Stochastic Programming - Index of

NETWORK PROBLEMS 305
speaking, we are looking for the expected value of the objective function in
(6.1) with respect to the random variables q(˜ξ). There is a very large collection
of different methods for bounding PERT problems. Some papers are listed at
the end of this chapter. However, most, if not all, of them can be categorized
as belonging to one or more of the following groups.
6.6.3.1 Series reductions
If there is a node with only one incoming and one outgoing arc, the node is
removed, and the arcs replaced by one arc with a duration equal to the sum
of the two arc durations. This is an exact reformulation.
6.6.3.2 Parallel reductions
If two arcs run in parallel with durations ˜ξ 1 and ˜ξ 2 then they are replaced
with one arc having duration max{˜ξ 1 , ˜ξ 2 }. This is also an exact reformulation.
6.6.3.3 Disregarding path dependences
Let ˜π i be a random variable describing when event i takes place. Then we can
calculate
˜π j =
max {˜π i + q k (˜ξ)},
i∈B + (i)\{i}
with k ∼ (i, j),
as if all these random variables were independent. However, in a PERT
network, the ˜π’s will normally be dependent (even if the q’s are independent),
since the paths leading up to the nodes usually share some arcs. Not only will
they be dependent, but the correlation will always be positive, never negative.
Hence viewing the random variables as independent will result in an upper
bound on the project duration. The reason is that E max{˜ξ 1 , ˜ξ 2 } is smaller
if the nonnegative ˜ξ 1 and ˜ξ 2 are (positively) correlated than if they are not
correlated. A small example illustrates this
Example 6.3 Assume we have two random variables ˜ξ 1 and ˜ξ 2 ,withjoint
distribution as in Table 1. Note that both random variables have the same
marginal distributions; namely, each of them can take on the values 1 or 2,
each with a probability 0.5. Therefore E max{˜ξ 1 , ˜ξ 2 } =1.7 from Table 1, but
0.25(1 + 2 + 2 + 2) = 1.75 if we use the marginal distributions as independent
distributions. Therefore, if ˜ξ 1 and ˜ξ 2 represent two paths with some joint arc,
disregarding the dependences will create an upper bound.
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