this paper the concept of chaotic attractor is used only as an example of a dynamic behavior that could a#ect the search process, we summarize the main characteristics and refer to [13] for a detailed theoretical analysis. Chaotic attractors are characterized by a "contraction of the areas", so that trajectories starting with di#erent initial conditions will be compressed in a limited area of the configuration space, and by a "sensitive dependence upon the initial conditions", so that di#erent trajectories will diverge. For an analytical characterization of this sensitive dependence, it is convenient to introduce the concept of Lyapunov exponent. Let us consider the function g that maps the point at step n

The field of metaheuristics for the application to combinatorial optimization problems is a rapidly growing field of research. This is due to the importance of combinatorial optimization problems for the scientific as well as the industrial world. We give a survey of the nowadays most important metaheuristics from a conceptual point of view. We outline the different components and concepts that are used in the different metaheuristics in order to analyze their similarities and differences. Two very important concepts in metaheuristics are intensification and diversification. These are the two forces that largely determine the behaviour of a metaheuristic. They are in some way contrary but also complementary to each other. We introduce a framework, that we call the I&D frame, in order to put different intensification and diversification components into relation with each other. Outlining the advantages and disadvantages of different metaheuristic approaches we conclude by pointing out the importance of hybridization of metaheuristics as well as the integration of metaheuristics and other methods for optimization.

We describe an artificial ant colony capable of solving the travelling salesman problem (TSP). Ants of the artificial colony are able to generate successively shorter feasible tours by using information accumulated in the form of a pheromone trail deposited on the edges of the TSP graph. Computer simulations demonstrate that the artificial ant colony is capable of generating good solutions to both symmetric and asymmetric instances of the TSP. The method is an example, like simulated annealing, neural networks and evolutionary computation, of the successful use of a natural metaphor to design an optimization algorithm.

. A new hybrid procedure that combines genetic operators to existing heuristics is proposed to solve the Quadratic Assignment Problem (QAP). Genetic operators are found to improve the performance of both local search and tabu search. Some guidelines are also given to design good hybrid schemes. These hybrid algorithms are then used to improve on the best known solutions of many test problems in the literature. 1. Introduction The quadratic assignment problem (QAP) can be stated as: min OE2P (n) n X i=1 n X j=1 a ij b OE(i)OE(j) ; where A = (a ij ) and B = (b kl ) are two n \Theta n matrices and P (n) is the set of all permutations of f1; :::; ng. Matrix A is often referred to as a distance matrix between sites, and B as a flow matrix between objects. In most cases, the matrices A and B are symmetrical with a null diagonal. A permutation may then be interpreted as an assignment of objects to sites with a quadratic cost associated to it. There are many applications that can be fo...

Ant System, the first Ant Colony Optimization algorithm, showed to be a viable method for attacking hard combinatorial optimization problems. Yet, its performance, when compared to more fine-tuned algorithms, was rather poor for large instances of traditional benchmark problems like the Traveling Salesman Problem. To show that Ant Colony Optimization algorithms could be good alternatives to existing algorithms for hard combinatorial optimization problems, recent research in this ares has mainly focused on the development of algorithmic variants which achieve better performance than AS. In this article, we present ¨�©� � –¨��� � Ant System, an Ant Colony Optimization algorithm derived from Ant System. ¨�©� � –¨��� � Ant System differs from Ant System in several important aspects, whose usefulness we demonstrate by means of an experimental study. Additionally, we relate one of the characteristics specific to ¨� ¨ AS — that of using a greedier search than Ant System — to results from the search space analysis of the combinatorial optimization problems attacked in this paper. Our computational results on the Traveling Salesman Problem and the Quadratic Assignment Problem show that ¨�©� � – ¨��� � Ant System is currently among the best performing algorithms for these problems.

The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using a state-of-the-art branch-and-bound algorithm running on a federation of geographically distributed resources known as a computational grid. Solution of QAPs of unprecedented complexity, including the nug30, kra30b, and tho30 instances, is reported.

In this paper, a fitness landscape analysis for several instances of the quadratic assignment problem (QAP) is performed and the results are used to classify problem instances according to their hardness for local search heuristics and meta-heuristics based on local search. The local properties of the tness landscape are studied by performing an autocorrelation analysis, while the global structure is investigated by employing a fitness distance correlation analysis. It is shown that epistasis, as expressed by the dominance of the flow and distance matrices of a QAP instance, the landscape ruggedness in terms of the correlation length of a landscape, and the correlation between fitness and distance of local optima in the landscape together are useful for predicting the performance of memetic algorithms - evolutionary algorithms incorporating local search - to a certain extent. Thus, based on these properties a favorable choice of recombination and/or mutation operators can be found.

Introduction The quadratic assignment problem (QAP) is an important problem in theory and practice. Many practical problems like backboard wiring [33], campus and hospital layout [8, 14], typewriter keyboard design [4], scheduling [18] and many others [13, 22] can be formulated as QAPs. The QAP can best be described as the problem of assigning a set of facilities to a set of locations with given distances between the locations and given flows between the facilities. The goal then is to place the facilities on locations in such a way that the sum of the product between flows and distances is minimal. More formally, given n facilities and n locations, two n \Theta n matrices A = [a ij ] and B = [b rs ], where a ij is the distance between locations i and j and b rs<F

We present HAS-SOP, a new approach to solving sequential ordering problems. HAS-SOP combines the ant colony algorithm, a population-based metaheuristic, with a new local optimizer, an extension of a TSP heuristic which directly handles multiple constraints without increasing computational complexity. We compare different implementations of HASSOP and present a new data structure that improves system performance. Experimental results on a set of twenty-three test problems taken from the TSPLIB show that HAS-SOP outperforms existing methods both in terms of solution quality and computation time. Moreover, HAS-SOP improves most of the best known results for the considered problems.

This paper presents HAS-QAP, a hybrid ant colony system coupled with a local search, applied to the