3. A MULTI-OBJECTIVE STOCHASTIC PROGRAMMING APPROACH
3.1 Basics of Stochastic Programming
3.1.2 Scenario Generation in Stochastic Programming Models
A different second stage decision is made for each realization of the random vector, ws ∈Ω, If this is denoted by y , the resulting second stage problem can be s
expressed as: Min )g(ys,ws (3.16) s.t. W(ws)ys =h(ws)−T(ws)x (3.17) 1 m s R y ∈ + (3.18)
Combining the above, the deterministic equivalent formulation of the two- stage model turns out to be as follows:
Min
∑
= + N l s s sg y w p x f 1 ) , ( ) ( (3.19) s.t Ax=b (3.20) x w T w h y w W( s) s = ( s)− ( s) for all ws ∈Ω (3.21) 0 m R x∈ + (3.22) 1 m s R y ∈ + (3.23)3.1.2 Scenario Generation in Stochastic Programming Models
A deterministic equivalent stochastic programming model is based on a scenario tree (or event tree) representation of the movement of stochastic variables in time. Each branch of the tree denotes a different path of evolution for the relevant random variables. The scenario tree has some decision nodes that represent the stages where the decision maker(s) decide on the courses of action to be pursued. The branches of the scenario tree disseminate from these decision nodes and correspond to alternative states of nature for the stochastic variables after each decision stage. The model is solved on this discretization and the solution determines an optimal
decision for each node based on the information set available at that point. Therefore, constructing a “good” scenario tree that approximates the real stochastic process is a key issue for the success of the SP model.
3.1.2.1 Overview of Scenario Generation Methods
Scenario generation has been an active field of research within the SP context and several alternative methods have been proposed for creating “good” scenario trees. Yu et al (2003) and Kaut and Wallace (2003) provide brief overviews of some common methods available for scenario tree generation.
The simplest approach for generating scenario is to use historical data regarding random variables without any modeling and claim that future will replicate the past (e.g. sampling from past yields from different points in time for generating scenarios for bond returns). This method allows for scenario generation without assuming any specific distributional form for the random variables. Bootstrapping historical data is a common method employed in Value-at-Risk analysis known as “Historical Simulation”. A drawback is that the approach is backward looking and does not represent expectations for the future. Thus, the results may be dominated by a “single, recent, specific crises and it is very difficult to test other assumptions” (Marrison, 2002, pp. 118)
Another approach that does not rely on distributional assumptions is to use the empirical characteristics of random variables and try to create scenarios that replicate those such as the moment matching method of Hoyland and Wallace (2001). In this approach, a scenario tree that matches the specified target values for the random variables, including correlations in-between, is generated. The users are allowed to specify the statistical properties (moments) that are relevant and the idea is to minimize some distance measure between these specified properties and the properties of the generated outcomes on the scenario tree. Hoyland et al (2003) propose an algorithm to speed up this scenario generation method.
A more sophisticated approach requires statistical or econometrical modelling that would capture the characteristics of the movements (and co-movements) of
random variables in time. Boender (1997) uses a vector autoregressive (VAR) time series model to generate asset returns and wage increase scenarios for Dutch pension funds. Villaverde (2003) presents two VAR models including US, European and Japanese assets and exchange rates.
In Pflug (2001), the method to generate the discrete scenario tree is based on the objective of minimizing the “approximation error”. This “optimal discretization method” tries to generate the discrete approximation in such a way that the “approximation error”, i.e. the “difference between the optimal value of the underlying problem and the value found by inserting the solution of approximate problem” is smallest.
Research efforts in the field of scenario generation has also concentrated on reducing the number of scenarios in a given scenario tree to control model complexity while preserving the degree of approximation. The approach of Heitsch and Römisch (2005) is to bundle and delete some scenarios repeatedly from a pre- supplied multivariate scenario tree generated from historical or simulated data series. They employ a certain distance metric and proceed by uniting or deleting scenarios that are “close” to each other to obtain a tree, smaller than the given scenario fan, which maintains to be a “good” approximation.
Since a scenario tree representation contains a limited number of branches, the problem solved is only an approximation of the real problem and thus the “quality” of the scenario tree is extremely important for the “quality” of the solution. The model solutions can be hardly relied if the scenario tree we use is far from representing the true stochastic process. Naturally, the higher the number of scenarios on the scenario tree, the better is the degree of representation. However, that comes along with an amplification in the complexity of the model, i.e. an increase in solution times, and thus, we need to restrict the size of the tree in order to preserve the ability to solve the model. Here lies a trade-off between having a good approximation of the real stochastic process and controlling the dimension of the SP model.
3.1.2.2 Evaluation of Scenario Tree Generation Methods
Despite the importance of scenario tree generation in the SP framework, to the best of our knowledge, there has been little research on the assessment of the representative capacity of scenario trees. Kaut and Wallace (2003) focus on this issue and discuss the evaluation of the quality of scenario generation methods, defining some minimal requirements. Specifically, they propose two measures to test the suitability of a certain generation method for a given SP model: one related with the robustness of the tree generator (stability) and the other regarding the bias it contains.
If the scenario tree generation method is stochastic, it can generate different instances in different runs. In that case, we need to ensure that solving the SP model on different trees, generated by the same method, yields similar optimal values. Thus, by what they define as “in-sample” stability, Kaut and Wallace (2003) propose that the optimal objective values obtained in the SP model based on different scenario tree instances should be approximately identical. While “in-sample stability” is concerned with the variability of the optimal objective function value, “out-of-sample stability” is related with the performance of the optimal solutions in the decision space. In this regard, the authors propose the evaluation of the solutions of the SP model in the “true” problem and test whether solutions obtained on different scenario trees yield similar results when plugged in the real problem. However, this is not always possible since we may not have full information about the actual distributions that drive our stochastic variables.
To ensure that the scenario generation method contains no bias, we need to compare the optimal values in the scenario based problem to that of the true problem and see whether or not they are close to each other. This is again impossible in most cases, since this requires solving the true problem optimally. As a proxy, Kaut and Wallace (2003) recommend the employment of a larger “reference tree” which is believed to have a better representation of the true stochastic process and use the results from this as a benchmark to test for a possible bias.