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A Two Phase Method for Solving the Distribution Problem in a Fuzzy Setting

Department of Computer Science, Czestochowa University of Technology, Dabrowskiego 73, 42-201 Czestochowa, Poland
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Entropy 2019, 21(12), 1214; https://doi.org/10.3390/e21121214
Received: 15 October 2019 / Revised: 5 December 2019 / Accepted: 7 December 2019 / Published: 11 December 2019
(This article belongs to the Section Multidisciplinary Applications)
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 α -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. View Full-Text
Keywords: fuzzy extension; simplex method; fuzzy multiobjective distribution problem; aggregation of aggregating modes fuzzy extension; simplex method; fuzzy multiobjective distribution problem; aggregation of aggregating modes
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Kaczmarek, K.; Dymova, L.; Sevastjanov, P. A Two Phase Method for Solving the Distribution Problem in a Fuzzy Setting. Entropy 2019, 21, 1214.

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