Stochastic program

All conventional approaches (mathematical and stochastic programming, parametric and nonparametric regression analysis) adopt as a common solution format real vectors, x and as performance criterion,... 

In the first part of this chapter (Section 9.2), we present an uncertainty conscious scheduling approach that combines reactive scheduling and stochastic scheduling by using a moving horizon scheme with an uncertainty conscious model. In this approach, it is assumed that decisions are made sequentially and that the effect of the revealed uncertainties can be partially compensated by later decisions. The sequence of decisions and observations is modeled by a sequence of two-stage stochastic programs. 

Mathematical optimization models that explicitly consider such a multi-stage structure belong to the class of multi-stage stochastic programs. A deterministic optimization model with uncertain parameters is extended to a multi-stage model by three measures ... 

A stochastic program is a mathematical program (optimization model) in which some of the problem data is uncertain. More precisely, it is assumed that the uncertain data can be described by a random variable (probability distribution) with sufficient accuracy. Here, it is further assumed that the random variable has a countable number of realizations that is modeled by a discrete set of scenarios co = 1,..., 2. 

In a stochastic program with recourse, some corrective decisions or recourse actions can be taken after the uncertainty is disclosed. Each point in time where a decisions is made is called a stage. The two-stage stochastic program is the most... 

The concept of the value of the stochastic solution (VSS) measures the advantage of using a two-stage stochastic program over using a deterministic one, in other words, it measures the cost of ignoring the uncertainty. 

In the deterministic program that corresponds to a stochastic program with discrete scenarios, the uncertain parameters are replaced by their mean values ... 

Plugging the first-stage solution of the EV problem xEV into the stochastic program (2S-MILP) gives the expected result of using the EV solution (EEV problem). The solution of the EEV problem is not necessarily optimal for the original 2S-MILP. Consequently, the optimal objective value of the EEV problem is always greater than (or at least equal to) the optimal objective value of the 2S-MILP, such that the objective of EEV is an upper bound for the optimal solution of the 2S-MILP ... 

An uncertainty conscious scheduling approach for real-time scheduling was presented in this chapter. The approach is based on a moving horizon scheme where in each time period a two-stage stochastic program is solved. For the investigated example it was found that the stochastic scheduler improved the objective on average by 10% compared to a deterministic scheduler. 

Fora recent survey of reactive and stochastic chemical batch scheduling approaches, the reader is referred to Floudas and Lin [2], For a survey of the different types of probabilistic mathematical models that explicitly take uncertainties into account, see Sahinidis [12]. For detailed information about stochastic programming and its applications, the reader is referred to the books of Birge and Louveaux [9], Ruszczynski and Shapiro [10], or Wallace and Ziemba [26]. 

Markert, A. (2004) User s guide to ddsip — AC Package for the Dual Decomposition of Stochastic Programs with Mixed-Integer Recourse. 

Wallace, S.W. and Ziemba, W.T. (eds.) (2005) Applications of Stochastic Programming. MPS-SIAM series in optimization. SIAM, Philadelphia, PA. 

Appendix A Two-Stage Stochastic Programming 183 Appendix B Chance Constrained Programming 185 Appendix C SAA Optimal Solution Bounding 187... 

All chapters are equipped with clear figures and tables to help the reader understand the included topics. Furthermore, several appendices are included to explain the general background in the area of stochastic programming, chance constraint programming and robust optimization. 

Birge, J.R. (1995) Models and model value in stochastic programming. Annals of Operations Research, 59, 1. 

Chemical process systems are subject to uncertainties due to many random events such as raw material variations, demand fluctuations, equipment failures, and so on. In this chapter we will utilize stochastic programming (SP) methods to deal with these uncertainties that are typically employed in computational finance applications. These methods have been very useful in screening alternatives on the basis of the expected value of economic criteria as well as the economic and operational risks involved. Several approaches have been reported in the literature addressing the problem of production planning under uncertainty. Extensive reviews surveying various issues in this area can be found in Applequist et al. (1997), Shah (1998), Cheng, Subrahmanian, and Westerberg (2005) and Mendez et al. (2006). 

Kristoffersen, T.K. (2005) Deviation measures in linear two-stage stochastic programming. Mathematical Methods of Operations Research, 62, 255. 

Linderoth, J., Shapiro, A., and Wright, S. (2002) The empirical behavior of sampling methods for stochastic programming. Optimization online, http // www.optimization-online.org/DB HTML/ 2002/01/424.html. 

Ruszczynski, R. and Shapiro, A. (2003) Handbooks in Operations Research and Management Science-Volume 10 Stochastic Programming, Elsevier Science, Amsterdam, Netherlands. 

Shapiro, A. and Homem-de-Mello, T. (1998) A simulation-based approach to two stage stochastic programming with recourse. Mathematical Programming, 81, 301. 

The stochastic model with recourse in the previous section takes a decision merely based on first-stage and expected second-stage costs leading to an assumption that the decision-maker is risk-neutral (Sahinidis, 2004). In order to capture the concept of risk in stochastic programming, Mulvey, Vanderbei and Zenios (1995) proposed the following amendment to the objective function ... 

Since stochastic programming adds computational burden to practical problems, it is desirable to quantify the benefits of considering uncertainty. In order to address this point, there are generally two values of interest. One is the expected value of perfect information (EVPI) which measures the maximum amount the decision maker is willing to pay in order to get accurate information on the future. The second is the value of stochastic solution (VSS) which is the difference in the objective function between the solutions of the mean value problem (replacing random events with their means) and the stochastic solution (SS) (Birge, 1982). 

The two-stage mixed-integer stochastic program with recourse that includes a total number of200 scenarios for each random parameter is considered in this section. All random parameters were assumed to follow a normal distribution and the scenarios for all random parameters were generated simultaneously. Therefore, the recourse variables account for the deviation from a given scenario as opposed to the deviation from a particular random number realization. 

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