First-stage decisions

Some decisions have to be taken before the uncertainty is disclosed. These are called the first-stage decisions and are denoted by the rt -dimensional vector x. The first-stage decisions cannot anticipate which scenario will realize and thus have to be the same for all scenarios. This is called non-anticipativity. 

Some of the production decisions of the aggregated scheduling problem are provided to the detailed scheduler. These decisions have to be taken before any observation of the outcome of the uncertain parameters is available. Thus, they correspond to the first-stage decisions of the two-stage stochastic problem. Consequently, the vector of first-stage decisions x consists of all production decisions of the short-term horizon N >rp and Zy, for i e 1,..., If. ... 

Deterministic Variables (First-Stage Decision Variables)... 

A solution based on perfect information would yield optimal first stage decisions for each realization of the random parameter Then the expected value of these decisions, known as wait-and-see (WS) can be written as (Madansky, 1960) ... 

The other quantity of interest is the VSS. In order to quantify it, we first need to solve the mean value problem, also referred to as the expected value (EV) problem. This can be defined as Min z(x, [) ]) where [ ] = f (Birge, 1982). The solution of the EV problem provides the first stage decisions variables evaluated at expectation of the random realizations. The expectation of the EV problem, evaluated at different realization of the random parameters, is then defined as (Birge, 1982) ... 

This indicates that the benefit of incorporating uncertainty in the different model parameters for the petrochemical network investment is 513 622. On the other hand, the EVP I can be evaluated by first finding the wait-and-see (WS) solution. The latter can be obtained by taking the expectation for the optimal first stage decisions evaluated at each realization . From (8.11), the EVPI is ... 

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... 

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. 

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. 

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. 

All previously discussed optimization problems are deterministic, that is, all the data required in those models is assumed to be perfectly known. In this section, we address stochastic programs in which some data may be considered imcertain. In this kind of problems, it is relevant to distinguish between two set of decisions (i.e., variables) the first stage decisions, and the recourse decisions. 

The most widely used and simplest stochastic program is the two-stage program. Here, the first stage decisions are represented by the vector x, while second stage decisions are represented by the vector y. The uncertain parameter is represented by f. Notice that the second stage decisions y are a function of both, the x first stage decisions as well as the uncertain event. In order to simplify the problem representation, the recourse function Q is introduced below. 

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