This paper deals with the evaluation of the expected shortfall or the conditional value-at-risk for the makespan in scheduling problems represented as temporal networks under incomplete and uncertain information. We consider temporal activity network representations of scheduling problems affected by uncertainties related to the activity durations and we assume that for these uncertainties only incomplete or imperfect information is available. More precisely, for each activity only the interval for its integer valued duration is known to the scheduler. We address the evaluation of the expected shortfall associated to a feasible schedule discussing its importance in scheduling applications. We propose lower and upper bounds, heuristics to determine a fast computational estimation of the expected shortfall, and an exact method for a class of activity networks. The experimental results show that the proposed method can enable to use the expected shortfall as optimization criterion for wide classes of scheduling approaches considering risk-aversion in different practical contexts.

中文翻译：

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在信息不完善的情况下，活动网络的预期完成时间不足

本文讨论了在信息不完整和不确定的情况下，以时间网络表示的调度问题中预期不足或条件风险值的评估。我们考虑了与活动持续时间相关的不确定性影响的调度问题的时间活动网络表示，并且我们假设对于这些不确定性，只有不完整或不完美的信息可用。更准确地说，对于每个活动，调度程序只知道其整数值持续时间的间隔。我们讨论了与可行时间表相关的预期短缺的评估，并讨论了其在调度应用程序中的重要性。我们提出上下限试探法，以确定对预期不足的快速计算估计，一类活动网络的精确方法。实验结果表明，该方法能够将期望的不足作为优化准则，用于考虑不同实际情况下的风险规避的大类调度方法。