Second stage decision variables

Stochastic Recourse Variables (Second-Stage Decision Variables)... 

Basically, there are two different ways to decompose a 2S-MILP (see Figure 9.10). The scenario decomposition separates the 2S-MILP by the constraints associated to a scenario, whereas the stage decomposition separates the variables into first-stage and second-stage decisions. For both approaches, the resulting subproblems are MILPs which can be solved by standard optimization software. 

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). 

All remaining decisions can be made after the observation of the outcome of uncertain parameters, either in the detailed scheduler or by decisions of the aggregated problem which are taken later. Thus, these decisions are considered as second-stage decisions. Consequently, the vector of second-stage decisions ya consists of all production decisions of the periods i > h and all continuous variables of the cost model for all periods. 

This is the basic idea of a two-stage stochastic program with recoiurse. At the first stage, before a realization of the random variables first-stage decision variables X to optimize the expected value g x) = t[G x, >)] of an objective fimction G(x, to) that depends on the optimal second stage objective function. 

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 stochastic problem is characterised by two essential features the uncertainty in the problem data and the sequence of decisions. In our case, the demand is considered as a random variable with a certain probability distribution. The binary variables associated to the opening of a plant/warehouse as well as the continuous variables that represent the capacity of plants/warehouses are considered as first stage decisions. The fluxes of materials and the sales of products are taken as second stage or recourse variables. The objective hinctions are therefore the expected net present value and the expected consumer satisfaction. 

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. 

In Equation (2), is a coefficient vector and W, h and T are matrices whose elements in principle might depend on the random variables u. The matrix IV is known as the recourse matrix. Fixed recourse means that the recourse matrix, W, is independent on u, whereas complete recourse means that any set of values that we choose for the first stage decisions, x, leaves us with a feasible second stage problem. 

Generating all scenarios for p potential products, each one with two outcomes, results in 2 scenarios. Each individual scenario is a fairly small deterministic problem. The demand and its associated probability for the different outcomes of each product are assumed to be known. If a product fails in the clinical trials, the demand is consequently zero over all remaining time periods. The multi-site investment strategy is common to ail possible scenarios present in the second stage. However, due to the different product demand patterns, every scenario has its own characteristic production, inventory and sales profile. The operational decisions reflect the scenario-dependant decisions made upon completion of the clinical trials and resolution of the uncertainty (wait-and-see) and they include timings of scale-up and qualifications runs (binary variables), allocation of products to manufacturing suites (binary variables), detailed production plans at each production site (continuous variables), inventory profiles (continuous variables), sales profiles at each sales region (continuous variables). 

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