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101
Optimization under uncertainty: Stateoftheart and opportunities
 Computers and Chemical Engineering
, 2004
"... A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemi ..."
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Cited by 92 (0 self)
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A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very largescale optimization models. Decisionmaking under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multiperiod or multistage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recoursebased stochastic programming, robust stochastic programming, probabilistic (chanceconstraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the stateoftheart in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomialtime approximation schemes for multistage stochastic programs and the application of global optimization algorithms to twostage and chanceconstraint formulations.
Sustainable Development
, 2006
"... O perations management methods have been applied profitably to a wide range of technology portfolio managementproblems, but have been slow to be adopted by governments and policy makers. We develop a framework that allows us to apply such techniques to a large and important public policy problem: en ..."
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Cited by 55 (1 self)
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O perations management methods have been applied profitably to a wide range of technology portfolio managementproblems, but have been slow to be adopted by governments and policy makers. We develop a framework that allows us to apply such techniques to a large and important public policy problem: energy technology R&D portfolio management under climate change. We apply a multimodel approach, implementing probabilistic data derived from expert elicitations into a novel stochastic programming version of a dynamic integrated assessment model. We note that while the unifying framework we present can be applied to a range of models and data sets, the specific results depend on the data and assumptions used and therefore may not be generalizable. Nevertheless, the results are suggestive, and we find that the optimal technology portfolio for the set of projects considered is fairly robust to different specifications of climate uncertainty, to different policy environments, and to assumptions about the opportunity cost of investing. We also conclude that policy makers would do better to overinvest in R&D rather than underinvest. Finally, we show that R&D can play different roles in different types of policy environments, sometimes leading primarily to cost reduction, other times leading to better environmental outcomes.
A multistage stochastic integer programming approach for capacity expansion under uncertainty
 J. Global Opt
, 2003
"... This paper addresses a multiperiod investment model for capacity expansion in an uncertain environment. Using a scenario tree approach to model the evolution of uncertain demand and cost parameters, and fixedcharge cost functions to model the economies of scale in expansion costs, we develop a mul ..."
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Cited by 52 (3 self)
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This paper addresses a multiperiod investment model for capacity expansion in an uncertain environment. Using a scenario tree approach to model the evolution of uncertain demand and cost parameters, and fixedcharge cost functions to model the economies of scale in expansion costs, we develop a multistage stochastic integer programming formulation for the problem. A reformulation of the problem is proposed using variable disaggregation to exploit the lotsizing substructure of the problem. The reformulation significantly reduces the LP relaxation gap of this large scale integer program. A heuristic scheme is presented to perturb the LP relaxation solutions to produce good quality integer solutions. Finally, we outline a branch and bound algorithm that makes use of the reformulation strategy as a lower bounding scheme, and the heuristic as an upper bounding scheme, to solve the problem to global optimality. Our preliminary computational results indicate that the proposed strategy has significant advantages over straightforward use of commercial solvers. 1
Stochastic Lagrangian relaxation applied to power scheduling in a hydrothermal system under uncertainty
 ANNALS OF OPERATIONS RESEARCH
, 2000
"... A dynamic (multistage) stochastic programming model for the weekly costoptimal generation of electric power in a hydrothermal generation system under uncertain load is developed. The model involves a large number of mixedinteger (stochastic) decision variables and constraints linking time period ..."
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Cited by 45 (7 self)
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A dynamic (multistage) stochastic programming model for the weekly costoptimal generation of electric power in a hydrothermal generation system under uncertain load is developed. The model involves a large number of mixedinteger (stochastic) decision variables and constraints linking time periods and operating power units. Astochastic Lagrangian relaxation scheme is designed by assigning (stochastic) multipliers to all constraints coupling power units. It is assumed that the stochastic load process is given (or approximated) by a nite number of realizations (scenarios) in scenario tree form. Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single (thermal or hydro) unit subproblems. The stochastic thermal and hydro subproblems are solved by astochastic dynamic programming technique and by a speci c descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the rst stage (primal) decisions starting from the optimal (stochastic) multipliers is developed. Numerical results are presented for realistic data from a German power utility andfornumbers of scenarios ranging from 5 to 100 and a time horizon from 7 to 9 days. The sizes of the corresponding optimization problems go up to 200.000 binary and 350.000 continuous variables, and more than 500.000 constraints.
Optimal Power Generation under Uncertainty via Stochastic Programming
 in: Stochastic Programming Methods and Technical Applications (K. Marti and P. Kall Eds.), Lecture Notes in Economics and Mathematical Systems
, 1997
"... : A power generation system comprising thermal and pumpedstorage hydro plants is considered. Two kinds of models for the costoptimal generation of electric power under uncertain load are introduced: (i) a dynamic model for the shortterm operation and (ii) a power production planning model. In both ..."
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Cited by 29 (10 self)
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: A power generation system comprising thermal and pumpedstorage hydro plants is considered. Two kinds of models for the costoptimal generation of electric power under uncertain load are introduced: (i) a dynamic model for the shortterm operation and (ii) a power production planning model. In both cases, the presence of stochastic data in the optimization model leads to multistage and twostage stochastic programs, respectively. Both stochastic programming problems involve a large number of mixedinteger (stochastic) decisions, but their constraints are loosely coupled across operating power units. This is used to design Lagrangian relaxation methods for both models, which lead to a decomposition into stochastic single unit subproblems. For the dynamic model a Lagrangian decomposition based algorithm is described in more detail. Special emphasis is put on a discussion of the duality gap, the efficient solution of the multistage single unit subproblems and on solving the dual problem...
A Class of Stochastic Programs with Decision Dependent Uncertainty
 MATHEMATICAL PROGRAMMING
, 2005
"... The standard approach to formulating stochastic programs is based on the assumption that the stochastic process is independent of the optimization decisions. We address a class of problems where the optimization decisions influence the time of information discovery for a subset of the uncertain para ..."
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Cited by 26 (11 self)
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The standard approach to formulating stochastic programs is based on the assumption that the stochastic process is independent of the optimization decisions. We address a class of problems where the optimization decisions influence the time of information discovery for a subset of the uncertain parameters. We extend the standard modeling approach by presenting a disjunctive programming formulation that accommodates stochastic programs for this class of problems. A set of theoretical properties that lead to reduction in the size of the model is identified. A Lagrangean duality based branch and bound algorithm is also presented.
A Finite Branch and Bound Algorithm for TwoStage Stochastic Integer Programs
, 2000
"... This paper addresses a general class of twostage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the c ..."
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Cited by 20 (6 self)
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This paper addresses a general class of twostage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Our computational results indicate superior performance of the proposed algorithm in comparison to the existing literature. Keywords: stochastic integer programming, branch and bound, finite algorithms. 1 Introduction Under the twostage stochastic programming paradigm, the decision variables of an optimization problem under uncertainty are partitioned into two sets. The first stage variables are those that have to be decided before the actual realization of the uncertain parameters. Subsequently, once the random events have presented themselves, further design or operational ...
Conditional ValueatRisk in Stochastic Programs with MixedInteger Recourse
, 2004
"... In classical twostage stochastic programming the expected value of the total costs is minimized. Recently, meanrisk models studied in mathematical finance for several decades have attracted attention in stochastic programming. We consider Conditional ValueatRisk as risk measure in the framewo ..."
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Cited by 20 (7 self)
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In classical twostage stochastic programming the expected value of the total costs is minimized. Recently, meanrisk models studied in mathematical finance for several decades have attracted attention in stochastic programming. We consider Conditional ValueatRisk as risk measure in the framework of twostage stochastic integer programming. The paper addresses structure, stability, and algorithms for this class of models. In particular, we study continuity properties of the objective function, both with respect to the firststage decisions and the integrating probability measure. Further, we present an explicit mixedinteger linear programming formulation of the problem when the probability distribution is discrete and finite. Finally, a solution algorithm based on Lagrangean relaxation of nonanticipativity is proposed.