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In this paper, we consider an optimization problem involving crashing an activity network under a single disruption. A disruption is an event whose magnitude and timing are random. When a disruption occurs, the duration of an activity that has yet to start—or alternatively, yet to complete—can change. We formulate a two-stage stochastic mixed-integer program, in which the timing of the stage is random. In our model, the recourse problem is a mixed-integer program. We prove the problem is NP-hard, and using simple examples, we illustrate properties that differ from the problem's deterministic counterpart. Obtaining a reasonably tight optimality gap can require a large number of samples in a sample average approximation, leading to large-scale instances that are computationally expensive to solve. Therefore, we develop a branch-and-cut decomposition algorithm, in which spatial branching of the first stage continuous variables and linear programming approximations for the recourse problem are sequentially tightened. We test our decomposition algorithm with multiple improvements and show it can significantly reduce solution time over solving the problem directly. [ABSTRACT FROM AUTHOR]

Dupacová's early work in minimax solutions of stochastic programs [[6], [24]] predated by half a century a recent surge of work in distributionally robust optimization. Two others major lines of Dupacová's research contributions concern scenario generation and scenario reduction [[15]], with precursor work including [[5], [21]], and her work on asymptotics [[10], [23]]. Jitka Dupacová was the first woman vice dean at the Faculty of Mathematics and Physics at Charles University in 1979. Stochastic programming has seen recent advances with far-reaching impact involving risk measures, distributionally robust optimization, and applications in areas ranging from energy and natural resources to economics and finance to statistics and machine learning. [Extracted from the article]

Community mitigation strategies to combat COVID-19, ranging from healthy hygiene to shelter-in-place orders, exact substantial socioeconomic costs. Judicious implementation and relaxation of restrictions amplify their public health benefits while reducing costs. We derive optimal strategies for toggling between mitigation stages using daily COVID-19 hospital admissions. With public compliance, the policy triggers ensure adequate intensive care unit capacity with high probability while minimizing the duration of strict mitigation measures. In comparison, we show that other sensible COVID-19 staging policies, including France's ICU-based thresholds and a widely adopted indicator for reopening schools and businesses, require overly restrictive measures or trigger strict stages too late to avert catastrophic surges. As proof-of-concept, we describe the optimization and maintenance of the staged alert system that has guided COVID-19 policy in a large US city (Austin, Texas) since May 2020. As cities worldwide face future pandemic waves, our findings provide a robust strategy for tracking COVID-19 hospital admissions as an early indicator of hospital surges and enacting staged measures to ensure integrity of the health system, safety of the health workforce, and public confidence. Selection of COVID-19 mitigation measures requires balancing health outcomes with economic impacts. Here, the authors derive a system to set triggers for increasing mitigation measures to preserve healthcare capacity, and describe how it has been used to support public health decision making in Austin, Texas. [ABSTRACT FROM AUTHOR]

We analyze scheduling a hybrid wind-conventional generator system to make it dispatchable, with the aim of profit maximization. Our models ensure that with high probability we satisfy the day-ahead power promised by the model, using combined output of the conventional and wind generators. We consider two scenarios, which differ in whether the conventional generator must commit to its schedule prior to observing the wind-power realizations or has the flexibility to adapt in near real-time to these realizations. We investigate the synergy between the conventional generator and wind farm in these two scenarios. Computationally, the non-adaptive model is relatively tractable, benefiting from a strong extended-variable formulation as an integer program. The adaptive model is a two-stage stochastic integer program with joint chance constraints. Such models have seen limited attention in the literature because of the computational challenges they pose. However, we develop an iterative regularization scheme in which we solve a sequence of sample average approximations under a growing sample size. This reduces computational effort dramatically, and our empirical results suggest that it heuristically achieves high-quality solutions. Using data from a wind farm in Texas, we demonstrate that the adaptive model significantly outperforms the non-adaptive model in terms of synergy between the conventional generator and the wind farm, with expected profit more than doubled. [ABSTRACT FROM AUTHOR]

Over the last two decades, coherent risk measures have been well studied as a principled, axiomatic way to characterize the risk of a random variable. Because of this axiomatic approach, coherent risk measures have a number of attractive features for computation, and they have been integrated into a variety of stochastic programming algorithms, including stochastic dual dynamic programming (SDDP), a common class of data-driven solution methods for multistage stochastic programs. Coherent risk measures and SDDP are tools used to manage risk while solving data-driven problems. Perhaps the most prominent example involves informing operations and deriving electricity prices in power systems with significant hydro-electric power, including the Brazilian interconnected power system. We focus on incorporating the more general class of convex risk measures into an SDDP algorithm, exemplifying our approach with the entropic risk measure. It is well-known that coherent risk measures lead to an inconsistency if agents care about their state at the end of the time horizon, but control risk in a stage-wise fashion. The entropic risk measure does not have this shortcoming. We illustrate the advantages of the entropic risk measure with two small examples from transportation and finance, and test the numerical viability of our adaptation of the SDDP decomposition scheme in a large-scale hydro-thermal scheduling problem using data from the Brazilian system. [ABSTRACT FROM AUTHOR]

Service organizations that operate outside the normal 8-hour day and face wide fluctuations in demand constantly struggle to optimize the size and composition of their workforce. Recent research has shown that improved personnel scheduling methods that take demand uncertainty into account can lead to significant reductions in labor costs. This paper addresses a staff planning and scheduling problem that arises at United States Postal Service (USPS) mail processing & distribution centers (P&DCs) and develops a two-stage stochastic integer program with recourse for the analysis. In the first stage, before the demand is known, the number of full-time and part-time employees is determined for the permanent workforce. In the second stage, the demand is revealed and workers are assigned to specific shifts during the week. When necessary, overtime and casual labor are used to satisfy demand. This paper consists of two parts: (1) the analysis of the demand distribution in light of historical data, and (2) the development and analysis of the stochastic integer programming model. Using weekly demand for a three-year period, we first investigate the possibility that there exists an end-of-month effect, i.e., the week at the end of month has larger volume than the other weeks. We show that the data fail to indicate that this is the case. In the computational phase of the work, three scenarios are considered: high, medium, and low demand. The stochastic optimization problem that results is a large-scale integer program that embodies the full set of contractual agreements and labor rules governing the design of the workforce at a P&DC. The usefulness of the model is evaluated by solving a series of instances constructed from data provided by the Dallas facility. The results indicate that significant savings are likely when the recourse problem is used to help structure the workforce. [ABSTRACT FROM AUTHOR]

Determining whether a solution is of high quality (optimal or near optimal) is fundamental in optimization theory and algorithms. In this paper, we develop Monte Carlo sampling-based procedures for assessing solution quality in stochastic programs. Quality is defined via the optimality gap and our procedures' output is a confidence interval on this gap. We review a multiple-replications procedure that requires solution of, say, 30 optimization problems and then, we present a result that justifies a computationally simplified single-replication procedure that only requires solving one optimization problem. Even though the single replication procedure is computationally significantly less demanding, the resulting confidence interval might have low coverage probability for small sample sizes for some problems. We provide variants of this procedure that require two replications instead of one and that perform better empirically. We present computational results for a newsvendor problem and for two-stage stochastic linear programs from the literature. We also discuss when the procedures perform well and when they fail, and we propose using ɛ-optimal solutions to strengthen the performance of our procedures. [ABSTRACT FROM AUTHOR]

Focuses on mathematical programming models for the location and routing problems where the customer demands are random parameters and the arrival time of the demand is known. Methods used to handle uncertainty in the parameters of a problem; Properties of stochastic linear programming and stochastic integer programs; Application to stochastic discrete location.

Handling uncertainty in natural inflow is an important part of a hydroelectric scheduling model. In a stochastic programming formulation, natural inflow may be modeled as a random vector with known distribution, but the size of the resulting mathematical program can be formidable. Decomposition-based algorithms take advantage of special structure and provide an attractive approach to such problems. We develop an enhanced Benders decomposition algorithm for solving multistage stochastic linear programs. The enhancements include warm start basis selection, preliminary cut generation, the multicut procedure, and decision tree traversing strategies. Computational results are presented for a collection of stochastic hydroelectric scheduling problems. [ABSTRACT FROM AUTHOR]