Michal Kaut

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.

Stability analysis of portfolio management with conditional value-at-risk

We examine the stability of a portfolio management model based on the conditional value-at-risk (CVaR) measure; the model controls risk exposure of international investment portfolios. We use a moment-matching method to generate discrete distributions (scenario sets) of asset returns and exchange rates so that their statistical properties match corresponding values estimated from historical data. First, we establish that the scenario generation procedure does not bias the results of the optimization program, and we determine the required number of scenarios to attain stable solutions. We then investigate the sensitivity of the CVaR model to mis-specifications in the statistics of stochastic parameters: mean, standard deviation, skewness, kurtosis, as well as correlations. The results are most sensitive to estimation errors in the means of the stochastic parameters (asset returns and currency exchange rates). Mis-specifications in the standard deviation, skewness and correlations of the random parameters also have considerable impact on the solutions. The effect of mis-specifications in the values of kurtosis, although less than that of the other statistics, is still not negligible.

Shape-based scenario generation using copulas

The purpose of this article is to show how the multivariate structure (the “shape” of the distribution) can be separated from the marginal distributions when generating scenarios. To do this we use the copula. As a result, we can define combined approaches that capture shape with one method and handle margins with another. In some cases the combined approach is exact, in other cases, the result is an approximation. This new approach is particularly useful if the shape is somewhat peculiar, and substantially different from the standard normal elliptic shape. But it can also be used to obtain the shape of the normal but with margins from different distribution families, or normal margins with for example tail dependence in the multivariate structure. We provide an example from portfolio management. Only one-period problems are discussed.

Stochastic optimization models for a single-sink transportation problem

In this paper, we study a single-sink transportation problem in which the production capacity of the suppliers and the demand of the single customer are stochastic. Shipments are performed by capacitated vehicles, which have to be booked in advance, before the realization of the production capacity and the demand. Once the production capacity and the demand are revealed, there is an option to cancel some of the booked vehicles against a cancellation fee; if the quantity shipped from the suppliers using the booked vehicles is not enough to satisfy the demand, the residual quantity is purchased from an external company. The problem is to determine the number of vehicles to book in order to minimize the total cost. We formulate a two-stage and a multistage stochastic mixed integer linear programming models to solve this problem and test them on a real case provided by Italcementi, the primary Italian cement producer and the fifth largest cement producer in the world. We test the influence of different scenario-tree structures on the solutions of the problem, as well as sensitivity of the results with respect to the cancellation fee.

Multi-horizon stochastic programming

Infrastructure-planning models are challenging because of their combination of different time scales: while planning and building the infrastructure involves strategic decisions with time horizons of many years, one needs an operational time scale to get a proper picture of the infrastructure’s performance and profitability. In addition, both the strategic and operational levels are typically subject to significant uncertainty, which has to be taken into account. This combination of uncertainties on two different time scales creates problems for the traditional multistage stochastic-programming formulation of the problem due to the exponential growth in model size. In this paper, we present an alternative formulation of the problem that combines the two time scales, using what we call a multi-horizon approach, and illustrate it on a stylized optimization model. We show that the new approach drastically reduces the model size compared to the traditional formulation and present two real-life applications from energy planning.

Single source single-commodity stochastic network design

Stochastics affects the optimal design of a network. This paper examines the single-source single-commodity stochastic network design problem. We characterize the optimal designs under demand uncertainty and compare with the deterministic counterparts to outline the basic structural differences. We do this partly as a basis for developing better algorithms than are available today, partly to simply understand what constitutes robust network designs.

A copula-based heuristic for scenario generation

This paper presents a new heuristic for generating scenarios for two-stage stochastic programs. The method uses copulas to describe the dependence between the marginal distributions, instead of the more common correlations. The heuristic is then tested on a simple portfolio-selection model, and compared to two other scenario-generation methods.

A heuristic for generating scenario trees for multistage decision problems

In stochastic programming models we always face the problem of how to represent the uncertainty. When dealing with multidimensional distributions, the problem of generating scenarios can itself be difficult. We present an algorithm for efficient generation of scenario trees for single or multi-stage problems. The presented algorithm generates a discrete distribution specified by the first four marginal moments and correlations. The scenario tree 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. Our testing shows that the new algorithm is substantially faster than a benchmark algorithm. The speed-up increases with the size of the tree and is more than 100 in a case of 20 random variables and 1000 scenarios. This allows us to increase the number of random variables and still have a reasonable computing time.

Single-commodity network design with random edge capacities

This paper examines the single-commodity network design problem with stochastic edge capacities. We characterize the structures of the optimal designs and compare with the deterministic counterparts. We do this partly to understand what constitutes robust network designs, but also to construct a heuristic for the stochastic problem, leading to optimality gaps of about 10%. In our view, that is a rather good result for problems that otherwise cannot be solved at all.