WATER POLLUTION CONTROL RESEARCH SERIES
16110 FZE 09/70
Use of New Analytical Methods
in Water Resource Development
U.S. DEPARTMENT OF THE INTERIOR • FEDERAL WATER QUALITY ADMINISTRATION
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Reports describe
the results and progress in the control and abatement
of pollution in our Nation's waters. They provide a
central source of information on the research, develop-
ment, and demonstration activities in the Federal Water
Quality Administration, in the U. S. Department of the
Interior, through inhouse research and grants and con-
tracts with Federal, State, and local agencies, research
institutions, and industrial organizations.
Inquires pertaining to Water Pollution Control Research
Reports should be directed to the Head, Project Reports
System Planning and Resources Office, Office of Research
and Development, Department of the Interior, Federal
Water Quality Administration, Room 1108, Washington, D. C,
202U2.
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Use of New Analytical Methods
in Water Resource Development
by
Department of Mechanical Engineering
The University of Texas at Austin
Austin, Texas 78712
for the
FEDERAL WATER QUALITY ADMINISTRATION
DEPARTMENT OF THE INTERIOR
Program #16110 FZE
Formerly WP-01186
September, 1970
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FWQA Review Notice
This report has been reviewed by the Federal
Water Quality Administration and approved for
publication. Approval does not signify that
the contents necessarily reflect the views and
policies of the Federal Water Quality Adminis-
tration, nor does mention of trade names or
commerical products constitute endorsement or
recommendation for use.
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ABSTRACT
Studies were made to determine the feasibility of applying recently
developed analtyical techniques to the problem of planning for
optimal water resource development. An optimal plan with regard to
the size and location of proposed reservoirs was taken to be that
physical configuration which provided the greatest net economic
benefit to the people of the region while meeting future water needs.
This research was aimed at providing an improvement over the sub-
optimal planning techniques of the past which did not take into
account the interacting effects which exist between each reservoir
and all downstream reservoirs. These interacting effects are actually
the predominant characteristics to be considered in the design of a
system of reservoirs in any river basin, and cannot be ignored.
For the purpose of formulating a mathematical model for this type of
problem, a number of recently developed optimization techniques were
studied. The best formulation was found to be that which treated the
reservoir system design as a branched multistage decision process. In
such processes, the decisions made at each stage (potential reservoir
site) affect the circumstances under which all subsequent decisions in
the sytem must be made. The intercoupled staging structure of this
process was then exploited by means of the nonserial dynamic program-
ming algorithms developed in the course of this research project. These
algorithms are effectively decomposition techniques which break down the
nonserial (i.e. branched) process into an equivalent serial process that
is amenable to solution through ordinary dynamic programming methods.
This report was submitted in fulfillment of Grant No. 16110 FZE, between
the Federal Water Quality Administration and the University of Texas at
Austin, directed by Dr. Charles S. Beightler.
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CONTENTS
Section Page
I Conclusions 1
II Recommendations ia-
III Introduction 1
IV Discussion and Results 4
V References H
VI Publications and Patents 13
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SECTION I
CONCLUSIONS
1. An accurate appraisal of the worth of any single water
resource development project can no longer be made on an
individual basis. Rather, each project for resource
development can be planned and designed correctly only
when it is considered as part of a larger system of projects
in the region of development. When this is done, the
expenditure of funds for project development can be made on
the basis of how the individual project will contribute to
the overall development of the entire region.
2. Sub-optimal planning in the determination of the size and
location of reservoirs in a river basin system results when a
reservoir design does not take into account the interacting
effects between each reservoir and all downstream reservoirs.
To avoid such sub-optimization, a mathematical model was
developed which takes into account not only the staged structure
of the problem, but also the effects of the non-serial
(branched) inputs characteristic of a river basin system.
3. A new solution technique was developed for finding the
optimal solution to the system as described by the mathematical
model. This algorithm is based upon the branch compression
method which was also developed during the course of this project.
4. This new method for finding the optimal reservoir system
design has been coded for a digital computer, but owing to
the early termination of this research project, the computer
program has not been up-dated to include all the optimization
techniques developed during the project.
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SECTION II
RECOMMENDATIONS
This project was limited to the development of
mathematical models and optimization techniques which
could be applied to the design of river basin systems.
Early and unexpected termination of the project did not
permit the production of a computer program with the
ability to handle the large number of converging and
diverging branches which are typical of river basin systems.
Development of such a computer program would permit
the solution of large-scale systems met in practice. This
would allow for the evaluation of existing systems as well
as the optimal design of new systems.
Further studies should be made of the Geometric
Programming algorithm, which makes provision for the handling
of non-linear inequality constraints that constrain the
system. Another important advantage of this algorithm is
that it gives the optimum system cost without necessarily
finding the optimal values of the parameters. Thus, a
planner can evaluate the economic desirability of a proposed
project immediately without working out the complete design.
11
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SECTION III
INTRODUCTION
Water resource development projects in the past ten
years have become rapidly more numerous and costly. It
soon had become apparent that an accurate appraisal of the
worth of a single project could no longer be made on an
individual basis. Rather, each project for resource develop-
ment can be planned and designed correctly only when it is
considered as a part of a larger system of projects in the
region of development. When this is done, the expenditure of
funds for project development can be made on the basis of how
the individual project will contribute to the overall develop-
ment of the entire region.
Because of the shortness of water supply, the inter-
action between supplies, and the high cost of project development,
it is necessary for planning decisions regarding the size and
scope of reservoir development to be the best possible ones.
Since water is a limited resource, and our populations and
industries are expanding rapidly, this planning represents
a vital problem of immense proportions; Texas alone will
spend several billion dollars on reservoir construction in
the next two decades. Development decisions of this type are
made largely by subjective means at the present time, and
there is little uniformity in reservoir planning. The inter-
action analyses employed in this research provide methods
of solution to water development problems which should allow
for uniform and objective decisions.
The earliest pioneering work in water resource systems
and optimization work was carried out in the Harvard Water
Program (Maass, 1962). Attempts have been made to use
optimization techniques in water resources planning and system
design; the greatest success was obtained using a simulation
technique. Indications of reported research of the Harvard
Water Program (Huffschnidt,(11)) imply that simulation still
was being used in the detailed, final stage optimization of a
given water resource system.
Research also was being directed toward use of other
techniques such as linear programming and "rough simulation"
for preliminary planning studies and to provide starting points
for simulation operations. Hall and Buras (9) were first to
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propose the application of dynamic programming to the
optimization of reservoir systems. Dynamic programming
has the advantage of effecting the decomposition of a
complex problem into a series of less complex problems.
It has been applied by Hall and others (see for example
Hall, 7, 8) to a number of water resource optimization
problems. Particularly in preliminary planning studies
involving the sizing of proposed reservoirs within a
river system, dynamic programming would appear to be the
most useful optimization technique. Unfortunately, it
cannot be applied directly, and hence, important modifications
of the original algorithm were necessary, as described later
in this report.
Most of the prior work attempted in the analysis of
water resource systems had made use either of simulation or
linear programming, although there are many drawbacks to
gaming, and the system functions which occur are actually
highly non-linear. Simulation complicates the problem by
attempting to solve say a twenty variable problem, whereas
the research described in this report employs newer techniques
to decompose such a previous problem into ten 2-variable
problems. Some attempts had been made to apply dynamic
programming to this type of problem, using time as the sole
state variable. This meant that no state variables were
available for coupling the reservoirs into an interdependent
system of interacting stages. No analysts had tried to
handle more than one decision variable at each stage of the
system, nor had anyone attempted the analysis of any branched
river basin systems. However, any realistic problem will
certainly require more than one decision variable at each
stage, and will contain several branched inputs, so that
serial optimization methods are not directly applicable.
The rationale behind the approach made in this project
was that of avoiding sub-optimization in the determination
of the size and location of reservoirs in a river basin system.
Such suboptimal planning results when a reservoir design does
not take into account the interacting between each reservoir
and all downstream reservoirs. These interacting effects
are actually the predominant characteristics to be considered
in the analysis of any river basin system. If all couplings
are ignored, and each reservoir is treated as though it
were isolated from all others in the system, a serious
suboptimization will result.
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The specific aims of this research were first, to
develop further the nonserial multistage techniques which
are now the mathematical optimization methods most applicable
to the field of water resource planning. Secondly, it was
proposed to investigate the potential application of two
other new methods. These are the discrete maximum principle,
and geometric programming, the latter being the most recent
optimization technique yet developed. These appeared to
offer the most powerful solution procedures available when
the number of state variables exceeds the limit which can
be handled by the recursive dynamic programming approach.
Both methods, however, have certain important limitations
which were not obvious at the onset of this project, and
which have not been discussed in the literature.
407-496 0-70-2
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SECTION IV
DISCUSSION AND RESULTS
This research project was originally approved for
the three year period, September 1, 1967 through August
31, 1970, and procedures for carrying out the proposed
research were planned accordingly. Unfortunately, funding
for the project was terminated on August 31, 1968, after
only one year of work had been underway. At that time, the
position of the work was reviewed in order to determine
what might be done to salvage as much as possible out of
the original proposed three-year research program. Although
significant progress had been made along the lines suggested
in the application for continuation support, the loss of
funding prevented two graduate student assistants from
continuing on the project.
Fortunately, uncommitted funds from the initial year
were used to support the project for one additional year.
Although this unexpended balance was employed to
minimize the dislocation resulting from the unexpected grant
termination, the funds available were clearly insufficient
to do more than tie up some loose ends.
The work proceeded in the direction initially established
by the research reported in the paper, "Superposition in
Branching Allocation Problems", by C. S. Beightler, D. B.
Johnson, and D. J. Wilde, published in the August 1965 issue
of the Journal of Mathematical Analysis and Applications.
This work was prompted in turn by an earlier paper,
"Optimization of Multistage Cyclic and Branching Systems by
Serial Procedures", by R. Aris, G. L. Nemhauser, and D. J. Wilde,
(A. I. Ch. E. Journal. November, 1964), which presented the
first practical computational method for optimizing branched
multistage systems. Since all river basin systems are
actually comprised of branched inputs, this paper had particular
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importance for those who deal specifically with the real
problem of determining the size and location of proposed
reservoir sites in such systems. The algorithm had the
unfortunate characteristic, however, of requiring that an
optimization be carried out simultaneously over two variables
at each junction point where a branch joins the system
network. Most of the real problems in this field require
that tabular data be used in original form when solving the
problem, and this requirement usually makes the two-variable
optimization steps totally infeasible.
In general, multistage optimization problems arise
from a study of systems which are composed of a sequence
of interrelated stages, at each of which a decision must be
made. The decision at each stage determines the return
(usually a profit of cost) at that stage, as well as the
input to the next stage, thus affecting the circumstances
under which all subsequent systems decisions must be made.
The multistage problem is to find the optimal sequence of
decisions which extremizes (maximizes or minimizes) the sum
of all the returns. This is a difficult problem because the
decisions made at any stage, although they may optimize the
return at that particular stage, may affect the inputs to
all subsequent stages in a disadvantageous way, leading to a
non-optimal return for the system as a whole. Thus the
stages are coupled together by their dependence on one another,
and the optimal policy (sequence of decisions) for an N
stage system involves the solution of an N dimensional problem.
For a system having a simple serial structure, in which
the stages are linked in a straight line, the method of dynamic
programming has been used to reduce the dimensionality of
the problem and render it amenable to analysis. This method,
developed by Bellman (4), is a decomposition technique which
converts one N-dimensional problem into N one-dimensional
problems that can then be solved sequentially. These N
problems, taken together, are equivalent to the original
problem, and their solution determines the optimal set of
decisions for the more complex problem. Dynamic programming
thus takes advantage of the structure of the problem and is
an efficient solution algorithm whenever that structure
involves simple units in series having no more than two decision
variables associated with each unit. The discrete version of
Pontryagin's maximum principle (14) is another method quite
similar to dynamic programming, as are suitable adaptations of
the calculus of variations. Modifications of these techniques
can often be used to eliminate some of the difficulties
associated with high dimensionality either in the number of
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decision variables at each stage or in the number of
parameters required to specify the input to the next
stage.
Powerful as these methods are, however, they
generally apply only to serial systems; if any feedback
or crosslinkage from one stage to another destroys the
serial structure, these techniques cannot be applied
without considerable care. As a result, some researchers
had attempted to extend the partial optimization approach
exemplified by dynamic programming and the maximum principle
to systems with branches and loops, thus greatly widening
the scope of application of optimization theory. Fan and
Wang (6) had shown how to adapt the maximum principle to
cyclic structures, and Jackson (12) had extended variational
methods to both cyclic and branching systems. Aris,
Nemhauser, and Wilde (1) introduced the concept of a cut
state to make possible the decomposition of cyclic and
branched systems into equivalent serial ones solvable by
serial procedures. Under favorable circumstances, their cut
states could be directed tab their optimum values by efficient
optimum seeking methods, a^procedure that is not possible for
ordinary state variables. Beightler, Johnson, and Wilde (2)
developed a superposition principle which is valid for linear
allocation problems. This principle permitted superposition
of optimal policies for ordinary serial dynamic programming
problems formed from the branches of the larger problem, but
was applicable only to systems which could be represented by
linear functions at every stage. Efforts to extend this
superposition principle to more general functions have not
met with success. In fact, our work here at The University
of Texas, using convex return functions and linear transitions,
has proved that superposition cannot be employed on even this
weak an extension of the fully linear case.
It was evident that an entirely new approach was
needed to the solution of the general non-serial multistage
problem. The research supported by this grant produced just
such a new approach, in the form of an algorithm which replaces
the non-serial system with an equivalent serial structure,
which can then be solved by the standard dynamic programming
algorithm. This method is called branch compression (see i),
since it effectively compresses an entire branch, consisting
of several stages, into one psuedo-stage. The importance of
this result is that any number of converging branches can be
compressed into psuedo-stages which are then combined to
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form an equivalent serial system. Therefore, this method
of decomposition permits an entire converging branch multi-
stage river basin system to be solved as a sequence of
one-state, one-decision problems, just as if the system
were serial. Furthermore, these results hold for any return
or transition functions whatsoever and, in particular, they
permit these functions to be given in tabular form. The
resulting reduction in dimensionality makes possible the
solution of the complex problems of reservoir design which
were previously not amenable to analysis.
The fundamental problem in determining the size and
location of reservoirs in a river basin system is that of
taking into account the interacting effects between a given
reservoir and all those downstream. In this system, each
stage corresponds to a potential reservoir site, and the
stage decision is the size (possibly zero) of the reservoir
to be built at that site. Each decision affects not only
the costs and economic benefits to be derived at the given
site, but also the circumstances (flow rate, which is the
state of the system at each stage) under which all decisions
at downstream sites must be made. It is this latter effect
that makes the dynamic programming approach so attractive,
since it takes these stage couplings into account at each
step of the optimization process. If all couplings are
ignored, and each reservoir treated as though it were isolated
from all others in the system, a serious suboptimization
would result. With the development of the branch compression
technique, dynamic programming can now be applied directly
to all branched multistage systems.
Publication (iv) reports the successful introduction
of stochastic variables into a serial multistage system.
This will allow the handling of flow data in probabilistic
form, when combined with the branch compression method for
serializing a river basin system. Previously, only deterministic
inputs were used in the dynamic programming formulation, so
that flow data had to be introduced in the form of average
rates. Indeed, the algorithm described in this paper appears
to be the correct technique to use for all stochastic flow
problems, and should replace the numerous simulation methods
which have been proposed. These methods unduly complicate
the problem, and can produce answers which are in error by
large, but unknown, amounts. It is hoped that this new technique
will be widely adopted and will supplant much of the
questionable simulation studies now appearing in the water
resources literature.
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The theoretical work developed in the above
publications was extended and applied to an example
problem in the paper, "Design of an Optimal Branched
Allocation System", which appeared in the Journal of
Industrial and Engineering Chemistry (see ii). Using
non-linear functions and numerical data, an example
problem was presented and solved analytically. Then,
this same problem was solved using a direct-search
optimization method at each stage of the non-serial
dynamic programming example problem. This method was
incorporated into a computer program which was detailed
in the paper through a flow chart. This program can be
used to solve any non-serial multistage decision problem
and is currently being rewritten to take advantage of
recent advances in computational efficiency provided by
the CDC 6600 computer system at The University of Texas .
The new program will also have the ability to handle a
larger number of both converging and diverging branches,
so as better to describe a typical river basin system.
Hopefully, running times will be decreased using this new
program, for it is our intention to apply this code to
the solution of some rather large-scale systems. The
older program did have the disadvantage of consuming large
amounts of expensive computer time when a fine grid (search
step size) was employed. If the new code lives up to
expectations, it will provide the most practical and
efficient solution technique available for analyzing large
water resource systems.
Finally, this project produced the research reported
in the paper, "Dynamic Programming in Water Resources
Development" (see iii) which was presented at the National
Symposium on the Analysis of Water Resource Systems in
Denver, July 1968. This work described the application of
non-serial dynamic programming to the optimal selection of
a system interconnected reservoirs in a river basin system.
In particular, the branch compression method was described in
detail, and shown to be the best optimization technique
available for this type of problem. The "cut-state" method
of Aris, Nemhauser and Wilde for solving this same problem
was also presented, and the numerical efficiencies of the
two methods were compared. The paper pointed out the
necessity for a mathematical model to take into account
the intercoupled staging structure of the system, in which
the output from each reservoir provides an input to all
downstream reservoirs. The result of this work has been
the development of a quite realistic mathematical model of a
8
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fairly general river basin system, into which current and
historical flow data can be introduced to check the
optimality of a presently existing system, as well as the
nature of any modifications which must be made to the .
computer program. The ultimate purpose of the program is
the prediction of optimal reservoir dimensions and locations
needed to produce an overall optimum water resource system.
For this purpose, projected flow data would be used in the
model, for both average and stochastic flows. Due to the
abrupt termination of this project, the computer program
which carries out the computational work of the mathematical
model was not finished, so that a test of the model was
not made. Hopefully, this program can be completed in the
near future as research support monies become available.
Also uncompleted was the projected work of applying
Geometric Programming to water resource systems. The
operating or capital cost of a single component of a system
is often proportional to powers of certain key design
parameters. Although changing technology, economic conditions,
and geographic factors may affect the proportionality constants
as time passes, the exponents often are fixed by invariant
physical laws. Suppose the total cost of a river basin system
to be the sum of such cost terms. The conventional way to
find the optimum design parameter values would be to solve
the simultaneous non-linear equations obtained by setting
the first partial derivatives of the cost function equal to
zero? a long, complicated, and not always successful procedure.
Duffin and Zener (5, 15, 16) discovered a faster, more elegant
approach which requires the optimization only of a much
simpler "dual function." In favorable cases the method
(called Geometric Programming) gives the optimum by inspection.
But more important even than this considerable computational
advantage is the fact that Geometric Programming shows that
at the optimum, the fractional contribution of each term to
the total cost depends only on the exponents of the parameters—
not at all on the cost coefficients. Thus, even in the
face of changing conditions one can still develop rigorous
invariant rules for the optimal fractions of total cost to be
allocated to various system components. A final advantage of
Geometric Programming is that it gives the optimum cost without
necessarily finding the optimal values of the parameters.
Thus, a planner can evaluate the economic desirability of a
proposed project immediately without working out the complete
design. The technique has recently been expanded so that polynomial
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inequalities can be handled.
It had been proposed to study this powerful theory
to see where it could be applied to water resource system
design. There are bound to be certain places where the
model is not immediately applicable, and research is needed
either to develop suitable approximations or to revise
and extend the method. Where applications are possible,
rigorous design rules would be assembled which would apply
to all similar river basin systems. Work is still proceeding
towards a further development of the basic theory of
Geometric Programming, but at present, applications of the
method to water resource systems have not been satisfactorily
accomplished. The major disadvantage of Geometric Programming
and the Discrete Maximum Principle, another modern optimization
method studied in this project, is that the objective
function and all system constraints must be expressed as
analytic functions. Indeed, these functions must all be
continuous and possess continuous first partial derivatives.
Such functions are not normally available to the analyst,
who must then attempt to fit such a function to the historical
data that describe the system. The strongest recommendation
for the use of the dynamic programming algorithms is that
such data need not be approximated by fitted functions, but
rather may be used directly in either graphical or tabular
form.
10
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SECTION V
REFERENCES
1. Aris, R., G. L. Nemhauser, and D. J. Wilde, "Optimization
of Multistage Cyclic and Branching Systems by Serial
Procedures," A.I.Ch. E.Journal, 10, pp. 913-919 (Nov. 1964)
2. Beightler, C.S., D.B. Johnson, and D. J. Wilde,
"Superposition in Branching Allocation Problems,"
journal of Mathematical Analysis and Applications,
10, (May, 1965).
3. Beightler, C.S., and L. G. Mitten, "Design of an Optimal
Sequence of Interrelated Sampling Plans," Journal of the
American Statistical Association, 59, pp. 96-104,
(March, 1964).
4. Bellman, R., and S. Dreyfus, "Applied Dynamic Programming,"
Princeton, N. J.; Princeton University Press, 1962.
5. R. Duffin, "Dual Programs and Minimum Cost," Soc. Ind.
Appl. Math, journal 10, pp. 119-123.
6. Fan, L. T., and C. S. Wang, "Optimization of Multistage
Processes with Product Recycle," Chemical Engineering
Science, 19, pp. 86-87 (Jan. 1964) .
7. Hall, Warren A., "Aqueduct Capacity Under an Optimum
Benefit Policy," Trans. Am. Soc. Civil Engrs., 128,
part III, pp. 162-172 (1963).
8. Hall, Warren A., "Optimum Design of Multiple-Purpose
Reservoirs," J. Hydraulics Div. Am. Soc. Civil Engr.,
90 (HY4), pp. 141-149 (1964).
9. Hall, Warren A., and Nathan Buras, "The Dynamic Programming
Approach to Water Resource Development," J. Geophys. Res.,
66(2), pp. 517-5.20 (1961).
10. Howard, Ronald A., "Dynamic Programming," Man ag emen t Sci.,
12 (5), pp. 317-348 (1966).
11. Huffschmidt, Maynard M., "Field Level Planning of Water
Resource Systems," Water Resources Res., 1(2), pp. 147-163
(1965).
11
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12. Jackson, R. "Some Algebraic Properties of Optimization
Problems in Complex Chemical Plants", Chem. Eng. Sci.,
9, pp. 19-31 (1964).
13. Maass, Arthur, et al, Design of Water Resource Systems,
Harvard University Press, Cambridge (1962).
14. Pontryagin, L. S., V. G. Boltyanskii, R. v. Gamkrelidze,
and E. F. Mischenko, "The Mathematical Theory of Optimal
Processe," translated by K. N. Trirogoff, New York:
Interscience,(1962).
15. C. Zener, "A Mathematical Aid in Optimizing Engineering
Designs," Proc. Nat. Acad. Sci.47, pp. 537-539 (1961).
16. C. Zener, and R. Duffin, "Optimization of Engineering
Problems," Westinghouse Engineer, pp. 154-160 (Sept. 1964)
12
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SECTION VI
PUBLICATIONS AND PATENTS
i. Meier, W. L., and C. S. Heightler, "Branch Compression
and Absorption in Nonserial Multistage Systems,"
Journal of Mathematical Analysis and Applications,21,
No. 2, pp. 426-430 (1968).
ii. Beightler, C. S., and W. L. Meier, "Design of an Optimal
Branched Allocation System," Industrial and Engineering
Chemistry, 60, No. 2, pp. 44-49 (1968).
iii. Beightler, C. S., and W. L. Meier, "Dynamic Programming
in Water Resources Development," Proceedings of the
National Symposium on the Analysis of Water-Resource
Systems, held in Denver, Colorado, July 1-3, 1968; pp.
64-71.
iv. Beebe, J. H., C. S. Beightler, and J. P. Stark, "Stochastic
Optimization of Production Planning," Operations Research,
16, No. 4, pp. 799-818 (1968).
13
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.Accession Number
Subject
Field i Groi>r>
06 A
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
c Organization*
Federal Water Quality Administration
.AT";
Use of New Analytical Methods for Water Resource Development
10
Authors)
C. S. Beightler
22
, , Date
9/70
12
Pages
13
ix 1 Project Number
16110 FZE
21
i c Contract Number
Note
Citation
The University of Texas at Austin
23
Descriptors (Starred First)
*River Basin Development, *0ptimum Development Plans, ^Dynamic Programming Reservoir
Design, Reservoir Sites
25 Identifiers (Starred First)
27
Abstract
Studies were made to determine the feasibility of applying recently developed analtyical
techniques to the problem of planning for optimal water resource development. An optimal
plan with regard to the size and location of proposed reservoirs was taken to be that
physical configuration which provided the greatest net economic benefit to the people of
the region while meeting future water needs.
This research was aimed at providing an improvement over the sub-optimal planning
techniques of the past which did not take into account the interacting effects which
exist between each reservoir and all downstream reservoirs. These interacting effects
are actually the predominant characteristics to be considered in the design of a system
of reservoirs in any river basin, and cannot be ignored.
For the purpose of formulating a mathematical model for this type of problem, a number
of recently developed optimization techniques were studied. The best formulation was
found to be that which treated the reservoir system design as a branched multistage
decision process. In such processes, the decisions made at each stage (potential
reservoir site) affect the circumstances under which all subsequent decisions in the
sytem must be made. The intercoupled staging structure of this process was then
exploited by means of the nonserial dynamic programming algorithms developed in the
course of this research project. These algorithms are effectively decomposition
techniques which break down the nonserial (i.e. branched) process into an equivalent
serial process that is amenable to solution through ordinary dynamic programming methods.
Abstractor
Dr. C. S. Beightler
Institution
The University of Texas at
WRjt02 (REV. OCT. 1968)
WRSIC
SEND TO: WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U S. DEPARTMENT OF THE INTERIOR
WASHINGTON, D.C. 20240
U. S. GOVERNMENT PRINTING OFFICE : 1970 O - 401-486
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