Multi-depot Vehicle Routing Problem with Pickup and Delivery Requests

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1 Multi-depot Vehicle Routing Problem with Pickup and Delivery Requests Pandhapon Sombuntham a and Voratas Kachitvichyanukul b ab Industrial and Manufacturing Engineering, Asian Institute of Technology, Klongluang, Pathumthani, THAILAND a pandhapon.s@gmail.com, b voratas@ait.ac.th Abstract. This paper considers a multi-depot vehicle routing problem with pickup and delivery requests. In the problem of interest, each location may have goods for both pickup and delivery with multiple delivery locations that may not be the depots. These characteristics are quite common in industrial practice. A particle swarm optimization algorithm with multiple social learning structures is proposed for solving the practical case of multi-depot vehicle routing problem with simultaneous pickup and delivery and time window. A new decoding procedure is implemented using the PSO class provided in the ETLib object library. Computational experiments are carried out using the test instances for the pickup and delivery problem with time windows (PDPTW) as well as a newly generated instance. The preliminary results show that the proposed algorithm is able to provide good solutions to most of the test problems. Keywords: vehicle routing problem, pickup and delivery, multiple depot, particle swarm optimization. PACS: 00 General INTRODUCTION The delivery of products from depots to customers is one of the key activities that play an important role in the effectiveness of business. In general, there is a set of customers to be served by a set of vehicles from a depot. The objective is to determine the optimal sequence of customers visited by each vehicle, called vehicle route, which satisfies such criteria as distance, time, and cost involved in the operation. This problem is generally known as vehicle routing problem (VRP). Many variants of VRP are studied to address the variety of conditions in real world applications. For example: the capacitated VRP (CVRP), the VRP with time windows (VRPTW), the heterogeneous fleet VRP (HVRP), the VRP with pickup and delivery (VRPPD), and multiple depots VRP (MDVRP). VRPPD is the generalized version of VRP which involves not only the deliveries but also the pickups of commodities from customers [1]. The VRPPD can be further subdivided into delivery-first and pickup-seconds, mixed pickups and deliveries, and simultaneous pickups and deliveries (see [1].) The pickup and delivery problems (PDP) were classified into three different groups based on the pickup-delivery relation [2]. The first is many-to-many problem. This is the case that any node can serve as a

2 source or destination for any commodity. A commodity may be picked up from one of many locations, and also delivered to one of many locations. The second is one-tomany-to-one problem. In this case, commodities are initially available at a depot and must be delivered to the customers, and, in addition, the commodities available at the customers must be delivered to the depot. This characteristic is usually found in many VRPPD studies. The last is one-to-one problem. Each commodity, can be called a request, has a given origin and destination. The variants of VRP and PDPs involve many constraints and are known to be NP-hard problems which consume a great deal of computational time to find optimal solutions for large problems. The heuristic approach such as tabu search algorithm, ant colony optimization, genetic algorithm (GA), and particle swarm optimization algorithm (PSO) are commonly adopted to solve these problems. These methodologies do not guarantee optimal solutions, but they generally promise a near optimal solution within a reasonable solution time. An approach called reactive tabu search was developed in [3] to solve PDP with time window (PDPTW). Some classical heuristics for Traveling Salesman Problem (TSP) were adapted to deal with PDP which has a single vehicle [4]. Three sets of heuristics algorithm were developed to deal with the VRPPD which cover the case of mixed and simultaneous pickup and delivery problem which can be viewed as one-tomany-to-one PDP [1]. An adaptive large neighborhood search (ALNS) heuristics to handle PDPTW was proposed in [5] to minimize total distance; times spent by each vehicle, and to maximize fulfilled demand. In each cycle during iteration, three removals and two insertions are used to rearrange some of the requests. A real-value version of PSO for solving CVRP, VRP with simultaneous pickup and delivery, and VRPTW were proposed in [6], [7], and [8]. The PSO algorithm used a solution representation which is consisted of the priority list of customer and its preferred vehicle. The particle is converted into the problem specific solution through the decoding procedure. The studies in [6], [7], and [8] demonstrated through the use of benchmark test problems that PSO can be very effective for solving generalized vehicle routing problems. Problem Description In many study, a customer is usually refers to as a place that must be served by a vehicle from a depot. A more general term location is used here instead of customer as a location could have both pickup and delivery requirements. The literatures regarding VRPPD and PDP are usually focused on the case for which each location, excluding depot, has only one destination for the commodities picked up at its location. It can also be found that many studies in VRP with pickup and delivery (VRPPD) had made rigid assumptions on the destination of the pickup commodities. The products are generally destined to be brought back to the depot. For PDP, a location is usually either a pickup or delivery location of a request. However, in real-world situation, a certain location can be a pickup location, a delivery location or both. Moreover, goods pickup from that location can be destined to several other locations as shown in FIGURE 1. This situation is very common in cluster of Small and Medium-sized Enterprises (SME) where the firms may not have sufficient fund to invest in

3 transportation resources and it is essential to form a business alliance to pool their resources. The common resources they may share are the fleet of vehicles which are used to transport goods among their alliance members. As a result, some locations may act like depots without owning fleet of vehicles. Pickup and Delivery locations of each requests Vehicle Routes Depot Location Request (From-To) Vehicle Route FIGURE 1. The VRP with many-to-many requests. This study presents an approach to handle the practical case of pickup and delivery of goods as described earlier. Each location can have many associated requests destined to be delivered to various locations. Additional features considered include time window, heterogeneous fleet of vehicles, and multiple depot characteristics. Hence, the problem is denoted here as the generalized vehicle routing problem for multi-depot with pickup and delivery requests (GVRP-MDPDR). The objectives are to minimize both the total distance and the number of vehicles used while simultaneously maximizing the number of fulfilled requests. The proposed method is based on the particle swarm optimization algorithm GLNPSO which is a version of PSO algorithm with multiple social learning structures [10, 11]. The remainder of this paper is organized as follow. First, the problem formulation is given in terms of input variables, decision variables, and mathematical model. Second, the PSO-based algorithm based on [7], [10], and [11] is given via the discussion of particle representation, encoding and decoding method, route construction procedure, and procedure to reduce the number of required vehicles. Third, computational experiments using benchmark PDPTW problems from Li and Lim [12] are discussed to demonstrate the effectiveness of the proposed algorithm. Finally, the results of this study are summarized along with suggestions for further research. PROBLEM FORMULATION The objectives considered in this study are the total distance, the number of vehicles used, and the number of fulfilled request. Vehicle routes are formed in such a way that (1) the total routing cost is minimized; (2) the number of vehicle is minimized; (3) the fulfilled demand is maximized.

4 In addition, the following restrictions must be met: (4) each request is served exactly once by a vehicle; (5) the load of a vehicle never exceeds its capacity; (6) each route starts and ends at the indicated terminals; (7) the number of vehicles used do not exceed maximum number of available vehicles; (8) the total duration of each route (including travel and service time) does not exceed a preset limit. The mathematical model for the problem is based on the model found in [5]. The main differences lie in the objective function and all vehicles are allowed to serve any requests if the assigned request does not exceed the vehicle s capacity. Moreover, the location can have multiple requests, i.e., pickup nodes and delivery nodes can share the same x-y coordination. The model given below served as a formal description of the problem. Input Parameters P : set of pickup nodes, {1 n}. D : set of delivery nodes, {n n}. N : set of all pickup and delivery nodes N= P D H i : a penalty cost if the request i is not served, i P. K : set of all vehicles, K = m. C k : capacity of vehicle k K. f k : fixed cost of vehicle k K if it is used g k : variable cost per distance unit of vehicle k K : nodes that represents the start station of vehicle k, k K. : nodes that represents end station of vehicle k, k K. V : set of all nodes. V = N { } { } A : set of (i, j) which is an arc from node i to node j,where i, j V. d ij, t ij : distance and travel time between node i and node j,for i and j N. Travel times satisfy the triangle inequality; t i j t il +t lj for all i, j, l V; and are nonnegative. s i : fixed service time when visiting node i. e i : variable service time per item units of node i. [a i,b i ] : time windows when the visit at the particular location must start; a visit to node i can only take place between time a i and b i l i : the quantity of goods to be loaded onto the vehicle at node i for i P and l i = l i n for i D. In this mathematical formulation, request i is represented by nodes i and i+n,where P and D, and any nodes may have same x-y coordinate as it is possible for a node to be both pickup and delivery location in this study.

5 Decision Variables The main decision variables in the model are described below: x ijk : a binary variable with the value 1 if vehicle k travels from node i to node j and is zero otherwise, where i, j V, k K. S ik : a nonnegative integer that indicates when vehicle k starts the service at location i, i V, k K L ik : a nonnegative integer that is an upper bound on the amount of goods on vehicle k after servicing node i, where i V, k K. S ik and L ik are defined only when vehicle k actually visits node i. z i : a binary variable that indicates if request i is placed in the request bank, where, i P. The value is one if the request is placed in the request bank and zero otherwise. The Mathematical Model The mathematical formulation is given below: Minimize (1) Subject to: (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

6 (15) (16) In the objective function, are the weights that reflect the relative importance of each cost component. The objective is to minimize the weighted sum of the distance traveled, the sum of the fix cost for each used vehicle, and the penalty cost associated with number of requests not scheduled. Equation (2) ensures that each pickup location is visited or that the corresponding request is placed in the request bank. Equation (3) ensures that the delivery location is visited if the pickup location is visited and that the visit is performed by the same vehicle. Equations (4) and (5) ensure that a vehicle leaves every start terminal and a vehicle enters every end terminal. Together with equation (6) this ensures that consecutive paths between and are formed for each vehicle. Equations (7) and (8) ensure that S ik is set correctly along the paths and that the time windows are obeyed. These constraints also make sub tours impossible. Equation (9) ensures that each pickup occurs before the corresponding delivery. Equations (10), (11) and (12) ensure that the load variable is set correctly along the paths and that the capacity constraints of the vehicles are enforced. This study focuses on the case when one location has many associated items that must be shipped to several different locations. The pickup and delivery nodes of each request in the mathematical model are modeled separately and the constraints are not in integer-linear form. This makes it much easier to understand the problem description and the formulation. Since the problem will be solved via metaheuristic, the integer-linear form of the model may not be necessary. Moreover, the model can support many variants of VRP such as VRPTW, CVRP, HVRP, VRPPD, and PDPTW. PSO FOR THE PROBLEM Particle swarm optimization (PSO) is an evolutionary computation technique proposed by Kennedy and Eberhart [9]. In PSO, a solution of a specific problem is represented (directly or indirectly) as a particle positioned in an n-dimensional space. The main algorithm used in this paper is GLNPSO which is the variant of PSO algorithm with multiple social learning terms as proposed in [10] and [11]. The VRP specific part of the procedure is based on [6], [7], and [8]. In a PSO algorithm for VRP problem, the values of position, velocity, and personal best of each particle are first set in the initialization step. In each cycle during iteration, the position of each particle is decoded into the vehicle route and its fitness value is evaluated. The personal best and social learning terms will be updated prior to the updates of the velocity and position of each particle. The multiple social learning terms used in the GLNPSO algorithm are global best (gbest), local best (lbest), and near neighbor best (nbest). The algorithm repeats until the stopping criterion, which is generally a fixed number of iterations, is met.

7 Solution Representation The particle representation described here is based on [6], [7], and [8]. The particle is composed of two parts. For n locations and m vehicles problem, the particle will be a vector of n+2m elements. The first n dimensions of a particle represent the priority of locations. The next 2m dimensions represent the x-y coordinate of the orientation points of m vehicles. The point serves as the center of the area that the vehicle prefers to perform its duty. The vehicle with its orientation point closer to a location will be preferable when considering that location. This solution representation has the same structure as those in [6], [7], and [8] and is shown in FIGURE 2 below. FIGURE 2. Particle representation Decoding Method In this research, there are three steps to convert a particle into a solution of the problem. These steps are revised and updated from those given in [14]. In the first step, the decoding considers the position value of the first n dimension of a particle as the corresponding position values of locations. Then, the location indices are sorted based on its corresponding position value; i.e., the smaller value of the position, the higher priority of that location. The sorted list of customer indices is considered as the location priority list. In the second step, the position values of the next 2m dimension, n+1 to 2m, are considered as the vehicle orientation points. For each location in the location priority list, Euclidean distance between the location and the vehicle orientation points are computed. The vehicle indices are then sorted based on Euclidean distance in ascending order. The sorted list of vehicle indices is considered as the corresponding row for the location in the vehicle priority matrix. The third step is to construct the vehicle routes. The key idea is simply to insert a location into the sequence of visits for a vehicle such that it incurs as many requests assigned to that vehicle as possible. This part starts by selecting a pickup location with highest priority in location priority list. This pickup location is inserted into the route of the most preferred vehicle from the vehicle priority matrix. The destination of items picked from the location is also considered for insertion into the route. The pickup location being considered and the destinations of the items are inserted at the positions which lead to maximum number of assigned requests. A request which has not been served by a vehicle is denotes as an unfulfilled request. For a problem with m vehicle, and r requests, decoding Particle Position θ lh (position of l th particle at the h th dimension) into Vehicle Operation sequences, O lj,

8 and Route, R lj (operation sequence and route of j th vehicle and the l th particle); the steps are given in Algorithm SD1 below. Algorithm SD1 1. Construct Location Priority List( N) a. Build set S = {1, 2 n} and N =. b. Select c from set S where c. Remove c from set S. d. Repeat step 1.b until S =. 2. Construct Vehicle Priority Matrix (W) a. Set the vehicle orientation point. For j =1 m, set xref j = θ l,n+2j-1 and yref j = θ l,n+2j. b. For each location i, i=1,, n i. Calculate the Euclidean distance between location i and vehicle orientation points using the formula: ii. Build set S = {1, 2 m} and W i =. iii. Select c from set S where iv. Add c to the last position in set W i v. Remove c from set S. vi. Repeat step 2.b.iii until S =. 3. Route Construction Start with i = 0 and j = 0, where i = 0, 1, 2 n and j = 0, 1, 2 m a. Set p = N i,c = W ij, R c ' = R c, M c ' = M c, and U p ' = U p b. Set k = last position of the R c ' before the station depot. c. Make a new candidate route by inserting p in the k th position of the R c '. i. Assign unfulfilled request to Ac' based on R c ' if feasible. Update U p '. ii. Consider request l in U p ' 1. Insert the D l into the R c ' which yield the lowest increase in distance (Cheapest insertion) and is feasible. 2. Remove request l from U p ' and assign request l to M c '. 3. Assign unfulfilled request based on R c ' if feasible. Update U p '. iii. Repeat 3.c.ii until all requests in U p ' are considered d. Improve route by eliminate repeating visited location in R c '. i. Set S =. ii. Identify locations which the vehicle visits more than once, and add locations number to S. iii. For each p in S, and for each repeating position of p in R c ', remove one position of location p from R c ' and move all operations which take place at the removed position to another position which is also location p and still exists in R c '. If it is feasible and total distance is reduced, save R c '.

9 e. Repeat 3.c until all positions are considered. f. For R c 'with highest number of assigned requests, Set R c = R c ' and U p = U p '. If there is no more requests in U p, move to next N i and repeat 3.a. Otherwise, consider the next W ij and repeat 3.a. An example problem with 4 locations and 3 vehicles is shown in FIGURE 3. A particle is then decoded into a priority matrix of location and vehicle as shown in FIGURE 4. The location priority list and the vehicle priority matrix are used in the route construction step Location Vehicle Orientation Vehicle 3 Location Location Vehicle 1 30 Vehicle Location 3 Location FIGURE 3. The example of vehicle orientation point In the route construction step, it can be seen that there are neighborhood moves applied to improve the route. Step 3.c is inserting a location into the route and considers the request that must be picked up from that location. Then the delivery locations of the requests are inserted into the route. It is noted that the insertion position of pickup location may also be moved to different position to yield shorter distance or more number of requests served. Another concept applied here is that the vehicle should be loaded with as much goods as possible when it visits a certain location. In steps 3.c.i and 3.c.ii.3, the unfulfilled requests will be assigned to the vehicle if they have both pickup and delivery locations contained in the existing vehicle route. It is enforced that the insertion must not violate the capacity constraints. Following this decoding method, there is a possibility that the same location may be inserted in to the same route more than once. This means that vehicles are allowed to go back for re-stocking or unloading before continue their delivery.

10 Location priority Vehicle Orientation point Particle Location Distance b/w Vehicle Orientation points and locations Veh1 Veh2 Veh3 Location Priority List Vehicle Priority Matrix Location Vehicle priority 1st 2nd 3rd Location Vehicle priority 1st 2nd 3rd FIGURE 4. Decoding a particle into a location and vehicle priority matrix. Reduce the Number of Vehicles In addition to the decoding method, an improvement procedure is also applied to reduce the number of vehicles used. The procedure is applied to all particles during the initialization steps and to the global best at some iteration. During the initialization step, every particle is decoded and evaluated. Here, the number of vehicle used is reduced by one each time. The particle is then decoded again. If the removal leads to better results, the number of vehicle used is reset for all particles. The reduction continues until the particle is decoded into an infeasible solution before moving to the next particle. The same procedure of reduction is also applied to the global best at certain iteration. If the number of vehicles used for the global best particle is reduced, those of all other particles are also reset. The results of this method are the reduction of vehicles used and the corresponding reduction of the dimension of particles when the number of vehicle is reduced.

11 COMPUTATIONAL RESULTS This section describes computational results to assess the performance of the proposed algorithm. Algorithm SD1 is implemented in C# using Visual Studio.Net. The PSO specific procedures are implemented using the PSO Class from the object library for evolutionary techniques ETLib, (see Nguyen, et.al. [15]). Algorithm SD1 is applied to solve 100-location instances constructed by Li and Lim [12]. The instances are single depot pickup and delivery problem with time windows. The primary objective is to minimize number of homogenous vehicle used and the secondary objective to minimize the total travel distance. PDPTW Test Instances There are 3 types of instances being solved here, clustered locations with short schedule horizon (lc101 to lc109), clustered locations with long schedule horizon (lc201 to lc208), and randomly distributed locations with short schedule horizon (lr101 to lr112). In each case, the problem is solved 5 times to compute average number of vehicle used and average total travel distance. Since the primary objective of the instances is to minimize the number of vehicle used, the fix cost of a vehicle is set to 10,000. The weights are set as and the penalty cost if a request is not served, H i, is set to a high value. The experiments are performed with the following PSO parameters. All experiments are run with 100 particles and 1000 iterations. All other PSO parameters are set as reported in [7]. The number of neighbors is five. The inertia weight is linearly decreasing from 0.9 to 0.4. The acceleration constants for personal best, global best, local best, and near neighbor best are 0.5, 0.5, 1.5, and 1.5. Each instance is run 5 times. The summary of the experiments on PDPTW instances is given in TABLE 1. Algorithm SD1 found the best known solution in 15 out of 56 cases. TABLE 1. Summary of experiment on the PDPTW instances Average % Minimum % Best known Deviation Deviation solutions found NV Distance NV Distance 15 of 56 cases 11.19% 8.26% 6.12% 6.13% Note: NV = Number of vehicles used. The detailed results for all problem instances are shown in TABLE 2. For problems with short and long schedule horizon and clustered locations, Algorithm SD1 shows promising results. The use of vehicle orientation point to construct the route from closer location may be a contributing factor. However, the result for the cases with randomly-distributed-geographical locations shows that there is still a gap for improvement. From observations, it takes longer computation time for the instances with random and mixed locations. For long horizon instances, many available

12 combinations lead to longer computing time. It can be seen that the randomlydistributed locations cases with long horizon schedule and half-clustered-half-random instances are not presented here as further investigations are necessary. It seems that the concept of orientation point of vehicle might not be appropriate when the locations of customers are randomly distributed. TABLE 2. Computational result for Li & Lim s PDPTW instances Best of 5 Best known solution Case Replications Average NV Distance NV Distance NV Distance lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lr lr lr lr lr lr lr lr lr lr lr lr Note: NV = Number of vehicles used, Bold value in column 4 and 5 indicates that it has same value as the best known solution.

13 New Instances Since the proposed approach is more general and can handle multiple destinations for requests from a pickup location, additional instances are generated to represent the cases of interest based on the set C101 found in [13]. This new problem instance considers all features of GVRP-MDPDR. It is generated here to test the proposed methodology. The original C101 instance is a vehicle routing problem with time windows and clustered locations. The instance has 100 locations without vehicles and 4 locations with 40 vehicles in total. The new instance of GVRP-MDPDR is generated based on the geographical data in C101 of Solomon s data set and it is denoted as MC1_1. The newly generated data include 200 requests with quantity and origindestination pair. The weights are set as and the penalty cost, H i, is set to a high value. The details of the instance are summarized in TABLE 3. TABLE 3. Details of the MC1_1 instance Description Range No. of Location 100 No. of depot 4 No. of Requests 200 No. of Available Vehicle 40 Vehicle capacity 200 Fix cost of vehicle 75, 90 or 100 Range of Quantity of order 10 to 50 Min time windows range 85 Avg. time windows range Max time windows range 1123 The experiment is carried out with the same PSO parameters as those in the previous section except the population size and the number of iterations which are 50 and 100, respectively. The results are shown in TABLE 4. TABLE 4. Computational result of MC1_1 VALUE REPLICATION NV TOTAL DISTANCE OBJECTIVE FUNCTION It is noted that there is no unfulfilled requests in any replications as the penalty cost, H i, has high value. With the same number of iteration and parameters, the solution obtained from each replication is consistent. It can be seen that the proposed algorithm is quite robust. Nevertheless, this is a preliminary result and more test instances are required before a general statement can be made.

14 CONCLUSION This paper explored the practical case of generalized vehicle routing problem that does not limit the goods from a pickup location to be shipped to only one location. A real-value PSO algorithm with some neighbor moves is proposed to solve the problem. The proposed approach is tested on some benchmark PDPTW instances. The result shows that it is effective for cases where the customer locations are clustered. More appropriate decoding method for random-location instances should be further investigated. Since Algorithm SD1 is proposed to handle requests from a pickup location with multiple delivery destinations, it can handle more general cases than the PDPTW. Since no publicly available benchmark problem instance exists for the GVRP- MDPDR, a new problem instance that considered all features of GVRP-MDPDR is generated to test the proposed methodology. The preliminary results demonstrated that the proposed algorithm can provide good solutions for cases that consider factors often found in industrial practices. Tests with more benchmark problem instances are required before a general statement could be made on the proposed algorithm. The improvement in route construction algorithm and decoding method for various forms of geographical data should be further investigated. ACKNOWLEDGMENT This work was supported by Thailand Research Fund under Grant MLSC The development of the object library for evolutionary techniques (ETLib) received financial support from the 2008 Royal Thai Government Joint Research Program for Visiting Scholars. REFERENCES 1. G. Nagy and S. Salhi. Heuristics algorithm for single and multiple depot vehicle routing problem with pickups and deliveries, in European Journal of Operational Research, Vol. 162, 2005, pp G. Berbeglia, J.-F. Cordeau, and G. Laporte. Dynamic pickup and delivery problems, in European Journal of Operational Research, 2009 In Press, Corrected Proof, DOI: /j.ejor W.P. Nanry, J.W. Barbes.(2000) Solving the pickup and delivery problem with time windows using reactive tabu search, in Transportation Research B Vol. 34, 2000, pp J. Renaud, F.F. Boctor, J. Ouenniche (2000). A heuristics for the pickup and delivery traveling salesman problem, in Computer & Operation Research, Vol. 27, 2000, pp S. Ropke, D. Pisinger. An adaptive large neighborhood search heuristic for the pickup and delivery with time windows, in Transportation Science, Vol. 40, 2006, pp Available : 6. T. J. Ai and V. Kachitvichyanukul. Particle swarm optimization and two solution representations for solving the capacitated vehicle routing problem, in Computer & Industrial Engineering Vol. 56, No. 1, 2009, pp

15 7. T. J. Ai and V. Kachitvichyanukul. A particle swarm optimization for the vehicle routing problem with simultaneous pickup and delivery in Computers & Operations Research, Vol. 36, 2009, pp T. J. Ai and V. Kachitvichyanukul. A particle swarm optimization for vehicle routing problem with time windows, in International Journal Operation Research. Vol. 9, No. 4, pp J. Kennedy and R Eberhart. Particle Swarm Optimization, in Proceedings of IEEE international Conference on Neural Networks, Volume 4, 1995, pp P. Pongchairerks and V. Kachitvichyanukul. A non-homogenous particle swarm optimization with multiple social structures, in Proceedings of International Conference on Simulation and Modeling, (Paper A5-02). 11. P. Pongchairerks and V. Kachitvichyanukul. Particle swarm optimization algorithm with multiple social learning structures, in International Journal Operational Research, Vol.6, No. 2, pp H. Li and A. Lim. Pickup and delivery problem with time windows Instances, Available : M.M. Solomon. Vehicle Routing Problem with Time windows Benchmarks Problems, Available : P. Sombuntham and V. Kachitvichayanukul. A Particle Swarm Optimization Algorithm for Multidepot Vehicle Routing problem with Pickup and Delivery Requests, Lecture Notes in Engineering and Computer Science: Proceedings of The International MultiConference of Engineers and Computer Scientists 2010, IMECS 2010, March, 2010, Hong Kong, pp S. Nguyen, T.J. Ai, and V. Kachitvichyanukul. Object Library for Evolutionary Techniques ETLib: User s Guide, Asian Institute of Technology, Thailand, 2010.

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