Stochastic Programming - Index of

282 STOCHASTIC PROGRAMMING
2
1
a
b
d
f
4 5
g
c
3
e
Figure 3 Example network 1.
least partially see the relevance of the last proposition. The following three
inequalities are examples of inequalities describing feasibility for the example
network:
β(2) ≤ γ(d) + γ(f),
β(3) ≤ γ(e),
β(2) + β(3) ≤ γ(d) + γ(e) +γ(f).
Proposition 6.2 states that the latter inequality is not needed, because
G({2, 3}) is not connected. From the inequalities themselves, we easily see
that if the first two are satisfied, then the third is automatically true. It is
perhaps slightly less obvious that, for the very same reason, the inequality
β(1) + β(4) + β(5) ≤ γ(a)+γ(c)
is also not needed. It is implied by the requirement that total supply must
equal total demand plus the companions of the first two inequalities above.
(Remember that each node set gives rise to two inequalities). More specifically,
the inequality can be obtained by adding the following two inequalities and
one equality (representing supply equals demand):
β(1) + β(2) + β(4) + β(5) ≤ γ(c),
β(1) + β(3) + β(4) + β(5) ≤ γ(a),
− β(1) − β(2) − β(3) − β(4) − β(5) = 0.
✷
Once you have looked at this for a while, you will probably realize that the
part of Proposition 6.2 that says that if G(Y )orG(N \Y ) is disconnected