Stein W. Wallace

Generating Scenario Trees for Multistage Decision Problems

In models of decision making under uncertainty we often are faced with the problem of representing the uncertainties in a form suitable for quantitative models. If the uncertainties are expressed in terms of multivariate continuous distributions, or a discrete distribution with far too many outcomes, we normally face two possibilities: either creating a decision model with internal sampling, or trying to find a simple discrete approximation of the given distribution that serves as input to the model. This paper presents a method based on nonlinear programming that can be used to generate a limited number of discrete outcomes that satisfy specified statistical properties. Users are free to specify any statistical properties they find relevant, and the method can handle inconsistencies in the specifications. The basic idea is to minimize some measure of distance between the statistical properties of the generated outcomes and the specified properties. We illustrate the method by single- and multiple-period problems. The results are encouraging in that a limited number of generated outcomes indeed have statistical properties that are close to or equal to the specifications. We discuss how to verify that the relevant statistical properties are captured in these specifications, and argue that what are the relevant properties, will be problem dependent.

Scenarios for Multistage Stochastic Programs

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.

Evaluation of scenario-generation methods for stochastic programming

In this paper, we discuss the evaluation of quality/suitability of scenario-generation methods for a given stochastic programming model. We formulate minimal requirements that should be imposed on a scenario-generation method before it can be used for solving the stochastic programming model. We also show how the requirements can be tested. The procedure of testing a scenario-generation method is illustrated on a case from portfolio management. In addition, we provide a short overview of the most common scenario-generation methods.

A Heuristic for Moment-Matching Scenario Generation

In stochastic programming models we always face the problem of how to represent the random variables. This is particularly difficult with multidimensional distributions. We present an algorithm that produces a discrete joint distribution consistent with specified values of the first four marginal moments and correlations. The joint distribution is constructed by decomposing the multivariate problem into univariate ones, and using an iterative procedure that combines simulation, Cholesky decomposition and various transformations to achieve the correct correlations without changing the marginal moments.With the algorithm, we can generate 1000 one-period scenarios for 12 random variables in 16 seconds, and for 20 random variables in 48 seconds, on a Pentium III machine.

Stochastic Programming Models in Energy

We give the reader a tour of good energy optimization models that explicitly deal with uncertainty. The uncertainty usually stems from unpredictability of demand and/or prices of energy, or from resource availability and prices. Since most energy investments or operations involve irreversible decisions, a stochastic programming approach is meaningful. Many of the models deal with electricity investments and operations, but some oil and gas applications are also presented. We consider both traditional cost minimization models and newer models that reflect industry deregulation processes. The oldest research precedes the development of linear programming, and most models within the market paradigm have not yet found their final form.

Decision Making Under Uncertainty: Is Sensitivity Analysis of Any Use?

Sensitivity analysis, combined with parametric optimization, is often presented as a way of checking if the solution of a deterministic linear program is reliable--even if some of the parameters are not fully known but are instead replaced by a best guess, often a sample mean. It is customary to claim that if the region over which a certain basis is optimal is large, one is fairly safe by using the solution of the linear program. If not, the parametric analysis will provide us with alternative solutions that can be tested. This way, sensitivity analysis is used to facilitate decision making under uncertainty by means of a deterministic tool, namely parametric linear programming. We show in this note that this basic idea of stability has little do with optimality of an optimization problem where the parameters are uncertain.

Hedging Electricity Portfolios via Stochastic Programming

Electricity producers participating in the Nordic wholesale-level market face significant uncertainty in inflow to reservoirs and prices in the spot and contract markets. Taking the view of a single risk-averse producer, we propose a stochastic programming model for the coordination of physical generation resources with hedging through the forward and option market. Numerical results are presented for a five-stage, 256 scenario model that has a two year horizon.

A Study of Demand Stochasticity in Service Network Design

The objective of this paper is to investigate the importance of introducing stochastic elements into service network design formulations. To offer insights into this issue, we take a basic version of the problem in which periodic schedules are built for a number of vehicles and where only the demand may vary stochastically. We study how solutions based on uncertain demand differ from solutions based on deterministic demand and provide qualitative descriptions of the structural differences. Some of these structural differences provide a hedge against uncertainty by using consolidation. This way we get consolidation as output from the model rather than as an a priori required property. Service networks with such properties are robust, as seen by the customers, by providing operational flexibility.

The performance of stochastic dynamic and fixed mix portfolio models

The purpose of this paper is to demonstrate how to evaluate stochastic programming models, and more specifically to compare two different approaches to asset liability management. The first uses multistage stochastic programming, while the other is a static approach based on the so-called constant rebalancing or fixed mix. Particular attention is paid to the methodology used for the comparison. The two alternatives are tested over a large number of realistic scenarios created by means of simulation. We find that due to the ability of the stochastic programming model to adapt to the information in the scenario tree, it dominates the fixed mix approach.

Sensitivity Analysis and Uncertainty in Linear Programming

Linear programming (LP) is one of the great successes to emerge from operations research and management science. It is well developed and widely used. LP problems in practice are often based on numerical data that represent rough approximations of quantities that are inherently difficult to estimate. Because of this, most LP-based studies include a postoptimality investigation of how a change in the data changes the solution. Researchers routinely undertake this type of sensitivity analysis (SA), and most commercial packages for solving linear programs include the results of such an analysis as part of the standard output report. SA has shortcomings that run contrary to conventional wisdom. Alternate models address these shortcomings.