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17 USC 105 interim-entered record; under review. The article of record as published may be found at http://dx.doi.org/10.1016/j.ejor.2021.01.011 Underground mine schedules seek to determine start dates for activities related to the extraction of ore, often with an objective of maximizing net present value; constraints enforce geotechnical precedence between activities, and restrict resource consumption on a per-time-period basis, e.g., development footage and extracted tons. Strategic schedules address these start dates at a coarse level, whereas tactical schedules must account for the day-to-day variability of underground mine operations, such as unanticipated equipment breakdowns and ground conditions, both of which might slow production. At the time of this writing, the underground mine scheduling literature is dominated by a deterministic treatment of the problem, usually modeled as a Resource Constrained Project Scheduling Problem (RCPSP), which precludes mine operators from reacting to unforeseen circumstances. Therefore, we propose a stochastic integer programming framework that: (i) characterizes uncertainty in duration and economic value for each underground mining activity; (ii) formulates a new stochastic variant of the RCPSP; (iii) suggests an optimization-based heuristic; and, (iv) produces implementable, tactical schedules in a practical amount of time and provides corresponding managerial insights. National Institute of Occupational Safety and Health National Agency for Research and Development (ANID)

International audience; We model the relation between an aggregator and consumers joining a coalition to reduce the risk resulting from the unpredictability of their base load demand, as a Stackelberg game formulated as a mathematical bilevel program with private information on the consumers' reservation prices. At the upper-level of the Stackelberg game, the aggregator optimizes his daily price profile so as to reach a net targeted profit which is the maximum value guaranteeing that no consumer will leave the coalition - to contract with a conventional retailer considered here as a fixed alternative - while meeting fairness criterion imposed by the cost-sharing mechanism. At the lower-level, the consumers are asked to provide in day ahead an estimate of their base load hourly demand profile and to schedule their shiftable loads depending on the price signal sent by the aggregator. We provide algorithms that determine the unique price profile and consumer shiftable load schedules as functions of the reservation price estimates. The Stackelberg game between the aggregator and the consumers being repeated for a period of time, the aggregator has the possibility to update his estimates of the reservation prices relying on a feedback function which depends on the percentage of activated loads. A randomized algorithm for consumers' reservation price learning based on regret minimization is provided. For four cost-sharing mechanisms such as uniform allocation, stand-alone cost, Shapley value, separable and non-separable costs, we determine the closed form of the aggregator's optimal net targeted profit guaranteeing the stability of the coalition. We also determine conditions guaranteeing the core non-emptiness and prove that for a profit-maximizing aggregator, the stand-alone cost is always preferable to the Shapley value, which coincides with the uniform allocation. Furthermore, the optimal size of the coalition - in terms of the aggregator's profit - can be determined analytically when the Shapley value is implemented as cost-sharing mechanism. The results are illustrated on a case study where we show that there exists an optimal net targeted profit below which the consumers energy bill is lower when joining the aggregator than with the conventional retailer. Coalition dynamics is also analyzed numerically depending on the consumer inertia in their energy supplier choice process, for each cost-sharing mechanism.

We discuss the incorporation of risk measures into multistage stochastic programs. While much attention has been recently devoted in the literature to this type of model, it appears that there is no consensus on the best way to accomplish that goal. In this paper, we discuss pros and cons of some of the existing approaches. A key notion that must be considered in the analysis is that of consistency, which roughly speaking means that decisions made today should agree with the planning made yesterday for the scenario that actually occurred. Several definitions of consistency have been proposed in the literature, with various levels of rigor; we provide our own definition and give conditions for a multi-period risk measure to be consistent. A popular way to ensure consistency is to nest the one-step risk measures calculated in each stage, but such an approach has drawbacks from the algorithmic viewpoint. We discuss a class of risk measures—which we call expected conditional risk measures—that address those shortcomings. We illustrate the ideas set forth in the paper with numerical results for a pension fund problem in which a company acts as the sponsor of the fund and the participants’ plan is defined-benefit.

We consider the problem of determining the minimal requirement one must establish in order to meet a series of future random payments. It is shown in a very general setting that this problem can be recast as a chance constrained model and how the technique of Sample Average Approximation can be employed to find solutions. We also use comonotonic theory to analyze analytical approximations in a restricted Gaussian setting. Our numerical illustrations demonstrate that the Sample Average Approximation is a viable and efficient way to solve the stated problem generally and outperforms the analytical approximations. In passing we present a result that is related to Stein’s famous lemma ( Stein, 1981 ) and is of interest in itself.