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Importance Sampling in stochastic programming: A Markov Chain Monte Carlo Approach

INFORMS JOURNAL ON COMPUTING 2015年第2期27卷 358-377页

作者： Parpas, Panos Ustun, Berk Webster, Mort Quang Kha Tran Univ London Imperial Coll Sci Technol & Med Dept Comp Sci London SW7 2AZ England MIT Dept Elect Engn & Comp Sci Cambridge MA 02139 USA MIT Engn Syst Div Cambridge MA 02139 USA Univ London Imperial Coll Sci Technol & Med Dept Comp Sci London SW7 2AZ England

stochastic programming models are large-scale optimization problems that are used to facilitate decision making under uncertainty. Optimization algorithms for such problems need to evaluate the expected future costs of current decisions, often referred to as the recourse function. In practice, this calculation is computationally difficult as it requires the evaluation of a multidimensional integral whose integrand is an optimization problem. In turn, the recourse function has to be estimated using techniques such as scenario trees or Monte Carlo methods, both of which require numerous functional evaluations to produce accurate results for large-scale problems with multiple periods and high-dimensional uncertainty. In this work, we introduce an importance sampling framework for stochastic programming that can produce accurate estimates of the recourse function using a small number of samples. Our framework combines Markov chain Monte Carlo methods with kernel density estimation algorithms to build a nonparametric importance sampling distribution, which can then be used to produce a lower-variance estimate of the recourse function. We demonstrate the increased accuracy and efficiency of our approach using variants of well-known multistage stochastic programming problems. Our numerical results show that our framework produces more accurate estimates of the optimal value of stochastic programming models, especially for problems with moderate variance, multimodal, or rare-event distributions.

关键词： Benders' decomposition cutting plane algorithms stochastic optimization stochastic programming importance sampling variance reduction Monte Carlo Markov chain Monte Carlo kernel density estimation nonparametric

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A stochastic programming approach for integrated nurse staffing and assignment

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IIE TRANSACTIONS 2013年第10期45卷 1059-1076页

作者： Punnakitikashem, Prattana Rosenberber, Jay M. Buckley-Behan, Deborah F. Mahidol Univ Coll Management Bangkok 10400 Thailand Univ Texas Arlington Dept Ind & Mfg Syst Engn Arlington TX 76019 USA Univ Texas Arlington Sch Nursing Arlington TX 76019 USA

The shortage of nurses has attracted considerable attention due to its direct impact on the quality of patient care. High workloads and undesirable schedules are two major reasons for nurses to report job dissatisfaction. The focus of this article is to find non-dominated solutions to an integrated nurse staffing and assignment problem that minimizes excess workload on nurses and staffing cost. A stochastic integer programming model with an objective to minimize excess workload subject to a hard budget constraint is presented. Three solution approaches are applied, which are Benders' decomposition, Lagrangian relaxation with Benders' decomposition, and a heuristic based on nested Benders' decomposition. The maximum allowable staffing cost in the budget constraint is varied in the Benders' decomposition and nested Benders' decomposition approaches, and the budget constraint is relaxed and the staffing cost is penalized in the Lagrangian relaxation with Benders' decomposition approach. Non-dominated bicriteria solutions are collected from the algorithms. The effectiveness of the model and algorithms is demonstrated in a computational study based on data from two medical-surgical units at a Northeast Texas hospital. A floating nurses policy is also evaluated. Finally, areas of future research are discussed.

关键词： Nurse staffing nurse assignment stochastic programming healthcare nurses hospitals

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A dynamic spare parts ordering and pricing policy based on stochastic programming with multi-choice parameters

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MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 2017年第2期23卷 196-221页

作者： Li, Ling Liu, Min Shen, Weiming Cheng, Guoqing Tongji Univ Sch Elect & Informat Engn Lab Intelligent Maintenance Syst & Serv Shanghai Peoples R China Jingdezhen Ceram Inst Sch Informat Engn Jingdezhen Peoples R China

This paper studies the uncertain and random factors in real-life spare parts supply networks which are abstract and complex dynamical systems, then quantifies these factors in a mathematical model. To seek a dynamic spare parts ordering and pricing policy from a distributor's viewpoint, stochastic programming with multi-choice parameters is applied to formulate this objective optimization problem. In our model, the optimal objective is to maximize the total expected profit of the members of the spare parts supply network, and the decision variables are distributor's selling price and ordering quantity in different periods. By using the methods of expectation operator of the fuzzy variable, Lagrange interpolating polynomial and global criteria, the model is solved, and the optimal ordering and pricing policy is obtained. The results of the numerical example and contrast experiments validate the feasibility and efficiency of the proposed model. Some significant conclusions drawn from the results of parameter sensitivity analysis can be referred by management practitioners. This general model can be applied in other fields of supply chain management, where random and uncertain factors need to be considered.

关键词： Dynamic ordering and pricing policy spare parts stochastic programming multi-choice programming

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Decomposition methods in stochastic programming

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MATHEMATICAL programming 1997年第1-3期79卷 333-353页

作者： Ruszczynski, A Department of Industrial Engineering University of Wisconsin-Madison Madison USA

stochastic programming problems have very large dimension and characteristic structures which are tractable by decomposition. We review basic ideas of cutting plane methods, augmented Lagrangian and splitting methods, and stochastic decomposition methods for convex polyhedral multi-stage stochastic programming problems. (C) 1997 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.

关键词： stochastic programming decomposition primal methods dual methods stochastic methods

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A stochastic programming approach for the optimal management of aggregated distributed energy resources

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COMPUTERS & OPERATIONS RESEARCH 2018年 96卷 199-211页

作者： Beraldi, P. Violi, A. Carrozzino, G. Bruni, M. E. Univ Calabria Dept Mech Energy & Management Engn Arcavacata Di Rende CS Italy

The paper focuses on the optimal management of distributed energy resources aggregated within a coalition. The problem is analyzed from the viewpoint of an aggregator, seen as an entity called to optimize the available resources so to satisfy the aggregated demand by eventually trading in the Day-Ahead Electricity Market. Both a full and a residual perspective in the management of the integrated resources is investigated and compared. The inherent uncertainty affecting the optimal decision problem, mainly related to the demand profile, electricity prices and production from renewable sources, is dealt by adopting the two-stage stochastic programming paradigm. The proposed models (different for the full and residual case) present a bi-objective function, integrating the expected profit and a risk measure, the Conditional Value at Risk, to control undesirable effects caused by the random variations of the uncertain parameters. A broad numerical study has been carried out on real case study. The analysis of the results clearly shows the benefits deriving from the stochastic optimization approach and the effect of considering different levels of risk aversion. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Energy market stochastic programming Distributed energy resources Aggregator Risk management

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A stochastic programming approach for the Bayesian experimental design of nonlinear systems

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COMPUTERS & CHEMICAL ENGINEERING 2015年 72卷 312-324页

作者： Lainez-Aguirre, Jose M. Mockus, Linas Reklaitis, Gintaras V. SUNY Buffalo Dept Ind & Syst Engn Amherst NY 14260 USA Purdue Univ Sch Chem Engn W Lafayette IN 47907 USA

Several approaches for the Bayesian design of experiments have been proposed in the literature (e.g., D-optimal, E-optimal, A-optimal designs). Most of these approaches assume that the available prior knowledge is represented by a normal probability distribution. In addition, most nonlinear design approaches involve assuming normality of the posterior distribution and approximate its variance using the expected Fisher information matrix. In order to be able to relax these assumptions, we address and generalize the problem by using a stochastic programming formulation. Specifically, the optimal Bayesian experimental design is mathematically posed as a three-stage stochastic program, which is then discretized using a scenario based approach. Given the prior probability distribution, a Smolyak rule (sparse-grids) is used for the selection of scenarios. Two retrospective case studies related to population pharmacokinetics are presented. The benefits and limitations of the proposed approach are demonstrated by comparing the numerical results to those obtained by implementing a more exhaustive experimentation and the D-optimal design. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Design of experiments stochastic programming NLP MINLP

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An Uncertain Optimization Method Based on Adaptive Discrete Approximation Rejection Sampling for stochastic programming with Incomplete Knowledge of Uncertainty

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ARABIAN JOURNAL FOR SCIENCE AND ENGINEERING 2023年第2期48卷 1399-1425页

作者： Fang, Bopeng Dong, Zhurong Zhao, Chen Liu, Zhi Wang, Jing Shenzhen Polytech Shenzhen 518055 Peoples R China Shenzhen Polytech Res Inst New Energy Vehicle Technol Shenzhen 518055 Peoples R China Chinese Acad Sci Shenzhen Inst Adv Technol Shenzhen 518055 Peoples R China

stochastic programming has been widely used in various application scenarios and theoretical research works. However, these excellent methods depend on specific explicit probability modeling with complete knowledge of uncertainty, which is very limited in practical problem since there is usually no way to abstract complex uncertainties into the commonly used known probability models. In this paper, a novel generative model named the Adaptive Discrete Approximation Rejection Sampling is proposed for stochastic programming with incomplete knowledge of uncertainty, which can not only simulate uncertain scenarios from a complex explicit probability model that cannot meet the constraints of existing sampling methods, but also even simulate scenarios from a sample set related to uncertainty when the specific explicit probability model of uncertainty is missing or unavailable. The method is to establish the easy-to-sample proposal distribution by approximately transforming the complex hard-to-sample target probability model, to make the proposal distribution close enough to the target distribution, so as to achieve an efficient sampling while ensuring the performance of the model. On this basis, combining the Monte Carlo method and heuristic optimization, an uncertain optimization model for stochastic programming with incomplete knowledge of uncertainty is further constructed, to solve the unavailability of the existing stochastic programming methods in the absence of explicit probability model of uncertainty. Experimental results show the advantages of the proposed method in terms of applicability, adaptability, accuracy, efficiency and model performance.

关键词： stochastic programming Rejection sampling Adaptive proposal distribution Monte Carlo method Machine learning

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Chance-constrained stochastic programming for integrated services network management

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ANNALS OF OPERATIONS RESEARCH 1998年第s期81卷 213-229页

作者： Medova, EA Univ Cambridge Judge Inst Management Studies Cambridge CB2 1AG England

The design, dimensioning and traffic management of an Asynchronous Transfer Mode (ATM) network may be modelled as a hierarchical planning problem at different time-scales - capacity provision/expansion with intervals of weeks or months and real-time call routing at the time scale of the connection, which is itself treated in hierarchical order of transmitting calls (sec), bursts (msec) or cells (mu sec). Taking the network topology as given, a chance constrained stochastic programme for integrated services network dimensioning and traffic management is formulated using Hui's notion of effective bandwidth. This allows the removal of probabilistic constraints at the call level and leads to consideration of the network management problem as a linear deterministic multicommodity flow problem. To allow flexibility and easy problem formulation with various objectives and network specifications the decision support tool MODLER is employed. The model described can be used for a prototype software system for future network design and management.

关键词： ROUTERS (Computer networks) COMPUTER programming DATA transmission systems COMPUTER networks stochastic programming INTEGRATED services digital networks ASYNCHRONOUS transfer mode ELECTRIC network topology

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Minimax and risk averse multistage stochastic programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2012年第3期219卷 719-726页

作者： Shapiro, Alexander Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA

In this paper we study relations between the minimax, risk averse and nested formulations of multistage stochastic programming problems. In particular, we discuss conditions for time consistency of such formulations of stochastic problems. We also describe a connection between law invariant coherent risk measures and the corresponding sets of probability measures in their dual representation. Finally, we discuss a minimax approach with moment constraints to the classical inventory model. (C) 2011 Elsevier B.V. All rights reserved.

关键词： stochastic programming Dynamic equations Robust optimization Coherent risk measures Risk averse stochastic optimization Problem of moments

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Risk-Averse stochastic programming vs. Adaptive Robust Optimization: A Virtual Power Plant Application

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INFORMS JOURNAL ON COMPUTING 2022年第3期34卷 1795-1818页

作者： Lima, Ricardo M. Conejo, Antonio J. Giraldi, Loic Le Maitre, Olivier Hoteit, Ibrahim Knio, Omar M. King Abdullah Univ Sci & Technol Comp Elect & Math Sci & Engn Div Thuwal 239556900 Saudi Arabia Ohio State Univ Dept Integrated Syst Engn Columbus OH 43210 USA Ohio State Univ Dept Elect & Comp Engn Columbus OH 43210 USA Res Inst Nucl Syst Low Carbon Energy Prod Fuel Dept Energy Div French Alternat Energies & Atom Energy Commiss F-13108 St Paul Les Durance France Ecole Polytech Ctr Natl Rech Sci Ctr Math Appl INRIA F-91128 Palaiseau France King Abdullah Univ Sci & Technol Phys Sci & Engn Div Thuwal 239556900 Saudi Arabia

This paper compares risk-averse optimization methods to address the selfscheduling and market involvement of a virtual power plant (VPP). The decision-making problem of the VPP involves uncertainty in the wind speed and electricity price forecast. We focus on two methods: risk-averse two-stage stochastic programming (SP) and two-stage adaptive robust optimization (ARO). We investigate both methods concerning formulations, uncertainty and risk, decomposition algorithms, and their computational performance. To quantify the risk in SP, we use the conditional value at risk (CVaR) because it can resemble a worst-case measure, which naturally links to ARO. We use two efficient implementations of the decomposition algorithms for SP and ARO;we assess (1) the operational results regarding first-stage decision variables, estimate of expected profit, and estimate of the CVaR of the profit and (2) their performance taking into consideration different sample sizes and risk management parameters. The results show that similar first-stage solutions are obtained depending on the risk parameterizations used in each formulation. Computationally, we identified three cases: (1) SP with a sample of 500 elements is competitive with ARO;(2) SP performance degrades comparing to the first case and ARO fails to converge in four out of five risk parameters;(3) SP fails to converge, whereas ARO converges in three out of five risk parameters. Overall, these performance cases depend on the combined effect of deterministic and uncertain data and risk parameters.

关键词： stochastic programming robust optimization risk management virtual power plant

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