198792

ℎ(119909))119871119877

According toLemma 3 we can get

Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 lArrrArr 119888(120579)119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909)

ge 119908

(20)

So for predetermined level 120575 120577 isin [0 1]

Pos 120579 | Pos 119888(120579)119879ℎ (119909) ge 119908 ge 120575 ge 120577

lArrrArr Pos 120579 | 119888(120579)119879ℎ (119909) ge 119908 minus 119877minus1 (120575) 1205731198881198791

ℎ (119909) ge 120577

lArrrArr 119888119879

ℎ (119909) + 119877minus1

(120575) 120573119888119879

1

ℎ (119909) + 119877minus1

(120577) 120573119888119879

2

ℎ (119909) ge 119908

(21)

Theorem 7 Assume that 119890119903119895

(120579) and 119887119903

(120579) are 119871119877 bifuzzyvariables denoted by 119890

119903119895

(120579) = (119890119903119895

(120579) 1205721198901199031198951

1205731198901199031198951

)119871119877

with fuzzy

119890119903119895

(120579) = (119890119903119895

1205721198901199031198952

1205731198901199031198952

)119871119877

and 119887119903

(120579) = (119887119903

(120579) 1205721198871199031

1205731198871199031

)119871119877

withfuzzy 119887

119903

(120579) = (119887119903

1205721198871199032

1205731198871199032

)119871119877

for 119903 = 1 2 119901 119895 = 1 2 119897and 119892

119903119895

(119909) ge 0 Then 119875119900119904120579 | 119875119900119904119890119903

(120579)119879

119892119903

(119909) ge119887119903

(120579) ge

120579119903

ge 120578119903

is equivalent to

119877minus1

(120579119903

) 120573119887

1199031

+ 119871minus1

(120579119903

) 120572119890119879

1199031

119892119903

(119909) minus 119890119879

119903

119892119903

(119909) + 119887119903

+ 119871minus1

(120578119903

) (120572119890119879

1199032

119892119903

(119909) + 120573119887

1199032

) ge 0

(22)

Proof The proof is similar to Theorem 6

3 Notations and Assumptions

The Notations and Assumptions of the proposed models aregiven below

31 Assumption

(i) The brand-new product and remanufactured productare sold at same selling price

(ii) The return rate of used product is infinite

International Journal of Mathematics and Mathematical Sciences 5

SupplierRejection

Inspection

Y gt 120574

Raw material flowUsed product flow

Returned stock

Y lt 1205740

CustomerRemanufacturing

Manufacturing

Remanufacturableinventories

Finished product flow

1205740 le Y lt 120574

Serviceable stock

Figure 1 The frame work of the closed-loop system

(iii) Remanufacturing cost depends on the quality of usedproduct

(iv) Storage capacities of raw material used productmanufactured product and remanufactured productare infinite

(v) Lead time is zero

4 Model Description

In this paper we study a closed-loop system involving amanufacturing facility and a remanufacturing facility Themanufacturer satisfies stochastic market demand by remanu-facturing the used product into ldquoas-newrdquo one and producingnew products from raw material The manufacturer collectsused items by offering low price to customer In reality thecollected used items are of different qualities In the reman-ufacturing facility the used products are inspected carefullyand sorted with respect to the quality of the used productThe products with good quality require less remanufacturingeffort than the products the quality level of which is below 120574but above 120574

0

and the remaining used products are rejectedAfter sorting process the used product is remanufacturedinto as-new product and stocked at serviceable inventorythat is used to satisfy market demand Most of the timethe remanufactured products can not satisfy all the demandThe manufacturer purchases raw material at high price Newproducts are produced at the manufacturing facility Ourobjective is to maximize the manufacturerrsquos expected profitby jointly determining the collected quantity of used productand the ordered quantity of raw material The frame work ofthe system is presented in Figure 1

5 Models Formulation in Crisp Environment

To formulate the problem we first present the unconstrainedmodel to evaluate the optimal order quantities Following thatwe analyze the model with a fill rate constraint and a budgetconstraint separately and then with both the constraints

51 Unconstrained Model (Model-1) Initially when collectedlot of used items confirms good quality standard that isquality of the used items satisfies GQL (with probability 119866

1

)the associate expected profit per period TP

1

(119876 1198760

) is

TP1

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

1119876+119891

0119876

0

[119901 (1198911

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198911

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

1119876+119891

0119876

0

0

[119901119909 + 119903 (1198911

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199011

119876 minus 1199010

1198760

(23)

In the above expected profit function the first term representsthe acquisition price of used product The second termis purchasing cost of raw material The third term is theexpected revenue minus the shortage cost when the demandis higher than production quantity that is 119909 gt 119891

1

119876 + 1198910

1198760

The fourth term is the expected revenue plus the salvage costwhen the demand is lower than production quantity that is119909 lt 1198911

119876+1198910

1198760

the surplus stock can either be offered with adiscount or sold to a secondary market at a unit salvage cost

6 International Journal of Mathematics and Mathematical Sciences

119903 with 119903 lt 119901 The fifth term is remanufacturing cost of useditems The sixth term is manufacturing cost of raw material

When the quality of collected lot of used items is belowGQL but above RQL (with probability 119866

2

) the associateexpected profit per period TP

2

(119876 1198760

) is

TP2

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

2119876+119891

0119876

0

[119901 (1198912

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198912

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

2119876+119891

0119876

0

0

[119901119909 + 119903 (1198912

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199012

119876 minus 1199010

1198760

(24)

Similarly when the quality of collected lot of used itemsis below RQL (with probability 119866

3

) the associate expectedprofit per period TP

3

(119876 1198760

) is

TP3

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

0119876

0

[1199011198910

1198760

minus 119904 (119909 minus 1198910

1198760

)] 119891 (119909) 119889119909

+ int119891

0119876

0

0

[119901119909 + 119903 (1198910

1198760

minus 119909)] 119891 (119909) 119889119909 minus 1199010

1198760

(25)

Therefore combining all possible qualities of collected useditems the total weighted expected profit per period becomes

TP (119876 1198760

) = 1198661

TP1

(119876 1198760

) + 1198662

TP2

(119876 1198760

)

+ 1198663

TP3

(119876 1198760

) (26)

The value of 1198661

1198662

and 1198663

can be estimated from previousquality history Our problem is

Max TP (119876 1198760

) (27)

Proposition 8 (a) The maximum value of 119879119875(1198761198760

) for theproblem (27) is attained for119876lowast and119876lowast

0

by solving the followingsystem of equations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

(28)

(b)The total expected profit TP(119876 1198760

) is concave in119876 and1198760

Proof (a) The first order partial derivatives of (26) withrespect to 119876 and 119876

0

are the following

120597TP (119876 1198760

)

120597119876= 0

120597TP (119876 1198760

)

1205971198760

= 0 (29)

which give (28)(b) The second order partial derivatives of (26) with

respect to 119876 and 1198760

are given below

1205972TP (119876 1198760

)

1205971198762

= minus (119901 + 119904 minus 119903)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

(1198912

119876 + 1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

12059711987620

= minus (119901 + 119904 minus 119903) 1198912

0

times [1198661

119891 (1198911

119876 + 1198910

1198760

) + 1198662

119891 (1198912

119876 + 1198910

1198760

) + 1198663

119891 (1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

1205971198761205971198760

= minus (119901 + 119904 minus 119903) 1198910

times [1198661

1198911

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119891 (1198912

119876 + 1198910

1198760

)]

le 0

(30)

Next the first (|1198671

|) and second (|1198672

|) order determinants ofHessian matrix are

100381610038161003816100381611986711003816100381610038161003816 =

1205972TP (119876 1198760

)

1205971198762lt 0

100381610038161003816100381611986721003816100381610038161003816

=1205972TP (119876 119876

0

)

1205971198762

1205972TP (119876 1198760

)

12059711987620

minus [1205972TP (119876 119876

0

)

1205971198761205971198760

]

2

= 1198912

0

(119901 + 119904 minus 119903)2

times [(1198911

minus 1198912

)2

1198661

1198662

119891 (1198911

119876 + 1198910

1198760

) 119891 (1198912

119876 + 1198910

1198760

)

+ 1198663

119891 (1198910

1198760

)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]]

gt 0

(31)

Therefore TP(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

International Journal of Mathematics and Mathematical Sciences 7

52 Model with Fill Rate Constraint (Model-2) The marketdemand and quality of returned products are uncertain inthe above model thus the manufacturer has to face twotypes of overstocking and understocking risks Under thesecircumstances we analyze the problem of maximizing theexpected overall profit of the hybrid system subject to a fillrate-type customer-service level Fill rate 120573measures the partof stochastic demand that is met from finished new brandproduct Consider

120573 = 1 minusExpected number of stockout unit

mean demand

= 1 minus119878 (119876119876

0

)

120583

(32)

where

119878 (119876 1198760

) = 1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(33)

Hence the resulting optimization model which representsthe maximization of total weighted expected profit subjectedto a fill rate constraint is

Max TP (119876 1198760

)

subject to 120573 ge 1205730

(34)

Proposition 9 (a) The maximum value for problem (34) isattained for 119876lowast 119876lowast

0

and 120582lowast1

by solving the following system ofequations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

(35)

(b)The problem (34) is a convex programming problem in119876 and 119876

0

Proof (a) The Lagrangian relaxation of the problem (1198751

) isthe following

Max 1198711

(119876 1198760

1205821

) (36)

where 1198711

(119876 1198760

1205821

) = TP(119876 1198760

) + 1205821

1198651

(119876 1198760

) and 1205821

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

1205971198711

120597119876= 0

1205971198711

1205971198760

= 0 1205821

1205971198711

1205971205821

= 0 (37)

gives (35)(b) The model (119875

1

) will be a convex programming prob-lem in 119876 and 119876

0

if the objective function to be maximizedshould be concave while the fill rate constraint (119865

1

= 120573 minus

1205730

) should be concave The objective function TP(119876 1198760

) isconcave in 119876 and 119876

0

The first (|119863

1

|) and second (|1198632

|) order determinants ofHessian matrix nabla2119865

1

(119876 1198760

) are

100381610038161003816100381611986311003816100381610038161003816 =

12059721198651

(119876 1198760

)

1205971198762

= minus1

120583

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]

lt 0

100381610038161003816100381611986321003816100381610038161003816 =

1205972

1198651

(119876 1198760

)

1205971198762

1205972

1198651

(119876 1198760

)

12059711987620

minus [1205972

1198651

(119876 1198760

)

1205971198761205971198760

]

2

=1

1205832[1198663

1198912

0

119891 (1198910

1198760

) (1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

)

+ 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

))

+ 1198661

1198662

1198912

0

119891 (1198911

119876 + 1198910

1198760

)

times 119891 (1198912

119876 + 1198910

1198760

) (1198911

minus 1198912

)2

] gt 0

(38)

As a result 1198651

(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

therefore the model 1198751

is a convex programmingproblem

53 Model with Budget Constraint (Model-3) In this sectionwe maximize the manufacturerrsquos expected profit by jointlydetermining the collected quantity of used product andthe ordered quantity of raw material subject to a budgetconstraint If 119861 is the available budget amount for purchasingused product and raw material and converting them to newproduct then problem is reduced to

Max TP (119876 1198760

)

subject to 119888119876 + 119894119876 + 1198880

1198760

+ 1198661

1199011

119876 + 1198662

1199012

119876 + 1199010

1198760

le 119861

(39)

8 International Journal of Mathematics and Mathematical Sciences

Proposition 10 The maximum value for problem (39) isattained for 119876lowast 119876lowast

0

and 120582lowast2

by solving the following system ofequations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205822

(119861 minus 119894119876 minus 119888119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(40)

Proof The Lagrangian relaxation of the problem (39) is thefollowing

Max 1198712

(119876 1198760

1205822

) (41)

where 1198712

(119876 1198760

1205822

) = TP(119876 1198760

) + 1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus

1198661

1199011

119876minus1198662

1199012

119876minus1199010

1198760

) and 1205822

is the Lagrangian multiplierMaximizing the above problem using Kuhn-Tucker condi-tions

1205971198712

120597119876= 0

1205971198712

1205971198760

= 0 1205822

1205971198712

1205971205822

= 0 (42)

gives (40)

54 Model with Both Fill Rate and Budget Constraint (Model-4) In this section we maximize the manufacturerrsquos expectedprofit subject to a fill rate constraint and a budget constraintand the problem reduces to

Max TP (119876 1198760

)

subject to 120573 minus 1205730

ge 0

119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876

minus1198662

1199012

119876 minus 1199010

1198760

ge 0

(43)

Proposition 11 (a) The maximum value for problem (43) isattained for119876lowast119876lowast

0

1205821

and 120582lowast2

by solving the following systemof equations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(44)

Proof The Lagrangian relaxation of the problem (43) is thefollowing

Max 11987112

(119876 1198760

1205821

1205822

) (45)

where 11987112

(119876 1198760

1205821

1205822

) = TP(119876 1198760

) + 1205821

(120573 minus 1205730

) + 1205822

(119861 minus

119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) and 1205822

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

12059711987112

120597119876= 0

12059711987112

1205971198760

= 0 1205821

12059711987112

1205971205821

1205822

12059711987112

1205971205822

= 0

(46)

gives (44)

6 Models in Bifuzzy Environments

To handle the imprecise nature of the parameters 120574 1205740

1205730

and 119861 of the above models we have developed the models inbifuzzy environment

And the bifuzzy variables are triangular 119871119877 bifuzzynumbers and denoted by

120574 (120579) = (120574 (120579) 12057211

12057311

)119871119877

with 120574 (120579) = (120574 12057212

12057312

)

1205740

(120579) = (1205740

(120579) 12057221

12057321

)119871119877

with 1205740

(120579) = (1205740

12057222

12057322

)

1205730

(120579) = (1205730

(120579) 12057231

12057331

)119871119877

with 1205730

(120579) = (1205730

12057232

12057332

)

119861 (120579) = (119861 (120579) 120572

41

12057341

)119871119877

with 119861 (120579) = (119861 12057242

12057342

)

(47)

61 Unconstraint Model (Model-5) In this section 120574 and 1205740

are assumed to be bifuzzy variablesThen the constraints and

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 5

SupplierRejection

Inspection

Y gt 120574

Raw material flowUsed product flow

Returned stock

Y lt 1205740

CustomerRemanufacturing

Manufacturing

Remanufacturableinventories

Finished product flow

1205740 le Y lt 120574

Serviceable stock

Figure 1 The frame work of the closed-loop system

(iii) Remanufacturing cost depends on the quality of usedproduct

(iv) Storage capacities of raw material used productmanufactured product and remanufactured productare infinite

(v) Lead time is zero

4 Model Description

In this paper we study a closed-loop system involving amanufacturing facility and a remanufacturing facility Themanufacturer satisfies stochastic market demand by remanu-facturing the used product into ldquoas-newrdquo one and producingnew products from raw material The manufacturer collectsused items by offering low price to customer In reality thecollected used items are of different qualities In the reman-ufacturing facility the used products are inspected carefullyand sorted with respect to the quality of the used productThe products with good quality require less remanufacturingeffort than the products the quality level of which is below 120574but above 120574

0

and the remaining used products are rejectedAfter sorting process the used product is remanufacturedinto as-new product and stocked at serviceable inventorythat is used to satisfy market demand Most of the timethe remanufactured products can not satisfy all the demandThe manufacturer purchases raw material at high price Newproducts are produced at the manufacturing facility Ourobjective is to maximize the manufacturerrsquos expected profitby jointly determining the collected quantity of used productand the ordered quantity of raw material The frame work ofthe system is presented in Figure 1

5 Models Formulation in Crisp Environment

To formulate the problem we first present the unconstrainedmodel to evaluate the optimal order quantities Following thatwe analyze the model with a fill rate constraint and a budgetconstraint separately and then with both the constraints

51 Unconstrained Model (Model-1) Initially when collectedlot of used items confirms good quality standard that isquality of the used items satisfies GQL (with probability 119866

1

)the associate expected profit per period TP

1

(119876 1198760

) is

TP1

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

1119876+119891

0119876

0

[119901 (1198911

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198911

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

1119876+119891

0119876

0

0

[119901119909 + 119903 (1198911

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199011

119876 minus 1199010

1198760

(23)

In the above expected profit function the first term representsthe acquisition price of used product The second termis purchasing cost of raw material The third term is theexpected revenue minus the shortage cost when the demandis higher than production quantity that is 119909 gt 119891

1

119876 + 1198910

1198760

The fourth term is the expected revenue plus the salvage costwhen the demand is lower than production quantity that is119909 lt 1198911

119876+1198910

1198760

the surplus stock can either be offered with adiscount or sold to a secondary market at a unit salvage cost

6 International Journal of Mathematics and Mathematical Sciences

119903 with 119903 lt 119901 The fifth term is remanufacturing cost of useditems The sixth term is manufacturing cost of raw material

When the quality of collected lot of used items is belowGQL but above RQL (with probability 119866

2

) the associateexpected profit per period TP

2

(119876 1198760

) is

TP2

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

2119876+119891

0119876

0

[119901 (1198912

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198912

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

2119876+119891

0119876

0

0

[119901119909 + 119903 (1198912

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199012

119876 minus 1199010

1198760

(24)

Similarly when the quality of collected lot of used itemsis below RQL (with probability 119866

3

) the associate expectedprofit per period TP

3

(119876 1198760

) is

TP3

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

0119876

0

[1199011198910

1198760

minus 119904 (119909 minus 1198910

1198760

)] 119891 (119909) 119889119909

+ int119891

0119876

0

0

[119901119909 + 119903 (1198910

1198760

minus 119909)] 119891 (119909) 119889119909 minus 1199010

1198760

(25)

Therefore combining all possible qualities of collected useditems the total weighted expected profit per period becomes

TP (119876 1198760

) = 1198661

TP1

(119876 1198760

) + 1198662

TP2

(119876 1198760

)

+ 1198663

TP3

(119876 1198760

) (26)

The value of 1198661

1198662

and 1198663

can be estimated from previousquality history Our problem is

Max TP (119876 1198760

) (27)

Proposition 8 (a) The maximum value of 119879119875(1198761198760

) for theproblem (27) is attained for119876lowast and119876lowast

0

by solving the followingsystem of equations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

(28)

(b)The total expected profit TP(119876 1198760

) is concave in119876 and1198760

Proof (a) The first order partial derivatives of (26) withrespect to 119876 and 119876

0

are the following

120597TP (119876 1198760

)

120597119876= 0

120597TP (119876 1198760

)

1205971198760

= 0 (29)

which give (28)(b) The second order partial derivatives of (26) with

respect to 119876 and 1198760

are given below

1205972TP (119876 1198760

)

1205971198762

= minus (119901 + 119904 minus 119903)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

(1198912

119876 + 1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

12059711987620

= minus (119901 + 119904 minus 119903) 1198912

0

times [1198661

119891 (1198911

119876 + 1198910

1198760

) + 1198662

119891 (1198912

119876 + 1198910

1198760

) + 1198663

119891 (1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

1205971198761205971198760

= minus (119901 + 119904 minus 119903) 1198910

times [1198661

1198911

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119891 (1198912

119876 + 1198910

1198760

)]

le 0

(30)

Next the first (|1198671

|) and second (|1198672

|) order determinants ofHessian matrix are

100381610038161003816100381611986711003816100381610038161003816 =

1205972TP (119876 1198760

)

1205971198762lt 0

100381610038161003816100381611986721003816100381610038161003816

=1205972TP (119876 119876

0

)

1205971198762

1205972TP (119876 1198760

)

12059711987620

minus [1205972TP (119876 119876

0

)

1205971198761205971198760

]

2

= 1198912

0

(119901 + 119904 minus 119903)2

times [(1198911

minus 1198912

)2

1198661

1198662

119891 (1198911

119876 + 1198910

1198760

) 119891 (1198912

119876 + 1198910

1198760

)

+ 1198663

119891 (1198910

1198760

)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]]

gt 0

(31)

Therefore TP(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

International Journal of Mathematics and Mathematical Sciences 7

52 Model with Fill Rate Constraint (Model-2) The marketdemand and quality of returned products are uncertain inthe above model thus the manufacturer has to face twotypes of overstocking and understocking risks Under thesecircumstances we analyze the problem of maximizing theexpected overall profit of the hybrid system subject to a fillrate-type customer-service level Fill rate 120573measures the partof stochastic demand that is met from finished new brandproduct Consider

120573 = 1 minusExpected number of stockout unit

mean demand

= 1 minus119878 (119876119876

0

)

120583

(32)

where

119878 (119876 1198760

) = 1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(33)

Hence the resulting optimization model which representsthe maximization of total weighted expected profit subjectedto a fill rate constraint is

Max TP (119876 1198760

)

subject to 120573 ge 1205730

(34)

Proposition 9 (a) The maximum value for problem (34) isattained for 119876lowast 119876lowast

0

and 120582lowast1

by solving the following system ofequations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

(35)

(b)The problem (34) is a convex programming problem in119876 and 119876

0

Proof (a) The Lagrangian relaxation of the problem (1198751

) isthe following

Max 1198711

(119876 1198760

1205821

) (36)

where 1198711

(119876 1198760

1205821

) = TP(119876 1198760

) + 1205821

1198651

(119876 1198760

) and 1205821

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

1205971198711

120597119876= 0

1205971198711

1205971198760

= 0 1205821

1205971198711

1205971205821

= 0 (37)

gives (35)(b) The model (119875

1

) will be a convex programming prob-lem in 119876 and 119876

0

if the objective function to be maximizedshould be concave while the fill rate constraint (119865

1

= 120573 minus

1205730

) should be concave The objective function TP(119876 1198760

) isconcave in 119876 and 119876

0

The first (|119863

1

|) and second (|1198632

|) order determinants ofHessian matrix nabla2119865

1

(119876 1198760

) are

100381610038161003816100381611986311003816100381610038161003816 =

12059721198651

(119876 1198760

)

1205971198762

= minus1

120583

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]

lt 0

100381610038161003816100381611986321003816100381610038161003816 =

1205972

1198651

(119876 1198760

)

1205971198762

1205972

1198651

(119876 1198760

)

12059711987620

minus [1205972

1198651

(119876 1198760

)

1205971198761205971198760

]

2

=1

1205832[1198663

1198912

0

119891 (1198910

1198760

) (1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

)

+ 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

))

+ 1198661

1198662

1198912

0

119891 (1198911

119876 + 1198910

1198760

)

times 119891 (1198912

119876 + 1198910

1198760

) (1198911

minus 1198912

)2

] gt 0

(38)

As a result 1198651

(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

therefore the model 1198751

is a convex programmingproblem

53 Model with Budget Constraint (Model-3) In this sectionwe maximize the manufacturerrsquos expected profit by jointlydetermining the collected quantity of used product andthe ordered quantity of raw material subject to a budgetconstraint If 119861 is the available budget amount for purchasingused product and raw material and converting them to newproduct then problem is reduced to

Max TP (119876 1198760

)

subject to 119888119876 + 119894119876 + 1198880

1198760

+ 1198661

1199011

119876 + 1198662

1199012

119876 + 1199010

1198760

le 119861

(39)

8 International Journal of Mathematics and Mathematical Sciences

Proposition 10 The maximum value for problem (39) isattained for 119876lowast 119876lowast

0

and 120582lowast2

by solving the following system ofequations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205822

(119861 minus 119894119876 minus 119888119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(40)

Proof The Lagrangian relaxation of the problem (39) is thefollowing

Max 1198712

(119876 1198760

1205822

) (41)

where 1198712

(119876 1198760

1205822

) = TP(119876 1198760

) + 1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus

1198661

1199011

119876minus1198662

1199012

119876minus1199010

1198760

) and 1205822

is the Lagrangian multiplierMaximizing the above problem using Kuhn-Tucker condi-tions

1205971198712

120597119876= 0

1205971198712

1205971198760

= 0 1205822

1205971198712

1205971205822

= 0 (42)

gives (40)

54 Model with Both Fill Rate and Budget Constraint (Model-4) In this section we maximize the manufacturerrsquos expectedprofit subject to a fill rate constraint and a budget constraintand the problem reduces to

Max TP (119876 1198760

)

subject to 120573 minus 1205730

ge 0

119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876

minus1198662

1199012

119876 minus 1199010

1198760

ge 0

(43)

Proposition 11 (a) The maximum value for problem (43) isattained for119876lowast119876lowast

0

1205821

and 120582lowast2

by solving the following systemof equations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(44)

Proof The Lagrangian relaxation of the problem (43) is thefollowing

Max 11987112

(119876 1198760

1205821

1205822

) (45)

where 11987112

(119876 1198760

1205821

1205822

) = TP(119876 1198760

) + 1205821

(120573 minus 1205730

) + 1205822

(119861 minus

119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) and 1205822

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

12059711987112

120597119876= 0

12059711987112

1205971198760

= 0 1205821

12059711987112

1205971205821

1205822

12059711987112

1205971205822

= 0

(46)

gives (44)

6 Models in Bifuzzy Environments

To handle the imprecise nature of the parameters 120574 1205740

1205730

and 119861 of the above models we have developed the models inbifuzzy environment

And the bifuzzy variables are triangular 119871119877 bifuzzynumbers and denoted by

120574 (120579) = (120574 (120579) 12057211

12057311

)119871119877

with 120574 (120579) = (120574 12057212

12057312

)

1205740

(120579) = (1205740

(120579) 12057221

12057321

)119871119877

with 1205740

(120579) = (1205740

12057222

12057322

)

1205730

(120579) = (1205730

(120579) 12057231

12057331

)119871119877

with 1205730

(120579) = (1205730

12057232

12057332

)

119861 (120579) = (119861 (120579) 120572

41

12057341

)119871119877

with 119861 (120579) = (119861 12057242

12057342

)

(47)

61 Unconstraint Model (Model-5) In this section 120574 and 1205740

are assumed to be bifuzzy variablesThen the constraints and

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Mathematics and Mathematical Sciences

119903 with 119903 lt 119901 The fifth term is remanufacturing cost of useditems The sixth term is manufacturing cost of raw material

When the quality of collected lot of used items is belowGQL but above RQL (with probability 119866

2

) the associateexpected profit per period TP

2

(119876 1198760

) is

TP2

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

2119876+119891

0119876

0

[119901 (1198912

119876 + 1198910

1198760

) minus 119904 (119909 minus 1198912

119876 minus 1198910

1198760

)]

times 119891 (119909) 119889119909

+ int119891

2119876+119891

0119876

0

0

[119901119909 + 119903 (1198912

119876 + 1198910

1198760

minus 119909)] 119891 (119909) 119889119909

minus 1199012

119876 minus 1199010

1198760

(24)

Similarly when the quality of collected lot of used itemsis below RQL (with probability 119866

3

) the associate expectedprofit per period TP

3

(119876 1198760

) is

TP3

(119876 1198760

)

= minus119888119876 minus 119894119876 minus 1198880

1198760

+ intinfin

119891

0119876

0

[1199011198910

1198760

minus 119904 (119909 minus 1198910

1198760

)] 119891 (119909) 119889119909

+ int119891

0119876

0

0

[119901119909 + 119903 (1198910

1198760

minus 119909)] 119891 (119909) 119889119909 minus 1199010

1198760

(25)

Therefore combining all possible qualities of collected useditems the total weighted expected profit per period becomes

TP (119876 1198760

) = 1198661

TP1

(119876 1198760

) + 1198662

TP2

(119876 1198760

)

+ 1198663

TP3

(119876 1198760

) (26)

The value of 1198661

1198662

and 1198663

can be estimated from previousquality history Our problem is

Max TP (119876 1198760

) (27)

Proposition 8 (a) The maximum value of 119879119875(1198761198760

) for theproblem (27) is attained for119876lowast and119876lowast

0

by solving the followingsystem of equations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

(28)

(b)The total expected profit TP(119876 1198760

) is concave in119876 and1198760

Proof (a) The first order partial derivatives of (26) withrespect to 119876 and 119876

0

are the following

120597TP (119876 1198760

)

120597119876= 0

120597TP (119876 1198760

)

1205971198760

= 0 (29)

which give (28)(b) The second order partial derivatives of (26) with

respect to 119876 and 1198760

are given below

1205972TP (119876 1198760

)

1205971198762

= minus (119901 + 119904 minus 119903)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

(1198912

119876 + 1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

12059711987620

= minus (119901 + 119904 minus 119903) 1198912

0

times [1198661

119891 (1198911

119876 + 1198910

1198760

) + 1198662

119891 (1198912

119876 + 1198910

1198760

) + 1198663

119891 (1198910

1198760

)]

le 0

1205972TP (119876 1198760

)

1205971198761205971198760

= minus (119901 + 119904 minus 119903) 1198910

times [1198661

1198911

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119891 (1198912

119876 + 1198910

1198760

)]

le 0

(30)

Next the first (|1198671

|) and second (|1198672

|) order determinants ofHessian matrix are

100381610038161003816100381611986711003816100381610038161003816 =

1205972TP (119876 1198760

)

1205971198762lt 0

100381610038161003816100381611986721003816100381610038161003816

=1205972TP (119876 119876

0

)

1205971198762

1205972TP (119876 1198760

)

12059711987620

minus [1205972TP (119876 119876

0

)

1205971198761205971198760

]

2

= 1198912

0

(119901 + 119904 minus 119903)2

times [(1198911

minus 1198912

)2

1198661

1198662

119891 (1198911

119876 + 1198910

1198760

) 119891 (1198912

119876 + 1198910

1198760

)

+ 1198663

119891 (1198910

1198760

)

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]]

gt 0

(31)

Therefore TP(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

International Journal of Mathematics and Mathematical Sciences 7

52 Model with Fill Rate Constraint (Model-2) The marketdemand and quality of returned products are uncertain inthe above model thus the manufacturer has to face twotypes of overstocking and understocking risks Under thesecircumstances we analyze the problem of maximizing theexpected overall profit of the hybrid system subject to a fillrate-type customer-service level Fill rate 120573measures the partof stochastic demand that is met from finished new brandproduct Consider

120573 = 1 minusExpected number of stockout unit

mean demand

= 1 minus119878 (119876119876

0

)

120583

(32)

where

119878 (119876 1198760

) = 1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(33)

Hence the resulting optimization model which representsthe maximization of total weighted expected profit subjectedto a fill rate constraint is

Max TP (119876 1198760

)

subject to 120573 ge 1205730

(34)

Proposition 9 (a) The maximum value for problem (34) isattained for 119876lowast 119876lowast

0

and 120582lowast1

by solving the following system ofequations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

(35)

(b)The problem (34) is a convex programming problem in119876 and 119876

0

Proof (a) The Lagrangian relaxation of the problem (1198751

) isthe following

Max 1198711

(119876 1198760

1205821

) (36)

where 1198711

(119876 1198760

1205821

) = TP(119876 1198760

) + 1205821

1198651

(119876 1198760

) and 1205821

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

1205971198711

120597119876= 0

1205971198711

1205971198760

= 0 1205821

1205971198711

1205971205821

= 0 (37)

gives (35)(b) The model (119875

1

) will be a convex programming prob-lem in 119876 and 119876

0

if the objective function to be maximizedshould be concave while the fill rate constraint (119865

1

= 120573 minus

1205730

) should be concave The objective function TP(119876 1198760

) isconcave in 119876 and 119876

0

The first (|119863

1

|) and second (|1198632

|) order determinants ofHessian matrix nabla2119865

1

(119876 1198760

) are

100381610038161003816100381611986311003816100381610038161003816 =

12059721198651

(119876 1198760

)

1205971198762

= minus1

120583

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]

lt 0

100381610038161003816100381611986321003816100381610038161003816 =

1205972

1198651

(119876 1198760

)

1205971198762

1205972

1198651

(119876 1198760

)

12059711987620

minus [1205972

1198651

(119876 1198760

)

1205971198761205971198760

]

2

=1

1205832[1198663

1198912

0

119891 (1198910

1198760

) (1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

)

+ 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

))

+ 1198661

1198662

1198912

0

119891 (1198911

119876 + 1198910

1198760

)

times 119891 (1198912

119876 + 1198910

1198760

) (1198911

minus 1198912

)2

] gt 0

(38)

As a result 1198651

(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

therefore the model 1198751

is a convex programmingproblem

53 Model with Budget Constraint (Model-3) In this sectionwe maximize the manufacturerrsquos expected profit by jointlydetermining the collected quantity of used product andthe ordered quantity of raw material subject to a budgetconstraint If 119861 is the available budget amount for purchasingused product and raw material and converting them to newproduct then problem is reduced to

Max TP (119876 1198760

)

subject to 119888119876 + 119894119876 + 1198880

1198760

+ 1198661

1199011

119876 + 1198662

1199012

119876 + 1199010

1198760

le 119861

(39)

8 International Journal of Mathematics and Mathematical Sciences

Proposition 10 The maximum value for problem (39) isattained for 119876lowast 119876lowast

0

and 120582lowast2

by solving the following system ofequations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205822

(119861 minus 119894119876 minus 119888119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(40)

Proof The Lagrangian relaxation of the problem (39) is thefollowing

Max 1198712

(119876 1198760

1205822

) (41)

where 1198712

(119876 1198760

1205822

) = TP(119876 1198760

) + 1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus

1198661

1199011

119876minus1198662

1199012

119876minus1199010

1198760

) and 1205822

is the Lagrangian multiplierMaximizing the above problem using Kuhn-Tucker condi-tions

1205971198712

120597119876= 0

1205971198712

1205971198760

= 0 1205822

1205971198712

1205971205822

= 0 (42)

gives (40)

54 Model with Both Fill Rate and Budget Constraint (Model-4) In this section we maximize the manufacturerrsquos expectedprofit subject to a fill rate constraint and a budget constraintand the problem reduces to

Max TP (119876 1198760

)

subject to 120573 minus 1205730

ge 0

119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876

minus1198662

1199012

119876 minus 1199010

1198760

ge 0

(43)

Proposition 11 (a) The maximum value for problem (43) isattained for119876lowast119876lowast

0

1205821

and 120582lowast2

by solving the following systemof equations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(44)

Proof The Lagrangian relaxation of the problem (43) is thefollowing

Max 11987112

(119876 1198760

1205821

1205822

) (45)

where 11987112

(119876 1198760

1205821

1205822

) = TP(119876 1198760

) + 1205821

(120573 minus 1205730

) + 1205822

(119861 minus

119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) and 1205822

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

12059711987112

120597119876= 0

12059711987112

1205971198760

= 0 1205821

12059711987112

1205971205821

1205822

12059711987112

1205971205822

= 0

(46)

gives (44)

6 Models in Bifuzzy Environments

To handle the imprecise nature of the parameters 120574 1205740

1205730

and 119861 of the above models we have developed the models inbifuzzy environment

And the bifuzzy variables are triangular 119871119877 bifuzzynumbers and denoted by

120574 (120579) = (120574 (120579) 12057211

12057311

)119871119877

with 120574 (120579) = (120574 12057212

12057312

)

1205740

(120579) = (1205740

(120579) 12057221

12057321

)119871119877

with 1205740

(120579) = (1205740

12057222

12057322

)

1205730

(120579) = (1205730

(120579) 12057231

12057331

)119871119877

with 1205730

(120579) = (1205730

12057232

12057332

)

119861 (120579) = (119861 (120579) 120572

41

12057341

)119871119877

with 119861 (120579) = (119861 12057242

12057342

)

(47)

61 Unconstraint Model (Model-5) In this section 120574 and 1205740

are assumed to be bifuzzy variablesThen the constraints and

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

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Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 7

52 Model with Fill Rate Constraint (Model-2) The marketdemand and quality of returned products are uncertain inthe above model thus the manufacturer has to face twotypes of overstocking and understocking risks Under thesecircumstances we analyze the problem of maximizing theexpected overall profit of the hybrid system subject to a fillrate-type customer-service level Fill rate 120573measures the partof stochastic demand that is met from finished new brandproduct Consider

120573 = 1 minusExpected number of stockout unit

mean demand

= 1 minus119878 (119876119876

0

)

120583

(32)

where

119878 (119876 1198760

) = 1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(33)

Hence the resulting optimization model which representsthe maximization of total weighted expected profit subjectedto a fill rate constraint is

Max TP (119876 1198760

)

subject to 120573 ge 1205730

(34)

Proposition 9 (a) The maximum value for problem (34) isattained for 119876lowast 119876lowast

0

and 120582lowast1

by solving the following system ofequations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= 119888 + 119894 + 1198661

1199011

+ 1198662

1199012

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= 1198880

+ 1199010

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

(35)

(b)The problem (34) is a convex programming problem in119876 and 119876

0

Proof (a) The Lagrangian relaxation of the problem (1198751

) isthe following

Max 1198711

(119876 1198760

1205821

) (36)

where 1198711

(119876 1198760

1205821

) = TP(119876 1198760

) + 1205821

1198651

(119876 1198760

) and 1205821

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

1205971198711

120597119876= 0

1205971198711

1205971198760

= 0 1205821

1205971198711

1205971205821

= 0 (37)

gives (35)(b) The model (119875

1

) will be a convex programming prob-lem in 119876 and 119876

0

if the objective function to be maximizedshould be concave while the fill rate constraint (119865

1

= 120573 minus

1205730

) should be concave The objective function TP(119876 1198760

) isconcave in 119876 and 119876

0

The first (|119863

1

|) and second (|1198632

|) order determinants ofHessian matrix nabla2119865

1

(119876 1198760

) are

100381610038161003816100381611986311003816100381610038161003816 =

12059721198651

(119876 1198760

)

1205971198762

= minus1

120583

times [1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

)]

lt 0

100381610038161003816100381611986321003816100381610038161003816 =

1205972

1198651

(119876 1198760

)

1205971198762

1205972

1198651

(119876 1198760

)

12059711987620

minus [1205972

1198651

(119876 1198760

)

1205971198761205971198760

]

2

=1

1205832[1198663

1198912

0

119891 (1198910

1198760

) (1198661

1198912

1

119891 (1198911

119876 + 1198910

1198760

)

+ 1198662

1198912

2

119891 (1198912

119876 + 1198910

1198760

))

+ 1198661

1198662

1198912

0

119891 (1198911

119876 + 1198910

1198760

)

times 119891 (1198912

119876 + 1198910

1198760

) (1198911

minus 1198912

)2

] gt 0

(38)

As a result 1198651

(119876 1198760

) is negative definite and thus concave in119876 and 119876

0

therefore the model 1198751

is a convex programmingproblem

53 Model with Budget Constraint (Model-3) In this sectionwe maximize the manufacturerrsquos expected profit by jointlydetermining the collected quantity of used product andthe ordered quantity of raw material subject to a budgetconstraint If 119861 is the available budget amount for purchasingused product and raw material and converting them to newproduct then problem is reduced to

Max TP (119876 1198760

)

subject to 119888119876 + 119894119876 + 1198880

1198760

+ 1198661

1199011

119876 + 1198662

1199012

119876 + 1199010

1198760

le 119861

(39)

8 International Journal of Mathematics and Mathematical Sciences

Proposition 10 The maximum value for problem (39) isattained for 119876lowast 119876lowast

0

and 120582lowast2

by solving the following system ofequations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205822

(119861 minus 119894119876 minus 119888119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(40)

Proof The Lagrangian relaxation of the problem (39) is thefollowing

Max 1198712

(119876 1198760

1205822

) (41)

where 1198712

(119876 1198760

1205822

) = TP(119876 1198760

) + 1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus

1198661

1199011

119876minus1198662

1199012

119876minus1199010

1198760

) and 1205822

is the Lagrangian multiplierMaximizing the above problem using Kuhn-Tucker condi-tions

1205971198712

120597119876= 0

1205971198712

1205971198760

= 0 1205822

1205971198712

1205971205822

= 0 (42)

gives (40)

54 Model with Both Fill Rate and Budget Constraint (Model-4) In this section we maximize the manufacturerrsquos expectedprofit subject to a fill rate constraint and a budget constraintand the problem reduces to

Max TP (119876 1198760

)

subject to 120573 minus 1205730

ge 0

119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876

minus1198662

1199012

119876 minus 1199010

1198760

ge 0

(43)

Proposition 11 (a) The maximum value for problem (43) isattained for119876lowast119876lowast

0

1205821

and 120582lowast2

by solving the following systemof equations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(44)

Proof The Lagrangian relaxation of the problem (43) is thefollowing

Max 11987112

(119876 1198760

1205821

1205822

) (45)

where 11987112

(119876 1198760

1205821

1205822

) = TP(119876 1198760

) + 1205821

(120573 minus 1205730

) + 1205822

(119861 minus

119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) and 1205822

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

12059711987112

120597119876= 0

12059711987112

1205971198760

= 0 1205821

12059711987112

1205971205821

1205822

12059711987112

1205971205822

= 0

(46)

gives (44)

6 Models in Bifuzzy Environments

To handle the imprecise nature of the parameters 120574 1205740

1205730

and 119861 of the above models we have developed the models inbifuzzy environment

And the bifuzzy variables are triangular 119871119877 bifuzzynumbers and denoted by

120574 (120579) = (120574 (120579) 12057211

12057311

)119871119877

with 120574 (120579) = (120574 12057212

12057312

)

1205740

(120579) = (1205740

(120579) 12057221

12057321

)119871119877

with 1205740

(120579) = (1205740

12057222

12057322

)

1205730

(120579) = (1205730

(120579) 12057231

12057331

)119871119877

with 1205730

(120579) = (1205730

12057232

12057332

)

119861 (120579) = (119861 (120579) 120572

41

12057341

)119871119877

with 119861 (120579) = (119861 12057242

12057342

)

(47)

61 Unconstraint Model (Model-5) In this section 120574 and 1205740

are assumed to be bifuzzy variablesThen the constraints and

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Mathematics and Mathematical Sciences

Proposition 10 The maximum value for problem (39) isattained for 119876lowast 119876lowast

0

and 120582lowast2

by solving the following system ofequations

(119901 + 119904) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904) 1198910

+ (119903 minus 119901 minus 119904) 1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205822

(119861 minus 119894119876 minus 119888119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(40)

Proof The Lagrangian relaxation of the problem (39) is thefollowing

Max 1198712

(119876 1198760

1205822

) (41)

where 1198712

(119876 1198760

1205822

) = TP(119876 1198760

) + 1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus

1198661

1199011

119876minus1198662

1199012

119876minus1199010

1198760

) and 1205822

is the Lagrangian multiplierMaximizing the above problem using Kuhn-Tucker condi-tions

1205971198712

120597119876= 0

1205971198712

1205971198760

= 0 1205822

1205971198712

1205971205822

= 0 (42)

gives (40)

54 Model with Both Fill Rate and Budget Constraint (Model-4) In this section we maximize the manufacturerrsquos expectedprofit subject to a fill rate constraint and a budget constraintand the problem reduces to

Max TP (119876 1198760

)

subject to 120573 minus 1205730

ge 0

119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876

minus1198662

1199012

119876 minus 1199010

1198760

ge 0

(43)

Proposition 11 (a) The maximum value for problem (43) isattained for119876lowast119876lowast

0

1205821

and 120582lowast2

by solving the following systemof equations

(119901 + 119904 +1205821

120583) (1198661

1198911

+ 1198662

1198912

) + (119903 minus 119901 minus 119904 minus1205821

120583)

times [1198661

1198911

119865 (1198911

119876 + 1198910

1198760

) + 1198662

1198912

119865 (1198912

119876 + 1198910

1198760

)]

= (1 + 1205822

) (119888 + 119894 + 1198661

1199011

+ 1198662

1199012

)

(119901 + 119904 +1205821

120583)1198910

+ (119903 minus 119901 minus 119904 minus1205821

120583)1198910

times [1198661

119865 (1198911

119876 + 1198910

1198760

) + 1198662

119865 (1198912

119876 + 1198910

1198760

) + 1198663

119865 (1198910

1198760

)]

= (1 + 1205822

) (1198880

+ 1199010

)

1205821

(1 minus1

120583[1198661

intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198662

intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

+ 1198663

intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909] minus 1205730

) = 0

1205822

(119861 minus 119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) = 0

(44)

Proof The Lagrangian relaxation of the problem (43) is thefollowing

Max 11987112

(119876 1198760

1205821

1205822

) (45)

where 11987112

(119876 1198760

1205821

1205822

) = TP(119876 1198760

) + 1205821

(120573 minus 1205730

) + 1205822

(119861 minus

119888119876 minus 119894119876 minus 1198880

1198760

minus 1198661

1199011

119876 minus 1198662

1199012

119876 minus 1199010

1198760

) and 1205822

is theLagrangian multiplier Maximizing the above problem usingKuhn-Tucker conditions

12059711987112

120597119876= 0

12059711987112

1205971198760

= 0 1205821

12059711987112

1205971205821

1205822

12059711987112

1205971205822

= 0

(46)

gives (44)

6 Models in Bifuzzy Environments

To handle the imprecise nature of the parameters 120574 1205740

1205730

and 119861 of the above models we have developed the models inbifuzzy environment

And the bifuzzy variables are triangular 119871119877 bifuzzynumbers and denoted by

120574 (120579) = (120574 (120579) 12057211

12057311

)119871119877

with 120574 (120579) = (120574 12057212

12057312

)

1205740

(120579) = (1205740

(120579) 12057221

12057321

)119871119877

with 1205740

(120579) = (1205740

12057222

12057322

)

1205730

(120579) = (1205730

(120579) 12057231

12057331

)119871119877

with 1205730

(120579) = (1205730

12057232

12057332

)

119861 (120579) = (119861 (120579) 120572

41

12057341

)119871119877

with 119861 (120579) = (119861 12057242

12057342

)

(47)

61 Unconstraint Model (Model-5) In this section 120574 and 1205740

are assumed to be bifuzzy variablesThen the constraints and

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 9

objective function can be expressed in bifuzzy in nature asfollows

Max TP (119876 1198760

) =119887

119887 minus 119886TP1

(119876 1198760

) minus119886

119887 minus 119886TP3

(119876 1198760

)

minus120574

119887 minus 119886(TP1

(119876 1198760

) minus TP2

(119876 1198760

))

minus1205740

119887 minus 119886(TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(48)

611 Equivalent Crisp Model In order to solve model (48)we use bifuzzy CCM based on Pos measure then we can getthe following model

Max 119908 (49)

subject to Pos120579 | Pos 119887

119887 minus 119886TP1

(119876 1198760

)

minus119886

119887 minus 119886TP3

(119876 1198760

)

times120574 (120579)

119887 minus 119886

times (TP1

(119876 1198760

)

minus TP2

(119876 1198760

))

minus1205740

(120579)

119887 minus 119886

times (TP2

(119876 1198760

)

minus TP3

(119876 1198760

))

ge 119908 ge 120575 ge 120577

(50)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (51)

subject to 119887TP1

(119876 1198760

) minus 119886TP3

(119876 1198760

)

ge (119887 minus 119886)119908

+ (120574 minus 12057211

119871minus1

(120575) minus 12057212

119871minus1

(120577))

times (TP1

(119876 1198760

) minus TP2

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(120575) minus 12057222

119871minus1

(120577))

times (TP2

(119876 1198760

) minus TP3

(119876 1198760

))

(52)

62 Model with Fill Rate Constraint (Model-6) In this sec-tion 120574 120574

0

and 1205730

are assumed to be bifuzzy variables Then

the constraints and objective function can be expressed inbifuzzy in nature as follows

Max (48)

(53)

subject to 1 minus119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

minus120574

120583 (119887 minus 119886)(SU2

(119876 1198760

) minus SU1

(119876 1198760

))

minus1205740

120583 (119887 minus 119886)(SU3

(119876 1198760

) minus SU2

(119876 1198760

))

minus1205730

ge 0

(54)

where

SU1

(119876 1198760

) = intinfin

119891

1119876+119891

0119876

0

(119909 minus 1198911

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU2

(119876 1198760

) = intinfin

119891

2119876+119891

0119876

0

(119909 minus 1198912

119876 minus 1198910

1198760

) 119891 (119909) 119889119909

SU3

(119876 1198760

) = intinfin

119891

0119876

0

(119909 minus 1198910

1198760

) 119891 (119909) 119889119909

(55)

621 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (56)

subject to (50) (57)

Pos120579 | Pos1 minus 119887

120583 (119887 minus 119886)SU1

(119876 1198760

)

+119886

120583 (119887 minus 119886)SU3

(119876 1198760

)

ge120574 (120579)

120583 (119887 minus 119886)

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+1205740

(120579)

120583 (119887 minus 119886)

times (SU3

(119876 1198760

)

minus SU2

(119876 1198760

))

+1205730

(120579) ge 1205791

ge 1205781

(58)

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Mathematics and Mathematical Sciences

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (59)

subject to (52) (60)

120583 (119887 minus 119886) minus 119887SU1

(119876 1198760

) + 119886SU3

(119876 1198760

)

ge (120574 minus 12057211

119871minus1

(1205791

) minus 12057212

119871minus1

(1205781

))

times (SU2

(119876 1198760

)

minus SU1

(119876 1198760

))

+ (1205740

minus 12057221

119871minus1

(1205791

) minus 12057222

119871minus1

(1205781

))

times (SU3

(119876 1198760

) minus SU2

(119876 1198760

))

+ 120583 (119887 minus 119886)

times (1205730

minus 12057231

119871minus1

(1205791

) minus 12057232

119871minus1

(1205781

))

(61)

63 Model with Budget Constraint (Model-7) In this section120574 1205740

and 119861 are assumed to be bifuzzy variables Then theconstraints and objective function can be expressed in bifuzzyin nature as follows

Max (48) (62)

subject to 119861 minus 119888119876 minus 119894119876 minus 119888

0

1198760

minus119887

119887 minus 119886

times1199011

119876 minus120574

119887 minus 119886119876 (1199012

minus 1199011

)

+1205740

119887 minus 1198861199012

119876 minus 1199010

1198760

ge 0

(63)

631 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908 (64)

subject to (50) (65)

Pos120579 | Pos119861 (120579) +1205740

(120579)

119887 minus 1198861199012

119876

ge 119888119876 + 119894119876 + 1198880

1198760

+119887

119887 minus 1198861199011

119876 + 1199010

1198760

+120574 (120579)

119887 minus 119886119876 (1199012

minus 1199011

)

ge 1205792

ge 1205782

(66)

850

500

1000

1500

Ord

ered

qua

ntiti

es

65 70 75 805

56

54

52 Expe

cted

pro

ft

times105

GQL 120574

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 2 Effect of 120574 on the optimal policy

50 55 60 650

500

1000

1500O

rder

ed q

uant

ities

5

55

6

65

Expe

cted

pro

ft

Qlowasttotal Qlowast

0

Qlowast

RQL 1205740

times105

TPlowast

Figure 3 Effect of 1205740

on the optimal policy

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908 (67)

subject to (52) (68)

(119887 minus 119886) (119861 + 12057341

119877minus1

(1205792

) + 12057342

119877minus1

(1205782

))

+ 1199012

119876(1205740

+ 12057321

119877minus1

(1205792

) + 12057322

119877minus1

(1205782

))

ge (119887 minus 119886) (119888119876 + 119894119876 + 1198880

1198760

+ 1199010

1198760

)

+ 1198871199011

119876 + 119876 (1199012

minus 1199011

)

times (120574 minus 12057211

119871minus1

(1205792

) minus 12057212

119871minus1

(1205782

))

(69)

64 Model with Fill Rate Constraint and Budget Constraint(Model-8) In this section 120574 120574

0

119861 and 1205730

are assumed to be

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 11

09 092 094 096 098 10

1000

2000

3000

Ord

ered

qua

ntiti

es

0

2

4

6

Expe

cted

pro

ft

times105

Qlowasttotal Qlowast

0

Qlowast TPlowast

Fill rate 1205730

Figure 4 Effect of 1205730

on the optimal policy

0

500

1000

1500

Ord

ered

qua

ntiti

es

Budget B

15 2 25 33

4

5

6Ex

pect

ed p

rofit

times105

times106

Qlowasttotal Qlowast

0

Qlowast TPlowast

Figure 5 Effect of 119861 on the optimal policy

bifuzzy variablesThen the constraints and objective functioncan be expressed in bifuzzy in nature as follows

Max (48)

subject to (54) (63) (70)

641 Equivalent Crisp Model In order to solve the abovemodel we use bifuzzy CCM based on Pos measure then wecan get the following model

Max 119908

subject to (50) (58) (66) (71)

Then by Theorems 6 and 7 based on Pos-Pos measure weconvert the above model to equivalent crisp model

Max 119908

subject to (52) (61) (69) (72)

7 Numerical Examples

We have solved the above models using gradient basednonlinear soft computing optimization technique (LINGO-140)

245 25 255 26

525

53

535

Expe

cted

pro

fit

Budget B1205730 = 0950

1205730 = 0952

1205730 = 0954

1205730 = 0956

1205730 = 0958

1205730 = 0960

times106

times105

Figure 6 Effect of budget 119861 for different values of 1205730

on the totalprofit

Table 1 The optimal results of the models

Problem 119876lowast 119876lowast0

119876lowasttotal TPlowast

Model-1 36588 86955 123543 536382Model-2 35930 87987 123917 536313Model-3 36092 85125 121218 535867Model-4 18358 94754 113112 528142

Table 2 Effect of changing 120574 on the optimal policy

120574 119876lowast 119876lowast0

119876lowasttotal TPlowast

65 44058 82885 126943 55139470 36588 86955 123543 53638275 29113 91057 120171 52390880 21342 95330 116672 51411585 12909 99962 112872 507288

Table 3 Effect of changing 1205740

on the optimal policy

1205740

119876lowast

119876lowast

0

119876lowast

total TPlowast

50 139203 10285 149488 629928525 58206 73362 131569 57168255 36588 86955 123543 536381575 21396 95685 117081 51609160 8810 102494 111303 505960625 000 107063 107063 50373965 000 107063 107063 503739

Table 4 Effect of changing 1205730

on the optimal policy

1205730

119876lowast 119876lowast0

119876lowasttotal TPlowast090 36588 86955 123543 536381092 36588 86955 123543 536381094 36588 86955 123543 536381096 33065 92741 125806 534172098 26771 105224 131996 513907100 9552 191228 200780 93471

71 Crisp Models (Model-1 to -4) Themathematical behaviorof proposed models is illustrated with the parameters 119901 =

3500 1199011

= 500 1199012

= 550 1199010

= 530 1198911

= 084 1198912

= 070

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 International Journal of Mathematics and Mathematical Sciences

Table 5 Effect of changing 119861 on the optimal policy

119861 119876lowast 119876lowast0

119876lowasttotal TPlowast

1500000 87956 16625 104581 3299761750000 61026 41975 103001 4107162000000 43720 62100 105820 4796852250000 36574 76709 113284 5232892500000 36588 86955 123543 5363822750000 36588 86955 123543 5363823000000 36588 86955 123543 536382

Table 6 Effect of changing119861 for different values of1205730

on the optimalpolicy

119861 TPlowast

1205730

= 0950 0952 0954 0956 0958 0960245000 531706 524776 mdash mdash mdash mdash250000 536293 535792 534568 532052 525886 mdash255000 536312 536138 535847 535430 534702 533153260000 536312 536138 535847 535430 534876 534172

Table 7 The optimal results of the bifuzzy models

Problem 119876lowast

119876lowast

0

119876lowast

total TPlowast

Model-5 39755 85091 124846 541570Model-6 39616 85295 112491 541568Model-7 39229 82838 122066 540840Model-8 25520 90260 115780 536394

1198910

= 088 1198880

= 1800 119888 = 850 119894 = 50 119903 = 2000 119904 = 15001205730

= 095 and 119861 = 2440000 119883 follows normal distributionwithmean 800 and standard deviation 220119884 follows uniformdistribution over the interval [40 90] 120574

0

= 55 120574 = 70Therefore 119866

1

= 119875(119884 ge 120574) = 040 1198662

= 119875(1205740

le 119884 lt

120574) = 030 1198663

= 119875(119884 lt 1205740

) = 030 The optimal results of themodels are given in Table 1

711 Sensitivity Analysis In this section we perform sen-sitivity analysis by changing the values of some importantparameters

Model-1 The optimal solutions for different values of 120574 aredisplayed in Table 2

Figure 2 shows that as 120574 increases while all other param-eters remain unchanged the optimal ordered quantity ofused product 119876lowast total ordered quantity of raw material andused product 119876lowasttotal and the expected total profit TPlowast havedecreased but there is increase in optimal ordered quantityof raw material 119876lowast

0

This is an expected result because forlarge value of the 120574 the probability of nonconformance toused product of good quality increases so that manufacturerdecides to order more raw material (119876lowast

0

) at high cost thanused product (119876lowast) resulting in a smaller value of profit (TPlowast)

The behavior of 119876lowast 119876lowast0

119876lowasttotal and TPlowast with respect to1205740

(see Figure 3) is found to be similar to that obtainedwith respect to 120574 as 120574

0

increases 119876lowast 119876lowasttotal and TPlowast havedecreased but there is increase in 119876lowast

0

Table 3 indicates that

when 1205740

is increased above 525 then the ordered quantity ofraw material 119876

0

is greater than the ordered quantity of usedproduct 119876

Model-2 Figure 4 presents how the ordering policy119876lowast 119876lowast

0

119876lowasttotal and the expected total profit TPlowast change asfill rate 120573

0

increases while all other parameters remain sameFrom Table 4 we find that for values of 120573

0

from 0 to94 the optimal ordering policy (119876lowast 119876lowast

0

) and the total profitTPlowast remain unchanged and their values are equal to theoptimal values of unconstrained model When the value of1205730

increases above 094 TPlowastlowast decreases rapidly and leadsto a negative profit It is observed that as 120573

0

increases themanufacturer in order to meet market demand increasesthe order quantity of raw material and decreases the orderquantity of used product

Model-3 Table 5 indicates that when 119861 lies in the interval (02500000) TPlowast increases as 119861 increases and the manufacturerproduces more quantities of new product from the rawmaterial and less from the used product to meet the demandThemarginal benefit of additional budget tends to zero as thebudget amount increases above 2500000 and expected profitapproaches the expected profit for unconstrained Model-1(see Figure 5)

Model-4 In Table 6 we investigate the effect of the availablebudget amount119861 for different values of120573

0

The pattern of TPlowastversus 119861 plot in Figure 6 is similar to Figure 5

72 FuzzyModel For fuzzy model we consider the same dataas in crisp model except the following bifuzzy numbers

120574(120579) = (120574(120579) 6 4)119871119877

with 120574(120579) = (70 3 2)119871119877

1205740

(120579) = (1205740

5 4)119871119877

with 1205740

(120579) = (55 2 15)119871119877

1205730

(120579) = (1205730

(120579) 004 006)119871119877

with 1205730

(120579) = (081 006005)119871119877

119861(120579) = (119861(120579) 200000 100000)

119871119877

with 119861(120579) =

(2400000 300000 400000)119871119877

and we set 120575 = 120577 = 1205791

= 1205781

= 1205792

= 1205782

= 095 The optimalresults of fuzzy models are given in Table 7

721 Sensitivity Analysis In Table 8 we investigate the effectof changes in the predetermined levels 120575 120577 120579

1

1205781

1205792

1205782

on119876lowast 119876lowast

0

119876lowasttotalTPlowast in the bifuzzy models Table 8 indicates

that Model-5 to Model-7 are significantly influenced by thepredetermined levels whereas Model-8 is highly sensitive tothe changes in predetermined levels

8 Conclusion

In this paper we have studied a closed-loop supplychain model where manufacturer satisfies stochastic marketdemand by remanufacturingrecycling used product andmanufacturing new product We assume that the qualityof the used product is random and remanufacturing cost

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 13

Table 8 Effect of predetermined level on the optimal policy

(120575 120577 1205791

1205781

1205792

1205782

) Problem 119876lowast

change in119876lowast

total TPlowast119876lowast0

(093 093 093 093 093 093)

Model-5 32993 minus09157 04265 04079Model-6 31311 minus07988 04476 04073Model-7 35737 minus03689 08981 04600Model-8 193072 minus24600 23379 08497

(095 095 093 093 093 093)

Model-5 00000 00000 00000 00000Model-6 03502 minus02386 minus00518 00005Model-7 01800 05002 03973 00442Model-8 193065 minus24598 23378 05345

( 095 095 097 097 097 097)

Model-5 00000 00000 00000 00000Model-6 minus07014 04743 01014 minus00036Model-7 minus01544 minus05067 minus03935 minus00532Model-8 minus276676 37477 minus31768 minus11617

(097 097 097 097 097 097)

Model-5 minus32323 08923 minus04211 minus03936Model-6 minus31029 07975 minus04736 minus03933Model-7 minus34820 03474 minus08832 minus04584Model-8 minus276697 37480 minus31770 minus13526

depends on the quality of used product We first derivethe proposed base case unconstrained model in Model-1Following that we investigate the model in the presenceof fill rate constraint and budget constraint (Model-2 to-4) Next to overcome uncertainty in some parameterswe develop the models in bifuzzy environment (Model-5 to -8) We perform a comparison of optimal results ofthe models through numerical examples Analysis resultshows that in low-budget scale industry manufacturer pro-duces more quantities of new product from the usedproduct and less from the raw material to meet thedemand but in order to prevent shortage the manufactureradopts manufacturing policy rather than remanufacturingpolicy

This paper is limited in the sense that (i) brand-newproducts and as-new products are absolutely substitutedby each other and sold at the same price (ii) the returnrate of used product is infinite This limitation suggests aninteresting extension to our research work

Notations

119883 A random variable denoting the marketdemand

119891(sdot) Probability density function of the marketdemand

119865(sdot) Cumulative density function of the marketdemand

119884 A random variable denoting the qualitycharacteristic of used item

119892(sdot) Probability density function of the qualitycharacteristic of used item

120574 The good quality level (GQL) of the usedproduct

1205740

The rejected quality level (RQL) of the usedproduct

1198661

the probability that the quality of collectedlot of used items satisfies GQL1198661

= 119875(119884 ge 120574)

1198662

The probability that the quality of collectedlot of the used items is below GQL but aboveRQL 119866

1

= 119875(1205740

le 119884 le 120574)

1198663

The probability that quality of collected lot ofthe used items is below RQL 119866

3

= 119875(119884 lt 1205740

)

119876 Collected quantity of used product1198760

Ordered quantity of raw material119876total Total ordered quantity of raw material and

used product119901 Unit selling price of product119888 Unit purchasing cost of used product1198880

Unit purchasing cost of raw material where1198880

gt 119888

119894 Unit inspection cost of used product1199010

Unit manufacturing cost of raw material1199011

Unit remanufacturing cost of used item thequality of which satisfies GQL

1199012

Unit remanufacturing cost of used item thequality of which is below GQL but aboveRQL where 119901

1

le 1199012

1198910

Conversion factor of raw material to finishednew product

1198911

Conversion factor of used item (the quality ofwhich satisfies GQL) to finished new product

1198912

Conversion factor of used item (the qualityof which is below GQL but above RQL) tofinished new product where 119891

1

ge 1198912

119903 Unit salvage cost where 119903 gt 119901119904 Unit shortage cost

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 International Journal of Mathematics and Mathematical Sciences

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E van der Laan M Salomon R Dekker and L van Wassen-hove ldquoInventory control in hybrid systems with remanufactur-ingrdquoManagement Science vol 45 no 5 pp 733ndash747 1999

[2] H Krikke ldquoImpact of closed-loop network configurations oncarbon footprints a case study in copiersrdquo Resources Conserva-tion and Recycling vol 55 no 12 pp 1196ndash1205 2011

[3] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004

[4] I Dobos and K Richter ldquoAn extended productionrecyclingmodel with stationary demand andreturn ratesrdquo InternationalJournal of Production Economics vol 90 pp 311ndash323 2004

[5] D-W Choi H Hwang and S-G Koh ldquoA generalized orderingand recovery policy for reusable itemsrdquo European Journal ofOperational Research vol 182 no 2 pp 764ndash774 2007

[6] S Rubio and A Corominas ldquoOptimal manufacturing-remanufacturing policies in a lean production environmentrdquoComputers and Industrial Engineering vol 55 no 1 pp234ndash242 2008

[7] C A Yano and L H Lee ldquoLot sizing with random yields areviewrdquo Operations Research vol 43 no 2 pp 311ndash334 1995

[8] A Hsu and Y Bassok ldquoRandom yield and random demand ina production system with downward substitutionrdquo OperationsResearch vol 47 no 2 pp 277ndash290 1999

[9] S Bollapragada and T E Morton ldquoMyopic heuristics for therandom yield problemrdquo Operations Research vol 47 no 5 pp713ndash722 1999

[10] B Kazaz ldquoProduction planning under yield and demand uncer-tainty with yield-dependent cost and pricerdquoManufacturing andServiceOperationsManagement vol 6 no 3 pp 209ndash224 2004

[11] N Steven ldquoInventory control subject to uncertain demandrdquo inProduction and Operations Analysis pp 255ndash261 McGraw-HillIrwin New York NY USA 5th edition 2005

[12] P H Zipkin Foundations of Inventory Management McGraw-Hill New York NY USA 2000

[13] S Axsater ldquoA simple procedure for determining order quanti-ties under a fill rate constraint and normally distributed lead-time demandrdquo European Journal of Operational Research vol174 no 1 pp 480ndash491 2006

[14] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[15] D K Jana B Das and T K Roy ldquoA partial backlogginginventory model for deteriorating item under fuzzy inflationand discounting over random planning horizon a fuzzy geneticalgorithm approachrdquo Advances in Operations Research vol2013 Article ID 973125 13 pages 2013

[16] D K Jana K Maity B Das and T K Roy ldquoA fuzzy sim-ulation via contractive mapping genetic algorithm approachto an imprecise production inventory model under volumeflexibilityrdquo Journal of Simulation vol 7 no 2 pp 90ndash100 2013

[17] D K Jana B Das and M Maiti ldquoMulti-item partial back-logging inventory models over random planning horizon in

Random Fuzzy environmentrdquo Applied Soft Computing vol 21pp 12ndash27 2014

[18] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[19] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[20] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIIrdquo Information Sciencesvol 9 no 1 pp 43ndash80 1975

[21] J MMendel John and RI B ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[22] L A Zadeh ldquoQuantitative fuzzy semanticsrdquo Information Sci-ences vol 3 no 2 pp 159ndash176 1971

[23] S Gottwald ldquoSet theory for fuzzy sets of higher levelrdquo Fuzzy Setsand Systems vol 2 no 2 pp 125ndash151 1979

[24] J Xu and X Zhou Fuzzy Link Multiple-Objective DecisionMaking Springer Berlin Germany 2009

[25] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[26] B LiuTheory and Practice of Uncertain Programming PhysicaHeidelberg Germany 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of