Multiperiod and stochastic formulations for a closed loop supply chain with incentives

Transcription

1 Multiperiod and stochastic formulations for a closed loop supply chain with incentives L. G. Hernández-Landa, 1, I. Litvinchev, 1 Y. A. Rios-Solis, 1 and D. Özdemir2, 1 Graduate Program in Systems Engineering, Universidad Autónoma de Nuevo León (UANL), Mexico 2 Faculty of engineering, Yasar University, Universite Caddesi, Turkey Reverse logistics network design problem we focus on is about locating distribution centers, inspection centers and remanufacturing facilities, and determining the acquisition price as well as the amount of returned goods to be collected depending on the unit cost savings and competitor s acquisition price. We introduce the multiple periods setting and stochastic demand formulated by scenarios. We develop two mathematical programming models to determine the pricing strategy of the recovered products together with the optimal network that must be designed to be the most profitable closed cycle. Our methodology is based on a Golden Section Search with some flexibility that enable us to fix the used product acquisition price and then solve the model as an integer linear programming. Moreover, we establish dependent size fixed costs of opening a distribution, an inspection, and a remanufacturing centers and show that they have a strong impact on the Golden Section search behavior. INTRODUCTION Reverse Logistics or Closed Loop Supply Chains are the networks designed with the goal of addressing a full process of product delivery and at the same time the product recovery at the end of its usable life for remanufacturing or disposal.? ] define a reverse logistics network as the relationship between the used product market and the market for The article is published in the original. Electronic address: leonardo@yalma.fime.uanl.mx Electronic address: deniz.ozdemir@yasar.edu.tr

2 2 new products: when these two markets coincide then we have a closed-loop network. In this context? ] divide into two groups the members of the closed loop supply chain: the members of the traditional logistics chain, including raw material suppliers, manufacturers, retailers and demand markets, and members of the reverse logistics chain, including market demand, recovery centers and manufacturers. The closed loop supply chain management has had an increasing interest due to the actual focus on ecologically sustainable development, e.g., in? ] the authors consider that good management of a logistics plan demonstrates the commitment of the company with the environment. Moreover, the economic benefits of reusing serviceable waste is substantial compared to the exclusive use of new raw materials. Currently, the European Union has a legislation to deal with the pollution caused by electronic wastes: Waste of Electric and Electronic Equipment (WEEE) Directive (2002/96/EC WEEE) [? ]. The general purpose of these directives is to reduce the electrical and electronic waste and to promote its reuse, recycling and the other forms of recovery in order to reduce the final disposal. The WEEE directive covers a wide range of products: small or large household appliances, telecommunications equipment, lighting equipment, electric tools, toys, sport and leisure equipment, medical devices, monitoring and controlling devices, and automated devices. Producers must now recover and recycle a predetermined fraction of sold products. These activities involve collection of used products, inspection and separation to determine whether the product is recoverable or not, reuse, recycling, remanufacturing or repairing the product, disposal of the unrecoverable products, and redistribution of recovered remanufactured products [? ]. We focus on a reverse logistics network design problem where distribution centers, inspection centers and remanufacturing facilities must be located, the acquisition price as well as the amount of returned goods to be collected depending on the unit cost savings and competitors acquisition price must be determined. The objective of this study is to determine the used product collection strategies in two different reverse logistics networks extended from the model proposed in? ]. The first network considers multiple periods along the time while the second one takes into account an stochastic demand formulated by scenarios. We develop two mathematical programming models to determine the pricing strategy of the recovered products together with the optimal network that must be designed to be the most

3 3 Factories I Distribution/ Inspection Centers J Clients K U ij X jk Y j =1 R H i =1 V j 0 i T j 0 =1 W kj 0 Figure 1: Closed loop schema. profitable closed cycle. More precisely, we have a closed loop supply chain of three levels (Fig. 1), where the manufactured product is shipped to distribution centers and from there it is distributed to the clients. From those clients, the used product is recovered and sent to the inspection centers where it is decided if it would be discarded or sent to the manufacturing plants (dashed lines of Fig. 1). Moreover, we must chose the best distribution centers, the best inspection centers (distribution center with a green mark in the figure), and the remanufacturing factories (factory marked with an R in the figure) in order to meet the levels of demand in a multiperiod or stochastic demand environment, such as minimizing transportation costs between the three levels of the chain. Furthermore, the solution should offer the best acquisition price of the used product as we consider that there is competition with other organizations as in? ]. Our methodology is as follows. First, we propose two new non linear integer programming, one for the multiperiod case and the other for the stochastic demand. Then, we adapt a Golden Section Search that enable us to fix the used product acquisition price and then solve the model as an integer linear programming. In this stage we explore the trade-off between quality and computational time. Moreover, we establish a new manner of determining the fixed costs of opening a distribution, an inspection, and a remanufacturing centers. We show that these new cost have a strong impact on the Golden Section search behavior. The rest of this work is structured as follows. In Section we show a brief literature review to underscore some work done on closed loop supply chain and customer incentive models. In Section I the multiperiod non linear model is presented together with the stochastic

4 4 non linear model. Section II exhibits our Golden Search Method we tune the accuracy of the branch-and-bound method for the solution of the subproblem. In this same section we introduce a new manner of computing the fixed costs of the problem. Section III is related to the experimental results on random generated instances to show that our methodology is efficient. A final section concludes this work. Literature review The reverse logistic network problem has been broadly studied: [? ], [? ], [? ], [? ], [? ], and [? ]. Moreover, there are some excellent reviews by? ] and? ]. With respect to reverse logistics that consider a multiperiod setting we can mention the work of? ] where the authors present a genetic algorithm. In [? ], a multiperiod reverse logistics network is designed under risk by a stochastic mixed integer linear programming. In [? ], a multicommodity formulation is presented to use a reverse bill of materials by integer linear programming. The stochastic demand point of view has been studied by [? ], [? ], [? ] by integer linear stochastic programmings. In [? ], the authors integrate a sampling strategy with an accelerated Benders decomposition. While [? ] propose a robust optimization model for handling the inherent uncertainty of input data.? ] mention that there are few pricing models for acquiring used products. We can cite the works of [? ] where the remaining value in the used products that can recovery is the companys main motivation for the collection operation. They use nested heuristics based on a tabu and Fibonacci searches.? ] it is presented a simulation model to calculate the collection costs. Our methodology is as follows. First, we propose two new non linear integer programming, one for the multiperiod case and the other for the stochastic demand. Then, we adapt a Golden Section Search that enable us to fix the used product acquisition price and then solve the model as an integer linear programming. In this stage we explore the trade-off between quality and computational time. Moreover, we establish a new manner of determining the fixed costs of opening a distribution, an inspection, and a remanufacturing centers. We show that these new cost have a strong impact on the Golden Search behavior. Our work is based on the work of? ], where the aim is locating distribution centers,

5 5 inspection centers and remanufacturing facilities, determining the acquisition price as well as the amount of returned goods to be collected depending on the unit cost savings and competitor s acquisition price to minimize the transportation costs, fixed costs and used product acquisition costs. She proposes a mixed-integer nonlinear programming problem that becomes a mixed integer programming when the acquisition price is set to a given value. The best value of the acquisition price is determined by the Golden Section search. Our work extends the one of [? ] since we introduce the multiperiod framework and include stochastic demand. Moreover, we improve the Golden Section search by introducing some flexibility that improves the execution times without loosing the quality of the solutions. I. NON LINEAR MIXED INTEGER PROGRAMMING We consider that a single type of product is produced, sent to the distribution centers, distributed to the clients, returned to the inspection centers, and returned to a remanufacturing plant. Our models must be an effective tool to decide which distribution and inspection centers must be established and which factories should have a remanufacture module to receive used product. Moreover, we also consider the decision of the price offered by the company to recover used product from the clients. In Section I A the multiperiod non linear mixed integer programming is presented while in Section I B the demand is considered stochastic. An interesting result is that both models represent different settings but have a similar mathematical structure that will be used by our solution methodology in Section II. A. Multiperiod Model Let I be the set of possible manufacturing plants, J the set of the potential distribution centers, K the set of clients, and T the set of the periods to be considered. Notice that the distribution centers and inspection centers have unlimited capacity. A client k has demand d t k for each period t of time. The proportion of returned products τ t for period t. When the returned products pass through the inspection center only a proportion α is useful and can be sent to the remanufacture plant. If the location of a distribution center j is set then

6 6 it incurs on a fixed cost of f j while if it set to be an inspection center then the cost is g j. The same happens with the manufacturing plants: with an extra cost of h i a plant i can remanufacture used products. We have the following sets of binary variables to make the decisions about the locations of the distribution centers, inspection centers, and the remanufacturing units. 1 if factory i is expanded with a remanufactoring unit, H i = 0 otherwise, 1 if location j is used as a distribution center, Y j = 0 otherwise, 1 if location j is used as an inspection center, T j = 0 otherwise. The amount of product sent from factory i to distribution center j at period t is represented with variable U t ij. Similarly, the amount of product sent from distribution center j to client k at period t is Xjk t. These are usually called as the forward flow of the network. The amount sent from client k to inspection center j at t is Wkj t, while the amount sent from inspection center j to remanufacturing factory i at t is V t ji. These are the reverse flows of the network. The cost per unit from factory i to distribution center j is c ij, from distribution center j to client k is e jk, from client k to inspection center j is ep kj, while from inspection center j to remanufacturing factory i is cp ji. We can now state the mathematical model for the multiperiod inverse logistic network that we name as multi-nlmip. min (f j Y j + g j T j ) + h i H i + ( c ij Ui,j t + e jk Xjk t + i I t T i I k K + cp ji Vji t + Wkj(ep t kj + L b)) (1) i I k K

7 Xjk t = d t k k K, t T (2) Uij t = Xjk t j J, t T (3) k K i I α Wkj t = k K i I V t ji j J, t T (4) U t ij V t ji s i i I, t T (5) V t ji U t ij 0 i I, t T (6) V t ji a i H i i I, t T (7) X t jk d t ky j j J, k K, t T (8) Wjk t τ t T j d t k j J, k K, t T (9) Wkj t = L L + l τ t d t k k K, t T (10) X t j,k, W t k,j, U t i,j, V t j,i, L 0 i I, j J, k K, t T Y j, T j, H i {0, 1} i I, j J. Objective function (1) considers in its first term the fixed costs of opening distribution and inspection centers. Last term considers the acquisition cost of the used product L and the benefit of remanufacturing these used products b. The rest of the terms are related to the forward and reverse flows. All this costs and benefits are considered for all the periods. Demand of each client must be satisfied for each one of the periods, this is acheived by constraints (2). Notice that a client can be supplied by several distribution centers. Balance constraints (3) ensure that the amount of products that arrive to a distribution center is the same that is sent to the clients. Similarly, balance constraints (4) consider the reverse flow between the clients and the inspection center multiplied by α which the expected rate of return of the used product. Constraints (5) imply that all that is produced in plant i (minus the recuperated products) does not violate the capacity s i of factory i at each period. Constraints (6) bound the amount of reused products by the amount of new products for each plant at each period. Constraints (7) bound the amount of reused products to the remanufacturing capacity of the plant a i if this plan has been set with a remanufacturing unit. Constraints (8) bound the amount of products that can be sent to a client from a distribution center that has been open. 7

8 8 Constraints (9) bound the amount of recuperated product that is inspected in center j (if it is chosen) by a proportion of the demand. Non-linear constraints (10) express the amount of used product as a function of the proportion between the recuperation price the company offers L by this same price plus the price l that the competitor is offering for recovering the products multiplied by the amount of product that can be recovered. B. Stochastic Formulation A stochastic formulation can be established by using almost the same model we just presented. Indeed, instead of having t periods, we would have s scenarios together with a probability for each scenario. We call this problems as stoch-nlmip. The structure of the solution spaces are very similar in the multiperiod case and in the stochastic one (with scenarios). The scenarios for the stochastic formulation stoch-nlmip are obtained from the uncertain demand of the clients and from the return rate of the product. We consider a set S of four scenarios: high demand and low return rate, low demand and low return rate, high demand and low return rate, low demand and high return rate. For each scenario, we have a probability of p s. The amount of product sent from factory i to distribution center j under scenario s is U s ij. The amount of product sent from distribution center j to client k at scenario s is X s jk. The amount sent from client k to inspection center j under s is Wkj s, while the amount sent from inspection center j to remanufacturing factory i under s is V s ji. The model with stochastic demand and stochastic return rate, stoch-nlmip, is as follows. min (f j Y j + g j T j ) + i I h i H i + p s ( c ij Ui,j s + e jk Xjk s + s S i I k K + cp ji Vji s + Wkj(ep s kj + L b)) (11) i I k K

9 Xjk s = d s k k K, s S (12) Uij s = Xjk s j J, s S (13) k K i I α Wkj s = k K i I V s ji j J, s S (14) U s ij V s ji s i i I, s S (15) V s ji U s ij 0 i I, s S (16) V s ji a i H i i I, s S (17) X s jk d s ky j j J, k K, s S (18) Wjk s τ s T j d s k j J, k K, s S (19) Wkj s = L L + l τ s d s k k K, s S (20) X s j,k, W s k,j, U s i,j, V s j,i, L 0 i I, j J, k K, s S Y j, T j, H i {0, 1} i I, j J As it can be seen, models multi-nlmip and stoch-nlmip are analogous in their structure but different in the nature of the modeling. In Section II we present a method for solving the multi-nlmip. This same methodology can be used for the stoch-nlmip as it will experimentally shown in Section III. 9 II. GS-MIP METHOD In model multi-nlmip (and stoch-nlmip) the only non linear restrictions are the ones of (10) (or (20)). Once the value we are offering for recovering the products L is fixed, then multi-nlmip (and stoch-nlmip) is linear an can be solved by a linear branch-andbound method. Nevertheless, the amount of time taken by a linear solver that implements a branch-and-bound for a fixed L can still be long. The key point is to establish fixed cost for the location of a distribution center j, f j, for an inspection center,g j, and for a remanufacturing plant, h i. Indeed, we propose to make them instance size dependent as follows. h i = F i( J + K ) I, f j = gj = F j( K ). (21) I

10 10 By parametrizing variable L we observe a somewhat convex structure of the solutions when we use the fixed costs (21). We take advantage of this characteristic to reduce the computation times by introducing some flexibility that guarantees near optimal solutions. In the following we explain in more details the methodology we call Golden Section search for Mixed Integer Programming (GS-MIP) for solving multi-nlmip and the stoch-nlmip. 1: L i 0, L u b, ɛ > 0 Algorithm 1 GS-MIP(γ%). 2: Golden number R = 3: repeat 5 1 2, D = L u L i 4: L 1 L i + D 5: L 2 L u D 6: F (L k ) value of multi-nlmip(l k ) with gap=γ%, k = 1, 2 7: if F (L 1 ) < F (L 2 ) then 8: L i L 2 9: else 10: L u L 1 11: end if 12: D = R(L u L i ) 13: until D ɛ Let multi-nlmip(l) be the mixed integer linear programming obtained by fixing the value of L. We apply the golden section search [? ] to find the value of L that makes multi- NLMIP(L) to be minimum by successively narrowing the range of values. This methodology is schematized in Algorithm 1 where we maintain the function values for triples of points whose distances form a golden ratio. We initialize L i, L u, ɛ, the golden number R, and set D. In steps 4 and 5 we establish the new upper and lower bound on L and in step 6 we evaluate the solutions by using an integer linear programming solver for multi-nlmip(l k ) with a tolerance gap (best integer solution - best relaxation solution)/best integer solution) of γ%. This flexibility is based on the somewhat convex shape of the parametric function value of multi-nlmip(l) and is the key point of the efficiency of our method. In steps 7-10 we update the bounds while in step 12, D is reduced by the golden number. We iterate until the interval D is smaller that the given ɛ.

11 11 In [? ] a golden search method is presented but without the important characteristic of the gap variation. Indeed, the novelty of having convex shape parametrized functions multi-nlmip(l) allow this flexibility of the gap without loosing quality in the solution. We show in Section III that our method GS-MIP(γ%) is efficient in different types of instances. Notice that the same methodology of Algorithm 1 can be applied to stoch-nlmip. In the next section we show experimental results of our propose methodology. III. EXPERIMENTAL RESULTS In this section we prove that the GS-MIP(30%) is efficient for solving the multi-nlmip and the stoch-nlmip. We generated a set of instances as follows based on the ones of [? ]. The coordinates of the factories, the distribution and inspection centers, and the clients where uniformly drawn from [0,1]. The cost between factories, distribution centers, inspection centers, and clients were given by the euclidean distance of the points. The size of the instance is denoted as ( I, J, K ) that corresponds to the number of factories, distribution centers, and clients, respectively. We categorize the instances into three sets: the small instances are of size (5, 10, 40), the medium size instances are of (10, 30, 60), while the large ones are of (15, 50, 80). The coordinates of an instance with 10 factories, 30 possible distribution centers, and 60 clients is presented in Figure 2. The demands of the clients are uniformly chosen from interval [0,100] for the multi-nlmip. We consider two periods. For the stoch- NLMIP the client demand has two ranges, interval [0,50] for low demand scenario and [50,100] interval for high demand scenario; τ = 0.4 for low return rate and τ = 0.8 for high return rate. Therefore, we have four scenarios. The fixed costs required by the opening of a remanufacturing unit, a distribution center, and an inspection center based on [? ] are 5000, 7500, and 10000, respectively. We recall that we propose to use dependent instance size fixed costs with equations (21). The acquisition price of the competitor l is fixed to 10 while the benefit obtained by remanufacturing a product is b = 20. For computing the capacity of the remanufacturing units a i and the capacity of the plants s i, for all i I, we used the following equations to

12 12 Y remanufacturing Remanufacturing units units distribution Distribution centers inspection Inspection centers Customer Clients zone X Figure 2: Coordinates of a large instance of size (15, 30, 80). ensure feasibility of the multi-nlmip instance [? ] with recovery rate α = 0.8: a i = α k t τ t d t k k t, s i = dt k. K K Analogously is done for the stoch-nlmip where these parameters will now depend on each scenario s S. In the case where an instance does not verify these equations it is easy to insert a dummy plant or client. The multi-nlmip and stoch-nlmip models were implemented in GAMS with the non linear solver DICOPT 1. Both models were also solved with GS-MIP(30%) implemented in GAMS with the integer linear solver of CPLEX We use the two possible alternatives given by CPLEX s solver: the branch-and-bound algorithm (B&B) and the interior point method (Barrier method). All the experiments were executed in a Sun Fire V440 with 4 Ultra SPARC III of 1602 Hhz, and 8 GB RAM. Table I presents the experimental results of models multi-nlmip and stoch-nlmip solved with our methodology GS-MIP(30%) (with ɛ = 0.001) and with DICOPT. Each row is the average of 10 different instances. The table has two sections, one for multi-nlmip 1 This program is based on the extensions of the outer-approximation algorithm for the equality relaxation strategy ( 2

13 13 multi-nlp GS-MIP(30%) and B&B GS-MIP(30%) and Barrier DICOPT 30% Instance Objective L Time Objective L Time Objective L Time (5,10,40) (10,20,60) (15,30,80) stoch-nlp GS-MIP(30%) and B&B GS-MIP(30%) and Barrier DICOPT 30% Instance Objective L Time Objective L Time Objective L Time (5,10,40) (10,20,60) (15,30,80) Table I: Performance of GS-MIP(30%) for multi-nlmip and stoch-nlmip solved with the B&B and the Barrier method versus DICOPT. and another for stoch-nlmip. First column indicates the size of the instance. The following three columns are for GS-MIP(30%) with the B&B, the other three columns are for GS- MIP(30%) with the Barrier method, and finally the last three columns are for DICOPT. For each method the mean objective time, the mean value of the acquisition price L, and the time in seconds are reported ( - means that DICOPT could not find a solution in less than seconds). The non lineal solver DICOPT is not able to find solutions for the large instances. Moreover, the quality of the solutions DICOPT offers is worst than the solutions obtained by GS-MIP(30%) in around 25% for multi-nlmip and around 4% for stoch-nlmip. Even if both problems have the same mathematical structure, DICOPT solves easily the stochastic problem. Notice that the quality of the solutions obtained by GS-MIP(30%) is the same for both methods B&B and Barrier (we are comparing averages but all the solutions had the same value for both methods). Worthwhile to mention is that the Barrier method for GS-MIP(30%) is faster than the B&B for the larger instances in both problems. In general, the times are reasonable for the planning problem we consider in this work. Since multi-nlp considers in this work two periods and stoch-nlmip has four scenarios for the demand, it

14 14 is a normal behavior that stoch-nlmip is more time consuming than multi-nlmip. In Table II we present the results of the implementation of GS-MIP(γ%) for the medium instances of multi-nlmip when the value of γ and ɛ vary for the B&B and the Barrier methods. The first column shows the gap precision γ, followed by the variation in the accuracy of the golden section method ɛ. Then there are two sections, one for GS-MIP(γ) with the B&B and the other for the Barrier method. First column of each section is the average number of iterations done by GS-MIP(γ). Second column is the average value of the acquisition price L, third column is the average objective function while last one is the execution time in seconds. The average values of the iterations, L, and objective function are only for showing similarities or discrepancies. Variations GS-MIP(γ) with B&B GS-MIP(γ) with Barrier γ% ɛ Iter L Objective Time Iter L Objective Time Table II: Variations of gap precision γ and in the golden section search precision epsilon for medium size instances of multi-nlmip. We can notice that the Barrier method overcomes the B&B one in time without reducing the solution quality. It is clear that the convergence of the B&B is slower when the gap is close to the optimum but when we give it some flexibility, the B&B is faster than the barrier method. We can also observe that the computation times improve as we reduce the

15 15 (5,10,40) (5,10,40) Objective Objective L value L value Figure 3: Impact of the fixed cost when they are set independently of the size of the instance (leftt panel) and when they are size dependent of the instance as we propose (right panel) for multi-nlmip on a small size instance. precision of the golden search. Moreover, it is notable that the quality of the solutions are at an acceptable level even if we allow some flexibility. For purposes of finding a high quality solution in an efficient time we suggest then to use the combination of accuracy of 30% of gap with ɛ =0.1 for the golden section method. Figure 3 shows the values of the recovery price L on the x-axis and the value of the objective function of multi-nlmip(l) on the y-axis for a small instance when using the fixed cost we propose (21) (right panel) and when using the independent fixed cost values of[? ] (left panel). The continuous line is obtained by partitioning the range of the L into 50 equidistant points and executing a the branch-and-bound with a gap optimality close to 0. The dots are the solutions obtained by the GS-MIP(30%). We can observe the convex shape of the objective function when using the fixed cost we propose that are dependent on the size of the instance. This convex shape strongly helps the GS-MIP(30%) since it is easier to avoid local optimums. The jumps on the objective function are related to the decisions of expanding a plant with a remanufacturing module, or with the decisions of opening a new inspection or distribution center. Figure 4 is analogous to Figure 3 but for medium size instances. We notice the same behavior. We perform the same analysis on the behavior of the objective function for the stoch-

16 16 Objective (10,20,60) Objective (10,20,60) L value L value Figure 4: Impact of the fixed cost when they are set independently of the size of the instance (right panel) and when they are size dependent of the instance (left panel) for multi-nlmip on a medium size instance. (5,10,40) (5,10,40) 8000 Objective Objective L value L value Figure 5: Impact of the fixed cost when they are set independently of the size of the instance (left panel) and when they are size dependent of the instance (right panel) for stoch-nlmip on a small size instance. NLMIP model just to validate that we have the same behavior as before. It can be observed in Figures 5 and 6.

17 17 (10,20,60) (10,20,60) Objective Objective L value L value Figure 6: Impact of the fixed cost when they are set independently of the size of the instance (left panel) and when they are size dependent of the instance (right panel) for stoch-nlmip on a medium size instance. CONCLUSIONS We focus on a reverse logistics network design problem where distribution centers, inspection centers and remanufacturing facilities must be located, the acquisition price as well as the amount of returned goods to be collected depending on the unit cost savings and competitor s acquisition price must be determined. Moreover, we introduce the multiple periods setting and stochastic demand formulated by scenarios. We develop two mathematical programming models to determine the pricing strategy of the recovered products together with the optimal network that must be designed to be the most profitable closed cycle. We adapt a Golden Section Search with some flexibility that enable us to fix the used product acquisition price and then solve the model as an integer linear programming. Moreover, we establish dependent size fixed costs of opening a distribution, an inspection, and a remanufacturing centers and show that they have a strong impact on the Golden Section search behavior. Experimental results show that the execution times are reasonable for our methodology and find that the Barrier method of the integer linear programming solver is competitive in many instances.

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