106 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
gation water its usable output will depend on local hydrology
and on the relation of the timing of the local hydrologic cycle
to the timing of irrigation demand in its market area. And
so on. The design typically determines capacity, but economic
and social factors determine output.
Be that as it may, the ancillary factors are usually not
subject to decision (or if they are they should be regarded as
part of the design), so that the economic performance of the
project is determined (leaving uncertainty aside) once the de-
sign specifications have been settled. We can denote the rela-
‘ionship between design and output by writing fi(x,, ..., x,)
for the extent of performance with respect to the ith objective
or target, where x, ..., x, is the design vector. Similarly we
can write c(y, ..., x,) as the cost of meeting the design spe-
cifications, x4, ..., x,. In this notation the designers are
charged with the task of choosing x, ..., x,, so as to minimize
c(x;, ..., x,) while satisfying fi(x,, ..., x,)>T, i=1, ..., k,
T; being the target level of the ith objective, and % the number
of objectives considered.
You will leap at once to several objections. One is: where
can we find these ambitious production functions, fi(x,, ..., x,,)?
On this, I hope that you will be willing to suspend -your dis-
belief; I want to discuss that topic after we have seen what
we can do with these functions if we have them. A second
objection is: the design specifications do not determine the
outputs. By varying the way in which a given structure is
operated, one form of output can be substituted for another.
A third is: how are the various target levels, T,, to be estab-
lished? I shall deal with the second and third obiections im-
mediately, and more or less together.
Let us reformulate slightly our model of a design problem
to create room for inserting some more complicated considera-
tions. Let Fi(xy, ..., x,, uy, ..., u,) denote the production or
performance function with respect to the ith objective, where,
in addition to the previous notation, #,, ..., #, describe a par-
"3] Dorfman - pag. 10
