A new crossover approach for solving the multiple travelling salesmen problem using genetic algorithms

Abstract

This paper proposes a new crossover operator called two-part chromosome crossover (TCX) for solving the multiple travelling salesmen problem (MTSP) using a genetic algorithm (GA) for near-optimal solutions. We adopt the two-part chromosome representation technique which has been proven to minimise the size of the problem search space. Nevertheless, the existing crossover method for the two-part chromosome representation has two limitations. Firstly, it has extremely limited diversity in the second part of the chromosome, which greatly restricts the search ability of the GA. Secondly, the existing crossover approach tends to break useful building blocks in the first part of the chromosome, which reduces the GA’s effectiveness and solution quality. Therefore, in order to improve the GA search performance with the two-part chromosome representation, we propose TCX to overcome these two limitations and improve solution quality. Moreover, we evaluate and compare the proposed TCX with three different crossover methods for two MTSP objective functions, namely, minimising total travel distance and minimising longest tour. The experimental results show that TCX can improve the solution quality of the GA compared to three existing crossover approaches.

Introduction

The multiple travelling salesman problem (MTSP) is a variation of the well-known travelling salesman problem (TSP), which is classed as being NP-hard. The MTSP is more difficult than the TSP, because it aims to resolve a group of Hamiltonian circuits without sub-tours for m (m > 1) salesmen to serve a set of n (n > m) cities. This leads to the optimum solution of MTSP becoming more computationally infeasible as the problem size increases. Compared to the TSP, the MTSP is more suitable for modelling many practical situations because it is capable of handling more than one salesman. As such, many practical problems have been modelled as a MTSP, such as print press scheduling (Gorenstein, 1970), crew scheduling (Svestka and Huckfeldt, 1973), hot rolling scheduling (Tang et al., 2000), mission planning (Ryan et al., 1998) and vehicle scheduling (Park, 2001). Additionally, there are several variations of the classical MTSP. Based on the number of depots, the MTSP can have a single depot or multiple depots. It can also have open or closed tours, where the difference is whether the salesmen need to return to their depot(s). Furthermore, if the salesmen need to pick up loads, it can be cast as the MTSP pickup and delivery problem (MTSPPDP) (Wang and Regan, 2002). If the tasks have time-related constraints, the MTSP is cast with time windows (MTSPTW) (Kim and Park, 2004). There are other variations of the MTSP in real world applications; however, in all these MTSP applications, job planning and vehicle scheduling are the most commonly researched areas (Bektas, 2006).

Due to the combinatorial complexity of the MTSP, many researchers have tried to relax the MTSP to the canonical TSP and use exact algorithms to solve it, yet the results have always been unsatisfactory (Bektas, 2006). Most of the research related to the use of GAs for the vehicle scheduling problem have focused on using two different chromosome designs (i.e. one chromosome representation and two chromosome representation) for the MTSP. Both of these chromosome designs can be manipulated using classic GA operators developed for the TSP; however, they are also prone to produce redundant solutions to the problem. Carter and Ragsdale (2006) proposed a GA for the MTSP that uses a two-part chromosome representation which they proved could effectively reduce such redundant solutions in the search space. The crossover operation for the two-part chromosome is separated into two independent operations. The first operation uses an ordered crossover operator, while the second operation uses an asexual crossover operator to ensure that the second part of the chromosome remains feasible. However, due to the nature of the two-part chromosome, Carter (2003) has suggested that further research into “more effective crossover operators” would be both important and necessary. To our knowledge, no specific crossover operator for the two-part chromosome in GAs has so far been proposed for solving MTSP. In this paper, we adapt the two-part chromosome representation method to encode the solution for the MTSP, and introduce a new crossover operator which dramatically improves the solution quality.

The paper is organised as follows. Section 2 provides a brief overview of the related work on solving the MTSP using GAs. Section 3 introduces the existing two-part chromosome encoding approach and associated crossover operation. The limitations of the existing method are presented in Section 4. In Section 5, we introduce a new crossover operator for two-part chromosomes and mathematically analyse the advantages of the proposed approach. Section 6 introduces the testing methodology and Section 7 presents the experimental results. Finally, Section 8 provides a conclusion and summary of the study.