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 ...

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a b s t r a c tThe high school timetabling is a classical combinatorial optimization problem that takes a large number of variables and constraints into account. Due to its combinatorial nature, solving medium and large instances to optimality is a challenging task. When resources are tight, it is often difficult to find even a feasible solution. Among the different requirements that are considered in Brazilian schools, two compactness requirements must be met on a teacher's schedule: the minimization of working days and the avoidance of idle timeslots. In this paper, we present a mixed integer linear programming model and a fix-and-optimize heuristic combined with a variable neighborhood descent method. Our method uses three different types of decompositions -class, teacher and day -in order to solve the high school timetabling problem. The method is able to find new best known solutions for seven instances, including three optimal ones. A comparison with results reported in the literature shows that the proposed fix-and-optimize heuristic outperforms state-of-the-art techniques for the resolution of the problem at hand.

This paper deals with the container loading problem which involves the selection of a subset of boxes, each box with a given volume, such that they fit in a single container and maximize its volume utilization subject to orientation and stability constraints. We propose a multi-start random constructive heuristic with a load arrangement that is based on maximal cuboids that fit in given empty spaces. Each instance is adaptively evaluated by a set of criteria, and at each step of the construction process one maximal cuboid is chosen probabilistically from a restricted list of candidates. In order to enhance the flexibility in the construction of a solution, a probabilistic reduction on such cuboids is allowed. Computational tests on several instances from the literature show that the proposed method performs better than other approaches.Keywords: container loading; cuboid arrangement; multi-start random constructive heuristic. ResumoNeste trabalho abordamos o problema de carregamento de contêiner que trata da seleção de um subconjunto de caixas, cada caixa com um dado volume, de forma a maximizar o volume ocupado de um único contêiner sujeito a restrições de orientação e estabilidade. Propomos uma heurística construtiva aleatória com múltiplos inícios que utiliza um arranjo de carga baseado em cubóides que maximizam a ocupação de espaços vazios. Cada instância é avaliada de forma adaptativa por um conjunto de critérios, e em cada passo do processo construtivo um cubóide é selecionado probabilisticamente de uma lista restrita de candidatos. Para aumentar a flexibilidade na construção de uma solução, permite-se uma redução probabilística no tamanho dos cubóides. Resultados computacionais em instâncias da literatura mostram que o método proposto apresenta um desempenho superior a outros enfoques sugeridos na literatura.

Problems of scheduling batch-processing machines to minimize the makespan are widely exploited in the literature, mainly motivated by real-world applications, such as burn-in tests in the semiconductor industry. These problems consist of grouping jobs in batches and scheduling them on machines. We consider problems where jobs have non-identical sizes and processing times, and the total size of each batch cannot exceed the machine capacity. The processing time of a batch is defined as the longest processing time among all jobs assigned to it. Jobs can also have non-identical release times, and in this case, a batch can only be processed when all jobs assigned to it are available. This paper discusses four different versions of batch scheduling problems, considering a single processing machine or parallel processing machines, and considering jobs with or without release times. New mixed integer linear programming formulations are proposed as enhancements of formulations proposed in the literature, and symmetry breaking constraints are investigated to reduce the size of the feasible sets. Computational results show that the proposed formulations have a better performance than other models in the literature, being able to solve to optimality instances only considered before to be solved by heuristic procedures.

This paper addresses the problem of scheduling jobs in a single machine with sequence dependent setup times in order to minimize the total tardiness with respect to job due dates. We propose variants of the GRASP metaheuristic that incorporate memorybased mechanisms for solving this problem. There are two mechanisms proposed in the literature that utilize a long-term memory composed of an elite set of high quality and sufficiently distant solutions. The first mechanism consists of extracting attributes from the elite solutions in order to influence the construction of an initial solution. The second one makes use of path relinking to connect a GRASP local minimum with a solution of the elite set, and also to connect solutions from the elite set. Reactive GRASP, which probabilistically determines the degree of randomness in the GRASP construction throughout the iterations, is also investigated. Computational tests for instances involving up to 150 jobs are reported, and the proposed method is compared with heuristic and exact methods from the literature.

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