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Showing papers on "Stochastic programming published in 1996"


Book
30 Nov 1996
TL;DR: This paper presents four approaches to handle Uncertainty in Decision Making using a Robust Discrete Optimization Framework and results show how these approaches can be applied to real-world problems.
Abstract: Preface. 1. Approaches to Handle Uncertainty In Decision Making. 2. A Robust Discrete Optimization Framework. 3. Computational Complexity Results of Robust Discrete Optimization Problems. 4. Easily Solvable Cases of Robust Discrete Optimization Problems. 5. Algorithmic Developments for Difficult Robust Discrete Optimization Problems. 6. Robust 1-Median Location Problems: Dynamic Aspects and Uncertainty. 7. Robust Scheduling Problems. 8. Robust Uncapacitated Network Design and International Sourcing Problems. 9. Robust Discrete Optimization: Past Successes and Future Challenges.

1,463 citations


Book
25 Apr 1996
TL;DR: In this paper, the authors present a survey of Fuzzy multiple objective decision-making techniques and their application in various aspects of the real world, such as: 1.1 Introduction.2 Goal Programming.
Abstract: 1 Introduction.- 1.1 Objectives of This Study.- 1.2 (Fuzzy) Multiple Objective Decision Making.- 1.3 Classification of (Fuzzy) Multiple Objective Decision Making.- 1.4 Applications of (Fuzzy) Multiple Objective Decision Making.- 1.5 Literature Survey.- 1.6 Fuzzy Sets.- 2 Multiple Objective Decision Making.- 2.1 Introduction.- 2.2 Goal Programming.- 2.2a A Portfolio Selection Problem.- 2.2b An Audit Sampling Problem.- 2.3 Fuzzy Programming.- 2.3.1 Max-Min Approach.- 2.3.1a A Trade Balance Problem.- 2.3.1b A Media Selection Problem.- 2.3.2 Augmented Max-Min Approach.- Example.- 2.3.2a A Trade Balance Problem.- 2.3.2b A Logistics Planning Model.- 2.3.3 Parametric Approach.- Example.- 2.4 Global Criterion Approach.- 2.4.1 Global Criterion Approach.- 2.4.1a A Nutrition Problem.- 2.4.2 TOPSIS for MODM.- 2. .2a A Water Quality Management Problem.- 2.5 Interactive Multiple Objective Decision Making.- 2.5.1 Optimal System Design.- 2.5.1a A Production Planning Problem.- 2.5.2 KSU-STEM.- 2.5.2a A Nutrition Problem.- 2.5.2b A Project Scheduling Problem.- 2.5.3 ISGP-II.- 2.5.3a A Nutrition Problem.- 2.5.3b A Bank Balance Sheet Management Problem.- 2.5.4 Augmented Min-Max Approach.- 2.5.4a A Water Pollution Control Problem.- 2.6 Multiple Objective Linear Fractional Programming.- 2.6.1 Luhandjula's Approach.- Example.- 2.6.2 Lee and Tcha's Approach.- 2.6.2a A Financial Structure Optimization Problem.- 2.7 Multiple Objective Geometric Programming.- Example.- 2.7a A Postal Regulation Problem.- 3 Fuzzy Multiple Objective Decision Making.- 3.1 Fuzzy Goal Programming.- 3.1.1 Fuzzy Goal Programming.- 3.1.1a A Production-Marketing Problem.- 3.1.1b An Optimal Control Problem.- 3.1.1c A Facility Location Problem.- 3.1.2 Preemptive Fuzzy Goal Programming.- Example: The Production-Marketing Problem.- 3.1.3 Interpolated Membership Function.- 3.1.3.1 Hannan's Method.- Example: The Production-Marketing Problem.- 3.1.3.2 Inuiguchi, Ichihashi and Kume's Method.- Example: The Trade Balance Problem.- 3.1.3.3 Yang, Ignizio and Kim's Method.- Example.- 3.1.4 Weighted Additive Model.- 3.1.4.1 Crisp Weights.- 3.1.4.1a Maximin Approach.- Example: The Production-Marketing Problem.- 3.1.4.1b Augmented Maximin Approach.- 3.1.4.1c Supertransitive Approximation.- Example: The Production-Marketing Problem.- 3.1.4.2 Fuzzy Weights.- Example: The Production-Marketing Problem.- 3.1.5 A Preference Structure on Aspiration Levels.- Example: The Production-Marketing Problem.- 3.1.6 Nested Priority.- 3.1.6a A Personnel Selection Problem.- 3.2 Fuzzy Global Criterion.- Example.- 3.3 Interactive Fuzzy Multiple Objective Decision Making.- 3.3.1 Werners's Method.- Example: The Trade Balance Problem.- 3.3.1a An Aggregate Production Planning Problem.- 3.3.2 Lai and Hwang's Method.- 3.3.3 Leung's Method.- Example.- 3.3.4 Fabian, Ciobanu and Stoica's Method.- Example.- 3.3.5 Sasaki, Nakahara, Gen and Ida's Method.- Example.- 3.3.6 Baptistella and Ollero's Method.- 3.3.6a An Optimal Scheduling Problem.- 4 Possibilistic Multiple Objective Decision Making.- 4.1 Introduction.- 4.1.1 Resolution of Imprecise Objective Functions.- 4.1.2 Resolution of Imprecise Constraints.- 4.2 Possibilistic Multiple Objective Decision Making.- 4.2.1 Tanaka and His Col1eragues' Methods.- Example.- 4.2.1.1 Possibilistic Regression.- Example 1.- Example 2.- 4.2.1.2 Possibilistic Group Method of Data Handling.- Example 28.- 4.2.2 Lai and Hwang's Method.- 4.2.3 Negi's Method.- Example.- 4.2.4 Luhandjula's Method.- Example.- 4.2.5 Li and Lee's Method.- Example.- 4.2.6 Wierzchon's Method.- 4.3 Interactive Methods for PMODM.- 4.3.1 Sakawa and Yano's Method.- Example.- 4.3.2 Slowinski's Method.- 4.3.2a A Long-Term Development Planning Problem of a Water Supply System.- 4.3.2b A Land-Use Planning Problem.- 4.3.2c A Farm Structure Optimization Problem.- 4.3.3 Rommelranger's Method.- Example.- 4.4 Hybrid Problems.- 4.4.1 Tanaka, Ichihashi and Asai's Method.- Example.- 4.4.2 Inuiguchi and Ichihashi's Method.- Example.- 4.5 Possibilistic Multiple Objective Linear Fractional Programming.- 4.6 Interactive Possibilistic Regression.- 4.6.1 Crisp Output and Crisp Input.- Example.- 4.6.2 Imprecise Output and Crisp Input.- Example.- 4.6.3 Imprecise Output and Imprecise Input.- Example.- 5 Concluding Remarks.- 5.1 Future Research.- 5.2 Fuzzy Mathematical Programming.- 5.3 Multiple Attribute Decision Making.- 5.4 Fuzzy Multiple Attribute Decision Making.- 5.5 Group Decision Making under Multiple Criteria.- Books, Monographs and Conference Proceedings.- Journal Articles, Technical Reports and Theses.- Appendix: Stochastic Programming.- A.1 Stochastic Programming with a Single Objective Function.- A.1.1 Distribution Problems.- A.1.2 Two-Stage Programming.- A.1.3 Chance-Constrained Programming.- A.2 Stochastic Programming with Multiple Objective Functions.- A.2.1 Distribution Problem.- A.2.2 Goal Programming Problem.- A.2.3 Utility Function Problem.- A.2.4 Interactive Problem.- References.

1,168 citations


Journal ArticleDOI
TL;DR: In this article, the authors developed a model and a solution technique for the problem of generating electric power when demands are not certain, and provided techniques for improving the current methods used in solving the traditional unit commitment problem.
Abstract: The authors develop a model and a solution technique for the problem of generating electric power when demands are not certain. They also provide techniques for improving the current methods used in solving the traditional unit commitment problem. The solution strategy can be run in parallel due to the separable nature of the relaxation used. Numerical results indicate significant savings in the cost of operating power generating systems when the stochastic model is used instead of the deterministic model.

593 citations


Journal ArticleDOI
TL;DR: A multinomial approximation of correlated exchange rate processes is proposed that leads to a consistent and tractable lattice model for this compound option valuation problem.
Abstract: In this paper, we develop a stochastic dynamic programming formulation for the valuation of global manufacturing strategy options with switching costs. Overall, we adopt a hierarchical approach. First, exchange rates are modeled as stochastic diffusion processes that exhibit intercountry correlation. Second, the firm's global manufacturing strategy determines options for alternative product designs as well as supply chain network designs. Product options introduce international supply flexibility. Supply chain network options determine the firm's manufacturing flexibility through production capacity and supply chain network linkages. Third, switching costs determine the cost of operational hedging, i.e., the costs associated with reducing downside risks. Overall, the firm maximizes its expected, discounted, global, after-tax value through the exercise of product and supply chain network options and/or through exploitation of operational flexibility contingent on exchange rate realizations. In this environment, the firm must trade off fixed operating costs, switching costs, and the economic benefits derived from exploiting differentials in factor costs and corporate tax rates. A multinomial approximation of correlated exchange rate processes is proposed that leads to a consistent and tractable lattice model for this compound option valuation problem. We then demonstrate how the global manufacturing strategy planning model framework can be utilized to analyze financial and operational hedging strategies.

470 citations


Journal ArticleDOI
TL;DR: Valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems are presented and incorporated within the branch-and-bound framework to result in a branch- and-reduce global optimization algorithm.
Abstract: This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.

343 citations


Journal ArticleDOI
TL;DR: A tabu search heuristic is developed for a version of the stochastic vehicle routing problem where customers are present at locations with some probabilities and have random demands and produces an optimal solution in 89.45% of cases.
Abstract: This paper considers a version of the stochastic vehicle routing problem where customers are present at locations with some probabilities and have random demands. A tabu search heuristic is developed for this problem. Comparisons with known optimal solutions on problems whose sizes vary from 6 to 46 customers indicate that the heuristic produces an optimal solution in 89.45% of cases, with an average deviation of 0.38% from optimality.

305 citations


Journal ArticleDOI
TL;DR: In this article, a unified process design framework for obtaining integrated process and control systems design, which are economically optimal and can cope with parametric uncertainty and process disturbances, is described.
Abstract: Fundamental developments of a unified process design framework for obtaining integrated process and control systems design, which are economically optimal and can cope with parametric uncertainty and process disturbances, are described. Based on a dynamic mathematical model describing the process, including path constraints, interior and end-point constraints, a model that describes uncertain parameters and time-varying disturbances (for example, a probability distributions or lower/upper bounds), and a set of process design and control alternatives (together with a set of control objectives and types of controllers), the problem is posed as a mixed-integer stochastic optimal control formulation. An iterative decomposition algorithm proposed alternates between the solution of a multiperiod “design” subproblem, determining the process structure and design together with a suitable control structure (and its design characteristics) to satisfy a set of “critical” parameters/periods (for uncertainty disturbance) over time, and a time-varying feasibility analysis step, which identifies a new set of critical parameters for fixed design and control. Two examples are detailed, a mixing-tank problem to show the analytical steps of the procedure, and a ternary distillation design problem (featuring a rigorous tray-by-tray distillation model) to demonstrate the potential of the novel approach to reach solutions with significant cost savings over sequential techniques.

265 citations


Journal ArticleDOI
TL;DR: This paper provides a mathematical justification for sample-path optimization by showing that under certain assumptions, the method will almost surely find a point that is, in a specified sense, sufficiently close to the set of optimizers of the limit function.
Abstract: Sample-path optimization is a method for optimizing limit functions occurring in stochastic modeling problems, such as steady-state functions in discrete-event dynamic systems It is closely related to retrospective optimization techniques and to M-estimation The method has been computationally tested elsewhere on problems arising in production and in project planning, with apparent success In this paper we provide a mathematical justification for sample-path optimization by showing that under certain assumptions---which hold for the problems just mentioned---the method will almost surely find a point that is, in a specified sense, sufficiently close to the set of optimizers of the limit function

264 citations


BookDOI
01 Jan 1996
TL;DR: The problem of optimal decisions can be seen as getting simulation and optimization effectively combined, and Optimization of Stochastic Models: The Interface Between Simulation andoptimization is suitable as a text for a graduate level course on Stochastics, or as a secondary text for an undergraduate level course in Operations Research.
Abstract: Stochastic models are everywhere. In manufacturing, queuing models are used for modeling production processes, realistic inventory models are stochastic in nature. Stochastic models are considered in transportation and communication. Marketing models use stochastic descriptions of the demands and buyer's behaviors. In finance, market prices and exchange rates are assumed to be certain stochastic processes, and insurance claims appear at random times with random amounts. To each decision problem, a cost function is associated. Costs may be direct or indirect, like loss of time, quality deterioration, loss in production or dissatisfaction of customers. In decision making under uncertainty, the goal is to minimize the expected costs. However, in practically all realistic models, the calculation of the expected costs is impossible due to the model complexity. Simulation is the only practicable way of getting insight into such models. Thus, the problem of optimal decisions can be seen as getting simulation and optimization effectively combined. The field is quite new and yet the number of publications is enormous. This book does not even try to touch all work done in this area. Instead, many concepts are presented and treated with mathematical rigor and necessary conditions for the correctness of various approaches are stated. Optimization of Stochastic Models: The Interface Between Simulation and Optimization is suitable as a text for a graduate level course on Stochastic Models or as a secondary text for a graduate level course in Operations Research.

228 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated a production planning problem in a periodic review environment with variable production capacity, random yields, and uncertain demand, and they proved that the objective function is quasi-convex and that the structure of the optimal policy is characterized by a single critical point for the initial stock level at each period.
Abstract: We investigate a production planning problem in a periodic review environment with variable production capacity, random yields, and uncertain demand. The implications of random yields and variable capacity for lot sizing previously have been explored separately, but not jointly. Many production environments are likely to be subject to both types of uncertainties. To minimize the total discounted expected costs production, holding, and shortage costs, we formulate the problem as a stochastic dynamic program. For the finite-horizon problem, we prove that the objective function is quasi-convex and that the structure of the optimal policy is characterized by a single critical point for the initial stock level at each period. That is, if the initial stock is greater than this critical point, the optimal planned production is zero; otherwise, it is greater than zero. Expressions for solving the critical point and the optimal planned production are obtained. We further show that the solution for the finite-horizon problem converges to that of the infinite-horizon problem.

225 citations


Book
29 Feb 1996
TL;DR: The aim of this monograph is to provide a scaffolding for future studies of Stochastic Decomposition, as well as some guidelines for computer implementation, to aid in the development of such a system.
Abstract: Preface. 1. Two Stage Stochastic Linear Programs. 2. Sampling Within Stochastic Linear Programming. 3. Foundations of Stochastic Decomposition. 4. Stabilizing Stochastic Decomposition. 5. Stopping Rules for Stochastic Decomposition. 6. Guidelines for Computer Implementation. 7. Illustrative Computational Experiments. Glossary.

Journal ArticleDOI
TL;DR: This paper develops a two-stage stochastic programming approach for process planning under uncertainty by extending a deterministic mixed-integer linear programming formulation to account for the presence of discrete random parameters and devise a decomposition algorithm for the solution of the Stochastic model.
Abstract: This paper develops a two-stage stochastic programming approach for process planning under uncertainty. We first extend a deterministic mixed-integer linear programming formulation to account for t...

Journal ArticleDOI
TL;DR: Two variants of a new method for solving discrete stochastic optimization problems by generating a sequence taking values in the set of feasible alternatives, which is shown to converge almost surely to a globally optimal solution of the underlying optimization solution.
Abstract: This paper is concerned with the problem of optimizing the performance of a stochastic system over a finite set of alternatives in situations where the performance of the system cannot be evaluated analytically, but must be estimated or measured, for instance, through simulation. We present two variants of a new method for solving such discrete stochastic optimization problems. This new method uses global search to look for the optimal solution. It generates a sequence taking values in the set of feasible alternatives, where each new element of the sequence is generated by comparing the current element with another candidate alternative and letting the next element of the sequence be the one of the current and candidate alternatives that appears to yield better performance. For both versions of the proposed method, the element of the set of feasible alternatives that the generated sequence visits most often is shown to converge almost surely to a globally optimal solution of the underlying optimization pr...

Journal ArticleDOI
TL;DR: This article introduces the first implementation of general purpose methods for finding good solutions to multistage, stochastic mixed-integer (0, 1) programming problems and introduces the notion of integer convergence for progressive hedging.
Abstract: Many problems faced by decision makers are characterized by a multistage decision process with uncertainty about the future and some decisions constrained to take on values of either zero or one (for example, either open a facility at a location or do not open it). Although some mathematical theory exists concerning such problems, no general-purpose algorithms have been available to address them. In this article, we introduce the first implementation of general purpose methods for finding good solutions to multistage, stochastic mixed-integer (0, 1) programming problems. The solution method makes use of Rockafellar and Wets' progressive hedging algorithm that averages solutions rather than data. Solutions to the induced quadratic (0,1) mixed-integer subproblems are obtained using a tabu search algorithm. We introduce the notion of integer convergence for progressive hedging. Computational experiments verify that the method is effective. The software that we have developed reads standard (SMPS) data files.

Journal ArticleDOI
TL;DR: This paper extends the applicability of optimization models to situations where both fuzzy and random data are in the state of affairs, and provides some possible avenues for further fruitful developments.

Journal ArticleDOI
TL;DR: In this paper, a dynamic programming (DP) model was used to improve the operation and efficient management of available water for the Aliyar Dam in Tamil Nadu, India, using a neural network procedure (DPN) and using a multiple linear regression procedure (DPR) model.
Abstract: Reservoir operating policies are derived to improve the operation and efficient management of available water for the Aliyar Dam in Tamil Nadu, India, using a dynamic programming (DP) model, a stochastic dynamic programming (SDP) model, and a standard operating policy (SOP). The objective function for this case study is to minimize the squared deficit of the release from the irrigation demand. From the DP algorithm, general operating policies are derived using a neural network procedure (DPN model), and using a multiple linear regression procedure (DPR model). The DP functional equation is solved for 20 years of fortnightly historic data. The field irrigation demand is computed for this study by the modified Penman method with daily meteorological data. The performance of the DPR, DPN, SDP, and SOP models are compared for three years of historic data, using the proposed objective function. The neural network procedure based on the dynamic programming algorithm provided better performance than the other mo...

Journal ArticleDOI
TL;DR: This paper proposes a method for optimizing convex performance functions in stochastic systems, which can include expected performance in static systems and steady-state performance in discrete-event dynamic systems; they may be nonsmooth.
Abstract: In this paper we propose a method for optimizing convex performance functions in stochastic systems. These functions can include expected performance in static systems and steady-state performance in discrete-event dynamic systems; they may be nonsmooth. The method is closely related to retrospective simulation optimization; it appears to overcome some limitations of stochastic approximation, which is often applied to such problems. We explain the method and give computational results for two classes of problems: tandem production lines with up to 50 machines, and stochastic PERT (Program Evaluation and Review Technique) problems with up to 70 nodes and 110 arcs.

Journal ArticleDOI
TL;DR: Methodology for sharing cuts in decomposition algorithms for stochastic programs that satisfy certain interstage dependency models enable sampling-based algorithms to handle a richer class of multistage problems, and may also be used to accelerate the convergence of exact decompose algorithms.
Abstract: Multistage stochastic programs with interstage independent random parameters have recourse functions that do not depend on the state of the system. Decomposition-based algorithms can exploit this structure by sharing cuts (outer-linearizations of the recourse function) among different scenario subproblems at the same stage. The ability to share cuts is necessary in practical implementations of algorithms that incorporate Monte Carlo sampling within the decomposition scheme. In this paper, we provide methodology for sharing cuts in decomposition algorithms for stochastic programs that satisfy certain interstage dependency models. These techniques enable sampling-based algorithms to handle a richer class of multistage problems, and may also be used to accelerate the convergence of exact decomposition algorithms.

Journal ArticleDOI
TL;DR: In this paper, the convergence of ordinal comparison has been studied in the context of regenerative simulations and it has been shown that ordinal contrast converges monotonically in the case of averaging normal random variables.
Abstract: Recent research has demonstrated that ordinal comparison has fast convergence despite the possible presence of large estimation noise in the design of discrete event dynamic systems. In this paper, we address the fundamental problem of characterizing the convergence of ordinal comparison. To achieve this goal, an indicator process is formulated and its properties are examined. For several performance measures frequently used in simulation, the rate of convergence for the indicator process is proven to be exponential for regenerative simulations. Therefore, the fast convergence of ordinal comparison is supported and explained in a rigorous framework. Many performance measures of averaging type have asymptotic normal distributions. The results of this paper show that ordinal comparison converges monotonically in the case of averaging normal random variables. Such monotonicity is useful in simulation planning.

Book
18 Jul 1996
TL;DR: Stochastic Programming Models with Probability and Quantile Objective Functions and Methods and Algorithms for Solving Probabilistic Problems.
Abstract: Stochastic Programming Models with Probability and Quantile Objective Functions. Basic Properties of Probabilistic Problems. Estimates and Bounds for Probabilities and Quantiles. Methods and Algorithms for Solving Probabilistic Problems. Notation List. Index.

Journal ArticleDOI
TL;DR: Stochastic integer programming is more complicated than stochastic linear programming, as will be explained for the case of the two-staged Stochastic Programming model in this article, and a survey of the results accomplished in this recent field of research is given.
Abstract: Stochastic integer programming is more complicated than stochastic linear programming, as will be explained for the case of the two-stage stochastic programming model. A survey of the results accomplished in this recent field of research is given.

Journal ArticleDOI
TL;DR: A successive convex approximation approach is proposed that produces an approximation to the expected recourse function which captures the future effects of current decisions under uncertainty and decomposes the network in each stage into tree subproblems, whose expected recourse functions are easy to obtain.
Abstract: We consider the class of multistage dynamic networks with random arc capacities a framework that is well suited to model dynamic fleet management problems. We propose a successive convex approximation approach that produces an approximation to the expected recourse function which captures the future effects of current decisions under uncertainty. This method decomposes the network in each stage into tree subproblems, whose expected recourse functions are easy to obtain. We also compare this method with two alternative methods on a set of dynamic fleet management problems. The numerical results show that this method is superior to the two alternative methods.

Journal ArticleDOI
TL;DR: In this paper, the advantages of such parallel implementations over serial implementations and compared alternative sequencing protocols for parallel processors are explored. But they require careful attention to processor load balancing, which may not be optimal.
Abstract: Multistage stochastic linear programs can represent a variety of practical decision problems. Solving a multistage stochastic program can be viewed as solving a large tree of linear programs. A common approach for solving these problems is the nested decomposition algorithm, which moves up down the tree by solving nodes and passing information among nodes. The natural independence of subtrees suggests that much of the computational effort of the nested decomposition algorithm can run in parallel across small numbers of fast processors. This paper explores the advantages of such parallel implementations over serial implementations and compares alternative sequencing protocols for parallel processors. Computational experience on a large test set of practical problems with up to 1.5 million constraints and almost 5 million variables suggests that parallel implementations may indeed work well, but they require careful attention to processor load balancing.


Journal ArticleDOI
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...

Journal ArticleDOI
TL;DR: In this article, the authors developed several comparative dynamics results for multidimensional dynamic optimization problems in economics, and provided sufficient conditions for the value function to be monotone and supermodular.

Journal ArticleDOI
TL;DR: A nonlinear multivariable fitting model to decompose the optimal policies obtained by dynamic programming of a unique aggregated reservoir is presented and the CPU time is reduced by a factormore of 15 to 20 compared with the back propagation technique.
Abstract: This paper presents a nonlinear multivariable fitting model to decompose the optimal policies obtained by dynamic programming of a unique aggregated reservoir. The nonlinear functions are generated using radial basis functions (RBF) neural networks. In this method the potential energy of all the reservoirs in the hydropower system is added to form one equivalent reservoir. The operating policy of the equivalent reservoir is determined by stochastic dynamic programming, and finally the operating rules of each reservoir are determined using RBF neural networks. To improve the multivariable representation of the data, a series of piecewise RBF neural networks is determined using clustering analysis. A fuzzy clustering approach is used to determine the RBF`s parameters. This approach has the advantages of being fast and simple to implement with well-established convergence properties. It also has a good representation of the covariance matrix, since all the data belong to all the classes at the same time with different membership grades. A comparison with the back propagation learning and principal components techniques is also reported for Quebec`s La Grande River installations. As a result, the proposed approach gives satisfactory operating rules compared with principal component analysis, and the CPU time is reduced by a factormore » of 15 to 20 compared with the back propagation technique.« less

Journal ArticleDOI
TL;DR: An enhanced Benders decomposition algorithm is developed for solving multistage stochastic linear programs and includes warm start basis selection, preliminary cut generation, the multicut procedure, and decision tree traversing strategies.
Abstract: 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.

Journal ArticleDOI
TL;DR: Convergence of the approximate solutions is proven under the stated assumptions and sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem.
Abstract: This work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support. Based on earlier results, sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem. Error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation. Convergence of the approximate solutions is proven under the stated assumptions. Preliminary computational results are outlined.

Journal ArticleDOI
TL;DR: A combined multiperiod/stochastic optimization formulation is proposed along with a decomposition-based algorithmic procedure for its solution and is illustrated with a process synthesis/planning example problem.