A hybrid algorithm for vehicle routing problem with time windows

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Abstract

Vehicle routing problem with time windows (VRPTW) is a well-known combinatorial problem. Many researches have presented meta-heuristics are effective approaches for VRPTW. This paper proposes a hybrid approach, which consists of ant colony optimization (ACO) and Tabu search, to solve the problem. To improve the performance of ACO, a neighborhood search is introduced. Furthermore, when ACO is close to the convergence Tabu search is used to maintain the diversity of ACO and explore new solutions. Computational experiments are reported for a set of the Solomon’s 56 VRPTW and the approach is compared with some meta-heuristic published in literature. Results show that considering the tradeoff of quality and computation time, the hybrid algorithm is a competitive approach for VRPTW.

Introduction

Vehicle routing problem (VRP) is found to be widely applicable to logistics distribution, school bus routing and mail etc. It has been studied for the last 40 years. A typical VRP can be described as a weighted graph. Fig. 1 shows an example of VRP, where the depot is denoted as 0 and the customers are denoted as 1 through 10. The solution includes three routes: 0–2–1–5–7–0, 0–3–4–6–0 and 0–8–9–10–0.

One of the most common extensions is vehicle routing problem with time windows (VRPTW) in practice. In VRPTW, vehicles service all the customers during a given interval without violating capacity constraints. The objective of the problem is to find a set of minimum cost routes for vehicles from a central supply depot. Many researches have presented meta-heuristic approaches are strong tools to solve VRPTW, such as Simulated Annealing (Czech and Czarnas, 2002, Li and Lim, 2003), genetic algorithms (Alvarenga et al., 2007, Berger and Barkaoui, 2004, Bräysy and Gendreau, 2001, Chen et al., 2001, Cheng and Wang, 2009, Potvin and Bengio, 1996, Tan et al., 2001a, Tan et al., 2001b, Ting and Huang, 2004), Tabu search (Chiang and Russell, 1997, Ho and Haugland, 2004, Potvin et al., 1996, Taillard et al., 1997), and ant colony optimization (Gambardella et al., 1999, Gong et al., 2007). Compared with Tabu search and genetic algorithm (GA), ACO is less applied in VRPTW. However, ACO has successfully been applied to solve capacitated vehicle routing problems, such as (Bullnheimer et al., 1999, Doerner et al., 2002, Doerner et al., 2004, Mazzeo and Loiseau, 2004, Yao and Yao, 2007, Yu et al., 2009).

The focus of this paper is to apply ACO in VRPTW. ACO finds good solution depending on the experience of proceeding ants. Sometimes, ACO tramps into local optimality. Therefore, it is fair and reasonable to introduce a method to prevent ACO into tramping into local optimality. Furthermore, the diversity of ACO is low near the convergence. Tabu search is a meta-heuristic based on recency memory. Tabu search can explore good solutions around a given solution by using a local or neighborhood search procedure and a Tabu list. The ideas between ACO and Tabu search are very different. Tabu search does not rely on the “history” information from the preceding search but randomly to select the search direction which can enlarge search space. Meantime, determining the initial solution of Tabu search is difficult. Considering the natures of ACO and Tabu search, this paper presents a hybrid algorithm integrating ACO and Tabu search to solve VRPTW. This paper is organized as follows. The problem formulation of VRPTW is introduced in Section 2. Section 3 describes the hybrid algorithm. Section 4 validates the proposed hybrid algorithm by Solomon’s benchmark test problems (Solomon, 1987). Lastly, conclusions are given in Section 5.

Section snippets

Problem formulation

The difference between VRPTW and VRP is the introduction of the time windows constraint. The time windows constraint is denoted a time interval between the earliest arrival time and the latest arrival time. In this study, if a vehicle arrives before the earliest arrival time the service cannot start until the time windows begins. The formulation of VRPTW is described as the following:Mini=0Nj=0,jiNk=1Kcijxijks.t.k=1Kj=1NxijkKfori=0(a)j=1Nxijk=j=1Nxjik1fori=0k{1,,K}(b)k=1Kj=0,jiNx

Improved ACO for VRPTW

Ant colony optimization is a meta-heuristic technique inspired from the behavior of real ants searching food. ACO was firstly presented by Dorigo, Maniezzo, and Colorni (1996) and it has successfully been applied as a solution tool to some classic compounding optimization problems (Colorni et al., 1994, Dorigo et al., 1996, Gambardella et al., 1999, Schoonderwoerd et al., 1996, Yu et al., 2009, Yu et al., 2009).

Numerical analysis

The heuristics described in the previous sections is coded in Visual C++.Net 2003 and executed on a PC equipped with 512 MB of RAM and a Pentium processor running at 1000 MHz. It is estimated by some well-known Solomon’s VRPTW instances (Solomon, 1987), which are divided into six data sets R1, R2, C1, C2, RC1 and RC2. In sets R1 and R2 customers are uniformly distributed, in sets C1 and C2 customers are clustered in groups and in sets RC1 and RC2 customers are semi-clustered. In addition, there

Conclusions

This paper presents a hybrid approach based on ACO and Tabu search. ACO first searches solutions. When ACO is close to optima or local optima, Tabu search, which uses the current best solution from ACO as the initial solution, is used to maintain the diversity of ACO and explore new solutions. Solomon’s 56 VRPTW is used to validate our algorithm. Results show that compared with some meta-heuristic published in literature ACO–Tabu is a effective tool for VRPTW.

Acknowledgments

This research is financed by the National Science Foundation for Post-doctoral Scientists of China 20080440168, the Doctoral Program Foundation for Young Scholar of Institutions of Higher Education of China through project 20070151013 and the Special Fund for Basic Scientific Research of Central Colleges, Dalian maritime university 2009QN094. The authors would also like to thank Dr. SUN Jian from School of Transportation Engineering, Tongji University for supporting their efforts for this paper.

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