STOCHASTIC

296 REVIEW OF ECONOMIC STUDIES
where the rules D are to be selected in accordance with certain prescribed properties,
e.g. those that depend linearly on the random variables. 1
A wider class of objectives has been studied or used for chance-constrained programming
problems than for linear programming under uncertainty. The principal ones
have been (a) the so-called £-type, where the functional is in the form of an expected
value, (6) the K-type, where the functional has a quadratic form, for example, in connection
with minimization of the variance, (c) the .P-type, where the functional expresses the goal
of maximizing the probability that at least a certain level of the functional is obtained. 2
The most common way of solving chance-constrained programming problems is to
transform the probabilistic constraints to their so-called deterministic equivalents. To
illustrate, assume that we have a single probabilistic constraint of the form
p(f, a,x,.Sfc,j2a. ...(2.4)
Assume further that 6, is normally distributed with mean p. and variance a 2 . We can
then write (2.4) as follows
where
or
i+F y~ l J g a, ...(2.5)
(271)* J„
t a,Xt-H>F-\*-i)e. -(2.6)
If we compare (2.4) and (2.6) we can see that the effect is to transform the probabilistic
expression to a deterministic one. Moreover, in this special case the constraint is linear,
and with a linear functional we have obtained a linear programming problem. This
method of transforming more complicated probabilistic constraints will often give
non-linear constraints and one has then to use some technique in the field of non-linear
programming. 3
Another method, described in Naslund and Whinston [13], makes use of the fact
that under certain conditions constraints similar to (2.4) can be transformed to an integral
whereupon variational methods can be used to solve the chance-constrained programming
problem.
In the remainder of the paper several transformations of probabilistic constraints
to their so-called deterministic equivalents will be made. We shall therefore conclude
this section with a discussion of the method in detail.
The constraints that we are dealing with in this paper are of the form
P(x>B)^«. ...(2.7)
This means that the probability that the random variable x is greater than B must be
greater than a.
In order to obtain a meaningful problem we need a>0-5 (see Charnes, Cooper and
Thompson [7]).
1 For an example of an explicit form of (2.3) see Naslund and Whinston [12].
2 For a discussion of these see A. Charnes and W. W. Cooper [4].
3 For detailed description of this method for solving chance-constrained programming problems
see [5] and [12].
320 PART III. STATIC PORTFOLIO SELECTION MODELS