Second-stage decisions

Some decisions have to be taken after the uncertainty has been disclosed. These are called the second-stage decisions and are denoted by a -dimensional vector... 

The result is a deterministic program, where the original second-stage decisions are not a function of the realized scenario, i.e., it is assumed the there is a single scenario problem and all decisions xEV and yEV have to be made before the observation. The corresponding optimization problem is called the expected value problem, (TV problem) and can be written as follows ... 

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. 

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

The optimum second-stage decision D2 is the output of optimizer M — 1 for this optimal state. D2 and S2 together give optimal state S3. The entire optimal sequence of states and decisions is found in this way, as shown below. 

In practice wacc is more complex to calculate because there are several debts incurred at different times and they require common equity as well as preferred equity. In addition, new capital may be raised through new stock offerings. Finally, one is faced with the problem of calculating a return of a project that has multiple decisions at different times, with uneven and uncertain revenues. Clearly, this simple formula needs some expansion, to add the complexities of projects containing multiple first- and second-stage decisions through time. 

Engineers have not yet caught up in relating these concepts with their models. As usual, the mix includes some second-stage decisions, but most of them are first-stage here... 

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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