Chapter
3
In Chapter 2 the problem of the optimal expansion of a water resources
system has been formulated in a way designed to facilitate solution of the
problem. More complicated problem statements could be written, of course,
but the existing optimization methods that can be applied to solve such
problem statements are inadequate either because they do not work or
because they take an inordinate amount of computer time or storage. In
this chapter we shall describe a strategy for solving the problem posed in
Chapter 2.
The problem as summarized in Section 2.4 will be designated for ease
of description as Problem I. As it stands it comprises the objective function
[expression (2.1)3
an ^ various types of constraints [expressions (2.2)(2.17)], and is a 0-1 mixed integer programming problem. Consequently,
all the feasible solutions will contain a mixture of integer and noninteger
variables; the integer variables are restricted to the values 0 or 1. We want
to specify (1) if and when each dam should be built, and also (2) a sequence of reservoir releases such that the objective function is maximized.
Several nonlinear terms appear in the problem statement that prevent
the use of linear integer programming as a tool for solution. Note that
constraints (2.7) and (2.10) are nonlinear because of the interaction between Si+ijt and Qimt. The objective function is nonlinear in two respects:
(1) the terms containing the double sum of Xijt are nonlinear, and (2)
interaction takes place between the pairs of variables /Sy* and Xw, λ/* and
Cjty and ajt and K jt . Because of the nonlinear terms in the problem statement and because of the discreteness of several of the independent and
dependent variables only a limited number of solution techniques can be
considered for the solution of Problem I.
66
3
In Chapter 2 the problem of the optimal expansion of a water resources
system has been formulated in a way designed to facilitate solution of the
problem. More complicated problem statements could be written, of course,
but the existing optimization methods that can be applied to solve such
problem statements are inadequate either because they do not work or
because they take an inordinate amount of computer time or storage. In
this chapter we shall describe a strategy for solving the problem posed in
Chapter 2.
The problem as summarized in Section 2.4 will be designated for ease
of description as Problem I. As it stands it comprises the objective function
[expression (2.1)3
an ^ various types of constraints [expressions (2.2)(2.17)], and is a 0-1 mixed integer programming problem. Consequently,
all the feasible solutions will contain a mixture of integer and noninteger
variables; the integer variables are restricted to the values 0 or 1. We want
to specify (1) if and when each dam should be built, and also (2) a sequence of reservoir releases such that the objective function is maximized.
Several nonlinear terms appear in the problem statement that prevent
the use of linear integer programming as a tool for solution. Note that
constraints (2.7) and (2.10) are nonlinear because of the interaction between Si+ijt and Qimt. The objective function is nonlinear in two respects:
(1) the terms containing the double sum of Xijt are nonlinear, and (2)
interaction takes place between the pairs of variables /Sy* and Xw, λ/* and
Cjty and ajt and K jt . Because of the nonlinear terms in the problem statement and because of the discreteness of several of the independent and
dependent variables only a limited number of solution techniques can be
considered for the solution of Problem I.
66