First stage decision variables

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

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 is the basic idea of a two-stage stochastic program with recoiurse. At the first stage, before a realization of the random variables 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. 

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. 

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. 

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. 

For acyclic supply chains, in this section, we propose MINLP models for the FDI-Outsourcing decision problem. First, we propose a model termed as the weighted base model. We propose an extension of this model by incorporating tax. This model is referred as the tax integrated model. In the proposed models we associate a decision variable, x (without the subscripts taken into consideration), for the production/procurement and inventory activities. This implies that the models have a decision variable x, associated with the stage i, which could be interpreted as the production (procurement or inventory activity) of a subcomponent i. The decision variable associated to the transport activity between two production stages i and j, is expressed as a function of the decision variables associated to i and j, that is X, and Xj, and a decision variable y associated to the transport modes. 

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