Abstract
In this paper, a new method for the solution of distribution problem in a fuzzy setting is presented. It consists of two phases. In the first of them, the problem is formulated as the classical, fully fuzzy transportation problem. A new, straightforward numerical method for solving this problem is proposed. This method is implemented using the -cut approximation of fuzzy values and the probability approach to interval comparison. The method allows us to provide the straightforward fuzzy extension of a simplex method. It is important that the results are fuzzy values. To validate our approach, these results were compared with those obtained using the competing method and those we got using the Monte–Carlo method. In the second phase, the results obtained in the first one (the fuzzy profit) are used as the natural constraints on the parameters of multiobjective task. In our approach to the solution of distribution problem, the fuzzy local criteria based on the overall profit and contracts breaching risks are used. The particular local criteria are aggregated with the use of most popular aggregation modes. To obtain a compromise solution, the compromise general criterion is introduced, which is the aggregation of aggregating modes with the use of level-2 fuzzy sets. As the result, a new two phase method for solving the fuzzy, nonlinear, multiobjective distribution problem aggregating the fuzzy local criteria based on the overall profit and contracts breaching risks has been developed. Based on the comparison of the results obtained using our method with those obtained by competing one, and on the results of the sensitivity analysis, we can conclude that the method may be successfully used in applications. Numerical examples illustrate the proposed method.
Highlights
The problem of optimal distribution and the transportation problem are similar mathematical formulations and may be considered as the particular cases of the general linear programming problem
The fully fuzzy transportation problem with triangular fuzzy numbers was solved in the case of three sources and four destinations
The α-cut representation of fuzzy values, fuzzy operation rules, and the probability approach to the intervals and fuzzy values comparison are used as the main mathematical tools
Summary
The problem of optimal distribution and the transportation problem are similar mathematical formulations and may be considered as the particular cases of the general linear programming problem. Based on the above literature analysis and especially on the comprehensive literature review presented recently in [12], we can say that the most informative solutions of fuzzy linear transportation problem and fuzzy linear distribution problem having similar mathematical formulations that may be obtained in the framework of fully fuzzy linear transportation problem FFLTP In this framework, all or part of parameters are fuzzy values and the decision variables are fuzzy values too. Since there are many possible methods for aggregating the fuzzy local criteria proposed in the literature, the problem of finding a compromise solution arises, which in the framework of this approach, for the first time, is solved on the basis of aggregation of aggregating modes with the use of level-2 fuzzy sets.
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