For a dynamic traveling salesman problem (DTSP), the weights (or traveling times) between two cities (or nodes) may be subject to changes. Ant colony optimization (ACO) algorithms have proved to be powerful methods to tackle such problems due to their adaptation capabilities. It has been shown that the integration of local search operators can significantly improve the performance of ACO. In this paper, a memetic ACO algorithm, where a local search operator (called unstring and string) is integrated into ACO, is proposed to address DTSPs. The best solution from ACO is passed to the local search operator, which removes and inserts cities in such a way that improves the solution quality. The proposed memetic ACO algorithm is designed to address both symmetric and asymmetric DTSPs. The experimental results show the efficiency of the proposed memetic algorithm for addressing DTSPs in comparison with other state-of-the-art algorithms.

The improvement in the performance of computers and mathematical programming techniques has led to the development of a new class of algorithms called matheuristics. Associated with an improvement of Mixed Integer Programming (MIP) solvers, these methods have successfully solved plenty of combinatorial optimization problems. This paper presents a matheuristic approach that hybridizes local search based metaheuristics and mathematical programming techniques to solve the capacitated p-median problem. The proposal considers reduced mathematical models obtained by a heuristic elimination of variables that are unlikely to belong to a good or optimal solution. In addition, a partial optimization algorithm based on the reduction is proposed. All mathematical models are solved by an MIP solver. Computational experiments on five sets of instances confirm the good performance of our approach.The first work on the CPMP appeared in scientific literature in the 1980s (Mulvey and Beck, 1984;Pirkul, 1987). Osman and Christofides (1994) used a hybrid approach that combines simulated annealing and tabu search and randomly generated 20 instances with size ranging from 50 to 100 customers to test the proposed methods. Maniezzo et al. (1998) presented an evolutionary method and an effective local search technique to solve the CPMP. Computational results showed the effectiveness of the proposed approach on five sets of instances, including those proposed by Osman and Christofides. More recently, Baldacci et al. (2002) proposed a new method based on a set partitioning formulation. The authors presented computational results on instances from the literature and proposed new sets of test problems with additional constraints: bounds on the cluster cardinality and incompatibilities between entities. Senne (2002, 2004) presented a column-generation method integrated to Lagrangean/surrogate relaxation to calculate lower bounds. Their proposed method identifies new productive columns, accelerating the computational process. Computational results were presented on instances generated based on a geographic database from the city São José dos Campos. Ahmadi and Osman (2005) proposed a combination of metaheuristics in a framework called GRAMPS (greedy random adaptive memory search method). A scatter search approach was proposed by Scheuerer and Wendolsky (2006), who evaluated it on instances from the literature, obtaining several new best solutions. Díaz and Fernández (2006) presented a hybrid scatter search and path relinking algorithm. The authors have run a series of computational experiments evaluating the proposed methods on instances from the literature, including instances corresponding to 737 cities in Spain. Both algorithms were evaluated separately; however, the combination of path relinking and scatter search gave the best results. Fleszar and Hindi (2008) solved the CPMP using variable neighborhood search to define sets of medians and the CPLEX package to solve assignment problems. Chaves et al. (2007) presented a hybrid heuristic ...

This article proposes algorithms for the Minmax version of the m-Traveling Salesman Problem in which the objective is to minimize the length of the longest route. A tabu search heuristic and two exact search schemes are developed. Problems involving up to 50 vertices are solved to optimality.

Ant colony optimization (ACO) algorithms have proved to be able to adapt for solving dynamic optimization problems (DOPs). The integration of local search algorithms has also proved to significantly improve the output of ACO algorithms. However, almost all previous works consider stationary environments. In this paper, the MAX -MIN Ant System, one of the best ACO variations, is integrated with the unstringing and stringing (US) local search operator for the dynamic travelling salesman problem (DTSP). The best solution constructed by ACO is passed to the US operator for local search improvements. The proposed memetic algorithm aims to combine the adaptation capabilities of ACO for DOPs and the superior performance of the US operator on the static travelling salesman problem in order to tackle the DTSP. The experiments show that the MAX -MIN Ant System is able to provide good initial solutions to US and the proposed algorithm outperforms other peer ACObased memetic algorithms on different DTSPs.

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Ant colony optimization (ACO) algorithms have proved to be suitable for solving dynamic optimization problems (DOPs). The integration of local search operators with ACO has also proved to significantly improve the output of ACO algorithms. However, almost all previous works of ACO in DOPs do not utilize local search operators. In this work, the MAX-MIN Ant System (MMAS), one of the best ACO variations, is integrated with advanced and effective local search operators, i.e., the Lin-Kernighan and the Unstringing and Stringing heuristics, resulting in powerful memetic algorithms. The best solution constructed by ACO is passed to the operator for local search improvements. The proposed memetic algorithms aim to combine the adaptation capabilities of ACO for DOPs and the superior performance of the local search operators. The travelling salesperson problem is used as the base problem to generate both symmetric and asymmetric dynamic test cases. Experimental results show that the MMAS is able to provide good initial solutions to the local search operators especially in the asymmetric dynamic test cases.

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