Showing papers on "Stochastic programming published in 2000"

Stochastic ranking for constrained evolutionary optimization

Abstract: Penalty functions are often used in constrained optimization. However, it is very difficult to strike the right balance between objective and penalty functions. This paper introduces a novel approach to balance objective and penalty functions stochastically, i.e., stochastic ranking, and presents a new view on penalty function methods in terms of the dominance of penalty and objective functions. Some of the pitfalls of naive penalty methods are discussed in these terms. The new ranking method is tested using a (/spl mu/, /spl lambda/) evolution strategy on 13 benchmark problems. Our results show that suitable ranking alone (i.e., selection), without the introduction of complicated and specialized variation operators, is capable of improving the search performance significantly.

Scenario reduction in stochastic programming: An approach using probability metrics

Abstract: Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics. Efficient algorithms are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty. For instance, it turns out that after 50% reduction of the scenario tree the optimal reduced tree still has about 90% relative accuracy.

An inexact two-stage stochastic programming model for water resources management under uncertainty

Abstract: An inexact two-stage stochastic programming (ITSP) model is proposed for water resources management under uncertainty. The model is a hybrid of inexact optimization and two-stage stochastic programming. It can reflect not only uncertainties expressed as probability distributions but also those being available as intervals. The solution meth od for ITSP is computationally effective, which makes it applicable to practical problems. The ITSP is applied to a hypothetical case study of water resources system operation. The results indicate that reasonable solutions have been obtained. They are further analyzed and interpreted for generating decision alternatives and identifying significant factors that affect the system's performance. The information obtained through these post-optimality analyses can provide useful decision support for water managers.

Scenarios for Multistage Stochastic Programs

Abstract: A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative scenarios (or a representative scenario tree) relative to the problem being modeled.

Stochastic Lagrangian Relaxation Applied to Power Scheduling in a Hydro-Thermal System under Uncertainty

Abstract: A dynamic (multi-stage) stochastic programming model for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain demand (or load) is developed. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. A stochastic 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 finite 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 a stochastic dynamic programming technique and by a specific descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the first stage (primal) decisions starting from the optimal (stochastic) multipliers is developed. Numerical results are presented for realistic data from a German power utility and for numbers of scenarios ranging from 5 to 100 and a time horizon of 168 hours. The sizes of the corresponding optimization problems go up to 200 000 binary and 350 000 continuous variables, and more than 500 000 constraints.

On the Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Stochastic Programs

Abstract: In this paper we discuss Monte Carlo simulation based approximations of a stochastic programming problem. We show that if the corresponding random functions are convex piecewise linear and the distribution is discrete, then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size. Moreover, by using the theory of large deviations, we show that the probability of such an event approaches one exponentially fast with increase of the sample size. In particular, this happens in the case of linear two- (or multi-) stage stochastic programming with recourse if the corresponding distributions are discrete. The obtained results suggest that, in such cases, Monte Carlo simulation based methods could be very efficient. We present some numerical examples to illustrate the ideas involved.

Probabilistic constrained optimization : methodology and applications

Abstract: Preface. Introduction to the Theory of Probabilistic Functions and Percentiles S. Uryasev. Pricing American Options by Simulation Using a Stochastic Mesh with Optimized Weights M. Broadie, et al. On Optimization of Unreliable Material Flow Systems Y. Ermoliev, et al. Stochastic Optimization in Asset & Liability Management: A Model for Non-Maturing Accounts K. Frauendorfer, M. Schurle. Optimization in the Space of Distribution Functions and Applications in the Bayes Analysis A.N. Golodnikov, et al. Sensitivity Analysis of Worst-Case Distribution for Probability Optimization Problems Y.S. Kan, A.I. Kibzun. On Maximum Realiability Problem in Parallel-Series Systems with Two Failure Modes V. Kirilyuk. Robust Monte Carlo Simulation for Approximate Covariance Matrices and VaR Analyses A. Kreinin, A. Levin. Structure of Optimal Stopping Strategies for American Type Options A.G. Kukush, D.S. Silvestrov. Approximation of Value-at-Risk Problems with Decision Rules R. Lepp. Managing Risk with Expected Shortfall H. Mausser, D. Rosen. On the Numerical Solution of Jointly Chance Constrained Problems J. Mayer. Management of Quality of Service through Chance-constraints in Multimedia Networks E.A. Medova, J.E. Scott. Solution of a Product Substitution Problem Using Stochastic Programming M.R. Murr, A. Prekopa. Some Remarks on the Value-at-Risk and the Conditional Value-at-risk G.Ch. Pflug. Statistical Inference of Stochastic Optimization Problems A. Shapiro.

Concavity and Efficient Points of Discrete Distributions in Probabilistic Programming

Abstract: We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples.

Optimal release strategies for biological control agents: an application of stochastic dynamic programming to population management

Abstract: 1. Establishing biological control agents in the field is a major step in any classical biocontrol programme, yet there are few general guidelines to help the practitioner decide what factors might enhance the establishment of such agents. 2. A stochastic dynamic programming (SDP) approach, linked to a metapopulation model, was used to find optimal release strategies (number and size of releases), given constraints on time and the number of biocontrol agents available. By modelling within a decision-making framework we derived rules of thumb that will enable biocontrol workers to choose between management options, depending on the current state of the system. 3. When there are few well-established sites, making a few large releases is the optimal strategy. For other states of the system, the optimal strategy ranges from a few large releases, through a mixed strategy (a variety of release sizes), to many small releases, as the probability of establishment of smaller inocula increases. 4. Given that the probability of establishment is rarely a known entity, we also strongly recommend a mixed strategy in the early stages of a release programme, to accelerate learning and improve the chances of finding the optimal approach.

Computational solution of capacity planning models under uncertainty

Abstract: The traditional supply chain network planning problem is stated as a multi-period resource allocation model involving 0–1 discrete strategic decision variables. The MIP structure of this problem makes it fairly intractable for practical applications, which involve multiple products, factories, warehouses and distribution centres (DCs). The same problem formulated and studied under uncertainty makes it even more intractable. In this paper we consider two related modelling approaches and solution techniques addressing this issue. The first involves scenario analysis of solutions to “wait and see” models and the second involves a two-stage integer stochastic programming (ISP) representation and solution of the same problem. We show how the results from the former can be used in the solution of the latter model. We also give some computational results based on serial and parallel implementations of the algorithms.

A Two-Stage Modeling and Solution Framework for Multisite Midterm Planning under Demand Uncertainty

Abstract: A two-stage, stochastic programming approach is proposed for incorporating demand uncertainty in multisite midterm supply-chain planning problems. In this bilevel decision-making framework, the production decisions are made “here-and-now” prior to the resolution of uncertainty, while the supply-chain decisions are postponed in a “wait-and-see” mode. The challenge associated with the expectation evaluation of the inner optimization problem is resolved by obtaining its closed-form solution using linear programming (LP) duality. At the expense of imposing the normality assumption for the stochastic product demands, the evaluation of the expected second-stage costs is achieved by analytical integration yielding an equivalent convex mixed-integer nonlinear problem (MINLP). Computational requirements for the proposed methodology are shown to be much smaller than those for Monte Carlo sampling. In addition, the cost savings achieved by modeling uncertainty at the planning stage are quantified on the basis of a r...

Robust Modeling of Multi-Stage Portfolio Problems

Abstract: In the paper, we develop, discuss and illustrate by simulated numerical results a new model of multi-stage asset allocation problem. The model is given by a new methodology for optimization under uncertainty — the Robust Counterpart approach.

Nested Partitions Method for Stochastic Optimization

Abstract: The nested partitions (NP) method is a recently proposed new alternative for global optimization. Primarily aimed at problems with large but finite feasible regions, the method employs a global sampling strategy that is continuously adapted via a partitioning of the feasible region. In this paper we adapt the original NP method to stochastic optimization where the performance is estimated using simulation. We prove asymptotic convergence of the new method and present a numerical example to illustrate its potential.

Option Methods for Incorporating Risk into Linear Capacity Planning Models

Abstract: Manufacturing and service operations decisions depend critically on capacity and resource limits. These limits directly affect the risk inherent in those decisions. While risk consideration is well developed in finance through efficient market theory and the capital asset pricing model, operations management models do not generally adopt these principles. One reason for this apparent inconsistency may be that analysis of an operational model does not reveal the level of risk until the model is solved. Using results from option pricing theory, we show that this inconsistency can be avoided in a wide range of planning models. By assuming the availability of market hedges, we show that risk can be incorporated into planning models by adjusting capacity and resource levels. The result resolves some possible inconsistencies between finance and operations and provides a financial basis for many planning problems. We illustrate the proposed approach using a capacity-planning example.

Exploration of the effectiveness of physical programming in robust design

Abstract: Computational optimization for design is effective only to the extent that the aggregate objective function adequately captures designer's preference. Physical programming is an optimization method that captures the designer's physical understanding of the desired design outcome in forming the aggregate objective function. Furthermore, to be useful, a resulting optimal design must be sufficiently robust/insensitive to known and unknown variations that to different degrees affect the design's performance. This paper explores the effectiveness of the physical programming approach in explicitly addressing the issue of design robustness. Specifically, we synergistically integrate methods that had previously and independently been developed by the authors, thereby leading to optimal-robust-designs. We show how the physical programming method can be used to effectively exploit designer preference in making tradeoffs between the mean and variation of performance, by solving a bi-objective robust design problem. The work documented in this paper establishes the general superiority of physical programming over other conventional methods (e.g., weighted sum) in solving multiobjective optimization problems. It also illustrates that the physical programming method is among the most effective multicriteria mathematical programming techniques for the generation of Pareto solutions that belong to both convex and non-convex efficient frontiers.

Planning logistics operations in the oil industry

Abstract: In this paper we apply stochastic programming modelling and solution techniques to planning problems for a consortium of oil companies. A multiperiod supply, transformation and distribution scheduling problem—the Depot and Refinery Optimization Problem (DROP)—is formulated for strategic or tactical level planning of the consortium's activities. This deterministic model is used as a basis for implementing a stochastic programming formulation with uncertainty in the product demands and spot supply costs (DROPS), whose solution process utilizes the deterministic equivalent linear programming problem. We employ our STOCHGEN general purpose stochastic problem generator to ‘recreate’ the decision (scenario) tree for the unfolding future as this deterministic equivalent. To project random demands for oil products at different spatial locations into the future and to generate random fluctuations in their future prices/costs a stochastic input data simulator is developed and calibrated to historical industry data. The models are written in the modelling language XPRESS-MP and solved by the XPRESS suite of linear programming solvers. From the viewpoint of implementation of large-scale stochastic programming models this study involves decisions in both space and time and careful revision of the original deterministic formulation. The first part of the paper treats the specification, generation and solution of the deterministic DROP model. The stochastic version of the model (DROPS) and its implementation are studied in detail in the second part and a number of related research questions and implications discussed.

A Stochastic Branch-and-Bound Approach to Activity Crashing in Project Management

Abstract: Many applications such as project scheduling, workflow modeling, or business process re-engineering incorporate the common idea that a product, task, or service consisting of interdependent time-related activities should be produced or performed within given time limits. In real-life applications, certain measures like the use of additional manpower, the assignment of highly-skilled personnel to specific jobs, or the substitution of equipment are often considered as means of increasing the probability of meeting a due date and thus avoiding penalty costs. This paper investigates the problem of selecting, from a set of possible measures of this kind, the combination of measures that is the most cost-efficient. Assuming stochastic activity durations, the computation of the optimal combination of measures may be very expensive in terms of runtime. In this article, we introduce a powerful stochastic optimization approach to determine a set of efficient measures that crash selected activities in a stochastic activity network. Our approach modifies the conventional Stochastic Branch-and-Bound, using a heuristic--instead of exact methods--to solve the deterministic subproblem. This modification spares computational time and by doing so provides an appropriate method for solving various related applications of combinatorial stochastic optimization. A comparative computational study shows that our approach not only outperforms standard techniques but also definitely improves conventional Stochastic Branch-and-Bound.

Multireservoir System Optimization using Fuzzy Mathematical Programming

Abstract: For a multireservoir system, where the number of reservoirs is large, the conventional modelling by classical stochastic dynamic programming (SDP) presents difficulty, due to the curse of dimensionality inherent in the model solution. It takes a long time to obtain a steady state policy and also it requires large amount of computer storage space, which form drawbacks in application. An attempt is made to explore the concept of fuzzy sets to provide a viable alternative in this context. The application of fuzzy set theory to water resources systems is illustrated through the formulation of a fuzzy mathematical programming model to a multireservoir system with a number of upstream parallel reservoirs, and one downstream reservoir. The study is aimed to minimize the sum of deviations of the irrigation withdrawals from their target demands, on a monthly basis, over a year. Uncertainty in reservoir inflows is considered by treating them as fuzzy sets. The model considers deterministic irrigation demands. The model is applied to a three reservoir system in the Upper Cauvery River basin, South India. The model clearly demonstrates that, use of fuzzy linear programming in multireservoir system optimization presents a potential alternative to get the steady state solution with a lot less effort than classical stochastic dynamic programming.

Convergence properties of two-stage stochastic programming

Abstract: This paper considers a procedure of two-stage stochastic programming in which the performance function to be optimized is replaced by its empirical mean. This procedure converts a stochastic optimization problem into a deterministic one for which many methods are available. Another strength of the method is that there is essentially no requirement on the distribution of the random variables involved. Exponential convergence for the probability of deviation of the empirical optimum from the true optimum is established using large deviation techniques. Explicit bounds on the convergence rates are obtained for the case of quadratic performance functions. Finally, numerical results are presented for the famous news vendor problem, which lends experimental evidence supporting exponential convergence.

Optimal Irrigation Allocation: A Multilevel Approach

Abstract: Optimal resources allocation strategies for a canal command in the semiarid region of Indian Punjab are developed in a stochastic regime, considering the competition of the crops in a season, both for irrigation water and area of cultivation. The proposed strategies are divided into two modules using a multilevel approach. The first module determines the optimal seasonal allocation of water as well as optimal cropping pattern. This module is subdivided into two stages. The first stage is a single crop intraseasonal model that employs a stochastic dynamic programming algorithm. The stochastic variables are weekly canal releases and evapotranspiration of the crop that are fitted to different probability distribution functions to determine the expected values at various risk levels. The second stage is a deterministic dynamic programming model that takes into account the multicrop situation. An exponential seasonal crop-water production function is used in this stage. The second module is a single crop stochastic dynamic programming intraseasonal model that takes the output of the first module and gives the optimal weekly irrigation allocations for each crop by considering the stress sensitivity factors of crops.

A framework for response surface methodology for simulation optimization

Abstract: textWe develop a framework for automated optimization of stochastic simulation models using Response Surface Methodology. The framework is especially intended for simulation models where the calculation of the corresponding stochastic response function is very expensive or time-consuming. Response Surface Methodology is frequently used for the optimization of stochastic simulation models in a non-automated fashion. In scientific applications there is a clear need for a standardized algorithm based on Response Surface Methodology. In addition, an automated algorithm is less time-consuming, since there is no need to interfere in the optimization process. In our framework for automated optimization we describe all choices that have to be made in constructing such an algorithm.