Location and selective routing problem with pricing for the collection of ...

Location and Selective Routing Problem with Pricing for the Collection of Used Products Necati Aras

Deniz Aksen

Tu˘grul Tekin

Department of Industrial Eng. Bo˘gazic¸i University 34342, ˙Istanbul, Turkey Email: [email protected]

College of Administrative Sciences and Economics Koc¸ University 34450, Sarıyer, ˙Istanbul, Turkey Email: [email protected]

Department of Industrial Eng. Bo˘gazic¸i University 34342, ˙Istanbul, Turkey Email: [email protected]

Abstract—One of the key concerns of the companies involved in product recovery is used product (core) acquisition. In this paper, we consider a firm that aims to locate facilities which serve as collection centers (CCs) for the cores to be collected from the dealers. The dealer addresses as well as the amount of cores piled up at each dealer are known. Also, each dealer has a reservation price for the cores, and if the offered acquisition price by the firm is less than this reservation price, the dealer will not return its cores. The collection operation is performed by vehicles that must start and end their routes at the same CC without visiting another one in the route. The objective of the firm is to maximize its profit by determining the locations of the CCs, the acquisition price offered for each core, the number of vehicles allocated to each opened CC and the route of each vehicle. The source of the revenue is the cost savings that results from using the components of the cores in remanufacturing as-good-as-new products. For the solution of this problem, we develop a mixedinteger linear programming formulation, which is an extension of the classical location-routing problem where (i) there is a profit associated with each dealer visited and not all dealers have to be visited, and (ii) an acquisition price has to be paid to the dealers for each core. Since this problem is N P-hard, we develop a tabu search heuristic for solving large instances.

I. I NTRODUCTION Environmentally conscious manufacturing, waste reduction and product recovery have emerged as alternative means of coping with the significant societal problem of environmental sustainability. This has motivated both the researchers and practitioners to focus on closed-loop supply chains, which comprise both the traditional (forward) supply chain and the reverse supply chain. The management of the reverse supply chain, which is called reverse logistics, consists of the collection, inspection, classification and proper recovery of the products used by consumers. One aspect of effectively handling the reverse logistics activities is to solve the network design problem, which is concerned with determining the number and locations of facilities to be opened and finding the amount of product flows between these facilities. In some of the models, the location decisions are made only for facilities in the reverse distribution channel performing the collection, inspection, and remanufacturing operations. There are also papers which deal with both the forward and reverse distribution channels in a coordinated way, where decisions are made not only for the facilities in the reverse channel, but also for those that perform

operations for the forward flow of products. A recent and comprehensive review of the network design papers within the reverse logistics context can be found in [1] and [5]. One of the key concerns of the companies involved in product recovery is used product acquisition (collection) as mentioned in [8]. It is indeed the first activity of product recovery, and triggers the other activities of the recovery system. The quantity and quality of returns can be increased by using buy-back campaigns and offering financial incentives to product holders. Accepting all end-of-use products in the waste stream is not a viable strategy for most companies since a high percentage of these will have poor quality, and hence will not be recoverable. As a consequence, adopting a proactive approach and implementing a used product acquisition strategy by offering an appropriate incentive is crucial for a company engaged in product recovery. There are only a few papers which incorporate the core acquisition explicitly in network design problems formulated as facility locationallocation models. An uncapacitated collection center location problem (CCLP) for incentive- and distance-dependent returns is analyzed in [3]. The product holders’ decision whether or not to participate in this buy-back campaign is affected by the distance traveled to the nearest collection center (CC) and the financial incentive that depends on the quality of the core. Two mixed-integer nonlinear programming models are proposed for the fixed-charge and p-median versions of the CCLP. The p-median version of the same CCLP is considered in [4] under the pickup collection policy in which dedicated vehicles depart from a CC, visit a single customer zone, and bring the collected cores back to the CC. The aim is to determine the locations of the CCs, the level of the financial incentive as well as the number and load mix of the vehicles. In both papers, the authors utilize tabu search-based solution procedures. In the present paper, we extend this last work by relaxing the restriction that a vehicle visits only a single customer (dealer) zone and incorporating the routing of the vehicles. Hence, the problem we tackle becomes a special type of the locationrouting problem (LRP) in which the acquisition price paid by the firm to take back cores is also a decision variable. Furthermore, it is not necessary to visit all the customer zones. We refer to this problem as the location and selective routing problem with profits (LSRPP). Since the LRP is known to be

N P-hard [9], and it is a special case of the LSRPP in which all dealers are visited and there is no acquisition price involved, the LSRPP is also N P-hard. This implies that large LSRPP instances are unlikely to be solved to global optimality by commercial mixed-integer linear programming (MILP) solvers within reasonable computation times. Therefore we develop a heuristic solution procedure based on tabu search. II. LITERATURE SURVEY A recent review of the LRP can be found in Nagy and Salhi [9]. This newest literature survey reviews and classifies past studies based on four criteria: hierarchical structure, type of input data, planning period and solution method. It should be noted that to the best of our knowledge there is no study within the LRP context where only some of the customers are visited and pricing is involved. Therefore, in this section we review the most recent studies on the classical LRP. Barreto et al. [6] solve the capacitated multi-depot LRP (CMDLRP) with a sequential clustering analysis-based heuristic algorithm. This algorithm uses several hierarchical and non-hierarchical grouping techniques and proximity measures. The same problem is considered by Prins et al. [10] who develop a memetic algorithm with population management. In another study, the same authors devise a greedy randomized adaptive search procedure complemented by a learning process and path relinking [11]. They also propose a Lagrangian relaxation method hybridized with a granular tabu search heuristic to yield a cooperative metaheuristic with information exchange between location and routing phases [12]. Marinakis and Marinaki [13] introduce a genetic algorithm to solve the bilevel programming-based formulation of the CMDLRP for a wood products distribution company. For the same problem, Marinakis and Marinaki [14] develop a hybrid metaheuristic based on a particle swarm optimization algorithm with path relinking. Wang et al. [15] propose an integrated two-phase hybrid metaheuristic based on a tabu search and ant colony system for the multi-depot LRP (MDLRP). A very large neighborhood search technique for the MDLRP with regional fleet assignment is considered in [2]. A tabu search-based adaptive memory procedure is outlined in [7] for solving the multiobjective MDLRP. A tabu search-based adaptive memory procedure is developed in [7] for the same problem and an application to a real-world problem is reported for the region of Andalucia in Spain. Yu et al. [16] focus on solving the LRP with capacitated depots and routes. The proposed simulated annealing-based heuristic is shown to be competitive with other well-known algorithms. III. THE LOCATION AND SELECTIVE ROUTING PROBLEM WITH PROFITS We can define the LSRPP as follows. A firm engaged in reverse logistics wants to locate facilities with unlimited capacity which will serve as CCs for the cores that can be collected from the dealers. The locations of the dealers as well as the amount of cores piled up at these dealers

are known. Moreover, each dealer has a reservation price (minimum amount of money he would be willing to receive) per core, and if the offered acquisition price by the firm is less than this reservation price, the dealer will not return any core. It is important to note that the acquisition price offered by the firm is the same for all dealerships since a pricing policy discriminating with respect to dealers would not be perceived as a fair attitude. Besides, dealers could easily share among each other the pricing information of the firm. This, in turn, would render a price discrimination policy impractical. Each dealer can only be visited by a single vehicle, hence split pickups are not permitted. All vehicles are identical with respect to load capacity and speed. They must start and end their routes at the same CC without visiting any other CC in the route. The objective of the firm is to maximize its profit by determining the locations of the CCs, the acquisition price offered for each unit of core collected from the dealers, the number of vehicles allocated to each opened CC and the route of each vehicle. Note that it is not necessary to visit all dealers. The source of the revenue is the cost savings that results from using the components of the cores in remanufacturing like-new products. For the solution of this problem, we develop an MILP formulation. Let ID and IC be the set of the dealers and candidate CC locations, respectively. Also let I = IC ∪ ID. The parameters are defined as follows: g=the revenue from each core, ai =the number of cores at dealer i, fi =the fixed cost of opening a CC at site i, c1 =unit vehicle operating cost, c2 =cost per unit distance traveled, q=vehicle capacity, ri =reservation price of dealer i per core, dij = distance between sites i and j. The decision variables are given as follows. Yi =1 if a CC is located as site i, it is zero otherwise; Xij =1 if dealer j is visited after dealer i, it is zero otherwise; Aik =1 if dealer i is assigned to CC k, it is zero otherwise; R=the acquisition price for each core collected, Ui =the load of the vehicle departing from site i (this variable is used to eliminate the subtours), Bik =auxiliary variable. The MILP formulation of the LSRPP is then obtained as

max

 

(ai gAik − ai Bik ) − c1

i∈ID k∈IC

− c2

 i∈I j∈I

 

Xij

i∈IC j∈ID

dij Xij −



fi Yi

(1)

i∈IC

subject to 

Xij ≤ 1

i ∈ ID

(2)

Xji ≤ 1

i ∈ ID

(3)

j∈I

(4)

j∈I,j=i



j∈I,j=i



i∈I,i=j

Xij =

 i∈I,i=j

Xji

q

 

 

Xji ≥

j∈IC i∈ID

ai Aik

(5)

k∈IC i∈ID

Ui − Uj + qXij + (q − ai − aj ) Xji ≤ q − aj ai ≤ Ui

i = j ∈ ID i ∈ ID

(6) (7)

q ≥ Ui Aik ≤ Yk  Aik ≤ 1

i ∈ ID i ∈ ID, k ∈ IC

(8) (9)

k∈IC



Xij ≥

j∈I,j=i





i ∈ ID

(10)

Aik

i ∈ ID

(11)

Aik

i ∈ ID

(12)

k∈IC

Xji ≥

j∈I,j=i



k∈IC

Xik + Aij − Akj ≤ 1 Xik + Akj − Aij ≤ 1 Xji ≤ Aij Xij ≤ Aij  Aik R ≥ ri

i = k ∈ ID, j ∈ IC (13) i = k ∈ ID, j ∈ IC (14) i ∈ ID, j ∈ IC i ∈ ID, j ∈ IC

(15) (16)

i ∈ ID

(17)

k∈IC

Bik ≥ R + max {rj }Aik j∈ID

− max {rj } j∈ID

Bik ≤ max {rj } j∈ID

i ∈ ID, k ∈ IC (18) i ∈ ID, k ∈ IC (19)

Xij , Aik , Yik ∈ {0, 1} Ui , Ri , Bik ≥ 0 Constraints (2) and (3) ensure that each dealer is visited at most once. Constraints (4) guarantee that the number of vehicles entering and leaving a site is the same. Constraint (5) makes sure that the total number of vehicles is sufficient to carry all the collected cores. Constraints (6), (7), and (8) are the subtour elimination constraints of Miller-Tucker-Zemlin type. Constraints (9) are used to assign a dealer to an open CC only. Constraints (10) ensure that a dealer can be assigned to at most one CC. Constraints (11) and (12) imply that if a dealer is assigned to a CC, then a vehicle must visit that dealer. Constraints (13) and (14) guarantee the assignment of two dealers to the same CC if they are visited consecutively by a vehicle. Constraints (15) say that if a dealer is visited right after a CC, then that dealer must be assigned to this CC. Constraints (16) are similar in nature to constraints (15). They ensure that if a dealer is visited right before a CC, then that dealer must be assigned to this CC. Constraints (17) simply make sure that the acquisition price must be at least equal to the reservation price of the visited dealers. Constraints (18) and (19) are used for defining auxiliary variable Bik that is utilized to linearize the nonlinear term R · Aik involving the product of continuous variable R and binary variable Aik that appears in the objective function.

Since LSRPP is N P-hard, large instances can only be solved by metaheuristic procedures within reasonable times. Therefore, we develop two tabu search (TS) heuristics for this problem. IV. SOLUTION PROCEDURE TS performs an exploration of the solution space by moving from a solution St identified at iteration t to the best solution St+1 in a subset of the neighborhood of St . Since St+1 does not necessarily improve upon St , a tabu mechanism is put in place to prevent the process from cycling over a sequence of solutions. This is done by declaring some attributes of the past solutions or past moves as tabu such that any solution or move possessing these attributes may not be considered for the next κ iterations. This mechanism is often referred to as short term memory with the number κ called tabu duration or tabu list size. Other features such as diversification and intensification are often implemented. The purpose of diversification is to ensure that the search process will not be restricted to a limited portion of the solution space. It keeps track of past solutions and penalizes frequently performed moves. This is often called long term memory. Intensification consists of performing a greedy local search around the best known solution. The proposed TS heuristics are called TSH and TSHLPO. Both of them use strategic oscillation, i.e., solutions in which one or more routes violate the vehicle capacity constraint are also admitted. We refer to these as capacity infeasible solutions. In calculating the objective value of such a solution, we multiply the total excess capacity (the amount of demand carried in excess of vehicle capacity summed over all infeasible routes) with a dynamically changing penalty, and subtract the resulting infeasibility cost from the total profit. At each iteration, both of the heuristics explore a number of neighboring solutions of the current solution including the infeasible ones and selects the one with the highest objective value as the new current solution (recall that LSRPP has a maximization objective). The critical components of the heuristics are the following: (i) The initial solution generation method; (ii) The neighborhood structures and associated tabu conditions; (iii) The selection of a neighboring solution as the new current solution; (iv) Updating the penalty value for strategic oscillation; (v) The termination conditions. A. Generating an initial solution The initial solution is obtained by putting all dealers to a single route in a random fashion. This route is then assigned to a randomly chosen CC. Note that this initial solution is almost certain to be infeasible; thus, its objective value will bear a significant penalty cost for the excess capacity usage of the vehicle. B. Neighborhood structures and tabu conditions The neighborhood structures (moves) of TSH and TSH-LPO affect the trajectory of the solutions which are designated as the current solution during the iterations. There exist two types of moves in our implementation: routing moves used to change

the routes of the vehicles and dealer selection moves employed to determine the set of visited dealers. The six routing moves are 1-0 Move, 1-1 Exchange, 2-2 Exchange, 2-Opt, 1-Split, and Inter-route Exchange (IrE). The dealer selection moves are 1-Add, 2-Add, 1-Drop, and 2-Drop. Below we explain each move separately. 1-0 Move: Given two dealers, the first one is removed from its current position and is inserted after the second one. 1-1 Exchange: Given two dealers, they swap their positions. 2-2 Exchange: Given two dealers, the first one and its successor swap their positions with the second one and its successor. 2-Opt: Given two dealers, if they are on the same route, then the two arcs connecting them with their successors are removed, the dealers are connected, their successors are connected, and finally the chain between the successor of the first dealer and the second dealer is reversed. If the given dealers are on different routes, the respective chains following the dealers are swapped. 1-Split: Given a dealer on an existing route, the chain from the CC until and including that dealer is retained, whereas the rest of the route is introduced as a new route. The new route can either start at the same CC or it can be assigned to another CC. 1-Split neighborhood always increases the total number of routes by one. IrE: There are three cases of this neighborhood structure. Case a. A chain of dealers is extracted from a given route and inserted as a new route for the same or some other CC. Like 1-Split move, this case of IrE increases the number of routes by one. Case b. A chain of dealers in a given route is moved from its current position into another route that belongs either to the same or a different CC. Case c. Two dealer chains of arbitrary sizes which belong to two different routes are swapped. 1-Add: Given a dealer not included in the routes, all possible insertion positions are examined, and the dealer is inserted in the best possible position. The best possible position is either the one promising the highest increase in the current solution’s objective value (total profit), or if no such position exists, then it is the one which decreases the objective value the least. 2-Add: Given two dealers not included in the routes, all possible insertion positions are examined and the given dealers are inserted successively in the best possible position as defined for the Add-1 move. 1-Drop: Given a dealer included in a route, it is removed from the route and its predecessor and successor are connected. 2-Drop: Given a dealer included in a route, the dealer and its successor are removed from the route, and the predecessor of the given dealer is connected to the successor of its successor. The heuristics utilize tabu conditions to enforce exploration of different regions of the solution space. To this end, whenever the solution is updated according to a certain move, a tabu condition is created associated with that move. It is forbidden to override the tabu condition for the next κ iterations (tabu duration) unless an immediate improvement in the best feasible objective value Z best is realized. The tabu condition is lifted

at the end of the tabu duration κ. In our algorithm, κ is randomly determined in the interval [7, 24] independent of the problem size. The tabu conditions for the six routing moves are listed below, while those for the dealer selection moves are not given since they are trivial (i.e., the added dealer(s) cannot be dropped from the solution, and the dropped dealer(s) cannot be added to the solution). 1-0 Move: Dealer whose position is changed cannot be relocated by 1-0 Move. 1-1 Exchange: Dealers who switch their positions cannot be re-swapped by 1-1 Exchange. 2-2 Exchange: Dealers who were chosen for switching their positions in the 2-2 Exchange cannot be re-swapped by this move. However, it is allowed to re-swap their successors who were also involved in the 2-2 Exchange. 2-Opt: Dealers involved in the 2-Opt move cannot be used together in another 2-Opt move. 1-Split: Dealer chosen for the splitting position of the 1Split cannot be selected again in another 1-Split move. IrE: Dealer(s) constituting the starting node(s) of the relocated (exchanged) chain(s) cannot be selected again in another move of Inter-route Exchange. C. Selecting a neighboring solution as the new current solution In each iteration, the current solution’s neighborhood N (St ) is searched for the new current solution St+1 by fully exploring all moves except IrE. In other words, the candidate subset Cand N (St ) of N (St ) that is to be explored exhaustively and N (St ) itself are equivalent in five of the six routing moves. The neighborhoods with respect to Case (b) and Case (c) of the IrE move are only partially generated and explored due to their computational burden. The neighborhood of Case (a), on the other hand, is fully explored. In fact, IrE is only then performed when the other moves cannot provide a better neighboring solution Pt than the current solution St , where the basis of comparison is the objective function value inclusive of the infeasibility penalty. To better understand the partial exploration procedure in the last two cases of IrE, we should first give the following three definitions: The distance between a node and a route: The distance / T , is the distance bebetween a node N1 and a route T , N1 ∈ tween N1 and the node N2 ∈ T which is closest to N1 among all nodes in T . We denote this distance by Dist(N1 , T ). The closest route to a node: The closest route to a node N1 is that particular route T not containing N1 whose distance to N1 , namely Dist(N1 , T ), is the smallest among all other routes not containing that node. We denote the closest route / T ∗. to N1 by T ∗ (N1 ) where N1 ∈ The closest route set of a route T: It is the set of the routes closest to each individual node in T . We denote the closest route set of T by T (T ). For example, let route T contain only three nodes, N1 , N2 , and N3 . Let the closest route of N1 be T  and that of both N2 and N3 be T  . Then, T (T ) will be {T  , T  }. Note that T  being in T (T ) does not imply that T is going to be in T (T  ).

With the above definitions, it is possible to explain the partial exploration procedure of the last two cases of IrE. Given a route T from which a chain is selected, the routes to be examined are only the ones in the closest route set of T , namely T (T ). This is, it is sufficient to determine and check the routes in the set T (T ) for each individual route T ∈ St . There is no need to consider the other routes not in T (T ), when Case (b) and Case (c) neighborhoods of the IrE move are explored for a particular route T in the current solution St . Moreover, in Case (c), if two routes T1 and T2 were considered before while the IrE neighborhood for T1 was being explored, then checking this couple again during the exploration of the neighborhood for T2 is unnecessary, and should be avoided to save computational time. A final note should be made regarding the diversification rule of the heuristics. It works as follows. While we explore the candidate neighborhood solutions, not only do we record the best neighboring solution Ptbest , but also the worst one having the lowest total profit minus the infeasibility penalty cost. Let Ptworst denote the worst solution in the neighborhood. Now, if Ptbest is ineffective meaning that it bears exactly the same route profit and infeasibility penalty cost as the current solution St , then the new current solution St+1 is set to Ptworst . If Ptbest is not ineffective, however, then St+1 is set to Ptbest as usual. The jump from St to Ptworst instead of Ptbest is made in a sense to diversify the search space of our algorithm. By this diversification scheme we intend to break the possible cycling among multiple local optima all having the same objective value. D. Updating the penalty value for strategic oscillation The penalty value policy for the strategic oscillation aims to have as much an equal number of visits to feasible and infeasible solutions as possible in the hope of having enough diversity while exploring the solution space. Too high a penalty value may prevent the algorithm from visiting infeasible solutions at all, as a result of which S best may be trapped at some local optimum. On the other hand, too low a penalty value may fail to identify any feasible solution. Therefore, the value of the penalty is to be changed dynamically to adapt to unequal number of feasible and infeasible solutions. For a given number of iterations (penaltyControlCount), the number of infeasible solutions visited (numInfeasible) is kept track of. If that number is greater than 60% of penaltyControlCount, then the penalty value (penaltyValue) is increased. If it is less than 40% of penaltyControlCount, then the penalty value is decreased. Each time penaltyValue gets updated, numInfeasible is reset to zero. E. Termination conditions We terminate the heuristics as soon as any of the following two types of termination conditions is met: (i) The maximum number of non-improving iterations (MaxNonimprovIter), and (ii) The time limit (CPUtimeLimit). The only difference between TSH and TSH-LPO is the application of local post optimization (LPO) in the latter.

Whenever the incumbent solution S best is updated, we apply LPO to improve it, which is implemented with the moves 1-1 Exchange, 1-0 Move, 2-Opt, 2-2 Exchange and again 2-Opt respecting this order. Each move is applied repeatedly as long as it produces an improvement in the incumbent solution. Finally, notice that the acquisition price offered per core is determined according to the maximum of the reservation prices of the visited dealers. This is, R = maxi:i is visited {ri }. In the computation of the objective value of any solution, this value of R should be used. V. C OMPUTATIONAL R ESULTS Here we first describe how we generated random problem instances and give information about the parameter selection in the TSH and TSH-LPO heuristics. Then, we report the results of our computations which include performance comparisons among three solution methods: solving the mathematical model by CPLEX 11.2, TSH, and TSH-LPO. By assigning five distinct values to the number of candidate CCs (|IC|=1,2,3,4,5) and ten distinct values to the number of dealers (|ID|=10,20,30,40,50,60,70,80,90,100), we obtain 40 instances where each instance is labeled as (|IC| , |ID|). The (x,y) coordinates of the candidate CC sites and dealer locations are sampled independently from a discrete uniform distribution in the interval [0, 500]. The travel distances dij between locations are calculated using the Euclidean distance. The reservation price ri of each dealer is sampled from a uniform random variable in [2.5, 7.5]. The number of cores ai at each dealer is generated from a discrete uniform distribution in the interval [5, 15]. The fixed cost fi of opening a CC at site i also follows the same distribution in the interval [150, 200]. The fixed vehicle operating cost c1 is equal to 100 and the variable vehicle cost c2 per distance traveled is taken as one. The vehicle capacity q is set to 5-fold of the number of cores at the dealer with the maximum number of cores, i.e., q = 5 maxi {ai }. The revenue g per unit core is equal to 15. It is worth mentioning that in generating the random instances we ensured not to obtain instances whose best solution is to visit all the dealers and collect all the cores. TSH and TSH-LPO were coded in Java and compiled with Java Server Virtual Machine on a computer having 16 GB of RAM and two Intel Quad Core Xeon 3.2 GHz processors. TSH and TSH-LPO were implemented with the following parameter values: penaltyControlCount=100, penaltyValue=0, penaltyStepSize=1, MaxNonimprovIter=2000, CPUtimeLimit=3 hr. CPLEX is also allowed a time limit of three hours for solving the mathematical model. The results are presented in Table I where “LB” stands for the best feasible solution provided by CPLEX, TSH, and TSH-LPO. This value constitutes a lower bound for the optimal profit for each instance considered. CPLEX can only give an optimal solution for the first two instances, which is indicated by an asterisk. Results indicate that TSH and TSH-LPO outperform CPLEX and yield better solutions within shorter CPU time in most of the instances. When we compare the average profit values, we observe that TSH-LPO provides on average a higher profit than TSH.

TABLE I P ERFORMANCE OF THE TSH AND TSH-LPO HEURISTICS

Inst. (1,10) (1,20) (2,20) (1,30) (2,30) (3,30) (1,40) (2,40) (3,40) (4,40) (1,50) (2,50) (3,50) (4,50) (5,50) (1,60) (2,60) (3,60) (4,50) (5,60) (1,70) (2,70) (3,70) (4,70) (5,70) (1,80) (2,80) (3,80) (4,80) (5,80) (1,90) (2,90) (3,90) (4,90) (5,90) (1,100) (2,100) (3,100) (4,100) (5,100) Avg.

CPLEX LB 0∗ 200.1∗ 539.5 553.1 821.5 105.8 0.0 7.3 111.3 191.8 95.3 138.8 464.6 255.1 3.4 170.8 358.2 308.9 293.3 54.4 0.0 658.2 247.5 0.0 0.0 382.3 65.7 202.9 341.8 206.0 140.6 354.5 337.7 417.4 55.3 0.0 249.7 0.0 0.0 241.7 214.4

TSH LB CPU 0 0.2 193.8 1.6 666.6 1.3 790.3 6.1 919.6 4.2 137.1 3.1 0.0 4.2 82.6 5.6 150.0 6.2 229.4 7.5 197.5 9.6 221.0 9.0 549.6 9.6 523.3 9.8 189.0 9.8 234.8 12.8 611.2 15.0 572.0 24.5 424.5 12.3 433.2 15.0 501.8 23.2 536.3 25.1 1246.8 19.5 510.4 22.7 711.4 27.3 812.2 35.7 634.5 35.0 1140.6 65.7 1343.5 28.9 1685.1 56.9 1205.1 91.6 726.4 77.7 1402.2 43.3 1349.4 85.0 964.8 75.7 962.4 124.5 600.8 78.0 1057.8 144.1 1355.1 109.6 1385.0 57.2 681.4 34.9

TSH-LPO LB CPU 0.0 0.2 200.1 1.8 666.6 1.5 790.3 5.0 919.6 3.6 137.1 3.5 0.0 3.9 80.9 5.2 150.0 5.4 229.4 8.8 197.5 9.1 222.3 8.8 549.6 9.7 523.3 9.5 189.0 10.5 234.8 16.8 608.6 20.8 588.3 24.9 427.8 15.4 433.2 15.5 567.1 59.4 546.3 19.5 1280.0 19.2 506.0 26.5 711.4 27.2 848.9 35.6 311.8 43.0 793.1 48.7 1324.4 45.7 1695.2 29.5 1064.3 47.4 833.6 42.1 1390.0 43.3 1455.0 82.5 1121.9 91.7 1009.2 80.2 951.4 86.4 1779.3 83.0 1341.4 75.9 1405.9 61.0 702.1 30.7

However, there are also instances where TSH is better than TSH-LPO. ACKNOWLEDGMENT ¨ ˙ITAK (The We gratefully acknowledge the support of TUB Scientific and Technological Research Council of Turkey) under grant no. 107M258. R EFERENCES ¨ [1] E. Akc¸alı, S. C¸etinkaya, H. Uster, “Network design for reverse and closedloop supply chains: an annotated bibliography of models and solution approaches”, Networks, Vol. 53, No. 3, pp. 231–248, 2009. [2] D. Ambrosino, A. Sciomachen, M.G. Scutell`a, “A heuristic based on multi-exchange techniques for a regional fleet assignment location-routing problem”, Computers and Operations Research, Vol. 36, No.,pp. 442– 460, 2009. [3] Aras, N., D. Aksen, “Locating collection centers for distance and incentive-dependent returns”, International Journal Production Economics, Vol. 11, No. 2, pp. 316–333, 2008. [4] Aras, N., D. Aksen, A.G. Tanu˘gur, “Locating collection centers for incentive-dependent returns under a pick-up policy with capacitated vehicles”, European Journal of Operational Research, Vol. 191, No. 3, pp. 1223–1240, 2008.

[5] Aras N., T. Boyacı, V. Verter, “Designing the reverse logistics network, in Closed Loop Supply Chains: New Developments to Improve the Sustainability of Business Practices, Ferguson M., Souza G. (Eds.), CRC Press, 2010. [6] Barreto S, C. Ferreira, J. Paix˜a, B.S. Santos, “Using clustering analysis in a capacitated location-routing problem”, European Journal of Operational Research, Vol. 179, No. 3, pp. 968–977, 2007. [7] Caballero, R., M. Gonzalez, F.M. Guerrero, J. Molina, C. Paralera, “Solving a multiobjective location routing problem with a metaheuristic based on tabu search: application to a real case in Andalusia”, European Journal of Operational Research, Vol. 177, No. 3, pp. 1751–1763, 2007. [8] Guide, V.D.R., R. Teunter, L.N. van Wassenhove, “Matching demand and supply to maximize profits from remanufacturing”, Manufacturing and Service Operations Management, Vol. 5, No. 4, pp. 303–316, 2003. [9] Nagy, G., S. Salhi, “Location-routing: issues, models and methods”, European Journal of Operational Research, Vol. 177, No. 2, pp. 649– 672, 2007. [10] Prins, C., C. Prodhon, R. Wolfler Calvo, “A memetic algorithm with population management (MA—PM) for the capacitated location-routing problem”, In: Gottlieb J, Raidl GR (Eds.). Evolutionary computation in combinatorial optimization. (Lecture Notes in Computer Science 3906). Berlin/Heidelberg: Springer, pp. 183–194, 2006. [11] Prins, C., C. Prodhon, R. Wolfler Calvo, “Solving the capacitated location-routing problem by a GRASP complemented by a learning process and a path relinking”, 4OR, Vol. 4, No. 3, pp. 221–238, 2006. [12] Prins, C., C. Prodhon, R. Wolfler Calvo, “Solving the capacitated location-routing problem by a cooperative Lagrangean relaxation-granular tabu search heuristic”, Transportation Science, Vol. 41, No. 4, pp. 470– 483, 2007. [13] Marinakis, Y., M. Marinaki, “A bilevel genetic algorithm for a real life location routing problem”, International Journal of Logistics: Research and Applications, Vol. 11, No. 1, pp. 49–65, 2008. [14] Marinakis, Y., M. Marinaki, “A particle swarm optimization algorithm with path relinking for the location routing problem”, Journal of Mathematical Modelling and Algorithms, Vol. 7, No. 1, pp. 59–78, 2008. [15] Wang, X., X. Sun, Y. Fang, “A two-phase hybrid heuristic search approach to the location-routing problem”, In: Proceedings of IEEE International Conference on Systems, Man and Cybernetics, Vol. 4, pp. 3338–3343, 2005. [16] Yu, V.F., S.-W. Lin, W. Lee, C.-J. Ting , “A simulated annealing heuristic for the capacitated location routing problem”, Computers and Industrial Engineering, Vol. 58, pp. 288–299, 2010.