Stochastic Programming - Index of

294 STOCHASTIC PROGRAMMING
[-1,1]
[1,3]
1
2
2
1
(0,[2,4]) 3
(0,[2,4])
(0,[2,6])
1
3
3
4 6
4
(0,[2,4])
2
(0,[4,8])
8
1
(0,[2,6])
4 7 5
5 [-2,0]
2
1
(0,[6,8])
2
(0,[0,2])
Slack
Figure 11
[3,3]
Network illustrating the different bounds.
6.5 Bounds
We discussed some bounds for general LPs in Chapter 3. These of course
also apply to networks, since networks are nothing but special cases of linear
programs. The Jensen lower bound can be found by replacing each random
variable (external flow or arc capacity) by its mean and solving the resulting
deterministic network flow problem. The Edmundson–Madansky upper bound
is found by evaluating the network flow problem at all extreme points of the
support. (If the randomness sits in the objective function, the methods give
opposite bounds, just as we discussed for the LP case.)
Figure 11 shows an example that will be used in this section to illustrate
bounds. The terminology is as follows. Square brackets, for example [a, b], are
used to denote supports of random variables. Placed next to a node, they
show the size of the random external flow. Placed in a setting like (c, [a, b]),
the square bracket shows the support of the upper bound on the arc flow for
the arc next to which it is placed. In this setting, c is the lower bound on
the flow. It can become negative in some of the methods. The circled number
next to an arc is the unit arc cost, and the number in a square on the arc is
the arc number. For simplicity, we shall assume that all random variables are
independent and uniformly distributed.
Figure 12 shows the set-up for the Jensen lower bound for the example from
Figure 11. We have now replaced each random variable by its mean, assuming
that the distributions are symmetric. The optimal flow is
f =(2, 0, 0, 0, 2, 2, 1, 1) T ,
with a cost of 18.
Although the Edmundson–Madansky distribution is very useful, it still has
the problem that the objective function must be evaluated in an exponential