Stochastic Programming - Index of

284 STOCHASTIC PROGRAMMING
procedure CreateIneq(Y : set of nodes);
begin
if A(Y ) ≠ ∅ then begin
create the inequality b(A(Y )) T β ≤ a(Q + ) T γ;
create the inequality b(A(N \Y )) T β ≤ a(Q − ) T γ;
end
else begin
create the inequality b(A(N )) T β ≤ 0;
create the inequality −b(A(N )) T β ≤ 0;
end;
end;
Figure 4
Algorithm for generating inequalities—capacitated case.
Figure 5 Example network used to illustrate Proposition 6.3.
The only set Y where i ∈ Y but j ∉ Y , at the same time as both G(Y )
and G(N \Y ) are connected, is the set where Y = {i}. The reason is that
node j blocks node i’s connections to all other nodes. Therefore, after calling
CreateIneq({i}), we can safely collapse node i into node j. Examplesofthis
can be found in Figure 5, (see e.g. nodes 4 and 5). This result is easy to
implement, since all we have to do is run through all nodes, one at a time,
and look for nodes satisfying B + (i) ∪ F + (i) ={i, j}. Whenever collapses take
place, F + and B + (or, alternatively, F ∗ and B ∗ ) must be updated for the
remaining nodes.
By repeatedly using this proposition, we can remove from the network all
trees (and trees include “double arcs” like those between nodes 2 and 5). We
are then left with circuits and paths connecting circuits. The circuits can be
both directed and undirected. In the example in Figure 5 we are left with