6A.
Optimization
of the Model Including Water
Quality
155
forward dynamic programming algorithm, with an NLP code used as a
subroutine. The capital budgeting problem would still be solved by the
branch and bound algorithm. Clearly, the repeated use of an NLP code can
be exceedingly time consuming because even the best current NLP codes
require times of the order of 1-10 sec for a single execution, whereas the
OKA requires times of the order of 0.01-0.1 sec for a single execution.
A more viable optimization method for the combined large-scale water
management problem that includes conjunctive management of water
quantity and w
r ater quality is a hierarchical approach similar to that described by Lasdon (1970) and Haimes (1972). The hierarchical approach
requires that the management problem set up in Sections 6.3 and 6.4 be
decomposed into two subproblems:
1. The water quantity problem, which is simply the problem described
in Chapter 2.
2. The water quality problem, which can be stated as follows:
Maximize the comprehensive objective function [Eq. (2.1) less Eq.
(6.1)] subject to expressions (6.2)-(6.19).
The flows used in subproblem 2 are the outputs of subproblem 1, while the
revenue from subproblem 2 is fed back into subproblem 1 as illustrated in
Fig. 6.4.
In solving subproblem 1 appropriate operation of the reservoirs (releases)
assists in maintaining waste quality in the streams. Thus constraints (lower
bounds) would need to be placed on the arcs in which quality must be
maintained; a wide variety of combinations of water quality constraints
exist. A six-step optimization strategy can be outlined:
1. Pick a set of water quality constraints.
2. Solve subproblem 1.
3. Introduce the net return from subproblem 1 and the flows generated
from subproblem 1 into subproblem 2, and solve subproblem 2.
4. Determine whether the combinations of water quality constraints
have been searched or eliminated. If so, stop. Otherwise, go to step 5.
5. Use a heuristic and/or branch and bound method to pick a new
combination of water quality constraints.
6. Go to step 2.
The coupling variables and functions are shown in Fig. 6.4.
O'Laoghaire (1974) has solved a simpler version of the water quality
problem. The formulated model is similar to the model presented in this
chapter without the option of low-flow augmentation. It includes the mini-
Optimization
of the Model Including Water
Quality
155
forward dynamic programming algorithm, with an NLP code used as a
subroutine. The capital budgeting problem would still be solved by the
branch and bound algorithm. Clearly, the repeated use of an NLP code can
be exceedingly time consuming because even the best current NLP codes
require times of the order of 1-10 sec for a single execution, whereas the
OKA requires times of the order of 0.01-0.1 sec for a single execution.
A more viable optimization method for the combined large-scale water
management problem that includes conjunctive management of water
quantity and w
r ater quality is a hierarchical approach similar to that described by Lasdon (1970) and Haimes (1972). The hierarchical approach
requires that the management problem set up in Sections 6.3 and 6.4 be
decomposed into two subproblems:
1. The water quantity problem, which is simply the problem described
in Chapter 2.
2. The water quality problem, which can be stated as follows:
Maximize the comprehensive objective function [Eq. (2.1) less Eq.
(6.1)] subject to expressions (6.2)-(6.19).
The flows used in subproblem 2 are the outputs of subproblem 1, while the
revenue from subproblem 2 is fed back into subproblem 1 as illustrated in
Fig. 6.4.
In solving subproblem 1 appropriate operation of the reservoirs (releases)
assists in maintaining waste quality in the streams. Thus constraints (lower
bounds) would need to be placed on the arcs in which quality must be
maintained; a wide variety of combinations of water quality constraints
exist. A six-step optimization strategy can be outlined:
1. Pick a set of water quality constraints.
2. Solve subproblem 1.
3. Introduce the net return from subproblem 1 and the flows generated
from subproblem 1 into subproblem 2, and solve subproblem 2.
4. Determine whether the combinations of water quality constraints
have been searched or eliminated. If so, stop. Otherwise, go to step 5.
5. Use a heuristic and/or branch and bound method to pick a new
combination of water quality constraints.
6. Go to step 2.
The coupling variables and functions are shown in Fig. 6.4.
O'Laoghaire (1974) has solved a simpler version of the water quality
problem. The formulated model is similar to the model presented in this
chapter without the option of low-flow augmentation. It includes the mini-