Generating Scenario Trees Research Articles

A Review of Stochastic Programming Methods for Optimization of Process Systems Under Uncertainty

Uncertainties are widespread in the optimization of process systems, such as uncertainties in process technologies, prices, and customer demands. In this paper, we review the basic concepts and recent advances of a risk-neutral mathematical framework called “stochastic programming” and its applications in solving process systems engineering problems under uncertainty. This review intends to provide both a tutorial for beginners without prior experience and a high-level overview of the current state-of-the-art developments for experts in process systems engineering and stochastic programming. The mathematical formulations and algorithms for two-stage and multistage stochastic programming are reviewed with illustrative examples from process industries. The differences between stochastic programming under exogenous uncertainty and endogenous uncertainties are discussed. The concepts and several data-driven methods for generating scenario trees are also reviewed.

Data-driven multi-stage scenario tree generation via statistical property and distribution matching

This paper brings systematic methods for scenario tree generation to the attention of the Process Systems Engineering community. We focus on a general, data-driven optimization-based method for generating scenario trees that does not require strict assumptions on the probability distributions of the uncertain parameters. Using as a basis the Moment Matching Problem (MMP), originally proposed by Høyland and Wallace (2001), we propose matching marginal (Empirical) Cumulative Distribution Function information of the uncertain parameters in order to cope with potentially under-specified MMP formulations. The new method gives rise to a Distribution Matching Problem (DMP) that is aided by predictive analytics. We present two approaches for generating multi-stage scenario trees by considering time series modeling and forecasting. The aforementioned techniques are illustrated with a production planning problem with uncertainty in production yield and correlated product demands.

Generating scenario trees: A parallel integrated simulation–optimization approach

A crucial issue for addressing decision-making problems under uncertainty is the approximate representation of multivariate stochastic processes in the form of scenario tree. This paper proposes a scenario generation approach based on the idea of integrating simulation and optimization techniques. In particular, simulation is used to generate outcomes associated with the nodes of the scenario tree which, in turn, provide the input parameters for an optimization model aimed at determining the scenarios’ probabilities matching some prescribed targets. The approach relies on the moment-matching technique originally proposed in [K. Høyland, S.W. Wallace, Generating scenario trees for multistage decision problems, Manag. Sci. 47 (2001) 295–307] and further refined in [K. Høyland, M. Kaut, S.W. Wallace, A heuristic for moment-matching scenario generation, Comput. Optim. Appl. 24 (2003) 169–185]. By taking advantage of the iterative nature of our approach, a parallel implementation has been designed and extensively tested on financial data. Numerical results show the efficiency of the parallel algorithm and the improvement in accuracy and effectiveness.

Horizon and stages in applications of stochastic programming in finance

To solve a decision problem under uncertainty via stochastic programming means to choose or to build a suitable stochastic programming model taking into account the nature of the real-life problem, character of input data, availability of software and computer technology. In applications of multistage stochastic programs additional rather complicated modeling issues come to the fore. They concern the choice of the horizon, stages, methods for generating scenario trees, etc. We shall discuss briefly the ways of selecting horizon and stages in financial applications. In our numerical studies, we focus on alternative choices of stages and their impact on optimal first-stage solutions of bond portfolio optimization problems.

Scenario-Based Management of Air Traffic Flow: Developing and Using Capacity Scenario Trees

Recent studies of the single-airport ground-holding problem use static or dynamic optimization to manage uncertainty about future airport capacities. Scenario trees of airport capacity profiles provide the basis for formulating multistage recourse problems. In this paper, methodologies are presented for generating scenario trees from empirical data, and the performance of scenario-based models is examined with the scenario trees. Most U.S. airports have capacity profiles that can be classified into some nominal scenarios, and for many airports, these scenarios can be naturally combined into scenario trees. The delay costs yielded from using dynamic optimization are consistently and considerably lower than from static optimization. The results illustrate the benefit of the wait-and-see strategy in a real-world setting and suggest the need for further research on implementing scenario-based dynamic strategies.

Scenario-Based Management of Air Traffic Flow

Recent studies of the single-airport ground-holding problem use static or dynamic optimization to manage uncertainty about future airport capacities. Scenario trees of airport capacity profiles provide the basis for formulating multistage recourse problems. In this paper, methodologies are presented for generating scenario trees from empirical data, and the performance of scenario-based models is examined with the scenario trees. Most U.S. airports have capacity profiles that can be classified into some nominal scenarios, and for many airports, these scenarios can be naturally combined into scenario trees. The delay costs yielded from using dynamic optimization are consistently and considerably lower than from static optimization. The results illustrate the benefit of the wait-and-see strategy in a real-world setting and suggest the need for further research on implementing scenario-based dynamic strategies.

Simulation and optimization approaches to scenario tree generation

In this paper, three approaches are presented for generating scenario trees for financial portfolio problems. These are based on simulation, optimization and hybrid simulation/optimization. In the simulation approach, the price scenarios at each time period are generated as the centroids of random scenario simulations generated sequentially or in parallel. The optimization method generates a number of discrete outcomes which satisfy specified statistical properties by solving either a sequence of non-linear optimization models (one at each node of the scenario tree) or one large optimization problem. In the hybrid approach, the optimization problem is reduced in size by fixing price variables to values obtained by simulation. These procedures are backtested using historical data and computational results are presented.

Applications of stochastic programming: Achievements and questions

When solving a decision problem under uncertainty via stochastic programming it is essential to choose or to build a suitable stochastic programming model taking into account the nature of the real-life problem, character of input data, availability of software and computer technology. Besides a brief review of history and achievements of stochastic programming, selected modeling issues concerning applications of multistage stochastic programs with recourse (the choice of the horizon, stages, methods for generating scenario trees, etc.) will be discussed.