An overview of a heuristic for vehicle routing problem with time windows

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P E R G A M O N C o m p u t e r s & Industrial E n g i n e e r i n g 37 (1999) 3 3 1 - 3 3 4

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An overview of a heuristic for vehicle routing problem with time windows

Fuh-hwa Franklin Liu* and Sheng-yuan Shen

Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu, Taiwan, Republic of China

*Addressee for all correspondence. Email:[email protected].

ABSTRACT

In this paper, a two-stage metaheuristic based on a new neighborhood structure is proposed to solve the vehicle routing problem with time windows. Our neighborhood construction focuses on the relationship between route(s) and node(s). Unlike the conventional methods for parallel route construction, we construct mutes in a nested parallel manner to obtain higher solution quality. Computational results for 60 benchmark problems are reported. The results indicate that our approach is highly competitive with all existing heuristics, in particular very promising for solving problems with large size. © 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Routing; Time Windows; Neighborhood; Heuristics INTRODUCTION

Routing and scheduling problems have been an intensive research for example surveys in (Bodin et al., 1983; Fisher, 1995). The purpose of this paper is to develop a method in solving the vehicle routing problem with time windows (VRPTW) so that either the obtainment of high quality fast-solution or an attempt in finding the best solution can be achieved.

The VRPTW can be formally stated as follows. Let G = (V, ,4) be a graph with node set V = N u {0} and arc set.4 = { (i,J)l i~ V , j ~ V, i ¢ j } , where N = { 1, 2,..., n} represents the customer set, and node 0 refers to the central depot. Each node i~ V is associated with a customer demand qi (q0 = 0), a service time si (So = 0), and a service-time window

[ei, li].

For every arc (i, j) ~ `4, a non-negative distance d/j and a non- negative travel time t O are known. Moreover, vehicles housed at the central depot are identical. Without loss of generality, each customer demand is assumed to be less than the vehicle capacity Q. In addition, the demand of each customer can not be split. The VRPTW is to find an optimal set of routes in such a way that:(i) all routes start and end at the depot;(ii) each customer in N is visited exactly once within its time window;(iii) the total of customer demands for each route can not exceed the vehicle capacity Q. Our primary objective is to minimize the number of vehicles used, and our secondary objective is to minimize the total distance traveled.

It is obvious that the VRPTW is NP-hard due to the NP-hardness of VRP. The exact algorithms recently developed for solving the VRPTW can be found in (Kolen et al., 1987; Desrochers et al., 1992; Fisher et al., 1997; Kohl and Madsen, 1997). However, on the fifty-six 100-customer benchmark problems by Solomon (1987), only a total of 11 problems were solved to optimality. Optimization approach can refer to the surveys by Desrochers et al. (1988) and Desrosiers et al. (1995).

Solomon (1987) was among the first to generalize VRP heuristics for solving the VRPTW. A parallel route building algorithm was contributed by Potvin et al. (1993). Kontoravdis and Bard (1995). Russell (1995) embedded a local improvement procedure into the route construction phase. Literature focusing on the investigation of improvement procedures based on the node-exchange and edge-exchange can be found in (Savelsbergh, 1985, 1992; Thompson and Psaraftis, 1993; Potvin and Rousseau, 1995).

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332 Proceedings of the 24th International Conference on Computers and Industrial Engineering In this paper, we propose a Route-NEighborhood-based Two-Stage methaheuristic (RNETS). In consid- eration of the geographical characteristics for VRPs, an appropriate solving approach is to spend time on efficient local operations than on inefficient global operations. Our RNETS provides the flexibility by ranging from a very small search domain to the entire domain.

The main contribution of this paper is the development of a new neighborhood structure based on a relationship between routes and nodes. Moreover, two new concepts, nested parallel route construction and end-effect handling, are presented in order to enhance solution quality.

AN OVERVIEW OF RNETS

A typical approach in solving VRPs consists of the following two phases: route construction phase and route improvement phase. Thus, in addition to an improvement procedure designed for the route improvement phase, a local improvement procedure is also considered during our route construction phase. Moreover, previous results have shown that the parallel route construction method is in general superior to the sequential one. RNETS considers a mixed strategy consisting of the following two stages. In the first stage, we construct routes in a nested parallel manner from the lower bound direction. With a solution obtained in stage L stage II is then to construct routes in parallel from an upper bound direction.

Route Construction

Phase

Improvement P~ase

Initialization LIR: _{EET: End-Effect Threshold}Local Improvement Rule

~[ Parallel Route L. v I Construction

_F

n o Parallel Route 1 Reinitialization Embedded Local Improvement End-Effect

Handling

Route

L

Improvement ]~

Figure 1. The core framwork for each stage of RNETS

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Proceedings of the 24th International Conference on Computers and Industrial Engineering 333

the construction of an initial set of mute neighborhoods. The parallel route construction block is responsi- ble for selecting a candidate and inserting it into a specific mute. The parallel route reinitialization block is to eliminate a number of constructed mutes and reconstruct an appropriate set of partial mutes. What we called the nested parallel mute construction is accomplished through the parallel route construction block

and the parallel route reinitialization block. The purpose of end-effect handling block is to apply a spe- cially designed procedure on "a few" unmuted customers who are left with no feasible vehicles with respect to the current set of routes during the route construction phase. Reasons in addressing the concepts of mute neighborhoods, nested parallel route construction, and end-effect handling are now described as follows.

Route Neighborhoods

To efficiently deal with large scale VRPs, an intuitive idea is to decompose the original hard problem into several easier subproblems or to divide the original domain into several smaller subdomains.

For VRPs, the term "neighborhood" usually refers to a set of nodes. To be clear, we discuss the neighborhood in the following two cases. Case 1: Before the start of route construction, we may divide the customer set into several smaller subsets or neighborhoods. Usually, these neighborhoods are disjoint parti- tions. Case 2: Having finished mute construction, we can apply local search techniques on the current solution. In both the cases, the construction of neighborhoods primarily focused on the relationship between node(s) and node(s). The relationship between mute(s) and node(s) is almost ignored.

Consequently, from a different point of view, we introduce the concept of route neighborhood. It is a rea- sonable conjecture that a customer holds higher probability of being served by near-by routes than by far- ther mutes. Under this observation, we attempt to construct a route neighborhood containing a set of "routes" for each node. In terms of the timing in constructing neighborhoods, our mute neighborhood con- stmction has higher similarities to the neighborhoods described in Case 1. However, our mute neighborhoods also play a role like the neighborhoods in Case 2. There are at least two benefits resulting from the construction of route neighborhoods. The first is that the weakness in handling time window constraints faced by the methods like "cluster-first and route-second" can be easily improved; this is indeed why we can generate high quality fast-solution by focusing on a smaller search domain. The second is that to thor- oughly explore solutions on the search space can be easily designed by adjusting the number of mutes to be contained in a mute neighborhood from one to a large enough number.

Nested Parallel Route Construction

Because of the presence of time window constraints, it seems hard to estimate a lower bound tight enough to the number of vehicles actually required. We otien encounter that many customers can not be feasibly served by using only those vehicles. Therefore, we perhaps need to generate more new mutes to accom- modate the remaining unrouted customers. Seeing that the parallel construction method had better per- formance in average, however, we introduce the concept of nested parallel route construction to obtain better solution quality for VRPTW. The nested parallel mute construction repeats the following two steps until all customers are muted or a stopping rule is satisfied. At the first step, we estimate a lower bound to the number of vehicles required for the unrouted customers, and construct a corresponding set of partial mutes. For the mutes just generated, the second step is using these mutes to service the unrouted customers until no feasible insertion locations can be found. We will describe the stopping rules in a later.

End-Effect Handling

In the presence of time window constraints, to eliminate a scheduled route entirely might not be an easy thing if we only rely on a refinement procedure in the route improvement phase. When there are only "a few" unmuted customers left after a number of parallel construction runs, we do not immediately create a new mute or a new set of mutes for them. Since our primary objective is to minimize the number of mutes, we design a special procedure to handle the remaining "a few" unrouted customers. We will say

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334 Proceedings of the 24th International Conference on Computers and Industrial Engineering

that there are only "a few" unrouted customers if the total of unmuted demands is smaller than a specified

threshold value. Three measurement criteria, called end-effect thresholds, are suggested for the threshold

value: vehicle capacity, maximal utilization rate x vehicle capacity, and average utilization rate x vehicle capacity. Here, the maximal and average utilization rates are computed with respect to the mutes con- stmcted in all previous parallel construction runs. These thresholds are the stopping rules mentioned ear- lier.

COMPUTATIONAL STUDIES

To assess the performance of RNETS, we first solved four real world problems obtained from George Kontoravids. Among them, two 249-customer problems, D249 and E249, were generated from a 249- customer data set reported in Baker and Schaffer (1988) and two 417-customer problems, D417 and E417, were reported in Russell (1995). In addition, RNETS was tested on fifty-six benchmark problems by Solomon (1987); each is 100-customer Euclidean problem.

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