Stochastic Programming - Index of

304 STOCHASTIC PROGRAMMING
First, it is implicit in the model that, while the original activity durations
are random, the changes a k x k are not. In terms of probability distributions,
therefore what we have done is to reduce the means of the distributions
describing activity durations, but without altering the variances. This might
or might not be a reasonable model. Clearly, if we find this unreasonable, we
could perhaps let a k be a random variable as well, thereby making also the
effect of the investment x k uncertain.
The above discussion is more than anything a warning that whenever we
introduce randomness in a model, we must make sure we know what the
randomness means. But there is a much more serious model interpretation if
we see (6.2) as a two-stage problem. It means that we think we are facing
a project where, before it is started, we can make investments, but where
afterwards, however badly things go, we shall never interfere in order to fix
shortcomings. Also, even if we are far ahead of schedule, we shall not cut back
on investments to save money. We may ask whether such projects exist—
projects where we are free to invest initially, but where afterwards we just sit
back and watch, whatever happens.
From this discussion you may realize (as you have before—we hope) that the
definition of stages is important when making models with stochasticity. In
our view, project scheduling with uncertainty is a multistage problem, where
decisions are made each time new information becomes available. This makes
the problem extremely hard to solve (and even formulate—just try!) But this
complexity cannot prevent us from pointing out the difficulties facing anyone
trying to formulate PERT problems with only two stages.
We said earlier that there were two ways of interpreting (6.2) in a setting
of uncertainty. We have just discussed one. The other is different, but has
similar problems. We could interpret (6.2) with uncertainties as if we first
observed the values of q and then made investments. This is the “wait-andsee
solution”. It represents a situation where we presently face uncertainty, but
where all uncertainty will be resolved before decisions have to be made. What
does that mean in our context It means that before the project starts, all
uncertainty related to activities disappears, everything becomes known, and
we are faced with investments of the type (6.2). If the previous interpretation
of our problem was odd, this one is probably even worse. In what sort of
project will we have initial uncertainty, but before the first activity starts,
everything, up to the finish of the project, becomes known This seems almost
as unrealistic as having a deterministic model of the project in the first place.
6.6.3 Bounds on the Expected Project Duration
Despite our own warnings in the previous subsection, we shall now show
how the extra structure of PERT problems allows us to find bounds on the
expected project duration time if activity durations are random. Technically