Robust Optimal Solution Based on Stochastic Ordering under Uncertainty

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  • 不確実環境下での確率論的順序関係に基づく頑健な最適解
  • フカクジツ カンキョウ カ デノ カクリツロンテキ ジュンジョ カンケイ ニ モトズク ガンケンナ サイテキカイ

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Abstract

Various models have been proposed for dealing with uncertainties in production planning or scheduling. Some kinds of uncertainties can be formulated as scenarios with occurrence probabilities, and either the expected criterion value based on occurrence probabilities or the worst criterion value is usually minimized to find a 'robust' optimal solution. In the latter case, scenarios are treated equally no matter how high the occurrence probabilities are, resulting in a worse criterion value for scenarios with high occurrence probabilities. The former case contrarily tends to provide a much worse criterion value for scenarios with very low occurrence probabilities. To overcome this drawback, we introduce a cumulative distribution function (c.d.f.) of closeness under a set of scenarios with occurrence probabilities, and propose a method for finding a "robust optimal solution according to stochastic ordering." Introducing a set of basic cumulative density functions with various ranks, we define the rank of closeness for a solution as the smallest value of k so that the c.d.f. of closeness is not smaller at any value of closeness than any basic cumulative density function with a rank value greater than or equal to k in the meaning of "stochastic ordering." A robust optimal solution is generated as a solution with the smallest rank value among the feasible solutions. If we have several candidates for such a solution, either illustrating the c.d.f. of closeness for each solution or providing the rank value as a detailed real value helps a decision maker to select the final robust optimal solution from among the candidate solutions. A numerical example in a single-machine scheduling problem to minimize mean flow time is shown to demonstrate that the proposed robust optimal solution is superior to the other robust optimal solutions derived through expected criterion value minimization and min-max optimization methods.

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