An artificial bee colony algorithm with a Modified Choice Function for the traveling salesman problem

Abstract

The Artificial Bee Colony (ABC) algorithm is a swarm intelligence approach which has initially been proposed to solve optimisation of mathematical test functions with a unique neighbourhood search mechanism. This neighbourhood search mechanism could not be directly applied to combinatorial discrete optimisation problems. In order to tackle combinatorial discrete optimisation problems, the employed and onlooker bees need to be equipped with problem-specific perturbative heuristics. However, a large variety of problem-specific heuristics are available, and it is not an easy task to select an appropriate heuristic for a specific problem. In this paper, a hyper-heuristic method, namely a Modified Choice Function (MCF), is applied such that it can regulate the selection of the neighbourhood search heuristics adopted by the employed and onlooker bees automatically. The Lin-Kernighan (LK) local search strategy is integrated to improve the performance of the proposed model. To demonstrate the effectiveness of the proposed model, 64 Traveling Salesman Problem (TSP) instances available in TSPLIB are evaluated. On average, the proposed model solves the 64 instances to 0.055% from the known optimum within approximately 2.7 min. A performance comparison with other state-of-the-art algorithms further indicates the effectiveness of the proposed model.

Introduction

A computational optimisation methodology involves finding feasible solutions from a finite set of solutions, and identifying only the optimal solution(s). Swarm intelligence algorithms constitute a sub-class of computational optimisation methodology [1]. Swarm intelligence algorithms are developed based on emergence of collective behaviours pertaining to a population of interacting individuals in adapting to the local and/or global environments. Examples of swarm intelligence algorithms include Particle Swarm Optimisation (PSO) [2], Ant Colony Optimisation (ACO) [3], Bat Algorithm (BA) [4], Firefly Algorithm (FA) [5], Cuckoo Search Algorithm (CSA) [6], and bee-inspired algorithms [[7], [8], [9]].

Bees are highly organised social insects. Their survival relies on assigning an important task to each bee in a cooperative mode. The tasks include reproduction, foraging, and constructing hive. Within these tasks, foraging is one of the most important tasks, because the bee colony must ensure an undisrupted supply of food to survive. The food foraging behaviours of bees can be computationally realised as algorithmic tools to solve various optimisation problems.

The Artificial Bee Colony (ABC) algorithm is one of the popular bee-inspired algorithms. Proposed by Karaboga [7], it is inspired by the foraging behaviours of honey bees in a colony. In the ABC algorithm, a food source represents a possible solution to the optimisation problem in the search space, and the nectar amount of the food source represents the fitness of that solution. The ABC algorithm defines three types of bees: employed bees, onlooker bees, and scout bees. An employed bee looks for new food sources around the neighbourhood of the food source that it has previously visited. An onlooker bee observes dances and selects a food source to visit. It tends to select good food sources from those found by the employed bees. A scout bee searches for new food sources randomly.

The mechanism of the ABC algorithm is as follows. The employed bees first perform a neighbourhood search nearby the food source in their memory (i.e. solution). Then, they go back to the hive and perform dances. The dances inform the onlooker bees about the fitness of each solution. Each onlooker bee observes and selects a food source to perform another neighbourhood search based on a probability proportional to the food source fitness (i.e. a roulette wheel selection). The onlooker bees tend to select good food sources from those found by the employed bees. The employed and onlooker bees perform neighbourhood search by perturbing an existing solution to produce a new solution. A greedy approach is applied to decide whether to accept the newly perturbed solution. If a solution could not be improved after a pre-determined number of trails (denoted as the limit), it is abandoned. The employed bee associated to that non-improving solution (i.e. local optimum) is abandoned, and it becomes a scout bee. The scout bee explores the search space at random and looks for a new solution.

This ABC algorithm has been used to solve optimisation of mathematical test functions [7]. Promising results have been reported by using a number of ABC variants [[10], [11], [12]]. To find the optimum solution of the mathematical test functions, the neighbourhood search performed by the employed and onlooker bees is formulated as follows (Eq. (1)):

$$v_{\mathit{ij}}=x_{\mathit{ij}}+\varphi \text{(}{x}_{\mathit{ij}}-x_{\mathit{kj}}\text{)}$$

in which x_i is the solution associated to the i-th employed bee, x_ij is j-th element (i.e. dimension) of solution x_i, v_i is the new solution produced based on x_i, v_ij is j-th element of solution v_i, j is a random integer between 1 and dim (the dimensionality of the problem), ϕ is a random real number between −1 and 1, and k is a random integer between 1 and n (the number of employed bees).

In recent years, the ABC algorithm has been modified to solve combinatorial discrete optimisation problems, such as quadratic assignment problem [13], p-median problem [[14], [15], [16]], minimum spanning tree problem [[17], [18], [19]], clustering problem [[20], [21], [22]], uncapacitated facility location problem [[23], [24], [25]], Knapsack Problem (KP) [[26], [27], [28]], Job Shop Scheduling Problem (JSSP) [[29], [30], [31]], Vehicle Routing Problem (VRP) [32,33], and Traveling Salesman Problem (TSP) [34,35]. However, Eq. (1) cannot be directly applied when solving this set of problems. The employed and onlooker bees are prescribed with a perturbative heuristic (or a set of perturbative heuristics) to generate new solutions. These heuristics are problem-specific, for instance, the neighbourhood search heuristics for TSP include insertion mutation, swap mutation, random 2-opt, etc. In view of the availability of a large variety of problem-specific heuristics, the key question concerning the selection of a particular heuristic has been posed in the literature in recent years. This leads to the motivation of using hyper-heuristics for tackling such problem, which is the focus of our research in improving the ABC algorithm.

A hyper-heuristic is a high-level automated methodology for selecting or generating a set of heuristics [36]. The term “hyper-heuristic” was coined by Denzinger et al. [37]. There are two main hyper-heuristic categories, i.e. selection hyper-heuristic and generation hyper-heuristic [38]. These two categories can be defined as ‘heuristics to select heuristics’ and ‘heuristics to generate heuristics’, respectively [36]. Both selection and generation hyper-heuristics can be further divided into two categories based on the nature of the heuristics to be selected or generated [38], namely either constructive or perturbative hyper-heuristics. A constructive hyper-heuristic incrementally builds a complete solution from scratch. On the other hand, a perturbative hyper-heuristic iteratively improves an existing solution by performing its perturbative mechanisms. The heuristics to be selected or generated in a hyper-heuristic model are known as the low-level heuristics (LLHs).

A typical selection hyper-heuristic model consists of two levels [36]. The low level contains a problem representation, evaluation function(s), and a set of problem specific LLHs. The high level manages which LLH to use for producing a new solution(s), and then decides whether to accept the solution(s). Therefore, the high-level heuristic performs two separate tasks i.e. (i) LLH selection and (ii) move acceptance [39]. The LLH selection method is a strategy to select an appropriate LLH from a set of available alternatives during the search process. The available LLH selection methods include simple random [40], choice function [[41], [42], [43]], tabu search [44], harmony search [45], backtracking search algorithm [46], and a set of reinforcement learning variants [47,48]. The move acceptance method decides whether to accept the new solution generated by the selected LLH. Examples of move acceptance methods include Only Improvement [49], All Moves [43], Simulated Annealing [50], Late Acceptance [40], and some variants of threshold-based acceptance.

Striking a balance between intensification and diversification is important for a hyper-heuristic [36,51]. Intensification encourages a hyper-heuristic to focus on the promising LLHs, which leads to a good performance. On the other hand, diversification serves as a forgive-and-forget policy which encourages attempts on those rarely used LLHs. Both intensification and diversification are crucial components as the capability of an LLH varies during different phases of the search process [52,53]. An LLH with a good performance in one phase should not dominate the subsequent search process, while a poor performance in one phase should not lead to a permanent discrimination of an LLH in the later phases. In this study, an LLH selection method which is based on a choice function, namely the Modified Choice Function (MCF) [42], is integrated with the ABC algorithm. Specifically, MCF is used to select the neighbourhood search heuristic deployed by the employed and onlooker bees. The reason of choosing MCF is because it is able to adaptively control the weights of its intensification and diversification components during different phases of the search process. Besides that, to enhance the performance of the proposed MCF-ABC model, it is integrated with the Lin-Kernighan (LK) local search strategy [54]. The proposed model is denoted as MCF-ABC. It is tested using benchmark TSP instances provided in TSPLIB [55].

This article starts with a description of the related work in Section 2. Section 3 presents the proposed MCF-ABC model. The results and findings including performance comparison are presented in Section 4. Finally, concluding remarks are presented in Section 5.