Topic
Stochastic programming
About: Stochastic programming is a research topic. Over the lifetime, 12343 publications have been published within this topic receiving 421049 citations.
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TL;DR: A bilevel stochastic programming problem (BSPP) model of the decision-making of an energy hub manager is presented and conditional value at risk is used to reduce the unfavorable effects of the uncertainties.
Abstract: A bilevel stochastic programming problem (BSPP) model of the decision-making of an energy hub manager is presented. Hub managers seek ways to maximize their profit by selling electricity and heat. They have to make decisions about: 1) the level of involvement in forward contracts, electricity pool markets, and natural gas networks and 2) the electricity and heat offering prices to the clients. These decisions are made under uncertainty of pool prices, demands as well as the prices offered by rival hub managers. On the other hand, the clients try to minimize the total cost of energy procurement. This two-agent relationship is presented as a BSPP in which the hub manager is placed in the upper level and the clients in the lower one. The bilevel scheme is converted to its equivalent single-level scheme using the Karush–Kuhn–Tucker optimality conditions although there are two bilinear products related to electricity and heat. The heat bilinear product is replaced by a heat price-quota curve and the electricity bilinear product is linearized using the strong duality theorem. In addition, conditional value at risk is used to reduce the unfavorable effects of the uncertainties. The effectiveness of the proposed model is evaluated in various simulations of a realistic case study.
107 citations
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TL;DR: How still sharper bounds may be generated based on the simple idea of sequentially applying the classic bounds to smaller and smaller subintervals of the range of the random variable is indicated.
Abstract: This paper is concerned with the determination of tight lower and upper bounds on the expectation of a convex function of a random variable. The classic bounds are those of Jensen and Edmundson-Madansky and were recently generalized by Ben-Tal and Hochman. This paper indicates how still sharper bounds may be generated based on the simple idea of sequentially applying the classic bounds to smaller and smaller subintervals of the range of the random variable. The bounds are applicable in the multivariate case if the random variables are independent. In the dependent case bounds based on the Edmundson-Madansky inequality are not available; however, bounds may be developed using the conditional form of Jensen's inequality. We give some examples to illustrate the geometrical interpretation and the calculations involved in the numerical determination of the new bounds. Special attention is given to the problem of maximizing a nonlinear program that has a stochastic objective function.
107 citations
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TL;DR: The results show that the formulation proposed by Bertsimas and Sim is the most appropriate model for uncertain scheduling problems, because it has the following advantages: the model has the same size as the other formulations, it preserves its linearity, and the ability to control the degree of conservatism for every constraint is controlled.
Abstract: This paper addresses the uncertainty problem in process scheduling using robust optimization. Compared to the traditional-scenario-based stochastic programming method, robust counterpart optimization method has a unique advantage, in that the scale of the corresponding optimization problem does not increase exponentially with the number of the uncertain parameters. Three robust counterpart optimization formulations―including Soyster’s worst-case scenario formulation, Ben-Tal and Nemirovski’s formulation, and a formulation proposed by Bertsimas and Sim―are studied and applied to uncertain scheduling problems in this paper. The results show that the formulation proposed by Bertsimas and Sim is the most appropriate model for uncertain scheduling problems, because it has the following advantages: (i) the model has the same size as the other formulations, (ii) it preserves its linearity, and (iii) it has the ability to control the degree of conservatism for every constraint and guarantees feasibility for the r...
107 citations
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TL;DR: In this paper, a stochastic programming framework for solving the scheduling problem faced by an industrial customer with cogeneration facilities, conventional power production system, and heat only units is presented.
107 citations
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TL;DR: It is shown that the proposed formulation captures the various decision-making policies toward demand satisfaction in a unified way, and the employed feasibility criterion for the incorporation of the uncertainty enables the exact reformulation of the two-stage model as a single large-scale optimization model.
Abstract: The paper addresses the problem of including aspects of uncertainty in process parameters and product demands at the design stage of multiproduct/multipurpose batch plants. A conceptual two-stage stochastic programming formulation is proposed with an objective function comprising investment costs, expected revenues from product sales, and a penalty term accounting for expected losses due to unfilled orders. It is shown that (i) the proposed formulation captures the various decision-making policies toward demand satisfaction in a unified way, (ii) the employed feasibility criterion for the incorporation of the uncertainty enables the exact reformulation of the two-stage model as a single large-scale optimization model, (iii) for the case of discrete equipment sizes and despite the use of general continuous probability distribution functions to describe the uncertainty, linearity of the model is preserved, allowing detailed scheduling models to be included directly in the optimization model, and (iv) for th...
107 citations