Computational Intelligence and Neuroscience

Volume 2016 (2016), Article ID 8932896, 13 pages

http://dx.doi.org/10.1155/2016/8932896

## Annealing Ant Colony Optimization with Mutation Operator for Solving TSP

Department of Computer Science, Faculty of Computing and Information Technology, University of Science and Technology, Sana’a, Yemen

Received 22 June 2016; Revised 15 October 2016; Accepted 19 October 2016

Academic Editor: Elio Masciari

Copyright © 2016 Abdulqader M. Mohsen. 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

Ant Colony Optimization (ACO) has been successfully applied to solve a wide range of combinatorial optimization problems such as minimum spanning tree, traveling salesman problem, and quadratic assignment problem. Basic ACO has drawbacks of trapping into local minimum and low convergence rate. Simulated annealing (SA) and mutation operator have the jumping ability and global convergence; and local search has the ability to speed up the convergence. Therefore, this paper proposed a hybrid ACO algorithm integrating the advantages of ACO, SA, mutation operator, and local search procedure to solve the traveling salesman problem. The core of algorithm is based on the ACO. SA and mutation operator were used to increase the ants population diversity from time to time and the local search was used to exploit the current search area efficiently. The comparative experiments, using 24 TSP instances from TSPLIB, show that the proposed algorithm outperformed some well-known algorithms in the literature in terms of solution quality.

#### 1. Introduction

One of the most popular combinatorial optimization problems is the traveling salesman problem (TSP) [1]. Given a set of cities, a salesman attempts to find the shortest or at least near to the shortest tour by visiting each city only once and turning back to the starting city. TSP is a representative of variety of combinatorial problems. It has been studied for the last 40 years. It has many real world applications such as the movement of people, postal delivery, school bus routes, garbage collection, design of hardware devices and radio electronic systems, machine scheduling, integrated circuits, and computer networks [2–4].

Metaheuristic algorithms are formally defined as algorithms that inspired by nature and biological behaviors. They produce high-quality solutions by applying a robust iterative generation process for exploring and exploiting the search space efficiently and effectively. Recently, metaheuristic algorithms seem to be a hot and promising research areas [5]. They can be applied to find near-optimal solutions in a reasonable time for different combinatorial optimization problems [6].

Metaheuristic algorithms such as genetic algorithms (GAs) [7], particle swarm optimization (PSO) [8], tabu search (TS) [7], simulated annealing (SA) [9], and ant colony optimizations (ACO) [10] are widely used for solving the TSP. Ant colony optimization proposed by Dorigo et al. in 1996 [10] simulates the intelligent behavior of real ants seeking for the food in nature. It has been successfully applied to solve many optimization problems such as TSP [10], quadratic assignment [11], job-shop scheduling [12], and load balancing in telecommunications networks [13].

In applying standalone metaheuristic algorithms, there is possibility of losing the diversity of the population through premature convergence and thus the algorithm gets stuck in local optima. Therefore, maintaining the diversity and making tradeoff between diversification and intensification by combining two or more algorithms to produce high-quality solutions and speed up the execution time is indispensable [14].

For hybrid ACO, the earliest study was conducted by McKendall and Shang [15]. They presented a hybrid ant system algorithm to solve dynamic facility layout problem. Another research was a hybrid ant system algorithm for solving TSP in which ant colony, genetic algorithm, and simulated annealing are hybridized [16]. For the hybrid ant colony system (ACS), many researches were conducted including the work by Huang and Liao [17], Yoshikawa and Otani [18], Xing et al. [19], Liao et al. [20], Lin et al. [21], Hajipour et al. [22], and Min et al. [16]. The research by Katagiri et al. [23] is an example for hybrid MAX-MIN Ant System. To solve TSP problems, several hybrid ACO variants with other metaheuristic algorithms such as SA, PSO, ACO, ABC, and ANN were proposed. Bontoux and Feillet [24] proposed a hybrid ACO algorithm with local search procedures to solve TSP. Tsai et al. [25] presented a hybrid ACO called ACOMAC algorithm for solving TSP. Beam-ACO algorithm is a hybrid ACO with beam search for solving TSP [26]. Chen and Chien presented a hybrid algorithm, called the genetic simulated annealing ant colony system with particle swarm optimization techniques, for solving TSP [27]. Junqiang and Aijia proposed a hybrid ant colony algorithm (HACO), which combined ACO with delete-cross to overcome the shortcoming of slow convergence speed of ACO [28]. Dong et al. [29] proposed an algorithm, called cooperative genetic ant system (CGAS) for solving TSP, which hybridized both GA and ACO to improve the performance of ACO. Recently, Gündüz et al. [30] presented a hybrid ACO with ABC for solving TSP. In addition, Mahi et al. [31] proposed a new algorithm in which ACO was hybridized with PSO and 3-Opt for solving small TSP instances. The PSO was used to determine the optimum values of the two main parameters of ACO which affected algorithm performance and the 3-Opt was used to escape from the local optima found by ACO algorithm. Furthermore, Yousefikhoshbakht et al. [32] proposed REACSGA for solving small TSP instances which employed the modified ACS for generating initial diversified solutions and GA for intensification mechanisms.

As noted above, previous studies show that ACO still has drawbacks. The performance of these studies was dramatically decreased when dealing with large-scale instances. To the best of my knowledge, no research has been done to hybridize elitist ant system with SA, mutation, and local search. Therefore, in this research a new hybrid elitist ant system with SA, mutation operator, and local search procedure is introduced for solving TSP. Introducing SA can help ACO to escape from the local optima. On the other hand, determining initial solution of SA is almost difficult. Therefore, the use of the ACO is useful in the generation of SA initial solution. While introducing the mutation operation to ACO algorithm will enhance the algorithm performance, expand the diversity of population, and inhibit the premature convergence. Applying either SA or mutation is based on the diversity level of the population. After applying SA or mutation, elitist ant system goes through a local search procedure to speed up the convergence.

The rest of the paper is structured as follows. Section 2 presents the TSP formulation. Section 3 describes the hybrid algorithm. The experimental results are presented in Section 4. Conclusions and future work are given in Section 5.

#### 2. Traveling Salesman Problem

TSP is an active field of research in computer science. It demonstrates all the aspects of combinatorial optimization and comes under the set of NP-hard problems which cannot be solved optimally in a polynomial time [33]. Solving TSPs is an important part of applications in many practical problems within daily live [2–4].

TSP is represented as a connected graph , consisting of a set of vertices , an edges set , and a set of distances associated with each edge in and stored in a distance matrix . The value is the cost of traversing from vertex to vertex and the diagonal elements are zeros. Given this information, a tour in TSP is formulated as a cyclic permutation, called Hamiltonian cycle of visiting each vertex in the graph exactly once, of , where is the city, on the tour, following city . The aim in solving TSP is to find a permutation that minimizes the length of the tour as shown inIt is worth mentioning that the total number of possible distinct feasible routes covering all cities is . This produced a problem which is very hard to solve (NP-hard problem).

#### 3. Algorithm Design

An overview of the ACO, SA, mutation operator, and the proposed algorithm is presented in the following subsections.

##### 3.1. Ant Colony Optimization

ACO is a population-based metaheuristic algorithm which was inspired by the foraging behavior of the real ants when searching for the shortest path from the food source to their nest. Analogically, the artificial ants search for good solutions iteratively in several generation. In each generation, every ant constructs its feasible solution path step by step guided by a transition rule that is a function of artificial pheromone and distance between two cities (heuristic information) [34] as shown in (2). After that, the ant lays down an amount of pheromone trail on the edges of its completed tour. In the next generation, ants are attracted by the pheromone trail. Therefore, this will guide the search in the search space towards good quality solutions.

TSP is identical to the foraging behaviors of real ants in nature. Therefore, applying ant colony optimization to solve TSP will be very simple. Equation (2) is used to calculate the probability of selecting city by ant to be visited after city .where denotes the amount of pheromone between city and city , indicates the distance between city and city (priori available heuristic information), is the parameter that represents the relative importance of the pheromone (dynamic evaluation) versus the heuristic value (static evaluation), and is a set of cities which ant has not yet visited. Therefore, the selection probability is proportional to the product of the static and dynamic evaluation.

In the dynamic evaluation, two pheromone update rules are used to calculate the amount of pheromone on each edge between cities. The first rule is called the local update rule as shown in where is the pheromone trail evaporation rate in local update rule and is the number of ants. Thus, the local update rule is decreasing the pheromone trails by a constant factor (pheromone evaporation). The second rule is the global update rule which adds extra amount pheromone trail to the best route in the population. It is worth mentioning that the best route is the shortest route as in elitist strategy [10], the extended version of original ant system algorithm. Equation (4) shows the definition of the global update rule in elitist ant system:where is the best route, is the distance of the best route, and is a positive integer. This means that the edges belonging to the global-best tour get an additional amount of pheromone each time the pheromone is updated.

The pseudocode of the basic ACO is illustrated in Algorithm 1.