ABSTRACT In this paper, we propose a new heuristic strategy to solve linear integer mathematical problems. The strategy begins by finding the optimal solution of the continuous associated problem and the simplex directions from that optimal solution. A total of n − m problems are generated and solved by the strategy, where m is the number of constraints and n is the number of variables of the problem. The best solution to those problems is the solution to the original problem. Several instance problems were randomly generated to validate the strategy, and our strategy finds good solutions in 80% of the instances. We also tested the proposed strategy with instances of the MIPLIB 2017. We compare a total of 39 instances and on average our strategy performed 47% fewer iterations than the solver, in 34 of 39 instances, the strategy found good solutions.

ABSTRACT In this paper, we present the integration of two problems related to the operations in a port terminal: the Berth Allocation Problem (BAP) integrated with the Machine Assignment Problem. We present a mixed-integer linear programming (MILP) formulation, capable of assigning and scheduling incoming vessels to berthing positions and the assignment of machines for handling the vessels. The machines can be quay cranes, mobile cranes, straddle carriers, forklifts, trucks, and any other machine. The problem aims to minimize the waiting time plus the handling time of the vessels. To solve the problem, we developed a heuristic algorithm, capable of solving a problem instance in seconds. To compare the results, we generate several instance problems based on real data and solve them with our MILP formulation implemented in a solver, our heuristic, and a First In First Out (FIFO) algorithm. The solver was able to find solutions only in small-scale instances, and the heuristic was able to find good solutions for all instances.

ABSTRACT Maritime shipping is vital to worldwide commerce. Due to the high flow in ports throughout the world, the efficient allocation of vessels in berths has become a problem. A new mathematical model and several algorithms are proposed in this paper to planning the allocation of the vessels in berths and the allocation of resources to the service of each vessel. Those resources, in general, are machines to load or unload vessels. The mathematical model was implemented on Cplex and can solve small scale instances, due to its high complexity. To solve larger instances, a genetic algorithm-based metaheuristic, a first-in first-out heuristic, and a machine allocation algorithm are also proposed in this paper. The model and the algorithms produce very useful and interesting results. Comparing, the results produced by the GA are, on average, 94% better than the results of the Cplex and 26% better than the results of FIFO.

This article presents improvements to the hybrid preconditioner previously developed for the solution through the conjugate gradient method of the linear systems which arise from interior-point methods. The hybrid preconditioner consists of combining two preconditioners: controlled Cholesky factorization and the splitting preconditioner used in different phases of the optimization process. The first, with controlled fill-in, is more efficient at the initial iterations of the interior-point methods and it may be inefficient near a solution of the linear problem when the system is highly ill-conditioned; the second is specialized for such situation and has the opposite behavior. This approach works better than direct methods for some classes of large-scale problems. This work has proposed new heuristics for the integration of both preconditioners, identifying a new change of phases with computational results superior to the ones previously published. Moreover, the performance of the splitting preconditioner has been improved through new orderings of the constraint matrix columns allowing savings in the preconditioned conjugate gradient method iterations number. Experiments are performed with a set of large-scale problems and both approaches are compared with respect to the number of iterations and running time.

Neste trabalho são desenvolvidos o estudo e implementação de métodos de pontos interiores específicos para o problema de planejamento do tratamento de câncer por radioterapia. Este é um problema de grande porte que contém uma estrutura matricial particular. Esta estrutura é explorada de forma eficiente obtendo um sistema linear de dimensão muito menor. Além disso, o sistema obtido é esparso, simétrico e definido positivo. Resultados numéricos em Matlab mostram que problemas de grande porte podem ser resolvidos em poucas iterações e baixo tempo computacional por esta abordagem.

In this work, a specialized interior point method is developed for planning cancer treatment by radiotherapy. This is a large-scale problem with a specific matrix structure. That structure is explored in an efficient way reducing the dimension of the linear system, which must be solved at each iteration. Moreover, the system obtained is sparse, symmetric and positive definite. Matlab numerical results show that large-scale problems can be solved in few iterations and short computational time by this approach.

Neste trabalho consideramos o problema combinado, que acopla os problemas de dimensionamento de lotes e de corte de estoque, incluindo uma formulação matemática deste problema. Consideramos algumas propriedades da matriz de restrições deste modelo e como construir uma base esparsa para ela, utilizando um reordenamento estático das colunas. Resultados numéricos de uma implementação que realiza trocas de colunas básicas e verifica sua esparsidade, simulando o método simplex são apresentados. Experimentos numéricos também comprovam a robustez desta abordagem. Concluímos que a proposta de construção da base estática esparsa leva a bons resultados computacionais com relação à velocidade e robustez em comparação com abordagens que não consideram a estrutura esparsa da matriz.

In this work the combined problem is considered, which solves simultaneously the lot sizing and the cutting stock problems. We study some properties of the matrix of constraints and how to factorize the base without losing sparsity in the simplex method context, by a static reordering of the basic columns. Numerical results simulating simplex iterations and verify the sparsity of the factorizations are presented. Numerical experiments had also proven the robustness of this strategy. We conclude that the approach of constructing of the static sparse base reordering leads to very good computational results for both: speed and robustness, in comparison with approaches which do not consider the sparse structure of the matrix of constraints.