Divide and Conquer Optimization for Closed Loop Supply Chains

Article pubs.acs.org/IECR

Divide and Conquer Optimization for Closed Loop Supply Chains T. Sundar Raj,† S. Lakshminarayanan,*,† and J. F. Forbes‡ †

Chemical and Biomolecular Engineering, National University of Singapore, 117576, Singapore Chemicals and Materials Engineering, University of Alberta, Alberta T6G 2 V4, Canada

‡

ABSTRACT: Environmental concerns and government regulations have encouraged/forced real-world supply chains to take back used product for recycle. A smart supply chain should therefore leverage significantly on the returned “used products” to produce new products at nominal cost and time. In such closed loop supply chains, the crucial challenge is to synchronize the used products recycling system, the production facility and the new product distribution system in the presence of uncertain customer demands and used product returns. The main motivation of this study is to improve the performance of closed loop supply chains using a divide and conquer optimization scheme. In this paper, it is shown that the manner in which the closed loop supply chain is divided into various subsystems, the way in which interactions between the subsystems are handled and the optimization sequence adopted can help to improve the profitability of the supply chain with minimal computational load. Comparisons between the proposed method and traditional single objective optimization are made to illustrate the advantages of the divide and conquer approach to closed loop supply chain optimization. for the benefit of other industries.9−11 General Electric company reuses the PET bottles to generate PBT copolymer which can be used for automotive and consumer applications.12 Numerous opportunities are available to generate value from used electronic products and other product types,6,13 etc. Recycling also indirectly reduces the utilization of virgin raw materials and disposal of used products. The influence of recycling phenomena on the cost and time of remanufactured product varies with the quality (condition) of the recycled used product. Remanufacturing can be carried out by utilizing: (a) the inferior used-products as a substitute for fresh raw material in the remanufacturing firms and (b) acceptable used-products renewed into new products in the refurbishment firms (Figure 2). The manner in which used products are collected from market customers and transported to the reproduction facilities are important in closed loop supply chains.14 Other than remanufacturing and refurbishment, the manufacturing firms also continue production of products using virgin raw material.15 Therefore, closing the product cycle has the potential to improve the performance of the overall supply chain. The crucial challenge in closed loop supply chains is in identifying the right tactical decisions to synchronize the collection centers (reverse channel), distribution networks (forward channel), and the production facilities.16,17 In this work, we devise an efficient method to derive tactical supply chain decisions for the forward channel, reverse channel, and the production system for the benefit of the overall closed loop supply chain network in the presence of uncertain demand and used product returns.

1. INTRODUCTION Supply chain is the collaborative association between inbound logistics, production firms, and outbound logistics (Figure 1). Inbound logistics purchases raw material from suppliers which is transformed to the value-added product (in production firms) and distributed to the market customers through various distribution echelons (outbound logistics).1,2 Traditional supply chains have mainly focused on performance metrics such as resource minimization and/or output (customer satisfaction) maximization.2,3 In recent times, government regulations and environmental concerns have necessitated the supply chain practitioners to recycle the used products.4−6 New environmental legislations have been driving the industrial ecology concept in both North American and European manufacturing companies.6 The idea of industrial ecology is to mimic ecosystems by having all the used products act as inputs for other processes.7 Hence, recycling is considered as one of the crucial performance metrics in modern supply chains. As a result, the closed loop system (as shown in Figure 1) collects the used products to reuse as substitute for raw materials. As a general rule, in closed loop supply chains, “new product” flows in the forward direction to the market customers and “used product” flows in the opposite direction to the refabrication facilities. There is plenty of evidence confirming the shift in supply chain operation from traditional to closed loop scenario. Major manufacturers in Japan, United States, and Europe have shown the way. The Japanese automotive industry recycles almost 82% (by weight) of used cars.8 Both Hewlett-Packard and InkCycle reduced the need for landfills by remanufacturing used toners and cartridges. The efforts made by InkCycle kept 225 tons of waste material out of landfills. General Motors converted 50 000 tons of waste materials to energy to facilitate economic benefitsnearly $1 billion was salvaged in annual revenue from recycled metal scraps. Johnson and its partners recapture more than 95% of the lead weight and 70% of the resin from spent batteries to reuse in new batteries. Other battery components and sulfuric acids are either treated or processed © 2013 American Chemical Society

Received: Revised: Accepted: Published: 16267

March 7, 2013 September 30, 2013 October 16, 2013 October 16, 2013 dx.doi.org/10.1021/ie400742s | Ind. Eng. Chem. Res. 2013, 52, 16267−16283

Industrial & Engineering Chemistry Research

Article

Figure 1. Schematic representation of traditional and closed loop supply chain.

Figure 2. Schematic representation of forward channel, reverse channel, and production facilities in closed loop supply chain.

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dx.doi.org/10.1021/ie400742s | Ind. Eng. Chem. Res. 2013, 52, 16267−16283


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2. CLOSED LOOP SUPPLY CHAINS A shift from traditional to closed loop scenario includes the additional uncertainty with the used-product returns due to factors such as product end-of-life period, leasing contracts, customer’s willingness to return, availability of collection locations, and the manner in which the recycled used-products are transported to the production facility.14,16,18,19 Other than end-of-life of product, the product return rate is further complicated by commercial returns, warranties, repairs during the product life cycle, etc.20,21 Possible collection methods for used-products are (a) retailers collect used products from the customer and sells to the manufacturer, (b) manufacturer directly collects used products from the customer, and (c) third party collects used products from the customers and later sells them to the production facility.22−24 The remanufacturing facility decides the appropriate collection method depending on the product type and return value. The collected used products are reprocessed/reused in the refurbishment and remanufacturing facility to produce new products. While usedproducts that need reconditioning (refurbishment) may be considered as high value recovery, used-products that are recycled, and or sent to waste management facilities are to be considered as low or no-value recovery (Figure 2). A resourceful supply chain should exploit more benefit from refurbishment facility so as to minimize manufacturing cost and time. Next to refurbishment, remanufacturing option is preferred to produce products at cheaper cost. Finally, the manufacturing firm that uses fresh raw materials to make products serves to take care of the deficiency (if any) in the required production volume. In practice, all production facilities adopt make-to-target strategy in diverse ways subject to their business goals.25,26 The production pattern for a reproduction facility is truly based on the availability of used products that can be converted into new products, whereas the production patterns at a manufacturing facility cover up the production deficiency caused by the reproduction firms. Shifting supply chain from conventional to closed loop scenario involves various managerial complications such as investment, business uncertainty, and risk.27 Many companies still find it difficult to manage performance at high standards because they are still at an early stage of development in recovery, reconditioning, and remanufacturing processes. Frank et al.28 modeled the reverse logistics characteristic of closed loop supply chain based on end-of-life vehicle (ELV) treatment in Germany. Vehicle routing planning is considered as an important design options to efficiently integrate reverse material flows into their genuine supply chains. Other recent research works21,29,30 focused in the area of optimal network design for closed loop supply chains. With an integrated perspective, Chung22 revised the tactical decisions of production and replenishment policy to maximize the profit of closed loop supply chain. The integrated approach results with a win−win situation for all supply chain entities and increases the joint profit of the supplier, the manufacturer, the retailer, and the third-party recycling dealer. A multiechelon, multiperiod, multiproduct closed loop supply chain is formulated as a mixed integer linear programming (MILP) model and optimized using a genetic algorithm.11 The crucial decisions optimized are in the area of material procurement, production, distribution, recycling, and disposal. The closed loop supply chain strategic decisions (design) and tactical decisions (operational parameters) are purely based on the exogenous parameters such as changes in the product demand, used product returns, transportation capacity, and

emission rates. Any changes in the exogenous parameters will have effects on the performance measures. Robust closed loop supply chain design and decisions are the other ways of reducing the influence of exogenous parameters on the supply chain performance. A robust optimization technique is developed to design a closed loop supply chain which faces uncertainty in the returned products quantity, the demands of second market customers, and the transportation costs between facilities.31 The advantage of robust design is compared with the supply chain design by replacing uncertain parameters with the nominal value. As expected, computational results show the superiority of the solution obtained from the robust model compared to the deterministic model. As most of the studies focused on overall performance of the closed loop supply chain, an interesting observation noticed in product transportation operations is that reusable products reduce the operational costs of the chain but increase the burden on the environmental costs, that is, CO2 emission because of additional transportation in the reverse channel.32 Additional research is needed to understand the performance associated with these investments, including economic (supply chain cost), reliability (customer satisfaction), and environmental (recovery) issues. Supply chain performance has been described as a combination of resources minimization and output maximization.33 Well-established supply chains seek to maintain desired customer satisfaction (output) as a constraint and reduce the supply chain cost (resources). Recent research has advanced various control strategies34 and optimization algorithms to solve supply chain problems as a single-objective35−37 or multiobjective3,38,39 case to reach optimal and nondominated solutions respectively. Dealing with real world supply chains results in a large scale optimization problem with an enormous number of decision variables. The difficulty of solving the optimization problem increases with the number of objectives, constraints, and the decision variables. Therefore, structuring the supply chain performance as a single-objective optimization problem (minimizing the supply chain cost) is not practical and optimal in all circumstances. The solution may get trapped in the local minima and/or violate the desirable (customer satisfaction) constraints due to heavy interaction between the supply chain entities. Even though recent developments in computational capabilities allow the solution of larger optimization problems, decomposition algorithms provide a practical approach for solving large scale optimization for complex systems40 Transformation of the original problem to a mixed integer linear programming (MILP) problem or a reduction in the number of decision variables by only optimizing the crucial ones and fixing the rest at some reasonable values are some possible approaches to solve the large scale problem. Another option is to use appropriate decomposition algorithms to break up the problem into manageable parts and solving them to obtain the solution for the original problem without compromising significantly on the quality of the solution. Such an approach can benefit supply chain problems involving large numbers of decision variables. 2.1. Decomposition Based Optimization. “Divide and conquer” (DC) is a well-established optimization technique to handle large scale supply chain network which requires proper cooperation and coordination between all entities in the network for enhanced overall performance.41 In cooperative systems, each subsystem has a separate objective and the collection of all such objectives form the objective of the overall system.42 While solving a cooperative system, a balance of competition and 16269



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transportation plan obtained from the first stage. A two-step methodology is proposed by Tasan45 where the first step allocates the production plant to the distribution centers along with the vehicles, and the second step attempts to lower the transportation cost by optimizing the vehicle route. Benders decomposition (BD) approach was adapted by Uster et al.46 to design a closed loop supply chain that has minimal processing time, transportation, and fixed location costs by optimally choosing the location of collection centers and the remanufacturing facilities. As opposed to multi-index formulation, the network was formulated separately in stages to improve the convergence rate by adopting the BD approach. An automotive industry that provides service parts for vehicle maintenance and repair was considered as a case study to illustrate the workability of the BD approach. Lazzaretto et al.47 identified three possible decomposition approaches for an energy system decomposition where the overall complex system is decomposed into subsystems, while the functional links to capture the coupling between the subsystems are managed. The highlights and the effect of three decomposition approaches are discussed with suitable examples. Terrazas-Moreno et al.40 proposed a novel hybrid decomposition approach which combines bilevel and spatial Lagrangean decomposition methods to investigate the simultaneous scheduling and planning problems in a continuous multiproduct plant that involves different temporal and spatial scales. Numerical results indicate that (a) for large-scale problems, decomposition methods outperform the full space solution and (b) as problem size increases the hybrid decomposition method becomes faster than the bilevel decomposition method. The purpose of the decomposition approach is not merely to decrease the problem size. A more important reason to adopt the decomposition approach is to reduce the problem difficulty by decomposing the original problem into less difficult subproblems. This advantage motivates the DC optimization approach as an interesting option to optimize closed supply chains with multiple subproblems/objectives and huge number of decision variables. In this study, we enhance the performance of a closedloop supply chain using DC optimization. We do this by partitioning the overall supply chain into subsystems such as forward channel, reverse channel, and the production units. Each channel is optimized individually in a sequential fashion and logically aggregated to reconstruct the results for the original system. The result obtained by DC optimization is validated against the results of other optimization methods (singleobjective and multiobjective optimization). DC optimization produces better results with less computational effort as compared to single-objective and multiobjective optimization methods, respectively. As a result, optimization of large scale supply chain networks can be achieved using DC optimization. A case study of a multiproduct multiechelon closed loop supply network is provided to illustrate the workability of DC optimization method. This paper is organized as follows. Section 3 describes the closed-loop supply chain network consisting of multiple tiers of decision-makers. The assumptions made regarding the network are also outlined. In section 4, the proposed methodology to optimize the behavior of closed loop supply chains is described. The efficacy of traditional optimization, multiobjective optimization, and decomposition optimization, and the comparison with respect to solution accuracy, optimization time, sensitivity, and desirability against uncertain inputs are addressed in section 5. Finally, the paper ends with the conclusions derived from the case study problem.

cooperation between the subsystems is necessary to attain best solutions for the overall network and possibly for the subsystems. Subsystems lacking in proper competitive strategies may fail to pursue better solutions, and those that lack in proper cooperative strategies may conflict with each other and deteriorate the solution for the overall system and the subsystems. Therefore competition and cooperation are both necessary to achieve good solutions.43 The DC technique is suitable for large scale optimization problems having huge number of mixed integer decision variables. The principle of DC suggests that (1) complex decision problems should be decomposed into smaller, more manageable parts, (2) these smaller parts are solved independently, and (3) the subproblem solutions are combined to generate reasonable solutions that are as close as possible to optimal solutions. These steps are repeatedly solved in an iterative fashion until the convergence to the final solution (Figure 3). Breaking or decoupling the original problem into

Figure 3. Decomposition optimization framework.

various subproblems and the task of aggregating solutions of subproblems to regain the solution for the original system are very crucial aspects in DC optimization. Decomposition heuristics have been adapted by various researchers for diverse supply chain problems. Lee et al.44 divided the multilevel supply chain into two subproblems using decomposition heuristics and optimized them in a sequential manner. In the first stage, cost flow is minimized by optimizing the transportation route, and in the second stage, the replenishment plan is optimized by integration with the 16270



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Figure 4. Schematic representation of overall supply chain network.

3. MODEL INFORMATION AND SUPPLY CHAIN DESCRIPTION 3.1. Model Information. The mathematical model for the supply chain considered here is based on the following assumptions: (1) a planning horizon of 300 days (2) all entities are reviewed on unit time (eight hour) basis (3) demand rates and the return rates (forecast) available for all the products. The used product return rate is expected to be less than the demand rate due to incomplete return and waste disposal (specifically return rate is assumed as an uniform distribution of demand rate between 60 and 70%)48 (4) no deterioration occurs either in the new product or in the returned used product (5) in the forward channel (pull driven), material flows in a flexible transportation capacity to improve customer satisfaction.37 (6) in the reverse channel (push driven), material flow is limited to fixed transportation capacity to reduce transportation cost (7) fixed manufacturing and remanufacturing rates and constant transportation lead-times (8) inventory space is unlimited to hold new product and used products48 (9) the quality of remanufactured products is comparable to newly manufactured products and cannot be distinguished from one another49 (10) to meet the distribution network demand, remanufactured products are preferred over the manufactured products 3.2. Problem Description. The closed loop supply chain considered in this study is a multiechelon multiproduct distribution network37,39,50,51 with multiple production facilities. The modeling aspects of the distribution network, collection centers, and production facilities have been explained in detail in

the Appendix. The forward supply chain includes raw material suppliers, production facility, distribution network, and the market customers. The reverse supply chain consists of market customers, collection centers, and reproduction facilities. Figure 4 is a schematic representation of the two-echelon distribution network. It contains four distributor centers (DC1 to DC4) and 10 retailer nodes (R1 to R10) serving 20 different market customers (C1 to C20). Nine different products (p ∈ A to I) are featured in this supply chain. All entities in the distribution network also act as collection centers for handling used product returns. All distribution nodes have separate inventory facility to accommodate new product and used-product return. New products flow in the forward direction from the production system to the market customer through various distribution centers and retailers. In contrast to new products, the used products flow in the reverse direction from market customers to the remanufacturing plant. New products are replenished on a unit time basis to satisfy downstream customers and maintain optimal inventory. Used products are collected and stock-piled at the retailer echelon and transferred to the upstream nodes (i.e., managed in push pattern) at full truck capacity to reduce the transportation and inventory holding cost. This large scale supply chain network consists of three production facilities namely manufacturing, remanufacturing, and refurbishment (Figure 4). All facilities produce homogeneous commodity (product) and operate with the production goal of attaining finished product at desired target. All production facilities have a separate warehouse facility for raw material/used product and finished product. The finished product inventory of all production facilities is connected to the centralized managed warehouse to serve downstream nodes. Among the three production facilities, the manufacturing unit is the most expensive in cost and time because it exploits fresh raw material and requires more lead time to convert the raw material to finished product (Table A1). Compared to manufacturing, remanufacturing requires the same lead time but is cheaper in 16271



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replenishment policy parameters for all the product nodes in the forward channel, (b) 87 used-product transportation capacity for all the product nodes in the reverse channel and (c) 30 decision variables in the production facilities. The key decisions in the production facilities are the desired inventory target for all new products at three production facilities (3 × 9 = 27). The number of production machineries at the manufacturing, remanufacturing and refurbishment facilities constitutes the 3 (= 3 × 1) crucial decisions. In total, this moderate scale case study problem has 206 decisions to optimize out of which 117 are discrete decisions. Real world supply chains are large scale problems associated with more complexities where the optimization is very challenging especially with huge number of mixed (continuous and discrete) decision variables and constraints. Therefore, breaking the large scale problem into subproblems and solving them independently would be an ideal way. Even though decomposition approaches are more appropriate for a large-scale system, the workability of the divide and conquer optimization method is discussed on the basis of the case study problem defined in Figure 4.

cost because it utilizes the recycled material (used product) as the raw material. The refurbishment unit has advantage both in cost and time because it refurbishes the used product (which is at a relatively good condition), therefore the cost and time requirement for conversion to new product is significantly less than that of the other production facilities. The details of distribution network, collection centers, production facilities, and the uncertain inputs (e.g., uncertain demand, uncertain used product returns) are provided in the Table A1. The multiechelon multiproduct closed loop supply chain shown in Figure 4 is modeled as mixed integer nonlinear program (MINLP) in GAMS optimization software. The distribution network (forward channel) contains 54 product nodes at retailer and 33 product nodes at distribution centers. At each product node the replenishment parameter is the main decision variable. The collection centers (reverse channel) also consist of 87 (=54 + 33) product nodes where the transportation capacity of the used product is the decision variables. Equation A15 introduces an additional binary variable to characterize Ifthen condition in GAMS. For multiperiod system, the problem size and the complexity depend on the number of time intervals. The characteristics of the developed multiperiod optimization model based on the 300 days time horizon are shown in Table 1.

4. PROPOSED METHODOLOGY The main objective here is to improve the performance of the closed-loop supply chain via application of the proposed DC optimization methodology which partitions the overall supply chain into three subsystems, namely, forward channel (distribution network), reverse channel (collection network), and the production systems (Figure 5a). The manner in which the supply chain network is partitioned and the manner in which the optimization is carried out for each subsystem are crucial aspects of this methodology. The subsystems are decoupled by eliminating the interactions between them using predefined expectations (interactions). Here, the forward channel is separated from the overall network by assuming the supplier’s capability to satisfy the distributor’s order; the reverse channel is isolated from the original system and optimized independently. Then, the production system is optimized after synchronizing with the revised forward and reverse channels. To explain further, consider the schematic shown in Figure 5b. In that panel, “a” denotes the input of the reverse channel (product returns input), “b” denotes the output of the reverse channel (product returns to the production system), “c” denotes the demand, “d” denotes the order placed, and “e” denotes the products. When the forward channel is isolated from the original system, “e” is unknown so “e” is the expectation required to solve the forward channel separately (note that we know “c” and “d”). The reverse channel does not have any inward (material or information) flow from other systems−therefore there is no need to have any predefined expectations (since we know “a” and “b”). The production system is solved after solving forward channel and reverse channel. Well-defined inputs “b” and “d” are obtained from forward and reverse channel solutions. Therefore the production system also does not have any predefined expectations. 4.1. Forward Channel. In the forward channel, the main goal is to minimize the cost associated with excess inventory (resources, IHr,p and IHdc,p) at all distribution nodes (eq 1). The primary constraint is that the customer satisfaction of the distribution network must be greater than 95%; that is, CSDN ≥ 95% (eq 2).

Table 1. Characteristics of the Multiperiod Optimization Model forward and reverse channel

value

number of continuous variables number of discrete variables used product transportation capacity @ retailer used product transportation capacity @ distribution center binary variables used to implement eq A15 number of constraints customer satisfaction @ retailer customer satisfaction @ distribution center used product transportation constraint, eq A15

Cr,p

1174848 78387 54

r × pa

Cd,p

33

d × pa

78300

t × (r + d) × pa

78300 54 33

r × pa d × pa

78300

t × (r + d) × pa

CSr,p CSd,p

a r = {R1−R10), d = {DC1−DC4}, p = refer to figure 3, t = {300 days, 1 unit time = 8 h} = 900.

The model statistics of the forward and reverse channel in the closed loop supply chain are shown in Table 1. Production facilities have number of production machineries at the manufacturing, remanufacturing, and refurbishment facilities. The scheduling and batch operations of the machineries (eq A16 to A24) are associated with more “if then” conditional statements and discrete variables (refer to Figure A1). GAMS failed to optimize even the moderate size case study problem due to huge discrete variables and conditional statements. In the case of a large scale problem, optimization is very challenging especially with mixed (continuous and discrete) decision variables and constraints. An alternative way is developing a closed loop supply chain model in MATLAB (described in Figure 4) with relatively less complexity; that is, conditional statements are easily implemented in the MATLAB/Simulink model without any additional binary variables which can be optimized using simulation-based optimization methods. The total number of decision variables in the developed MATLAB model is reduced to 204 which include (a) 87 16272



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Figure 5. (a) Closed loop supply chain (divide and conquer) optimization framework; (b) information flow for the divide and conquer optimization. tH

cost1 =

t

∑ ∑ ∑ (IH r,p,t − dr,p,t) t=0

r

CSr = CSr,p =

p

∑t H= 0 Yrc,p t

∑t H= 0 dcr,p

tH

+

t

∑ ∑ ∑ ∑ (IHdc,p,t − ddc,p,t) t=0

;

r

dc

p

CSDN = average(CSr , CSdc) ≥ 0.95

CSdc = CSdc,p =

(1)

∑t H= 0 Ydcr,p t

∑t H= 0 drdc,p

(2)

Replenishment parameter is the crucial decision variable in the forward channel as it influences the material purchase from their suppliers. Product replenishment should be optimal to maintain

(2)

where 16273



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Table 2. Summary of Decomposition Approaches forward channel− distribution network

criteria objective: minimize decision variable no. of decision variables bound constraint

reverse channel−collection centers

excess inventory replenishment parameter 87 (continuous) 0.01 to 1 CSDN ≥ 0.95

production firms

inventory cost + transportation cost transportation capacity 87 (discrete)

used product inv cost + new product ei cost + plant idle cost + excess inventory + inventory cost + transportation cost production units @M,RM,RF product (A, B, ..., F) inventory targets @M, RN, RF 30 (discrete)

50, 100, ..., 500

m ∈ [1, 2 ..., 35], sp ∈ [1,2, ..., 16] × 500 CSW ≥ 0.95

4.3. Production Facility. After revising the decisions in forward and reverse channels, proper scheduling and synchronizing of the production and reproduction systems is important to produce new products at nominal cost and time while achieving at least 95% customer satisfaction (CSw). Achieving the CSw at least 95% is important to maintain the interaction (assumed or predefined) between the forward channel and production facility. Material recovery and fresh manufacturing are the two key adjustments necessary to meet the orders from the distribution network. As stated earlier, the priority in the production system is (a) refurbishment unit (to process recovered high value used products), (b) remanufacture unit (to handle recovered low value used products), and (c) manufacturing units (from fresh raw material). The used product will go to the refurbishment unit or remanufacturing unit depending on its condition. Ideally, the refurbishment and reproduction units should dominate the production to satisfy market customers. Because of inefficient used product collection and handling, an additional production unit that uses fresh raw material is necessary to satisfy shortages in meeting customer orders and the fluctuations in the market demand. The production facility is improved (with the help of the revised forward and reverse channels) by choosing the right number of machineries in the remanufacturing, refurbishment, and manufacturing systems. The goal is to provide at least 95% CSW to the distribution network (eq 4), while minimizing the fresh product inventory cost (IHmn,p, IHrm,p, and IHrfm,p), used product inventory cost (IHUrm,p and IHUrfm,p) and plant idle cost (Idmn,m, Idrm,m and Idrfm,m) (eq 5).

desired CS and less excess inventory. A straightforward way to decouple the forward channel from the overall network is by assuming or predefining the warehouse capability to deliver product with respect to the replenishment order. In this study, we assumed that the warehouse is capable to satisfy 95% (on an average) of replenishment orders from the distributor center. Therefore the interaction between the warehouse and distribution centers is predefined as CS w = 95% (=U[90%,100%]). Predefined interaction with an uncertainty range is more relevant for decentralized supply chains where the material delivery purely depends on the product availability or inventory at-hand. Next to customer satisfaction, environmental issues are crucial (this will be addressed in reverse channel optimization). 4.2. Reverse Channel. Recovering value from returned used products and making the supply chains environmentally friendly is the objective of reverse channel. Recovery also has indirect benefits like reducing waste disposal and the usage of virgin raw material. In this case study problem, it is assumed that 65% of used product is returned by market customers to their corresponding retailers. All retailers and distribution centers will push the returned used products to their respective upstream nodes, and the production plant. Unnecessary retention of used products at the collection centers is the bottleneck in the reverse channel. Such unnecessary stagnation of used products at the distribution network affects the operation of reproduction and refurbishment units, increases the used product inventory holding cost, and decreases the return on investment (ROI). Furthermore, for perishable products, depreciation comes into play (note that this is not considered in this work). Therefore, the crucial task is to reduce the retention of used products at the distribution nodes to utilize the used products and to lessen the inventory holding cost (IHUr,p and IHUdc,p). At the same time, the manner in which used products are transported to the upstream nodes (utilizing the full truck capacity) is also important to reduce the transportation cost (TCr,p and TCdc,p) (eq 3). tH

cos t 2 =

t

CSw,p =

t = 0 dc

p

tH

+

+

p

t = 0 dc

t=0

p

p tH

∑ ∑ IH rfm,p,t + ∑ ∑ IHUrm,p,t t=0

p

p

t=0

tH

tH r

(4) tH

tH

+

∑ ∑ ∑ TCr,p,t + ∑ ∑ ∑ TCdc,p,t t=0

≥ 0.95

∑ ∑ IH mn,p,t + ∑ ∑ IH rm,p,t t=0

tH r

t

∑t H= 0 ddc → w,p tH

cost 3 =

∑ ∑ ∑ IHUr,p,t + ∑ ∑ ∑ IHUdc,p,t t=0

∑t H= 0 Yw → dc,p

p

(3)

+

The isolated reverse channel aims to optimize the reverse channel cost by pushing the used product to the production facility in an optimal way by identifying the right truck capacity to balance the transportation and inventory holding cost. For the reverse channel, determining the right transportation/truck capacity from the set of available choices is the key decision to be made.

p

tH

mn

∑ ∑ IHUrfm,p,t + ∑ ∑ Id mn,x ,t t=0

p

tH

rm

t=0 x=1 tH

rfm

∑ ∑ Id rm,x ,t + ∑ ∑ Id rfm,x ,t t=0 x=1

t=0 x=1

(5)

In all these stages, the optimization is carried out by considering the interaction implicitly as a CSW constraint rather than simultaneous (explicit) optimization. The objectives, decision variables, and the constraints for all subsystems are listed in Table 2. 16274



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4.4. Optimization Tools. The overall closed loop supply chain is modeled in the Matlab-Simulink environment and optimized using simulation-based optimization by integrating it with the TOMLAB optimization toolbox. In TOMLAB, the combination of glcDirect,52 and SNOPT,53 optimization algorithm were extensively used to optimize the overall supply chain system and subsystems depending on the type of decision variables (continuous or integer) in the problem. SNOPT applies a sparse sequential quadratic optimization (SQO) for nonlinear problems, whereas the glcDirect routine is the extended version of DIRECT algorithm,54 (modification of the standard Lipschitzian approach) that handles problems with nonlinear and integer constraints. The above-mentioned algorithm is used to solve the closed-loop supply chain using the traditional optimization method and DC optimization method. The Simulation models and the optimization codes can be made available to interested readers upon request. Nondominated Solution Genetic Algorithm II. NSGA-II is a well-established multiobjective optimization algorithm which provides nondominated (Pareto) solutions with less computational cost. Our past success in solving multiobjective problems with NSGA-II encouraged us to adopt it over other multiobjective optimization methods available in the public domain.39 However, for better solution accuracy and fast convergence, superior optimization algorithms may be necessary when dealing with large problems. NSGA-II starts with random initialization of populations followed by repetitive steps which involve function evaluation and Pareto solution (population) improvement until final convergence. At each generation, the quality of Pareto solution is improved by sorting nondominated solutions and selecting best parents from the population to generate offspring (intermediate population). Nondominated solutions are sorted with respect to rank and crowding distance. Crowding distance is the measure that preserves diversity in the Pareto solutions. The best parents are selected from the population using tournament selection. Crossover and mutation of the identified best parents lead to offspring generation. Finally, the populations are updated (to improve the Pareto quality) by choosing best chromosomes from population and offspring. The derived Pareto solution is presumed to be best after searching the solution space by diverse chromosomes in the populations for a specific number of generations. We employ this multiobjective optimization method to derive Pareto solutions (supply chain cost and customer satisfaction) for our case study. The supply chain cost is the sum of forward channel cost (cost1, eq 1), reverse channel cost (cost2, eq 3) and production facility cost (cost3, eq 5), whereas the customer satisfaction of the overall supply chain is an average of customer satisfaction at retailers, distribution centers, and warehouse (eq 6b). cost = cost1 + cost 2 + cost 3

(6a)

CST = avg(CSR , CSDC , CS W )

(6b)

continuous decision variables which is significantly less compare to the 204 decision variables in the original problem. The reverse channel is optimized in the second conquering stage. All collection centers aim to reduce the total reverse channel cost by balancing the inventory holding cost and the transportation cost. Our objective, in the second stage, is to find the best transportation capacity to push the right amount of used product to reproduction firms to facilitate refurbishment and remanufacturing operations. This subproblem has only 87 discrete decision variablesagain, the number of decision variables is significantly less compared to the original problem. In the third conquering stage, allocating the right number of machineries and the inventory target are the necessary decisions required for the production facilities. In this final stage, the total cost (=cost1 + cost2 + cost3) is optimized (Figure 5). The decisions derived from stage 1 and stage 2 are utilized in the third conquering stage; therefore, this particular step has only 30 decision variables compared to 204 decisions in the original problem. 5.1. Comparison with Single/Multi Objective Optimization. Once the optimization problem is defined, it has either one or multiple global optima. An interesting issue is to examine which optimization technique is superior in taking us close to the global minima. In the case of large scale systems which can be separated into various subsystems, the decomposition optimization has been found to be superior to the single-objective traditional optimization method.43,55 Also, for large scale complex problems with a combination of continuous and discrete decisions, the single-objective traditional optimization is known to fall short.56 The solutions obtained by DC optimization are compared with those obtained from the traditional single-objective optimization technique and the computationally intensive multiobjective optimization technique. NSGA-II (described in section 4.4) is employed to solve the multiobjective optimization problem. Figure 6 shows the nondominated solutions (supply

Figure 6. Nondominated solution obtained by NSGA-II.

chain cost vs customer satisfaction) obtained by optimization of the closed loop supply chain in a multiobjective fashion. From the Pareto front, the nondominated solution corresponding to 95% CS (desired) is taken to compare with the single-objective optimization and DC optimization methods. With respect to the computation (CPU) time, the decomposition optimization technique has the upper hand over other two methods (Table 3). This is due to the advantage gained by breaking the original large scale problem into smaller subproblems and optimization of the less difficult subproblems. In this case study, forward and

5. RESULTS AND DISCUSSION In the first conquering stage, the forward channel is optimized. All distribution nodes are assumed to be responsive toward the uncertain demand and practice proportional replenishment policy. Our objective is finding the best replenishment parameter to replenish right amount of products to provide 95% CSDN to the market customer and maintain inventory at optimal condition. The customer satisfaction constraint is implemented using dynamic penalty function. This subproblem has 87 16275



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Table 3. Results of Decomposition Approaches system 1

single-objective approach

decomposition approach

multiobjective approach

computation time (h) cost1 (×107) cost2 (×107) cost3 (×107) total cost (×107) Hessian *E+6 Hessian eigenvalues

18.7403 1.0681 1.5167 6.8913 9.4761 3.5732e+006 8/87 (positive)

11.7515 0.5015 1.3620 5.3486 7.2121 1.4776e+097 69/87(positive)

66.4208 0.8004 1.6040 2.0465 4.4509 1.4615e+035 7/87 (positive)

Figure 7. Customer satisfaction at all product nodes (retailer 1 to 90, distribution center 91 to 126, and warehouse 127 to 135).

a single objective, decomposition, and multiobjective optimization provides solutions characterized by Hessians whose determinants are nonzero. As expected, the optimality test concludes that all optimized solutions gained from the above three optimization approaches fall in the saddle point region. In the decomposition approach, 79.3% of eigenvalues are positive, whereas single objective and multiobjective approaches result in only 9.2% and 8% positive eigenvalues, respectively. By comparison, the solution with the DC approach is relatively more attractive than those obtained with single objective and multiobjective optimization. This optimality condition is reflected in the cost1 measure (Table 3), where the cost at the distribution network is relatively less for the solution obtained from DC optimization compared to the single-objective and multiobjective optimization. In the reverse channel, the cost2 measure obtained from DC optimization is superior to single and multiobjective optimization approaches. This is mainly due to the reduction in problem size and complexity which helps to reach superior solutions. Table 3 shows that cost2 obtained using DC optimization are 10% and 15% less than single-objective and multiobjective solution respectively. With reference to production facilities, the cost3 obtained in the DC approach is again 22% less than single-

reverse channels are isolated and optimized in parallel. Finally, the production system is optimized by linking with optimized forward and reverse channels. Other than computation time, the decomposition approach produces improved results than the conventional single-objective approach and multiobjective approach. This is mainly because of the advantage acquired by dividing the original problem into subproblems, where the subproblems are relatively less complex compared to the original problem. Note that this case study has an the additional advantage in that all subproblems are free from mixed decision variables (Table 2). The optimality and the quality of the obtained solutions can also be evaluated using a Hessian value. Hessian corresponds to the value of the second-order partial derivative of objective function with respect to continuous decision variables. This value is utilized in large-scale optimization problems to test optimality conditions of the obtained solutions; however, computing the Hessian matrix is computationally intensive. If all the eigenvalues of the Hessian matrix are positive, then the obtained objective value is a local minimum. If all its eigenvalues are negative, then the objective function is a local maximum. If the Hessian has both positive and negative eigenvalues, then the solution represents a saddle point. In this work, the Hessian obtained from traditional 16276



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demand is significantly less than normal demand. In the reverse channel, the decrease in used product returns affects the material flow to the remanufacturing firms and its operations. Inefficient used product returns force manufacturers to use fresh raw materials to produce new products. Therefore, the investment made on the remanufacturing facility is not impressive resulting in reduced return on investment (ROI). Figure 8 shows the variations in supply chain cost for changes in uncertain demand and used product returns. For all situations (a to d), uncertain inputs have significant influence on supply chain cost. Figure 9 confirms that situations a and b face large changes in supply chain cost as compared to the nominal situation. The supply chain cost surface obtained using the decomposition method is significantly less than supply chain cost surface of a single-objective method for all market situations a−d (Figure 9left and 9right). To quantify the relative difference between the supply chain cost surface, we defined supply chain cost margin as the difference between highest supply chain cost (situation a) and lowest supply chain cost (situation b). The ratio of supply chain cost margin faced by traditional single-objective solutions and decomposition solutions is 1.4857 indicating that the supply chain cost margin is 48.57% higher for solutions obtained using traditional single-objective approach than those obtained with the decomposition approach. These observations confirm that the solution derived using decomposition approach is superior than the solution obtained from single-objective approach. 5.3. Desirability of the Solution with Respect to the Customer Satisfaction Constraint. Uncertain inputs also have significant influence on customer satisfaction as shown in Figure 10. Desirability is the permissibility of the derived decisions in managing customer satisfaction constraints against the uncertain inputs (customer demand and used product returns). We studied the effectiveness of supply chain decisions derived from single-objective and decomposition optimization in managing the CS constraints at the distribution network and warehouse facility. The MOO solution fails to meet the desired CS of 95% at all product nodes (as shown in Figure 7); therefore we exclude MOO results for this analysis. The basic idea is assigning a score (eqs 7 and 8) to a set of constraints with respect to the desired limits (degree of constraint violation, LV = 0.9, HV = 0.95).

objective. In contrast, the multiobjective optimization provides lower cost3 than DC optimization. The reason for low cost3 in the multiobjective solution is because of the customer satisfaction constraint (eq 6b). Figure 7 shows the customer satisfaction for all the product nodes in the retailers, distribution centers, and warehouses. The CS for the multiobjective method solution (corresponding to CST ≈ 95%) distributed widely compared to the conventional and DC method (Figure 7). Especially, at the warehouse the CSW is significantly less than 95% due to low inventory and results in low inventory cost at the production sites. Therefore, the multiobjective solution is not comparable with other methods as it falls short to satisfy the CS constraint at all product nodes in the supply chain system. 5.2. Robustness of the Derived Solution (Obtained by Three Optimization Methods) With Respect to Uncertain Demand and Used Product Returns. Robustness of the derived solutions (the stability of derived decisions against the uncertain inputs) is also an important issue to study. Robustness is measured for the changes (mean shift) in inputs, for example, uncertain demand and uncertain used-product returns. We studied the sensitivity of performance metrics with respect to uncertainty in the demand and used product returns in the range of −15% to +15% (Figure 8).

Figure 8. Influence of uncertain inputs (demand/used product return) on supply chain cost.

(a) Increase in product return and decrease in demand increases the supply chain cost. Increase in product returns increases the used product inventory holding cost in the reverse channel. Decrease in demand increases the excess inventory at the forward channel and the production facility. (b) Decrease in product return and increase in demand decreases the supply chain cost. Decrease in product returns reduces the used product inventory holding cost in the reverse channel (as shown in Figure 8). Increase in demand depletes the excess inventory at the forward channel and production facility. (c) Increase in product return and demand is the best situation for supply chains interested in establishing their products and extending their business in the market. Though a slight increase in reverse channel cost is to be expected, this situation has an advantage in producing more products at the cheaper cost and time using remanufacturing facility. (d) The situation where both demand and product returns decrease than expected is a critical scenario. This situation increases the excess inventory of new products because actual

DCS,DN

DCS,WH

D=

⎧ 0, CSDN ≤ LV ⎪ ⎪ CSDN − LV =⎨ , LV ≤ CSDN ≤ HV ⎪ HV − LV CSDN ≥ HV ⎪ 1, ⎩

(7)

⎧ 0, CS WH ≤ LV ⎪ ⎪ CS WH − LV =⎨ , LV ≤ CS WH ≤ HV ⎪ HV − LV CS WH ≥ HV ⎪ 1, ⎩

(8)

DCS,DN × DCS,WH

(9)

The overall desirability measure (eq 9) is represented as the geometric mean of individual desirability values.57 Figure 10 shows that the overall desirability of managing the constraints is greatly improved by adopting the decomposition approach rather than the traditional optimization technique. All the above results confirm the advantage of practicing DC optimization to derive superior, reliable (against uncertain 16277



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Figure 9. Sensitivity of supply chain cost for −15% to +15% uncertainties in market demand and used-product returns decomposition method (left); conventional method (right).

Figure 10. Desirability of customer satisfaction constraint for −15% to +15% uncertainties in market demand and used-product returns (left) decomposition method; (right) conventional method.

lends itself to parallel execution and iterative convergence to improve solution quality.

inputs) and desirable (in managing constraints) supply chain decisions with less computational effort. With the increase in problem size, the benefit gained by DC optimization is relatively far improved (in terms of solution quality and computation time) than single and multiobjective optimization approaches. This DC approach can also be extended to resolve bottleneck(s) in existing supply chains. If the bottleneck resides in one of the subsystems, the optimal solution can be determined by solving only the relevant subproblem. One possible limitation of the DC approach is that it is only suited to systems which can be divided into small parts and have decision variables with no/minimal overlap. For large scale systems, the conventional approach may limit solution quality because of problem complexity and/or the initial point; however a careful selection of optimization algorithm and/or tuning method might provide feasible solution. DC optimization is powerful and scalable because it naturally

6. CONCLUSIONS Periodic revision in supply chain decisions is mandatory to maintain the performance at high standards and remain proactive against heavy competition and uncertainties. Model-based optimization strategies help to derive supply chain decisions with reference to forecasted inputs. Real world supply chains are large, complex, and have to make several decisions to improve performance. Traditional optimization methods may be overwhelmed by the problem size and complexity leading to inferior solutions while using up a lot of computing time. Therefore in this work, we considered the supply chain as a cooperative system, which can be divided into subsystems and conquered locally for the benefit of the overall network and the subsystems. The results obtained using the DC approach is much improved 16278



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signify the time taken by the supplier to process the order and for the distributor j to update the material p received. With the accurate estimation of Lj,p, it is possible to set appropriate set points that are responsive to the demand. An exponential forecaster with α = 0.111 (eq A8) was used in all distribution nodes practicing responsive strategy to forecast the downstream demand as suggested in the literature.58 The rate at which downstream orders are satisfied by node j depends on the inventory level of product p at-hand. Whenever inventory athand is high, the distribution node can satisfy all downstream customer orders (mj,p = 1); when it has limited inventory, the distributor has a policy of satisfying equal proportion of all downstream orders (0 ≤ mj,p ≤ 1). This order processing is modeled by (eq A9). All distribution nodes replenish the products from their suppliers using the proportional policy (eq A10) where SIPj,p(t) is given by (eq A7); Kj,p is the replenishment parameter. Choosing the right replenishment parameters in relation to the demand pattern and business goals is a challenging task for each component of the network and consequently for the overall network. The performance of supply chains can be judged by measuring or examining suitable indicators. Choosing the right performance indicator or a combination of performance indicators depends on the characteristics of the supply chain system. For a distribution network, the resource indicators represent the supply chain costs (e.g., excess inventory, eq A11), the output indicators represent the outcome of the distribution node (customer satisfaction, eq A12). Excess inventory (EIj) is indicative of the cost incurred due to stocking more goods than the required level. Customer satisfaction (CSj) is the percentage of downstream orders satisfied by the distribution node. For retailer node: [i → dc, j → r, k → c] and for distributor node: [i → wh, j → dc, k → r]

than that with traditional optimization techniques. The obtained solution with the DC approach is more robust against the uncertain inputs and close to optimality compared to traditional single-objective optimization technique.

■

APPENDIX Highly integrated manufacturing, product distribution, and recycle systems have resulted in more and more large-scale operations. Large-scale engineering has garnered a great amount of interest from academic scholars and industrial practitioners to ensure high-level operations. This section will elaborate the modeling aspects and the dynamic behavior of the distribution network, collection centers, manufacturing facilities, and their internal (control) strategies. The dynamic behavior of the supply chain is modeled using material and information balances (see Figure 2 for the schematic representation of a multiechelon network). The discrete model is adopted to model multiproduct multiechelon supply chain and described next. A.1. Forward Channel Formulation

The main objective of the distributor node “j” is to organize its inventory position IPj,p(t) of product “p” (p = A to I) at discrete time t, at the desired target level. For node j, let Yij,p denote the material flow from its supplier i and Yjk,p denote the material flow from node j to a downstream node k. The inventory position at time t depends on the inventory position at time t − 1, the materials received, and the materials dispatched from node j (eq A1−A3). The inventory position IPj,p(t) is also the sum of the inventory at-hand IHj,p(t) (eq A4−A5) and the inventory onroad IRj,p(t) of that particular product p. Equation A6 notes that the inventory on-road at time t is the sum of the orders satisfied by the supplier during the past Lj,p time periods (but not received at node j at time t due to the transportation delay). Yij,p(k) is the material shipped by the supplier at time k against an order placed by the node j Uji,p(k). The lead time (Lj,p) is the time taken by the supplier to satisfy the orders placed by the downstream nodes. In other words, it is the time delay faced by the node j to receive the material after placing the replenishment order. It includes the time taken by the retailer node to place the order, the time taken by the supplier to process the order, and the product transportation time. The order is assumed to be communicated instantaneously using advanced information technologies. Therefore, the lead time mainly corresponds to the time taken by the supplier to process the downstream order and the transportation time. The lead time depends on the geographical location of the supplier and customer, modes of transportation available, and the product availability. The lead time information can be obtained from the authorities of the distribution node or estimated from time-series data gathered from the supply chain. Note that, in this work, we consider the lead times to be fixed and unchanging (Lj,p = Li,p = 12) over the time horizon of interest. In general, the decentralized node prefers to become more responsive to the market demand by maintaining a flexible inventory position to minimize inventory holding cost, excess inventory, and back order cost. The flexibility in inventory position is achieved by setting desired inventory position target SIPj,p(t) in response to the forecasted demand for Lj,p time periods. Equation A7 indicates that the optimum desired inventory target is equal to the responsiveness factor (i.e., Lj,p + 2) times the forecasted demand (i.e., the forecasted demand for Lj,p + 2 time periods). Lj,p represents the lead time faced due to transportation time, and the extra two discrete time periods

IPj , p(t ) = IPj , p(t − 1) + Yij , p(t ) −

∑ Yjk ,p(t ) k

1 1 Yij , p(z −1) − 1 − z −1 1 − z −1

IPj , p(z −1) =

(A1)

∑ Yjk ,p(z−1) k

(A2)

IPj , p(t ) = IHj , p(t ) + IR j , p(t )

(A3)

IHj , p(t ) = IHj , p(t − 1) + Yij , p(t − Lij , p) −

∑ Yjk ,p(t ) k

(A4)

IHj , p(z −1) =

z −Lij ,p 1 Y (z −1) − −1 ij , p 1−z 1 − z −1

∑ Yjk ,p(z−1) k

(A5)

IR j , p(z −1) =

1 − z −Lij ,p Yij , p(z −1)→IR j , p(t ) 1 − z −1

t

=

∑

Yjk , p(k)

k = t − Lij , p

(A6)

SIPj , p(z −1) = (Lij , p + 2) ∑ dk̅ , p(z −1) k

dk̅ , p(z −1) = 16279

αk , p 1 − (1 − αk , p)z −1

(A7)

dk , p(z −1) (A8)



Article

Yjk , p = z −1 ∑ mj , pdk , p

Table A1. Closed Loop Supply Chain Information

(A9)

parameters

Uji , p(t ) = Kj , p(SIPj , p(t ) − IPj , p(t ))

EIj = avg[max(IHj , p(t ) −

(A10)

∑ Ukj ,p(t ), 0)]

lead time (WH to DC) lead time (DC to R) IH (at r and dc) market demand

12 unit time 12 unit time 1 $/ unit product/unit time ∼ normal (45, 8) product units/unit time Collection Centers lead time (DC to WH) 12 unit time lead time (R to DC) 12 unit time IHU (at r and dc) 1 $/unit product/unit time TC (at r and dc) 1*no of units $/truck used product returns ∼ uniform (0.6, 0.7) × market demand Production Facility manufacturing time 5 unit time/machine remanufacturing time 5 unit time/machine refurbishment time 3 unit time/machine production capacity 200 product units/machine used product for refurbishment ∼ uniform (0.5,0.6) × used product facility returns used product for remanufacturing ∼ (1-uniform (0.5, 0.6)) × used product facility returns IH (at mn, rm and rfm) 1 $/unit product/unit time IHU (at rm and rfm) 1 $/unit product/unit time Id (at mn, rm and rfm) ∼2000, 2000, and 2500 $/unit time refurbished product cost 50 $/unit product remanufacturing product cost 75 $/unit product fresh product 90 $/unit product product sale price 150 $/unit product

∀ j, p

k

(A11)

⎡ ∑t H ∑ Y ⎤ t=0 j jk , p ⎥ CSj = avg⎢ t ⎢ ∑ H ∑ Ukj , p ⎥ ⎣ t=0 k ⎦

∀ j, p (A12)

A.2. Reverse Channel Formulation

The main idea of a distributor node acting as a collection center ‘j’ is to recover maximum amount of value by integrating the used product returns into the supply chain. The common processes involved in the reverse channel are the acquisition of used products, their inspection, testing and disposition, reverse logistics, remanufacturing, and distribution. Although the common activities are well established, the managerial importance and understanding of these activities are different in different scenarios.20 Usually, at all collection centers, the used products (YUkj,p) are collected, stockpiled, and pushed to the upstream nodes. They are used at the remanufacturing facilities for reproduction. Equations A13 and A14 indicate that used product inventory is the balance of used product influx from the downstream customers and used product moved to the upstream nodes. The primary concern in reverse channel is the manner in which used product inventory is reviewed and transferred to the upstream nodes. According to eq A15, the used product inventory is reviewed at each time period and the material is pushed to the upstream nodes only when the availability of the collected used product is more than the truck capacity (Cj,p). The transportation time and cost associated with IHUj,p is given in Table A1. For retailer node: [i → dc, j → r, k → c]; for distributor node: [i → wh, j → dc, k → r] IHUj , p(t ) = IHUj , p(t − 1) +

the machineries m and the product transferred to the distribution centers to satisfy the distributor orders (eq A17). IHpf, p(t ) = IHpf, p(t − 1) + ∀ pf ∈ {mn, rm, rfm}, p

∑ YUkj ,p(t ) − YUji ,p(t )

∑ YUkj ,p(t ) − k

1 YUji , p(t ) 1 − z −1 (A14)

⎧ ⎪ 0, IHUj , p(t ) < Cj , p YUji , p(t ) = ⎨ ⎪ ⎩Cj , p , IHUj , p(t ) ≥ Cj , p

(A15)

A.3. Production Facility Formulation

The generic closed loop supply chain will include three types of production facilities, namely manufacturing “mn”, remanufacturing “rm” and refurbishment facility “rfm”. Each production facility has own inventory target (SPpf,p, pf = [mn, rm, rfm]) for each product p. The discrepancy in the desired inventory target (epf.p) drives the production machinery at all production facilities (eq A16−A21). epf, p(t ) = SPpf, p − IHpf, p(t )

pf ∈ {mn, rm, rfm}

(A17)

All products produced from the manufacturing, remanufacturing, and refurbishment units are considered to have identical quality.49 With respect to production cost and time, the products produced from refurbishment facility have advantage over remanufacturing and manufacturing facilities. Therefore, refurbished product is preferred ahead of product from other facilities to satisfy the distributor order (eq A18). Next to refurbished products, remanufactured products have cost advantage over products manufactured using fresh raw material (eq A19). Finally, product produced from virgin raw material is used to cover the backlog caused because of insufficient product from remanufacturing and refurbishment facilities (eq A20).

(A13)

1 1 − z −1

∑ Ppf,p,m(t ) − Ypf,p(t ) m

k

IHUj , p(z −1) =

values Distribution Network

⎧ ⎪ Udw, p(t ), IH rfb, p(t ) ≥ Udw, p(t ) Yrfb, p(t ) = ⎨ ⎪ ⎩ IH rfb, p(t ), IH rfb, p(t ) ≤ Udw, p(t )

(A18)

⎧U (t ) − Y (t ), dw, p rfb, p ⎪ ⎪ Yrm, p(t ) = ⎨ IH rm, p(t ) ≥ Udw, p(t ) − Yrfb, p(t ) ⎪ ⎪ IH rm, p(t ), IH rm, p(t ) ≤ Udw, p(t ) − Yrfb, p(t ) ⎩

(A16)

IHpf,p is the inventory of new product available at all production facilities. It is the balance of sum of products produced (Pm,p) in

(A19) 16280



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Figure A1. Scheduling for multiproduct production system.

⎧U (t ) − Y (t ) − Y (t ), rfb, p rm, p ⎪ dw, p ⎪ IH ( ) ( ) t ≥ U t − Yrfb, p(t ) − Yrm, p(t ) mn, p dw, p ⎪ Ymn, p(t ) = ⎨ ⎪ IH mn, p(t ), ⎪ ⎪ IH mn, p(t ) ≤ Udw, p(t ) − Yrfb, p(t ) − Yrm, p(t ) ⎩

(e) Repeat step a−d until all machineries are assigned for production or until all discrepancies in product inventory are eliminated via production. Manufacturing Unit: Pmn, p*,m(t + Lmn)

(A20)

Ypf, p(t ) = Yrfb, p(t ) + Yrm, p(t ) + Ymn, p(t )

⎧ emn, p * ≥ pc m , statusmn, m = idle ⎪ pc(t ), =⎨ ⎪ otherwise ⎩ 0,

(A21)

Each production facility has multiple production resources (machineries) m. All machineries have identical production capacity. All machineries have the capability to produce any product A, B, ..., I. The production rules followed at all production facilities for each review period are described in Figure A1 and eqs A22−A24. Here, we describe them step by step. (a) Spot the unemployed (idle) machineries from the set m1 to mn (one by one). (b) Identify the products p having high inventory discrepancies. Check whether the discrepancy is greater than the production capacity. If yes proceed to step c, else switch to next product facing high discrepancy. (c) Make sure the raw material/used product is available to produce the product identified in the step b. If yes proceed to step d, else switch to step b to identify other product facing high discrepancy and having enough raw material/ used product to produce new product. (d) Start production and update the status of machinery, raw materials.

(A22)

Remanufacturing Unit: ⎧ pc(t ), e rm, p * ≥ pcm , rwrm, p * ≥ pcm , ⎪ ⎪ Prm, p*, m(t + Lrm) = ⎨ statusrm, m = idle ⎪ ⎪ 0, otherwise ⎩ (A23)

Refurbishment Unit: Prfm, p*, m(t + Lrf ) ⎧ pc(t ), e rfm, p * ≥ pcm , rwrfm, p * ≥ pcm , ⎪ ⎪ =⎨ statusrm, m = idle ⎪ ⎪ 0, otherwise ⎩

(A24)

where p* = p corresponds to max(ew,p). 16281



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Nomenclature: Variables

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C = transportation truck capacity CS = customer satisfaction d = product demand D = desirability e = deviation of product inventory from the desired inventory at pf EI = excess Inventory IH = inventory at hand of fresh products IHU = inventory at hand of used products IR = inventory on-road IP = inventory position Id = plant Idle cost K = order replenishment parameter L = product lead time pc = production capacity P = product produced SIP = set point for inventory position in the forward channel SP = desired inventory for the production facilities TC = transportation cost U = order placed to maintain inventory position Y = product shipped by the supplier against an order placed by the node YU = used product shipped by the retailer and distribution nodes Symbols

α = demand forecast parameter

Subscripts

c = customers dc = distribution centers p = products A to I pf = production facility pf ∈ {mn, rm, rfm} r = retailer t = time tH = simulation time horizon mn = manufacturing facilities rm = remanufacturing facilities rfm = refurbishment facilities wh = warehouse

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +65-65168484. Fax: +6567791936. Notes

The authors declare no competing financial interest.

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REFERENCES

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Divide and Conquer Optimization for Closed Loop Supply Chains

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