A Game Theoretic Approach to Multi-Period Newsvendor Problems with Censored Markovian Demand

Farzaneh Mansourifard, Parisa Mansourifard, B. Krishnamachari
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Abstract

This paper studies the Newsvendor problem for a setting in which (i) the demand is temporally correlated, (ii) the demand is censored, (iii) the distribution of the demand is unknown. The correlation is modeled as a Markovian process. The censoring means that if the demand is larger than the action (selected inventory), only a lower bound on the demand can be revealed. The uncertainty set on the demand distribution is given by only the upper and lower bound on the amount of the change from a time to the next time. We propose a robust approach to minimize the worst-case total cost and model it as a min-max zero-sum repeated game. We prove that the worst-case distribution of the adversary at each time is a two-point distribution with non-zero probabilities at the extrema of the uncertainty set of the demand. And the optimal action of the decision-maker can have any of the following structures: (i) a randomized solution with a two-point distribution at the extrema, (ii) a deterministic solution at a convex combination of the extrema. Both above solutions balance over-utilization and under-utilization costs. Finally, we extend our results to uni-model cost functions and present numerical results to study the solution.
含删节马尔可夫需求的多时期报贩问题的博弈论方法
本文研究了一种情况下的报贩问题,其中:(1)需求是时间相关的,(2)需求是审查的,(3)需求的分布是未知的。这种相关性被建模为马尔可夫过程。审查意味着如果需求大于行动(选定的库存),则只能显示需求的下限。需求分布的不确定性仅由从一个时间点到下一个时间点的变化量的上界和下界给出。我们提出了一种鲁棒方法来最小化最坏情况下的总成本,并将其建模为最小-最大零和重复博弈。我们证明了每次对手的最坏情况分布是在需求不确定性集的极值处具有非零概率的两点分布。决策者的最优行为可以具有以下任意结构:(i)在极值处具有两点分布的随机解,(ii)在极值的凸组合处具有确定性解。上述两种解决方案都平衡了过度利用和未充分利用的成本。最后,我们将结果推广到单模型成本函数,并给出数值结果来研究解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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