Mathematical Problems in Engineering

Volume 2015, Article ID 640231, 13 pages

http://dx.doi.org/10.1155/2015/640231

## Minimization of the Total Traveling Distance and Maximum Distance by Using a Transformed-Based Encoding EDA to Solve the Multiple Traveling Salesmen Problem

Department of Information Management, Cheng Shiu University, No. 840, Chengcing Road, Niaosong District, Kaohsiung 83347, Taiwan

Received 15 May 2015; Revised 7 August 2015; Accepted 18 August 2015

Academic Editor: Jason Gu

Copyright © 2015 S. H. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Estimation of distribution algorithms (EDAs) have been used to solve numerous hard problems. However, their use with in-group optimization problems has not been discussed extensively in the literature. A well-known in-group optimization problem is the multiple traveling salesmen problem (mTSP), which involves simultaneous assignment and sequencing procedures and are shown in different forms. This paper presents a new algorithm, named , which is based on self-guided genetic algorithm with a minimum loading assignment (MLA) rule. This strategy uses the transformed-based encoding approach instead of direct encoding. The solution space of the proposed method is only . We compare the proposed algorithm against the optimal direct encoding technique, the two-part encoding genetic algorithm, and, in experiments on 34 TSP instances drawn from the TSPLIB, find that its solution space is . The scale of the experiments exceeded that presented in prior studies. The results show that the proposed algorithm was superior to the two-part encoding genetic algorithm in terms of minimizing the total traveling distance. Notably, the proposed algorithm did not cause a longer traveling distance when the number of salesmen was increased from 3 to 10. The results suggest that EDA researchers should employ the MLA rule instead of direct encoding in their proposed algorithms.

#### 1. Introduction

Estimation of distribution algorithms (EDAs) use the learning while optimizing principle [1]. Two review articles have suggested that EDAs have emerged as a prominent alternative to evolutionary algorithms [2, 3]. In contrast to genetic algorithms (GAs), which employ crossover and mutation operators to generate solutions, EDAs explicitly extract global statistical information from the previous search to build a posterior probability model of promising solutions from which new solutions are sampled [4, 5]. This crucial characteristic distinguishes EDAs from GAs [6, 7].

Numerous studies aimed at using EDAs to solve nondeterministic polynomial-time hard (NP-hard) scheduling problems have shown that EDAs are able to perform effectively in terms of the solution quality [2, 8, 9]. Ceberio et al. [2], in particular, extensively tested 13 famous permutation-based EDAs on four combinatorial optimization problems, including the quadratic assignment problem, traveling salesman problem (TSP), permutation flowshop scheduling problems (PFSPs), and linear ordering problem. Their paper provides a good basis for comparison.

Although EDAs are effective in solving various hard problems, EDA studies seldom extensively discuss a problem. To our knowledge, only one EDA, namely, that is proposed by Shim et al. [10], can solve in-group optimization problems such as the multiple traveling salesmen problem (mTSP) and parallel machine scheduling problems (PMSPs) [11]. In-group optimization problems involve assigning and sequencing procedures simultaneously. Take the mTSP, for example: a number of cities are assigned to salesmen and these cities are visited only once by a salesman, where . Thus, this appears to be an NP-hard problem.

Because only one EDA could solve in-group optimization problems, there is much room for additional research. In-group optimization problems are relevant in industry, such as in the application of the mTSP. This research developed a new EDA, named , dealt with by using a self-guided genetic algorithm (SGGA) [12] with the minimum loading assignment (MLA) rule to solve the mTSP. As opposed to direct encoding, the proposed strategy is called the transformed-based encoding approach. The solution space of the MLA is only . We compare the proposed algorithm against the optimal direct encoding technique, the two-part encoding genetic algorithm (TPGA) [13]. Notably, the solution space of the two-part encoding approach is . The proposed MLA method, consequently, is superior to the two-part encoding technique, and an improved solution quality is expected when the SGGA works with the MLA method.

This paper is organized as follows: Section 2 primarily reviews the literature on in-group optimization problems, encoding techniques, and EDAs. In Section 3, the core MLA method is presented to dispatch cities to salesmen. This assignment rule is further employed by the SGGA in Section 4. Section 5 reveals the effectiveness of the proposed algorithm, which is compared with the existing famous direct encoding methods, including the one-chromosome and two-part chromosome encoding. Finally, we draw conclusions in Section 6.

#### 2. Background Information

The mTSP is a well-known in-group optimization problem. We review mTSP studies and their variants in Section 2.1. To solve in-group optimization problems, numerous encoding techniques could be applied in evolutionary algorithms. Solution representations fall into two classes: direct and indirect encoding methods [11], relevant studies about which are presented in Sections 2.2 and 2.3, respectively. The final section illustrates combinatorial-based EDAs.

##### 2.1. In-Group Optimization Problems

Bektas [11] reviewed the seven types of in-group optimization problems, which we detail in Table 1. Among the variants of in-group optimization problems, the most well-known form is the mTSP because it models daily activities and exists in every enterprise [13]. The problem properties of the mTSP include assignment and sequence optimization procedures. For instance, we must optimize the traveling sequence for the route of each salesman. Both procedures directly lead to the traveling cost and time of the trip after assigning salesmen to visit places every day. A detailed definition of the mTSP can be found in [11].