Stochastic Programming - Index of

290 STOCHASTIC PROGRAMMING
Since this must be true for all possible values of ˜ξ, we get one such inequality
for each ξ. IfT (ξ) ≡ T 0 , we can make this more efficient by calculating only
one cut, given by the following inequality:
{
− b[A(Y )] T T0 1 + a(Q + ) T T0
2 }
x
∑
≤ min
ξ∈Ξ
i
{
− b[A(Y )] T h 1 i + a(Q+ ) T h 2 }
i ξi − b[A(Y )] T h 1 0 + a(Q+ ) T h 2 0 .
The minimization is of course very simple in the independent case, since
the minimization can be moved inside the sum. When facets have been
transformed into inequalities in terms of x, we might find that they are linearly
dependent. We should therefore subject them, together with the constraints
Ax = b, to a procedure that removes redundant constraints. We have discussed
this subject in Chapter 5.
The above results have two applications. Both are related to preprocessing.
Let us first repeat the one we briefly mentioned above, namely that, after the
inequalities have been added to Ax = b, we have relatively complete recourse,
i.e. any x satisfying the (expanded) first-stage constraints will automatically
produce a feasible recourse problem for all values of ˜ξ. This opens up the
avenue to methods that require this property, and it can help in others where
this is really not needed. For example, we can use the L-shaped decomposition
method (page 171) without concern about feasibility cuts, or apply the
stochastic decomposition method as outlined in Section 3.8.
Another—and in our view more important—use of these inequalities is in
model understanding. As expressions in x,theyrepresentimplicit assumptions
made by the modeller in terms of the first-stage decisions. They are implicit
because they were never written down, but they are there because otherwise
the recourse problem can become infeasible. And, as part of the model, the
modeller has made the requirements expressed in these implicit constraints. If
there are not too many implicit assumptions, the modeller can relate to them,
and either learn about his or her own model, or might decide that he or she
did not want to make these assumption. If so, there is need for a revision of
the model.
It is worth noting that the inequalities in terms of β and γ are also
interesting in their own right. They show the modeller how the external
flow and arc capacities must combine in order to produce a feasible recourse
problem. Also, this can lead to understanding and/or model reformulation.
6.4 An Investment Example
Consider the simple network in Figure 10. It represents the flow of sewage (or
some other waste) from three cities, represented by nodes 1, 2 and 3.