Multi-depot Vehicle Routing Problem with Pickup and Delivery Requests Pandhapon Sombunthama and Voratas Kachitvichyanukulb ab
Industrial and Manufacturing Engineering, Asian Institute of Technology, Klongluang, Pathumthani, 12120 THAILAND Email:
[email protected],
[email protected] 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
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
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 11 12 10
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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.
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 D N Hi K Ck fk gk ′ V A dij , tij
si ei [ai,bi] li
: set of pickup nodes, {1… n}. : set of delivery nodes, {n+1... 2n}. : set of all pickup and delivery nodes N= PD : a penalty cost if the request i is not served, i P. : set of all vehicles, |K| = m. : capacity of vehicle k K. : fixed cost of vehicle k K if it is used : 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. : set of all nodes. V = N { … }{ … } : set of (i, j) which is an arc from node i to node j ,where i, j V. : distance and travel time between node i and node j ,for i and j N. Travel times satisfy the triangle inequality; ti j til +tlj for all i, j, l V; and are nonnegative. : fixed service time when visiting node i. : variable service time per item units of node i. : time windows when the visit at the particular location must start; a visit to node i can only take place between time ai and bi : the quantity of goods to be loaded onto the vehicle at node i for i P and li = −li−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.
Decision Variables The main decision variables in the model are described below: : a binary variable with the value 1 if vehicle k travels from node i to node j xijk and is zero otherwise, where i, j V, k K. Sik : a nonnegative integer that indicates when vehicle k starts the service at location i, i V, k K : a nonnegative integer that is an upper bound on the amount of goods on Lik vehicle k after servicing node i, where i V, k K. Sik and Lik are defined only when vehicle k actually visits node i. zi : 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 Subject to:
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