A hybrid variable neighborhood tabu search heuristic for the vehicle routing problem with multiple time windows
Introduction
The purpose of this paper is to present a new variable neighborhood-tabu search heuristic for the Vehicle Routing Problem with Multiple Time Windows (VRPMTW). The VRPMTW arises, for example, in the delivery operations of furniture and electronic retailers, where customers are sometimes offered a choice of delivery periods. The problem also occurs in long-haul transportation (see e.g., Rancourt et al. [15]). The VRPMTW has received relatively little attention in the operations research literature. To our knowledge, it has only been investigated by Favaretto et al. [7] who considered variants with single or multiple visits. These authors have proposed an ant colony heuristic for the problem. In contrast, several publications are available on the related Traveling Salesman Problem [12], [13], the Team Orienteering Problem with Multiple Time Windows [21], [19], and the Multi-Visits Multi-interdependent Time Windows VRP [6]. In this paper we describe a hybridized variable neighborhood-tabu search heuristic for the VRPMTW. The proposed heuristic embeds the concept of adaptive memory (Rochat and Taillard [16]), sometimes used in tabu search (see also Bozkaya et al. [3]), into the standard variable neighborhood search framework. The remainder of this paper is organized as follows. The VRPMTW is formulated in Section 2. The mathematical model is presented in Section 3. The heuristic is described in Section 4. The route minimization algorithm is presented in Section 5 followed by computational results in Section 6, and by conclusions in Section 7.
Section snippets
Problem description
The VRPMTW is defined on a directed graph , where V is the vertex set and A is the arc set. The vertex set is partitioned into , where is a set of customers and D is a set of depots. We consider a set R of vehicles and we denote by m the number of vehicles. We denote by Qk the capacity of vehicle k, where . Every customer has a non-negative demand qi, a non-negative service time si, a set of pi time windows. The travel time associated with
Mathematical programming formulation
The VRPMTW can be formulated as a 0– mixed integer linear program, along the lines of Favaretto et al. [7]. We define in Table 1 the variables and the parameters used in our formulation.
Description of the variable neighborhood heuristic
Variable Neighborhood Search (VNS) was introduced by Mladenović and Hansen [10] as a generic local search methodology. It has since been successfully applied to a variety of contexts, including graph theory [2], packing problems [11] and location-routing [9]. The basic idea of VNS is to apply a systematic change of neighborhoods within a local search. In our implementation of VNS, we consider a number of different neighborhood structures instead of a single one, as is the case of many local
Minimizing route duration
Once a route's order is obtained, it is often possible to delay the departure time from the depot without violating the time windows constraints of customers, which can reduce the total route duration. Some infeasible routes can also become feasible. Savelsbergh [17] has introduced the concept of forward time slack to postpone the beginning of service at a given customer. Cordeau et al. [5] have used this concept to improve their results on multi-depot and other VRPTW instances with maximum
Computational results
This section presents our experimental results on benchmark instances of the Vehicle Routing Problem with Multiple Time Windows (VRPMTW) [7]. We report our best and average computational results on different sets of instances. Using a C++ implementation, these experimental results were obtained under MS Windows on workstations with 3.3 GHz Intel Core i5 vPro processors, and 3.2 GB RAM.
Conclusions
>We have presented a hybridization of the variable neighborhood search metaheuristic using a tabu search memory concept in order to solve the vehicle routing problem with multiple time windows. The paper also described a minimum backward time slack algorithm for this problem. The algorithm records the minimum waiting time and the minimum delay during route generation and adjusts the arrival and departure times backward during their computation. While the procedure proposed by Tricoire et al.
Acknowledgments
This work was partially supported by KFUPM Deanship of research under Grant IN101038 and by the Canadian Natural Sciences and Engineering Research Council under Grants 105574-12 and 39682-10. This support is gratefully acknowledged. Thanks are due to the referees for their valuable comments.
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