Abstract

Ant Colony Optimization (ACO) Despite being one of the youngest metaheuristics, the number of applications of ACO algorithms is very large. In principle, ACO can be applied to any combinatorial optimization problem for which some iterative solution construction mechanism can be conceived. Most applications of ACO deal with N P-hard combinatorial optimization problems, that is, with problems for which no polynomial time algorithms are known. ACO algorithms have also been extended to handle problems with multiple objectives, stochastic data and dynamically changing problem information. There are, as well, extensions of the ACO metaheuristic for dealing with problems with continuous decision variables. This chapter provides a concise overview of several noteworthy applications of ACO algorithms. This overview is necessarily incomplete because the number of currently available ACO applications goes into the hundreds. Our description of the applications follows the classification used in the 2004 book on ACO by Dorigo & Stützle [3] but extending the list there with many recent examples. 1 Applications to N P-hard problems ACO was primarily intended for solving combinatorial optimization problems, among which N P-hard problems are the most challenging ones. In fact, no polynomial-time algorithms are known for such problems, and therefore heuristic techniques such as ACO are often used for generating high-quality solutions in reasonable computation times. 1 Routing problems Routing problems involve one or more agents visiting a predefined set of locations, and the objective function and constraints depend on the order in which the locations are visited. Perhaps the best-known example is the traveling salesman problem (TSP) ACO algorithms have been successful in tackling various variants of the vehicle routing problem (VRP). The first application of ACO to the capacitated VRP was due to Bullnheimer et al. [15] used an ACO algorithm for a problem that combines the two-dimensional packing and the capacitated vehicle routing problem (2L-CVRP), showing that it outperforms a tabu search algorithm. In this problem, items of different sizes and weights are loaded in vehicles with a limited weight capacity and limited two-dimensional loading surface, and then they are distributed to the customers. Other variants of VRP with different loading constraints have also been tackled by means of ACO Scheduling problems Scheduling problems concern the assignment of jobs to one or various machines over time. Input data for these problems are processing times but also often additional setup times, release dates and due dates of jobs, measures for the jobs' importance and precedence constraints among jobs. Scheduling problems have been an important application area of ACO algorithms, and the currently available ACO applications in scheduling deal with many different job and machine characteristics. The single-machine total weighted tardiness problem (SMTWTP) has been tackled 4 by both den Besten et al. Finally, state-of-the-art results have been obtained in the car sequencing problem by the ACO algorithm proposed by Solnon [34], and these results have been further improved by Morin et al. [35] by means of a specialized pheromone model. The car sequencing problem has also been used as an example application by Khichane et al. [33] to explore the integration of constraint programming techniques into ACO algorithms. Subset problems The goal in subset problems is, generally speaking, to find a subset of the available items that minimizes a cost function defined over the items and that satisfies a number of constraints. This is a wide definition that can include other classes of problems. There are, however, two characteristic properties of the solutions to subset problems: The order of the solution components is irrelevant, and the number of components of a 5 solution may differ from solution to solution. An important subset problem is the set covering problem (SCP). Lessing et al. [41] compared the performance of a number of ACO algorithms for the (SCP), with and without the usage of a local search algorithm based on 3-flip neighborhoods A subset problem closely related to the capacitated VRP (CVRP) is the capacitated minimum spanning tree problem (CMST), which has been effectively tackled by a hybrid ACO algorithm [44] based on a previous ACO algorithm for the CVRP Finally, a class of problems for which ACO has recently shown competitive results is that of multi-level lot-sizing with Assignment and layout problems In assignment problems, a set of items has to be assigned to a given number of resources subject to some constraints. Probably, the most widely studied example is the quadratic assignment problem (QAP), which was among the first problems tackled by ACO algorithms The ANTS algorithm has also been applied to the frequency assignment problem (FAP), in which frequencies have to be assigned to links and there are constraints on the minimum distance between the frequencies assigned to each pair of links. ANTS showed good performance on some classes of FAP instances in comparison with other approaches Another notable example is the generalized assignment problem, where a set of tasks have to be assigned to a set of agents with a limited total capacity, minimizing the total assignment cost of tasks to agents. The MMAS algorithm proposed by Lourenço & Serra [61] tackled a real-world problem related to ambulance locations in Austria by means of an ACO algorithm; and Blum Machine learning problems Bioinformatics problems Computer applications to molecular biology (bioinformatics) have originated many N P-hard combinatorial optimization problems. We include in this section general problems that have attracted considerable interest due to their applications to bioinformatics. This is the case of the shortest common supersequence problem (SCSP), which is a well-known N P-hard problem with applications in DNA analysis. Michel & Middendorf [72, An important problem in bioinformatics is protein folding, that is, the prediction of a protein's structure based on its sequence of amino acids. A simplified model for protein folding is the two-dimensional hydrophobic-polar protein folding problem Interesting is also the work of Blum et al. Other problems in bioinformatics have been successfully tackled by means of ACO algorithms: Korb et al. [77] on a selection problem in biomarker identification, which combines ACO with support vector machines (SVM). Applications to problems with non-standard features We review in this section applications of ACO algorithms to problems having additional characteristics such as multiple objective functions, time-varying data and stochastic information about objective values or constraints. In addition, we mention applications of ACO to network routing and continuous optimization problems. Multi-objective optimization In many real-world problems, candidate solutions are evaluated according to multiple, often conflicting objectives. Sometimes the importance of each objective can be exactly weighted, and hence objectives can be combined into a single scalar value by using, for example, a weighted sum. This is the approach used by Doerner et al. When there is no a priori knowledge about the relative importance of objectives, the goal usually becomes to approximate the set of Pareto-optimal solutions-a solution is Pareto optimal if no other solution is better or equal for all objectives and strictly better in at least one objective. Iredi et al. Stochastic optimization problems In stochastic optimization problems, data are not known exactly before generating a solution. Rather, because of uncertainty, noise, approximation or other factors, what is available is stochastic information on the objective function value(s), on the decision variable values, or on the constraint boundaries. The first application of ACO algorithms to stochastic problems was to the probabilistic TSP (PTSP). In the PTSP, each city has associated a probability of requiring a visit, and the goal is to find an a priori tour of minimal expected length over all cities. Bianchi et al. [91] and Bianchi & Gambardella [92] proposed an adaptation of ACS for the PTSP. Very recently, this algorithm was improved by Balaprakash et al. [93], resulting in a state-of-the-art algorithm for the PTSP. Other applications of ACO to stochastic problems include vehicle routing problems with uncertain demands Dynamic optimization problems Dynamic optimization problems are those whose characteristics change while being solved. ACO algorithms have been applied to such versions of classical N P-hard problems. Notable examples are applications to dynamic versions of the TSP, where the distances between cities may change or where cities may appear or disappear Several other routing algorithms based on ACO have been proposed for a variety of wired network scenarios Continuous optimization problems Continuous optimization problems arise in a large number of engineering applications. Their crucial difference from combinatorial problems, which were the exclusive application field of ACO in the early research efforts, is that decision variables in such problems have a continuous, real-valued domain. Recently, various proposals have been made of how to handle continuous decision variables within the ACO framework 10 Industrial applications While most research is done on academic applications, commercial companies have started to use ACO algorithms for real-world applications Conclusions Nowadays, ACO is a well established metaheuristic with hundreds of successful implementations applied to a wide range of optimization problems. Several of these implementations have shown to be, at least at the time of their publication, the state of the art for the respective problems tackled, including problems such as vehicle routing, sequential ordering, quadratic assignment, assembly line balancing, open-shop scheduling, and various others. Applications of ACO to dynamic routing problems in telecommunication networks have been particularly successful, probably because several algorithm characteristics match well the features of the applications. By analysing the many available ACO implementations, one can identify ingredients necessary for the successful application of ACO. Firstly, an effective mechanism for iteratively constructing solutions must be available. Ideally, this construction mechanism exploits problem-specific knowledge by using appropriate heuristic information. Secondly, the best performing ACO algorithms have specialized features that allow to carefully control the balance between the exploration of new solutions and the intensification of the search around the best solutions. Such control mechanisms are offered by advanced ACO algorithms such as ACS or MMAS. In fact, the original Ant System has been abandoned by now in favor of better performing variants. Thirdly, the usage of local search algorithms for improving the solutions constructed by the ants is very successful in practice. Finally, the integration of other techniques such as constraint programming, tree search techniques or multi-level frameworks often yields a further improvement in performance or increases the robustness of the algorithms. Further information on ACO and related topics can be obtained by subscribing to the moderated mailing list aco-list, and by visiting the Ant Colony Optimization web page (www.aco-metaheuristic.org).