26
1.
Introduction
projection algorithms. For T m&x = 5 the problem comprises 45 variables,
15 equality constraints, and 90 inequality constraints.
1.3.2. Dynamic
Programming
An important feature of most mathematical models of real systems is
that even if all the subsystems are optimized in separate phases, the whole
system is not necessarily optimal. However, a water resources system has a
unique characteristic, namely that all the water in the river basin flows
downhill! Therefore, unless there is recycling of water back upstream, say
by pumping, another class of optimization techniques is applicable—those
that make use of the characteristic information flow in the system, such as
dynamic programming [Beveridge and Schechter, 1970, pp. 679-702].
Dynamic programming is able to exploit the structure of a problem by
decomposing it into sequence of optimal subproblems, each of which is of
a smaller scale than the original problem. Hall and Buras [1961] first
pointed out the efficacy of dynamic programming in finding operating rules
for reservoirs. Since then, much work on refining operating rules using
dynamic programming has been reported in the literature [Amir, 1967;
Butcher, 1968; Hall, 1964; Hall et al, 1968; Hall and Howell, 1963;
Mobasheri and Harboe, 1970; Schwerg and Cole, 1968; Young, 1967].
Dynamic programming proceeds to optimize the elements of a system
in the inverse direction to the information flow in the system. For example, in Fig. 1.9 reservoir 3 would be optimized first, then reservoirs 2
plus 3, and finally 1 plus 2 plus 3. Note that whatever the choice will be
for the irrigation water for reservoir 3 in the optimization of reservoir 3,
the choice cannot affect the optimization of any of the upstream reservoirs
because of the information flow. Hence Dz t is chosen to optimize the return
from reservoir 3, Du can be selected to optimize the return from reservoirs
1 plus 2 independently of the choice of Dzt, and so on. Although dynamic
programming can be used to handle problems with nonlinear objective
functions and constraints, if too many (more than two) decision variables
such as Djt exist at each stage, the degree of effort required for the search
for the optimum at each stage becomes excessive.
The dynamic programming problem can be formally written as follows.
For each stage of the serial system there is a nonlinear difference equation
that represents the "state" equation
χ<+ι = Φι(χ*, y«, t)
(1.7)
where x t is the η-dimensional state (dependent) variable vector comprising
j = 1,..., η stages, y* the η-dimensional decision (independent, control)
1.
Introduction
projection algorithms. For T m&x = 5 the problem comprises 45 variables,
15 equality constraints, and 90 inequality constraints.
1.3.2. Dynamic
Programming
An important feature of most mathematical models of real systems is
that even if all the subsystems are optimized in separate phases, the whole
system is not necessarily optimal. However, a water resources system has a
unique characteristic, namely that all the water in the river basin flows
downhill! Therefore, unless there is recycling of water back upstream, say
by pumping, another class of optimization techniques is applicable—those
that make use of the characteristic information flow in the system, such as
dynamic programming [Beveridge and Schechter, 1970, pp. 679-702].
Dynamic programming is able to exploit the structure of a problem by
decomposing it into sequence of optimal subproblems, each of which is of
a smaller scale than the original problem. Hall and Buras [1961] first
pointed out the efficacy of dynamic programming in finding operating rules
for reservoirs. Since then, much work on refining operating rules using
dynamic programming has been reported in the literature [Amir, 1967;
Butcher, 1968; Hall, 1964; Hall et al, 1968; Hall and Howell, 1963;
Mobasheri and Harboe, 1970; Schwerg and Cole, 1968; Young, 1967].
Dynamic programming proceeds to optimize the elements of a system
in the inverse direction to the information flow in the system. For example, in Fig. 1.9 reservoir 3 would be optimized first, then reservoirs 2
plus 3, and finally 1 plus 2 plus 3. Note that whatever the choice will be
for the irrigation water for reservoir 3 in the optimization of reservoir 3,
the choice cannot affect the optimization of any of the upstream reservoirs
because of the information flow. Hence Dz t is chosen to optimize the return
from reservoir 3, Du can be selected to optimize the return from reservoirs
1 plus 2 independently of the choice of Dzt, and so on. Although dynamic
programming can be used to handle problems with nonlinear objective
functions and constraints, if too many (more than two) decision variables
such as Djt exist at each stage, the degree of effort required for the search
for the optimum at each stage becomes excessive.
The dynamic programming problem can be formally written as follows.
For each stage of the serial system there is a nonlinear difference equation
that represents the "state" equation
χ<+ι = Φι(χ*, y«, t)
(1.7)
where x t is the η-dimensional state (dependent) variable vector comprising
j = 1,..., η stages, y* the η-dimensional decision (independent, control)