The classical transportation problem is actually well known both in theory and numerical resolution. We are interested in the multi-subscripts capacitated transportation problem of axial sum launched by specialists some years ago. Our work deals with the capacitated problem with four subscripts for which we have established an existence criterion, an optimality condition and an algorithm of resolution.
In this paper, we introduce an approach for scheduling problems of n tasks on m identical parallel machines with unavailability periods. This problem is strongly NP-complete which makes finding an optimal solution looks impossible task. In this frame, we suggest a novel heuristic in which availability periods of each machine are filled with the highest weighted tasks. To improve the performance of this heuristic, we have used, on one hand, different diversification strategies with the aim of exploring unvisited regions of the solution space, and on the other hand, two well-known neighborhoods (neighborhood by swapping and neighborhood by insertion). The computational experiment was carried out on three identical parallel machines with different availability periods. It must be mentioned that tasks movement can be within one machine or between different machines. The performance criterion to optimize in this problem is the weighted sum of the end dates of tasks. Note that all data in this problem are integer and deterministic.
In this paper, we will solve the four index fully fuzzy transportation problem (\textit{FFTP$_{4}$}) with some adapted classical methods. All problem's data will be presented as fuzzy numbers. In order to defuzificate these data, we will use the ranking function procedure. Our method to solve the \textit{FFTP$_{4}$} composed of two phases; in the first one, we will use an adaptation of well-known algorithms to find an initial feasible solution, which are the least cost, Russell's approximation and Vogel's approximation methods. In the second phase, we will test the optimality of the initial solution, if it is not optimal, we will improve it. A numerical analysis of the proposed methods is performed by solving different examples of different sizes; it is determined that they are stable, robust, and efficient. A proper comparative study between the adapted methods identifies the suitable method for solving \textit{FFTP$_{4}$}.
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