Elizabeth Ferreira Gouvêa Goldbarg

Particle Swarm Optimization Algorithm for the Traveling Salesman Problem

Particle swarm optimization, PSO, is an evolutionary computation technique inspired in the behavior of bird flocks. PSO algorithms were first introduced by Kennedy & Eberhart (1995) for optimizing continuous nonlinear functions. The fundamentals of this metaheuristic approach rely on researches where the movements of social creatures were simulated by computers (Reeves, 1983; Reynolds, 1987; Heppner & Grenander, 1990). The research in PSO algorithms has significantly grown in the last few years and a number of successful applications concerning single and multi-objective optimization have been presented (Kennedy& Eberhart, 2001; Coello et al., 2004). This popularity is partially due to the fact that in the canonical PSO algorithm only a small number of parameters have to be tuned and also due to the easiness of implementation of the algorithms based on this technique. Motivated by the success of PSO algorithms with continuous problems, researchers that deal with discrete optimization problems have investigated ways to adapt the original proposal to the discrete case. In many of those researches, the new approaches are illustrated with the Traveling Salesman Problem, TSP, once it has been an important test ground for most algorithmic ideas. Given a graph

A Memetic Algorithm for the Car Renter Salesman Problem

The Traveling Salesman Problem (TSP) is a classic Combinatorial Optimization problem. Given a graph G=(N,M), where N={1,...,n} is the set of nodes and M={1,...,m} is the set of edges, and costs, cij, associated with each edge connecting vertices i and j, the problem consists in finding the minimum length Hamiltonian cycle. The TSP is NP-hard (Garey & Johnson, 1979) and one of the combinatorial optimization problems more intensively investigated. The size of the larger non trivial TSP instance solved by an exact method evolved from 318 cities in the 80’s (Crowder & Padberg, 1980), to 7397 cities in the 90’s (Applegate et al., 1994) and 24978 cities in 2004. The best mark was reached in 2006 with the solution of an instance with 85900 cities (Applegate et al., 2006). The TSP has several important practical applications and a number of variants (Gutin & Punnen, 2002). Some of these variants are classic such as the Peripatetic Salesman (Krarup, 1975) and the M-tour TSP (Russel, 1977), other variants are more recent such as the Colorful TSP (Xiong et al., 2007) and the Robust TSP (Montemanni et al., 2007), among others. A new TSP variant is introduced in this chapter named The Car Renter Salesman Problem (CaRS). It models important applications in tourism and transportation areas and represents a complex variant that challenges the state of the art. In this paper the new problem and some variations are presented, its complexity is analyzed and some related problems are briefly overviewed. A memetic algorithm is proposed for the problem and it is compared to a hybrid GRASP/VND algorithm. CaRS Problem is introduced in Section 2, where several conditions under which this variant can be presented are introduced. Section 3 presents two metaheuristic methods for the investigated problem. In order to compare the performance of the proposed approaches, a set of instances introduced for the new problem, named CaRSLib. This set contains Euclidean and non-Euclidean symmetric instances with number of cities ranging from 14 to 300 and number of cars between 2 and 5. A set of 40 instances is used in the computational experiments. The heuristics proposed in Section 3 establish the first upper limits for solutions of CaRSLib of instances. The results of computational experiments comparing the performance of the proposed approaches are presented in Section 4. Statistical tests are applied to support conclusions on the behavior of the proposed algorithms. According to

Evolutionary algorithms for a three-objectives oil derivatives network problem

To distribute oil derivatives by multi-product pipelines is an important problem faced by the petroleum industry. Some researchers deal with it as a discrete problem where batches of products flow in a network. Minimizing delivery time is a usual objective handled by engineers when scheduling products in pipeline networks. Nevertheless, other costs may also be considered such as losses due to interfaces and electrical energy. Losses due to interfaces occur when different products sent consecutively contaminate each other. Usually, no separation devices exist between batches of different products and losses due to interface can be significant. The price paid for electrical energy varies during the day, so it is important also to try to minimize this cost. These three minimization objectives are considered, simultaneously, i.e. delivery time at demand nodes, interface losses and electrical energy cost. A transgenetic algorithm is proposed and applied to thirty random instances. The results obtained with the proposed method are compared with those produced by an NSGAII algorithm.

Transgenetic Algorithms for the Multi-objective Quadratic Assignment Problem

The multi-objective Quadratic Assignment Problem (mQAP) is a hard optimization problem with many real-world applications, such as in hospital layouts. The main purposes of this paper are: (1) the investigation of hybrid algorithms combining Transgenetic Algorithms and Evolutionary Multi-objective Optimization (EMO) frameworks to deal with mQAP and (2) to compare the ability of EMO algorithms based on Pareto dominance with those based on decomposition to deal with the mQAP. Transgenetic Algorithms (TAs) are evolutionary algorithms based on cooperation as the main evolutionary strategy. Two hybrid algorithms are proposed to deal with the mQAP: NSTA (TA + NSGA-II) and MOTA/D (TA + MOEA/D). To analyze the performance of the proposed algorithms, non-parametric statistical tests and multi-objective quality indicators are used. The proposed algorithms are compared with NSGA-II and MOEA/D in 126 instances of the mQAP. The results demonstrate the superiority of decomposition and transgenetic based algorithms, particularly in MOTA/D.

Analyzing Limited Size Archivers of Multi-objective Optimizers

In the context of multi-objective optimization, where there may be many optimal incomparable solutions, most of the optimizers maintain a limited repository, to keep the objective vectors of the solutions found during the execution. There are several methods to decide which vectors remain in that limited size archive, and these different techniques may have properties that guarantee the diversity and quality of their outcomes. This paper examines some of those strategies, analyzing their properties, and comparing empirically their outputs based on two quality indicators, additive epsilon and hyper volume. Most of the archiving techniques studied in this work cannot ensure that at the end of the process their vectors are all optimal. Due to this fact, a new approach is presented, based on a second archive to store the points which would be discarded. The main idea is verify how much the recycled vectors could improve the generated set. In the realized tests, the method had not a significant time cost regardless the adopted archiving technique.