Fuzzy EOQ Model for Deteriorating Items with Price ... · an inventory model where shortages are not allowed. For defuzzi - cation, he used signed distance method in order to nd the

Fuzzy EOQ Model forDeteriorating Items with Price

Dependent Demand usingGraded Mean Integration Value

1 R.Varadharajan and 2 Fabian Sangma. A1 Assistant Professor, Dept.of Mathematics, SRM University

Kattankulathur, Tamilnadu, [email protected]

2Dept.of Mathematics, SRM UniversityKattankulathur, Tamilnadu, India

Abstract

It is found from the literature that most of the au-thors have considered inventory problems without shortagein fuzzy environment. In this paper we proposed an Eco-nomic Order Quantity (EOQ) inventory model for deterio-rating items with price dependent demand with shortage infuzzy environment. We consider various costs as a triangu-lar fuzzy numbers to deal with the uncertainty which is hap-pened in the inventory situation. The different fuzzy costsare defuzzified by using Graded Mean Integration ValueMethod. Finally, Sensitivity analysis for the fuzzy modelhas been illustrated with numerical example.AMS Subject Classification: 90B05; 90B06; 90B30Key Words: Inventory Model, Triangular fuzzy number,Defuzzification, Graded Mean Integration Value.

1 Introduction

In todays environment, everything is seen as a business and everybusiness is different from other which requires lot of managementskills to run it. In a business, we need to find out what is bestfor the business and how can we manage our resources efficiently

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International Journal of Pure and Applied MathematicsVolume 119 No. 9 2018, 351-361ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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in order to gain profit. Maximum profit and minimum lose is themain goal for todays business. Figuring out what will provide ourgoal needs lot of proficiency. For this we require a system whichcan deliver our requirements where we control our stocks and gainmaximum profit. For this purpose we have system called Inven-tory control system. Inventory control is the process of regulatingstocks and maximizing the profit with minimum total cost. It playsan important role in controlling inventories and cost in business en-vironment. There are many inventory models have been proposedby the researchers depending upon various factors affecting inven-tories. One of the important factors is deterioration of items in theinventory system. Developing an optimal inventory policy for de-teriorating items cannot be ignored. Lot of modification has gonethrough in these models for deteriorating items to become morerealistic.

1.1 Fuzzy Set Theory

A fuzzy set was initially presented by A. Zadeh and D. Klaua asthe extension of notion of set. In mathematics, fuzzy sets can bedefined as the sets where elements of these sets have membershipfunctions. These membership functions are valued in the interval[0,1]. Fuzzy set theory is widely used for one purpose which is ifthe data or information are vague.

1.2 Fuzzy Inventory Control

There are many uncertainties in the inventory system where it canbe solved by using probabilistic method. But still some of the un-certainties in the inventories cannot be solved by this method. So,in order to optimize these uncertainties we introduce fuzzy wherethe uncertainties are taken as fuzzy numbers. These fuzzy numbersare solved by fuzzification and defuzzification methods. In todaysenvironment, the purchasing prices have always been the reason forthe demand to fluctuate or decrease so demand is dependent onpurchasing price. In this paper we proposed a fuzzy EOQ modelfor deteriorating items with price dependent demand. Fuzzy settheory is applied to the model where triangular fuzzy number rep-resents the demand and purchasing price. Graded mean integrationvalue method is used for defuzzification. Sensitivity analysis for theproposed model is given.

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1.3 Fuzzy Set

A fuzzy set is a pair (X,A) where X is a set and A : X → [0, 1]. Afuzzy set is where the members are allowed to have partial mem-bership and hence the degree of membership varies from 0 to 1.

1.4 α - Cut

α - cut of a fuzzy set A is the crisp set where the members havetheir membership values greater than or at least equal to α.

1.5 Fuzzy Number

Fuzzy number is expressed as a fuzzy set defining a fuzzy intervalin the real number ℜ. Fuzzy number should be normalized andconvex.

1.6 Triangular Fuzzy Number

Definition 1. The triangular fuzzy number can be defined asA = (a, b, c) and the membership function of this fuzzy number canbe taken as follows:

µA(x) =

x − a

b − a, when a ≤ x ≤ b

c − x

c − b, when b ≤ x ≤ c

0, otherwise

α - cut interval for this shape is given as

Aα = [(a2 − a1)α + a1 − (a3 − a2)α + a3]

2 Literature Review

In inventory model, many researchers have stretched their ideas inthis field inventory and modified their results according to the cir-cumstances occur where deterioration of items are to be considered.Lot of researchers continuously extended their work in this area.Jaggi and Mittal [6] established a EOQ model where the systemworks for the deteriorating items with the stocks being imperfect.This model helps in determining a process of finding perfect qualitystocks or a new resource for the retailers. Bhowmick and Samauta

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[7] developed a deterministic inventory model where shortages areallowed, with different production cycle and different productionrates. This inventory model was proposed for deteriorating items.Kasthuri et al [8] developed a model in which storage capacity; num-ber of orders and manufacturing cost are considered as constraints.Unit cost and demand are considered to be the parameters withshortages. For defuzzification, a Kuhn-Tucker condition is used.Umap [5] considered warehouses with deterioration for which herecommended Economic Order Quantity model where holding anddeteriorating cost were measured in fuzzy. Syed et al [3] proposedan inventory model where shortages are not allowed. For defuzzifi-cation, he used signed distance method in order to find the optimalsolution. Maragatham and Lakshmidevi [9] proposed an inventorymodel in fuzzy environmental with shortages being allowed. Theyused trapezoidal fuzzy number for fuzzification and to defuzzify it,they used signed distance method and found the optimal solution.

3 Mathematical Model of Inventory Sys-

tem with Shortages

The Characteristics of the model given as

Figure 1: Inventory System

In this model, S is the inventory level where it starts at t = 0.As the time reaches time Ta the level of inventory decreases withdemand and deterioration. The total order cycle is Tl and it isdivided into two parts Ta and Tb such that Tl = Ta + Tb. Duringthe time Ta, the items are drawn from the inventory as needed andduring the time Tb, the orders for the item are being accumulated

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but not filled. Then at the end of the interval Tl, an Quantity Z isproduced. This quantity decreases to make up the shortages thataccumulated in the time interval [Ta, Tl] the shortage mount up asthe quantity reduces. Procedure is continued again.

3.1 Presumptions

• The lead is considered to be zero.

• Deterioration will be instantaneous.

• Considered cycle will have no replacement.

• Considered model will have shortages

• Fuzzy demand and fuzzy purchasing cost are considered.

• For defuzzification, graded mean integration value method isused.

• Demand is given as r = ACα3 as unit price is related to them.

3.2 Derivation for Crisp Inventory

In this inventory model with shortages, there is a change in themodel due to r and λ. Since the change in Q(t) is directly propor-tional to demand and deterioration, and we get

dQ(t)

dt= −λQ(t) − r, 0 < t < Ta

and with demand

dQ(t)

dt= −r, Ta < t < Tl (1)

where the boundary conditions(bc’s) are Q(0) = S,Q(Ta) = 0.In order to solve this equation bc’s is used in (1).

IndQ(t)

dt= −λQ(t) − r, boundary condition Q(0) = S is used and

Q(t) = − r

λ+

(S +

r

λ

)e−λt for the interval 0 < t < Ta.

IndQ(t)

dt= −r, boundary condition Q(Ta) = 0 is used and

Q(t) = −D(t − t1) for the interval Ta < t < Tl.

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The total cost of this model is found by the following:The total inventory cost (TIC) = PC + OC + HC + DC − SCPurchasing cost (PC) = CpqOrdering cost (OC) = C0.Holding (carrying) cost

(HC) = Ch

∫ Ta

0Q(t)dt = Ch

[1

λ

(S +

r

λ

)(1 − e− λS

r+λ

)− rS

λ(r + λ)

]

Shortage cost

(SC) = CS

∫ Tl

TaQ(t)dt = CSr

[TaTl − T 2

a

2− T 2

l

2

]=

CsZ2

2r.

Deterioration cost (DC) = D[S − rTa] = DS

[1 − r

r + λ

].

The lot size q = rTl + λTa = rTa + rTb + λTa = S + Z.

Ta =S

r + λ=

q − Z

r + λ, Tb =

Z

rand Tl = Ta + Tb.

TIC = Cpq +Co +Ch

[1

λ

(q − Z +

r

λ

) (1 − e− λ(q−Z)

r+λ

)− r(q − Z)

λ(r + λ)

]

+CSZ2

2r+ D(q − Z)

[1 − r

r + λ

]

3.3 Derivation for Fuzzy Inventory

The derivation for the crisp inventory for the price dependent de-mand is derived and the derivation for the fuzzy inventory for theprice dependent demand has to be derived. Considering the modelin fuzzy where the parameters demand and purchasing cost are tri-angular fuzzy numbersC̃p = Fuzzy Purchase cost.r̃ = ACα

p .

T̃ IC = C̃p q+Co+Ch

[1

λ

(q − Z +

r̃

λ

) (1 − e− λ(q−Z)

r̃+λ

)− r̃(q − Z)

λ(r̃ + λ)

]

+CSZ2

2r̃+ D(q − Z)

[1 − r̃

r̃ + λ

]

= (Cp1 , Cp2 , Cp3) q+Co+Ch

[1

λ

(q − Z +

ACpα1

λ, q − Z +

ACpα2

λ, q − Z +

ACpα3

λ

)

−Ch

[1

λ

(q − Z +

ACpα1

λ

)(e

− λ(q−Z)ACpα

3+λ

),

(q − Z +

ACpα2

λ

)(e

− λ(q−Z)ACpα

2+λ

),

(q − Z +

ACpα3

λ

)(e

− λ(q−Z)ACpα

1+λ

)]

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+

(ACpα

1(q − z)

λ(ACpα3

+ λ),ACpα

2(q − z)

λ(ACpα2

+ λ),ACpα

3(q − z)

λ(ACpα1

+ λ)

)

+CSZ2

2

(1

ACpα3

,1

ACpα2

,1

ACpα1

)+ D(q − Z)

−D(q − Z)

[(ACpα

1

ACpα3

+ λ,

ACpα2

ACpα2

+ λ,

ACpα3

ACpα1

+ λ

)]

From this fuzzy total inventory cost, triangular fuzzy numbers a1, a2

and a3 can be found.

a1 = (Cp1)q + Co + Ch

[1

λ

(q − Z +

ACpα1

λ

)]

−Ch

[1

λ

(q − Z +

ACpα1

λ

)(e

− λ(q−Z)ACpα

3+λ

)]+

(ACpα

1(q − z)

λ(ACpα3

+ λ)

)

+CSZ2

2

(1

ACpα3

)+ D(q − Z) − D(q − Z)

[(ACpα

1

ACpα3

+ λ

)]

a2 = (Cp2)q + Co + Ch

[1

λ

(q − Z +

ACpα2

λ

)]

−Ch

[1

λ

(q − Z +

ACpα2

λ

)(e

− λ(q−Z)ACpα

2+λ

)]+

(ACpα

2(q − z)

λ(ACpα2

+ λ)

)

+CSZ2

2

(1

ACpα2

)+ D(q − Z) − D(q − Z)

[(ACpα

2

ACpα2

+ λ

)]

a3 = (Cp3)q + Co + Ch

[1

λ

(q − Z +

ACpα3

λ

)]

−Ch

[1

λ

(q − Z +

ACpα3

λ

)(e

− λ(q−Z)ACpα

1+λ

)]+

(ACpα

3(q − z)

λ(ACpα1

+ λ)

)

+CSZ2

2

(1

ACpα1

)+ D(q − Z) − D(q − Z)

[(ACpα

3

ACpα1

+ λ

)]

3.4 Defuzzification

The fuzzy numbers a1, a2 and a3 can be reduced to crisp values us-ing defuzzification method. Graded mean integration value methodis used on the fuzzy total cost to defuzzify it.

Graded mean integration value =a1 + 4a2 + a3

6= d(T̃ IC, 0).

d(T̃ IC, 0) =1

6

[(Cp1 + 4Cp2 + Cp3)q + 6Co

+Ch

[1

λ

(6(q − Z) +

ACpα1

λ+

4ACpα2

λ+

ACpα3

λ

) ]

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−Ch

[1

λ

(6(q−Z)+

(ACpα

1

λ

)(e

− λ(q−Z)ACpα

3+λ

)+

(4ACpα

2

λ

)(e

− λ(q−Z)ACpα

2+λ

)+

(ACpα

3

λ

) (e

− λ(q−Z)ACpα

1+λ

)]]+

Ch(q − Z)

λ

(ACpα

1

ACpα3

+ λ+

4ACpα2

ACpα2

+ λ+

ACpα3

ACpα1

+ λ

)

+CSZ2

2

(1

ACpα3

+4

ACpα2

+1

ACpα1

)+ 6D(q − Z)

−D(q − Z)[ (

ACpα1

ACpα3

+ λ+

4ACpα2

ACpα2

+ λ+

ACpα3

ACpα1

+ λ

)]

dF (Z)

dZ=

1

6

[− 6Ch

λ−Ch

λ

(−6+

ACpα1

ACpα3

+ λ+

4ACpα2

ACpα2

+ λ+

ACpα3

ACpα1

+ λ

)

−Ch

λ

( ACpα1

ACpα3

+ λ+

4ACpα2

ACpα2

+ λ+

ACpα3

ACpα1

+ λ

)

+CSZ( ACpα

1

ACpα3

+ λ+

4ACpα2

ACpα2

+ λ+

ACpα3

ACpα1

+ λ

)

+D( ACpα

1

ACpα3

+ λ+

4ACpα2

ACpα2

+ λ+

ACpα3

ACpα1

+ λ

)− 6D

]

4 Numerical Example

In order to show there is unique method to optimize inventorymodel using fuzzy set theory, a numerical example is given. Fromthis example it can be shown that the output of this model has un-certainties which can be allocated to source of input uncertainties.Input data :Deterioration rate λ = 0.6, Ordering cost Co = Rs. 750 per order,Lot size Q = 350, D = Rs. 12 per unit, A = 75, α = 0.05,Demand r = 101.9658.

The following table shows the Sensitivity Analysis forincrease in holding cost and shortage cost

S.No. Ch CS Cp Z S Ta Tb Tl TIC1 2 4 (39,40,41) 150.9414 199.0586 1.9408 1.4716 3.4124 16297.32 3 5 (39,40,41) 180.4180 169.5820 1.6534 1.7591 3.4125 17397.93 4 6 (39,40,41) 200.0691 149.9309 1.4618 1.9506 3.4125 18060.7

The following table shows the Sensitivity Analysis forincrease in holding cost alone


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The following table shows the Sensitivity Analysis forincrease in shortage cost alone


5 Conclusion

An inventory model with fuzzy environment is developed consid-ering price dependent demand. It is proved that there exists aunique method to solve the uncertainties which is caused due tothe deterioration. Even shortages are allowed to find the solutionby fuzzification with triangular fuzzy number and defuzzificationwith graded mean integration value method. Sensitivity Analy-sis is showed for different holding cost and shortage cost to showuniqueness of this model. For future study, this model can be im-proved with deteriorating items being allowed to be replaced. Evenhaving limitations in budget of the inventory can be allowed.

References

[1] Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility,Fuzzy Sets and Systems, (1), (1978), 3-28.

[2] Zimmermann, H. J., Fuzzy Set Theory and Its Applications,Kluwer Academic, Boston.

[3] J.K. Syed and L.A. Sahu, B.Bhaula and L.K.Raju, An inven-tory model without shortages using singed distance method,Applied Mathematics and Information Science, 1(2), (2007),200-203.

[4] Klir, G. J., and Bo Yuan, Fuzzy Sets and Fuzzy Logic: Theoryand Applications, Prentice hall of India Private Limited.

[5] H.P.Umap, Fuzzy EOQ Model for deteriorating items withtwo warehouses, Journal of Statistics and Mathematics, 1(2),(2010), 1-6.

[6] C. K. Jaggi and M. Mittal , Economic order quantity modelfor deteriorating items with imperfect quality, Revista Investi-gation Operational, 32(2), (2011), 107-113.

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[7] J. Bhowmick and G. P. Samutha , A deterministic inventorymodel of deteriorating items with two rates of production,shortages and variable production cycle, Internationally Schol-arly Research Network, ISRN Applied Mathematics, (2011) 1-16.

[8] R. Kasthri, P. Vasanthi, S.Ranganayagi and C.V. Seshaiah, Multi Item inventory model involving three constraints: AKarush-Kuhn- Tucker conditions approach, American Journalof Operational Research, (2011), 155-159.

[9] M. Maragatham and P.K. Lakshmidevi, A fuzzy inventorymodel for deteriorating items with price dependent demand,International journal of Fuzzy Mathematical Archive, (2014),39-47.

List of Notations:Q(t) - The level of inventory at time t; R - Rate of Demand;q - Order quantity; λ - Rate of deterioration in [0, Ta];Z - The inventory shortage level; S - The level of inventory at t = 0;Ta - Time where the inventory attains zero;Tb - Time where shortages exist; Tl - Total cycle timeCh - Holding cost per unit item; CS - Shortage cost per unit itemCp - Purchase cost per unit item; Co - Ordering cost per orderTIC - The total inventory cost for the period [0, Tl]C̃3 - Purchase cost per unit item in fuzzy; r̃ - Demand rate in fuzzy

T̃ IC - Total inventory cost for the period [0, Tl] in fuzzy

T̃ ICGmi - Defuzzified T̃ IC by graded mean integration valueT̃a - The fuzzy time where the inventory attains zeroT̃b - The fuzzy time where shortages are allowed

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Fuzzy EOQ Model for Deteriorating Items with Price ... · an inventory model where shortages are not allowed. For defuzzi - cation, he used signed distance method in order to nd the