Hybrid multiobjective evolutionary algorithm with fast sampling strategy-based global search and route sequence difference-based local search for VRPTW

Highlights

• HMOEA-GL combines FSS-GS and RSD-LS.

• FSS-GS quickly explores the entire solution space.

• RSD-LS further enhances the search ability of HMOEA-GL.

• Designing suitable coding method and proper genetic operators.

• Simple insertion search is used to reduce the number of vehicles.

Abstract

The vehicle routing problem with time windows (VRPTW) is an important and widely studied combinatorial optimization problems. This paper aims at VRPTW with the objectives of reducing the number of vehicles and minimizing the time-wasting during the delivery process caused by early arrival. To solve this NP-hard problem, a hybrid multiobjective evolutionary algorithm with fast sampling strategy-based global search and route sequence difference-based local search (HMOEA-GL) is proposed. Firstly, fast sampling strategy-based global search (FSS-GS) of HMOEA-GL extensively explores the entire solution space to quickly guide the search direction towards the center and edge areas of Pareto frontier. Secondly, route sequence difference-based local search (RSD-LS) is executed on the individuals with poor performance in the population obtained by FSS-GS to enhance the search ability of HMOEA-GL. In addition, the suitable coding method and proper genetic operators are designed, especially, a simple insertion search is used to reduce the number of vehicles in VRPTW. Comparing with NSGA-II, SPEA2, and MOEA/D, experimental results on 12 Solomon benchmark test problems indicate that the proposed HMOEA-GL is effective, and more excellent in convergence, while maintaining a satisfying distribution performance. HMOEA-GL could be an effective intelligent algorithm for expert and intelligent decision support system to help logistics companies to make decisions.

Introduction

Under the environment of economic globalization and information technology, people’s online shopping and long-distance ordering is more and more convenient and popular, and logistics plays an increasingly obvious role in modern economic activities, at the same time, the logistics problem becomes more complex. It is very necessary to design and use efficient intelligent decision support system (IDSS) to help decision-makers deal with logistics problem. IDSS is a decision support system that makes extensive use of artificial intelligence (AI) techniques, which works like a human consultant to support the decision-makers. In-depth analysis of the logistics problem, it can be found that the core is the vehicle routing problem (VRP). A good solution of VRP can take into account many aspects, not only can complete the customer service on time and on demand, but also can reduce the cost of distribution center consumption, to achieve a win-win. With the progress of the Internet, the type of customers of VRP is more likely to be transformed from large-volume customers like production base and supermarkets to small-volume customers such as family and individuals. Unlike large-volume customers who can receive service 24 hours a day, small-volume customers are generally only allowed to receive service within a few hours. Therefore, the consideration of time window is very necessary. At the same time, small-volume customers generally do not have a parking space, so the waiting cost of vehicles in the city may increase significantly, should not be ignored. With the efforts of researchers, great progress has been made in solving this problem, but due to the complexity of the problem and the different research background, the research on this problem is still very important and ongoing.

VRP is one of the most important and widely studied combinatorial optimization problems, which is relevant to transportation logistics such as post, parcel and distribution services. Its objective is to obtain the lowest-cost set of routes. In the beginning, truck dispatching problem was proposed by Dantzig and Ramser (1959) as a generalization of the travelling salesman problem (TSP), cost has mostly been associated with the travel distance. In 1964, Clarke and Wright summarized this problem as vehicle routing problem that is well-known: i.e., how to use a fleet of trucks to serve a set of customers which have different locations and needs. The reason VRP can be one of the most widely studied topics in the field of operations research is that VRP have important practical significance and with high complexity when this problem integrated into real-life. However, the current VRP model is very different from the original, because when VRP is combined with real-life, there are many factors that cannot be ignored. Like the number of vehicles, delivery time, parking cost, makespan, and workload balance, etc. are important to consider. Considering these realistic factors, VRP have several common variants that involve different constraints, like time-dependent travel times, with time windows, support delivery while pickup, can update customer information in real time, etc. Among them, the variant with time windows (VRPTW) has vehicles with limited capacity and the specific delivery time windows, and is particularly relevant to practical applications. The main problem studied in this paper is the vehicle routing problem with time windows which objective to minimize the number of vehicles and the waste time caused by the premature arrival.

Because the limitations of many constraints of VRP, so it is often to consider the VRP as a multiobjective problem. VRP is also considered as an NP-hard problem (Lenstra & Kan, 1981), when solving large-scale real-world VRPs, heuristics and meta-heuristics are often more suitable, exact algorithms are only efficient for small problem. The same is true for the variant VRPTW, optimal solutions for small instances of the VRPTW can be obtained by exact methods, but the computation time required increases considerably for larger instances (Desrochers, Desrosiers, & Solomon, 1992). For this reason, in recent years, various heuristic and meta-heuristics algorithm for solving VRPTW have become the hotspot of research. Algorithms such as simulated annealing (SA), tabu search (TS), ant colony optimization (ACO), genetic algorithm (GA), particle swarm optimization (PSO), differential evolution (DE), have proven to be effective in solving complex multiobjective problems. These algorithms are also used to solve complex VRPTW. Heuristic algorithms can greatly reduce the computational complexity and have many successful applications. However, with the increasing of the complexity and scale of the VRPTW, there may suffer some problems. For example, the speed of the algorithm or the final solution set is still not satisfactory. Therefore, research on VRPTW is still ongoing.

According to the No Free Lunch Theorem (Wolpert, Macready et al., 1997), no algorithm can solve all the problems. Each of the classic algorithms has its own unique advantages, and inevitably there are some drawbacks. Proper combination of different algorithms with different characteristics may further enhance the search effectiveness by adopting the advantages of each algorithm, and consequently may overcome the inherent limitations of single algorithm (Tang & Wang, 2013). Therefore, it is naturally to mix different techniques and utilize their advantages for solving complicated problems (Zhou et al., 2011).

As we know, GA is one of the most popular type of evolutionary algorithms (EA), which can easily be adapted to various types of problems (Gen, Cheng, 2000, Gen, Cheng, Lin, 2008, Holland, 1992, Tasan, Gen, 2012, Yu, Gen, 2010). DE is a direction-based stochastic search technique which can improve the convergence of the algorithm by lead the poor solution to close to the good solution (Storn & Price, 1997). But both of these algorithms have some drawbacks, which is easy to converge to local optima or even arbitrary points rather than the global optimum, resulting in premature convergence. This challenge is particularly hard when the dimension of problems is high and there are a lot of local optima.

Therefore, it is an interesting but challenging to hybridize GA and DE regarding the evolutionary behavior and potential advantages for VRPTW. Inspired by the successful hybrid strategies and application in previous research, in this paper, hybrid multiobjective evolutionary algorithm with fast sampling strategy-based global search and route sequence difference-based local search (HMOEA-GL) is proposed to solve complicated multiobjective VRPTW. The objectives of the problems one is to reduce the number of vehicles and another is to minimize the time-wasting during the delivery process caused by early arrival. HMOEA-GL combines fast sampling strategy-based global search (FSS-GS) and route sequence difference-based local search (RSD-LS). FSS-GS is designed as the global search strategy which includes center area-based elite sampling strategy and edge area-based sampling strategy. Center area-based elite sampling strategy according to Pareto dominating and dominated relationship-based fitness function (PDDR-FF) can fast picking out the individuals located in the central area of the Pareto frontier (Zhang, Gen, & Jo, 2014). Edge area-based sampling strategy based on vector evaluated genetic algorithm (VEGA) which prefers the individuals located in the edge area of the Pareto frontier (Schaffer, 1985). By mixing these two strategies and executing suitable genetic operators as well as simple insertion search (SIS), FSS-GS can rapidly improve the convergence and distribution performances toward the center and edge regions of Pareto frontier. As a follow-up step, drawing on the evolutionary strategy of DE, RSD-LS is used to further improve the quality of individuals. RSD-LS guides individuals with poor performance to move toward better performing individuals. By hybridizing FSS-GS and RSD-LS, HMOEA-GL can achieve fast convergence and sufficient distribution. As the core of IDSS, HMOEA-GL could be used to solve various scheduling problems such as VRPTW, which can provide multiple appropriate solutions for decision-makers by different weights of the objectives according to decision-makers’ needs. With HMOEA-GL, an IDSS can be established to effectively support decision-makers in a series of scheduling decision-making processes such as vehicle scheduling.

The rest of this paper is organized as follows. Section 2 introduces the related work to solve the vehicle routing problem with time windows. Section 3 provides definitions of VRPTW. A detailed description of the HMOEA-GL and the main strategies are shown in Section 4. Experimental design and the analysis of results are detailed in Section 5. Conclusions and future work are provided in Section 6.