Second-stage variables

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. 

When the second stage decisions are real-valued variables, the value function Qu(x) is piecewise-linear and convex in x. However, when some of the second stage variables are integer-valued, the convexity property is lost. The value function Qafx) is in general non-convex and non-differentiable in x. The latter property prohibits the use of gradient-based search methods for solving (MASTER). 

In order to evaluate the VSS we first solved the deterministic problem, as illustrated in the previous section, and fixed the petrochemical network and the production rate of the processes. We then solved the EEV problem by allowing the optimization problem to choose second stage variables with respect to the realization of the uncertain parameters From (8.12), the VSS is ... 

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... 

In addition, Guillen etal. consider uncertainty in the demand-price relation parameters. Thus, they build a stochastic model, in which processes are first-stage decisions, not parameters as is common in batch scheduling models, and sales are second-stage variables. The model renders different schedules and prices (Figure 12.28). The resulting schedule... 

The demand is modeled using normal distributions and sampling scenarios. The amount ordered in time Tq is considered as a first-stage variable, that is, a decision made before the uncertainty is revealed, whereas the amounts of materials ordered in the next periods, t l, t z and T 3, are considered second-stage variables, decisions made after the uncertainty materialization. 

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