A New Multi-criteria Scenario-based Solution Approach

7/27/2019 A New Multi-criteria Scenario-based Solution Approach

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Ann Oper ResDOI 10.1007/s10479-013-1435-z

A new multi-criteria scenario-based solution approachfor stochastic forward/reverse supply chain networkdesign

Hamed Soleimani Mirmehdi Seyyed-Esfahani Mohsen Akbarpour Shirazi

Springer Science+Business Media New York 2013

Abstract Analyzing current trends in supply chain management, lead to nd unavoidablesteps toward closing the loop of supply chain. In order to expect best performance of Closed-Loop Supply Chain (CLSC) network, an integrated approach in considering design and plan-ning decision levels is necessary. Further, real markets usually contain uncertain parameterssuch as demands and prices of products. Therefore, the next important step is consideringuncertain parameters.

In order to cope with designing and planning a closed-loop supply chain, this paperproposes a multi-period, multi-product closed-loop supply chain network with stochasticdemand and price in a Mixed Integer Linear Programming (MILP) structure. A multi cri-teria scenario based solution approach is then developed to nd optimal solution throughsome logical scenarios and three comparing criteria. Mean, Standard Deviation (SD), andCoefcient of Variation (CV), which are the mentioned criteria for nding the optimal solu-tion. Sensitivity analyses are also undertaken to validate efciency of the solution approach.The computational study reveals the acceptability of proposed solution approach for thestochastic model. Finally, a real case study in an Indian manufacturer is evaluated to ensureapplicability of the model and the solution methodology.

Keywords Closed-loop supply chain Mixed integer linear programming Reverselogistic Stochastic optimization Scenario-based solution

1 Introduction

The vast tendency toward closing the loop of supply chain originate in its capability andfeasibility in terms of economic criteria, which lead managers to think of prot maximiza-tion instead of cost minimization approaches (Guide and Van Wassenhove2009). Although

H. Soleimani M. Seyyed-Esfahani (B ) M.A. ShiraziAmirkabir University, Valiasr Crossroad, Tehran, Irane-mail:[email protected]

H. Soleimanie-mail:[email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Fig. 1 A generic form of forward/reverse logistics (Tonanont et al.2008)

the classical form of supply chain (forward supply chain) just tried to fulll customers re-quests (Chopra and Meindl2007), new denitions make supply chain responsible for End of Life (EOL) products too (reverse supply chain). For instance, in the US, 75 % of customersclaim that they consider environmental reputation of manufacturers in their purchasing, and

80 % of customers even pay more for environmental friendly goods (Lamming and Hamp-son 1996). In a CLSC, manufacturers have to be responsible for collecting used productsfrom customers and trying to reuse them in any possible forms or at least dispose them(Soleimani et al.2013). Various procedures can be undertaken for reused products (calledreturn products) including simple repairing and then reselling them to second markets, re-manufacturing the EOL products, recycling return products to use them as raw materials,and environmental-friendly disposing. A generic structure of CLSC is illustrated in Fig.1(Tonanont et al.2008). In this gure, forward, and reverse supply chain are presented bysolid lines and dashed lines, respectively.

The problem of CLSC design and planning is NP-hard and complicated (Krarup andPruzan1983 and Schrijver2003), which is hard to achieve optimal solutions for real-sizedinstances. On the other hand, the other important factor of real situations is the uncertainty of parameters. Hence, in this paper, bi-level important decisions of designing and planning of a multi-period, multi-product CLSC are undertaken in an uncertain environment. Demandsof rst and second customers, price of selling rst and second products, and the price of purchasing used products from customers are considered as nondeterministic parameters inthe study. In order to guarantee the applicability of the model, an efcient scenario-basedsolution approach is developed to cope with the proposed stochastic model. A case study of a plastic water cane products manufacturer in India is exploited to evaluate the model andthe solution approach.

The rest of this paper is arranged as follow. In Sect.2, a complete literature review ispresented. The model characteristics and formulation is demonstrated in Sect.3. Computa-tional analyses and the proposed scenario-based solution approach are illustrated in Sect.4.Section 5 is assigned to sensitivity analysis. Case study evaluation is presented in Sect.6.Finally, Sect.7 discusses conclusions of the study and future research.


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2 Literature review

Designing and planning a CLSC with stochastic parameters is a critical issue, which needsto be considered by researchers. There are few papers dealing with mentioned problem and

trying to solve it in an efcient and practical way. Recent review papers can shed morelight on this claim. Pokharel and Mutha (2009) reviewed the current advancements in re-verse logistics (RL) and they mentioned about necessity of generic models and stochasticapproaches in this area. Also Subramoniam et al. (2009) presented another review in re-verse logistics in automotive industry, which pointed out the lack of stochastic approachesin designing CLSC. Such points can also nd in Sasikumar and Kannan (2009).

Concentrating on design and planning problem of CLSC, and regarding stochastic ap-proaches, there are some papers, which can be discussed as follows (Table1). The charac-teristics of this study are claried at the last row of Table1.

Reviewing Table1 can be helpful to be convinced of the necessity of this study in three

points of view: In terms of model, there are few stochastic, multi-period, and multi-product papers (rows

6 and 13), which is regarded in this study. Indeed, the proposed model of this study isscenario-based, multi-period, and multi-product with various possible ows between net-work entities, which can construct a close-to-real network.

In terms of stochastic parameters, this study proposes a complete set of nondeterministicparameters, which are demands of the rst and second customers (return rate), price of rst products, price of return products, and purchasing price of EOL products. This paper,as regarding Table1, is the only, which considers such nondeterministic parameters in its

stochastic approach. In terms of nondeterministic solution approaches, there are various methodologies indealing with such problems. This paper tries to develop a new efcient scenario-basedapproach, which can rationally achieve optimal solutions through some criteria. It shouldbe pointed out that based on the difculties of other approaches such as two-stage stochas-tic optimization approaches specially in real world instances, scenario-based solutionmethodologies are mentioned as effective and well-behaved solution methods for stochas-tic problems (Dembo1991 and Kaut and Wallace2007).

On the other hand, using three criteria in selecting optimal point in scenario-based solu-tion methodologies is developed by this study to elevate the reliability of optimal deci-sions in uncertain environment.Finally, based on the above consideration, and the analyses of literature review in Table1,

the necessities of current study is revealed in terms of proposing and solving a stochasticmodel in an efcient and practical manner.

3 Model development

The model notations are based on the multi-products, multi-period, and multi echelon CLSC

network that presented in Soleimani et al. (2013) (Fig.2) except that current study is a non-deterministic research and some notations are also changed to improve the applicability of the model. There are also differences in the assumptions of the model, which are completelypresented as follows:

The model is scenario-based, multi-echelon, multi-product, and multi-period consists of suppliers, manufacturers, warehouses, distributors, and customers in its forward supply


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T a

b l e 1

L i t e r a t u r e s u r v e y o f C L S C d e s i g n a n d p l a n n i n g u n d e r u n c e r t a i n t y

R o w

P a p e r

P e r i o d

P r o d u c t

D e m a n d s

R e t u r n

P r i c e s

U n c e r t a i n a p p r o a c h

S o l u t i o n m e t h o d

1 .

z k r a n d B

a s l g i l ( 2 0 1 3 )

M u l t i

S i n g l e

U n c e r t a i n

F u z z y

C P L E X

2 .

P i s h v a e e e t a l . (

2 0 1 1 )

S i n g l e

S i n g l e

U n c e r t a i n

U n c e r t a i n

S t o c h a s t i c

C P L E X

3 .

P i s h v a e e a n d T o r a b i ( 2 0 1 0 )

M u l t i

S i n g l e

U n c e r t a i n

U n c e r t a i n

F u z z y

L I N G O

4 .

E l - S a y e d e t a l . ( 2 0 1 0 )

M u l t i

S i n g l e

U n c e r t a i n

U n c e r t a i n

S t o c h a s t i c - T w o s t a g e

D a s h o p t i m i z a t i o n

5 .

F r a n c a s a n d M i n n e r ( 2 0 0 9 )

S i n g l e

M u l t i

U n c e r t a i n

U n c e r t a i n


S i m u l a t i o n

6 .

z c e y l a n a n d P a k s o y ( 2 0 1 3 )

M u l t i

M u l t i

U n c e r t a i n

U n c e r t a i n

F u z z y

C P L E X

7 .

L i s t e s ( 2 0 0 7 )

S i n g l e

S i n g l e

U n c e r t a i n

U n c e r t a i n


C P L E X

8 .

A m i n a n d Z

h a n g ( 2 0 1 2 )

S i n g l e

M u l t i

F u z z y

G A M S

9 .

C h o u i n a r d e t a l . (

2 0 0 8 )

S i n g l e

S i n g l e

U n c e r t a i n

U n c e r t a i n


C P L E X

1 0 .

F a c c i o e t a l . ( 2 0 1 1 )

S i n g l e

M u l t i

U n c e r t a i n

S t o c h a s t i c

G e n e r a l s o l v e r s

1 1 .

Z h o u a n d M

i n ( 2 0 1 1 )

S i n g l e

M u l t i

U n c e r t a i n

S t o c h a s t i c

G e n e t i c a l g o r i t h m

1 2 .

A m i n a n d Z

h a n g ( 2 0 1 3 )

U n c e r t a i n

S t o c h a s t i c ( F u z z y w e i g h t s )

E x a c t

1 3 .

Z h u a n d X i u q u a n ( 2 0 1 3 )

M u l t i

M u l t i

U n c e r t a i n

S t o c h a s t i c

H y b r i d g e n e t i c a l g o r i t h m

1 4 .

R a m e z a n i e t a l . ( 2 0 1 3 )

S i n g l e

M u l t i


M u l t i c r i t e r i a a p p r o a c h e s

1 5 .

T h i s s t u d y

M u l t i

M u l t i

U n c e r t a i n

U n c e r t a i n

U n c e r t a i n

S t o c h a s t i c

N e w S c e n a r i o b a s e d s o l u t i o n


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chain and disassembly centers, redistributors, disposal centers, and second customers inits reverse logistics.

Dealing with used products can be undertaken in four alternatives: repairing by disassem-bly centers, remanufacturing by manufacturers, recycling by suppliers, and disposing by

disposal centers. Disassembly centers are responsible for collecting used products from rst customers, de-ciding the next-step alternative decisions for return products, and repairing some portionsof them.

Demands of rst customers and price of rst products are directly considered as non-deterministic parameters through some scenarios. Besides, return products rate, price of second products, and purchasing price of used products are also regarded as stochastic pa-rameters through factors related to demands of rst customers and price of rst productsrespectively.

In terms of designing decision variables, the maximum number of facilities could regard

nondeterministic and it could be different for each scenario. The potential locations, capacity of all facilities, and all cost parameters are predeter-mined.

Quality, demand, and price of returned products are different from rst customers andthey cannot be sold as new products.

In terms of network ows, manufacturers, warehouses, and distributers can supply rstcustomers and manufacturers, warehouses, disassembly centers, and redistributors cansupply second customers.

The formulation of the model is presented as follows:

Sets:

S: Set of scenarios, indexed by s.L: Potential number of suppliers, indexed by l.F: Potential number of manufacturers, indexed by f .W: Potential number of warehouses, indexed by w .

Fig. 2 The CLSC network structure (arrows show the possible ows) (Soleimani et al.2013)


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D: Potential number of distributors, indexed by d .C: Potential number of rst customers (retailers), indexed by c.A: Potential number of disassembly centers, indexed by a .R: Potential number of redistributors, indexed by r .

P: Potential number of disposal centers, indexed by p .K: Potential number of second customers, indexed by k.U: Set of products, indexed by u.T: Set of periods, indexed by t .

Parameters:

S s : Maximum number of suppliers in scenario s.F s : Maximum number of manufacturers in scenario s.W s : Maximum number of warehouses in scenario s.D s : Maximum number of distributors in scenario s.

A s : Maximum number of disassembly centers in scenario s.R s : Maximum number of redistributors in scenario s.P s : Maximum number of disposal centers in scenario s.M : a sufciently large constant.D cuts : Demand of product u of rst the customer c in period t in scenario s,D kuts : Demand of product u of the second customer k in period t in scenario s,P cuts : Unit price of product u at rst customer c in period t in scenario s,PU cuts : Purchasing cost of product u at rst customer c in period t in scenario s,P kuts : Unit price of product u at second customer k in period t in scenario s,F i : Fixed cost of activating location i . DS ij : Distance between location i and location j .SC lut : Capacity of supplier l of product u in period t ,SRC lut : Recycling capacity of supplier l of product u in period t ,FC f ut : Manufacturing capacity of manufacturer f of product u in period t , RFC f ut : Remanufacturing capacity of manufacturer f of product u in period t ,WC wut : Warehouse capacity of warehouse w of product u in period t , DC dut : Capacity of distributor d of product u in period t , AC au : Capacity of disassembly a of product u in period t , RDC rut : Capacity of redistributor r of product u in period t ,PC put : Capacity of disposal center p of product u in period t , MT lut : Material cost of product u per unit which is supplied by supplier l in pe-

riod t , RT sut : Recycling cost of product u per unit which is recycled by supplier l in pe-

riod t ,FT f ut : Manufacturing cost of product u per unit, which is undertaken by manufacturerf in period t ,

RFT f ut : Remanufacturing cost of product u per unit, which is undertaken by manufac-turer f in period t ,

DAT aut : Disassembly cost of product u per unit by disassembly center a in pe-riod t ,

RPT aut : Repairing cost of product u per unit that is repaired by disassembly center a in period t ,

PT aut : Disposal cost of product u per unit disposed by disposal center p in pe-riod t ,

NMT f ut : Non-utilized manufacturing capacity cost of product u of manufacturer f in period t ,


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NRMT f ut : Non-utilized remanufacturing cost of product u of manufacturer f in pe-riod t ,

ST ut : Shortage cost of product u per unit in period t ,Fh f u : Manufacturing time of product u per unit at manufacturer f , RFh

f u: Remanufacturing time of product u per unit at manufacturer f ,

RT sut : Recycling cost of supplier l of product u in period t ,WHT wut : Holding cost of product u per unit at the warehouse w in period t , DHT dut : Holding cost of product u per unit at the store of distributor d in period t ,B lu , B f u , B du , B au , B ru , B wu , B cu : Batch size of product u from supplier l manufac-turer f , distributor d , disassembly a , redistributor r , warehouse w and cus-tomer c respectively.

TRT ut : Transportation cost of product u per unit per kilometer in period t , RRut : Return ratio of product u at rst customers in period t , Rc: Recycling ratio,

Rm: Remanufacturing ratio, Rr : Repairing ratio, Rp: Disposal ratio,

First-stage decision variables:

L is : Binary variable equals 1 if location i is activated in scenario s and 0 other-wise.

Second-stage decision variables:

Q ijuts : Flows of product u from node (entity) i to node (entity) j in period t in

scenario s,R wuts : Residual inventory of product u for warehouse w in period t in scenarios,

R duts : Residual inventory of product u for distributor d in period t in scenario s.TLijs : Binary variable, which is equal to 1 if a transportation link is established betweennode i and node j in scenario s and 0 otherwise.

3.1 Objective function

Objective function is total prot, which can be calculated by total sales minus total costs fora scenario. Total sales:

First products sales (ows that start from distributors, manufacturers, and warehousesto customers):

d D cC uU t T

(Q dcuts Bdu P cuts ) +f F cC uU t T

(Q fcuts B f u P cuts )

+wW cC uU t T

(Q wcuts Bwu P cuts ) s S, (3.1)

Total sales of second products (return product ows, which start from redistributors,warehouses, and manufacturers to second customers):

rR kK uU t T

(Q rkuts B ru P kuts ) +f F kK uU t T

(Q fkuts B f u P kuts )

+wW kK uU t T

(Q wkuts Bwu P kuts ) s S (3.2)


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Total costs: Total costs are calculated for each scenario as follows:Total costs= xed costs+ material costs+ manufacturing costs+ non-utilized ca-

pacity costs+ shortage costs+ purchasing costs+ disassembly costs+ recycling costs+remanufacturing costs+ repairing costs+ disposal costs+ transportation costs+ inven-

tory holding costs.Each of the above mentioned costs are calculated for each scenario as follows:Fixed costs (location costs):

lL

F l L ls +f F

F f L f s +d D

F d L ds +aA

F a L as +rR

F r L rs +pP

F p L ps

+wW

F w L ws s S (3.3)

Material costs (return materials benets should be considered here):

lL f F uU t T

Q lf uts B lu MT lut aA ll uU t T

Q aluts Bau (MT lut RT lut ) s S

(3.4)

Manufacturing costs:

f F d D uU t T

(Q fduts B f u F T f ut ) +f F wW uU t T

(Q fwuts B f u F T f ut )

+f F cC uU t T

(Q fcuts B f u F T f ut ) +f F kK uU t T

(Q fkuts B f u F T f ut ) s S

(3.5)Non-utilized capacity costs for manufacturers:

f F uU t T

(F C f u /F h f u )L f s d D

(Q fduts B f u ) wW

(Q fwuts B f u )

cC

(Q fcuts B f u ) +wW rR

(Q wruts Bwu ) +wW kK

(Q wkuts Bwu ) NMT f u

+

f F uU t T

( RFC f u /RFh f u )L f s

rR

(Q fruts B f u )

kK

(Q fkuts B f u )

wW rR

(Q wruts Bwu ) +wW kK

(Q wkuts Bwu ) NRMT f u s S (3.6)

Shortage costs (for distributor):

Shortage uts = (cC

(uU

(t T

D cuts t T d D

(Q dcuts Bdu ) t T f F

Q fcuts B f u

t T wW

Q wcuts Bwu s S

Shortage costs =uU t T

Shortage uts ST ut s S

(3.7)

Purchasing costs:

cC aA uU t T

Q cauts P U cuts B cu s S (3.8)


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Disassembly costs:

cC aA uU t T

Q cauts Bcu DAT aut s S (3.9)

Recycling costs:

aA lL uU t T

Q aluts Bau RT sut s S (3.10)

Remanufacturing costs:

aA f F uU t T

Q afuts Bau RFT f ut s S (3.11)

Repairing costs:

aA rR uU t T

Q aruts Bau RPT aut s S (3.12)

Disposal costs:

aA pP uU t T

Q aputs Bau PT put s S (3.13)

Transportation costs:

t T uU lL f F

Q lf uts B su TRT ut DS lf +t T uU f F d D

Q fduts B f u TRT ut DS f d

+t T uU f F wW

Q fwuts B f u TRT ut DS f w

t T uU f F cC

Q fcuts B f u TRT ut DS f c +t T uU f F kK

Q fkuts B f u TRT ut DS f k

+t T uU wW cC

Q wcuts Bwu TRT ut DS wc

t

T u

U w

W k

K

Q wkuts Bwu TRT ut DS wk +

t

T u

U d

D c

C

Q dcuts Bdu TRT ut DS dc

+t T uU aA lL

Q aluts Bau TRT ut DS as

t T aA uU f F

Q afuts Bau TRT ut DS af +t T uU aA pP

Q aputs Bau TRT ut DS ap

+t T uU aA rR

Q aruts Bau TRT ut DS ar

t T uU f F rR

Q f ruts B f u TRT ut DS f r +t T uU wW rR

Q wruts Bwu TRT ut DS wr

+t T uU rR kK

Q rkuts B ru TRT ut DS ruk

t T uU cC aA

Q cauts Bcu TRT ut DS ca

+t T uU wW d D

Q wduts Bwu TRT ut DS wd s S

(3.14)


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Inventory holding costs:

wW uU t T

R wuts WHT wut +d D uU t T

R duts DHT dut s S (3.15)

3.2 Constraints

lL

Q lf uts B su =d D

Q fduts B f u +wW

Q fwuts B f u +cC

Q fcuts B f u ,

s S, u U, f F, t T (3.16)

f F

Q fwuts B f u = R wuts +d D

Q wduts Bwu +cC

Q wcuts Bwu ,

s S, u U, w W, t T (3.17)

f F

Q fduts B f u +wW

Q wduts Bwu = R dut +cC

Q dcuts Bdu ,

s S, u U, d D, t T (3.18)

d D

Q dcuts Bdu +f F

Q fcuts B f u +wW

Q wcuts Bwu + Shortage uts D cuts ,

s S, u U, c C, t T (3.19)

aA

Q cauts Bcu =

d D

Q dcuts Bdu +

f F

Q fcuts B f u +

wW

Q wcuts Bwu RRut ,

s S, u U, c C, t T (3.20)

cC

Q cauts Bcu =lL

(Q aluts Bau ) +f F

(Q af uts Bau ) +rR

(Q aruts Bau ) +pP

(Q aputs Bau ),

s S, u U, a A, t T (3.21)

cC

(Q cauts Bcu ) Rc =lL

(Q aluts Bau ), s S, u U, a A, t T (3.22)

cC

(Q cauts Bcu ) Rm =

f F

(Q af uts B au ), s S, u U, a A, t T (3.23)

cC

(Q cauts Bcu ) Rr =rR

(Q aruts Bau ), s S, u U, a A, t T (3.24)

cC

(Q cauts Bcu ) Rp =pP

(Q aputs Bau ), s S, u U, a A, t T (3.25)

Rc + Rm + Rr + Rp = 1 (3.26)

aA

(Q af uts Bau ) =rR

(Q f ruts B f u ) +kK

(Q fkuts B f u ),

s S, u U, f F, t T (3.27)

aA

(Q aruts Bau ) +f F

(Q fruts B f u ) =kK

(Q rkuts B ru ),

s S, u U, r R, t T (3.28)

rR

(Q rkuts B ru ) D kuts , s S, u U, k K, t T (3.29)


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Constraints (3.16) to (3.29) are balanced constraints. These constraints guarantee theequality of all entering ows to a network entity and all issuing ows of the same entityin a scenario. These constraints should be hold for all entities in all periods of a scenario.Precisely, constraints (3.16) are the balance constraints of manufacturers for a scenario,

constraints (3.17) to (3.21) are related to warehouses (3.17), distributors (3.18), customers(3.19), disassembly centers inputs (3.20), and disassembly centers output (3.21), respec-tively. Again, constraints (3.22) to(3.29) are recycling rate constraints (3.22), remanufactur-ing rate constraints (3.23), repairing rate constraints (3.24), disposal rate constraints (3.25),manufacturers reverse ows constraints (3.27), redistributors constraints (3.28), and ulti-mately, second customers constraints (3.29). Finally, sum of all alternative decision rates forused products should be equal to 1 for a scenario (constraint (3.26)).

f F

Q lf uts B su SC lut L ls , s S, u U, l L, t T (3.30)

d D

Q fduts B f u +wW

Q fwuts B f u +cC

Q fcuts B f u +kK

Q fkuts B f u Fh f u FC f ut L f s

s S, u U, f F, t T (3.31)

R wut WC wut L ws , s S, u U, w W, t T (3.32)

f F

Q fduts B f u +wW

Q wduts Bwu + R dut DC dut L ds ,

s S, u U, d D, t T (3.33)

lLQ aluts Bau +

f F Q af uts Bau +

rRQ aruts Bau +

pP Q aputs Bau AC aut L as ,

s S, u U, a A, t T (3.34)

kK

Q rkuts B ru RC rut L rs , s S, u U, r R, t T (3.35)

aA

Q aluts Bau SRC lut L ls , s S, u U, l L, t T (3.36)

aA

Q aputs Bau PC put L ps , s S, u U, p P , t T (3.37)

f F

Q fwuts B f u WC wut L ws , s S, u U, w W, t T (3.38)

Constraints (3.30) to (3.38) are capacity constraints. These constraints ensures that allentities entering and issuing ows be less than their capacities. Precisely, constraints (3.30)control all suppliers output capacity for each product in all periods for a scenario. Con-straints (3.31) to (3.38) are capacity limitations of manufacturers, warehouses, distributors,redistributors, suppliers, disposal centers, and warehouses respectively.

TLlfs uU t T

Q lf uts M TLlfs , l L, f F, s S (3.39)

TLf ds uU t T

Q fduts M TLf ds , f F, d D, s S (3.40)

TLf ws uU t T

Q fwuts M TLf ws , f F, w W, s S (3.41)


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TLf cs uU t T

Q fcuts M TLf cs , f F, c C, s S (3.42)

TLf ks uU t T

Q fkuts M TLf ks , f F, k K, s S (3.43)

TLf rs uU t T

Q fruts M TLf rs , r R, f F, s S (3.44)

TLwds uU t T

Q wduts M TLwds , w W, d D, s S (3.45)

TLwcs uU t T

Q wcuts M TLwcs , w W, c C, s S (3.46)

TLwks uU t T

Q wkuts M Liwks , w W, k K, s S (3.47)

TLwrs uU t T

Q wruts M Liwrs , w W, r R, s S (3.48)

TLdcs uU t T

Q dcuts M TLdcs , d D, c C, s S (3.49)

TLcas uU t T

Q cauts M TLcas , a A, c C, s S (3.50)

TLals uU t T

Q aluts M TLals , l L, a A, s S (3.51)

TLaf s uU t T

Q af uts M TLaf s , f F, a A, s S (3.52)

TLars uU t T

Q aruts M TLars , r R, a A, s S (3.53)

TLaps uU t T

Q aputs M TLaps , p P , a A, s S (3.54)

TLrks uU t T

Q rkuts M TLrks , k K, r R, s S (3.55)

Constraints (3.39) to (3.55) are related to shipping limitations between two entities. Forinstance, in constraints (3.39), if in a scenario there is no real shipping way between a sup-plier and a manufacturer, so there would be no ows between them and vice versa.

lL

L ls S s s S (3.56)

f F

L f s F s s S (3.57)

d D

L ds D s s S (3.58)

wW

L ws W s s S (3.59)

aA

L as A s s S (3.60)

rR

L rs R s s S (3.61)


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Table 2 Parameters range of thecomputational study Row Parameter Uniformly distributed or rates

1. Demand 030002. Second demand rate 50 % of demand

3. Price 15000200004. Second product price 50 % of price5. Purchasing cost 10 % of price6. Manufacturer capacity 6000140007. Remanufacturer capacity 50 % of Manufacturer capacity8. Supplier capacity 18000420009. Supplier recycling capacity 50 % of Supplier capacity10. Recycling cost 1010011. All other reverse cost 1010012. Others facilities capacities 60001400013. Material cost 100100014. Manufacturing cost 100100015. All other forward cost 100100016. Shortage cost 1000500017. Supplier xed cost 710 million18. Manufacturer xed cost 70150 million19. Distributor xed cost 12 million20. warehouse xed cost 0.11 million21. Disassembly xed cost 0.11 million22. Redistributors xed cost 0.11 million23. Disposal centers xed cost 0.11 million24. Batch size 1

pP

L ps P s s S (3.62)

Constraints (3.56) to (3.62) are the limitations to maximum number of allowable acti-vated locations for each scenario. These types of managerial limitations or budget relatedlimitations are necessary to complete the model.

4 Solution methodology and computational evaluation

In order to evaluate the effectiveness of the proposed stochastic model, a CLSC consists of ve units of each entity is considered (ve suppliers, ve manufacturers, ve warehousesetc.). The study is undertaken for 5 products in 5 periods and 11 various logically-generatedscenarios. The data are generated based on the uniform distributed functions on the basisof Table2. It should be mentioned that the recycling, remanufacturing, repair, and disposalrates are respectively 0.2, 0.4, 0.3, and 0.1 of return products and batch sizes are one for allentities.

The above-mentioned ranges in Table2 reveal the intervals, which parameters of themodel are generated. On the other hand, considering real market situations, prices and de-mands are two main uncertain parameters, which are regarded as nondeterministic scenario-based values here. Quality and suitability of scenario-generation methods for a given


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Table 3 Objective function values of 11 different scenarios (in millions)

Scenarios Deterministic S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 Worst-case Best-case

Prot 3309 3400 3263 3274 3126 3450 3276 3304 3165 2191 5453

stochastic programming models are discussed and proved in various studies such as Dembo(1991) and Kaut and Wallace (2007), and Listes (2007). They also pointed out that based onsimplicity and applicability characteristics of scenario-based approaches in comparison withother methods (like two-stage stochastic programming), they can be introduced as efcientand powerful methodologies to cope with stochastic programming problems. In this study,we develop a new multi-criteria scenario-based approach to solve the proposed stochasticproblem of CLSC design and planning.

Totally, 11 different scenarios are assigned logically and randomly. In the rst scenario,

which called deterministic scenario, mean values of demands and prices (1500 and 17500respectively) are considered. Then, 4 various possibilities for demands based on the range of the data in Table2 (0300) and 2 different possible values for prices are created. The combi-nation of these possibilities, leads to 8 various scenarios. In order to cover extreme situations,two more scenarios are considered: worst-case and best-case scenarios. The worst-case sce-nario considers demands at the lowest possible values (1000) and the best-case scenarioconsiders them at the highest possible values (2500). Finally, adding the last 2 scenarios, 11different scenarios are created for demands and prices, which affect return rates, return prod-uct demands, and purchasing price of used products. All computations are coded by IBMILOG CPLEX 12.2 optimization software and run with a Core 2 Duo-2.26 GHz processorlaptop. The complete solution steps are illustrated as follows: First, all scenarios are solved separately and the optimum points are reported and

recorded. These solutions are candidate solutions for nal optimal point. Indeed, decision-makers try to make best decision now, to ensure best performance in future. Therefore,these candidate solutions should be evaluated to nd best one in terms of regarding allscenarios.

Second, the candidate solutions are evaluated in all scenarios and their performance (objective function values) are recorded.

Third, three criteria are evaluated to analyze the overall performance of each candidatesolutions in facing with all scenarios. This paper presents mean, standard deviation, andcoefcient of variation as acceptable criteria to decide about best solution among all can-didate solutions.

Finally, optimal solution are selected based on the analyses of three criteria and appro-priate sensitivity analyses are undertaken to determine the reliability of the developedsolution approach.

Under the above consideration, the rst step is solving all scenarios to achieve optimalsolution of each one, which are mentioned as candidate solutions for nal decision. The re-sults are illustrated in Table3. Further, Fig.3 contains a schematic view of different objective

function (prot) values.Analyzing Table3 and Fig.3, lead to important conclusions to be convinced of suf-ciency of the number of scenarios:

The total mean of scenario objective values is 3282 million. There are 5 scenarios withgreater values, and 6 scenarios with lower values than mean value. This proves the fairrandom distributions of scenarios to consider all situations.


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Fig. 3 Objective function values (prot) of all 11 scenarios (in millions)

Mean Square Error (MSE) of results except the worst and the best cases is 101 million.Meanwhile, considering the worst and the best cases, MSE will be around 766 million(23 % of mean value). This point can prove the diversication of various scenarios. In-deed, we can nd different range of solutions and it can guarantee achieving the appropri-ate candidate solutions. Further, the range of results except the worst and the best cases is324 million and considering the worst and the best cases, range will be 3252 million (96 %of mean value). Again, this can reconrm the discussion about diversication issues.

Considering the above-mentioned points lead to conclude that the generated scenarioscan cover sufcient intervals of different situations. Now, the evaluation phase of differ-ent candidate solutions should be performed. Indeed, in this step, solutions are evaluatedthrough all scenarios. Here we have 11 candidate solutions and 11 different scenarios so themodel should be solved 121 times. The results are presented in Table4.

Under the analyses of Table4, all candidate solutions of all scenarios are evaluated interms of overall performance. Therefore, in each column, the performances of a candidatesolution are evaluated in 11 scenarios. Surely, each candidate solution reveals its best per-formance in its corresponding scenario (it is highlighted as the main diagonal of Table4).The last three rows are the criteria of making nal decision of the best solution in varioussituations. The performances of candidate solutions considering three types of criteria areillustrated in Figs.4, 5, and6.

Finally, analyzing these criteria in Table4 and Figs. 4, 5, and 6, lead to the optimalsolution, which are discussed as follows:

Solution 2, which is achieved by scenario 2, presents the mean objective function valueof 3276 million facing with all scenarios. Thus, regarding mean criterion, it is selectedas best-performed solution (maximum prot mean) among all situations (scenarios). Itmeans we can judge solution 2 as a well-behaved solution in various kinds of situationsin terms of mean criterion, which is illustrated in Fig.4.

Considering a risk criteria lead decision makers to ensure reliability of their decisions.Denitely, relying just on mean value without regarding uctuations is not a reliable wayof dealing with uncertain situations (Ogryczak2000). Thus, it is decided to consider asimple risk measure, we can focus on variance (here SD) as a risk criterion. It is importantfor us to have a risk-base optimal solution in different conditions. Therefore, the overall


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Table 4 Scenario-based solution approach for stochastic model (results in millions)

Scenarios SolutionsDet.solutions

Sol. 1 Sol. 2 Sol. 3 Sol. 4 Sol. 5 Sol. 6 Sol. 7 Sol. 8 Sol.WC

Sol.BC

Det. 3309 3285 3308 3256 3307 3284 3308 3256 3308 3124 3113S 1 3388 3400 3389 3358 3389 3340 3389 3358 3390 3047 3233S 2 3263 3199 3263 3193 3262 3198 3263 3193 3262 3042 3055S 3 3212 3217 3212 3274 3211 3217 3212 3274 3211 2944 3165S 4 3125 3116 3125 3111 3126 3115 3125 3111 3126 3002 3014S 5 3436 3450 3437 3409 3437 3450 3437 3409 3437 3088 3284S 6 3276 3212 3274 3207 3276 3211 3276 3207 3276 3052 3069S 7 3232 3244 3232 3304 3231 3244 3232 3304 3232 2957 3194S 8 3164 3156 3164 3152 3165 3155 3164 3152 3165 3037 3053WC 2135 2104 2135 2071 2133 2101 2135 2071 2133 2191 1912BC 4332 4496 4496 4617 4328 4501 4333 4617 4329 3376 5453Mean 3261.1 3262 3276 3268 3260 3256 3261 3268 3261 2987 3231SD 499.53 544.1 535.9 577 499.1 544.7 499.8 577 499.4 288.1 826.81/ CV = 1/ (SD / mean) 6.5283 5.995 6.113 5.6646.532 5.978 6.525 5.664 6.53 10.37 3.908

Sol.: solution. Det.: deterministic, WC: worst-case, BC: best-case

Fig. 4 Mean results of objective values for candidate solutions (in millions)

SD performances for all solutions are evaluated. Results of Table4 and Fig.5 reveal thatsolution 4 presents the minimum SD. Indeed, regarding risk criterion, the solution withlower SD will be more reliable in uctuated environment. Now we have two differentoptimal solutions through two criteria: mean and SD. Consequently, an integrated ap-proach is needed to make nal decision, which leads to the third criteria: coefcient of variation.

Finally, regarding mean and SD simultaneously, leads to the most important criteria,which is coefcient of variation. This is a simple approach to make an integrated deci-


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Fig. 5 Standard deviation results of objective values for candidate solutions (in millions)

Fig. 6 Coefcient of variation results of objective values for candidate solutions

sion considering mean-SD (mean-variance) criteria. Again, solution 4 is best one withthe minimum CV (actually, maximum 1/ CV) among all candidate solutions and we canselect it as the nal optimal solution (see Fig.6).

Regarding the special and rare kind of situation in the worst and the best cases, we shouldeliminate them in selecting the optimal solution. Indeed, they are incorporated just toconsidering all various reasonable scenarios.

A brief review of solution 2 and solution 4 in designing level are presented in Table5 inwhich the activated facilities are illustrated with value 1.

Reviewing Table5 reveals important strategic distinctions between solutions 2 and 4.In the mentioned table, the activated and not-activated locations are claried by one andzero respectively. Consequently, these two selected solutions can be compared in terms of different design decision variables, which show differences in locations of suppliers, disas-semblies, and redistributors.


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Table 5 A brief review of solution 2 and solution 4 in designing level

Facilities Node 1 Node 2 Node 3 Node 4 Node 5

Manufacturers in solution 2 0 0 1 1 1

Manufacturers in solution 4 0 0 1 1 1Suppliers in solution 2 1 0 1 1 0Suppliers in solution 4 1 0 0 1 1Warehouses in solution 2 0 0 1 1 0Warehouses in solution 4 0 0 1 1 0Distributors centers in solution 2 0 0 0 1 1Distributors centers in solution 4 0 0 0 1 1Disassembly in solution 2 1 1 1 1 0Disassembly in solution 4 1 0 1 1 0Disposal centers in solution 2 0 0 1 0 1Disposal centers in solution 4 0 0 1 0 1Redistributors in solution 2 1 0 0 1 1Redistributors in solution 4 1 0 1 1 1

5 Sensitivity analysis

A very important step to prove the reliability of the proposed solution approach will be anal-yses in sensitivity. If the candidate solutions (specially the optimal ones) can prove their

acceptable performance in these analyses, the results and the solution approach is more re-liable. On the other hand, in location-allocation type of the problems, xed costs are thehighest cost parameters and they play the most important roles in the results so they arechosen for sensitivity analysis in this paper. In this section, two different strategies are con-sidered in changing xed costs, which are presented as follows: 50 % increasing in all xed costs of all facilities and evaluating all candidate solutions in

11 scenarios for new xed-cost situation. 50 % decreasing in all xed costs of all facilities and evaluating all candidate solutions in

11 scenarios for new xed-cost situation.

The complete results of the above-mentioned strategies in sensitivity analyses are illus-trated in Tables6 and 7.Under consideration of Tables6 and 7, interesting points of the solution methodology

and the candidate solutions are claried, which are discussed as follows: Except the solutions of the worst-case and the best-case scenarios, other candidate so-

lutions are rarely could achieve best performance in the corresponding scenarios. Bestperformances of each scenario are highlighted in Tables6 and 7, which can prove theeffects of xed costs in the optimal solutions and the necessity of this sensitivity analysis.Since, the worst and the best cases scenarios are a special situation, so their associatecandidate solutions reveal best performance in the corresponding scenarios.

Regarding mean criterion, vague results are achieved, which cannot lead decision makersto nd the optimal solution under uncertainty. When the xed costs are increased 50 %,three candidate solutions present best performances, which are solutions 2, 6, and deter-ministic solution (see Table6). Vice a Versa, when the xed costs are decreased 50 %,two candidate solutions present best performance, which are solutions 3 and 7 (see Ta-ble 7). Results prove that although the candidate solution 2, reveals the best performance


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Table 6 Evaluating candidate solutions for+ 50 % xed-costs strategy (results in millions)



Sol.BC

Det. 3249 3218 3248 3177 3246 3214 3248 3177 3246 3087 2979S 1 3329 3332 3329 3280 3328 3329 3329 3280 3328 3010 3099S 2 3203 3132 3203 3115 3201 3127 3203 3115 3201 3005 2921S 3 3152 3149 3152 3196 3150 3146 3152 3196 3150 2907 3031S 4 3065 3048 3065 3032 3065 3044 3065 3032 3064 2965 2880S 5 3377 3383 3377 3330 3376 3379 3377 3330 3376 3051 3149S 6 3216 3144 3216 3128 3214 3140 3216 3128 3214 3015 2934S 7 3173 3176 3172 3226 3170 3173 3172 3226 3170 2920 3060S 8 3104 3088 3104 3073 3104 3084 3104 3073 3103 2999 2919WC 2075 2037 2074 1993 2072 2030 2074 1993 2072 2154 1778BC 4273 4428 4273 4538 4267 4430 4273 4538 4267 3339 5319Mean 3201 3194 3201 3190 3199 3190 3201 3190 3199 2950 3097SD 500 544 500 577 499 546 500 577 499 288 827CV = (SD / mean) 0.1561 0.1703 0.1562 0.18090.1560 0.1711 0.1562 0.1809 0.1561 0.0977 0.2670

Table 7 Evaluating candidate solutions for 50 % xed-costs strategy (results in millions)



Sol.BC

Det. 3368 3353 3368 3334 3369 3356 3368 3334 3370 3162 3248S 1 3448 3467 3449 3437 3451 3471 3449 3437 3452 3085 3368S 2 3322 3266 3323 3272 3324 3269 3323 3272 3324 3080 3190S 3 3272 3284 3272 3353 3272 3288 3272 3353 3273 2981 3300S 4 3185 3183 3186 3189 3188 3186 3186 3189 3188 3040 3149S 5 3496 3518 3497 3487 3499 3521 3497 3487 3499 3126 3418S 6 3336 3279 3337 3285 3337 3282 3337 3285 3338 3089 3203S 7 3292 3311 3292 3383 3293 3315 3292 3383 3294 2994 3329S 8 3224 3223 3225 3230 3227 3226 3225 3230 3227 3074 3188WC 2195 2172 2195 2150 2195 2172 2195 2150 2195 2228 2047BC 4392 4563 4394 4696 4390 4572 4394 4696 4391 3414 5588Mean 3321 3329 3322 3347 3322 3332 3322 3347 3323 3025 3366SD 500 544 500 577 499 546 500 577 499 288 827CV = (SD / mean) 0.1505 0.1634 0.1505 0.17240.1503 0.1638 0.1505 0.1724 0.1503 0.0953 0.2456

in Table 6, but it cannot preserve its superiorities in sensitivity analyses in second xedcosts strategy. Therefore, relying on the mean criterion cannot ensure nding the optimalsolutions based on the proposed scenario-based solution methodology, which is commonin earlier researches.

Considering the risk criterion (standard deviation) and the integrated mean-risk criteria(CV), interesting results of conrming the superiorities of candidate solution 4 in both


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sensitivity analysis strategies are achieved. The results of Tables6 and 7 prove that thecandidate solution 4, which could previously present best performance in Table4 of themain model, again, reveals best performances in objective function values in terms of achieving the lowest standard deviation and the lowest CV. Therefore, regarding the pro-

posed integrated mean-risk criteria of this paper could lead decision makers to nd reli-able optimal solutions under uncertain environment.

Finally, the analyses of the sensitivity studies prove the reliability of the proposedscenario-based solution methodology to achieve optimal solutions, while regarding multicriteria in decision-making procedure (integrated mean-risk approach).

6 Case study: the plastic water cane manufacturer

A well-known Indian company who produces plastic water cane products is selected tostudy the proposed model and the solution methodology. The company faced to demanduncertainty for the next 12 months, which considered as 12 periods here. The companyexpect 500 to 600 thousands unit of demand per period. The detail of the parameters ispresented in Table8.

In order to apply the model and the solution methodology to the selected case study,three scenarios of demands are considered: high demand, mid demand, and low demand.Based on the solution methodology, which is completely described in Sect.4, these scenariosare solved and then the results are regarded as candidate optimal solutions. The candidatesolutions are evaluated under various scenarios and then based on the suggested criteria;best ows of the network are assigned to achieve the highest prot. The results of objectivefunction values and criteria analysis are illustrated in Table9.

The results of Table9 reveal that the candidate solution, which is achieved by mid-demand scenario, presents best performance in terms of considering mean criterion of itsperformance in all scenarios. However, if we are supposed to consider SD and CV (as theintegrated mean-risk one), the candidate solution that is attained through low-demand sce-nario presents best performance. Thus, we can judge low-demand scenario solution as opti-mal one when regarding both mean and SD criteria simultaneously. Finally, in all cases, thereverse supply chain reveals a huge prot, which can make it reliable and protable to be

developed and invested by the managers of the company.

7 Conclusion and future researches

In this paper, we cope with a very important problem about designing and planning a closed-loop supply chain. A scenario-based, multi-echelon, multi-period, and multi-product modelincluding various ows between network entities is developed in this paper. This model hasmany characteristics of a generic model in both designing and planning stages. To solve theproposed model we have used IBM ILOG CPLEX 12.2 optimizer software. At the secondstep, we deal with non-deterministic demand and price parameters and then a multi criteriascenario based solution approach is developed to nd optimal solution through some log-ical scenarios and three comparing criteria. 11 various scenarios (one deterministic, eightrandom, one worst-case and one best-case scenarios) are generated and solved to achieve11 different candidate optimal solutions. Then, we have evaluated the performance of eachsolution in all scenarios. We have calculated mean, variance, and coefcient of variations


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Table 8 Parameters of the case study

Row Parameter Average or range permonth (period)

1. Demand (nondeterministic) 500000 to 6000002. Second demand rate 75 % of rst3. First product price 70 Rupees per unit4. Second products price 40 Rupees per unit5. Purchasing price 15 Rupees per unit6. Manufacturers capacity 800

Remanufacturing capacity 550Warehouses capacity 4250Suppliers capacity 1400Distributors capacity 3750Disassembly (collection) centers capacity 4250Recycling centers capacity 4100Redistributors capacity 2400Disposal centers capacity 3250

7. Costs of supplying a unit in suppliers 20 RupeesCosts of manufacturing a unit in manufacturers 5 RupeesCosts of holding a unit in warehouses for one period 2 RupeesCosts of distributing a unit in distributors 2 RupeesCosts of remanufacturing a unit in manufacturers 5 Rupees

Costs of collecting a unit in disassembly centers 2 RupeesCosts of recycling a unit in suppliers 5 RupeesCosts of distributing a unit in redistributors 2 RupeesCosts of disposing a unit of disposal centers 1 RupeesNot utilizing manufacturing costs 1 Rupees per unitPer unit cost for not covering customers demand (shortage costs) 3 Rupees per unit

8. Recycling rate (portion of return products) 70 %9. Remanufacturing rate (portion of return products) 20 %10. Repair rate (portion of return products) 5 %

11. Disposal rate (outsource) (portion of return products) 5 %

Table 9 The results of case study (in millions)

Scenario Optimum Mean of the optimumin various scenarios

SD of the optimumin various scenarios

CV of the optimumin various scenarios

High demand 221 198 24.58 0.12Mid demand 210 200 16.74 0.08

Low demand 195 198 2.517 0.01

as three proposed criteria to achieve optimal solution. The computational study and the cor-responding sensitivity analyses reveal reliability of the proposed solution approach for thestochastic model. Indeed, regarding integrated mean-risk criteria such as CV present reliable


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solutions in uncertain environment. Finally, a real case study in an Indian manufacturer isevaluated to ensure applicability of the model and the solution methodology.

Surely, there are some guides as future research; rst, in order to cope with large sizeinstances, some heuristics, meta-heuristics, or elevated exact methods like branch and bound

and column generation approaches can be utilized. Second, the risk considering methodand the scenario-based solution approach can be developed in terms of incorporating otherintegrated criteria of decision making under uncertainty.

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A New Multi-criteria Scenario-based Solution Approach