Resources, Conservation and Recycling 74 (2013) 156– 169
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Resources, Conservation and Recycling
jo ur n al hom epa ge : www.elsev ier .com/ locate / resconrec
n optimization model for product returns using genetic algorithmsnd artificial immune system
li Diabata,∗, Devika Kannanb,∗, Mathiyazhagan Kaliyanc,d, Davor Svetinovice
Engineering Systems & Management, Masdar Institute of Science & Technology, Abu Dhabi, United Arab EmiratesDepartment of Mechanical & Manufacturing Engineering, Aalborg University, DK-2450 Copenhagen, DenmarkDepartment of Business and Economics, University of Southern Denmark, DenmarkDepartment of Production Engineering, National Institute of Technology, Tiruchirappalli, IndiaComputing & Information Science, Masdar Institute of Science & Technology, Abu Dhabi, United Arab Emirates
a r t i c l e i n f o
rticle history:eceived 20 December 2010eceived in revised form 25 October 2012ccepted 16 December 2012
eywords:everse logisticsocation-allocationenetic algorithm (GA)rtificial immune system (AIS)
a b s t r a c t
Current environmental issues emerging in the world are reflected in the environmental legislation ofseveral countries. Because environmental issues are important, industries actively seek ways in whichto reduce their environmental footprint. One effective method is through the use of reverse logistics.Reverse logistics is the concept of reusing used products in order to reduce wastes and to increase anindustry’s environmental performance and resulting profits. Stock selection, transportation, centralizedcollection, data collection, refurbishing, and remanufacturing are some of the more commonly utilizedreverse logistic operations. An effective reverse logistics network is essential for increasing the flowof goods from customers to producers. The objective of this paper is to develop a multi-echelon reverselogistics network for product returns to minimize the total reverse logistics cost, which consists of renting,inventory carrying, material handling, setup, and shipping costs. Industries need to give more attention tothe task of collecting used products from customers and establishing collection facilities. In this study, a
mixed integer non-linear programming (MINLP) model is developed to find out the number and locationof initial collection points and centralized return centers required for an effective return and collectionsystem, and also the maximum holding time (collection frequency) for aggregation of small volumes ofreturned products into large shipments. Two solution approaches, namely genetic algorithm and artificialimmune system, are implemented and compared. The usefulness of the proposed model and algorithmare demonstrated via an illustrative example.
. Introduction
Supply chain management (SCM) is primarily concerned withntegrated purchasing strategy, integrated logistics, supplier inte-ration, buyer and supplier partnerships, supply base management,trategic supplier alliances (Tan, 2001) and has rarely consideredeverse logistics issues. The present day’s reverse logistics con-ept is becoming familiar to industries and there is an emergingrend to reduce waste (Thierry et al., 1995; Diabat and Govindan,011; Kannan et al., 2009). Kopicky et al. (1993) defined reverse
ogistics as, “Reverse Logistics is a broad term referring to the
ogistics management and disposal of hazardous or non-hazardous
aste from packaging and products. It includes reverse distribu-ion which causes goods and information to flow in the opposite
direction of normal logistics activities” (p. 7). Rogers and Tibben-Lembke(1999, p. 17) defined reverse logistics concepts from adifferent perspective: “Reverse logistics is the process of planning,implementing and controlling the efficient, effective inbound flowand storage of secondary goods and related information oppo-site to the traditional supply chain directions for the purpose ofrecovering value and proper disposal” (p. 17). Many researchershave found that economics, environmental laws, and the envi-ronmental consciousness of consumers are the driving factors foradopting reverse logistics concepts in industries (Guide and vanWassenhove, 2003; Kannan and Sasikumar, 2009; Louwers et al.,1999; Bloemhof-Ruwaard et al., 1999; Farrow and Jonhson, 2000;De Brito and Dekker, 2003; Diabat and Simchi-Levi, 2009; Abdallahet al., 2010, 2012a,b; Diabat and Govindan, 2011; Abdallah et al.,2012a; Kannan et al., 2012; Shen et al., 2013; Diabat et al., 2013).
The efficiency of service management depends heavily on the effec-tiveness of reverse logistics operations (De Brito and Dekker, 2003).Many activities are included in reverse logistics concept such asthe reuse of used products and the collection, disassembly, and
rocessing of excess inventory of products, parts, and/or materi-ls (Daugherty et al., 2005). Now, reverse logistics operations areocusing on environmental issues called “green” logistics (Rogersnd Lembke, 2001). In USA, Platt and Hyde (1997) found that everyear, 60 million computers entered the market and over 12 mil-ion computers were disposed. Of these, however, only 10% wereemanufactured or recycled. Rogers and Lembke (2001) found thatSA spent $37–$921 billion in reverse logistics operation per year.
Supply chain network design is classified into two divisions,amely forward logistics and reverse logistics. By definition, a for-ard logistics network moves from manufacturer to customer,hereas the focus of a reverse logistics network is from customer toanufacturer (Ramezani et al., 2012; Abdallah et al., 2012b). There-
ore, establishing an efficient product recovery network design is challenging and an emergent issue in reverse logistics (Lieckensnd Vandaele, 2007). Alumur et al. (2012) pointed out that reverseogistics network designs are motivated by an application-orientedpproach. A reverse logistics network encompasses the backwardetwork called recovery network which is necessarily integratedith the forward network in order to form a closed-loop network
Ramezani et al., 2012).The collection of used products is very difficult and needs a
ell-estimated structure in reverse logistics (Lee and Dong, 2009;rivastava, 2008; Fleischmann, 2003). This paper develops a math-matical model to optimize the reverse logistics network linkingustomers, initial collection points, and centralized return centers.t also develops a consolidation strategy by determining the maxi-
um holding time (collection frequency) for aggregation of smallolumes of returned products into large shipments.
This paper is structured as follows. In Section 2, the relevantiterature is reviewed. In Section 3, the problem environment,ssumptions, indices, model parameters, and model formulationre described in detail, and in Section 4, a case study is described.n Section 5, the solution methodology is proposed, and in Section, the results of the computational study are presented. Finally, inection 7, conclusions and suggestions for future research are given.
. Relevant literature
Many researchers in the field of supply chain managementave focused on the forward movement (the transformation of theaterials from the suppliers to the end consumer, or the forward
ogistic). Much less attention, however, has been devoted to theeld of reverse logistics (Anzi et al., 2007). Owing to an increasingwareness of environmental concerns, industries need to give morettention toward the management of waste streams and reductionf non-renewable resources usage (Gungor and Gupta, 1999; Karat al., 2003; Govindan et al., 2012; Kannan et al., 2012; Anonymous,000; Kannan, 2009). Due to the above reasons, industries havetarted to implement the reverse logistics concept (Diabat et al.,013). Kara et al. (2007) expressed that a vital environmentalanagement objective is to reduce waste and to adopt a prod-
ct recovery concept. Lee and Dong (2009) found that a businesspecializing in the recovery of used electronic personal computerroducts was estimated to have a value between $2 and $3 bil-
ion in 1996. Approximately 25 million obsolete PCs became readyor remanufacture or disposal in 1997. Fleischmann et al. (1997)lassified three dimensions of reverse logistics, namely: reverseistribution network, inventory control systems with return flows,nd production planning with the reuse of parts and materials.ue to the changes in government regulations about the envi-
onment, many industries have started to adopt reverse logistic
oncepts by means of going backwards from customers to recov-ry centers within their logistics systems (Lu and Bostel, 2007;amilton, 2001; Pohlen and Farris, 1992). Srivastava (2008) pointedut that reverse logistic plays a major role in service management
and Recycling 74 (2013) 156– 169 157
activities and take-back for products such as automobiles, refriger-ators and other white goods, cellular handsets, lead-acid batteries,televisions, and personal computers (PCs). The reverse logisticsconcept is getting special attention because of the environmen-tal issues as well as economic reasons (Ginter and Starling, 1978;Gupta and Veerakamolmal, 2000; Sasikumar et al., 2010; Haberlandet al., 1997; Pochampally et al., 2009; Kusumastuti et al., 2008;Kara et al., 2007; Sasikumar and Kannan, 2008a,b, 2009; Chenget al., 1998). Rogers and Tibben-Lembke (1999) and Alvarez-Gilet al. (2007) stated that reverse logistics adoption gives tremendouseconomic benefits. Reverse logistics provides an area for improvingprofitability and customer satisfaction by means of implemen-ting an appropriate structure by which to enact reverse logistics.Many models have been proposed which focus on aspects suchas product recycling and planning/distribution (Kroon and Vrijens,1995; Barros et al., 1998; Listes and Dekker, 2001; Giannikos, 1998;Gaspar and Collard, 2000; Jayaraman et al., 1999; Fleischmann et al.,2001; Salema et al., 2007; Rogers and Tibben-Lembke, 2011). Inrecent years, the recovery of used products in the electronics fieldis getting special attention. Fleischmann et al. (2001) proposed anintegrated solution approach for logistics network configurationsof electronic products remanufacturing involving both forward andreverse flows.
However, there is a need for an appropriate logistics infra-structure for the collection of used products (Hoshino et al., 1995;Min, 1989). This logistic infrastructure is called a network. Chopraand Meindl (2000) stated that network design is a strategic issuein conventional supply chain management. Fleischmann (2001)examined reverse-supply networks consisting of multiple collec-tion points and central facilities where returned products wereinspected, sorted, and refurbished. Fleischmann (2000) expressedthree important factors for designing the network design in reverselogistics, namely, the supply uncertainty, the degree of centraliza-tion of testing and sorting, and the interrelation between forwardand reverse flows. Reverse logistics network design is a premedi-tated concern of major importance for the economic feasibility ofproduct recovery activities (Cimino et al., 2010; Schultmann et al.,2006; Shih, 2001). Barros et al. (1998) conducted a case study aboutthe design of a logistics network for recycling sand coming fromprocessing construction waste in Netherlands. Jayaraman et al.(1999) developed a closed-loop logistic model for a recoverableproduct environment.
Jayant et al. (2011) have developed modeling and simulationin a reverse logistics network for collection of end of life productsfor XYZ Limited Company in North India. Mixed integer linear pro-gramming (MILP) models have become a standard approach fornetwork design (Fleischmann, 2000). Louwers et al. (1999) pro-posed a non-linear programming model procedure to determinethe location and size of regional recycling centers for a carpetwaste management network with a linear approximation solution.Corbacioglu and Van der laan (2007) analyzed two-product systemswith joint manufacturing and remanufacturing by traditional val-uation methodology. Pati et al. (2008) formulated a mixed integergoal programming (MIGP) model to assist in a proper managementof the paper recycling logistics system from the Indian perspec-tive. Realff et al. (1999) proposed a multi level capacitated facilitylocation mixed-integer programming model to support reverselogistics network design for carpet recycling. Lee et al. (2009) ana-lyzed integrated forward and reverse distribution network designin third party logistics. Lieckens and Vandaele (2007) proposed amixed integer linear program model (MILP) with some queuingcharacteristics using a G/G/m model. Jayaraman et al. (2003) for-
mulated a MILP model to determine an efficient strategy for thereverse logistics operations of hazardous products. Aras and Aksen(2007) also formulated a mixed-integer nonlinear facility location-allocation model to determine both the optimal locations of the
1 ation
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58 A. Diabat et al. / Resources, Conserv
ollection centers and the optimal incentive values for each returnype so as to maximize the profit from the returns. Kara et al.2007) proposed an Arena 7.0 simulation model for a reverse logis-ics network for white goods collection in the Sydney Metropolitanrea. Salema et al. (2007) proposed a mixed integer formulationsing standard branch and bound techniques in a reverse logisticsetwork with capacity limits, multi-product management, andncertainty on product demands and returns. Using Tabu search,
logistics network was designed for end-of-lease computer prod-ct recovery by developing a deterministic programming model forystematic management (Lee and Dong, 2008).
Alumur et al. (2012) formulated a mixed-integer linear pro-ramming model which was flexible to incorporate most of theeverse network structures reasonable in practice. Kooi et al. (1996)roposed a mixed integer linear programming model for the setupf a multi-echelon logistics network for product recovery with aiven supply and demand using a linear programming (LP) solver.rikke et al. (1999a,b) also proposed another MILP model for theulti-echelon product recovery network design which focused on
he remanufacturing of a certain type of photocopier. Jayaramant al. (1999) analyzed the logistic network of an electronic equip-ent remanufacturing company in the USA. Lu and Bostel (2007)
ave used a 0–1 mixed integer programming model for two-levelocation problems with three types of facilities in specific reverseogistics system in a remanufacturing network using a Lagrangianeuristics algorithm. Ko and Evans (2007) used a mixed integeronlinear programming model for the design of a dynamic inte-rated distribution network to account for the integrated aspect ofptimizing the forward and return network simultaneously usingenetic algorithms. Lieckens and Vandaele (2007) formulated anfficient design of a reverse logistics network using mixed inte-er linear program to determine which facilities should be openedn order to minimize the investment, processing, transportation,isposal and penalty costs while the supply, demand, and capac-
ty constraints are satisfied using a differential evaluation. Listes2007) formulated a generic stochastic model for the design ofetworks comprising both supply and return channels, organized in
closed loop system. Alshamrani et al. (2007) developed a heuristicrocedure for treating the route design-pickup strategy planning
roblem. Table 1 shows the literature sources for reverse logisticsetwork design with the proposed models.
Many of the researchers formulated network designs in reverseogistics for product recovery (reuse, remanufacturing, recycling,
able 1iterature sources for reverse logistics network design.
Sl.No Authors Model
1. Chen and Su (2010) Weighted fuzzy goal programming
2. Dehghanian and Mansour (2009) Three-objective mathematical program3. Das and Chowdhury (2012) Mixed integer programming (MIP)
4. Du and Evans (2006) Bi-objective optimization model
5. Mingyong and Erbao (2010) Mixed integer programming
6. Ghezavati et al. (2009) Nonlinear integer programming model7. Kannan et al. (2009) Mixed integer linear programs (MILP-m8. Kannan et al. (2010) Mixed-integer programming model
9. Ko and Evans (2007) Mixed integer nonlinear programming
10. Lee et al. (2009) Mixed-integer programming model
11. Lee and Dong (2009) Mixed Integer Programming (MIP)
12. Lieckens and Vandaele (2007) Mixed integer nonlinear program (MIN13. Listes (2007) Stochastic integer programming
14. Lu and Bostel (2007) Mixed integer programming model
15. Min et al. (2006a) Mixed integer non-linear programming16. Min et al. (2006b) Mixed-integer programming model
17. Pishvaee et al. (2010) Bi-objective mixed integer programmin18. Pishvaee et al. (2009) Stochastic programming model-mixed
19. Pishvaee et al. (2011) Deterministic mixed-integer linear pro20. Sayed et al. (2010) Stochastic mixed integer linear program21. Qin and Ji (2010) Fuzzy simulation and genetic algorithm22. Yongsheng and Shouyang (2008) Mixed integer formulation
and Recycling 74 (2013) 156– 169
and repair) by using mixed integer programming. Also, they pro-posed a mixed integer model with GA, differential equations, etc.From the literature survey, it is clearly evident that there is noresearch on network design using GA and artificial immune sys-tem. Table 1 is also one of the strong proofs for this research and itshows the research gap in the past literature.
Kannan and Sasikumar (2009) pointed out that the model pro-posed by Min et al. (2006a) is invalid, and that there is errors inthe calculation of the total reverse logistics cost. In this study, themodel proposed by Min et al. (2006a) is modified by altering therenting and shipping cost terms in the objective function, and theproposed mixed integer nonlinear programming problem is solvedusing genetic algorithm and artificial immune system.
3. Problem description
Due to the increasing environmental pressure from stricter gov-ernment regulations and customers, industries need to reduce theirwaste and improve their environmental performance (Srivastava,2008; Govindan et al., 2012; Muduli et al., 2012). Because ofthese reasons, industries have started to adopt the reverse logis-tics practices, using returned products and incorporating productrecovery activities in their production activities (Lee and Dong,2009). Reverse logistics has become increasingly important as aprofitable and sustainable business strategy (Du and Evans, 2008).Regarding reverse logistics, many researchers recognize the ben-efits of implementing reuse, recycling, remanufacturing logistics(Lee et al., 2007; Azevedo et al., in press). Krikke et al. (2003)proposed many examples to promote the use of modularity forimproved recovery and optimal reuse of products, as well as toobtain economic and higher ecological advantages.
In the reverse logistics concept, the centralized return centerplays an imperative role in linking initial collection points to themanufacturing or reprocessing facilities (Walker, 2000). This pro-cess includes carefully choosing the location of initial collectionpoints for product returns (i.e. collection points that are locatednearer to the customer can help to reduce the shipping time forproduct returns), establishing centralized return centers to aggre-gate small shipments into a large shipment, using the fastest
mode of transportation to reduce the in-transit inventory carryingcost, determining the optimal number of initial collection pointsand centralized return centers, and determining the holding timefor the consolidation of returned products. These considerations
Methodology used
Particle Swarm optimization (PSO)ming model Multi-objective genetic algorithm (MOGA)
g Memetic algorithminteger linear programming Scenario-based stochastic approachgramming model Novel robust optimization
ming Multi-stage stochastic program GA
B&B techniques
ation
waitiiplltfsaam
3
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•
•
•
3
3
Xkc
˛
A. Diabat et al. / Resources, Conserv
ere incorporated into the model proposed by Min et al. (2006b),nd the resulting model using genetic algorithms and artificialmmune system was used to find out the optimal location, andhe number and size of both initial collection points and central-zed return centers. But, a design of networks for used productss difficult and takes considerable time. In this study, severalroblem statements are required to develop a mixed integer non-
inear programming (MINLP) model to find out the number andocation of initial collection points and centralized return cen-ers and also the maximum holding time (collection frequency)or aggregation of small volumes of returned products into largehipments. Two solution approaches, namely genetic algorithmnd artificial immune system, are implemented and compared. Thelgorithms were implemented in C++, and a comparative study wasade between them.
.1. Assumptions
The various assumptions involved are adopted from Min et al.2006b):
The possibility of direct shipment from customers to a centralizedreturn center is ruled out due to insufficient volume.Given the small volume of individual returns from customers, aninitial collection point has sufficient capacity to hold returnedproducts during the consolidation process.The transportation costs between customers and their nearestcollection points are negligible.The location/allocation plan covers a planning horizon withinwhich no substantial changes occur in customer demands or thetransportation infrastructure.
.2. Indices
i index for customers; i ∈ Ij index for initial collection points; j ∈ Jk index for centralized return centers; k ∈ K
.3. Model parameters
aj annual cost of rent for initial collection point jb daily inventory carrying cost per unitw annual working daysri daily volume of products returned by customer ih cost of handling one unit of product per dayqk cost of establishing centralized return center kmk maximum capacity of centralized return center kdij distance between customer i and initial collection point kdjk distance between collection point j and centralized return cen-ter kl maximum allowable distance from customer i to initial collectionpoint jf (Xj0, dj0) = E˛ (=function of the freight rate)
˛ is a discount rate that may depend on the shipping volumejk between initial collection point j and centralized return center; is a penalty rate that may depend on the distance djk betweenollection point j and centralized return center k.⎧⎪⎨ 1 for Xj0 ≤ p1
⎧⎪⎨ 1 for dj0 ≤ q1
= ⎪⎩ ˛1 for p1 < Xj0 ≤ p2
˛2 for Xj0 > p2
= ⎪⎩ ˇ1 for q1 < dj0 ≤ q2
ˇ2 for dj0 > q2
and Recycling 74 (2013) 156– 169 159
E unit freight rate.p1, p2 volume breakpoints used in the calculation of ˛q1, q2 distance breakpoints used in the calculation of ˇz minimum number of established initial collection points.g minimum number of established centralized return centers.M arbitrarily large number.
3.4. Decision variables
Xjk = volume of products returned from initial collection point j tocentralized return center kTj = maximum holding (collection) time (in days) at initial collec-tion point jYij = 1, if customer i is allocated to initial collection point j; 0, oth-erwise.Zj = 1, if an initial collection point is established at site j; 0, other-wise.Gk = 1, if a centralized return center is established at site k; 0, oth-erwise.
3.5. Mathematical formulation
A MINLP is developed to minimize the total reverse logistics costcomprising renting, inventory carrying, setup, material handlingand shipping costs, and is modified from Kannan et al. (2009). Theobjective function (1) has a nonlinear form, because both inventorycarrying and shipping costs are affected by the length of a collectionperiod. The MINLP model is:
Minimize
⎡⎣∑
j
ajZj + bw∑
j
⎧⎨⎩
∑j
riYij(Tj + 1)
2
⎫⎬⎭ + hw
∑j
ri
+∑
k
qkGk +∑
K
{Gk
∑k
(Xjk
w
Tj + 1
)· f (Xjk, djk)
}⎤⎦ (1)
Subject to the following constraints:Constraint (2) assures that each customer is assigned to exactly
one initial collection point∑j
Yij = 1, ∀i ∈ I, (2)
Constraint (3) prevents any return flow from a closed initialcollection point.∑
i
Yij ≤ M.Zj, ∀j ∈ J, (3)
Constraint (4) requires the incoming flow to equal the outgoingflow at an initial collection point.∑
i
riYij(Tj + 1) =∑
k
Xjk, ∀j ∈ J, (4)
Constraint (5) ensures that the total volume of productsreturned from initial collection points does not exceed the max-imum capacity of a centralized return center.∑
j
Xjk ≤ mkGk, ∀k ∈ K, (5)
Constraint (6) ensures that each initial collection point shouldbe located within a certain allowable proximity to customers.
dijYij ≤ 1, ∀i ∈ I, ∀j ∈ J, (6)
160 A. Diabat et al. / Resources, Conservation and Recycling 74 (2013) 156– 169
Table 2Input coordinates of the initial collection points and the centralized return centers.
Potential sites for initial collection points Site coordinate Potential sites for centralized return centers Site coordinate
Constraint (7) ensures a minimum number of initial collectionoints for product returns.
≤∑
j
Zj, (7)
Constraint (8) ensures a minimum number of centralized returnenters for product returns.
≤∑
k
Gk, (8)
Constraint (9) preserves the non-negativity of decision variablesjk
jk ≥ 0, ∀j ∈ J, ∀k ∈ K, (9)
Constraint (10) ensures the integrality of decision variables Tj
j ∈ (0, 1, 2, 3, 4, 5, 6, 7), ∀j ∈ J, (10)
Constraint (11) ensures the binary integrality of decision vari-bles Yij, Zj, Gk
ij, Zj, Gk ∈ (0, 1), ∀i ∈ I, ∀j ∈ J, ∀k ∈ K (11)
. Description of the case study
The case study was adopted from Min et al. (2006b) withhe aim of validating the proposed methodology using a modi-ed mixed integer nonlinear programming model. Reta.com is annline fabricated e-tailer that sells various computer equipment
nd peripherals. Customers of Reta.com drop their used productsn the nearest initial collection points such as local supermarkets,omputer spare parts stores, etc. These collection points receivesed products and store them as “non-selling” items. This will incur
able 3ocations and daily demands of customers.
Customer number Site coordinate Daily demand
x y
1 16 4 12
2 19 25 43
3 2 59 34
4 9 2 21
5 49 54 19
6 33 11 10
7 29 50 37
8 25 59 22
9 3 36 35
10 33 22 29
11 45 27 22
12 46 6 21
13 25 33 11
14 28 33 27
15 3 0 44
the variable costs associated with rent of non-selling product stor-age. These collection points are called centralized return centers.(For more details, kindly refer to Min et al., 2006b.)
The potential locations of collection points and centralizedreturn centers are summarized in Table 2. Once established, thesefacilities would serve a total of 30 clusters of customers and thedaily demand of each cluster is shown in Table 3. The total dailydemand is 850 units and multiplying this by the number of annualworking days (250 days) gives a total annual demand of 212,500units. For simplicity, in this work the Euclidean distance is used formeasuring the travel distances. Based on the computational stud-ies, we set the optimal maximum allowable distance between thecustomers and initial collection points to 25 miles, a value that bal-anced the conflicting objectives of minimizing reverse logistics costand maximizing the customer service.
Other input parameters associated with initial collection pointsand centralized return centers are summarized in Table 4.
5. Solution methodology
Since our model belongs to a class of NP-complete problem, itis difficult to find out the optimal solution at a reasonable com-putational time. We therefore used the genetic algorithm (GA)and artificial immune system (AIS) approaches to solve our model.These approaches are described below.
5.1. Genetic algorithm
GA was first proposed by Holland (1975). GA finds a better solu-tion for complex mathematical problems within a reasonable time(Wang and Chen, 2012; Chen et al., 2013; Khan et al., 2010). GA isa stochastic search technique that mimics the principle of natural
Working days per year w 250Unit handling cost at the initial
collection pointh $0.1
Cost of establishing a centralizedreturn center
qk $3000
Capacity of a centralized returncenter
mk 1000 units
Service coverage l 25 milesUnit standard transportation cost E $1
Discount rate with respect toshipping volume
˛1 0.8
˛2 0.6p1 200 unitsp2 400 units
Penalty rate with respect toshipping distance
ˇ1 1.1
ˇ2 1.2q1 60 milesq2 25 miles
Minimum number of establishedinitial collection points
z 1
gKfie2oisom((atspsyriaA2L
mosomes to create their offspring (Pasandideh et al., 2011). A
Minimum number of establishedcentralized return centers
g 1
enetics to find a solution (Gen and Cheng, 2000; Goldberg, 1989;han and Govindan, 2011; Kannan et al., 2010). GA is always pre-
erred to solve problems in an effective manner by generation viats systematic operators, which provide improvements and vari-ty in the solution population (Tuzkaya et al., 2011; Kannan et al.,009). Generally, many different GA approaches exist dependingn the problems studied, and many researchers have used themn their work efficiently (Tasan and Gen, 2012). Min et al. (2006b)tated in their literature review that GA may be applied to a varietyf problems, namely, fixed charge location (Jaramillo et al., 2002),inimum spanning tree (Zhou and Gen, 1999), network design
Palmer and Kershenbaum, 1995), distribution inventory modelsHaq and Kannan, 2006a,b; Kannan et al., 2009) and warehousellocation (Zhou et al., 2003). Pasandideh et al. (2011) pointed outhe advantage of GA over conventional search techniques in theense that it starts with an initial set of random solutions calledopulation. Each individual in the population is called a chromo-ome, representing a solution to the problem at hand. In recentears, GA is getting special attention in supply chain managementesearch studies (Chen et al., 1998). Many researchers have used GAn reverse logistics network design (Ko and Evans, 2007; Bautistand Pereira, 2006; Min et al., 2006b; Aras et al., 2008; Aras and
ksen, 2008; Lee and Dong, 2009; Min and Ko, 2008; Kannan et al.,009; Lee et al., 2009; Lee and Chan, 2009; Lee and Dong, 2009;ieckens and Vandaele, 2007; Alshamrani et al., 2007).
0 0 0 0 0 0 0 0 1 0 1 1
DAYS
icp1 icp2 icp3
Fig. 1. A genetic representation of a chromosome, icp, ini
and Recycling 74 (2013) 156– 169 161
The procedure of GA is summarized as follows (Pasandideh et al.,2011):
(i) Population size (N): It is the number of the chromosomes orscenarios that are kept in each generation.
(ii) Crossover rate (Pc): This is the probability of performing acrossover in the GA method.
(iii) Mutation rate (Pm): This is the probability of performing muta-tion in the GA method.
In this paper, we have represented the chromosome using a one-dimensional array of binary values, representing decision variablesrelated to initial collection points, centralized return centers, andcollection periods (i.e., holding time for consolidation at the collec-tion point), as illustrated in Fig. 1. The chromosome consists of 10initial collection points, 7 days of collection periods at collectionpoints, and 5 centralized return centers. Each collection point isdescribed by four bits: the first bit is 1 if the initial collection pointis open, and 0 if it is closed; the remaining three bits representthe collection period for that collection point, so that eight possiblecollection periods (0–7) are possible. Each centralized return cen-ter point is described by one bit which is 1 if it is open and 0 if it isclosed.
5.1.1. Genetic operatorsWe have used four genetic operators in our genetic algorithm,
which are described below based on Min et al. (2006b).Cloning operator: The cloning operator keeps the best solutions
(Ko and Evans, 2006). In our genetic algorithm, the top 20 percentof the chromosomes in the current population are copied to thenext generation.
Parent selection operator: Many researchers have mentionedthe parent selection operator as an imperative process in GA(Wang and Chen, 2012; Gen and Cheng, 2000; Tuzkaya et al.,2011; Tasan and Gen, 2012; Min et al., 2006b). Two parents areselected from the current generation by a selection method thatassigns reproduction opportunities to each individual parent in thepopulation (Ko and Evans, 2006). Gen and Cheng (1997) stateddifferent parent selection methods including roulette wheel selec-tion, tournament selection, rank selection, elitism selection, andrandom selection. In this study, a binary tournament selectionmethod was used that begins with forming two teams of chromo-somes. Each team consists of two chromosomes randomly drawnfrom the current population. The two best chromosomes fromeach team are selected for the crossover operations. This processresults in two offspring, each of which enters into the new popula-tion.
5.1.2. Crossover operatorIn a crossover process, it is necessary to mate the pairs of chro-
crossover operator is used to recombine two strings to get a bet-ter string (Awan et al., 2008). There are several types of crossovers,which includes single-point crossover, multi-point crossover, and
0 0 0 0 1 1 0 0 0
icp10 crc 1 to 5
tial collection points; crc, centralized return center.
1 ation
ufoudlcictt
5
tattcisgrtttrrfltopwptrcca
5
aaotdTrseoa
62 A. Diabat et al. / Resources, Conserv
niform crossover (Gen and Cheng, 2007). Parents are selectedrom the current generation and considered for the crossoverperation to form their offspring (Tuzkaya et al., 2011). Here, wesed a two-point crossover in which the crossover points are ran-omly selected, subject to the constraints that the first point is
ocated within the region of the chromosome that describes theonfiguration of the initial collection points, and the second points located within the region of the chromosome that describes theonfiguration of the centralized return centers. The region betweenhe selected points is swapped between the two parents to produceheir children.
.1.3. Mutation operatorIn GA, mutation is the second operation for exploring new solu-
ions (Pasandideh et al., 2011). Mutation adds new information in random way to the genetic search process and ultimately helpso avoid getting trapped at local optima (Awan et al., 2008). Muta-ion is a background operator which produces spontaneous randomhanges in various chromosomes. Similar to crossover, mutations done to prevent the premature convergence and explores newolution space (Gen and Cheng, 2000; Lee et al., 2009). In ourenetic algorithm, mutation operates either on the bits within theegion of the chromosome that determines which initial collec-ion points are opened/closed, or on the bits within the region ofhe chromosome that determines which centralized return cen-ers are opened/closed. If the bit to be mutated lies within theegion of the chromosome, that will determine which centralizedeturn centers are opened/closed, and then those bits are simplyipped. However, if the bit to be mutated lies within the region ofhe chromosome that determines which initial collection points arepened/closed, then in addition to flipping that bit, the collectioneriod associated with that initial collection point is mutated asell. If the mutation has the effect of closing that initial collectionoint, the corresponding collection period is set to zero; whereas ifhe mutation has the effect of opening that initial collection point,andom values are selected for the three bits that determine theollection period associated with that initial collection point. Thisonfiguration achieves a high level of diversity among each gener-tion.
.1.4. Fitness functionThe fitness function is unique for each problem (Rao, 1999).GA imitates the survival of the fittest principle of nature to make
search process (Lee et al., 2009). In a biological sense, fitness is quality value which is a measure of the reproductive efficiencyf chromosomes (Awan et al., 2008). Based on the fitness func-ion, decoding the chromosome generates a candidate solution andetermines its fitness value (Min et al., 2006b; Ko and Evans, 2007).he fitness value is a measure of the goodness of a solution withespect to the original objective function and the “degree of infea-ibility” (Santos et al., 2010). Under the assumption that there isnough capacity at each collection point owing to the small volumef returns, this result can be accomplished by solving the followingssignment problem (Min et al., 2006b):
Minimize
⎡⎣∑
j
ajZj + bw∑
j
⎧⎨⎩
∑j
riYij(Tj + 1)
2
⎫⎬⎭
⎤⎦
Subject to∑
j
Yij = 1, ∀i ∈ I,
∑
i
Yij ≤ M Zj, ∀j ∈ J,
dijYij ≤ 1, ∀i ∈ I, ∀j ∈ J,Yij ∈ (0, 1), ∀i ∈ I, ∀j ∈ J,
and Recycling 74 (2013) 156– 169
The second procedure assigns open initial collection points to acentralized return center taking into account capacity limitations.This step can be accomplished by solving the following transporta-tion problem using the simplex method:
Minimize∑
k
qkGk +∑
K
⎧⎨⎩Gk
∑j
(Xjk
w
Tj + 1
)· f (Xjk, djk)
⎫⎬⎭
subject to∑
i
riYij(Tj + 1) =∑
k
Xjk, ∀j ∈ J,∑j
Xjk ≤ mkGk, ∀k ∈ K,
Xjk ≥ 0, ∀j ∈ J, ∀k ∈ K
The fitness function is formed by adding a penalty to the originalobjective function.
The following penalty function was used (Min et al., 2006b):
Penalty function =∑
j
∑k
p� × g(Xjk, mk, Gk)∑
i
∑j
p� ×
h(dij, Yij, lk),where
pv = penalty value;
g(Xjk, mk, Gk) = 1, if∑
j
Xjk > mkGk; 0, otherwise;
h(dij, Yij, l) = 1, if dijYij > l; 0, otherwise.
The penalty value is chosen to be considerably larger thanany possible objective function value corresponding to the cur-rent population of individuals. A fitness value is assigned to eachchromosome by applying two procedures in succession. The firstprocedure calculates the total daily demand of the opened collec-tion points by assigning customers to the nearest collection point.
5.2. Artificial immune system
During the last decade new computational intelligenceapproaches have been developed based on the principles ofimmune system, called artificial immune system (AIS) (Liu et al.,2006; Deng et al., 2007; Lu et al., 2007). The most importantfunction of an immune system is to save the body from unfamil-iar invaders (Kumar et al., 2011). AIS algorithm replicates humanbody’s defense system against viruses (Ada and Nossal, 1987; Kilicand Nguyen, 2010; Aickelin and Dasgupta, 2005). De Castro andVon Zuben (2002) defined AIS as “an abstract or metamorphiccomputational system using ideas gleaned from the theories andcomponents of immunology”. AIS is a nature-inspired algorithmwhose motivation comes from the vertebrate natural immune sys-tem (NIS) (Butler and Kazakov, 2010). In modern years, AIS isgetting tremendous attention from researchers due to the immunesystem and its powerful information processing capabilities (Awanet al., 2008). The AIS was inspired by theoretical immunology andobserved immune functions, principles, and models (De Castro andTimmis, 2002). Hart and Timmis (2005) found that AIS has beenused in recent years to solve artificial or benchmark problems andto tackle real-world applications using an equally diverse set ofimmune-inspired algorithms. AIS techniques have been summa-rized as (1) learning, (2) anomaly detection, and (3) optimization(Afshari and Sajedi, 2012). Cheng and Cheng (2011) determinedthat by using AIS we can develop a model to solve differentproblems including anomaly detection, clustering, and functionoptimization.
From the literature review, it is evident that various researchershave mentioned that AIS covers huge areas of complex compu-tational and engineering problems such as pattern recognition,fault and anomaly detection, data mining and classification,
Interaction of components Crossover RecognitionInteraction with environment Fitness function Recognition
function
A
sgcc1Cae2m2Wlttiawfi2stowaiTto
a
5
f
A
same antibody, let i and j be two randomly selected centralized
TC
dopted from Aickelin (2008).
cheduling, machine learning, autonomous navigation, robot navi-ation, clustering/classification, bio-informatics, image processing,ombinatoric optimization, numeric function optimization, andomputer security and optimization (Dasgupta and Forrest,999a,b; Engin and Doyen, 2004; Gonzalez and Dasgupta, 2003;arter, 2000; Hart and Timmis, 2005; Chen et al., 1996; De Castrond Von Zuben, 2002; Michelan and Von Zuben, 2002; Fukudat al., 1999; Endoh et al., 1998; Dasgupta, 1999; Gasper and Collard,000). Important parameters of AIS are clonal selection, immuneemory, affinity maturation, and receptor editing (Kumar et al.,
006; Chan et al., 2006; Chen et al., 1996; Engin and Doyen, 2004).hen artificial immune system is applied to an optimization prob-
em, the problem can be treated as the antigen and the solution tohe problem as the antibody (Liu et al., 2012). In order to utilizehe artificial immune system concept in optimization, the affinitys defined so that it depends on the value of the objective functionnd on the degree to which the constraints are satisfied. In otherords, the greater the degree to which the constraints are satis-ed, the higher the affinity will be (Chan et al., 2011; Prakash et al.,012). In addition, if two solutions satisfy the constraints to theame degree, the one with the smallest value of the objective func-ion will have a larger affinity or fitness value. Initially, a populationf random solutions is generated, representing a pool of antibodieshich undergo proliferation and maturation. The proliferation of
ntibodies is realized by cloning each member of the initial pool,.e., copying each of the initial solutions depending on their affinity.he proliferation rate is often chosen to be directly proportional tohe affinity so that the greater the affinity, the greater the numberf offspring.
Artificial immune system makes use of two fundamental mech-nisms called cloning selection and affinity maturation.
.2.1. Cloning selectionThe affinity value of each antibody is calculated from the affinity
unction, defined as
ffinity (z) = 1Total cost(z)
,
able 6omputational results of GA and AIS.
Using GA
Criterion Best Worst M
Total fitness value 204, 860 225,700 2Number of generation 172 194
and Recycling 74 (2013) 156– 169 163
where z is a given antibody. Thus, antibodies z with lower totalcost will have a higher affinity value. Since the cloning rate of anantibody is proportional to its affinity, in the next generation therewill be more clones of antibodies with lower cost than antibodieswith higher cost.
The number of clones of each antibody in the next generationcan be calculated as:
Number of clones = affinity value of antibodytotal of affinity value of antibodies in the population
× (Number of antibodies)
After calculating the average affinity value, each antibody withan affinity value greater than the average is cloned twice, whileeach antibody with a lesser affinity value owing to higher cost iscloned only once.
5.2.2. Affinity maturationAffinity maturation is achieved either through mutation or
receptor editing. Each of these two mechanisms is described below.
5.2.2.1. Mutation. For each antibody, two mutation procedures:inverse mutation and pairwise interchange mutation–are applied,resulting in two new antibodies. The cost values for all three anti-bodies are calculated. If the cost value of either of the mutatedantibodies (pairwise interchange and inverse mutation) is smallerthan that of the original antibody, then the original antibody isreplaced by the mutated antibody having the minimum cost. Theinverse mutation and pairwise interchange mutation proceduresare described below.
(a) Inverse mutation: For an antibody, let i and j be two randomlyselected initial collection points, 0 < i, j < J, where J is the numberof initial collection points and i /= j. For each bit in the antibodyassociated with an initial collection point that lies between iand j inclusive, flip the bits in the antibody that determine thestatus of that initial collection point (i.e. opened/closed), as wellas those that determine the collection period associated withthat initial collection point. Then, for that same antibody, let iand j be two randomly selected centralized return centers, 0 < i,j < K, where K is the number of centralized return centers andi /= j. For each bit in the antibody associated with a centralizedreturn center that lies between i and j inclusive, flip the bitsin the antibody that determine the status of that centralizedreturn center (i.e. opened/closed).
(b) Pairwise interchange mutation: For an antibody, let i and j betwo randomly selected initial collection points, 0 < i, j < J, whereJ is the number of initial collection points and i /= j. Swap thebits that determine the opened/closed status of initial collectionpoints i and j, as well as the bits that determine the collec-tion period of initial collection points i and j. Then, for that
return centers, 0 < i, j < K, where K is the number of central-ized return centers and i /= j. Swap the bits that determine theopened/closed status of centralized return centers i and j.
Using AIS
ean Best Worst Mean
15,830 192, 240 195,780 193,430182 135 156 143
1 ation
5aerbmrsTa
5
m
G
ch a
tibo
mue
dy 2)
ody 2
ody 1
E tion r
C opul
R
5
prT
64 A. Diabat et al. / Resources, Conserv
.2.2.2. Receptor editing. After the cloning and mutation processes, percentage of the antibodies in the antibody population areliminated (the worst B% of the population) and replaced by theandomly generated antibodies. This mechanism, which is a verte-rate immune system mechanism, is called receptor editing. Thisechanism has the effect of generating new antibodies that cor-
espond to new search regions of the search space. Exploring newearch regions may help the algorithm to escape from local optima.he resulting antibody population becomes the next generation ofntibodies.
.2.3. Artificial immune system algorithmThe artificial immune system solution procedure can be sum-
arized as:
enerate a population of P antibodies (P is the population size)
For each generation:
• Decode the antibodies and determine the affinity of ea
• Calculate the selection probabilities (rate of cloning)
• Perform the cloning process (generate copies of the an
For each generated clone:
• Perform inverse mutation (generate a new antibody 1)
• inversthe from obtained antibody new the Select
pairwise interchange mutation (generate a new antibo
• Find the Total cost of the two new antibodies:
if Total cost (new antibody 1) < Total cost (new antib
then, Clone = new antibody 1
if Total cost (new antibody 2) < Total cost (new antib
(Clone)
then, Clone = new antibody 2
else, Clone = Clone;
antibody = Clone;
liminate worst B% of antibodies in the population (B is the elimina
reate new antibodies to replace those that were eliminated (B% of p
epeat the above steps until a certain criterion is met.
.3. Genetic algorithm versus artificial immune system
Both artificial immune system and genetic algorithms areopulation-based evolutionary and biologically inspired algo-ithms. The difference between GA and AIS are summarized inable 5.
and Recycling 74 (2013) 156– 169
ntibody
dies)
tation perform and
) < Total cost (clone)
) < Total cost
ate of antibodies)
ation)
6. Computational study
Reverse logistics is a process of moving goods from the pointof consumption to the point of origin for the purpose of capturingthe value or for the proper disposal of products (Rogers, 1999). Thecomprehensive literature review shows that research contributionsdocumented so far in the broad domain of reverse logistics designproblems have overlooked the dynamic industrial scenario. To dealwith this problem, the authors have developed a mixed integer non-linear programming (MINLP) model to find out the number andlocation of initial collection points and centralized return centersand also the maximum holding time (collection frequency) for theaggregation of small volumes of returned products into large ship-ments. To solve the problem, both genetic algorithms and artificialimmune system are implemented.
6.1. Base case analysis
The parameter values were set based on extensive exper-
iments. The parameter values used for the GA were: popula-tion size = 50; maximum number of generations = 300; cloningrate = 20%; crossover rate = 80%; mutation rate = 5%. The parame-ter values used for the artificial immune system were: population
ation and Recycling 74 (2013) 156– 169 165
siaFmcntsb
hIncrci
sgto
ttett
uppppIaa
6
iarHtcfarfsta
TS
A. Diabat et al. / Resources, Conserv
ize = 50; maximum number of generations = 300; antibody elim-nation rate (B%) = 30%. Computational experiments were run on
3 GHz Intel Pentium IV computer equipped with 4 GB of RAM.or AIS algorithm parameters were: population size = 60; maxi-um number of generations = 400; No. of antibodies selected for
loning selection = 7; No. of randomly added antibodies = 8; Max.o. of clones/antibodies = 4. After the running the algorithm for 30imes independent for the base case and the results obtained arehown in Table 6. It is observed that, proposed AIS algorithm is givesetter solution than GA algorithm and Min et al. (2006b).
To expose the effectiveness of the proposed AIS algorithm, itas been compared with the GA algorithm as shown in Table 6.
t is noticeable that the proposed AIS algorithm requires a lesserumber of fitness functions leading toward a better solution, asompared to the GA. It is observed that the network design ofeverse logistics using AIS shows tremendous improvements in theollection of used products from the customer and is helpful toncrease the total reverse logistics cost.
Table 6 shows the total annual reverse logistics cost of the threeolutions: namely, the solution obtained by Min et al. (2006a), theenetic algorithm, and the artificial immune system. It is evidenthat the AIS solution ($187,200) is lower than both the solutionsbtained by Min et al. ($211,570) and the GA ($192,240).
A longer maximum holding period for consolidation increaseshe inventory carrying cost, but it can reduce the total reverse logis-ics cost due to the increased freight consolidation opportunities. Toxamine the trade-off between the maximum holding period andhe total reverse logistics cost, we conducted the following sensi-ivity analysis by changing the maximum holding period (Fig. 2).
The proximity of initial collection points to customer pop-lation centers can enhance the level of customer service byroviding customers with better access to the initial collectionoints. However, reducing the distances between initial collectionoints and customers may increase the number of initial collectionoints, which may in turn increase the total reverse logistics cost.
n order to determine the optimum maximum distance betweenn initial collection point and the customers it serves a sensitivitynalysis is conducted.
.2. Sensitivity analysis
Generally, finding the accurate values assigned to the numer-cal components of a model is a challenging task in the decisionnalysis. The collection of true values for real problems requireselevant data, which can sometimes be difficult (Yimsiri, 2009).ence, we employ, upon occasion, rough estimates. Because of
he inherent uncertainty about the true value of a numericalomponent, it is important to find out how the solution derivedrom the model would change (if at all) if the numerical valuessigned were changed to other plausible values. This process iseferred to as sensitivity analysis (http://www.doc.ic.ac.uk/∼frk/
rank/da/4.%20sensitivity%20analysis.pdf).Kara et al. (2007) usedensitivity analysis to investigate the outcome of incoming goods,he fixed and variable costs of transport, the load and unload times,nd the inventory cost in their assessment of the uncertainty and
able 7ensitivity analysis with varying nearness initial collection points.
Nearness of initial collection points: total cost in dollars ($)
Fig. 2. Comparison of total reverse logistics cost.
performance of a reverse logistics network for a white good col-lection. Du and Evans (2008) also conducted sensitivity analysis oninvestment costs, transportation costs, transportation times, thecustomer’s expected cycle time, and other parameters to see howthese parameters affect the objective function values and the non-dominated solution set. Similarly, Min et al. (2006b) also used asensitivity analysis to investigate the maximum holding period,location of initial collection points, and unit inventory carrying costin genetic algorithms. A similar type of sensitivity analysis is alsoapplied in this research to show the degree of sensitivity to the solu-tions and decision variables with respect to changes in the initialcollection points and maximum holding time.
6.2.1. Sensitivity analysis of the location of initial collection pointsIn reverse logistics network design, the location of the initial
collection points can highly influence the rate of return. Initial col-lection points must be located near enough to customers so thatthey are able to return their used products without many difficul-ties (Melachrinoudis et al., 1995). Lee and Chan (2009) mentionedthat location of initial collection points should be near to customersand easily accessible. Fleischmann et al. (2004) mentioned thatvolume of returns usually depends upon the location of collectionpoints and choosing an appropriate location is a primary functionin the network design. The distance between the customer and theinitial collection point is an important factor for reverse logisticsnetwork design (Alumur et al., 2012; Ramezani et al., 2012; Lee andDong, 2009; Min et al., 2006b; Kannan et al., 2009; Haq and Kannan,2007). In this work, the closeness of initial collection points to thecustomers are studied by using four different distances: 17 miles,21 miles, 25 miles, and 29 miles (Min et al., 2006b). The results ofeach distance have been summarized in Table 7. The results areobtained by running the algorithm thirty times independently foreach problem, and the results are categorized according to thebest, worst, and mean values. Out of the four different distances,
it is noticed that the AIS algorithm has provided results compa-rable to those given by the other GA and by Min et al. (2006b).As expected, as the maximum distance increased, the number ofrequired initial collection points decreased. As a result, the total
everse logistics cost decreased. We found that setting the max-mum distance to 25 miles achieved a good balance between theonflicting objectives of minimizing total reverse logistics cost andaximizing customer service.
.2.2. Sensitivity analysis of the maximum holding timeLowering the inventory cost mainly depends upon minimizing
he holding time. Jayant et al. (2012) has performed a sensitivitynalysis on the holding period to determine the optimal length ofolding time for consolidation at the collection centers. The model
nferred that as the increasing maximum holding period decreases,he reverse logistics cost over the entire network structure remainstable. Min et al. (2006b) conducted a sensitivity analysis on theolding period to determine the optimal length of holding time
or consolidation at the collection centers. Table 8 summarizes theesults of the sensitivity analysis of maximum holding period forour different time periods (1 day, 2 days, 3 days, and 4 days) usingA, AIS, and also the results of Min et al. (2006b). From Table 8, it cane inferred that the total reverse logistics cost decreases based onn increase in the holding time formulated by the other algorithms.lso, we infer that AIS gives lesser values compared to the other
wo proposed algorithms. This result shows that the AIS algorithmerforms a well compared to other algorithms.
.2.3. Sensitivity analysis with varying unit inventory carryingost
Day by day customers return their used products to the ini-ial collection point, but the return of used product volume is not
able 10etails of the parameters and results obtained using different algorithms for 15 problems
Problem no. Customer size Initial collection points (ICP)
constant and the amount of products collected varies every day.So, the initial collecting center has to store the used products forsome period and wait until a huge volume has been collected. Theaccumulation of a considerable volume saves in transportationcosts from initial collection point to the centralized return center. Atthe same time, inventory costs for each product occurs due to stor-age rent. Table 9 summarizes the sensitivity analysis of varying theunit inventory carrying costs using the proposed GA and AIS algo-rithms. From the result of the sensitivity analysis, it is inferred thatinventory control at the initial collection point can give a tremen-dous result. In this sensitivity analysis, results also demonstratethat the AIS algorithm obtains a lower inventory cost. After the col-lection of a certain quantity of products, it should be transported tothe next stage of the network. Otherwise, the inventory costs willincrease and, correspondingly, total reverse logistic costs will alsoincrease.
6.3. Parameters and results obtained using different algorithmsfor 15 problems
To show the robustness and reliability of the proposed algo-rithms, fifteen more problems are considered with increasingcomplexity. Table 10 shows the comparative results of proposed
algorithms of GA and AIS. In these fifteen problems, customer sizeis varied from 30 to 200, the initial collection point (ICP) is variedfrom 15 to 30, and the centralized return centers (CRC) is variedfrom the 6 to 15. From Table 10 below, it can be inferred that total
.
Centralized return centers (CRC) Total cost in dollars ($)
everse logistics cost increases based on an increase in customerize and initial collection points.
. Conclusions and future research
Now, many industries are adopting reverse logistics concept toulfill government regulations, sustainability expectations, and toain a business advantage from their recovered products (Das andhowdhury, 2012). In this paper, an attempt has been made toddress the reverse logistics network problem to find out the num-er and location of initial collection points and centralized returnenters and also to determine the maximum holding time (collec-ion frequency) for the aggregation of small volumes of returnedroducts into large shipments. We have proposed a mixed integeronlinear programming model to provide a minimum-cost solution
or the reverse logistics network design problem involving prod-ct returns, and it is solved using both GA and AIS algorithms. Theodel and solution procedures allow for the determination of the
ptimal proximity of initial collection points to customers and alsohe optimal holding time for consolidation at these initial collectionoints. The obtained results are compared with Min et al. (2006b).A solutions, and our proposed model give a better solution over
hese previous results. The model can be successfully implementedn industries for a modified network of reverse logistics.
Directions for future research include: (1) generalizing theodel to a multi-objective framework, so that in addition toinimizing the total reverse logistics cost, other objectives are con-
idered such as minimizing risk (transportation risk, storage risk,e-processing risk, and final disposal risk), minimizing responseime, and minimizing non-relevant waste collection; (2) general-zing the model to a multi-echelon hierarchical network that canandle both direct shipment from customers to centralized returnenters and indirect shipment through initial collection points; (3)llowing the demand to be stochastic.
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