Accelerated benders' decomposition for integrated forward/reverse logistics network design under uncertainty

Vahdat, Vahab; Vahdatzad, Mohammad Ali Article Accelerated benders' decomposition for integrated forward/reverse logistics network design under uncertainty Logistics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Vahdat, Vahab; Vahdatzad, Mohammad Ali (2017) : Accelerated benders' decomposition for integrated forward/reverse logistics network design under uncertainty, Logistics, ISSN 2305-6290, MDPI, Basel, Vol. 1, Iss. 2, pp. 1-21, https://doi.org/10.3390/logistics1020011 This Version is available at: https://hdl.handle.net/10419/310091 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ logistics Article Accelerated Benders’ Decomposition for Integrated Forward/Reverse Logistics Network Design under Uncertainty Vahab Vahdat 1,*ID and Mohammad Ali Vahdatzad 2 1Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02114, USA 2Department of Industrial Engineering, Yazd University, Yazd 89195-741, Iran; [email protected] *Correspondence: [email protected]; Tel.: +1-857-206-9311 Received: 16 October 2017; Accepted: 27 November 2017; Published: 9 December 2017 Abstract: In this paper, a two-stage stochastic programming modelling is proposed, to design a multi-period, multistage, and single-commodity integrated forward/reverse logistics network design problem under uncertainty. The problem involved both strategic and tactical decision levels. The first stage dealt with strategic decisions, which are the number, capacity, and location of forward and reverse facilities. In the second stage, tactical decisions, such as base stock level as an inventory policy, were determined. The generic introduced model consisted of suppliers, manufactures, and distribution centers in forward logistic and collection centers, remanufactures, redistribution, and disposal centers in reverse logistic. The strength of the proposed model is its applicability to various industries. The problem was formulated as a mixed-integer linear programming model and was solved by using Benders’ Decomposition (BD) approach. In order to accelerate the Benders’ decomposition, a number of valid inequalities were added to the master problem. The proposed accelerated BD was evaluated through small-, medium-, and large-sized test problems. Numerical results confirmed that the proposed solution algorithm improved the convergence of BD lower bound and the upper bound, enabling to reach an acceptable optimality gap in a convenient time. Keywords: integrated forward/reverse logistics network; accelerated Benders’ Decomposition; two-stage stochastic programming; valid inequalities 1. Introduction The main purpose of supply chain management (SCM) is to integrate entities including suppliers, manufacturers, distribution centers, and retailers, in order to acquire raw materials, transform raw materials to finished products, and distribute products to customers in an efficient way [ 1 ]. Achieving success in supply chain management involves several decisions relating to flow of information, products, and funds. The above-mentioned decisions fall into three levels: supply chain design, planning, and operations. In general, a Supply Chain Network Design (SCND) problem includes long-term decisions (strategic level), such as facility location, number, capacity level, and technology selection; mid-term decisions (tactical level), which usually contain the production quantity and the volume of transportation between entities; and finally short-term decisions (operational level) where all material flows are scheduled, based on decisions made in the two other levels [2]. Over the last decade, growing environmental awareness [ 3 ], social responsibility [ 4 ] and strict governmental regulations have changed the traditional management of supply chains, by showcasing the importance of integrating sustainable efforts to current management practices [ 5 , 6 ]. Sustainability efforts can be embedded from the first stages of product design and manufacturing to the recovery processes of product end-of-life stage [ 7 ]. Reverse Logistics (RL) is generally considered as the activities associated with returned products, such as collection, recovery, remanufacturing, refurbishing, Logistics 2017,1, 11; doi:10.3390/logistics1020011 www.mdpi.com/journal/logistics Logistics 2017,1, 11 2 of 21 and disposal of used or malfunctioned products [ 8 ]. In fact RL can be seen as part of sustainable development by ensuring the fact that “the society uses and reuses all the value which has been put into the product” [9]. The ways that organizations handle RL, in particular, product returns, depends on the nature of their products [ 10 ]. A study has shown that the chance of a sold product being returned is between 3 to 50, percent based on the industry [ 11 , 12 ]. For instance, the average return rate for the electronics industry is 8.5% and for the apparel and fashion industry it is 19.4%. The reason that RL is probably the most neglected section in supply chain practices [ 11 ] is the difficulties that come along with it. In comparison with forward supply chains that have uncertainties in customer demand, price [ 13 ] and resource capacity levels, RL operations are confronted with a higher degree of uncertainty, such as collection rates, availability of recycled production inputs, disposal, and recycling rates [ 14 ]. Greater uncertainty for forecasting returned products, many-to-one products movements [ 15 ] and associated transportation costs, due to lack of planning in transportation modes [ 16 ], as well as pricing of refurbished products [17–21] are examples of challenges related to RL. Some RL networks only deal with backward products flows, originating from customers or buyers to the collectors’ facilities. For instance, Kim, Kang [ 22 ] tackled the problem of backward food waste transportation from local collecting areas to the designated treatment facilities in South Korea. Ferri, Chaves [ 23 ] built a model to manage municipal solid waste by implementing selective collection and composting of organic materials in Brazil. While some research focuses on the planning and operation of backward logistics, a more complex, yet prevalent, showcase is exploring the supply chains that simultaneously have forward and backward logistics operations, generally called closed-loop supply chains (CLSC). Remanufacturing of returned products is one of the main activities associated with CLSC that has been practiced successfully in many industries, such as the printing industry [ 24 ], the electronic industry [ 25 ] (e.g., cellphones, cameras, and computers [ 26 ]), and the automotive industry [ 27 ]. De Brito, Dekker [ 28 ] provided a comprehensive review of case studies that have practiced RL and CLSC activities. Although there is a wealth of literature that has studied CLSC networks, most studies only consider one out of the many aspects of CLSCs [ 19 ]. New models need to be developed to help the manufacturer optimize the system in an integrated view [ 29 , 30 ]. The true value of the closed-loop supply chain networks can be fully gained, only if all aspects of a CLSC are optimized in a coordinated way. For this purpose, in this paper an integrated forward/reverse logistic network will be introduced where, in the forward direction, the raw materials for the manufactures are gained from different suppliers, i.e., whole sale contract, spot market, and recycled material. In the backward direction, returned products are transferred to collection centers for inspection and classification purposes. Based on the quality of the returned products, they can be conveyed to distribution centers for reselling the product, to remanufacturing plants for refurbishment processes, or to disposal centers for the end-of life cycle. The refurbished products are conveyed to customers by a second channel to distribution centers for second market customers. In this paper, we first developed a Mixed Integer Linear Programming (MILP) model for a multi-period, single-product, and capacitated integrated forward/reverse logistic network design. The model is formulated with a two-stage stochastic programming approach, with stochastic demand and returned product quantity. In the first stage, the number, capacity, and location of collections, plants and distribution centers, along with raw material acquisition are determined. In the second stage, tactical decisions such as base stock level for inventory management of distribution centers, are determined. The model is solved with Benders’ Decomposition (BD) approach where valid inequalities are added to accelerate the solution approach. In summary, the major contribution of this research is an integrated forward/reverse logistic network design, amenable for forward and reverse flow, so that medium-term and long-term decisions are made simultaneously. Two of such tactical (medium-term) decisions are: (1) An inventory policy for distribution centers, considering new, returned, and refurbished products. A risk pooling policy is Logistics 2017,1, 11 3 of 21 further analyzed at distribution centers for effective response to uncertainties in demand for new and refurbished products. (2) A raw material stocks policy for manufacturers, in which plants can provide the raw material from recycled product, spot market or long-term contract with certain suppliers. Selecting an appropriate inventory level for each facility (distribution centers and manufacturers) from predetermined capacity levels, based on the pull/push hybrid mechanism, is important in real-life applications, which is addressed in our model. The model is solved by a two-stage stochastic programming model, with an accelerate BD, where some valid inequalities are added to the master problem equations, in order to avoid infeasibility of problem solution space. The remainder of the paper is organized as follows. In the next section, a review of literature, pertaining to stochastic closed-loop supply chain is investigated. In Section 3, a mathematical formulation of the proposed CLSD design is presented. The solution method is introduced in Section 4, followed by an analysis of computational results in Section 5. Finally, in Section 6, we conclude by reviewing contributions of this research and offer some issues for future research. 2. Literature Review In recent years, a number of reviewing articles have been published on supply chains and reverse logistics (e.g., [ 31 , 32 ]). A recent paper by Govindan, Fattahi [ 33 ] provided a review of studies for both SCND and RL under uncertainty. They classified the papers, based on the planning decisions, network structure, and paradigms related to supply chain management, and discussed the stochastic optimization techniques used to solve the SCND problems. One of the areas that has been further investigated is the impact of disruption in SCND. Authors have concluded that future supply chains should be designed robustly to allow effective responses to disruptions made by both humans and nature (e.g., floods, earthquakes, terrorist attacks, and economic crises) [34–36]. Prior to this study, Govindan, Soleimani [ 37 ] reviewed papers on RL and CLSC, published between 2007 to 2013, with further investigation of papers that took sustainability and green issues into account. Ackali et al. [ 38 ] presented a critical review on RL and Integrated Forward/Reverse Logistic Network (IFRLN) problems, and discussed the main characteristics of models and solution methods proposed in the literature. Chanintrakul et al. [ 39 ] reviewed open-loop (forward supply chain) and closed-loop supply chain models by considering the impact of uncertainty in recent research. They argued the fact that few research studies have dealt with demand and return uncertainties, in terms of quality and quantity, and tactical decisions should be resolved, along with strategic decisions, which previous research has not effectively investigated. As the importance of RL has emerged in recent years, various conceptual, mathematical, and socio-economical models have been developed to assist the operation and management of RL (e.g., [ 40 – 44 ]). In the last decade, RL networks became globally large and complex and the necessity of mathematical models to be a good proxy of real-world problems became inevitable. Mathematical models should embed the future uncertainties or plausible sources of randomness, to showcase complexities of RL networks. Uncertainty has been addressed as a stochastic parameter that changes over time, for any decision level, including strategic and tactical level decisions. At the tactical level, there has been a great deal of research that has signified the importance of uncertainties in decisions pertaining to the distribution of products, raw material acquisition, and demand fulfilments [ 45 , 46 ]. At the strategic level, a number of researchers have used uncertainties in the facility location of SCND under uncertainty [ 47 ]. Snyder, Atan [ 35 ] further classified risks and uncertainties in SC as Yield uncertainty, capacity uncertainty, lead-time uncertainty and cost uncertainty, where the “boundaries among these forms of supply uncertainty are often blurry” [ 35 ]. Including each, or combining of multiple uncertainties, increases the complexity of supply chain modelling and planning. Demand for the products is one of the examples of uncertainties that much of the SCN design literature considers a known, simplified static, and deterministic demand [ 31 ]. It has been shown that product demand fluctuates over time, due to several reasons, such as seasonality and introduction of new products into the market. An example of demand uncertainty was shown by Liste¸s and Dekker [ 48 ], in the recycling Logistics 2017,1, 11 4 of 21 and re-use of sand originating from demolition waste in The Netherlands. Similarly, product return quantity varies, based on the expected life-cycle of a product, customer satisfaction, and quality of a product. In a case study presented by Salema, Barbosa-Povoa [ 49 ] demand for new products and return of products was shown to be stochastic in an office document company in the Iberian market. For an integrated forward/reverse logistic network design, one of the first stochastic models was presented by Listes [ 50 ] and later Listes et al. [ 48 ]. The model only considers one echelon forward network, combined with two echelon reverse networks. The uncertainty is handled in a stochastic formulation by means of discrete alternative scenarios. Matthew et al. [ 51 ] studied a network design problem for carpet recycling in the United States (US) where supply and demand parameters were stochastic. Later Salema et al. [49] extended Fleischmann’s model [52] to a capacitated multi-product stochastic CLSC, applied to an office document company in Spain. Most articles in stochastic IFRLN literature are single-period (e.g., [ 53 – 58 ]). Lee et al. [ 59 ] introduced a multi-period, multi-product dynamic location and allocation model, under demand uncertainty. To solve the model, an integrated sampling Average Approximation (SAA) method, with a simulated annealing (SA) algorithm was developed. The literature that studied stochastic IFRLN network design problems, considering inventory policies, are few. Lieckens et al. [ 60 ] extended a closed-loop supply Mixed-Integer Linear Programming (MILP) model, combined with queuing characteristics, using a G/G/m model, which increased dynamic aspects, like the lead time and inventory position of the basic model. Since combining RL with a queuing model intensifies the computational complexity of the model, they restricted it to a single-level, single-product network design problem that covered a single-period. The new MINLP was solved with the differential evolution technique (DE). El-Sayed et al. [ 61 ] proposed a MILP multi-period, multi-echelon forward and reverse logistic network design model under uncertainty. The problem was formulated to maximize the total expected profit under risk. To achieve a generic model of CLSC, the authors incurred various costs, such as transportation, materials, remanufacturing, recycling, disposal, non-utilized capacity, storage, shortage, recycling, and inventory holding cost. Tables 1and 2structure a systematic review of the literature for closed-loop supply chain and integrated forward/reverse logistic network design problem under uncertainty. Characteristics of networks are coded and demonstrated in Table 1and the review of existing studies are presented in Table 2. As shown in Table 2, most of the papers are those that are single-period and single-product. A few papers solve their model with an exact optimization approach, where utilizing commercial solvers are more common. In this paper, we will first develop a MILP model for a multi-period, single-product, and capacitated integrated forward/reverse logistic network design. Due to the uncertainty of various parameters in real problems, demand and return quantity of products are considered to be stochastic. The model will be formulated with a two-stage stochastic programming approach. When analyzing real-world problems, while a policy scenario examination is desired, two-stage stochastic programming can be utilized effectively, for the models in which system information is uncertain [ 55 ]. In the two-stage stochastic programming, first-stage decisions are made instantly without considering the future outcome of uncertainty, but in the second stage, decisions are delayed until uncertainty has been eliminated. Therefore, in the first stage of the proposed two-stage stochastic model, strategic decisions will be determined, which are the number, capacity, and location of collection, plants and distribution centers as well as the number of wholesale contracts. Tactical decisions will be made in the second stage (e.g., base stock level). We will utilize Latin Hypercube Sampling to make scenarios from input data by considering correlations between each market. The model will be solved with an accelerating Benders’ Decomposition (BD) approach. Numerical tests investigate the power of accelerated BD in handling with uncertainty and solving the problem with an acceptable optimal gap. Logistics 2017,1, 11 5 of 21 Table 1. Modeling approach codes. Category Detail Code Category Detail Code Model objectives Cost minimization CM Features of model Period Profit maximization PM Single-period S Responsiveness R Multi-period M Quality Q Facility capacity Other OT Un-capacitated U Features of model Stochastic parameters Capacitated C Quantity of demand D Capacity expansion CE Quantity of returns R Single sourcing SS Quality of returns RQ Model Mixed Integer Linear Programming MILP Recovery rate RR Recovery cost RC Mixed Integer Non-Linear Programming MINLP Transportation cost TC Lead time LT Decision variables of model Inventory decisions I Income In Facility capacity Fc Other OT Demand satisfaction D Product commodity Transportation values TV Single-commodity S Location/allocation LA Multi-commodity M Transportation mode selection TM Solution methodology Technology selection TS Exact solution method EX Heuristic solution method HE Table 2. Summary of stochastic integrated forward/reverse logistic network design. Ref. Model Obj. Stoch. Param. Product Com. Period Facility Cap. Model D.V. Sol. Method Solution Approach [50] PM R S S C MILP TV, LA EX B&C [51] PM D M M C MILP TV, LA, Fc, TM –aAIMMS [48] PM R, In M S C MILP TV, LA – aCPLEX [62] PM D, R S S C MILP TV, LA, SS EX Integer L-Shape Method [49] CM TC, D, RM S C MILP TV, LA, D – aCPLEX [60] PM LT S S C MINLP TV, LA, Fc, I HE Differential Evaluation (DE) [59] CM D, R M M C MILP TV, LA HE SAA with SA [54] CM, OT TC, R, OT M S C MILP TV, LA, TS – aCPLEX10 [58] CM TC, D, R, RQ S S C MILP TV, LA – aLINGO [55] PM D, R S S C MILP TV, LA – aCPLEX [57] CM D, R M S C MILP TV, LA EX SAA with CPLEX [53] CM RQ S S MILP LA EX SAA [63] CM, R, Q D, R, RC, OT M S C, SS MILP TV, LA, Fc – aCommercial Solver [56] CM D, R S S C MILP TV, LA – aCPLEX [61] PM D, R S M C MILP TV, LA, I – aXpressSp [20] PM OT M S C MILP TV, LA – aCPLEX [64] CM D, R, RQ S S C MILP TV, LA – a CPLEX/GAMS [4] PM R, RQ P S C MILP TV, LA EX SAA [65] PM D, R, TC S M C MILP TV, LA EX Accelerated BD Our paper CM D, R S M C MILP TV, LA, I EX Accelerated BD aCommercial solver packages are used or the solution method is not reported. Logistics 2017,1, 11 6 of 21 3. Problem Definition 3.1. Model Description The general structure of the proposed IFRLN is illustrated in Figure 1. In the forward direction, the new product is manufactured in plants from raw materials, provided from different suppliers, i.e., whole sale contract, spot market, and recycled materials. The product is conveyed from plants to customers through distribution centers within certain safety stock levels. In the backward direction, returned products are transferred from product sellers to collection centers for testing and inspecting. After classification, returned products are conveyed to distribution centers, remanufacturing plants, and disposal centers, depending on the amount of repair required. Remanufactured products are transferred to second market customers through certain distribution centers. This model is proposed with a generic nature, but it can encompass various industries, such as electronics, apparel, and automotive industries. In fact, the model is more appropriate for industries in which products can be highly remanufactured and sold in the second market as refurbished products. Logistics 2017, 1, 11 6 of 25 3. Problem Definition 3.1. Model Description The general structure of the proposed IFRLN is illustrated in Figure 1. In the forward direction, the new product is manufactured in plants from raw materials, provided from different suppliers, i.e., whole sale contract, spot market, and recycled materials. The product is conveyed from plants to customers through distribution centers within certain safety stock levels. In the backward direction, returned products are transferred from product sellers to collection centers for testing and inspecting. After classification, returned products are conveyed to distribution centers, remanufacturing plants, and disposal centers, depending on the amount of repair required. Remanufactured products are transferred to second market customers through certain distribution centers. This model is proposed with a generic nature, but it can encompass various industries, such as electronics, apparel, and automotive industries. In fact, the model is more appropriate for industries in which products can be highly remanufactured and sold in the second market as refurbished products. Figure 1. The proposed integrated forward/reverse logistic network model, consisting of suppliers, manufacturers, distribution centers, collection/inspection centers and disposal centers. The introduced model is a multi-stage, multi-period, capacitated, single commodity IFRLN under uncertainty. Our specifications for the model are listed below: • The periodic review policy is used for the distribution centers and manufacturers, in which the inventory levels are reviewed at certain intervals and the appropriate orders are placed after each review. The inventory level of raw material should meet a specific amount in each period. The production and shipment from the manufacturers to the distribution centers takes place, to raise the inventory level of distribution centers to the base-stock level (S) at the beginning of each period. This concept is referred to as the push strategy in the related literature. On the other hand, customer demands are met with the inventory kept by the distribution centers. The customers only place orders to the distribution centers. This system is known as a pull-based system. • A hybrid concept for production plants is considered. Due to the fact that locating manufacture and remanufacture plants in the same potential place will reduce fixed costs, we are interested in locating hybrid plants. • In distribution centers, a risk pooling strategy is considered, where both new and remanufactured products are held simultaneously. The “risk-pooling” strategy is an efficient way of managing demand uncertainty, for which inventory needs to be centralized at distribution centers (DC’s) arriving at a convenient service level. Each DC uses a base stock level Figure 1. The proposed integrated forward/reverse logistic network model, consisting of suppliers, manufacturers, distribution centers, collection/inspection centers and disposal centers. The introduced model is a multi-stage, multi-period, capacitated, single commodity IFRLN under uncertainty. Our specifications for the model are listed below: • The periodic review policy is used for the distribution centers and manufacturers, in which the inventory levels are reviewed at certain intervals and the appropriate orders are placed after each review. The inventory level of raw material should meet a specific amount in each period. The production and shipment from the manufacturers to the distribution centers takes place, to raise the inventory level of distribution centers to the base-stock level (S) at the beginning of each period. This concept is referred to as the push strategy in the related literature. On the other hand, customer demands are met with the inventory kept by the distribution centers. The customers only place orders to the distribution centers. This system is known as a pull-based system. • A hybrid concept for production plants is considered. Due to the fact that locating manufacture and remanufacture plants in the same potential place will reduce fixed costs, we are interested in locating hybrid plants. • In distribution centers, a risk pooling strategy is considered, where both new and remanufactured products are held simultaneously. The “risk-pooling” strategy is an efficient way of managing demand uncertainty, for which inventory needs to be centralized at distribution centers (DC’s) arriving at a convenient service level. Each DC uses a base stock level inventory policy to satisfy Logistics 2017,1, 11 7 of 21 demands from retailers, as well as safety stock to cope with the variability of customer demand at retailers, to achieve “risk-pooling” benefits. • As mentioned above, the inventory level of a raw material should meet a specific amount in each period. To this aim, raw material is provided through wholesale contracts, spot markets and recycled materials. A wholesale contract is a long term agreement with suppliers to convey a certain proportion of raw materials in the beginning of each period. If the amount of provided raw material from a wholesale contract and recycled material do not meet the base stock level in each period, the shortage of raw material is compensated for by buying from spot markets, but at a higher price. To specify the study scope, assumptions and limitations in the proposed model, the formula is as follows. •A single-product, multi-stage, multi-period supply chain network is given. • We assume a finite set of facilities (i.e., manufacturers and distribution centers) should be opened. •There is no limitation on the capacity of the material flow through the network. • We are faced with uncertainty for the demand of the customers to the distribution centers and return of used products to collection centers. •Transportation costs are linearly dependent on the distance between stages. • Distribution centers and raw material stock at manufactures incur inventory holding costs at the end of each period. • All of the returned products must be collected, but a shortage is allowed, to satisfy the demands of second market customers. •Customers’ locations are known and fixed. 3.2. Model Formulation According to Birge et al. [ 66 ], in a stochastic optimization model, decisions could be taken in two stages. In the first stage, strategic decisions are determined as here-and-now decisions, which should be made before the demand and return realization, and the tactical decisions should be made in the second stage as wait-and-see decisions. Moreover, the second stage in our model considers multi-periods, in which the tactical costs can be efficiently captured. This would be advantageous, specifically for those supply chain networks whose demands differ from one period to another period. The following notations are used for the mixed integer linear programming (MILP) of the proposed model: Sets: ISet of potential manufacturer locations i,i0∈I JSet of potential distribution center locations j∈J TSet of periods in planning horizon t,k∈T CSet of customers for new product c∈C C0Set of customers for used product c0∈C0 LSet of potential collection center locations l∈L DSet of disposal locations d∈D RSet of seller products r∈R SSet of scenarios s∈S Parameters, constants, and coefficients: Fixed costs: FM iFixed cost of locating manufacturer at location i FRM iFixed cost of locating remanufacturer at location i FDc jFixed cost of locating distribution center for new product at location j FDc0 jFixed cost of locating distribution center for used product at location j FCL lFixed cost of locating collection center at location l Logistics 2017,1, 11 8 of 21 Capacity costs and saving costs: sP iSaving cost of locating a hybrid manufacture/ remanufacture facility at location i sDcs jSaving cost of locating a hybrid distribution center facility at location j VcM iCost for capacity of manufacturer iper unit of product VcRM iCost for capacity of remanufacturer iper unit of product VcDc jCost for capacity of distribution center jper unit of new product VcDc0 jCost for capacity of distribution center jper unit of used product VcCl lCost for capacity of collection center lper unit of returned product Capacity of facilities: CapMax−M iMaximum available capacity of manufacturing at location i CapMax−RM iMaximum available capacity of remanufacturing at location i CapMax−Dc jMaximum available capacity for new products at distribution center j CapMax−Dc0 jMaximum available capacity for second hand products at distribution center j CapMax−Cl lMaximum available capacity of collection center at location l CapMax−P iMaximum available capacity for production facilities at location i CapMax−Dcs jMaximum available capacity for distributing center facilities at location j Transportation costs: TcM−Dc ij Cost of transporting, per unit of product, between manufacturer pand distribution center j TcDc−Cu jc Cost of transporting, per unit of new product, between distribution center jand customer c TcDc0−Cu0 jc0Cost of transporting, per unit of used product, between distribution center jand customer cu0 TcSr−Cl rl Cost of transporting, per unit of product, between seller rand collection center l TcCl−Di ld Cost of transporting, per unit of product, between collection center land disposal d TcDi−M di Cost of transporting, per unit of recycled product, between disposal dand manufacturer i TcCl−Dc0 lj Cost of transporting, per unit of product, between collection center land distribution center j TcCl−M li Cost of transporting, per unit of product, between collection center land manufacturer i TcM−Rm ii0Cost of transporting, per unit of product, between manufacturer iand remanufacturer i0 Inventory costs: IcDc jCost of holding, per unit of inventory, in distribution center j IcM iCost of holding, per unit of inventory, in manufacturer i Demand and return: DCu cst Product demand of customer cin scenario sat period t Rsrts Product returns of seller rin scenario sat period t Other parameters: PrsProbability of scenario s BOM The quantity of raw material needed for one unit of a product Csm Cost of buying raw material from spot market Coefficients and ratios: βRate of raw material shipping from disposal center to raw material stock λRate of new product shipping from manufacture centers to distribution centers γ1Rate of product shipping from collection centers to distribution centers γ2Rate of product shipping from collection centers to disposal centers M A large number NtNumber of periods Decision variables: Binary variables (relating to opening and locating facilities): xM iBinary variable equals 1 if a manufacturer is located at location i, 0 otherwise xRM iBinary variable equals 1 if a remanufacturer is located at location i, 0 otherwise yDc j Binary variable equals 1 if a distribution center for a new product is located at location j, 0 otherwise y0Dc j Binary variable equals 1 if a distribution center for a used product is located at location j, 0 otherwise xp ii Binary variable equals 1 if a manufacturer and remanufacturer are located at location i, 0 otherwise yDcs j Binary variable equals 1 if a new product distribution center and used product distribution center are located at location j, 0 otherwise zCl lBinary variable equals 1 if a collection center is located at location l, 0 otherwise Logistics 2017,1, 11 15 of 21 bM i≥BOM ×qpM ist →∑ i bM i≥BOM ×∑ i qpM ist ∑ i bM i/BOM ≥∑ i qpM ist ∀s∈S,∀t∈T(I) λ×qpM ist =∑ j fM−Dc ijst →λ×∑ i qpM ist =∑ j ∑ i fM−Dc ijst ∑ i qpM ist = ∑ j ∑ i fM−Dc ijst !/λ∀s∈S,∀t∈T (II) Since bDc j=t ∑ k=1 ∑ i fM−Dc ijsk −t−1 ∑ k=1 ∑ cfDc−Cu jcsk obviously it can be inferred that bDc j≥t ∑ k=1 ∑ i fM−Dc ijsk . (I) and (II) lead to constraint ∑ i bM i/BOM ≥ ∑ j bDc j!/λ . Since we showed that constraint (41) is constructed using the constraints of the SP, adding it to the mathematical formulation does not change the feasible space. Thus, the optimal value of the objective function remains unchanged. 5. Computational Results To evaluate the performance of the proposed Benders’ decomposition algorithm, in terms of the solution quality, numerical experiments on a set of randomly-generated problem instances were examined. The solution algorithm was implemented in GAMS 23.5 (General Algebraic Modeling System [ 77 ]) using ILOG-CPLEX 11.0 (GAMS Development Corporation, Washington, DC, USA). All experiments were run with an Intel Pentium IV dual core 2.1 GHz CPU PC at 1 GB RAM under a Microsoft Windows XP environment. Data Generation for Parameters and Settings The required data for the random generation of problem instances drawn from the probability distributions and equations are shown in Table 3. Using the generated parameters, twelve problem instances with different sizes were constructed. Table 4specifies the features of the problem instances used to evaluate the proposed solution approach. Table 3. Nominal values of the model parameters. For most of the parameters, a uniform distribution is utilized. For demand and return, an autoregressive time series (AR) is used. Parameter Range Parameter Range FM p~Uniform (1,000,000, 4,000,000) TcM−Rm p,p0~Uniform (10, 25) FRM p~Uniform (500,000, 1,500,000) TcCl−M i,p ~Uniform (10, 20) FDc dc ~Uniform (500,000, 2,500,000) IcDc dc ~Uniform (20, 25) FDc0 dc ~Uniform (400,000, 600,000) IcM p~Uniform (30, 40) FCl i~Uniform (300,000, 900,000) DCu cu,t,sc AR(1): DCu cu,t,sc = α+β1DCu cu,t−1,sc +εcu,t,sc VcM p~Uniform (1000, 1800) α~Uniform (20, 40) VcRM p~Uniform(2000, 2800) βi~Uniform (0.15, 0.2) VcDc dc ~Uniform (1500, 3000) εcu,t,sc ~N(0, Uniform (20, 35)) VcDc0 dc ~Uniform (900, 1500) DCu cu,t−1,sc ~Uniform (30, 50) CapMax−Dc dc ~Uniform (7000, 15,000) Rssr,t,sc AR(1): Rssr,t,sc = α+β1Rssr,t−1,sc +εcu,t,sc CapMax−Dc0 dc ~Uniform (1000, 2000) α~Uniform (10, 20) CapMax−Cl i~Uniform (1000, 5000) βi~Uniform (0.15, 0.2) TcM−Dc p,dc ~Uniform (10, 30) εcu,t,sc ~N(0, Uniform (10, 25)) Logistics 2017,1, 11 16 of 21 Table 3. Cont. Parameter Range Parameter Range TcDc−Cu dc,cu ~Uniform (15, 30) Rssr,t−1,sc ~Uniform (20, 30) TcDc0−Cu0 dc,cu0~Uniform (10, 30) M 60 TcCl−Di i,di ~Uniform (20, 35) β0.7 TcDi−M di,p ~Uniform (10, 30) λ0.95 TcSr−Cl sr,i ~Uniform (15, 30) γ10.4 TcCl−Dc0 i,dc ~Uniform (10, 20) γ20.4 Table 4. Characteristics of test problems. Four test cases are generated for each small, medium, and large test problems. Each test case has a specific distinction to the other cases. Size of Test Problems ID i j l C C0r d S T Small 1 4 8 8 10 15 10 2 20 12 2 4 8 8 10 15 10 2 40 12 3 5 10 10 12 15 12 2 20 12 4 5 10 10 12 15 12 2 40 12 Medium 5 8 18 12 18 15 15 2 20 12 6 8 18 12 18 15 15 2 40 12 7 10 20 12 20 15 15 2 20 12 8 10 20 12 20 15 15 2 40 12 Large 9 15 40 30 40 15 20 2 20 12 10 15 40 30 40 15 20 2 40 12 11 20 60 40 60 15 20 2 20 12 12 20 60 40 60 15 20 2 40 12 As shown in Table 4, in order to fully investigate the performance of the proposed solution algorithm, several test problems with different sizes were examined. These size differences led to a better understanding of accelerated BD power in comparison with classic BD. As the size of the instances increase, the number of binary variables increases exponentially, making the problem very hard to solve in reasonable time frame. Table 5demonstrates the number of binary and continuous variables of generated test problems. Table 5. Number of variables and constraints in each test problem. ID Number of Variables No. of Constraints No. of Scenarios Binary Continuous 1 44 117,213 35,116 20 2 44 234,333 70,156 40 3 55 169,316 43,532 20 4 55 338,516 86,972 40 5 90 358,747 67,586 20 6 90 717,307 135,026 40 7 102 433,183 75,682 20 8 102 866,143 151,364 40 9 195 1,439,176 143,584 20 10 195 2,877,976 287,167 40 11 280 2,750,921 202,516 20 12 280 5,501,321 405,032 40 Test problems were solved with accelerated BD, classic BD, and CPLEX solver. If the solution methodology finds a solution with optimality gap below a threshold value of 0.005, it will stop. If not, Logistics 2017,1, 11 17 of 21 the solution methodologies are stopped when they reach the time or BD iteration thresholds, defined based on the size of the problems. For small size problems, the time and iteration are 3 h and 40 iterations, for medium size, 5 h and 70 iterations, and for large scale problems, 10 h and 100 iterations. Table 6illustrates the optimality gap and the CPU time for solving each test problem with these methods. The optimality gap in Table 6is calculated using the following equation: Optimality gap =(Upper Bound −Lower Bound) Lower Bound ×100 (46) Table 6. A comparison of the proposed accelerated Benders’ Decomposition (BD) to classic BD and CPLEX, for small, medium, and large size test problems. CPLEX Classic BD Accelerated BD ID Optimality Gap (%) CPU (s) Optimality Gap (%) CPU (s) Optimality Gap (%) CPU (s) 1 0 210 4.231 330.12 0.8197 320.64 2 0 721.18 7.3141 645.56 0.4826 642.61 3 0 400.5 11.8911 400.5 0.5528 393.76 4–b>3 h 15.0164 779.74 0.8998 780.02 5 0 2751.16 11.4512 1312.51 1.3446 1268.44 6–b>5 h 14.7121 2669.98 1.5875 2618.37 7–b>5 h 15.1241 1591.56 2.6123 1540.67 8–b>5 h 16.0195 3090.12 3.4303 3089.33 9–b>10 h 15.9184 5093.42 4.9106 5009.21 10 –b>10 h 17.412 10,274.84 7.2837 10,121.71 11 –b>10 h 18.1027 7421.12 6.2287 7021.13 12 –b>10 h 19.8193 14,573.69 8.585 14,011.87 b The dashes mean that admissible time to solve the problem with CPLEX has reached without reaching to optimality. As illustrated in Table 6, the average optimality gaps for BD and proposed accelerated BD were 13.91% and 3.24%, respectively. Therefore, the solution algorithm performs well compared to the classic BD and CPLEX. Furthermore, in terms of CPU time, the accelerated BD approach is meaningfully better than the CPLEX solver, while performing similarly to classic BD. Accelerated BD solved the large-scale problems better than classic BD with an acceptable optimality gap. In small scale problems, the difference was not significant. CPLEX was only capable to solve three small scale and one medium size test problems in an admissible time. Note that, for all of the instances, the stopping criteria that have been reached, were the number of BD iterations for classic and accelerated BD. By comparing the proposed accelerated BD with the classic BD, one can realize that valid inequalities cause faster convergence of lower and upper bound factors. One of the underlying reasons is that classic BD was initialized from an empty subset sof extreme rays and extreme points, where valid inequalities provided an initial value for the lower bound factor of accelerated BD and led to faster convergence of the upper and lower bound factors. In order to understand the effectiveness of introduced valid inequalities, the set of valid inequalities constraints (39)–(45) was divided into two subsets and the effectiveness of each individual subset on the lower bounds, optimality gap, and computational times were compared. ABD-I (Accelerated Benders’ Decomposition-Initial state) shows the current accelerated BD approach, considering all valid inequalities, ABD-1 represents the accelerating Benders’ Decomposition, with only constraints 39 and 40 as valid inequality cuts, and ABD-2 considers constraints 41 to 45 as valid inequality cuts to the master problem. Table 7 presents the results of comparing ABD-I, ABD-1, ABD-2. When all valid inequality cuts, ABD-I, were considered in the solving of the master problem, the best results in both the CPU time and the optimality gap, were achieved. While ABD-1 and ABD-2 both improved the optimality gap and computational time compared to classic BD, ABD-2 provided stronger cuts, leading to a higher impact on the evaluation criteria. Logistics 2017,1, 11 18 of 21 Table 7. Effectiveness of valid inequality cuts in terms of lower bound, optimality gap, and CPU time. ABD-I ABD-1 ABD-2 ID Lower Bound Zlb Gap (%) CPU (s) Lower Bound Zlb Gap (%) CPU (s) Lower bound Zlb Gap (%) CPU (s) 1 127,007,212 0.81 320.64 126,982,020 0.83 327.21 126,994,615 0.82 325.16 2 129,448,338 0.48 642.61 129,422,577 0.50 643.28 129,448,338 0.48 643.12 3 151,583,859 0.55 393.76 151,312,985 0.73 396.30 151,508,519 0.6 396.74 4 137,039,534 0.89 780.02 136,998,797 0.92 780.53 137,012,373 0.91 781.46 5 210,742,541 1.34 1268.44 210,368,884 1.52 1296.98 210,700,958 1.36 1270.14 6 215,373,195 1.58 2618.37 215,140,222 1.69 2660.02 215,288,419 1.62 2622.63 7 230,643,712 2.61 1540.67 229,859,667 2.96 1573.89 229,881,994 2.95 1576.68 8 215,260,805 3.43 3089.33 213,485,714 4.29 3090.01 215,219,189 3.45 3090.81 9 418,327,768 4.91 5009.21 416,027,739 5.49 5064.46 418,287,897 4.92 5060.58 10 444,456,828 7.28 10,121.71 440,514,861 8.24 10,245.74 442,888,060 7.66 10,199.11 11 617,187,359 6.22 7021.13 615,680,328 6.48 7142.71 616,085,342 6.41 7130.65 12 644,450,944 8.58 14,011.87 636,073,843 10.01 14,315.41 642,026,640 8.99 14,149.29 6. Conclusions In today’s competitive business environment, the design and management of an integrated forward/reverse supply chain network is an important and difficult problem to solve. To this aim, we proposed a generic multi-stage, multi-period, single commodity and capacitated IFRLN design that considers both strategic and tactical decisions in one platform. The reason is that any decisions made at a strategic level will demarcate the scope of tactical decisions and in order to achieve a fully efficient supply chain, the integration of medium- and long-term decisions is unavoidable. The push/pull strategy, risk pooling strategy, and raw material acquisitions are examples of tactical decisions in different stages of a supply chain that are incorporated in the studied model. Moreover, the demand for products (new and recovered products) and the return of products from resellers are considered stochastic parameters to mitigate real-world problems. Benders’ Decomposition approach was used as a strong exact solution methodology to tackle the proposed two stage stochastic model. Due to the slow convergence of lower and upper bound factors in large scale problems, a number of valid inequalities were applied to the master problem. Test problem results showed that the accelerated BD had a dominant optimality gap in comparison with the classic BD in acceptable CPU time. The proposed model can be further extended for multi-commodity configuration; many supply chains procure several products, while each may have different patterns of returns, both in quantity and quality. Pricing of new and refurbished products and recovered raw materials impacts the gained profits from the first and second markets and should be considered in future models. Further investigation can be provided on non-linear inventory policies for base stock level, such as (S,S) and (R,Q). Acknowledgments: There is no funding or grants associated with this research. 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