A new multi-criteria scenario-based solution approach ...

Ann Oper Res DOI 10.1007/s10479-013-1435-z

A new multi-criteria scenario-based solution approach for stochastic forward/reverse supply chain network design 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 find unavoidable steps toward closing the loop of supply chain. In order to expect best performance of ClosedLoop Supply Chain (CLSC) network, an integrated approach in considering design and planning decision levels is necessary. Further, real markets usually contain uncertain parameters such as demands and prices of products. Therefore, the next important step is considering uncertain parameters. In order to cope with designing and planning a closed-loop supply chain, this paper proposes a multi-period, multi-product closed-loop supply chain network with stochastic demand and price in a Mixed Integer Linear Programming (MILP) structure. A multi criteria scenario based solution approach is then developed to find optimal solution through some logical scenarios and three comparing criteria. Mean, Standard Deviation (SD), and Coefficient of Variation (CV), which are the mentioned criteria for finding the optimal solution. Sensitivity analyses are also undertaken to validate efficiency of the solution approach. The computational study reveals the acceptability of proposed solution approach for the stochastic model. Finally, a real case study in an Indian manufacturer is evaluated to ensure applicability of the model and the solution methodology. Keywords Closed-loop supply chain · Mixed integer linear programming · Reverse logistic · Stochastic optimization · Scenario-based solution 1 Introduction The vast tendency toward closing the loop of supply chain originate in its capability and feasibility in terms of economic criteria, which lead managers to think of profit maximization instead of cost minimization approaches (Guide and Van Wassenhove 2009). Although

B

H. Soleimani · M. Seyyed-Esfahani ( ) · M.A. Shirazi Amirkabir University, Valiasr Crossroad, Tehran, Iran e-mail: [email protected] H. Soleimani e-mail: [email protected]

Ann Oper Res

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 fulfill customers’ requests (Chopra and Meindl 2007), new definitions make supply chain responsible for End of Life (EOL) products too (reverse supply chain). For instance, in the US, 75 % of customers claim that they consider environmental reputation of manufacturers in their purchasing, and 80 % of customers even pay more for environmental friendly goods (Lamming and Hampson 1996). In a CLSC, manufacturers have to be responsible for collecting used products from 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 (called return products) including simple repairing and then reselling them to second markets, remanufacturing 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 figure, forward, and reverse supply chain are presented by solid lines and dashed lines, respectively. The problem of CLSC design and planning is NP-hard and complicated (Krarup and Pruzan 1983 and Schrijver 2003), which is hard to achieve optimal solutions for real-sized instances. 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. Demands of first and second customers, price of selling first and second products, and the price of purchasing used products from customers are considered as nondeterministic parameters in the study. In order to guarantee the applicability of the model, an efficient scenario-based solution 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 and the solution approach. The rest of this paper is arranged as follow. In Sect. 2, a complete literature review is presented. The model characteristics and formulation is demonstrated in Sect. 3. Computational 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.

Ann Oper Res

2 Literature review Designing and planning a CLSC with stochastic parameters is a critical issue, which needs to be considered by researchers. There are few papers dealing with mentioned problem and trying to solve it in an efficient and practical way. Recent review papers can shed more light on this claim. Pokharel and Mutha (2009) reviewed the current advancements in reverse logistics (RL) and they mentioned about necessity of generic models and stochastic approaches in this area. Also Subramoniam et al. (2009) presented another review in reverse logistics in automotive industry, which pointed out the lack of stochastic approaches in designing CLSC. Such points can also find in Sasikumar and Kannan (2009). Concentrating on design and planning problem of CLSC, and regarding stochastic approaches, there are some papers, which can be discussed as follows (Table 1). The characteristics of this study are clarified at the last row of Table 1. Reviewing Table 1 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 is scenario-based, multi-period, and multi-product with various possible flows between network entities, which can construct a close-to-real network. • In terms of stochastic parameters, this study proposes a complete set of nondeterministic parameters, which are demands of the first and second customers (return rate), price of first products, price of return products, and purchasing price of EOL products. This paper, as regarding Table 1, is the only, which considers such nondeterministic parameters in its stochastic approach. • In terms of nondeterministic solution approaches, there are various methodologies in dealing with such problems. This paper tries to develop a new efficient scenario-based approach, which can rationally achieve optimal solutions through some criteria. It should be pointed out that based on the difficulties of other approaches such as two-stage stochastic optimization approaches specially in real world instances, scenario-based solution methodologies are mentioned as effective and well-behaved solution methods for stochastic problems (Dembo 1991 and Kaut and Wallace 2007). • On the other hand, using three criteria in selecting optimal point in scenario-based solution methodologies is developed by this study to elevate the reliability of optimal decisions in uncertain environment. Finally, based on the above consideration, and the analyses of literature review in Table 1, the necessities of current study is revealed in terms of proposing and solving a stochastic model in an efficient 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 nondeterministic 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 completely presented 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

Multi

Single

Pishvaee et al. (2011)

Pishvaee and Torabi (2010)

El-Sayed et al. (2010)

Francas and Minner (2009)

Özceylan and Paksoy (2013)

Liste¸s (2007)

Amin and Zhang (2012)

Chouinard et al. (2008)

Faccio et al. (2011)

Zhou and Min (2011)

Amin and Zhang (2013)

Zhu and Xiuquan (2013)

Ramezani et al. (2013)

This study

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

Multi

Single

Multi

–

Single

Single

Single

Single

Single

Multi

Multi

Single

Multi

Özkır and Ba¸slı gil (2013)

1.

Period

Paper

Row

Multi

Multi

Multi

–

Multi

Multi

Single

Multi

Single

Multi

Multi

Single

Single

Single

Single

Product

Uncertain

–

Uncertain

Uncertain

–

Uncertain

Uncertain

–

Uncertain

Uncertain

Uncertain

Uncertain

Uncertain

Uncertain

–

Demands

Table 1 Literature survey of CLSC design and planning under uncertainty

Uncertain

–

–

–

Uncertain

–

Uncertain

–

Uncertain

Uncertain

–

Uncertain

Uncertain

Uncertain

–

Return

Uncertain

–

–

–

–

–

–

–

–

–

Uncertain

–

–

–

Uncertain

Prices

Stochastic

Stochastic-Two stage

Stochastic

Stochastic (Fuzzy weights)

Stochastic

Stochastic

Stochastic-Two stage

Fuzzy

Stochastic-Two stage

Fuzzy

Stochastic-Two stage

Stochastic-Two stage

Fuzzy

Stochastic

Fuzzy

Uncertain approach

New Scenario–based solution

Multi criteria approaches

Hybrid genetic algorithm

Exact

Genetic algorithm

General solvers

CPLEX

GAMS

CPLEX

CPLEX

Simulation

Dash optimization

LINGO

CPLEX

CPLEX

Solution method

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• • •

• • • •

chain and disassembly centers, redistributors, disposal centers, and second customers in its reverse logistics. Dealing with used products can be undertaken in four alternatives: repairing by disassembly centers, remanufacturing by manufacturers, recycling by suppliers, and disposing by disposal centers. Disassembly centers are responsible for collecting used products from first customers, deciding the next-step alternative decisions for return products, and repairing some portions of them. Demands of first customers and price of first products are directly considered as nondeterministic parameters through some scenarios. Besides, return products rate, price of second products, and purchasing price of used products are also regarded as stochastic parameters through factors related to demands of first customers and price of first products respectively. 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 predetermined. Quality, demand, and price of returned products are different from first customers’ and they cannot be sold as new products. In terms of network flows, manufacturers, warehouses, and distributers can supply first customers and manufacturers, warehouses, disassembly centers, and redistributors can supply 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 flows) (Soleimani et al. 2013)

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D: C: A: R: P: K: U: T:

Potential number of distributors, indexed by “d”. Potential number of first customers (retailers), indexed by “c”. Potential number of disassembly centers, indexed by “a”. Potential number of redistributors, indexed by “r”. Potential number of disposal centers, indexed by “p”. Potential number of second customers, indexed by “k”. Set of products, indexed by “u”. Set of periods, indexed by “t ”.

• Parameters: Ss : Maximum number of suppliers in scenario “s”. Fs : Maximum number of manufacturers in scenario “s”. Ws : Maximum number of warehouses in scenario “s”. Ds : Maximum number of distributors in scenario “s”. As : Maximum number of disassembly centers in scenario “s”. Rs : Maximum number of redistributors in scenario “s”. Ps : Maximum number of disposal centers in scenario “s”. M: a sufficiently large constant. Dcuts : Demand of product “u” of first the customer “c” in period “t ” in scenario “s”, Dkuts : Demand of product “u” of the second customer “k” in period “t ” in scenario “s”, Pcuts : Unit price of product “u” at first customer “c” in period “t ” in scenario “s”, PU cuts : Purchasing cost of product “u” at first customer “c” in period “t ” in scenario “s”, Pkuts : Unit price of product “u” at second customer “k” in period “t ” in scenario “s”, Fi : Fixed cost of activating location “i”. DSij : Distance between location “i” and location “j ”. SClut : Capacity of supplier “l” of product “u” in period “t ”, SRClut : Recycling capacity of supplier “l” of product “u” in period “t ”, FCf ut : Manufacturing capacity of manufacturer “f ” of product “u” in period “t ”, RFCf ut : Remanufacturing capacity of manufacturer “f ” of product “u” in period “t ”, WCwut : Warehouse capacity of warehouse “w” of product “u” in period “t ”, DCdut : Capacity of distributor “d” of product “u” in period “t ”, ACau : Capacity of disassembly “a” of product “u” in period “t ”, RDCrut : Capacity of redistributor “r” of product “u” in period “t ”, PCput : 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 period “t ”, RT sut : Recycling cost of product “u” per unit which is recycled by supplier “l” in period “t ”, FT f ut : Manufacturing cost of product “u” per unit, which is undertaken by manufacturer “f ” in period “t ”, RFT f ut : Remanufacturing cost of product “u” per unit, which is undertaken by manufacturer “f ” in period “t ”, DAT aut : Disassembly cost of product “u” per unit by disassembly center “a” in period “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 period “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 period “t ”, ST ut : Shortage cost of product “u” per unit in period “t ”, Fhf u : Manufacturing time of product “u” per unit at manufacturer “f ”, RFhf 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 ”, Blu , Bf u , Bdu , Bau , Bru , Bwu , Bcu : Batch size of product “u” from supplier “l” manufacturer “f ”, distributor “d”, disassembly “a”, redistributor “r”, warehouse “w” and customer “c” respectively. TRT ut : Transportation cost of product “u” per unit per kilometer in period “t ”, RRut : Return ratio of product “u” at first customers in period “t ”, Rc: Recycling ratio, Rm: Remanufacturing ratio, Rr: Repairing ratio, Rp: Disposal ratio, • First-stage decision variables: Lis : Binary variable equals “1” if location “i” is activated in scenario “s” and “0” otherwise. • Second-stage decision variables: Qij uts : Flows of product “u” from node (entity) “i” to node (entity) “j ” in period “t ” in scenario “s”, Rwuts : Residual inventory of product “u” for warehouse “w” in period “t ” in scenario “s”, Rduts : Residual inventory of product “u” for distributor “d” in period “t ” in scenario “s”. TLij s : Binary variable, which is equal to “1” if a transportation link is established between node “i” and node “j ” in scenario “s” and “0” otherwise. 3.1 Objective function Objective function is total profit, which can be calculated by total sales minus total costs for a scenario. • Total sales: First products sales (flows that start from distributors, manufacturers, and warehouses to customers):   (Qdcuts Bdu Pcuts ) + (Qf cuts Bf u Pcuts ) d∈D c∈C u∈U t∈T

f ∈F c∈C u∈U t∈T

  + (Qwcuts Bwu Pcuts )

∀s ∈ S,

(3.1)

w∈W c∈C u∈U t∈T

Total sales of second products (return product flows, which start from redistributors, warehouses, and manufacturers to second customers):   (Qrkuts Bru Pkuts ) + (Qf kuts Bf u Pkuts ) r∈R k∈K u∈U t∈T

f ∈F k∈K u∈U t∈T

  + (Qwkuts Bwu Pkuts ) w∈W k∈K u∈U t∈T

∀s ∈ S

(3.2)

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• Total costs: Total costs are calculated for each scenario as follows: Total costs = fixed costs + material costs + manufacturing costs + non-utilized capacity costs + shortage costs + purchasing costs + disassembly costs + recycling costs + remanufacturing costs + repairing costs + disposal costs + transportation costs + inventory holding costs. Each of the above mentioned costs are calculated for each scenario as follows: Fixed costs (location costs):       Fl Lls + Ff Lf s + Fd Lds + Fa Las + Fr Lrs + Fp Lps l∈L



+

f ∈F

d∈D

a∈A

r∈R

p∈P

∀s ∈ S

Fw Lws

(3.3)

w∈W

Material costs (return materials benefits should be considered here):   Qlf uts Blu MTlut − Qaluts Bau (MTlut − RTlut ) l∈L f ∈F u∈U t∈T

∀s ∈ S

a∈A l∈l u∈U t∈T

(3.4)

Manufacturing costs:     (Qf duts Bf u F Tf ut ) + (Qf wuts Bf u F Tf ut ) f ∈F d∈D u∈U t∈T

f ∈F w∈W u∈U t∈T

  + (Qf cuts Bf u F Tf ut ) + (Qf kuts Bf u F Tf ut ) f ∈F c∈C u∈U t∈T

∀s ∈ S

f ∈F k∈K u∈U t∈T

(3.5)

Non-utilized capacity costs for manufacturers:    (Qf duts Bf u ) − (Qf wuts Bf u ) (F Cf u /F hf u )Lf s − f ∈F

u∈U

t∈T

d∈D

w∈W

     − (Qf cuts Bf u ) + (Qwruts Bwu ) + (Qwkuts Bwu ) NMT f u c∈C

+

 f ∈F

u∈U

w∈W r∈R

w∈W k∈K

(RFCf u /RF hf u )Lf s −

t∈T



(Qf ruts Bf u ) −

r∈R

    − (Qwruts Bwu ) + (Qwkuts Bwu ) NRMT f u w∈W r∈R

 (Qf kuts Bf u ) k∈K

∀s ∈ S

(3.6)

w∈W k∈K

Shortage costs (for distributor):      (Qdcuts Bdu ) − Qf cuts Bf u Shortageuts = ( ( ( Dcuts − c∈C u∈U t∈T

−





Qwcuts Bwu

t∈T f ∈F

t∈T d∈D

∀s ∈ S

(3.7)

t∈T w∈W

Shortage costs =



Shortageuts × STut

∀s ∈ S

u∈U t∈T

Purchasing costs:

 c∈C a∈A u∈U t∈T

Qcauts P Ucuts Bcu

∀s ∈ S

(3.8)

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Disassembly costs: 

∀s ∈ S

Qcauts Bcu DAT aut

(3.9)

c∈C a∈A u∈U t∈T

Recycling costs: 

Qaluts Bau RT sut

∀s ∈ S

(3.10)

a∈A l∈L u∈U t∈T

Remanufacturing costs: 

Qaf uts Bau RFT f ut

∀s ∈ S

(3.11)

Qaruts Bau RPT aut

∀s ∈ S

(3.12)

∀s ∈ S

(3.13)

a∈A f ∈F u∈U t∈T

Repairing costs:  a∈A r∈R u∈U t∈T

Disposal costs: 

Qaputs Bau PT put

a∈A p∈P u∈U t∈T

Transportation costs:   Qlf uts Bsu TRT ut DSlf + Qf duts Bf u TRT ut DSf d t∈T u∈U l∈L f ∈F

+

t∈T u∈U f ∈F d∈D

 

Qf wuts Bf u TRT ut DSf w

t∈T u∈U f ∈F w∈W



Qf cuts Bf u TRT ut DSf c +

t∈T u∈U f ∈F c∈C

+

  



Qwcuts Bwu TRT ut DSwc

t∈T u∈U w∈W c∈C

  

Qwkuts Bwu TRT ut DSwk +



t∈T u∈U w∈W k∈K

+



Qaluts Bau TRT ut DSas

Qaf uts Bau TRT ut DSaf +

t∈T a∈A u∈U f ∈F

+



Qf ruts Bf u TRT ut DSf r +



   t∈T u∈U w∈W r∈R

Qrkuts Bru TRT ut DSruk

t∈T u∈U r∈R k∈K



Qcauts Bcu TRT ut DSca

t∈T u∈U c∈C a∈A

+

(3.14) Qaputs Bau TRT ut DSap

t∈T u∈U a∈A p∈P

t∈T u∈U f ∈F r∈R

+



Qaruts Bau TRT ut DSar

t∈T u∈U a∈A r∈R



Qdcuts Bdu TRT ut DSdc

t∈T u∈U d∈D c∈C

t∈T u∈U a∈A l∈L



Qf kuts Bf u TRT ut DSf k

t∈T u∈U f ∈F k∈K

   t∈T u∈U w∈W d∈D

Qwduts Bwu TRT ut DSwd

∀s ∈ S

Qwruts Bwu TRT ut DSwr

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Inventory holding costs:    Rwuts WHT wut + Rduts DHT dut w∈W u∈U t∈T

∀s ∈ S

(3.15)

d∈D u∈U t∈T

3.2 Constraints  l∈L

Qlf uts Bsu =



Qf duts Bf u +

d∈D



Qf wuts Bf u +

w∈W



Qf cuts Bf u ,

c∈C

∀s ∈ S, ∀u ∈ U, ∀f ∈ F, ∀t ∈ T    Qf wuts Bf u = Rwuts + Qwduts Bwu + Qwcuts Bwu , f ∈F

d∈D

(3.16)

c∈C

∀s ∈ S, ∀u ∈ U, ∀w ∈ W, ∀t ∈ T    Qf duts Bf u + Qwduts Bwu = Rdut + Qdcuts Bdu , f ∈F

w∈W

(3.17)

c∈C

∀s ∈ S, ∀u ∈ U, ∀d ∈ D, ∀t ∈ T    Qdcuts Bdu + Qf cuts Bf u + Qwcuts Bwu + Shortageuts ≥ Dcuts , d∈D

f ∈F

w∈W

∀s ∈ S, ∀u ∈ U, ∀c ∈ C, ∀t ∈ T      Qcauts Bcu = Qdcuts Bdu + Qf cuts Bf u + Qwcuts Bwu RRut , a∈A

(3.18)

d∈D

f ∈F

(3.19)

w∈W

∀s ∈ S, ∀u ∈ U, ∀c ∈ C, ∀t ∈ T (3.20)      Qcauts Bcu = (Qaluts Bau ) + (Qaf uts Bau ) + (Qaruts Bau ) + (Qaputs Bau ), c∈C

f ∈F

l∈L

∀s ∈ S, ∀u ∈ U, ∀a ∈ A, ∀t ∈ T   (Qcauts Bcu )Rc = (Qaluts Bau ), c∈C



(Qcauts Bcu )Rm =

c∈C

(3.21) ∀s ∈ S, ∀u ∈ U, ∀a ∈ A, ∀t ∈ T



∀s ∈ S, ∀u ∈ U, ∀a ∈ A, ∀t ∈ T

(Qaf uts Bau ),

(Qcauts Bcu )Rr =

 (Qaruts Bau ),

∀s ∈ S, ∀u ∈ U, ∀a ∈ A, ∀t ∈ T

(3.24)

(Qcauts Bcu )Rp =

 (Qaputs Bau ),

∀s ∈ S, ∀u ∈ U, ∀a ∈ A, ∀t ∈ T

(3.25)

p∈P

r∈R

f ∈F

(3.27)

k∈K

∀s ∈ S, ∀u ∈ U, ∀r ∈ R, ∀t ∈ T  (Qrkuts Bru ) ≤ Dkuts , ∀s ∈ S, ∀u ∈ U, ∀k ∈ K, ∀t ∈ T r∈R

(3.26)

k∈K

∀s ∈ S, ∀u ∈ U, ∀f ∈ F, ∀t ∈ T    (Qaruts Bau ) + (Qf ruts Bf u ) = (Qrkuts Bru ), a∈A

(3.23)

r∈R

Rc + Rm + Rr + Rp = 1    (Qaf uts Bau ) = (Qf ruts Bf u ) + (Qf kuts Bf u ), a∈A

(3.22)

f ∈F

c∈C



p∈P

l∈L

c∈C



r∈R

(3.28) (3.29)

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Constraints (3.16) to (3.29) are balanced constraints. These constraints guarantee the equality of all entering flows to a network entity and all issuing flows of the same entity in 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), respectively. Again, constraints (3.22) to (3.29) are recycling rate constraints (3.22), remanufacturing rate constraints (3.23), repairing rate constraints (3.24), disposal rate constraints (3.25), manufacturers reverse flows constraints (3.27), redistributors constraints (3.28), and ultimately, second customers constraints (3.29). Finally, sum of all alternative decision rates for used products should be equal to 1 for a scenario (constraint (3.26)).  Qlf uts Bsu ≤ SClut Lls , ∀s ∈ S, ∀u ∈ U, ∀l ∈ L, ∀t ∈ T (3.30) f ∈F

 d∈D

Qf duts Bf u +



Qf wuts Bf u +

w∈W



Qf cuts Bf u +

c∈C



 Qf kuts Bf u Fhf u ≤ FCf ut Lf s

k∈K

∀s ∈ S, ∀u ∈ U, ∀f ∈ F, ∀t ∈ T

(3.31)

Rwut ≤ WCwut Lws , ∀s ∈ S, ∀u ∈ U, ∀w ∈ W, ∀t ∈ T   Qf duts Bf u + Qwduts Bwu + Rdut ≤ DCdut Lds , f ∈F

w∈W

∀s ∈ S, ∀u ∈ U, ∀d ∈ D, ∀t ∈ T     Qaluts Bau + Qaf uts Bau + Qaruts Bau + Qaputs Bau ≤ ACaut × Las , l∈L

(3.32)

f ∈F

r∈R

(3.33)

p∈P

∀s ∈ S, ∀u ∈ U, ∀a ∈ A, ∀t ∈ T  Qrkuts Bru ≤ RCrut × Lrs , ∀s ∈ S, ∀u ∈ U, ∀r ∈ R, ∀t ∈ T

(3.34) (3.35)

k∈K



Qaluts Bau ≤ SRClut × Lls ,

∀s ∈ S, ∀u ∈ U, ∀l ∈ L, ∀t ∈ T

(3.36)

Qaputs Bau ≤ PCput × Lps ,

∀s ∈ S, ∀u ∈ U, ∀p ∈ P , ∀t ∈ T

(3.37)

a∈A

 a∈A



Qf wuts Bf u ≤ WCwut × Lws ,

∀s ∈ S, ∀u ∈ U, ∀w ∈ W, ∀t ∈ T

(3.38)

f ∈F

Constraints (3.30) to (3.38) are capacity constraints. These constraints ensures that all entities entering and issuing flows be less than their capacities. Precisely, constraints (3.30) control all suppliers’ output capacity for each product in all periods for a scenario. Constraints (3.31) to (3.38) are capacity limitations of manufacturers, warehouses, distributors, redistributors, suppliers, disposal centers, and warehouses respectively.  Qlf uts ≤ M TLlf s , ∀l ∈ L, ∀f ∈ F, ∀s ∈ S (3.39) TLlf s ≤ u∈U t∈T

TLf ds ≤



Qf duts ≤ M TLf ds ,

∀f ∈ F, ∀d ∈ D, ∀s ∈ S

(3.40)

u∈U t∈T

TLf ws ≤

 u∈U t∈T

Qf wuts ≤ M TLf ws ,

∀f ∈ F, ∀w ∈ W, ∀s ∈ S

(3.41)

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TLf cs ≤

Qf cuts ≤ M TLf cs ,

∀f ∈ F, ∀c ∈ C, ∀s ∈ S

(3.42)

Qf kuts ≤ M TLf ks ,

∀f ∈ F, ∀k ∈ K, ∀s ∈ S

(3.43)

Qf ruts ≤ M TLf rs ,

∀r ∈ R, ∀f ∈ F, ∀s ∈ S

(3.44)

u∈U t∈T



TLf ks ≤

u∈U t∈T



TLf rs ≤

u∈U t∈T

TLwds ≤



Qwduts ≤ M TLwds ,

∀w ∈ W, ∀d ∈ D, ∀s ∈ S

(3.45)

Qwcuts ≤ M TLwcs ,

∀w ∈ W, ∀c ∈ C, ∀s ∈ S

(3.46)

Qwkuts ≤ M Liwks ,

∀w ∈ W, ∀k ∈ K, ∀s ∈ S

(3.47)

Qwruts ≤ M Liwrs ,

∀w ∈ W, ∀r ∈ R, ∀s ∈ S

(3.48)

Qdcuts ≤ M TLdcs ,

∀d ∈ D, ∀c ∈ C, ∀s ∈ S

(3.49)

Qcauts ≤ M TLcas ,

∀a ∈ A, ∀c ∈ C, ∀s ∈ S

(3.50)

Qaluts ≤ M TLals ,

∀l ∈ L, ∀a ∈ A, ∀s ∈ S

(3.51)

u∈U t∈T

TLwcs ≤

 u∈U t∈T

TLwks ≤

 u∈U t∈T

TLwrs ≤

 u∈U t∈T

TLdcs ≤

 u∈U t∈T

TLcas ≤

 u∈U t∈T

TLals ≤

 u∈U t∈T

TLaf s ≤



Qaf uts ≤ M TLaf s ,

∀f ∈ F, ∀a ∈ A, ∀s ∈ S

(3.52)

Qaruts ≤ M TLars ,

∀r ∈ R, ∀a ∈ A, ∀s ∈ S

(3.53)

Qaputs ≤ M TLaps ,

∀p ∈ P , ∀a ∈ A, ∀s ∈ S

(3.54)

Qrkuts ≤ M TLrks ,

∀k ∈ K, ∀r ∈ R, ∀s ∈ S

(3.55)

u∈U t∈T

TLars ≤

 u∈U t∈T

TLaps ≤

 u∈U t∈T

TLrks ≤

 u∈U t∈T

Constraints (3.39) to (3.55) are related to shipping limitations between two entities. For instance, in constraints (3.39), if in a scenario there is no real shipping way between a supplier and a manufacturer, so there would be no flows between them and vice versa.  Lls ≤ Ss ∀s ∈ S (3.56) l∈L



Lf s ≤ Fs

∀s ∈ S

(3.57)

Lds ≤ Ds

∀s ∈ S

(3.58)

f ∈F

 d∈D



Lws ≤ Ws

∀s ∈ S

(3.59)

w∈W



Las ≤ As

∀s ∈ S

(3.60)

Lrs ≤ Rs

∀s ∈ S

(3.61)

a∈A

 r∈R

Ann Oper Res Table 2 Parameters range of the computational study

Row

Parameter

Uniformly distributed or rates

1.

Demand

0–3000

2.

Second demand rate

50 % of demand

3.

Price

15000–20000

4.

Second product price

50 % of price

5.

Purchasing cost

10 % of price

6.

Manufacturer capacity

6000–14000

7.

Remanufacturer capacity

50 % of Manufacturer capacity

8.

Supplier capacity

18000–42000

9.

Supplier recycling capacity

50 % of Supplier capacity

10.

Recycling cost

10–100

11.

All other reverse cost

10–100

12.

Others facilities capacities

6000–14000

13.

Material cost

100–1000

14.

Manufacturing cost

100–1000

15.

All other forward cost

100–1000

16.

Shortage cost

1000–5000

17.

Supplier fixed cost

7–10 million

18.

Manufacturer fixed cost

70–150 million

19.

Distributor fixed cost

1–2 million

20.

warehouse fixed cost

0.1–1 million

21.

Disassembly fixed cost

0.1–1 million

22.

Redistributors fixed cost

0.1–1 million

23.

Disposal centers fixed cost

0.1–1 million

24.

Batch size

1



Lps ≤ Ps

∀s ∈ S

(3.62)

p∈P

Constraints (3.56) to (3.62) are the limitations to maximum number of allowable activated locations for each scenario. These types of managerial limitations or budget related limitations 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 five units of each entity is considered (five suppliers, five manufacturers, five warehouses etc.). The study is undertaken for 5 products in 5 periods and 11 various logically-generated scenarios. The data are generated based on the uniform distributed functions on the basis of Table 2. It should be mentioned that the recycling, remanufacturing, repair, and disposal rates are respectively 0.2, 0.4, 0.3, and 0.1 of return products and batch sizes are one for all entities. The above-mentioned ranges in Table 2 reveal the intervals, which parameters of the model are generated. On the other hand, considering real market situations, prices and demands are two main uncertain parameters, which are regarded as nondeterministic scenariobased values here. Quality and suitability of scenario-generation methods for a given

Ann Oper Res Table 3 Objective function values of 11 different scenarios (in millions) Scenarios Deterministic S 1 Profit

3309

S2

S3

S4

S5

S6

S7

S8

Worst-case Best-case

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 Liste¸s (2007). They also pointed out that based on simplicity and applicability characteristics of scenario-based approaches in comparison with other methods (like two-stage stochastic programming), they can be introduced as efficient and powerful methodologies to cope with stochastic programming problems. In this study, we develop a new multi-criteria scenario-based approach to solve the proposed stochastic problem of CLSC design and planning. Totally, 11 different scenarios are assigned logically and randomly. In the first scenario, which called deterministic scenario, mean values of demands and prices (1500 and 17500 respectively) are considered. Then, 4 various possibilities for demands based on the range of the data in Table 2 (0–300) and 2 different possible values for prices are created. The combination 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 scenario considers demands at the lowest possible values (1000) and the best-case scenario considers them at the highest possible values (2500). Finally, adding the last 2 scenarios, 11 different scenarios are created for demands and prices, which affect return rates, return product demands, and purchasing price of used products. All computations are coded by IBM ILOG CPLEX 12.2 optimization software and run with a Core 2 Duo-2.26 GHz processor laptop. 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 final optimal point. Indeed, decisionmakers try to make best decision now, to ensure best performance in future. Therefore, these candidate solutions should be evaluated to find best one in terms of regarding all scenarios. • 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 candidate solutions in facing with all scenarios. This paper presents mean, standard deviation, and coefficient of variation as acceptable criteria to decide about best solution among all candidate solutions. • Finally, optimal solution are selected based on the analyses of three criteria and appropriate sensitivity analyses are undertaken to determine the reliability of the developed solution approach. Under the above consideration, the first step is solving all scenarios to achieve optimal solution of each one, which are mentioned as candidate solutions for final decision. The results are illustrated in Table 3. Further, Fig. 3 contains a schematic view of different objective function (profit) values. Analyzing Table 3 and Fig. 3, lead to important conclusions to be convinced of sufficiency of the number of scenarios: • The total mean of scenario objective values is 3282 million. There are 5 scenarios with greater values, and 6 scenarios with lower values than mean value. This proves the fair random distributions of scenarios to consider all situations.

Ann Oper Res

Fig. 3 Objective function values (profit) 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 diversification of various scenarios. Indeed, we can find different range of solutions and it can guarantee achieving the appropriate candidate solutions. Further, the range of results except the worst and the best cases is 324 million and considering the worst and the best cases, range will be 3252 million (96 % of mean value). Again, this can reconfirm the discussion about diversification issues. Considering the above-mentioned points lead to conclude that the generated scenarios can cover sufficient intervals of different situations. Now, the evaluation phase of different candidate solutions should be performed. Indeed, in this step, solutions are evaluated through all scenarios. Here we have 11 candidate solutions and 11 different scenarios so the model should be solved 121 times. The results are presented in Table 4. Under the analyses of Table 4, all candidate solutions of all scenarios are evaluated in terms of overall performance. Therefore, in each column, the performances of a candidate solution are evaluated in 11 scenarios. Surely, each candidate solution reveals its best performance in its corresponding scenario (it is highlighted as the main diagonal of Table 4). The last three rows are the criteria of making final decision of the best solution in various situations. The performances of candidate solutions considering three types of criteria are illustrated in Figs. 4, 5, and 6. Finally, analyzing these criteria in Table 4 and Figs. 4, 5, and 6, lead to the optimal solution, which are discussed as follows: • Solution 2, which is achieved by scenario 2, presents the mean objective function value of 3276 million facing with all scenarios. Thus, regarding mean criterion, it is selected as best-performed solution (maximum profit mean) among all situations (scenarios). It means we can judge solution 2 as a well-behaved solution in “various kinds of situations” in terms of mean criterion, which is illustrated in Fig. 4. • Considering a risk criteria lead decision makers to ensure reliability of their decisions. Definitely, relying just on mean value without regarding fluctuations is not a reliable way of dealing with uncertain situations (Ogryczak 2000). Thus, it is decided to consider a simple risk measure, we can focus on variance (here SD) as a risk criterion. It is important for us to have a risk-base optimal solution in different conditions. Therefore, the overall

Ann Oper Res Table 4 Scenario-based solution approach for stochastic model (results in millions) Scenarios

Solutions Det. Sol. 1 Sol. 2 Sol. 3 Sol. 4 Sol. 5 Sol. 6 Sol. 7 Sol. 8 Sol. solutions WC

Sol. BC

Det.

3309

S1

3388

3400 3389 3358 3389 3340 3389 3358 3390 3047 3233

S2

3263

3199 3263 3193 3262 3198 3263 3193 3262 3042 3055

S3

3212

3217 3212 3274 3211 3217 3212 3274 3211 2944 3165

S4

3125

3116 3125 3111 3126 3115 3125 3111 3126 3002 3014

S5

3436

3450 3437 3409 3437 3450 3437 3409 3437 3088 3284

S6

3276

3212 3274 3207 3276 3211 3276 3207 3276 3052 3069

S7

3232

3244 3232 3304 3231 3244 3232 3304 3232 2957 3194

S8

3164

3156 3164 3152 3165 3155 3164 3152 3165 3037 3053

WC

2135

2104 2135 2071 2133 2101 2135 2071 2133 2191 1912

BC

4332

4496 4496 4617 4328 4501 4333 4617 4329 3376 5453

Mean

3261.1

3262 3276 3268 3260 3256 3261 3268 3261 2987 3231

SD

499.53

544.1 535.9 577

1/CV = 1/(SD/mean) 6.5283

3285 3308 3256 3307 3284 3308 3256 3308 3124 3113

499.1 544.7 499.8 577

499.4 288.1 826.8

5.995 6.113 5.664 6.532 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 Table 4 and Fig. 5 reveal that solution 4 presents the minimum SD. Indeed, regarding risk criterion, the solution with lower SD will be more reliable in fluctuated environment. Now we have two different optimal solutions through two criteria: mean and SD. Consequently, an integrated approach is needed to make final decision, which leads to the third criteria: coefficient of variation. • Finally, regarding mean and SD simultaneously, leads to the most important criteria, which is coefficient of variation. This is a simple approach to make an integrated deci-

Ann Oper Res

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

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

sion considering mean-SD (mean-variance) criteria. Again, solution 4 is best one with the minimum CV (actually, maximum 1/CV) among all candidate solutions and we can select it as the final optimal solution (see Fig. 6). • Regarding the special and rare kind of situation in the worst and the best cases, we should eliminate them in selecting the optimal solution. Indeed, they are incorporated just to considering all various reasonable scenarios. A brief review of solution 2 and solution 4 in designing level are presented in Table 5 in which the activated facilities are illustrated with value “1”. Reviewing Table 5 reveals important strategic distinctions between solutions 2 and 4. In the mentioned table, the activated and not-activated locations are clarified by one and zero respectively. Consequently, these two selected solutions can be compared in terms of different design decision variables, which show differences in locations of suppliers, disassemblies, and redistributors.

Ann Oper Res 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

1

Suppliers in solution 2

1

0

1

1

0

Suppliers in solution 4

1

0

0

1

1

Warehouses in solution 2

0

0

1

1

0

Warehouses in solution 4

0

0

1

1

0

Distributors centers in solution 2

0

0

0

1

1

Distributors centers in solution 4

0

0

0

1

1

Disassembly in solution 2

1

1

1

1

0

Disassembly in solution 4

1

0

1

1

0

Disposal centers in solution 2

0

0

1

0

1

Disposal centers in solution 4

0

0

1

0

1

Redistributors in solution 2

1

0

0

1

1

Redistributors 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 analyses 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 reliable. On the other hand, in location-allocation type of the problems, fixed costs are the highest cost parameters and they play the most important roles in the results so they are chosen for sensitivity analysis in this paper. In this section, two different strategies are considered in changing fixed costs, which are presented as follows: • 50 % increasing in all fixed costs of all facilities and evaluating all candidate solutions in 11 scenarios for new fixed-cost situation. • 50 % decreasing in all fixed costs of all facilities and evaluating all candidate solutions in 11 scenarios for new fixed-cost situation. The complete results of the above-mentioned strategies in sensitivity analyses are illustrated in Tables 6 and 7. Under consideration of Tables 6 and 7, interesting points of the solution methodology and the candidate solutions are clarified, which are discussed as follows: • Except the solutions of the worst-case and the best-case scenarios, other candidate solutions are rarely could achieve best performance in the corresponding scenarios. Best performances of each scenario are highlighted in Tables 6 and 7, which can prove the effects of fixed 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 associate candidate solutions reveal best performance in the corresponding scenarios. • Regarding mean criterion, vague results are achieved, which cannot lead decision makers to find the optimal solution under uncertainty. When the fixed costs are increased 50 %, three candidate solutions present best performances, which are solutions 2, 6, and deterministic solution (see Table 6). Vice a Versa, when the fixed costs are decreased 50 %, two candidate solutions present best performance, which are solutions 3 and 7 (see Table 7). Results prove that although the candidate solution 2, reveals the best performance

Ann Oper Res Table 6 Evaluating candidate solutions for +50 % fixed-costs strategy (results in millions) Scenarios

Solutions Det. Sol. 1 Sol. 2 Sol. 3 Sol. 4 Sol. 5 Sol. 6 Sol. 7 Sol. 8 Sol. solutions WC

Sol. BC

Det.

3249

3218

3248

3177

3246

3214

3248

3177

3246

3087

2979

S1

3329

3332

3329

3280

3328

3329

3329

3280

3328

3010

3099

S2

3203

3132

3203

3115

3201

3127

3203

3115

3201

3005

2921

S3

3152

3149

3152

3196

3150

3146

3152

3196

3150

2907

3031

S4

3065

3048

3065

3032

3065

3044

3065

3032

3064

2965

2880

S5

3377

3383

3377

3330

3376

3379

3377

3330

3376

3051

3149

S6

3216

3144

3216

3128

3214

3140

3216

3128

3214

3015

2934

S7

3173

3176

3172

3226

3170

3173

3172

3226

3170

2920

3060

S8

3104

3088

3104

3073

3104

3084

3104

3073

3103

2999

2919

WC

2075

2037

2074

1993

2072

2030

2074

1993

2072

2154

1778

BC

4273

4428

4273

4538

4267

4430

4273

4538

4267

3339

5319

Mean

3201

3194

3201

3190

3199

3190

3201

3190

3199

2950

3097

SD

500

544

500

577

499

546

500

577

499

288

827

CV = (SD/mean) 0.1561

0.1703 0.1562 0.1809 0.1560 0.1711 0.1562 0.1809 0.1561 0.0977 0.2670

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

Solutions Det. Sol. 1 Sol. 2 Sol. 3 Sol. 4 Sol. 5 Sol. 6 Sol. 7 Sol. 8 Sol. solutions WC

Sol. BC

Det.

3368

3353

3368

3334

3369

3356

3368

3334

3370

3162

3248

S1

3448

3467

3449

3437

3451

3471

3449

3437

3452

3085

3368

S2

3322

3266

3323

3272

3324

3269

3323

3272

3324

3080

3190

S3

3272

3284

3272

3353

3272

3288

3272

3353

3273

2981

3300

S4

3185

3183

3186

3189

3188

3186

3186

3189

3188

3040

3149

S5

3496

3518

3497

3487

3499

3521

3497

3487

3499

3126

3418

S6

3336

3279

3337

3285

3337

3282

3337

3285

3338

3089

3203

S7

3292

3311

3292

3383

3293

3315

3292

3383

3294

2994

3329

S8

3224

3223

3225

3230

3227

3226

3225

3230

3227

3074

3188

WC

2195

2172

2195

2150

2195

2172

2195

2150

2195

2228

2047

BC

4392

4563

4394

4696

4390

4572

4394

4696

4391

3414

5588

Mean

3321

3329

3322

3347

3322

3332

3322

3347

3323

3025

3366

SD

500

544

500

577

499

546

500

577

499

288

827

CV = (SD/mean) 0.1505

0.1634 0.1505 0.1724 0.1503 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 fixed costs strategy. Therefore, relying on the mean criterion cannot ensure finding the optimal solutions based on the proposed scenario-based solution methodology, which is common in earlier researches. • Considering the risk criterion (standard deviation) and the integrated mean-risk criteria (CV), interesting results of confirming the superiorities of candidate solution 4 in both

Ann Oper Res

sensitivity analysis strategies are achieved. The results of Tables 6 and 7 prove that the candidate solution 4, which could previously present best performance in Table 4 of the main model, again, reveals best performances in objective function values in terms of achieving the lowest standard deviation and the lowest CV. Therefore, regarding the proposed integrated mean-risk criteria of this paper could lead decision makers to find reliable optimal solutions under uncertain environment. Finally, the analyses of the sensitivity studies prove the reliability of the proposed scenario-based solution methodology to achieve optimal solutions, while regarding multi criteria 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 to study the proposed model and the solution methodology. The company faced to demand uncertainty for the next 12 months, which considered as 12 periods here. The company expect 500 to 600 thousands unit of demand per period. The detail of the parameters is presented in Table 8. 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 scenarios are solved and then the results are regarded as candidate optimal solutions. The candidate solutions are evaluated under various scenarios and then based on the suggested criteria; best flows of the network are assigned to achieve the highest profit. The results of objective function values and criteria analysis are illustrated in Table 9. The results of Table 9 reveal that the candidate solution, which is achieved by middemand scenario, presents best performance in terms of considering mean criterion of its performance in all scenarios. However, if we are supposed to consider SD and CV (as the integrated mean-risk one), the candidate solution that is attained through low-demand scenario presents best performance. Thus, we can judge low-demand scenario solution as optimal one when regarding both mean and SD criteria simultaneously. Finally, in all cases, the reverse supply chain reveals a huge profit, which can make it reliable and profitable 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 closedloop supply chain. A scenario-based, multi-echelon, multi-period, and multi-product model including various flows between network entities is developed in this paper. This model has many characteristics of a generic model in both designing and planning stages. To solve the proposed model we have used IBM ILOG CPLEX 12.2 optimizer software. At the second step, we deal with non-deterministic demand and price parameters and then a multi criteria scenario based solution approach is developed to find optimal solution through some logical scenarios and three comparing criteria. 11 various scenarios (one deterministic, eight random, one worst-case and one best-case scenarios) are generated and solved to achieve 11 different candidate optimal solutions. Then, we have evaluated the performance of each solution in all scenarios. We have calculated mean, variance, and coefficient of variations

Ann Oper Res Table 8 Parameters of the case study Row

Parameter

Average or range per month (period)

1.

Demand (nondeterministic)

500000 to 600000

2.

Second demand rate

75 % of first

3.

First product price

70 Rupees per unit

4.

Second products price

40 Rupees per unit

5.

Purchasing price

15 Rupees per unit

6.

Manufacturers capacity

800

Remanufacturing capacity

550

7.

Warehouses capacity

4250

Suppliers capacity

1400

Distributors capacity

3750

Disassembly (collection) centers capacity

4250

Recycling centers capacity

4100

Redistributors capacity

2400

Disposal centers capacity

3250

Costs of supplying a unit in suppliers

20 Rupees

Costs of manufacturing a unit in manufacturers

5 Rupees

Costs of holding a unit in warehouses for one period

2 Rupees

Costs of distributing a unit in distributors

2 Rupees

Costs of remanufacturing a unit in manufacturers

5 Rupees

Costs of collecting a unit in disassembly centers

2 Rupees

Costs of recycling a unit in suppliers

5 Rupees

Costs of distributing a unit in redistributors

2 Rupees

Costs of disposing a unit of disposal centers

1 Rupees

Not utilizing manufacturing costs

1 Rupees per unit

Per 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 optimum in various scenarios

SD of the optimum in various scenarios

CV of the optimum in various scenarios

High demand

221

198

24.58

0.12

Mid 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 corresponding sensitivity analyses reveal reliability of the proposed solution approach for the stochastic model. Indeed, regarding integrated mean-risk criteria such as CV present reliable

Ann Oper Res

solutions in uncertain environment. Finally, a real case study in an Indian manufacturer is evaluated to ensure applicability of the model and the solution methodology. Surely, there are some guides as future research; first, in order to cope with large size instances, some heuristics, meta-heuristics, or elevated exact methods like branch and bound and column generation approaches can be utilized. Second, the risk considering method and the scenario-based solution approach can be developed in terms of incorporating other integrated criteria of decision making under uncertainty.

References Chopra, S., & Meindl, P. (2007). Supply chain management: strategy, planning and operation. Upper Saddle River: Pearson/Prentice Hall. Amin, S. H., & Zhang, G. (2012). An integrated model for closed-loop supply chain configuration and supplier selection: multi-objective approach. Expert Systems with Applications, 39(8), 6782–6791. Amin, S. H., & Zhang, G. (2013). A three-stage model for closed-loop supply chain configuration under uncertainty. International Journal of Production Research, 51(5), 1405–1425. Chouinard, M., D’Amours, S., & Aït-Kadi, D. (2008). A stochastic programming approach for designing supply loops. International Journal of Production Economics, 113(2), 657–677. Dembo, R. S. (1991). Scenario optimization. Annals of Operations Research, 30(1), 63–80. El-Sayed, M., Afia, N., & El-Kharbotly, A. (2010). A stochastic model for forward-reverse logistics network design under risk. Computers & Industrial Engineering, 58(3), 423–431. Faccio, M., Persona, A., Sgarbossa, F., & Zanin, G. (2011). Multi-stage supply network design in case of reverse flows: a closed-loop approach. International Journal of Operational Research, 12(2), 157–191. Francas, D., & Minner, S. (2009). Manufacturing network configuration in supply chains with product recovery. Omega, 37(4), 757–769. Guide, V. D. R., & Van Wassenhove, L. N. (2009). OR forum—the evolution of closed-loop supply chain research. Operations Research, 57(1), 10–18. Kaut, M., & Wallace, S. W. (2007). Evaluation of scenario-generation methods for stochastic programming. Pacific Journal of Optimization, 3(2), 257–271. Krarup, J., & Pruzan, P. M. (1983). The simple plant location problem: survey and synthesis. European Journal of Operational Research, 12(1), 36–81. Lamming, R., & Hampson, J. (1996). The environment as a supply chain management issue. British Journal of Management, 7(s1), S45–S62. Liste¸s, O. (2007). A generic stochastic model for supply-and-return network design. Computers & Operations Research, 34(2), 417–442. Ogryczak, W. (2000). Multiple criteria linear programming model for portfolio selection. Annals of Operations Research, 97(1–4), 143–162. Özceylan, E., & Paksoy, T. (2013). Fuzzy multi objective linear programming approach for optimizing a closed-loop supply chain network. International Journal of Production Research, 51(8), 2443–2461. Özkır, V., & Ba¸slıgil, H. (2013). Multi-objective optimization of closed-loop supply chains in uncertain environment. Journal of Cleaner Production, 41, 114–125. Pishvaee, M. S., & Torabi, S. A. (2010). A possibilistic programming approach for closed-loop supply chain network design under uncertainty. Fuzzy Sets and Systems, 161(20), 2668–2683. Pishvaee, M. S., Rabbani, M., & Torabi, S. A. (2011). A robust optimization approach to closed-loop supply chain network design under uncertainty. Applied Mathematical Modelling, 35(2), 637–649. Pokharel, S., & Mutha, A. (2009). Perspectives in reverse logistics: a review. Resources, Conservation and Recycling, 53(4), 175–182. Ramezani, M., Bashiri, M., & Tavakkoli-Moghaddam, R. (2013). A new multi-objective stochastic model for a forward/reverse logistic network design with responsiveness and quality level. Applied Mathematical Modelling, 37(1–2), 328–344. Sasikumar, P., & Kannan, G. (2009). Issues in reverse supply chain, part III: classification and simple analysis. International Journal of Sustainable Engineering, 2(1), 2–27. Schrijver, A. (2003). Combinatorial optimization: polyhedra and efficiency. Berlin: Springer. Soleimani, H., Seyyed-Esfahani, M., & Shirazi, M. A. (2013). Designing and planning a multi-echelon multiperiod multi-product closed-loop supply chain utilizing genetic algorithm. The International Journal of Advanced Manufacturing Technology. doi:10.1007/s00170-013-4953-6.

Ann Oper Res Subramoniam, R., Huisingh, D., & Chinnam, R. B. (2009). Remanufacturing for the automotive aftermarketstrategic factors: literature review and future research needs. Journal of Cleaner Production, 17(13), 1163–1174. Tonanont, A., Yimsiri, S., Jitpitaklert, W., & Rogers, K. J. (2008). Performance evaluation in reverse logistics with data envelopment analysis. In Proceedings of the 2008 industrial engineering research conference (pp. 764–769). Zhou, G., & Min, H. (2011). Designing a closed-loop supply chain with stochastic product returns: a genetic algorithm approach. International Journal of Information Systems for Logistics and Management, 9(4), 397–418. Zhu, X., & Xiuquan, X. U. (2013). An integrated optimization model of a closed-loop supply chain under uncertainty. In LISS 2012 (pp. 1389–1395). Berlin: Springer.