Two-Stage Stochastic Integer Programming

In the previous section it was shown that the performance of a scheduler can be significantly improved by the use of stochastic models. In this section, we present the mathematical models that represent two-stage stochastic scheduling problems and algorithmic approaches to the optimization of the schedules. 

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 constraints of a two-stage stochastic linear program can be classified into constraints on the first-stage variables only (9.3.2) and constraints on the first and on the second-stage variables (9.3.3). The latter represent the interdependency of the stages. All constraints are represented as linear inequalities with the matrices Aer x , Tb6 R 2xn Wm Rm2x 2, and the vectors b Rm2 and K, e R 2. 

The model (DEP) covers the general case with parametric uncertainties in the objective function (q ), in the left-hand-side multipliers of x and yai (Tffl and W , respectively) and in the right-hand-side parameters (h ). 

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Till, J Engell, S. and Sand, G. (2005) Rigorous vs stochastic algorithms for two-stage stochastic integer programming applications. Inti. J. Inf. Technol., 11, 106-115. 

The hybrid algorithm is in general suitable for any two-stage stochastic mixed-integer linear program with integer requirements in the first-stage and in the... 

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. 

The above formulation is an extension of the deterministic model explained in Chapter 5. We will mainly explain the stochastic part of the above formulation. The above formulation is a two-stage stochastic mixed-integer linear programming (MILP) model. Objective function (9.1) minimizes the first stage variables and the penalized second stage variables. The production over the target demand is penalized as an additional inventory cost per ton of refinery and petrochemical products. Similarly, shortfall in a certain product demand is assumed to be satisfied at the product spot market price. The recourse variables V [ +, , V e)+ and V e[ in... 

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