1.3. Techniques for the Optimization
of a Water Resources System
21
investment to be highlighted, the solution yielding a minimum of the total
of the current joint construction costs plus the discounted sum of the operating expenses may be sought. Suppose we use the latter, discounted at
8%, as the criterion of optimality. The problem can then be reduced to one
of finding the minimum of the following linear objective function:
fix) = Σ kiXi + Σ [V(l + r)«] Σ *<**
where r is a discount factor, ki is the capital investment for project i, Sa
represents the operating costs for project i in year i, and T m&li is the planning horizon of 4 yr. With r = 0.08, we obtain
/(x) = (24 + (1/1.08) (14) + (1/1.08
2 )(14) + . ·
+ . · ·
Application of a linear programming code to minimize /(x) yields the
solution
xi = 694, x 2 = 825, x z = 0, x* = 0, x b = 0
f(x) = 1.24 Χ 10
5
As is clear from problem (1.1), the major handicap in the use of linear
programming is that all the functions and inequalities (and equations)
must be linear and the variables continuous. If they are not, then nonlinear
programming [Beveridge and Schechter, 1970, pp. 355-502] must be employed.
The nonlinear programming problem can be formally stated as
Minimize
f(x)
for
χ 6 E
n
(1.4a)
subject to m linear and/or nonlinear equality constraints
hj(x) =0
j = 1, . . . , m
(1.4b)
and (p — m) linear and/or nonlinear inequality constraints
<7;(x) <0
i = m + l, ...,p
(1.4c)
Figure 1.8 illustrates the nonlinear programming problem for two variables. The feasible region is identified by the hash marks along the equality
constraint. Although in some special cases the equality constraints can be
solved explicitly for selected variables and those variables eliminated from
the problem as independent variables, reducing the problem to one with
inequality constraints only, most often the equality constraints can only
be solved implicitly and must be retained.