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...ntroduction In the early nineties an algorithm called ant system was proposed as a novel heuristic approach for the solution of combinatorial optimization problems (Dorigo et al., 1991; Dorigo, 1992; =-=Dorigo et al., 1996-=-). Ant system (AS), which was first applied to the traveling salesman problem, was recently extended and/or modified both to improve its performance and to apply it to other optimization problems. Imp...

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...ng salesman problem, was recently extended and/or modified both to improve its performance and to apply it to other optimization problems. Improved versions of AS include, among others, ACS (Dorigo & =-=Gambardella, 1997-=-), MAX --MIN Ant System (St utzle & Hoos, 1998b), and AS rank (Bullnheimer et al., 1997b). All these algorithms have been applied to the TSP with varying degree of success, but always improving over A...

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...n general, it is important to note that they are an interesting approach only for those shortest path problems to which more classical algorithms like dynamic programming or label correcting methods (=-=Bertsekas, 1995-=-) cannot be efficiently applied. This is the case, for example, for the following types of shortest path problems: . NP-hard problems, for which the dimension of the full state-space graph is exponent...

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...wing ones: (i) it is relatively easy to adapt the ant colony metaphor to it, (ii) it is a very difficult problem (NP-hard), (iii) it is one of the most studied problems in combinatorial optimization (=-=Lawler et al., 1985-=-; Reinelt, 1994), and (iv) it is very easy to state and explain it, so that the algorithm behavior is not obscured by too many technicalities. In the following we will present Ant System, a paradigmat...

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...o that obtainable by more sophisticated methods. More research will be necessary to assess whether ACO algorithms can reach the performance of state-of-the-art algorithms like Iterated Lin-Kernighan (=-=Johnson & McGeoch, 1997-=-). As in the TSP case, the first application of an ACO algorithm to the QAP has been that of Ant System (Maniezzo et al., 1994). In the last two years several ACO algorithms for the QAP have been pres...

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...re performed. The triggering of pheromone evaporation happens after all ants have completed their tours. Of course, it would be easy to add a local optimization daemon action, like a 3-opt procedure (=-=Lin, 1965-=-); this has been done in most of the ACO algorithms for the TSP that have been developed as extensions of AS (see for example (Dorigo & Gambardella, 1997; St utzle & Hoos, 1998a)). The amount of phero...

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...re quadratic assignment (Maniezzo et al., 1994; St utzle & Hoos, 1998a), vehicle routing (Bullnheimer et al., 1997a; Gambardella et al., 1999), connection oriented and connectionless network routing (=-=Schoonderwoerd et al., 1996-=-; Di Caro & Dorigo, 1998; Di Caro & Dorigo, 1998), sequential ordering (Gambardella & Dorigo, 1997), graph coloring (Costa & Hertz, 1997), shortest common supersequence (Michel & Middendorf, 1998), si...

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...t application of an ACO algorithm to the QAP has been that of Ant System (Maniezzo et al., 1994). In the last two years several ACO algorithms for the QAP have been presented by Maniezzo and Colorni (=-=Maniezzo & Colorni, 1999-=-, in press), Maniezzo (Maniezzo, 1998), and Stützle (Stützle & Hoos, 1998a). Currently, ACO algorithms are among the best heuristics for attacking reallife QAP instances 3 . We refer to (Stützle & Dor...

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...nal AS, have also been applied to a diverse set of optimization problems. Examples are quadratic assignment (Maniezzo et al., 1994; Stützle & Hoos, 1998a), vehicle routing (Bullnheimer et al., 1997a; =-=Gambardella et al., 1999-=-), connection oriented and connectionless network routing (Schoonderwoerd et al., 1996; Di Caro & Dorigo, 1998; Di Caro & Dorigo, 1998), sequential ordering (Gambardella & Dorigo, 1997), graph colorin...

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...connection oriented and connectionless network routing (Schoonderwoerd et al., 1996; Di Caro & Dorigo, 1998; Di Caro & Dorigo, 1998), sequential ordering (Gambardella & Dorigo, 1997), graph coloring (=-=Costa & Hertz, 1997-=-), shortest common supersequence (Michel & Middendorf, 1998), single machine tardiness (Bauer et al., 1999), multiple knapsack (Leguizamón & Michalewicz, 1999), etc. For many of these problems, the re...

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...s performance and to apply it to other optimization problems. Improved versions of AS include, among others, ACS (Dorigo & Gambardella, 1997), MAX –MIN Ant System (Stützle & Hoos, 1998b), and ASrank (=-=Bullnheimer et al., 1997-=-b). All these algorithms have been applied to the TSP with varying degree of success, but always improving over AS performance. These improved versions, as well as the original AS, have also been appl...

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...7). They obtained excellent results and were able to improve the best known solutions for many benchmark instances. A first application of AS to the job-shop scheduling problem has been presented in (=-=Colorni et al., 1994-=-). Despite showing the viability of the approach, the computational results are not competitive with state-of-the-art algorithms. More recently, MMAS has been applied to the flow shop scheduling probl...

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...lications. 1 Introduction In the early nineties an algorithm called ant system was proposed as a novel heuristic approach for the solution of combinatorial optimization problems (Dorigo et al., 1991; =-=Dorigo, 1992-=-; Dorigo et al., 1996). Ant system (AS), which was first applied to the traveling salesman problem, was recently extended and/or modified both to improve its performance and to apply it to other optim...

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...roblems using ACO algorithms are the traveling salesman problem and the quadratic assignment problem (QAP). Results obtained by the application of ACO algorithms to the TSP are very encouraging (see (=-=Stützle & Dorigo, 1999-=-b) for an overview of applications of ACO algorithms to the TSP): they are often better than those obtained using other general purpose heuristics like evolutionary computation or simulated annealing....

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...sed an extension of AS to assignment type problems and present an application of their approach to the graph coloring prob3 Another very good algorithm inspired by the ant colony metaphor is HAS-QAP (=-=Gambardella et al., 1997-=-). It is not listed with the others above because it doesn’t follow the ACO meta-heuristic.slem obtaining results comparable to those obtained by other heuristic approaches. An application of AS rank ...

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...s been that of Ant System (Maniezzo et al., 1994). In the last two years several ACO algorithms for the QAP have been presented by Maniezzo and Colorni (Maniezzo & Colorni, 1999, in press), Maniezzo (=-=Maniezzo, 1998-=-), and St utzle (St utzle & Hoos, 1998a). Currently, ACO algorithms are among the best heuristics for attacking reallife QAP instances 3 . We refer to (St utzle & Dorigo, 1999a) for an overview of the...

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...roblems using ACO algorithms are the traveling salesman problem and the quadratic assignment problem (QAP). Results obtained by the application of ACO algorithms to the TSP are very encouraging (see (=-=Stützle & Dorigo, 1999-=-b) for an overview of applications of ACO algorithms to the TSP): they are often better than those obtained using other general purpose heuristics like evolutionary computation or simulated annealing....

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...ng salesman problem, was recently extended and/or modified both to improve its performance and to apply it to other optimization problems. Improved versions of AS include, among others, ACS (Dorigo & =-=Gambardella, 1997-=-), MAX --MIN Ant System (St utzle & Hoos, 1998b), and AS rank (Bullnheimer et al., 1997b). All these algorithms have been applied to the TSP with varying degree of success, but always improving over A...

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...s performance and to apply it to other optimization problems. Improved versions of AS include, among others, ACS (Dorigo & Gambardella, 1997), MAX –MIN Ant System (Stützle & Hoos, 1998b), and ASrank (=-=Bullnheimer et al., 1997-=-b). All these algorithms have been applied to the TSP with varying degree of success, but always improving over AS performance. These improved versions, as well as the original AS, have also been appl...

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...ded andor modified both to improve its performance and to apply it to other optimization problems. Improved versions of AS include, among others, ACS (Dorigo & Gambardella, 1997), MAX-MZN Ant System (=-=Stutzle & Hoos, 1998-=-b), and Asrank (Bullnheimer et al., 1997b). All these algorithms have been applied to the TSP with varying degree of success, but always improving over AS performance. These improved versions, as well...

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...owing the viability of the approach, the computational results are not competitive with state-of-the-art algorithms. More recently, MMAS has been applied to the flow shop scheduling problem (FSP) in (=-=Stützle, 1998-=-). The computational results have shown that MMAS outperforms earlier proposed Simulated Annealing algorithms, while it performs comparably to Tabu Search. Another very recent application is that to t...

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...dering (Gambardella & Dorigo, 1997), graph coloring (Costa & Hertz, 1997), shortest common supersequence (Michel & Middendorf, 1998), single machine tardiness (Bauer et al., 1999), multiple knapsack (=-=Leguizamón & Michalewicz, 1999-=-), etc. For many of these problems, the results obtained place the ant system based approaches among the best available heuristics. In this paper we present the Ant Colony Optimization (ACO) meta-heur...

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...g (Schoonderwoerd et al., 1996; Di Caro & Dorigo, 1998; Di Caro & Dorigo, 1998), sequential ordering (Gambardella & Dorigo, 1997), graph coloring (Costa & Hertz, 1997), shortest common supersequence (=-=Michel & Middendorf, 1998-=-), single machine tardiness (Bauer et al., 1999), multiple knapsack (Leguizamón & Michalewicz, 1999), etc. For many of these problems, the results obtained place the ant system based approaches among ...

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...relatively easy to adapt the ant colony metaphor to it, (ii) it is a very difficult problem (NP-hard), (iii) it is one of the most studied problems in combinatorial optimization (Lawler et al., 1985; =-=Reinelt, 1994-=-), and (iv) it is very easy to state and explain it, so that the algorithm behavior is not obscured by too many technicalities. In the following we will present Ant System, a paradigmatic example of h...

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...lgorithm based on the island model typical of many parallel genetic algorithms implementations. MMAS has recently been applied to the generalized assignment problem by Ramalhinho Lourençou and Serra (=-=Lourenço & Serra, 1998-=-), obtaining very promising computational results. In particular, their algorithm is shown to find optimal and near optimal solutions faster than a GRASP algorithm which was used for comparison. Anoth...

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...rview of existing applications. 1 Introduction In the early nineties an algorithm called ant system was proposed as a novel heuristic approach for the solution of combinatorial optimization problems (=-=Dorigo et al., 1991-=-; Dorigo, 1992; Dorigo et al., 1996). Ant system (AS), which was first applied to the traveling salesman problem, was recently extended and/or modified both to improve its performance and to apply it ...

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..., and Agazzi to vehicle routing problems with time windows improves the best known solutions for some benchmark instances (Gambardella et al., 1999). Michel and Middendorf (Michel & Middendorf, 1998; =-=Michel & Middendorf, 1999-=-) have proposed an application of AS to the shortest common supersequence. They introduce the use of a lookahead function during the solution construction by the ants. They also present a parallel ver...

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...ng the viability of the approach, the computational results are not competitive with state-of-the-art algorithms. More recently, MMAS has been applied to the flow shop scheduling problem (FSP) in (St =-=utzle, 1998-=-). The computational results have shown that MMAS outperforms earlier proposed Simulated Annealing algorithms, while it performs comparably to Tabu Search. Another very recent application is that to t...