Adjustability of a discrete particle swarm optimization for the dynamic TSP

Soft Comput DOI 10.1007/s00500-017-2738-9 METHODOLOGIES AND APPLICATION Adjustability of a discrete particle swarm optimization for the dynamic TSP Łukasz Stra˛ k1 · Rafał Skinderowicz1 · Urszula Boryczka1 © The Author(s) 2017. This article is an open access publication Abstract This paper presents a detailed study of the discrete particle swarm optimization algorithm (DPSO) applied to solve the dynamic traveling salesman problem which has many practical applications in planning, logistics and chip manufacturing. The dynamic version is especially important in practical applications in which new circumstances, e.g., a traffic jam or a machine failure, could force changes to the problem specification. The DPSO algorithm was enriched with a pheromone memory which is used to guide the search process similarly to the ant colony optimization algorithm. The paper extends our previous work on the DPSO algorithm in various ways. Firstly, the performance of the algorithm is thoroughly tested on a set of newly generated DTSP instances which differ in the number and the size of the changes. Secondly, the impact of the pheromone memory on the convergence of the DPSO is investigated and compared with the version without a pheromone memory. Moreover, the results are compared with two ant colony optimization algorithms, namely the MAX –MIN ant system (MMAS) and the population-based ant colony optimization (PACO). The results show that the DPSO is able to find high-quality solutions to the DTSP and its performance is competitive with the performance of the MMAS and the PACO algorithms. Moreover, the pheromone memory has a positive impact on Communicated by V. Loia. B Łukasz Stra˛k Rafał Skinderowicz Urszula Boryczka 1 Institute of Computer Science, University of Silesia in Katowice, Bȩdzińska 39, 41-205 Sosnowiec, Poland the convergence of the algorithm, especially in the face of dynamic changes to the problem’s definition. Keywords Discrete particle swarm optimization · Pheromone memory · Dynamic traveling salesman problem · Population-based ant colony optimization 1 Introduction A problem in which input data are variable (i.e., time dependent) is called a dynamic optimization problem (DOP). The aim of optimization in the DOP is to continuously track and adapt to changes and quickly find the best solution (Li 2011). DOPs can be divided into two groups (Mori and Kita 2000): online (direct adaptation) and offline (indirect adaptation). In the first case, the changes are hard to predict and could happen at any moment during the algorithm runtime. In the second case, the changes happen at specified intervals between which the problem definition does not change. Many other DOPs classification criteria were described by Yang and Yao (2013), including – time linkage— is the change affected by the current solution; – cyclicity—is the change to the problem cyclic or not; – factors—do the changes affect the number or values of variables, constraints, etc. DOPs were investigated mostly in the context of the continuous optimization. In this work, we focus on the discrete DOP that is the DTSP in the offline version. Computational intelligence methods are a set of techniques that are used to solve optimization problems. They include algorithms that were inspired by the collective behav- 123 Ł. Stra˛k et al. ior of animals which are able to solve complex problems in their natural environment by cooperating with one another. It is often beyond the capabilities of a single individual to solve these problems, i.e., to accomplish tasks such as foraging and nest building, yet this is not problematic for a group of animals. When a combination of relatively simple behaviors of particular individuals produces complex interactions, then we can talk about swarm or collective intelligence. The PSO is a good example of a computational intelligence algorithm inspired by the natural behavior of animal herds, e.g., a school of fish or a flock of birds. Both fish and birds are able to coordinate the movement of the flock without the need to communicate directly. The PSO was proposed by Kennedy and Eberhart (1995) in order to solve the function optimization problem. In the PSO, a swarm (population) of particles moves around in the solution search space. Each particle has a position that corresponds to a solution to the problem being solved and velocity that shows the direction and velocity of the movement. A movement of a particle is also influenced by its local best known position, typically denoted by p Best, and the best position found by the swarm (usually denoted by g Best). This indirect interaction of the particles is intended to guide the swarm toward solutions of good quality. The relative simplicity and efficiency of the PSO was a key to its many successful applications, also in solving the discrete combinatorial optimization problems. The behavior of certain species of ants that forage for food provided inspiration for creating ant colony algorithms (Dorigo and Stützle 2010). While searching for food, these ants lay a certain amount of pheromone as they move. At the beginning, they randomly choose the direction of the search process. With each subsequent trip from the nest to a food source and back, over time ants will deposit the largest amount of pheromone on the shortest path because much of the pheromone that has been laid on longer paths will evaporate before being reinforced by returning ants. After a while, a given pheromone trail will be reinforced to such an extent that when beginning its trip, each subsequent ant will follow that pheromone trail that has been laid down on the shortest path from the nest to a food source. In this case, pheromone acts as a means of indirect communication and in a way as collective memory. Ant colony algorithms and particle swarm optimization are the most popular methods that were inspired by natural collective animals behavior. It is easy to compare two algorithms based on computational experiments if these algorithms can be run on the same input data. As for the traveling salesman problem, the TSPLIB library facilitates such comparisons (Reinelt 1995). Apart from data themselves, this library also provides optimal solutions for most problems. However, there is no such library for the dynamic traveling salesman problem. One of the aims of this paper is to create a new approach for testing the quality of results that are obtained for the DTSP. 123 A library of the DTSP instances that have been generated contains information about optimal solutions for every subproblem which allows for a simple and precise evaluation of the performance of the algorithms. This library was used to compare the swarm intelligence algorithms investigated in this paper, i.e., the DPSO, the MMAS and the PACO. 1.1 Contributions Compared to our previous articles on the DPSO for the DTSP (described briefly in Sect. 2), the research has been extended in a few directions. (...truncated)