Expected shortfall for the makespan in activity networks under imperfect information

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.