Introduction
67
Possible techniques of solution include the generalized Lagrange multiplier technique [Everett, 1963; Kaplan, 1966], dynamic programming
[Bellman and Dreyfus, 1962], and mixed variable programming [Benders,
1962], but each method is rendered ineffective because of some factor or
characteristic of Problem I.
The difficulties with the application of the generalized Lagrange multiplier technique are twofold:
1. First, the method requires that the alternative new projects be independent. In Problem I the alternative projects are independent with respect to cost factors and required investment but are interrelated with
respect to benefits.
2. Second, the method may fail to provide an optimal solution [Cord,
1964; Weingartner, 1966].
Dynamic programming might be deemed to be suitable to solve Problem
I, but previous experience has shown that the computer core storage requirements would be prohibitively large because of the large number of
state variables involved in Problem I. Note that:
1. When the problem has more than one constraint, the number of state
variables increases.
2. Because of the physical configuration of the problem, a dynamic programming formulation would include converging and diverging branches,
with a consequent increase in the number of state variables.
For problems involving a large number of constraints Dantzig [1957] and
Nemhauser and Ullmann [1969] reported that dynamic programming was
not an efficient tool. Swanson [1970] reached the same conclusion for
water resource systems involving several state variables and more than two
stages.
A third possible method of optimization is Benders' [1962] algorithm
for mixed variable programming problems. This algorithm requires that
the objective function be separable with respect to the integer and continuous variables, i.e., the two sets of variables must be capable of being linearly
summed. However, there is interaction among variables in the objective
function so that Problem I does not meet the condition of separability.
In the Texas Water Development Board Study [1969] it was decided to
develop a screening technique that would be able to reduce drastically the
numbers of alternatives to be considered. This technique, which utilized a
variety of optimization routines to find "near optimum" solutions, embodied in four major phases: (1) initial element sizing and reservoir operating rules using an optimal allocation program, (2) initial screening of
67
Possible techniques of solution include the generalized Lagrange multiplier technique [Everett, 1963; Kaplan, 1966], dynamic programming
[Bellman and Dreyfus, 1962], and mixed variable programming [Benders,
1962], but each method is rendered ineffective because of some factor or
characteristic of Problem I.
The difficulties with the application of the generalized Lagrange multiplier technique are twofold:
1. First, the method requires that the alternative new projects be independent. In Problem I the alternative projects are independent with respect to cost factors and required investment but are interrelated with
respect to benefits.
2. Second, the method may fail to provide an optimal solution [Cord,
1964; Weingartner, 1966].
Dynamic programming might be deemed to be suitable to solve Problem
I, but previous experience has shown that the computer core storage requirements would be prohibitively large because of the large number of
state variables involved in Problem I. Note that:
1. When the problem has more than one constraint, the number of state
variables increases.
2. Because of the physical configuration of the problem, a dynamic programming formulation would include converging and diverging branches,
with a consequent increase in the number of state variables.
For problems involving a large number of constraints Dantzig [1957] and
Nemhauser and Ullmann [1969] reported that dynamic programming was
not an efficient tool. Swanson [1970] reached the same conclusion for
water resource systems involving several state variables and more than two
stages.
A third possible method of optimization is Benders' [1962] algorithm
for mixed variable programming problems. This algorithm requires that
the objective function be separable with respect to the integer and continuous variables, i.e., the two sets of variables must be capable of being linearly
summed. However, there is interaction among variables in the objective
function so that Problem I does not meet the condition of separability.
In the Texas Water Development Board Study [1969] it was decided to
develop a screening technique that would be able to reduce drastically the
numbers of alternatives to be considered. This technique, which utilized a
variety of optimization routines to find "near optimum" solutions, embodied in four major phases: (1) initial element sizing and reservoir operating rules using an optimal allocation program, (2) initial screening of