Final Report
ECONOMIC EVALUATION
OF WATER QUALITY
SANITARY ENGINEERING RESEARCH LABORATORY
COLLEGE OF ENGINEERING
AND
SCHOOL OF PUBLIC HEALTH
UNIVERSITY OF CALIFORNIA
BERKELEY
SERL REPORT No. 69-8
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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 of our Nation's
waters. They provide a central source of information on the research,
development, and demonstration activities of the Federal Water
Pollution Control Administration, Department of the Interior, through
inhouse research and grants and contracts with Federal, State, and
local agencies, research institutions, and industrial organizations.
Water Pollution Control Research Reports will be distributed to
requesters as supplies permit. Requests should be sent to the Plan-
ning and Resources Office, Office of Research and Development,
Federal Water Pollution Control Administration, Department of the
Interior, Washington, D. C. 20242.
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FINAL REPORT
ECONOMIC EVALUATION OF WATER
to the
Federal Water Pollution Control Administration
Department of the Interior
on
FWPCA Research Grant Project 16090DLU
(WP-00597-06)
"by the
University of California
Berkeley
<3.#;uo-4«^uy //'/my^/j/ /
/\ 7 I- (f/W^^^*M'i*rZ*~
P. H. McGauhey l.MA^fiddle^rooks
Faculty Investigator /Faculty Investigator
November 1969
Sanitary Engineering Research Laboratory
College of Engineering
and
School of Public Health
University of California
Berkeley
SEKL Report No. 69-8
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FWPCA Review Notice
This report has been reviewed by the Federal Water
Pollution Control Administration and approved for
publication. Approval does not signify that the con-
tents necessarily reflect the views and policies of
the Federal Water Pollution Control Administration,
nor does mention of trade names or commercial
products constitute endorsement or recommendation
for use.
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ABSTRACT
The tendency to set quality of the water resource rather than quality of
discharges as the objective of environmental control makes it necessary to
develop some relationship between concentration of individual pollutants in the
resource and in the discharge in terms of characteristics of the receiving
estuary. Moreover, the growing percentage of the water resource which is
degraded in quality through beneficial use together with the increasing investment
necessary to restore water quality, makes it important to minimize the cost of
achieving water quality objectives. The study makes use of modern mathematical
models, programming techniques, and input-output analysis to optimize quality
control systems; and illustrates the use of the models by examples drawn from
San Francisco Bay data and quality requirements.
This report was submitted in fulfillment of project 16090DLU (WP-00597-06)
under the partial sponsorship of the Federal Water Pollution Control Administration.
iii
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PREFACE
NEED FOR STUDY
A general need for studying the economic value of water quality derives from
the fact that the "burgeoning population of the United States and the economic growth
on which it depends for an ever-increasing standard of living requires a growing per
capita utilization of water. Inasmuch as the water resource of the nation is
relatively constant, this means that the consequences of beneficial use of water
"becomes more evident each year; and "because these consequences are a decrease in
water quality which can he offset only "by investment in expensive water purification
worksj the economic aspects cannot "be ignored.
The specific need for study, however, derives from the general need in a
variety of ways. In the practical sense "quality" has no absolute definition but
is measurable only in terms of the use water is expected to serve. Consider as an
extreme example 10,000 acre-feet of snow melt on the one hand and an equal amount of
sea water on the other. Presumably any citizen would consider the fresh water as
being of vastly higher quality than the sea water. Yet either might serve such
purposes as navigation or industrial cooling water equally well. The untreated sea
water would, of course, be useless for domestic supply, agriculture, and industrial
process water, whereas the fresh water might be catastrophic to aquatic life if
discharged suddenly into a saline estuary. To overcome either of such extremes
requires an economic investment in water quality management. Considering that
ranges of quality rather than absolute levels characterize the acceptability of
water in any beneficial use, it is at once evident that some spectrum of water
quality criteria can be arbitrarily set up to define decreasing degrees of accept-
ability of water, ranging from the highest aesthetic level to complete disaster.
The problem then becomes that of determining the economic value to the user of
having available water at one level of quality as against some other level, quality
being defined in terms of those characteristics of water of importance to the
specific use concerned. Thus the first specific need for study of the economic
value of water quality is to determine some means by which to calculate, or at least
to estimate, the value to the user of any specific quality of water in terms of
comparative economic return to him in his enterprise.
A second need for studies leading to methods for evaluating water quality
economically lies in the area of achievement of public policy. To an increasing
degree citizens and public officials are of the mind that high-quality environment,
pure water, and clean air are the heritages of all Americans and constitute the
objectives of public policy aimed at environmental control- At the same time,
unregulated population growth has persisted and society is insisting that each
individual in that population be enabled to enjoy a standard of living which only
further exploitation of resources will make possible. To the extent that deprecia-
tion of water quality is a by-product of economic growth, the disparity between
national objectives and the harsh realities of resource exploitation presents
several economic questions:
1. How great is the public investment needed to achieve our national
environmental goals of water quality?
2. How far can we in reality progress toward these national goals of
water quality with that fraction of our national wealth we feel
justified in dedicating to water resource management, considering
other demands for investment on a necessarily finite national
budget?
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3- What is the optimum way to invest available funds in water quality
management in order to enhance economic growth while at the same
time achieving water quality goals?
4. How can the water quality criteria established by regulatory
agencies for any given sector of the water resource be achieved
at a minimum cost to the water-using sector of the economy?
The foregoing questions generate yet a third need for studies such as herein
reported. This is for ways to adapt modern methods of mathematical modeling,
systems analysis, and operations research to the economic evaluation of water
quality- The questions involved are complex in the extreme; numerous alternative
decisions are possible; and optimum economic solutions are required. Thus the third
need in a logical sequence becomes the first need in research, i.e., the development
of mathematical approaches. Thereafter, the way to their application to practical
problems can be sought.
NATURE OF STUDY AND REPORT
The studies herein reported were undertaken in response to the need outlined
in the preceding paragraphs. The report itself is composed of two major sections.
In the first is summarized, in limited detail, work completed during the period
September 1962 to January 1969 and reported in four widely distributed reports [l-it-]
to the sponsoring agencies, which included: Resources for the Future, National
Institutes of Health, and the Federal Water Pollution Control Administration. The
first section of the report covers primarily those phases of the study in which
mathematical modeling and systems analysis were developed and tested for applicability
to specific problems -outlined in the 'Need For Study.' Four phases of the study are
reported in Section I- They include:
1. An engineering economic model for water quality management;
2. A mathematical model of dissolved oxygen concentrations in fresh water
streams;
3- A linear programming water quality control model; and
k. A multicomponent model of optimal quality control in estuarine
waters.
The first two of these four phases were, as hereinafter noted, concerned
with maintenance of the quality of water in flowing streams; the second two with
the quality of estuarine waters.
The fifth phase"of the study, which is the subject of the second section of
this report, concerns the application of the previous estuarine models to a real
situation. — the San Francisco Bay — for which results of several years of study by
other investigations, costing several millions of dollars, made specific data
available. In addition, advantage is taken in this section of an input-output model
of the economy of the San Francisco Bay area, likewise developed at considerable
cost in time and money to other projects and agencies. Prom it is evaluated the
constraint of waste water quality maintenance costs imposed on the economic growth
which might otherwise be expected to accrue from an optimum allocation of scarce
water to various beneficial uses. Inasmuch as the material in Section II of this
report has not previously been evaluated in the context of water quality, it is
developed in detail under the more appropriate heading, "A Multisectoral Programming
Analysis of the Waste Water Economy of the San Francisco Bay Region.'
Objectives, concepts and rationale, general methodology, and major con-
clusions are discussed later as they pertain to the various phases of the study.
The report, then, is the result of the cooperative effort of a number of
individuals representing several disciplines.
vi
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ORGANIZATION FOR STUDY
The study was begun in 1962 through a one-year predoctoral fellowship grant
to Mr. Richard Frankel from Resources for the Future, Inc. Early results indicated
that the subject of economic evaluation of water quality might be worth further
exploration. For that purpose NIH Grant No. WP-00597 was made in 1963 to Professors
P. H. McGauhey and Walter F. Rowland; and later to P. H. McGauhey and Dr. G. T. Orlob
of the Sanitary Engineering Research Laboratory of the University of California at
Berkeley. Under the guidance of these faculty investigators and the Department of
Natural Resources Economics of the University, Mr. Richard Frankel completed the
first phase of the study and developed the report [l] as his dissertation for the
Ph.D. dgree in Water Resources Engineering. Dr. Richard Frankel and Mr. William W.
Hansen, an advanced graduate student in Water Resources Engineering, reported [2]
phase 2 of the study.
The third phase of the study was conducted under Grant No. USDI WP-597 as
responsibility for the Water Pollution Control Program was transferred from the
Department of Health, Education, and Welfare to the USDI Federal Water Pollution
Control Administration. Mr. John P. Carew, an economist, joined the project staff
in 1965. With professional guidance in the field of economics from Professor B. K.
Ward of the University of California Department of Economics, and in the field of
operations research and systems analysis from Professor R. M. Van Slyke, also of the
University of California staff, the study produced a third report [J] under the
authorship of Mr. Carew and Professor McGauhey. Ultimately this report was expanded
by Mr. Carew independently to satisfy the thesis requirements for his Ph.D. in~
Economics.
Under a continuation of the USDI WP-597 Grant and with the continued
cooperation of Professor Van Slyke, the work of Dr. Carew was developed to a much
more sophisticated degree by Mr. Shishir K. Mukherjee. His report on the fourth
phase of the study [h] served both as the basis for his doctoral degree in Operations
Research Engineering and as one of the models needed for the fifth and final"phase
of the study.
To continue the study from the Mukherjee report [4] and to produce the added
material needed for the final report, use was made of both the mathematical models
previously developed and of competence gained in input-output studies by Dr. H-
Craig Davis, Dr. E. M. Lofting, and Mr. Jona Bargur during professional as well as
doctoral studies in the fields of Economics and Water Resources Engineering, under
grants to Professor P. H. McGauhey from the University of California Water Resources
Center, the OWRR, the Bureau of Mines, the Corps of Engineers, and the California
State Department of Water Resources. Section II of the report, which constitutes
phase 5 of the study,, is authored by Mr. J. S. Bargur, Dr. E. M. Lofting, and
Dr. H- Craig Davis.
Upon the retirement of Professor P. H. McGauhey from the University of
California, Dr. E. J. Middlebrooks was appointed by the University to serve as
Faculty Investigator for the remaining months of the contract, although emergency
arrangements were made with Professor McGauhey to participate in producing the
final report.
ACKNOWLEDGMENTS
The faculty investigators and project staff who served during the more than
six years the study was in progress are indebted to so many individuals for guidance,
counsel, programming, and technical assistance that almost no listing can be complete.
Attention has already been called, in the section 'Organization For Study,' to many
whose assistance made it possible for the designated Faculty Investigators in engi-
neering to continue with studies involving economics and systems. Others who might
be singled out for special thanks include Professors S. V. Wantrup, M. F. Brewer,
D. I- Jenkins, Ray B. Krone, David K. Todd, William F. Jewell, Roger K. Glassey, and
Robert E. Selleck of the University of California; and Professor Clarence J. Velz
of the University of Michigan; Mr. A. R. Blanch and Mr. William Pierce of the Federal
VII
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¥ater Pollution Control Administration; Dr. Robert A. Littleford of the Public Health
Service; Mr. Robert Russell of the U. S. Forestry Division; and numerous members of
the staff of the Sanitary Engineering Research Laboratory of the University of
California .
Particular gratitude is expressed to Resources for the Future, Inc., to the
National Institutes of Health (PHS), and. the Federal Water Pollution Control
Administration (USDI), without whose financial support the study would not have
been possible.
viii
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TABLE OF CONTENTS
PREFACE v
LIST OF TABLES xiii
SECTION I
SUMMARY OF PHASES I — k
Chapter
I. SUMMARY OF PHASE 1 1
Introduction 1
Concept and Rationale ' 1
Objectives of Study 1
Procedure 2
Conclusions 2
II. SUMMARY OF PHASE 2 k
Introduction k
Concept and Rationale k
Objectives of Study k
Procedure k
Results and Conclusions 5
III. SUMMARY OF PHASE J 6
Introduction 6
Concept and Rationale 6
Objectives of the Study 7
Procedure 7
Results and Conclusions 8
IV. SUMMARY OF PHASE k 9
Introduction 9
Concept and Rationale 9
Objectives of the Study 9
ix
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TABLE OF CONTENTS (Continued)
Chapter
Procedure ............................ 10
Results and Conclusions ..................... 10
SECTION II
PHASE 5
A MULTISECTORAL PROGRAMMING ANALYSIS FOR THE
MANAGEMENT OP THE WASTE WATER ECONOMY
OF THE SAN FRANCISCO BAY REGION
PREFACE TO SECTION II ....................... 13
V. SAN FRANCISCO BAY AREA ....................... 17
Important Characteristics .................... 17
Population Distribution and Projections ............. 17
General Economic Characteristics ...... .......... 19
VI. THE NINE -COUNTY BAY AREA MULTISECTOR MODEL FOR 1963 ........ 20
General Description of an Input -Output Model .......... 20
The Input-Output Model in Summary ................ 20
Developmsnt of the Model .................... 22
Final Demand Vectors and Projection to Target Years ....... 27
VII. WASTE WATER DISPOSAL INTO SAN FRANCISCO BAY ............ 29
Types of Waste and Specific Categories of Pollutants ...... 29
Waste Load Coefficients ..................... 29
Effluent Coefficients ...................... 36
VIII. QUALITY STANDARDS FOR SAN FRANCISCO BAY .............. ^0
Ideal Water Quality Objectives for the Bay-Delta Region ..... ^0
Permissible Levels of Concentration of Pollutants ........ ko
Steady State Linear Optimal Dispersion Model .......... kl
Optimal Annual Levels of Discharge of Pollutants
into the Bay ........................ H
IX. MULTICONSTITUENT WASTE WATER PROJECTION MODEL ........... ^9
Economic Multiplier Analysis .................. ^9
Waste Constituent Multiplier Analyses .............. 50
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TABLE OF CONTENTS (Continued)
Chapter Page
Waste Constituent Interactions Tables 51
Critical Time Period Analysis 52
X. LINEAR PROGRAMMING WATER QUALITY MANAGEMENT MODELS 56
Introduction 56
An Optimizing Approach to Water Quality Management 56
Linear Programming and Multisector Models . 56
Water Quality Management Models 58
Constraints 59
Objective Function 60
The Untreated Effluent Programming Model 6j
Primal Variables, Optimal Solutions, and
Infeasible Solutions 63
Empirical Aspects of the Input Data 63
Results of the Untreated Effluent Programming Model 6k
Primal Problem 6k
The Dual Problem - Shadow Prices 72
Multiphase Treatment Programming Models 7^
Formulation of the Programming Models 7^
A Model with Natural Runoff and Domestic
Waste Water Considerations 76
Treatment Cost Minimization Models
77
Summary of the Four Versions of Multiphase
Treatment Models 78
Input Data Aspects 79
Phases of Treatment — Primary, Secondary,
and Tertiary Processes 79
Unit Costs and Removal Efficiency 82
Results of the Multiphase Treatment Model
for the San Francisco Bay Qk
Primal Problems 84
Dual Problems 90
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TABLE OF CONTENTS (Continued)
Chapter
Extensions 91
Multiple Quality Standards 91
Interzonal Effluent Shipments 95
SECTION III
SIM4ARY AND CONCLUSIONS
PHASES 1-5
XI. SUMMARY AND CONCLUSIONS 99
Summary • • 99
Conclusions 100
APPENDIX -Effluent "Interactions" Tables 105
REFERENCES . . . 115
xii
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Table
I.
II.
III.
IV.
LIST OF TABLES
Title
Population of Bay Region Counties 1960-1966 and Estimated
Percentage Change 1965-2020
Population Projection in 9-County San Francisco Bay Region
1965-2020
Sector Classification for the Model
Interindustry Transactions Table (Flow of Goods and
Services by Industry of Origin and Destination) 1963
Nine Bay Area Counties
Interindustry Transactions — Technical Coefficient Matrix
(Direct Purchases per Dollar of Output) 1963
18
19
23
24
VI.
VII.
VIII.
IX.
X.
XI.
XII.
XIII.
XIV.
XV.
XVI.
XVII.
XVIII.
XIX.
XX.
XXI.
Interindustry Transactions (Direct and Indirect
Requirements per Dollar of Final Demand) 1963
Nine Bay Area Counties
Final Demand Projections — 3$ Growth Rate
Final Demand Projections — 6$ Growth Rate
Procedure of Calculation of Waste Load Coefficients
Calculation of Annual Production (Physical Units) for
Use in Calculation of Waste Load Coefficients
Calculation of Waste Load Coefficients
Industrial Waste Load Coefficients
Calculation of Agricultural Waste Load Coefficients
Natural Runoff Waste Production
Domestic Waste Production
Domestic Effluent - 1963
Procedure of Calculation of Industrial Effluent
Coefficients
Waste Load and Effluent Data Requirements for
Water Quality Model
Waste Load and Effluent Characteristics of Discharged
Effluent into San Francisco Bay (Except Industrial)
Water Quality Standards for San Francisco Bay
Hydrological Input Data for Linear Programming
Dispersion Model in North Bay
c-j
26
27
28
31
32
33
34
35
35
36
36
37
38
39
4l
45
xiii
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Table
LIST OF TABLES (Continued)
Title
Page
XXII. Hydro logical Input Data for Linear Programming
Dispersion Model in South Bay ................ ^6
XXIII. Optimal Quality Requirements of Waste Discharge
into San Francisco Bay ................... ^7
XXIV. Format of Steady State Linear Programming
Dispersion Model ...................... ^8
XXV. Simple and Weighted Effluent and Waste Multipliers
for the San Francisco Bay Region .............. 51
XXVI. Sectoral Weighted Effluent and Waste Multipliers
for San Francisco Bay Region ................ 5^
XXVII. Critical Time Periods for Pollutants Discharged into
San Francisco Bay with 3$ Average Annual Growth
Rate for Final Demand .................... 55
XXVIII. Levels of Untreated Domestic Waste ............... 65
XXIX. Procedure of Calculations of Right -Hand Side Values
for the Water Quality Management Programming
Model ............................ 66
XXX. Projection of Domestic Waste Load Coefficients ......... 6?
XXXI. Domestic Effluent Discharged into the San Francisco
Bay under Various Combinations of Levels of
Primary and Secondary Treatment ............... 68
XXXII. Results of Untreated Effluent Programming Model ......... 70
XXXIII. Optimal Activity Levels —Primal Variables Solution
to the Untreated Effluent Programming Model ......... 71
XXXIV. Critical Time Periods Resulting from Untreated
Effluent — Programming Model ................ 73
XXXV. Comparison of Shadow Prices for Treatment of
Pollutant with Actual Treatment Costs ............ 75
XXXVI. Unit Costs of Waste Water Treatment ............... 82
XXXVII. Waste Water Treatment Process Performance ............ 83
XXXVIII. Gross Regional Product of the Various Programming
Models ($106) ........................ 814-
XXXIX. Multiphase Treatment Model —Minimization of Treatment Costs
Results of Industrial Treatment Only ............ 85
XL- Multiphase Treatment Model - Minimization of Treatment Costs
Results of Industrial and Domestic Treatment ........ 86
YLT. Value of Flushing from Delta .................. 87
xiv
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LIST OF TABLES (Continued)
Table Title
XLII. Multiphase Treatment Model —Maximization of Gross
Regional Product, Results of Industrial
Treatment Only
XL-Ill. Multiphase Treatment Model —Maximization of Gross
Regional Product, Results of Industrial and
Domestic Treatment 89
XLIV. Dual Variables (Shadow Prices) for Treatment
Models Without Flushing ($10S) 92
XLV. Dual Variables (Shadow Prices) for Treatment
Models With Flushing ($106) 93
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SECTION I
Summary of Phases 1-4
by
P. H. McGauhey
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I. SUMMARY OP PHASE 1
INTRODUCTION
The initial phase of the study was published in January 1965 as SEEL Report
No. 65-3 of the University of California [l]. It was authored "by Mr. Richard -J.
Frankel and entitled "Economic Evaluation of Water Quality; An Engineering-Economic
Model for Water Quality Management; First Annual Report."
CONCEPT AND RATIONALE
The study was based on the concept that the water quality requirements
established for a given stream system by a regulatory agency might be met more
economically by engineered systems designed to accomplish a different spectrum of
objectives than Is commonly the case. To test this hypothesis a computer model v.-as
established to simulate a -water course of specified characteristics and subject tc
flow regulation at its headquarters. Within the upstream area, a community of
specified size was presumed to discharge domestic sewage effluent into the stream.
Downstream a second community, also of specified size, was presumed to draw upon
the stream for its domestic water supply. The reach of stream between the effluent
outfall and the water supply intake was of specified length, subject to natural
self-purification, and under the jurisdiction of a regulatory agency empowered tc
impose quality standards deemed appropriate to the protection of beneficial uses
it might deem important. No augmentation of the stream by influent springs or by
tributaries was assumed. This, however, was merely to simplify the model for the
initial study, as there is nothing in the concept to preclude such additions in
any amounts or of any quality characteristics.
The principal factors in the system which were subject to natural or
artificial variation included:
1. Discharge or flow of stream.
2. Quality of water in stream above sewer outfall.
3- Degree of self-purification of stream.
k. Degree of treatment of waste water.
5- Concentration and pollutant content of waste water.
6- Nature of beneficial use of the stream.
7- Water quality standards imposed on stream.
8. Degree of treatment of water supply of downstream community.
9- Division of treatment between the two communities.
OBJECTIVES OF STUDY
The specific objective of the study was to utilize computer techniques tc
determine the effects of the seven variables on the cost of water reuse. That is,
to determine how by process engineering the quality of water required in the stream
and by the domestic water supply could be achieved at a minimum cost. Conceivably
it might be cheaper in terms of public investment to vary from conventional treatment
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of sewage and water supply, take advantage of the self-purifying ability of the
stream, and so operate the system as to achieve all water quality objectives at a
reduced cost.
PROCEDURE
The study gave first consideration to factors which influence pollution-cost
relationships, including:
1. Quality factors in domestic waste water which affect water
purification plant operation.
2. Cost and removal efficiencies of waste water treatment
plants.
3> Cost of water treatment plants and processes.
4. Stream flow in its relation to water quality.
More than 30 pages of the report [l] is devoted to factors in cost-quality
relationships, with particular concern for:
1. Suspended solids and dissolved organic matter.
2. Coliform organisms (bacterial indicator of sewage pollution).
3- Nutrients such as nitrogen and phosphates.
k. Surface active agents.
Having established the basic cost functions needed in the economic analysis,
the study then turned to the development of computer models and an explanation of
the methodology involved. The following models were established for computer
programming:
1. A dynamic oxygen sag model, incorporating photosynthetic effects.
2. A biological purification (bacterial die-away) model.
3- A detergent degradation model.
The models were then programmed for the IEM 1620, the coliform die-away
being set up for both-chlorinated and unchlorinated effluents for numerous travel
times downstream under varying conditions. Computer programs are included in the
report [l].
Analyses were made both on the basis of laboratory studies, which became
a limited aspect of the project, and on actual reported costs and stream standards.
CONCLUSIONS
Phase 1 of the study reported some 21 conclusions drawn from results of
computer analysis of the engineering-economic model developed in the study. Typical
of the most important of these, in terms of the concepts on which the study was
postulated, is that: The amount of savings to water purification plants downstream
is a direct benefit of improved upstream treatment. The ratio of cost savings
During the period of phase 1 of the study (1962-196**), detergents were
considered a problem in water quality. The changeover to biodegradable detergents
occurred in June 1965*
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downstream to the costs of additional treatment upstream is quite small, varying
from 0 to 0.10. Maximum return per dollar investment is realized when secondary
treatment is added to primary.
The model itself represents a successful tool, capable of further development
and extension, to evaluate the effects of pollutional loads and water quality
management techniques on downstream beneficial uses. Implicit in the conclusions
reported, although not spelled out in detail, is the idea that the model might be
most useful when extended to more complex systems, possibly regional in nature.
.In the simple situation the problem of optimizing treatment costs by allocating
certain treatment functions to waste water, water purification, and natural processes
is complicated by the nature of treatment processes themselves. Specifically,
treatment processes represent large discrete variables, only a few of which comprise
a treatment plant- For example, the engineer does not design a waste water sedimen-
tation tank to remove 5, 10, 15, 20, etc. percent of the suspended solids. Either
none at all are removed or 50 to 60 percent are taken out. Thus the variability
which might be shown up by the model is obscured by the size of increments involved.
Moreover, even if percentage removals were highly variable, it seems unlikely that
costs would follow the same sensitivity scale. Certainly the cost ratio of providing
sedimentation, for instance, to remove 5 percent instead of 10 percent of solids is
by no means 1:2.
Nevertheless the study did report important conclusions and present discussions
of their subtleties too extensive to summarize here.
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II. SUMMARY OF PHASE 2
INTRODUCTION
Phase 2 of the study was both an outgrowth of, and a refinement of, phase 1.
Its results were published in August 1965 as SEEL Report No, 65-11. As noted in a
previous section of this report, Mr. William W. Hansen joined with Dr. Richard J.
Frankel in authoring it. It represented the Second Annual Report of the project
on Economic Evaluation of Water Quality and was subtitled "A Mathematical Model of
Dissolved Oxygen Concentration in Freshwater Streams."
CONCEPT AND RATIONALE
By far "the most common criterion of water quality which has historically
been applied to flowing streams is its oxygen resource in terms of dissolved oxygen
(DO). Reduction in BOD has therefore long been the principal objective of biological
treatment of waste water as well as the parameter by which the effectiveness of
treatment has been judged. Since Ellis [5] first observed in 1937 that 5 mg/Ji of
DO is needed to support a good mixed fish faunae, the tendency has grown to establish
an average DO content of 5 or 6 mg/& as the standard to be met in water quality
management. During the investigative work which led to the oxygen sag model
established for phase 1 of the study, evidence was found which indicated that the
diurnal variation in DO content and the rate of change of oxygen tension may not be
held below their critical limits for aquatic life by the simple imposition of a
standard criterion of 5 mg/-^ average DO. Should such prove to be the case, economic
conclusions based upon the engineering-economic model of phase 1 might become
unacceptable. Pursuant to this rationale, phase 2 was initiated for the following
purposes.
OBJECTIVES OF STUDY
Three objectives were established for phase 2 of the study:
1. To set up a mathematical model expressing the relationships between
oxygen-demanding fractions of a waste water and the responses of a
living stream.
2. To apply computer techniques to the model to determine the magnitude
and location-of maxima and minima in the oxygen resources of the stream.
3. To verify the mathematical model by applying it to data observed in
the field.
PROCEDURE
Pursuant to the foregoing objectives a new oxygen sag model was developed
to include oxygen consumption by anaerobic decomposition products of the benthos
and decay of the initial oxygen saturation deficit, as well as the oxygen consumed
in biodegradation of suspended and dissolved organic matter and the consumption of
oxygen by photosynthesis. This model was programmed for the computer as reported
in the Second Annual Report [2].
To test the program it was applied to a case study utilizing data for the
Flint River published by the Water Resources Commission of the State of Michigan.
Their data permitted description of the diurnal variation in DO at a number of
points along a critical reach of stream.
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RESULTS AMD CONCLUSIONS
Although certain adjustments and assumptions were made necessary "by some
conditions peculiar to the Flint River situation, the conditions existing in the
stream were found to "be describatole "by the theoretical model. It was therefore
concluded that the model could be used advantageously to determine what improve-
ments in the stream environment would result from greater dilution, and from
various changes in plant operating or streamflow management practices.
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III. SUMMARY OF PHASE 3
INTRODUCTION
By the time phase 2 of the project had been completed, a number of changes
in the public attitude toward resource conservation were emerging. Most significant
of these were an emphasis on quality "of the environment" and, in the case of water,
"quality of the resource itself" in contrast with a previous preoccupation with
"pollution" in terms of the quality of individual waste discharges. This meant
that quality objectives were to be set for bodies of water by regulatory agencies
and, although these objectives might be met only by regulating the quality of
individual waste discharges, there was no reason to believe that the most economical
way to achieve these objectives was by subjecting each waste discharger to exactly
the same discharge requirements.
From the viewpoint of the project this change in public objectives now
offered two possible directions of investigation- The model developed in phases 1
and 2 of the study might be expanded, or emphasis placed on the most economical way
to achieve any given water resource quality objectives. It was decided to adopt
the second alternative for a number of reasons:
1. Phases 1 and 2 of the study had already demonstrated that the model
could be extended to more complex systems where investment decisions
must be made. However, its resolving power was best suited to such
complex systems involving multiple installations and regional water
quality considerations rather than to a simple situation where water
treatment alternatives would approach a situation of indivisibility.
2. To expand the model to optimum proportions would mean a research
commitment at a greater than authorized budgetary level and probably
on a long-term basis because considerable time might be expected to
elapse between the present and the future date when water quality
management might indeed be practiced on a regional scale suited to
the model.
3> It seemed evident that the newer "resource quality" approach was the
one most certain to dominate public policy in the years ahead and
that, therefore, directing attention to its implications should be
the most productive of research results.
4. Data suited to the needs of the project were available as a result
of extensive studies of the San Francisco Bay System by other
investigators -
5- No change in the needs and objectives for which the research grant
was made was involved in the shift of emphasis.
CONCEPT AND RATIONALE
The concepts and rationale underlying phase 3 of the study are essentially
presented in the preceding paragraphs. Specifically, it was postulated that by
modern methods of systems analysis and programming techniques it should be possible
to determine the economic effects of changing the quality requirements of a water
resource in terms of the mini mum cost of waste water management to any specific
community.
-------
OBJECTIVES OF THE STUDY
The specific objective, as set forth by Dr. John Carew and Professor P. H.
McGauhey in the report on phase 3 [3] of the study was to develop a linear program-
ming water quality control model capable of minimizing the cost of achieving any
given water quality goal when the concentrations of various waste products were
reduced to imposed, or required, levels by presently-known technology.
PROCEDURE
The procedure utilized was to construct a mathematical linear programming
water quality control model and then to illustrate its use by applying it to data
available from local sources. First it was assumed for purposes of the study that
the quality of water in the basin to be analyzed was characterized by some 2J
chemical components which are released into the San Francisco Bay by three
municipalities and nine individual industries. However, for purposes of illustrating
the use of the model, only the eleven characteristics for which data were most
complete were utilized.
From data available from literature cited in the report [j]j the cost in
dollars per million gallons of water treated to each of the following six levels
was obtained for 17 common pollutants, including the eleven utilized:
0. No treatment.
1. Conventional primary treatment.
2. Conventional secondary treatment.
3- Coagulation with lime or alum, sedimentation, and rapid sand
filtration.
4. Sorption.
5. Electrodialysis.
6. Evaporation.
Capital costs of treatment systems were handled outside the model for reasons of
convenience.
For purposes of illustration it was assumed that all 12 dischargers were
presently required to treat wastes at primary level, although the model itself
could handle any combination of required treatments had data on this aspect been
available.
The model was used to determine what degree of treatment might be imposed
on each of the 12 dischargers if the objective was to reduce to concentration limits
(set by the local ¥ater Quality Control Board) pollutants reported as entering the
Suisun Bay —Lower San Joaquin River [6], including:
1. Phenol
2. Chlorides
3• Nitrogen
4. Phosphate
5- Dissolved silica
6. Total sulfide
-------
7. Total suspended solids
8. Oil and grease
9- COD
10. BOD
11. Flow volume.
RESULTS AMD CONCLUSIONS
Although the solutions presented in the report [3] for purpose of illustra-
tion depend upon inputs of somewhat restricted data, it is evident that substantial
savings to the 12 dischargers as a whole would result if a nonuniform spectrum of
requirements was imposed. That is, one particular discharger of a given pollutant
might not be required to treat his wastes, whereas another discharger of the same
material might have to treat to the third or fourth of the six levels noted. It
was also evident from the results that the linear programming model could readily
"be applied to wastes restricted to different standards for different quality factors
(pollutants). It could "be adapted to partial flows as well, should industry choose
to split its waste streams and apply different degrees of treatment to different
fractions of its discharge.
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IV. SUMMARY OF PHASE k
INTRODUCTION
One of the major limitations of the Linear Programming Water Quality Control
Model reported in phase 3 [3] is that the quality standards utilized in the solution
•are expressed as the maximum allowable quantities of each waste constituent dis-
charged to the estuary. Although this is in part due to the custom of setting
discharge standards prevailing at the time, it does not readily get at maximum
allowable concentration levels in the water resource itself. This limitation could
become increasingly significant because the tendency to place restrictions on water
resource quality was accompanied by a growing concern for ecosystems of estuarine
waters. Specifically, it was questioned that although the receiving water might
meet requirements once waste constituents were well dispersed, what would be the
situation in water masses during the interval between discharge and equilibrium
dispersal?
Phase 4 was initiated in an effort to refine phase 3 by the development of
dispersion models for optimum allocation of water quality and the integration of
such dispersion models with the water quality control model of phase 3- Dr. Shishir
K. Mukherjee conducted this phase of the study and prepared the report [k].
CONCEPT AND RATIONALE
As in phase 3> the underlying rationale of phase k was that:
1. Although economic grounds alone are not a sufficient basis for water
quality control, the objectives of control should be attained at a
minimum cost•
2. Operations research and systems analysis techniques can be applied
to optimizing the cost of achieving water quality objectives.
3- By internalizing various technological external diseconomies, water
quality objectives can be dealt with to a satisfactory degree; i.e.,
the dilemma of environmental degradation as the price of economic
prosperity can be resolved.
OBJECTIVES OF THE STUDY
Five specific objectives were listed in the report on phase h [k]. They
included:
1. To study the transportation, dispersion, and degradation of wastes
constituents in an estuarine basin — aimed at relating the resulting
concentration to the amounts and locations of waste discharges and
hydrological parameters.
2. To develop dispersion models of optimal allocation of water quality
in an estuary to achieve various related objectives, and solution
of the continuous versions of these models by the application of
optimal control theory to obtain an insight into the nature of
optimal discharge policies.
3. To develop discrete versions of the dispersion models as linear
programs, with a view to using the discrete data available from
sampling studies in an estuary.
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10
U. To develop an integrated multicomponent model of dispersion and
waste treatment which will simultaneously provide an optimal plan
of waste treatment, along with a plan of optimal waste discharge
along the estuary, given a prescribed quality standard, and solve
it as a linear program.
5- To study various modifications of the model and the economic effects
of various alternative quality-improvement projects such as flow
augmentation and transportation of waste by piping within the
estuary or directly into the ocean.
PROCEDURE
The procedure was to develop an optimization model based on a one-dimensional
description of the dispersion process which includes both longitudinal dispersion
and advection. Although the final model was developed with reference to an estuary,
it is regional in nature and applicable to a river basin terminating in an estuary.
A steady state model was likewise described in detail for quality conditions
averaged over a tidal cycle. To include seasonal variations in volume of flow,
waste generation and discharge rates, and other hydrological parameters, an unsteady
state model was also developed.
A duality theory for a continuous mathematical programming problem was
formulated, as was a linear optimal control theory and a continuous dispersion
model. After rigorous proving of these theories by mathematical methods, the
dispersion model was formulated for mathematical programming and a series of
problems examined.' Thereafter a waste treatment model was developed. It generated
an optimal waste treatment plan for all waste-producing agencies in the region by
minimizing the total waste-treatment costs. Finally a complete model of estuarine
water quality management was developed by a synthesis of the dispersion and waste
treatment models. Economic interpretations of the synthesized model are discussed
and the model illustrated by an example from the San Francisco Bay system.
RESULTS AMD CONCLUSIONS
The principal conclusions drawn by the author of the report on phase k [4]
are:
1. The boundaries of knowledge about water pollution, waste treatment,
and mixing processes in estuaries are sufficiently limited that much
is to be desired in terms of understanding of these processes before
they can be described in precise mathematical terms.
2. No reliable theoretical method is available for most estuarine
situations. Additional research is needed to develop a three-
dimensional description of the dispersion process along with a
method of estimating directional dispersion coefficients.
5- Movement and resulting concentrations of conservative substances
can be described adequately if a proper description of the dispersive
and advective processes is available. However, for nonconservative
constituents the accuracy of a model is dependent to a great extent
on the reliability of the estimates of the reaction rate constants.
Usually these estimates are arrived at by subjective judgments based
on the experience of the investigator with the particular environment
and waste discharges. There is a need to seek better mathematical
descriptions of nonconservative processes, particularly of the
biological reactions.
h. A need for continual search for more efficient, complete, and
inexpensive treatment processes cannot be overemphasized. The
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11
optimal operation and control of treatment processes is by itself
a challenging field.
The solutions of the water quality control model proposed is
sensitive to the water quality standards used. Multicomponent
water quality standards are still not available for many estuarine
basins. The harmful effects of many constituents by themselves or
in conjunction with other constituents on human health and marine
ecology are not yet completely understood in quantitative terms.
Further research and legislation are needed for establishing proper
water quality standards.
Quantitative information regarding harmful effects and social costs
of pollution and benefits from quality improvement projects are
lacking today so that these costs and benefits cannot be included
in optimization models. The author has developed deterministic
models and considered concentrations averaged over one or more
complete tidal cycles. In unsteady state models, only long-term
or seasonal variations were considered. Stochastic models may be
developed to study variations of water quality during a tidal cycle
and effects of random variations of volume of flow in the estuary.
There has been some progress in this direction.
Finally there is a need to consider environmental pollution as an
integral problem, whether it is in the water, in the air, or in
the form of solid wastes. Public awareness and public education
about the desirability of a congenial environment may be a necessary
step toward creating a better world for all of us.
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SECTION II
Phase 5
A MULTISECTORAL PROGRAMMING ANALYSIS FOR THE
MANAGEMENT OF THE WASTE WATER ECONOMY
OF THE SAN FRANCISCO BAY REGION
by
Jona S. Bargur
E. M. Lofting
H. Craig Davis
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PREFACE TO SECTION II
INTRODUCTION
Although the objectives of phase 4 of the study were not fully attainable
within the period of time allotted to that phase of the investigation, the work
was a major contribution and made possible the more complex phase 5, which is the
subject of this section (Section II) of the report. Inasmuch as phase 5 of the
study is herein reported for the first time, it is presented in considerable detail,
both in its relation to the preceding phases of the investigation and to the problem
of water quality management to which it pertains. Like phase 4 it deals specifically
with the estuarine waters of San Francisco Bay primarily because it is in such waters
that the problem of protection of ecosystems is most acute; the consequences of poor
dispersion of wastes most critical; and, as a result, the task of relating water
resource quality objectives to waste discharge criteria most difficult for regulating
agencies.
CONCEPTS AND RATIONALE
The basic concept underlying this phase of the study is that water quality
management procedures must somehow be made equal to the task of achieving national
goals of environmental quality.
Surface and subsurface supplies of water are increasingly being required to
meet the rapidly growing demands for domestic, industrial, agricultural, and
recreational uses. While water is generally considered to be a "renewable" resource
through the hydrologic cycle, the rapid depletion of many sources and severe quality
degradation of others, even such large bodies as the Great Lakes and most major
estuaries including San Francisco Bay, is cause for serious concern. Suddenly man
begins to fear that the entire ecological balance of major geographic regions is
irreversibly threatened.
Traditionally, water development, allocation, and management programs have
been based on engineering feasibility considerations and anticipated local and
regional demands for various uses via some form of benefit-cost analyses. The
present day problems relating to increasing concentrations of population, rapidly
expanding industrial and agricultural production, and the use of water bodies as
sumps for the associated wastes are causing attention to be focused on more precise
forms of economic and programming analyses through which the effects of alternative
water use patterns, including waste water disposal, can be more clearly revealed
and evaluated. Such analytical methods, if carried forward with some degree of
refinement, may force a more rational use of water and permit an optimal allocation
of regional resources in which a full potential for economic growth and development
may be realized.
Water quality problems relating to estuarine waters have received prime
attention in the United States in the past two decades. Indeed, by contrast, far
in advance of the efforts now being directed toward major limnological problems,
serious concern for the fate of the large East Coast estuaries such as the Delaware
and Potomac was in evidence by the late fifties. A comprehensive study of the
San Francisco Bay was begun in the early sixties and major policy issues for the
ultimate management of the quality of the waters of the Bay-Delta system are still
under consideration.
-------
"In the process of setting water quality standards for interstate
and coastal waters, many States further chose to adopt standards
for their intrastate waters as well. Thus, it is safe to state
that for the first time in history explicitly stated water
quality criteria and goals are being made applicable, and enforce-
able at various governmental levels, to the vast majority of the
Nation's waters —be these creeks, rivers, streams, lakes, Great
Lakes, or estuarine and coastal waters.
".... Major sources of pollution are return flows from municipal,
industrial, power, and agricultural withdrawals, and runoff and
sediment from lands used for urban, mining, agricultural, log-
ging, and other purposes .... The waste loads from municipal
systems alone are expected to increase nearly four times over the
next 50 years .... Even if municipal and industrial vaste loads
are substantially reduced through treatment, pollution problems
may continue to exist in densely populated and high industrialized
areas where assimilative capacity of receiving waters is exceeded.
".... The Nation's growing population will demand multipurpose
water resources development which could require a high degree of
structural control of river systems in many parts of the country-
On the other hand, future generations should be assured of sub-
stantial areas of preserved scenic and wild rivers, wilderness,
wetlands, natural estuaries, beaches, and shores and other areas
of natural beauty. Often there is an interdependence between
preservation and development. The benefits of development
projects that change nature's patterns must be weighed against
disruptions of the natural environment." [7]
An estuary is defined by Powell [8] as "that reach of a river where river
water mixes with a measurably dilute sea water." The factors governing the basic
water quality characteristics of a particular estuary are extremely complex and are
generally governed by the physical configuration of the estuary, the inflows from
the associated river system, tidal action, prevailing winds, stratification caused
by temperature, salinity, or some combination of these, and ambient temperature
conditions which may or may not cause seasonal overturns of the estuarine waters.
In general, the natural assimilative capacity of an estuary, i.e., the
ability of the water to receive waste discharge of all types, is considerably greater
than that of a fresh water body of similar magnitude. Under such circumstances an
optimal solution is sought in which the assimilative capacity of the estuary is
utilized to that maximum beyond which some predetermined quality standards will be
violated. The general level of treatment for waste water discharges which will
preserve these standards and yet still permit the assimilative capacity of the
estuary to be fully utilized will be viewed as the most "economic" level of treat-
ment. The practical problems of finding this "optimum" are difficult, but since
most industrialized societies have demonstrated, and continue to demonstrate, that
they wish to minimize their investment in waste treatment facilities, the solution
of the problems can be viewed in this context as being a highly desirable one.
Most river waters, and particularly the waters inflowing to San Francisco
Bay, contain high concentrations of nitrogen and phosphorus. These elements form
the basis for the rapid development of algal blooms, and hence are called "nutrients"
or biostimulants. There is some evidence to indicate that industrial and domestic
sewage effluent now being discharged in large quantities into the Bay may have an
inhibiting effect on the algal growth and thus if improved waste water discharge
practices were implemented at the present time the rapid growth of algae would
result [9].
Thus the complex and delicate balance of the water quality problem in an
estuary requires a total view of the waste water discharges of the surrounding
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15
region and an understanding of the prevailing sectoral interrelationships in addition
to the synergistic effects of the discharges as mixing and dispersion takes place
in the estuary itself.
The problem therefore has two major aspects, first the clear analysis of the
industrial structure of the surrounding region and the economic interrelationships
of the major waste water generating sectors, and secondly a dispersion model for
the estuary in which well-defined mixing and diffusion patterns for pollutants can
be traced. Without these two key aspects of the problem being set out independently
and then fused, meaningful analyses cannot be undertaken or overall treatment levels
determined relative to desired quality standards.
Traditionally, in-stream water quality problems have fallen primarily into
the realm of hydraulic and sanitary engineering, hence the basic research approach
to problems of this type has evolved around the mixing and dispersion concept of
fluid mechanics and the dissolved oxygen and BOD relationships of water and wastes.
However, more recently a multidisciplinary approach to water quality problems has
begun to characterize research in the water resources field.
OBJECTIVES OF STUDY
The general objective of phase 5 of the study, herein reported, was to
attempt to unify the various and diverse factors bearing on estuarine water quality
management into an operational analytical technique. The specific objective was
to develop a water quality management model through the conjunctive use of input-
output and linear programming techniques, making use of programs and concepts
developed in related research on the Economic Evaluation of Water; and, additionally,
making specific use of the steady state linear optimal dispersion model developed
by Shishir K. Mukherjee in phase 4 of the study [k].
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V. SAN FRANCISCO BAY ABEA
IMPORTANT CHARACTERISTICS
Some significant characteristics of San Francisco Bay are presented here to
give the study a level of self-contained continuity. Exhaustive studies of the Bay
region have "been carried out with specific objectives in view [10-12]. While much
of the material these prior studies generated is descriptive and may have teen a
by-product of particular objectives, it is extremely valuable and frequently of
high quality.
The main portion of the Bay, excluding the network of waterways known as the
Delta, extends generally from the City of Pittsburg on the north to the City of
San Jose on the south, a distance of approximately 85 miles. The approximate area
of the Bay at mean tide is about Uj5 square miles. Simple division would indicate
that its average width is somewhat over 5 miles. It is estimated that about 100
years ago the area of the Bay was of the order of 700 square miles, revealing the
extent to which land at its perimeter has been claimed for uses relating to urban,
agricultural, and industrial development [10].
The total volume of water in the Bay at mean tide is approximately 2J5
billion cubic feet.
"The tidal prism, or the volume between high and low tides, is
about 50 billion cubic feet or 21 percent of the average total
volume of water in the Bay. Fifteen to 20 percent of this tidal
prism is replaced by new ocean water during each tidal cycle.
This is the principal mechanism by which pollutants are ultimately
removed from the Bay."[10]
In general the Bay is quite shallow with an average water depth of about
20 feet. In such circumstances wind waves can disturb bottom sediments and increase
the turbidity of the water [10].
The average annual precipitation in the San Francisco Bay region is about
19 inches. The mean annual evaporation for the same area is approximately 48 inches.
This is double the annual precipitation which results in the loss of more than
650,000 acre-feet of vater each year by surface evaporation alone [10].
POPULATION DISTRIBUTION AND PROJECTIONS
For purposes of this study the San Francisco Bay region has been defined as
the nine counties region immediately surrounding the Bay, i.e.,
Alameda
Contra Costa
Mar in
Napa
San Francisco
San Mateo
Solano
Santa Clara
Sonoma.
Since waters inflowing to the Bay also originate in, or flow through, Yolo,
San Joaquin, and Sacramento counties it might be suggested that the term twelve-
county region could more properly have been chosen. This aspect was considered,
IT
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18
"but since the main focus of the study is on direct waste water discharges from the
region immediately surrounding the Bay, the nine counties listed above were chosen
as "being the representative group of counties in which the wastes associated with
productive activity and population would interact directly with the waters of
San Francisco Bay.
To seme considerable extent population concentrations and the level of
productive activities bear a high correlation in industrialized societies. Tourism,
recreation, or raw materials extraction might provide exceptions to the generalization-
is! some regions. For the Bay region, population and productivity are indeed highly
interrelated. Realising the relationship, population figures and projections
(presented in Table l) can provide an insight into the magnitude of the environmental
problems that will confront the region.
TABLE I
POPULATION OF BAY REGION COUNTIES 1960-19668
AND ESTIMATED PERCENTAGE CHANGS 1965-2020
Counties
A lame da
Contra Costa
Marin
Napa
San Francisco
San Ma tec
Sclanc
Santa Clara
Sonoma
Total
Percent
Increase
I960
908,20Q
409,050
1^6,820
65,890
740,316
444,387
134,597
642,315
1^7,375
3,638,939
1966
1,045,600
528, it 00
195,600
76,300
738,100
538,900
162,700
924, 500
183,000
^393,100
20.7$
Estimated Growth
1965-2020^
£ Change
111
169
19-
183
^.^
58
588
122
409
See Reference [13], Table B-7-
See Refere-.ce [ll], Table 1.
Total regional population figures and projections are shown in Table II
on the following page.
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TABLE II
POPULATION PROJECTION IN 9-COUNTY
SAN FRANCISCO BAY REGION
1965-2020
Population
(io3)
Year
1965
-^336a
1970
k,9kj°
1975
5,550b
1980
6,158a
1990
7,^71b
2000
8,785*
2020
io,8ooa
See Reference [ll].
Interpolated.
GENERAL ECONOMIC CHARACTERISTICS
In 1963 the State of California accounted for 11.7 percent of U- S. Gross
National Product [1^,15]. Of actual dollar value ($65.3 billion) the nine-county
Bay region produced approximately 17 "billion dollars* or about 25 percent of total
state product. A comparison of the relative sources of civilian income from
participation in current production for California and the United States shows that
for practically all major categories California proportions vary only slightly from
those of the U. S. [l6]. For 82 major categories of productive activity [17]
California lacked a Coal Mining sector and the Bay region additionally lacked the
following sectors: Chemical and Fertilizer Mineral Mining, Iron and Ferroalloy Ores
Mining, Nonferrous Metal Ores Mining, Tobacco Manufactures, Wooden Containers, and
Engines and Turbines. All other categories of productive activities were present in
their various forms.
See Table IV, this report.
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VI. THE NINE-COUNTY BAY AREA MULTISEGTOR
MODEL FOR 196?
GENERAL DESCRIPTION OF
AN INPUT-OUTPUT MODEL
Input-output analysis is an econometric technique which focuses upon the
interdependencies among the various industries or sectors of the economy and the
relations of these sectors as sellers to final consumers- The technique was
introduced in 1936 "by Wassily Leontief who modified and aggregated the Walrasian
general equilibrium system [l8] to produce a statistical description of the United
States economy [19], derived from empirical observations of interindustry trans -
actions. The Leontief formulation was the first operational economic model to
embody the essence of general equilibrium. Applications of the theory since that
time have resulted in the construction of input-output tables for more than fifty
national economies and for an expanding number of regions and urban areas.
The model derives its name from the method by which it categorizes economic
transactions. Each exchange of goods and services between sectors in the model is
recorded in double entry fashion as both a sale of output and a purchase of input-
The basic static version of the model is normally presented in the form of three
types of tables: Transactions Table, Technical Coefficient Table, and Table of
Direct and Indirect Requirements.
THE INPUT-OUTPUT
MODEL IN SUMMARY
The first of these three tables, the Transactions Table, is based upon a
set of accounting identities or balance equations which state that the aggregate
sales of a particular sector are equal to the total purchases from that sector.
Given an economy divided into n sectors, the balance equations take the form
Xi = Xil + "• + Xin + Yi (i = 1, .-., n) (1)
where
X. = the total output of industry ±,
x.'. = the amount of commodity i required by industry j, and
Y. = the exogenous or final demand for commodity i.
Once value added (factor payments) and input flows are also recorded, the
simple precept of the table that total inputs equal total outputs can be stated as
(i = j = I, ..., n)
20
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21
where
X. = total purchases made by sector j ,
J
v = value added of sector j, and
j
m. = purchases of imports by sector j.
J
The second table, the Table of Technical Coefficients, represents the
model's structural equations which state the assumption of fixed technical
coefficients a..., i.e., the assumption that inputs into each sector are a direct
function of the level of output of that sector. Compared to the n balance equations,
there are n2 structural equations of the form
x. = a X (i, J = 1, ..., n) (J)
-*- J ^J J
where
X. = the total output of industry j, and
J
a. . = the technical coefficient defined by x. . and X. above.
1 J -*- J d
The model's second table is thus the [a..] matrix.
To derive the third and most useful table, the Table of Direct and Indirect
Requirements, the structural Equation 3 is first substituted into the balance
Equation 1. This substitution results in
Xi = SilXl + '•• + aijXj + ••• + SinXn + Yi ^ = ^ "•' n) (4)
which may be written more compactly in matrix form as
X = AX + Y
or
X - AX = (I - A)X = Y
where A = [a..] and I is an identity matrix.* The general solution, by matrix
inversion**, 1^ay now be expressed as
X = (I - A)
-i
An identity matrix is a matrix whose principal (NW-SE) diagonal is composed
of unit values and whose off-diagonal elements are zeros.
Inversion is the counterpart of division in matrix algebra. The inverse of
any matrix A is defined as that matrix which when premultiplied or postmultiplied
by the original matrix A will yield an identity matrix. Thus AA^A = 1.
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22
The matrix A in the previous notation is the Table of Technical Coefficients
derived from the Transactions Table in the manner previously described. The
"Leontief inverse/' T(l - A)"1, is the third table and is customarily transposed*,
(I - A)"1, in order that the relevant information can be read along the rows rather
than down the columns.
The Table of Direct and Indirect Requirements answers the question of how
much output must be produced in order to satisfy final demands. If steel is the
commodity in question, the economy will have to produce:
as much steel as consumers will use directly;
plus as much steel as is needed to produce other final
consumer products;
plus as much steel as is needed to produce the inputs
for these final consumer products;
plus as much steel as is needed to produce the inputs
which are in turn used to manufacture those inputs
which go into those final products;
plus as much steel as is needed for the inputs to make
the inputs for the final products;
and so on, ad_ infinitum. [20]
Gross regional product, the regional counterpart of GNP, can "be calculated
from the input-output model by either of the two usual methods. The income or
factor payments approach can "be undertaken by summing the value added flows for
each of the intermediate and final demand sectors. On the other hand, the same
result can be obtained via the sales of final product approach by summing the total
purchases of each final demand sector, subtracting out imports.
DEVELOPMENT OF THE MODEL
The Nine-County Bay Area Model for 1963 was developed from secondary data
sources which were used to proportion or otherwise aid in the estimation of Gross
Output values of production for the major sectors of the regional economy [lM •
The elements of the final demand vectors were similarly proportioned on the basis
of available data [lU]. The interindustry transactions table was prepared by means
of a FORTRAN IV computer program developed for this purpose [21]. This program is
essentially a computerized version of the "Moore-Petersen" method described in their
article dealing with the construction of an input-output table for the economy of
the State of Utah [22]. The basic input-coefficients used were those of the
CEIR/Fortune study describing the structure of the U. S. economy for 1966 [23].
The sector aggregation is shown in Table III and was chosen to emphasize those
industries known to be waste-intensive and to discharge major waste constituents,
either conservative (nondegradable) or nonconservative (degradable), which result
in substantial quality degradation of the receiving waters.
The interindustry transactions table for the 9-county region is shown as
Table IV; the technical coefficient matrix, including value added coefficients
(Table V) succeeds the flow table; the table of direct and indirect requirements
per dollar of output to final demand is shown finally as Table VI.
*
A matrix is transposed by interchanging its rows and columns.
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23
TABLE III
SECTOR CLASSIFICATION FOR THE MODEL
Industry No. and Industry Title
Related 1966 National Input-Output
Sectors from CEIR Study [2J]
Rows
1 Agriculture, Forestry and Fisheries
2 Mining
3 Construction
k Food and Kindred Products
5 Paper and Allied Products
6 Chemicals and Chemical Products
7 Petroleum Refining and Related
Industries
8 Stone and Clay Products
9 Fabricated Metal Products
10 All Other Manufacturing
11 Transportation, Communication.,
Electric, Gas and Sanitary Services
12 Wholesale and Retail Trade
13 Finance, Insurance and Real Estate
it Services and Government Enterprises
15 Interindustry Inputs
16 Sectoral Imports
17 Foreign Imports
18 Total Sectoral Imports
19 Value Added
20 Adjustment
21 Total Gross Outlay
Columns
1-14 Same as above (Rows)
15 Intermediate Outputs - Total
16 Personal Consumption Expenditures
17 Gross Private Capital Formation
18 Federal Government Expenditures
19 State and Local Government
Expenditures
20 Bet Exports
21 Total Final Demand (Sum of Row
Elements of Columns 16-20)
22 Total Gross Outputs
23 Commodity Imports
1, 2, 3, k
5, 6, 7, 8, 9, 10
11, 12, 13, Ik, 15, 16, 17, 18
20, 21, 22, 23, 2k, 25, 26, 27, 28
38, 39
4i, 42, 43, 44
50
55, 56, 57, 58
19, 29, 30, 31, 32, '3, 3^, 35, 36,
37, to, 46, 47, 48, 119, 51, 52, 53,
54, 59, 60, 61, 62, 63, 6k, 65, 66,
67, 68, 69, 70, 71, 72, 73, Ik, 75,
76, 77, 78, 79, 80
8l, 82, 83, 84, 85, 86
87
90, 91, 92, 93, 9^, 95,
100, 101
5, 97, 99,
-------
TABIE IV
INTERINDUSTRY TBAHSACTIONS
(FLOW Of GOODS AND 35RVICES BY INDUSTRY OP
ORIGIN AND DESTINATION) 1963
SJME BW ABBA COOSTIBS
(All figures in millions of dollars)
Rov 1
Row 2
Row 3
Row 4
Rov 5
Row 6
Rov 7
Row 8
Row 9
Row 10
Row 11
Row 12
Row 13
Row 14
Row 15
Row 16
Row 17
Row 18
Row 19
Row 20
Rov 21
54.43
o.oS
6-59
35-69
0-34
9.91
9.30
0.24
1.03
6.05
12.76
18.00
26.51
12.71
193-64
io".8u
124.96
214.34
0.07
533-01
O.OO
0.40
0.05
0.00
0.11
0.72
0.65
0.41
0.36
5-15
5-59
1-73
7-77
2-34
21.27
6.14
7.63
13-77
46.47
0.00
81.51
3-26
2.64
0.45
0.54
14. 18
61 Ag
62.88
18S.79
288.96
459-69
123.83
304.91
41.27
172-26
1719-16
315-08
0.00
315-08
1435.81
0.05
3470.07 •
• 317.31
0.20
10.49
527-93
44.79
16.13
11.76
19-4?
66.99
23-04
141.60
107.53
28.50
185.83
1501-51
#5-28
108.85
792-11
777-95
-0.02
5071-56
0.00
0.21
1-25
1-59
71-25
9.07
5-08
1.07
2.89
22.44
17.60
12.26
3-73
7-12
153-55
31-87
20.90
52-78
120.84
0.00
327.17
0-31
1.11
0.87
9.98
15-25
121.02
18.27
4.26
8.38
17.12
27.80
15.90
10.24
41. lit
289-66
52.09
9-85
61-93
217-55
-0.01
569-13
0.00
62.63
2.40
1.12
6.54
45.82
116.67
2.73
83.89
2.82
114.86
17.32
23-75
44.13
464.68
786.55
51-56
857-91
315-22
0.00
1617.81
0.03
1.26
0.13
0.21
9.48
9-28
2.66
23-67
3.00
8.15
24.79
9-70
5.03
9.20
106.59
31.21
2. 77
33-99
112.63
42.03
295.24
Column
11
14
15
16
Row 1
Row 2
Row 3
Row 4
Hew 5
Row 6
Row 7
Row 8
Row 9
Row 10
Row 11
Row 12
Row 13
Row 14
Row 15
Row 16
Row 17
Row 18
Row 19
Row 20
Row 21
0.00
0.03
0.44
0.01
4.73
5.61
3.21
3.85
56.90
204.04
17-60
23.25
8.67
16.93
325.28
68.75
3-67
72.42
263.71
-0.01
661.40
25-75
4-71
8.98
2.82
74.60
98.20
14.41
31-22
124.02
1424.80
153-62
lSl-37
70-l8
165-45
2380.12
579-37
135-02
714-38
2046.65
-32.38
5108.78
0.49
5.90
89.31
4.24
2.26
5.50
72.87
1.05
7.10
72.57
297-67
54.67
91.60
261-95
964.97
96.40
51-26
147-65
1718.94
0.02
2831.58
1.85
0.02
28.75
22.24
22.93
7-54
27-49
6.63
6.42
50-47
121-57
57-33
245.27
340.27
938-77
29.61
1.08
30-69
2571.89
-4-35
3537.00
34.39
0.51
270-57
2.8l
5-69
5.18
21.87
0.86
0.83
55-18
77.89
61.04
523-61
225-18
1265-61
87.86
3-77
91-63
2T85.64
0.01
4142.89
12-14
0-79
111-51
124.54
31-05
42-55
20.88
8.16
8.37
551-29
425.20
132.48
260.22
427.15
2156.31
200.88
61.50
262.37
2516.16
0.02
4914.86
449.96
8o.48
531.79
735-71
301.19
456.01
536.00
286.36
579.13
286o.6o
1560.39
997-50
1346.35
1911.63
1246i. 11
3083.19
468.47
5551.66
15143.81
0.00
51162.01
58.77
0.74
0.00
1725-23
85-32
117.77
266.42
8.88
14.75
1294.10
748-36
2^00.30
2192.30
1560.61
10313-24
516.76
134.00
650.76
76.00
0.00
11040.00
Cclunn
17
18
19
20
21
23
Rov 1
Row 2
Roy 3
Row 4
Rov 5
Row 6
Row 7
Row 8
Row 9
Bow 10
Row 11
Row 12
How 15
Roy 14
Row 15
Rov 16
Rov 17
Row 18
Rev 19
Row 20
Rov 21
0.00
0.00
1533-84
0.00
0.00
0.00
0.00
0.00
61.59
26O.02
0.58
0.59
O.19
0.00
1856.41
85-87
-o.oo
85.87
-0.00
0.00
1942-.27
20.58
0.26
255-74
6.71
0.65
10.15
149-50
0.00
4.92
669-15
49.52
54.48
4.45
764.01
1967.72
S29-05
14}. 00
372.05
866.20
0.00
3305-95
3-90
0.04
650.20
8.47
0.00
5.20
14.94
0.00
1.01
24.90
44.71
7-19
17.08
81.2}
836.87
16.42
-0.00
16.42
949.50
0.00
1804.79
O.OO
0.00
520.50
597.44
0.00
0.00
801-15
0.00
0.00
0.00
426.22
197.14
582.82
597-38
3724.66
0.00
0.00
O.OO
0.00
0.00
5724.66
83.05
1.03
29VS.28
2537.85
25.98
133-12
1251.81
8.88
82.27
2248.18
1271-19
2539-50
2796.54,
3005.23
18700.90
848.07
277-00
1125.07
1891.70
0.00
21717-67
533-01
81.51
3470.07
3071-56
527-17
569-15
1617-81
295-24
661-40
5108.78
2831-58
3537-00
4142.89
4914,86
51162.01
3931.26
745.47
4676.73
17035.51
0.00
52879-68
-1066-97
-998.57
0.00
0.00
-92.61
-121.74
0.00
-100.84
-164.68
-1385.85
0.00
0.00
0.00
0.00
-3931.26
0.00
745.47
4676.73
0.00
0.00
0.00
-------
TABLE V
INTERINDUSTRY TRANSACTIONS - TECHNICAL COEFFICIENT MATRIX
(DIRECT PURCHASES PER DOLLAR OF OUTPUT) 1963
NINE BAY AREA COUNTIES
Column
Row
1
2
3
1+
5
6
7
8
9
10
11
12
13
11+
Value
Added
1
.102118
.000000
.000939
.103306
.000000
.00051+5
.000000
.oooioa
.000000
.00501+0
.000173
.000517
.008301
.0021+70
.1+02000
2
.000150
.001+907
.000761
.000065
.00061+2
.001950
.038713
.001+268
.00001+5
.000922
.002081+
.000006
.000125
.000161
. 570000
5
.012361+
.000613
.000130
.0031+15
.003821
.001529
.0011+83
.oooi+i+o
.000665
.001758
.03151+1
.008128
.065309
.022688
.1+13000
1+
.066959
.000000
.000156
.171877
.001+860
.017536
.000692
.000711
.000015
.000552
.0011*97
.006288
.000678
.025339
.253000
5
.000638
.001350
.001+086
.011+582
.217777
.023281
.ooi+oi+3
.032109
.007151
.011+602
.000798
.0061+83
.001373
.006318
.369000
6
.018593
.008833
.017720
.005251
.027723
.21261+0
.028522
.0511+32
.0081+82
.019222
.001236
.002132
.001250
.008657
.382000
7
.01741+8
.00797"+
.018121
.003829
.0091+1!+
.052102
.072116
.009010
.001+855
.002821
.025735
.007772
.005279
.001+21+8
.195000
8
.0001+50
.005030
.052676
.006326
.003270
.0071+85
.001687
.080172
.005821
.006111
.000371
.001871+
.000208
.001660
.381000
9
.001932
.001+1+ 17
.083272
.021810
.008833
.011+721+
.011+767
.010161
•055791
.021+276
.002507
.001815
.000200
.001703
.399000
10
.011351
.03861+6
.1321+73
.007501
.068588
.030081
.00171+3
.027605
.3081+97
.278892
•025558
.011+269
.0081+92
.108099
.1+00000
11
. 0239^0
.oi+W.
.035685
.01+6100
.053795
.01+881+6
.070997
.083966
.026610
.050070
.105125
.051+571
.018801
.086515
.607000
12
.033770
.021221+
.087869
.035008
.0571+75
.027937
.010706
•052855
•055155
•035502
.019307
.016209
.011+73*+
.026955
.727000
13
.01+9736
•095326
.011893
.009279
.0111+01
.017992
.Oll+68o
.017037
.013109
•013737
.03231+9
.06931+1+
.126388
.05291+6
.672000
11+
.02331+6
.028708
.01+961+2
.060500
.021762
.072286
.027278
.031161
•025597
.032585
.092505
.096205
.051+353
.086906
. 512000
ro
ui
-------
TABLE VI
INTERINDUSTRY TRANSACTIONS
(DIRECT AND INDIRECT REQUIREMENTS PER
DOLLAR OF FINAL DEMAHD) 1963
NINE BAY AREA COUNTIES
Column 1 z 5 4.5 6 7 8 9 10 11 12 1? I1*
Row
1 1.1257577 .OOlW$o .0227859 .0940684 .005914; .0310855 .0259002 .0054072 .0088859 .0411286 .0511295 .0510875 .0769752 .0572601
2 .0020415 1.0057558 .0122419 .002709; .0051056 .0154245 .0128501 .0071488 .0086716 .07;84o7 -0647559 -0;i5057 .1202800 -0543911*
5 .OOJ8486 .0025797 1.0090003 .0055509 .0165408 .0356059 .0268371 .0609736 .0980170 .2526979 .0772794 .1118169 .0398355 .0951910
4 .1418294 .0009754 .0144762 1.2235754 .0268957 .0169711 .0129889 .0105214 .0326736 .0564510 .0885285 .0602242 .0385312 .1088510
5 .0027524 .0020915 .0125941 .0115447 1.2847728 .0515907 .0195323 .0072379 .0189491 .1497220 .0982190 .0627345 .0339627 .0615460
6 .0059168 .0048674 .0121612 .0325249 .O4;246l 1.2790256 .0498335 .0128036 .0262243 .0968153 .0997091 .0521406 .0469916 .1298961
7 .0011541 .0425126 .0085765 .0039882 .0087106 .0419648 1.0834561 .0036276 .0199584 .0302615 .1001394 .0210553 .0335531 .0541112
8 .0017531 .0058365 .009204} .0051416 .0490855 .0492868 .0176986 1.0892344 .0166231 .0734180 .1223092 .0490701 .0379841 .0667549
9 -0037389 .0012000 .0081485 .0039876 .0213395 .0262098 .0113349 .0110643 1.0735284 .4783377 .0629105 .0622952 .OJ66452 .0663207
10 .0092652 .002009 .0095076 .0055465 .0300076 .0386790 .0095691 .0110449 .0392245 1.4253644 .0659165 .0607990 .0371659 .0725793
11 .0023813 .0039001 .0432239 .0068553 .0050465 .0080486 .0344758 .0040335 .0100216 .0749945 I.i4i6j73 .0350928 .0564145 .1305666
12 .0035446 .0007492 .0194331 .0123786 .0115349 .0073242 .0124605 .0040659 .0061442 .0503218 .0594225 1.0277406 .0935185 .1251845
13 .0118946 .0008097 .0793564 .0050444 .0050123 .0070868 .0107236 .0055575 .0093356 .0482349 .0414053 .0309178 1.1570775 .0844766
l4 .0092847 .0012189 .0362073 .0367358 .0153049 .0202290 .0121906 .0061391 .0121889 .1912230 -1270830 .0484802 .0828612 1.1324392
-------
27
FINAL DEMAND "VECTORS AMD
PROJECTION TO TAR(£T YEARS
The categories of final demand shown in the Transactions Table for the
9-ccmnty Bay region were added to form a single final demand vector. This vector
is a column vector of fourteen elements, corresponding to the l4 productive sectors
specified in the Transactions Table.
For projection purposes each element was assumed to increase "by 3 percent
and 6 percent per year to the selected target years shown in Tables VII and VIII.
The tables show the values resulting from the compounding of the elements of the
vector from the base year. The assumptions build in a fixed pattern of expenditures
and expected upper and lower bounds for the growth rate in real terms. Nevertheless;
in view of historical growth trends, such simple basic assumptions seemed to be
appropriate to the task of determining some ranges for effluent discharges in future
years. If effluent discharge conditions implied by these projections are in fact
worse, as indeed they may be, then additional measures for pollution abatement will
have to be implemented.
TAB1E VII
FINAL DEMAND PROJECTIONS - 3$ (EOWTH RATE
(Used as Minimum Expected Growth Rate in
Linear Programming Model, $10S)
^~"^\^^ Year
Sector ^^--^^
1
2
3
k
5
6
7
8
9
10
11
12
13
14
1963
83.05
1.03
2,9^8.28
2,337.85
25.98
133-12
1,231.81
8.88
82.27
2,248.18
1,271.19
2,539-50
2,796.54
3,003.23
1968
96.25
1.19
3,405-46
2,709-57
30.11
1511.28
1,1+27.66
10.29
95-35
2,605-64
1,^73.31
2,943.28
3,241.19
3,480.74
1970
102.15
1.27
3,614.08
2,875-55
31-95
163.74
1,515-13
10.92
101.19
2,765.26
1,563.56
3,123-58
3,459-74
3,693.97
1975
118.32
1-47
4,187.04
3,331.44
37-02
189.70
1,755-33
12.65
117-23
3,203.66
l,8n.44
3,618.78
3,985.07
4,279.60
1980
137-03
1.70
4,848.16
3,857.45
42.87
219-65
2,032.49
14.65
135-74
3,709.50
2,097.46
4,190.17
4,6l4.29
4,955-33
1990
183.54
2.28
6,493.60
5,166.65
57-41
294.19
2,722.30
19.62
181.82
4,968.48
2,809-33
5,612.29
6,180-35
6, 637-14
-------
28
TABLE VIII
FINAL DEMAND PROJECTIONS - 6% (2?OWTH RATE
(Used as Maximum Expected Growth Rate in
Linear Programming Model, $10S)
^\^^ Year
Sector ^"""--s^^^
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1963
83.05
1.03
2,938.28
2,337-85
25.98
133-12
1,231.81
8.88
82.27
2,21*8.18
1,271.19
2,539-50
2,796.5^
3,003.23
1968
111.29
1.38
3,937.29
3,132.72
34.81
178.38
1,650.62
11-90
110.24
3,012.56
1,703-39
3,402.93
3,747.36
4,024-33
1970
124.57
1.5^
4, 407-42
3,506.77
38-97
199.68
1,847-71
13.32
123.40
3,372.27
1,906.78
3,809.25
4,194.81
4,504.84
1975
167.76
2.08
5,935-32
4,722.46
52.48
268.90
2,488.25
17-9^
166.18
4,541-32
2,567.80
5,129-79
5,649.00
6,066.52
1980
224.23
2.78
7,933-35
6,312.19
70.14
359-42
3,325.88
23.98
222.13
6,070.09
3,432.21
6,856.65
7,550.65
8,108.72
1990
402.79
4.99
14,250.66
11,338-57
126.00
645-63
5,974.28
43-06
399-00
10,903.67
6,165.27
12,316-57
13,563.22
14,565.66
The principal merit of this method of projection is that the results obtained
are consistent with one another, given the technological structure of the regional
economy. It is highly unlikely that single industry projections based on the
extrapolation of trends or other methods will give a set of consistent projections
in which the underlying interrelationships of the regional economy are taken fully
into account.
-------
VII. WASTE WATER DISPOSAL INTO SAN FRANCISCO BAY
TYPES OF WASTE AND SPECIFIC
CATEGORIES OF POLLUTANTS
There are three major types of polluted waters entering San Francisco Bay.
These are: municipal and domestic wastes, industrial discharges, and return flows
from irrigation and drainage, including surface wash or natural runoff after periods
of precipitation.
Municipal and domestic wastes generally have high levels of "biological
oxygen demand (BOD) and can also pose health hazards if pathogens are present in
large numbers, as may frequently "be the case. Domestic wastes may also contribute
nitrogen and phosphorus in quantities large enough to cause excessive algal growth.
Agricultural return waters and surface wash may carry dissolved salts,
nutrients, or biostimulants from the leaching of fertilizers and soil constituents,
and may also contain pesticides and other conservative pollutants that may be toxic
to aquatic life.
Industrial waste waters can contain a spectrum of exotic pollutants about
which very little may be known concerning their combined (synergistic) toxic effects
in the receiving waters.
The major pollutants or waste constituents that are herein given attention
are as follows:
BOD — Biological oxygen demand, the capacity for a
waste to deplete the dissolved oxygen (DO)
from the receiving water.
Nitrogen — In combination with phosphorus and sunlight,
will cause algal growth, i.e., eutrophication.
Phosphorus — In combination with nitrogen, will react as
stated above •
Phenol —A caustic poisonous acidic compound; toxic
even when highly diluted.
TSS — Total suspended solids — material which might
be removed by settling, consisting of decomposable
organic matter or inorganic matter.
Oil and Grease —All oils, products of petroleum, and hydrocarbons.
Gross Heavy Metals — Metal particles in suspension, such as copper,
lead, cadmium, zinc, iron, chromium, etc.
WASTE LOAD COEFFICIENTS
"The idea of integrating the aspects of water resource requirements
with those of the economy as a whole for a given area, be it a
nation, a state, or an estuary, was pointed out in the early 1950's-
The advantages accruing to the evaluation of both water resources
and other resources of a region from the use of a combined accounting
system, which also incorporates regional income and product accounts
with the interindustry aspects of the intermediate transactions
within the area and with other regions with which it trades, were
since then recognized. The implementation of such models lagged
29
-------
behind the need for obvious reasons. One was the operational
difficulty in marshalling the vast quantities of consistent data
on both the traditional aspects of the economy and on water-related
accounts. Others were related to methodological difficulties in
processing the data in a consistent manner within the framework of
the input-output model.
".... The input-output model ... aims, in the long run, to relate
the economy of a ... region to its environment in as complete a
manner as possible. Such a model will cover in detail, the
various inputs -- air, water, and other materials — used by each
industry. It will also identify all the .various outputs furnished
to the environment, such as steel and other economic goods, and
discharged water and water pollutants.... The development of such
a model attempts to achieve a much more complete accounting
framework than is now possible. It would enable the evaluation
of national and regional policies concerned with the quality of
the environment as well as the state of the economy-"
In a manner similar to the construction of labor or water use coefficients,
water quality or waste load coefficients may also be developed. As a first step
it is necessary to develop for each economic sector the appropriate waste load
coefficient to express the production of the particular pollutant in terms of
commodity production. A linear relationship is assumed and such a coefficient is
determined by dividing the total amount of the pollutant produced by the sector
during the base period of the input-output table by the sector's gross output.
In the event a. nonlinear relationship is found to exist, piecewise linear approxi-
mations may be incorporated, as illustrated below.
VOLUME OF
POLLUTANT
GROSS OUTPUT
Empirically, waste load coefficients for the seven waste constituents under
consideration were calculated as shown in Tables IX - XII. The procedure of
calculation of these coefficients for the agricultural sector is presented in
Table XIII.
-------
TABLE IX
PROCEDURE OP CALCULATION OP WASTE LOAD COEFFICIENTS
Sectoral
Daily
Waste
Flow
mgd
(1)
Givenb
Annual
Sectoral
Waste
Flow
mg
(2)
(l) x 365
Annual Waste Flow
Annual Production
gal/ton
(3)
b
Given
Annual
Production
ton
(2) x 10s
(3)
Waste Load
Coefficient
of Pollutant
Ib/ton
(5)
Givenb
Total Waste
Production
of Pollutant
Ib
(6)
Gross
Output
$106
(7)
From
I/O Table
Waste Load
Coefficient
of Pollutant
lb/$10s
(8)
Transformation of waste load coefficients expressed in terms of Annual Waste Load/Annual Production
(Ib/ton) into waste load coefficients in terms of Annual Waste Load/Annual Gross Output (tons/$106).
See Reference [25].
-------
TABLE X
CALCULATION OF ANNUAL PRODUCTION (PHYSICAL UNITS) FOR
USE IN CALCULATION OF WASTE LOAD COEFFICIENTS
ro
Sector
Food
Paper
Petroleum
Stone and Clay
Chemicals
Fabricated Metals
Other Manufactures
Daily
Effluent
12 counties
mgd
67-3
14.0
72.4
17-8
Daily
Effluent
9 counties
mgd
53-2
47.9
197-7
12.4
66.5
16.4
Annual
Effluent
10e x gal/yr
19,500
17, 500
72,000
4,540
24,200
6,000
Annual
Production
10s x ton/yr
2.48
0.625
165-0
(barrels)
7-8
0.68
0.26
Gross
Output
$10e
3071.56
327-17
1617.81
295.24
569-13
661.40
5108.58
-------
33
TABLE XI
CALCULATION OF WASTE LOAD COEFFICIENTS
Sector
Food
Paper
Petroleum
Stone
Chemicals
Fabricated
Metals
Waste Load
Coefficient,
It/ton
Annual Waste
Production,
ton/yr x 103
Waste Load
Coefficient,
ton/$10e
Waste Load
Coefficient,
Ib/ton
Annual Waste
Production,
ton/yr x 103
Waste Load
Coefficient,
ton/$10s
Waste Load
Coefficient,
lb/103
barrel
Annual Waste
Production,
ton/yr x 103
Waste Load
Coefficient,
ton/$106
Waste Load
Coefficient,
Ib/ton
Annual Waste
Production,
ton/yr x 103
Waste Load
Coefficient,
ton/$10s
Waste Load
Coefficient,
It/ton
Annual Waste
Production,
ton/yr x 103
Waste Load
Coefficient,
ton/$106
Waste Load
Coefficient,
It/ton
Annual Waste
Production,
ton/yr x 103
Waste Load
Coefficient,
ton/$106
BOD
17-40
19.54
6.36
33-0
9-3
28.1*2
131.0
9.82
6.07
2.8o
2.44
0.75
1.31
0.91*
0.11
0.17
Nitrogen
2.30
2.60
0.84
0.87
0.24
0.73
74.2
5.56
3.44
0.008a
0.03
13.90
4.29
0.75
o.i4a
0.21
Phosphate
O.lf
0.04
0.233
0.72
0.87
0.065
0.04
0.15
o.o4
0.08
Oil and
Grease
3.05
3.44
1-12
31-5
2.36
1.46
O.OOOJ
0.001
1.58
0.48
0.84
TSS
38.60
^5-51
14. 16
51-50
14.63
4.47
111.3
8.34
5.16
1.26
4.1*5
15-07
146-50
45-28
79-56
Phenols
9-1
0.68
0.42
0.0001
O.OOOJ
0.001
0.004
0.0012
0.002
Gross
Heavy
Metals
1-57
0.12
0.07
2.15
0.66
1.16
Calculation of pollutant annual waste production is based on «aste load per employee coefficient
(lb/day/employee ) .
-------
TABLE XII
INDUSTRIAL WASTE LOAD COEFFICIENTS
SIC
Code
20
26
28
29
32
34
Industry Group
Pood
Paper
Chemicals
Petroleum
Stone and Glass
Fabricated Metals
All Other4
Manufacturing
Annual Waste Load/Annual Output
BODa
ton/$10e
6.36
28. 42
1-31
6.07
2.80
0.17
-
Nitrogen
ton/ $10 a
0.84
0.73
0.75
3.44
O.OJ
0.21
0.05
Phosphates
ton/ $10 8
0.04
0.72
0.08
0.0k
-
-
0.01
Oil and
Grease
ton/$108
1.12
-
0.84
1.46
0.001
-
-
TSSb
ton/ $10 s
14.16
4.47
79-56
5.16
15-07
-
0.62
Phenols
ton/$108
-
-
0.002
0.42
0.001
-
-
Gross
Heavy
Metals
ton/$10e
-
-
1.16
0.07
-
-
-
Total0
Effluent
acre -f t/$10e
7-26
88.70
33-46
18.97
22.76
10.84
1.01
aBiological Oxygen Demand.
bTotal Suspended Solids.
CTotal process effluent — excluding cooling purposes c
a,
Assuming "All Other Manufacturing" adds 5$ of the waste production in each polluting category.
-------
35
TABLE XIII
CALCULATION OP AGRICULTURAL WASTE LOAD COEFFICIENTS
Sector
Agriculture
Annual Waste Production
„. , a
Nitrogen
10s x Ib/yr
SJ.O
Effluent5
acre -f t/yr
40,000
Gross
Output
$10 e
533-01
Waste Load Coefficient
Nitrogen
ton/$10e
21.58
Effluent
acre-ft/$106
75-05
See Reference [10].
To this point, only waste resulting directly from economic activities has
teen considered. Additionally, however, estuarine waters customarily receive
pollutant flows from natural runoff waters and from discharge of domestic wastes.
Natural runoff pollutants evolve from the passage of tributary waters over
erodible soil surfaces such as unpaved streets and construction sites, and from
exposed earth surfaces — primarily agricultural lands containing various fertilizers
and animal waste products. The contribution to the deterioration of water quality
from natural runoff, however, is for most categories of pollutants quite limited
relative to other sources, as presented in Table XIV.
TABLE XIV
NATURAL RUNOFF WASTE PRODUCTION
Annual Waste
Production
103 Ib/yr
Annual Waste
Production
ton/yr
Annual Effluent
acre-ft/yr
BOD
18,900
8,000
Nitrogen
5,200
2.3
Phosphorus
250
0.11
Oil and
Grease
6,100
2.76
TSS
1, 660
752.8
Effluent
Coefficient
262,000b
See Reference [26].
South Bay
Central Bay
North Bay
Total
acre-ft/yr
1^9,000
48,000
65,000
262,000
-------
36
Domestic waste water flows emanating from sanitary fixtures, garbage
disposals, washers, air conditioners, etc. constitute a prime source of some
pollutant measures such as BOD (biological oxygen demand) and nitrogen. Waste
load coefficients for domestic waste are given in Table XV.
TABLE XV
DOMESTIC WASTE PRODUCTION
Q
Daily Waste Load
Coefficient,
Ib/capita/day
Annual Waste
Production,
tons/yr
BOD
0.17
121,711
Nitrogen
0.043
30,785
Phosphate
0.008
563
TSS
0.10
71, 5^
Oil and
Grease
0.025
17,907
Gross
Heavy
Metals
0.004
260
See Reference [27].
EFFLUENT COEFFICIENTS
Inasmuch as waste water treatment plants have a limited capability for total
removal of waste constituents, a certain upper limit for effluent from all sources
discharging to the Bay has been set. If this limit is exceeded regardless of the
general level of treatment being provided, then the permissible concentrations of
major waste constituents may be violated.
Effluent coefficients for major waste discharge sources are presented in
Tables XII — XIV. The method of calculation in each case is indicated directly in
Tables XVI and XVII.
TABLE XVI
DOMESTIC EFFLUENT - 1963
Single
Dwelling
Multiple
Dwelling
Total
Effluent
Coefficient
gal/ capita/ day
67
55
Population
106
3.02
1.05
4.07
Effluent
Coefficient
acre -ft/1, 000 pop.
75-1
61.6
Annual Domestic
Effluent
acre-ft/yr
227,779
64,600
291,379
See Reference [27].
-------
TABLE XVII
PROCEDURE OF CALCULATION OF INDUSTRIAL EFFLUENT COEFFICIENTS
Industrial
Category-
Food
Paper
Chemicals
Petroleum
Stone and Glass
Fabricated Metals
Other Manufacturing
Effluent8
Process
mgd
19-9
25-9
17-0
27.4
6.0
6.4
k.6
Total
mgd
67-3
1*7.9
72. 4
197-7
14.0
17.8
Total Annual Effluent
Process
acre ~f t/yr
22,294
29,016
19,045
30,696
6,721
7,170
5,153
Total
acre -f t/yr
75,396
53,366
81,110
221,485
15,684
19,9^1
Process Effluent
Coefficients
Output
$10e
3,071.56
327-17
569.13
1,617-81
295-24
661.40
5,108.78
Coefficients
acre -f t/$10e
7.26
88.7
33-46
18.97
22.76
10.84
1.01
aSee Reference [25].
-------
Thus the data requirements as presented in Table XVIII became available for
application to the water quality management model.
TABLE XVIII
WASTE LOAD AND EFFLUENT DATA REQUIREMENTS
FOR WATER QUALITY MODEL
Category
A.
B.
c.
D.
Domestic
Natural Runoff
Agriculture
Industry
(l) Six Primary Polluters
(2) All Other Manufactures
Effluent
Coefficient
X
X
X
X
X
Waste Load
Coefficient
X
X
X
X
X
A summary of waste load and effluent coefficients for agricultural return
flow, natural runoff, and domestic waste is given in Table XIX.
-------
TABLE XIX
WASTE LOAD AMD EFFLUENT CHARACTERISTICS OF DISCHAROED EFFLUENT
INTO SAN FRANCISCO BAY (EXCEPT INDUSTRIAL)
a
Agriculture
Return Flow
Natural Runoff
Domestic0
(1963)
BOD
ton/yr
-
8,000
121,711
Nitrogen
ton/yr
21.58
2.J
30,785
Phosphates
ton/yr
-
0.11
563
Oil and
Grease
ton/yr
-
2.76
17,907
TSS
t'on/yr
-
752.8
71,5^
Phenol
ton/yr
-
-
-
Gross
Heavy
Metals
ton/yr
-
-
260
Effluent
acre-ft/yr
75-05
262,000
291,379
See Table XIII, this report.
b
See Table XIV, this report.
"See Tables XV and XVI, this report.
VO
-------
VIII. QUALITY STANDARDS FOR SAN FRANCISCO BAY
IDEAL WATER QUALITY OBJECTIVES
FOR THE BAY-DELTA. REGION
The waters of the San Francisco Bay-Delta region provide innumerable
amenities to the population for an entire spectrum of socioeconomic and aesthetic
endeavors. Traditionally such endeavors have usually been considered "beneficial
uses." Increasingly, however, concern about the ultimate habitability of the
environment for both man and other forms of life is forcing the concept of ecological
balance to supersede the narrower notion of a well-ordered list of "beneficial"
attributes that stem from maintaining the natural qualities of estuarine waters.
Ideally the water quality objectives for San Francisco Bay should be to maintain
the ecological balance of the region in such a way as to permit the traditionally
ordered list of "beneficial" uses for the water to be met as a necessary but not
overriding objective. From this viewpoint, water quality management is imbued with
stronger positive overtones than might otherwise be the case while at the same time
giving consideration both to preservation of the environment and the urban-metropolitan
economic structure of society.
PERMISSIBLE LEVELS OF CONCENTRATION
OF POLLUTANTS
The fact that "water quality" has meaning only in terms of beneficial use
makes it difficult to define objectives that approach the ideal and suggests that
the foregoing concept of preservation of any given ecosystem while at the same time
discharging waste effluents may be unattainable. However, just as PHS Drinking Water
Standards permit certain concentrations of impurities which are evidently noninjurious
to human health, so may certain levels of pollutants in estuarine waters be acceptable
to a desirable ecosystem in the Bay; if not indeed to the ecosystem which might exist
in the complete absence of the activities of urban man. The critical problem is to
determine what quality constituents must be regulated and in what concentrations each
is compatible with the water quality objectives enunciated for the Bay, either alone
or in combination with other constituents.
Seven waste constituents are dealt with explicitly in this report. The
permissible levels shown in Table XX were arrived at either by using generally
established standards or, where necessary, by a variety of calculations modified or
adjusted in the light of the judgment of others* who have long worked directly with
problems of quality"of the waters of San Francisco Bay. These values, however,
represent concentrations of dispersed wastes that might currently be permissible in
Bay waters. What might happen to the ecosystem of the Bay between the moment of
discharge of any pollutant and its dispersion throughout the water mass is a matter
of great concern in water quality management, particularly because not all creatures
in the ecosystem are sufficiently mobile to protect themselves by simply avoiding a
particular area of the Bay. On the other hand, there is no real way to attain the
water quality objectives of the Bay except to limit the amounts of any pollutant
which any given discharger may release to the Bay and under what conditions, inasmuch
as the concept of permitting no discharge whatsoever is not yet compatible with the
realities of the economic base of the national standard of living. This means that
the mixing and dispersion concepts of basic hydraulics must be applied to the "Bay"
System in order to relate specific source loads to concentration levels permitted in
the receiving waters. "~
Most notably, Mr. Phillip N. Storrs, Engineering-Science, Inc.
-------
TABLE XX
WATER QUALITY STANDARDS FOR SAN FRANCISCO BAY
BOD
Nitrogen
Phosphates
Oil and Grease
TSS
Phenol
Gross Heavy3
Metals
v>
Effluent
(acre-ft/yr)
Current
Concentrations in
the Bay
Minimum
mg/.e
1.0
0.23
0.1
12.0
Maximum
mg/J
12.0
0.35
1.9
< 1.0
18.0
Recommended
Quality
Objectives
-J .4-1-,-,
in the
Bay
mg/.e
11. OP
2.0
0.1
2.0
12.0
0.01
0.33
7^5,000
leveIs.
See Reference [28].
of 1968 levels or maintaining projected 1980
Q
Based on calculations of rate of oxygen utilization
with a minimum DO (Dissolved Oxygen) level of 5-0
to maintain fish life.
STEADY STATE LINEAR OPTIMAL
DISPERSION MODEL
Mukherjee [k] recognized this tendency to express water quality standards in
terms of the maximum allowable concentration levels in the water resource itself, in
contrast with the traditional limiting of the maximum quantity of each waste constit-
uent that might be discharged to the body of water. To achieve the maximum allowable
concentration resulting from any discharge, Mukherjee applied the technique of linear
programming to basic hydraulic concepts. In describing this approach Mukherjee
noted :
"Specifically, an optimization model is presented (phase 4), based
on a one -dimensional description of the dispersion process which
includes both longitudinal dispersion and advection in an estuary.
It can be used for an estuary where tidal effects are pronounced
and downstream velocity is low, or for a fast -flowing river.
"A steady state model is described in detail for quality conditions
averaged over a tidal cycle . "
-------
From a theoretical viewpoint it may "be considered that waste constituents
either in solution or carried as suspended matter have the same advective velocity
as that of the flowing water. The time rate of advective transport of waste constit-
uent for any section of a water course is given by Q c, where Q is specified as the
volume flow rate for the section and c is the concentration of the waste constituent
expressed as mass per unit volume. If the waste constituent is of the conservative
type then the above concepts present a ready means for estimating the concentration
limits in any section of a water course if the original amounts discharged into the
flowing water are known. For nonconservative waste constituents the time factor
required for them to exhibit the degradable qualities must also be known. As the •
degradable (nonconservative) waste constituents are carried to a given section of the
water course the travel time from the original source of discharge is thus also
important if concentration limits of these types of constituents are to be calculated.
"Dispersion" is the generally-used term which implies the mixing which results
from molecular diffusion along with that due to other factors such as turbulence,
stratification, and other departures from one-dimensional flow due to shear flow.
The resultant mixing in long channels (i.e., water courses) due to the above phenomena
is usually termed "longitudinal dispersion."
With steady state assumptions the diffusion equation for a nonconservative
substance has the form: [4]
AE
_£ - AVC - / (S - k Ac) dx
dx \J
For a conservative substance, the form of the equation is:
AE £=• = Avc - / S dx
dx J
o
where:
A = the average cross-sectional area
v = the average advective velocity
c = the average waste constituent concentration
E = the average longitudinal dispersion coefficient
k = first-order decay rate constant of a degradable constituent
S = net rate of discharge of constituent mass per unit length
x = longitudinal distance along the estuary.
Given the maximum allowable concentration of various waste constituents (set
to achieve some predetermined goals) at different sections of an estuary the task
is to find a scheme which determines the conditions for the maximum permissible
total amount discharged for each constituent. This will yield conditions which
will minimize the total amount of waste constituents that must be removed from waste
discharge sources discharging directly to the receiving waters. This presumably
will minimize treatment costs.
-------
Mathematically:
Find D = max / S(x)dx. S e S
J S(x)dx,
X y
"*
and find the corresponding optimal discharge pattern S(x). S is a set of feasible
discharge patterns S which will satisfy the following equations and any other
constraints which may additionally be imposed:
0 s L (x) s S(x) S U (x) , x e [0 , L]
5 S
and
and
dc (x)
0 S c (x) S Uc(x) , x e [0 , L]
-- = a(x)c (x) - b(x) / S(y)dy - T b(x) , x e [0 , L]
CLA o
where
a(x) = v(x)/E
b(x) = l/A(x)E
T = act\ial amount of the constituent crossing into the
boundary at x = 0
x = longitudinal section [0 , L]
U (x) and
L (x) = the upper and lower bounds on the discharge rate
from basic physical consideration, essentially
to avoid high local concentrations near a specific
outfall.
Alternative goals and corresponding costs may be evaluated by substituting different
sets of values for U (x).
In solving for steady state conditions for conservative constituents (k = 0)
the equation
AE ===- = Avc - / S dx
dx J
can b.e modified to a finite difference form by substituting
- c. c.. + c.
and c. =
-------
and simplifying to
c. = d.c.
1+1 11-1
k=i
with
E.
(i = 1, ..., n-l)
(with n sections)
. (x.
di = si fi
x
An (En - Xn V
E
a =
n E - x v
n n n
where A., E., v., and c. are the values at sampling station i.
The finite difference form of the equation can be further modified to permit
its adaptation to a general programming model in which standard linear programming
techniques can he used to find the optimal solution limits of constituent discharges
as posed in the introductory portion of the report as being one of the two basic
tasks of the research.
The equation system developed in the foregoing paragraphs applies to the
case of a conservative constituent. A modification of the equations to include a
decay term permits the formulation of a linear program for nonconservative
constituents. (See Mukherjee [4] pp. 53 and 83.)
OPTIMAL ANNUAL LEVELS OF DISCHARCE
OF POLLUTANTS INTO THE BAY
The data inputs for the linear programming model outlined in the preceding
section are shown in Tables XXI and XXII. These data were taken from Volume VII of
"A Comprehensive Study of San Francisco Bay" [29]. Optimal solutions were found for
four nonconservative waste constituents and three conservative waste constituents.
Linear programming models were set up and solved for the North Bay and South Bay for
both summer and winter conditions (a total of k rotations) and for all seven
constituents. Weighted annual averages were developed from the solutions of the
linear programs as optimal quality requirements of waste discharge into San Francisco
Bay, presented in Table XXIII, which were then set out as constraints (or right-hand
sides) for a further overall programming model.
-------
TABLE XXI
HYDROLOGICAL INPUT DATA FOR LINEAR PROGRAMMING DISPERSION
MODEL IN NORTH BAY6-
Ho.
1
2
3
k
5
6
7
8
9
10
11
12
13
14
Section
Number
from
Report
16
17
18
19
20
21
22
25
24
25
26
27
28
29/31
Cross -
Sectional
Area
Ai
10s ft2
1.3^
2.52
1.44
1.00
1.15
1-15
0.50
0.32
0.33
0.4l
0.58
0.43
0.23
0.24
Diffusion
Coefficients
E.
ft2/sec
4;800
3,000
4,500
2,400
1,400
900
1,030
1,300
800
500
250
400
700
1,000
Length
of
Sections
X,
103 ft
4.5
12.2
19.6
12-3
22.6
25-7
18.9
32.6
14.1
9-5
14.1
15-6
17-2
26.0
Net Advective Velocities
Vi
Summer
10"3 ft/sec
- 0.149
0.108
0-341
0.4l4
0.914
3-76
7.62
6.38
5-76 .
5-75
6-35
10.9
11-7
Winter
10"3 ft/sec
2.24
4.60
6.65
6.80
5-39
9.90
14.0
11.2
9.44
9-01
7-10
11.2
11-7
See Reference [29], Tables XV - XVIII.
Flow Seaward +
Flow Landward
-------
46
TABLE XXII
HYDROLOdtCAL INPUT DATA FOR LINEAR PROO&MMING DISPERSION
MOEEL IB SOUTH
Section
1
2
3
4
5
6
1
8
9
10
11
12
13
14
15
Cross -
Sectional
Area
Ai
10s ft2
2.6
2-35
1.67
1.80
1-95
1*1
0.68
0.57
0.39
0.22
0.15
0.06
0.04
0.02
0.01
Diffusion
Coefficients
Ei
ft2/ sec
4,800
4,800
550
700
900
700
1,000
1,400
1,000
553
378
307
207
331
1,170
Length
of
Sections
Xi
10s ft
18.0
19-1
19-6
21-3
21.0
27-6
15.6
11-7
20.5
16.0
10.5
7-5
9-3
5^
16.5
Net Advective Velocities
Vi
Summer
10~3 ft/sec
- 1.02
- 1.19
- 1.46
- 1.36
- 1.17
- 0.963
- 1.48
- 1.23
- 1.18
- 1-71
- 0.363
1.22
1.99
6.72
10-3
Winter
10~3 ft/sec
3.98
1.38
1-55
1.17
1.06
0-977
1.60
1.12
1.24
2.45
1-95
4.28
4.45
8.16
12.1
aSee Reference [29], Tables XV -XVIII.
Plow Seaward +
Flow Landward -
-------
TABLE XXIII
OPTIMAL QUALITY REQUIREMENTS OF WASTE
DISCHARCE INTO SAN FRANCISCO BAY
47
Conservative
Constituents
Oil and Grease
TSS
Gross Heavy
Metals
Hone conservative
Constituents8
BOD
Nitrogen
Phosphates
Phenol
Total Effluent
(acre-ft/yr)
Recommended
Quality
Criteria
mg/^
2.0
12.0
0-33
11.0
2.0
0.1
0.01
Optimal Discharge into Bay
Summer
It/ day
195,200
1,170,000
32,600
993,340
469,130
9,930
2,3^6
Winter
Ib/day
193,500
1,160,000
32,300
558,700
263,886
5,588
1,320
Total
tons/yr
31,800
190,000
5,300
127,000
59,895
1,270
300
7^5,000
I)ecay factors used for summer
k = 0.10/day.
= 0.17/day and for winter
The formats of the two linear programming models for conservative and
nonconservative constituents are shown in Table XXIV-
-------
R16
PI
R?
R5
R*
R7
pq
"10
Rll
R12
"13
P14
TABLE XXIV
FORMAT OF STEADY STATE LINEAR PRO(2?AMCNG DISPERSION MODEL
CONSERVATIVE CONSTITUENT
ccccccc <; s s s
upper bound
N
E
F
F
E
E
F
p
E
E
F
E
F
F
L
CCC CCCCCClllllllS
I
-11 *
-* 1 *
-* 1 *
-* 1 *
-* 1 *
-* 1 *
* t A
— * 1 *
-* 1 *
-* 1 *
-* 1 *
*l ^
1 *
-* 1 *
-* 1 *
-* 1 *
****** **********
s
I
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*
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*
*
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*
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* <
* *
?SSsSlllll
>67"90l?'*^
I 1 1 I 1 1 I 1 I
*
* * *
*****
t********
k****'***~**
r*********
NONCONSERVATIVE CONSTITUENT
CCCCCC SSSSSS
ccr, cccccr. IIIIIISSSSSSSSSIIIIIIF
R16
»1
R2
"5
PiS
R9
P10
R14
P15
upper
N
E
E
F
F
E
E
F_
F
E
F
F
F
E
E
E
ll
_*
lllllllllllll
*
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* 1
* 4
* 4
* si
* ^
-*
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c ^
+
-*
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-*-*-*-*-*-*
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bound
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-*-*-*-<
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-*-*-*—*-*-*-*-*-*-*-*_*_*
•*-*-*-*-*-*-*-*-*-*-*—*-*_*_#
*-*-*-*-*-*-*-*-*-*-*-*_*_*_*_«
*
-------
IX. MULTICONSTITUENT "WASTE WATER PROTECTION MODEL
ECONOMIC MULTIPLIER ANALYSIS
The study cf the properties of an input -output model, as opposed to its use
in making projections and predictions, is customarily classified as "structural"
analysis . On the national level such analyses are often undertaken to compare
interindustry relationships of one country with those of another, the comparison
frequently being drawn "between an underdeveloped and a developed nation [JO]. Such
analyses are also undertaken in connection with studies of the nation's "balance of
payments [jl]. On the regional level, however, one of the major functions, if not
the primary function, of structural analysis has been multiplier analysis, the study
of the effects of changes in one part of the economic system upon other components
as well as upon the system as a whole. Almost all such multiplier analyses have
"been confined to regions.
The traditional Keynesian multipliers with their reliance upon the consump-
tion function have always "been defined in terras of highly aggregated variables.
Income is generally treated as a single variable and no attempt is made to dis-
aggregate en a sectoral basis.
"By the nature of macroeconomics, its income -multiplier analysis is
carried out strictly at the most general level. It does not ask
who will produce the extra output when final demand is increased,
cr in which sector of the economy the additional national product
is used. This kind cf analysis is not enough to enable us to find
cut what will happen in individual industries .... This short-
coming of macroanalysis can be eliminated if input-output method
is used instead. For input-output analysis deals with smaller
parts cf the economy than macroeconomics and its emphasis is on
individual sectors, not en the national total."
To illustrate the logic of the input -output multiplier, let us assume that
within a regional economy industry i increases its output due to an upward shift of
final demand for its product. As industry i expands production, it simultaneously
increases its purchases cf inputs. But since industry i's inputs are the outputs
of ether industries, these supplying industries also expand their production and
purchases of inputs in view of increased output demand. Thus a chain-reaction
expansion of production spreads throughout the economy.
Since the element r^. cf the matrix (i - A)"1 reveals the output directly
and indirectly required from industry j to support the delivery of one dollar of
production to final demand by industry i, a dollar change in final demand for the
product of industry i will result in a change in the output of industry j of r. ..
^-3
Denoting v. as the value added per dollar of gross output of industry j, the
total change in gross regional product resulting from the initial increase in the
final demand for the output cf industry i may be calculated as
v
AGRP = , v AX = ; v, ; r,, AY,
/—i J J
i
An exception is the Leontief and Hoffenberg study [52]. The effects of dis-
armament on a regional basis, however, were later studied by Leontief, Morgan,
Polenske, Simpson, and Tower [551-
-------
and the multiplier [55] may be expressed as
r. . AY.
n
)
/_,
M. =
AY.
WASTE CONSTITUENT MULTIPLIER ANALYSES
In a manner similar to the construction of economic multipliers, water
quality or pollutant load multipliers nay also be developed from the input-output
inverse matrix. As & first step it vas necessary to develop for each economic
sector the appropriate pollution coefficient to express the production of the
particular pollutant in terms of commodity production, as discussed in Chapter VII.
Once such coefficients are established, pollutant multipliers can be derived
through application, of the coefficients to the Leontief inverse- For a regional
economy of n sectors and a particular pollutant p, an n-element vector, -^p
(j = 1, ..., n), of load coefficients is constructed. For purposes of computation
the vector is transformed into a diagonal matrix, 1%^, by consecutively entering the
vector's elements along the principal diagonal of an n order matrix (i.e., a matrix
with n rows and n columns) whose off-diagonal elements are zeros. The resulting
matrix is then premultiplied by the transposed Leontief inverse of the input-output
model to yield the "pollutant content" matrix or table of pollutant production. Each
entry, P- •, in the table or matrix P reveals in units of volume the pollutant
associated with, or "pollutant content" of, the output of industry .3 reauired by
industry i for the latter industry to deliver one monetary unit of production to the
final demand sector.
V _ .P ( i = 1, . . . , n)
ij " L ik kj (3=1,
=, •--,
k=i
where !>ik] = [r ] .
The row sums of the above matrix P can then be summed to form pollutant
load multipliers. An element m^ in the vector Mip will represent the direct plus
indirect production of the pollutant p associated with a change in the final demand
of industry i.
„ '» X ^ *i •
-L - - "1 [V -1 L
The change in the final demand for industry i nay for purposes here be
grouped into three categories : l) relatively small changes which do not result
in alterations of the economy's structural relations; 2) larger changes neces-
sitating or caused by changes in the technical structure; and 3) the entrance of
new industries into the region. Pollutant load multipliers as formulated above are
most easily and accurately calculated for the first category in which changes in
final demand are not associated with technological change. Changes in either the
interindustry structural relations or in water -use technology require corresponding
adjustnents in the Table of Technical Coefficients and in the pollutant load
coefficients, respectively.
-------
51
The waste load coefficients developed in Chapter VII can be used, based on
the above concept, in conjunction with the "inverse matrix" of the input-output
table to further develop waste constituent multipliers. The matrices relating
direct and indirect changes of waste constituent output to unit changes in final
demand have been termed "waste constituent interactions" tables. This terminology
was adapted from the Moore and Petersen study of Utah [22] in which they develop
employment or labor coefficients and by a similar procedure develop what they chose
to call an 'employment interactions" table.
WASTE CONSTITUENT INTERACTIONS TABLES
In the appendix are presented the "constituent interactions" tables for
each of the 7 waste constituents and total effluent discharge dealt with in this
report. Each table has been formed by postmultiplying the transposed Leontief
inverse by a matrix of constituent discharges entered on the principal diagonal
of the matrix with all off-diagonal elements being zero. The matrices are shown
as the resulting product of the matrix multiplication and have not been transposed
after the multiplication was carried out.*
The waste constituent multiplier effects of each Bay industry can be read
directly from the tables. For example, if the final demand for the Paper and Allied
Products (sector 5) in the Bay increases by $1 million, the industry in meeting the
increase in final demand will be responsible for discharge (directly and indirectly)
of 3b.tt3 tons of BOD and 1.12 tons of nitrogen annually, etc. For convenience, the
total waste constituent discharges from all industries due to a simultaneous change
of $1 million in final demand in each industry are given in Table XXV.
TABLE XXV
SIMPLE AND WEIGHTED EFFLUENT AMD WASTE MULTIPLIERS
FOR THE SAN FRANCISCO BAY REGION
BOD
Nitrogen
Phosphates
Oil and Grease
TSS
Phenol
Gross Heavy Metals
Effluent
(acre-ft/$10s of
Total Final
Demand)
Simple Multipliers
ton/$106 of
Total Final Demand
5,1^5
2,665
98
352
15, 19^
0.0^1
1^2
26,500
Weighted Multipliers
ton/lfo of Total
Final Demand
M8.37
206.31
5-05
62.36
1,060.88
6.66
7-50
1,599-03
This is emphasized because "in-core" operations with the computer can frequently
lead to printout statements in which it is uncertain that a transpose has actually
resulted.
-------
CRITICAL TIME PERIOD ANALYSIS
The waste constituent multipliers developed in the previous section under
conditions of existing treatment levels can be utilized for projecting future loads
on the environment within the limitations outlined immediately above. This is
accomplished by applying the multipliers to the projected final demand vector.
t V t t
F = ) M? Y.
P /_, iP i
where F is the rate of discharge of effluent constituent or pollutant p through
economic; activity in year t. The accumulation of particular pollutant p within
the environment at a point k years from the present, CL, may now be expressed as
k
_t \ f _t ,Tt ,Tt t i „(>
G= ) I F + N + U -D J+F
P Z_,\P P P P / P
t=i
where
N = the flow of pollutant p from natural runoff (pollutant weight/time),
t
U = the flow of pollutant p emanating from domestic effluent (pollutant
p weight/time),
D = the dispersive flow of pollutant p from the body of water into which
it was expelled (pollutant weight/time),
F = the amount of pollutant p in the waters at time t = 0 (pollutant
p weight).
Once water quality standards for the estuary have been established as
discussed in Chapter VIII and waste load multipliers in terms of the standards have
been constructed, the multipliers can be applied to determine the critical time
period which will elapse under the conditions of existing treatment operations
before the quality standards are violated. Such a critical period T can be
calculated from the following:
T
s =
P
t=i
where s is the quality standard pertaining to pollutant p in terms of pollutant
weight.
An alternative method for calculating critical time periods employing
weighted multipliers rather than simple multipliers was developed and applied in
this report.
As defined previously, the multipliers do not reflect the relative importance
of various sectors with regard to final demands upon their production. A $1 million
increase in the level of final demand for the output of one sector in the Bay region
may represent a rather insignificant increase relative to the same increase in ~
demand for the production of another. The relative importance of each of the
different sectors may therefore be reflected by 'weighting' the multipliers in regard
to the total final demand for their outputs. This can be accomplished by calculating
-------
the total water withdrawal requirements resulting not from a $1 million increase in
final demand but from an increase of one percent. Thus the new multiplier — the
'weighted multiplier' as opposed to the 'simple multiplier' —may "be defined as
follows:
M. =
IP
V
r . . e .
(Y±/100)
A summary of simple and weighted multipliers is given in Table XXV and the
individual sectoral weighted multipliers are given in Table XXVI.
A inethod for calculating critical time periods using weighted multipliers
was developed and used as follows:
.t
Lp
M
100 q
Waste Discharge
Limit of
Constituent p
in year t
[tons]
Present Level
of Waste
Constituent
Discharge
[tons]
Weighted
Multiplier
for
Constituent p
Maximum Possible
Increase in
Discharge of
Pollutant p
[1/100]
This equation is solved for q and then substituted in the following equation
to solve for t'
1 + q = (1 + r)
t1
where r is the expected annual growth rate of final demand.
If t' ^ t, another calculation for a different t is performed. After several
trial and error calculations when t « t' it represents the critical time period at
which the expected discharge of constituent p is forecasted to violate the quality
standards.
The resulting critical time periods for the various constituents due to the
multiplier analysis is given in Table XXVII.
If one or more of the critical time periods calculated for the pollutants
under consideration prove to be sufficiently short to warrant concern, the regional
analyst may then turn to means of influencing the pollution process. The manner in
which such influences my be exerted is the focus of the succeeding sections of
this report.
-------
TABLE XXVI
SECTORAL WEIGHTED EFFLUENT AND WASTE MULTIPLIERS FOR SAN FRANCISCO BAY REGION
Sector
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Total
BOD
ton/l$ F.D?
0.82
0.003
3^-67
204.49
9.21
4.68
85.55
0.68
1.80
24.95
5.59
13-20
8.11
24.62
4l8.37
Nitrogen
ton/l$ F.D.
20.33
0.001
7.34
97.45
0.28
1.75
46.78
0.02
0.33
8.09
2.4l
3.55
8.67
9-31
206.31
Phosphates
ton/l$ F.D.
0.01
-
0.53
1.63
0.23
0.17
0.61
0.003
0.02
0.90
0.08
0.25
O.l4
0.48
5-05
Oil and
Grease
ton/l$ F.D.
0.09
-
2.06
32.72
O.O2
1-57
19-9^
0.006
0.04
1.12
0.81
0.96
0.75
2.28
62.06
TSS
ton/l# F.D.
3-35
0.01
123.10
445.20
2-57
136.88
112.02
1.83
2.26
98.69
13-22
24.37
25.21
7^.17
1,060.88
Phenol
ton/l$ F.D.
0.01
-
0.30
0.12
O.OO2
0.03
5-53
-
0.003
0.09
0.18
0.13
0.11
0.15
G.66
Gross Heavy
Metals
ton/l$ F.D.
0.03
-
1.17
0.47
O.Ol
1.97
1.47
0.004
0.02
1.01
0.15
0.23
0.25
0.72
7-50
Effluent
acre-ft/l$ F.D.
72-57
0.02
182.16
546.39
29.22
65-08
285.22
2.78
12.84
156.91
23.77
52.56
58.43
111.08
1,599-03
F.D. = fin"1 demand.
-------
TABLE XXVII
CRITICAL TIME PERIODS FOR POLLUTANTS DISCHARGED INTO SAN FRANCISCO BAY
WITH 5$ AVERAGE ANNUAL GROWTH RATE FOR FINAL DEMAND3-
^^"^•-^^^ Constituents
Critical"1- — .^^
Year ^^~~~--^^^
Without Flushing
With Flushing
BOD
1970
1978
Nitrogen
1972
1982
Phosphates
1965
1976
Oil and
Grease
> 2000
> 2000
TSS
1976
1990
Phenol
1965
1965
Gross
Heavy
Metals
> 2000
> 2000
Effluent
1971*
> 2000
Based on untreated constituents weighted multipliers; i.e., no treatment of industrial waste
is assumed.
-------
X. LINEAR PROGRAMMING WATER QUALITY
MANAGEMENT MODELS
INTRODUCTION
In order to achieve any specific estuarine water quality level with respect
to the various polluting constituents, either at minimum costs or to ensure maximum
regional economic growth, a linear programming water quality model can be formulated
and applied to set up optional guidelines for water quality management. In this
section of the report such a model is formulated and applied to the San Francisco
Bay. In order to.understand and evaluate its virtues and limitations, however, some
discussion of the theory and mathematics of an optimizing approach may be in order.
AN OPTIMIZING APPROACH TO
WATER QUALITY MANAGEMENT
One of the major objectives of resource management is the optimum utilization
of the resources, i.e., to construct a plan of efficient allocation of the resources
as related to a predetermined social objective. It is obvious that with a limited
availability of a certain resource the continuous depletion of it should follow a
plan of utilization which will ensure best results with respect to a certain objective
within the availability constraints. Capital, foreign exchange, mineral resources,
skilled labor, and water resources are some examples of scarce resources which" are
usually treated by this approach.
While the application of optimization techniques to water resources management
is usually applied where constraints are imposed by the regional availability of
either surface or ground water for beneficial uses (irrigation, municipal, and
industrial), it seems only logical to apply a similar approach to water quality
management where existing water bodies such as streams, lakes, estuaries, and ocean
fronts serve as receiving waters for natural and human wastes. Thus, in terms of
scarcity, quality of such water bodies may be considered as a natural resource useful
for waste disposal purposes. The fact that this resource is scarce and deserves
optimal allocation with respect to type and level of the pollutants discharged into
it was demonstrated by the analysis in the preceding chapter which indicated that
with a moderate growth of economic activity in the near future, and under present
practices of waste treatment of the various waste productive units adjacent to the
Bay, the minimum water quality criteria established for the Bay will soon be
violated.
LINEAR PROGRAMMING AND
MULTISECTOR MODELS
The general linear programming formulation and an iterative technique (the
simplex method) for its solution were devised by George Dantzig in 19^7- Since its
conception linear programming has proved to be a powerful tool for achieving optimal
solutions and has been successfully applied to a broad range of problems extending
from those in welfare and development economics to problems in engineering and
management operations.
Linear programming is a technique for optimization of a system in which all
the relationships are linear. Given a set of m contraint equalities or inequalities
in n variables, linear programming technique can find a set of nonnegative values
of the unknown variables which will maximize or minimize an objective function. The
objective function is an algebraic expression which is to be optimized, i.e., either
maximized or minimized depending on the nature of the problem, and the constraints
56
-------
57
are mathematical statements of limitations beyond which, or outside of•which, an
objective is not feasible. The constraints define geometrically a solution space
or feasible region within which is to be found the best, or optimal value, of the
objective function.
Associated with every linear programming maximization problem is a directly
related minimization problem, and vice versa. Choosing either the maximization or
minimization problem to be designated as the "primal," the remaining problem is
called the "dual." Duality is a mathematical property which relates the pair of
problems one to another in the following way:
1. If the primal has a solution, so does the dual.
2. The optimum solution of the primal is equal to the optimum solution
of the dual.
3- Solution of one problem leads to an immediate solution of the other.
The fundamental association between a maximization problem, arbitrarily
designated as the "primal" problem and its dual minimization problem can be seen
from the standard formulation:
Primal Problem Dual Problem
n m
Maximize Z = ) c. X. Minimize Y = ) b. W.
Z_i J J / i 1 l
j=l 1=1
" m
: } a.. X. §b. Subject to: ) a.; W. s;
L, ij J i /j ij i
j
X S 0 W. S 0
J 1
where
c. is an n x 1 vector of constants,
X. is an n x 1 vector of primal variables,
J
a.. is an m x n matrix of constants,
b. is an m x 1 vector of constants,
W. is an m x 1 vector of dual variables.
The symmetry of the two linear programming problems can be highlighted by
noting from the above formulation the following points :
1. If the primal problem involves maximization, the dual involves
minimization, and vice versa. •
2. If the primal constraints involve s signs, the dual constraints
involve & signs, and vice versa.
-------
3- The constants c- in the primal problem replace the constants
bj^ in the dual problem, and vice versa.
U. In the constraint inequalities the matrix of constants [a.^]
in the dual is the transpose of the matrix [a.^ ] of the
primal.
5- A new set of variables appears in the dual.
6. Neglecting the number of nonnegativity conditions, if there are
n variables and m inequalities in the primal problem, in the
dual there will be m variables and n inequalities.
An economic interpretation of the dual variables is presented subsequently.
The techniques of linear programming and input-output analyses form a basis
for the construction of economic models. Multisector models differ from L-P models,
however, in the same way that positive economics differs from normative economics.
The task of the former branch of economic science is to provide a system of
hypotheses that can be utilized to derive estimations concerning the effects of
changes in economic phenomena. The input-output multisector model can thus be
labelled a positive or explication model. It is a set of functional relationships
designed to reflect economic reality in such a way as to enable the analyst to make
predictions regarding the effects upon the economic system of an impulse occurring
somewhere within the system. The model does not incorporate target variables or
optimizing procedures.
The linear programming model, on the other hand, can be related to the
science of normative economics, that branch of economics which deals not with
"what is" or "what will be" but with "what ought to be." It is a decision rather
than an explication model and differs fundamentally from the input-output model in
that it explicitly introduces goals into the problem formulation. These goals
appear not only in the objective function but are included in the constraints as
well.
Both approaches exclude the possibility of negative outputs evolving from
productive processes and, due to the common assumption of linear economic relation-
ships, both exclude also the possibility of external economies and diseconomies.
The fundamental difference in the basic assumptions of the two models stems
from the fact that economic goals are incorporated into the linear programming
model. The attainment of such goals implies choice among economic alternatives and
it is this element of choice that is distinctive to programming. Even though it is
possible for sophisticated versions of the input-output model to assume different
linear production functions for different levels of output, the model must still
rely for a particular range of output upon the assumption that each commodity is
produced by a single production technique which produces no other commodities. The
linear programming formulation differs, however, in that a) a single commodity
may be the output of several different production processes, and b) a single
production technique may have several outputs.
The linear programming model is thus the more general in that its assumptions
are less restrictive. Moreover, its design is such that it can be applied within
the dec is ion-making process in such a manner as to incorporate the relevant information
yielded by the input-output model. How this conjunctive use of the two techniques
may be accomplished will be evident from the discussion in the following section.
WATER QUALITY MANAOMENT MODELS
The use of optimization techniques for achieving a certain stream or estuary
water quality level at minimum costs under certain economic and engineering constraints
-------
59
applying various pollution control methods is a relatively new development (since
1963). Models of controlling the BOD level in a river or an estuary to maintain a
required DO concentration have been developed by Thomann [36,37]- Dissolved oxygen
was also taken as the controlling parameter in the Delaware Estuary study by Sobel
[38], as well as the study of costs of alternative methods for DO management in the
Potomac River by Davis [39]. These studies applied linear programming model as
their technique for optimization assuming linear relationships among the various
parameters affecting the quality of these water bodies.
"A Linear Programming Water Quality Control Model" by J. Carew and P. H.
McGauhey [3] was developed to illustrate a methodology to minimize the cost of
achieving any water quality goal when the quantities of various wastes which were
predicted by a separate input-output model served as inputs to the multicomponent
water quality control model. The integration of a dispersion model with the model
of water quality control to express preferred quality standards as the maximum
allowable concentration levels in the water resource itself, was performed by S- K>
Mukherjee [t], as discussed in a preceding chapter. The following model attempts
to integrate these separate models into one programming model through the conjunctive
use of input-output and linear programming techniques. With such a programming
model as the basic tool, the various potential extensions and the empirical applica-
tion of the model to the San Francisco Bay water quality management can be explored.
CONSTRAINTS
The water quality management model is, of course, subject to various
constraints which establish the boundary conditions on the process of attaining
the program's objectives. Assuming a regional economy of n sectors which discharge
a total of m pollutants into the adjacent esturaine waters, the first set of
constraints derive from the input-output model. These constraints state that once
a portion of the gross output Xi of a particular sector i has been utilized to
satisfy intermediate demands, AX, upon that sector, there is sufficient output
remaining tc satisfy the final demands, Y.j_, of the region.
X. -
3=1
If for the moment Y^ is interpreted as being a vector of lower "bounds on the
final demands, i.e., a minimum bill of goods for the region, then by use of the
inequality rather than the equality sign, the minimum level of output of industry i
is restricted to the sum of its alternative uses, but the possibility that the
minimal final demands may be exceeded is not excluded.
The second set of inequalities relates to the water quality standards
pertaining to the m pollutants present in the estuarine waters. Given coefficients
eiT) which indicate the amount of pollutant p expelled per dollar of j ' s gross output,
the total flow of the pollutant into the waters under conditions of no treatment is
e- X-. Thus the second set of inequalities states that the total flow of pollutant
PJemenating from all n sectors of the economy must not exceed the quality standard
Sp established for this pollutant as the maximum computed by the discharge level
model noted in Chapter VIII.
Z
e x. S S (p = 1, ..., m)
JP J P
Waste treatment considerations are introduced into this basic formulation of the
model at a later stage .
-------
6o
OBJECTIVE FUNCTION
Subject to the foregoing three basic constraints, the objective of the
program might "be to select that combination of outputs and treatment levels that
would maximize some measure of regional welfare. With this objective one might
choose to maximize an approximate measure of gross regional product
Z (v X ) where v. is the value added of sector i.
3=1 3 J 3
In a fully employed economy, limitations put on the assimilative capacity
of the Bay will result in limitations on the potential growth of the regional
economy. Therefore, the objective of the program would be to give rise to those
productive sectors with minimum waste contribution while simultaneously satisfying
a set of final demands and the established pollutant standards.
In order to set a limit on the output levels of the more productive sectors
of the economy, an upper estimate of final demand levels would also be incorporated.
The addition of an upper limit iV would imply that no matter how large a particular
industry's relative contribution to regional -value added, the industry would not be
allowed to produce beyond the reasonable limits of effective demand for its product.
Such an upper bound on each sector would of course reflect the capacity constraints
of these sectors.
Y1 S X. - ) a.. X. S Y? (i = 1, ..., n) .
1 i L-i 2-J 0 i
Thus the basic version of the programming model focuses upon output restric-
tions as a result of maximum allowable levels of waste discharge to satisfy the
quality standard of the Bay.
The program is thus to:
n
Maximize Z = / v. X
L I 0 J
Subject to Y^ fe 2_, (5ij ' aij) Xj - *1 (i = 1, ..-, n)
3=1
n
e. X. •£ S (p = i, ••-,
JP J P
=1
XJ
J=l
X. a 0
(j
-------
61
or in matrix notation,
Maximize Z = v X
Subject to YU (I - A) x s Y1 (1=1,..., n)
X § Sp (p = 1, ..., m)
k X s
X £ 0
where
[a..] = matrix of technical coefficients a-y which represent the
amount of production from industry i required to support
one dollar of output at industry j ($/time/$/time)
(dimension n x n),
X. = a vector of gross output of industry j ($/time)(dimension n),
J
Y. = a vector of final demand for output of industry ($/time)
(dimension n),
Y. = upper bound to final demand,
Y7 = lower bound to final demand,
s = a vector of maximum allowable amount of pollutant p discharged
^ into the estuary (pollutant weight/time) (dimension m),
[e. ] = a matrix of amount of effluent component or pollutant p
JP produced per unit of gross output (pollutant weight/$)
(dimension n x m),
v. = a vector of value added coefficients ($/$/time) (dimension n),
j
k. = a vector of effluent produced per unit of gross output
J (effluent volume/time/$/time) (dimension n),
W = maximum allowable amount of total effluent to be discharged
into the Bay (effluent volume/time) (dimension l),
[6..] = Kronecker delta (identity matrix).
-^
-------
62
problem. While the primal problem can be regarded generally as an allocation
problem, the dual can be considered as a pricing problem. The relation between
the primal and its dual conforms to the relation between the system of allocation
of limited resources to the system of their pricing or imputed values . The dual
minimization problem as related to the above primal will be formulated as follows :
n
Minimize D = } ( itU - «L )
Z_j \ i i /
m
V
JL . • / / k.)
i Z-i P p
1=1 p=l
n m
Subject to }(jtU-:n:L)(o..-a..)+)
Z_i \ i i / \ ij ij /Li
p=i
or in matrix notation.
Minimize D=lirU-nLJY+yS+pW
Subject to In - n )(l-A)+re+pkav
where
it = the vector of "shadow prices" associated with the final
constraints. They represent the marginal increase in the
objective function due to unit change in the level of
final demand.
it = is the vector of "shadow prices" associated with the upper
bound constraints on final demand (it^ 5 0).
jr = is the vector of "shadow prices" associated with the lower
bound constraints on final demand ($/time)( dimension n)
7 = the vector of "shadow prices" associated with the maximum
discharge level of pollutants and thus these prices represent
the marginal change in the objective function due to a unit
change in the allowable discharge levels . Thus they may be
considered to be the imputed value to the treatment process,
i.e., the opportunity costs of removing a unit of the
constituent through treatment ($/pollutant weight) (dimension
m) (y § 0).
p = a scalar; the "shadow price" associated with the maximum
discharge level of effluent ($/volume of effluent)
(p ^ 0).
-------
63
Thus the dual problem may "be interpreted as a problem of minimizing the sum
of the imputed value of delivering a required amount to final demand and the imputed
values of treatment subject to a constraint which states that the accounting cost of
treatment for each sector minus the accounting value of exceeding the lower bound of
final demand (or plus the accounting value of not exceeding the upper bound of final
demand) should be greater or equal to the value added of this sector.
THE UNTREATED EFFLUENT
PROGRAMMING MODEL
Primal Variables, Optimal Solutions,
and Infeasible Solutions
The basic programming model as formulated in the preceding section may
actually be described as an untreated effluent programming model. Results of such
models, i.e., the level of the primal and dual variables, will reflect the effect
of the waste discharge constraints on the growth of the economy in general, and on
the growth of the various sectors in particular. In contrast to the pollution
multiplier model discussed in Chapter IX, which forecasts the critical time periods
under assumption of an uncontrolled nonefficient growth pattern of the regional
economy, the untreated effluent programming model in which still no treatment is
assumed will result in an efficient growth pattern which will maximize the gross
regional product and satisfy the projected requirements of deliveries to final
demand as well as the water quality constraints. This growth pattern will be
followed until the solution to the model becomes infeasible, i.e., again forecasting
a critical period in time at which the quality standards with respect to a certain
pollutant will be violated unless treatment processes are employed. Since the
solution to the programming model represents a growth of a controlled economy, it
may be expected that these critical time periods will occur at a later stage than
in the uncontrolled economy as expressed by the pollutant multiplier model. Thus
restrictions on the growth of certain sectors may postpone the critical time period
when the quality standards will be violated without the employment of treatment.
The levels of the sectoral outputs are the values of the primal variables at an
optimal solution to the programming model.
Empirical Aspects of the Input Data
The untreated effluent programming model as formulated requires the fol-
lowing input data for empirical application:
1. An input-output table of the San Francisco Bay Area (9 counties) of
l4 aggregated sectors as developed in Chapter VI•
2. A set of industrial waste load coefficients for 7 types of constituents,
the production, as polluting weight per unit of output as developed
in Chapter VII-
3. A vector of effluent coefficients specifying the level of effluent
for each sector per unit of output.
4. Projections of final demand requirements throughout the period
1963-1980 under assumptions of 3 percent and 6 percent annual
growth. These vectors will serve, respectively, as lower and
upper bounds on the final demand requirements. The final demand
data for selected years are given in Tables VII and VIII.
5- In this model it is assumed that the quality of water in the Bay
is expressed by the resulting concentration of certain polluting
constituents such as BOD, nitrogen, phosphates, TSS, phenol, etc.,
-------
which are discharged into the Bay. The transformation of maximum
allowable concentration levels into maximum allowable discharge
levels has been done using the Multicomponent Optimal Quality
Control Model [k], as described in Chapter VIII.
From these maximum discharge levels, time dependent amounts of
polluting constituents which are expected to exist in domestic
effluent and natural runoff as given in Table XVIII are subtracted
to result in the right-hand side values to the pollution constraints
of the model as given in Table XXIX.
Calculation of future domestic levels of pollutants as given in
Table XXVIII is based on population projections as given in Table
II, projections of domestic waste load coefficients as given in
Table XXX, and on the current levels of treatment of domestic
effluent at which 60 percent of the total volume is being treated
by primary process while 35 percent is treated by primary and
secondary processes. It is also assumed that the percent of
secondary treatment of the domestic effluent will increase in the
future to at least reach 50 percent of the total volume. The
forecasted amounts of pollutants from domestic effluent with three
different levels of primary and secondary treatment are given in
Table XXXI.
These values of right-hand side for the pollution constraints were
calculated for two alternative flow regimes from the Delta into
the Bay for flushing purposes — Alternative A - no flushing;
Alternative B -with flushing of 2,000 cfs from the Delta which
will result in relaxing the minimum quality standards by 25 percent
(the annual flushing volume adds 25 percent of fresh water to the
average volume of the Bay); i.e., the allowable amounts to be
discharged will be 125 percent of the amounts in Alternative A.
6. The volume of effluent allowable to be discharged into the Bay
as derived in Chapter VIII.
7- A vector of value added coefficients as given in the table of
direct coefficients of the I/O model of the Bay — Table V-
Once these data were derived and introduced into the programming model,
a linear programming computer program IBM MPS/360 was employed to arrive at the
optimal solutions of the various alternatives as shown by the accompanying format.
Results of the Untreated Effluent
Programming Model
Primal Problem. The results of the optimal solution to the primal problem
of the programming model may be analyzed with respect to the following aspects:
1. The value of the time dependent objective function.
2- The level of the primal variables.
3. Critical time periods.
4. The effect of flushing from the Delta on the solution.
5. The expected level of pollutant discharge into the Bay.
The numerical results of these items are given in Tables XXXII and XXXIII•
-------
TABLE XXVIII
LEVELS OF UNTREATED DOMESTIC WASTE
' " — — - ^____^ Year
Pollutant ^~-- ~~^^_^
Effluent; acre-ft/yr
BOD, ton/yr
TSS, ton/yr
Nitrogen, ton/yr
Phosphates, ton/yr
Oil and Grease, ton/yr
a ,
Gross Heavy Metals, ton/yr
1965
291,379
121,711
71,544
30,785
563
17,907
260
1970
332,169
138,750
81,559
35,095
642
20,4l4
296
1975
372,960
155,788
91,575
39,405
721
22,921
333
1980
513,817
213,497
121,990
51,850
985
25,432
369
1990
602,051
259,019
148,000
62,905
1,195
30,855
448
2000
690,352
348,149
203,109
85,565
1,757
36,282
527
Includes floatable materials.
-------
PROCEDURE OF CALCULATIONS OF MOHT-HMID SIDS VALUES FOE THE
WATEE QUALITY MANAGEMENT PROGRAMMING MODEL
Year
1965
1970
1975
I960
1990
Criteria
Dottier, tic
Runoff
Indus t •
R.H.S.
Criteria
Domestic
Runoff
R.H.S.
Criteria
Domestic
Runoff
R.H.S.
Criteria
Ilomectir
Runoff
R.H.S.
Criteria
Domestic
Rur.c ff
R.H.D.
BOD
Mean
Volume
127,000
59, '-1*0
8,000
1*0,000
59,3''o
127,000
£.7,970
8,000
51,030
127,000
76,51,0
8,000
1,2,660
127,000
89/70
8,000
29,530
1,':7,OQO
108,780
8 , 000
10,22')
.'", X M.V.
tons/yr
160,000
59,6Uo
8,000
1-0,000
92, 360
160,000
(-7,970
8,000
81,, 030
160,000
76 , ;-l*0
8,000
75,660
If 0, 000
89,670
8,000
52, ',30
160,000
108,780
8,000
>.•„:«>
Nitrm-cn
Mean
Volu™
ton=/yr
59,895
21,000
2,500
27,000
36,595
59,895
2U, 000
?, 500
33,595
59,895
27,000
2,300
30,595
59,395
"1*,099
2,500
•"3,595
59,395
58,000
' , '',00
19,595
.25 ,: M.V.
75, ''67
21,000
2,300
27,000
52,l.'7
75,1.67
2l*,000
1.9,095
75,'"7
27,000
2,500
45,095
75, '"-7
5)1,000
2,500
39,095
75,'"' 7
58,000
2, ',00
-,,v,
Pho1;ph,tn,
Mean
Volume
tono/yr
1,?70
563
no
530
597
1,270
61,2
110
518
1,270
7:.'i
110
1,59
1,2'|'0
985
110
175
1,270
1,195
]JO
.25 >; M.V.
tonc/yr
1,580
565
110
530
907
1,580
61|?
110
828
1,580
721
110
719
1,580
985
110
1.85
1 , 580
1,195
110
275
fill and Gre'Jf.c
Mfiin
Volume
tonr,/yr
31,800
•>,!*>
-
( ,500
18,750
31,800
6,,ltOO
25,llOO
51,800
7,500
2'., 500
'a, 800
8,100
2 ' ,700
59,895
9,800
50,095
.25 r M.V.
tonc/yr
39,800
5,790
-
6, '00
?.!•, 7 50
59,800
6,1.00
55,1*00
59,800
7, '-00
-
32,500
39,800
8,100
-
•U,700
75, 1.67
9,800
• ,'Vj
T
Mean
Volume
tono/yr
190, ooo
21,1*00
760
117,000
50,8)10
190,000
24,500
16,::
190, ')00
-
<-'/, '00
7^0
161,900
190,000
50,500
760
158,800
'59,000
57,000
760
5"i,:-i>o
SE
.25 -t M.V.
ton.-/yr
253,000
?1,!*00
760
117,000
98,840
258,000
2tt , 500
760
232,800
;",8,ooo
i
27,
76.0
229,900
:.:8,ooo
',0,500
7(:-0
226,800
h 55 , ooo
',7,000
'/• 0
1.17, :'io
Volume
ton,/yr
'00
-
-
6,90
-
500
-
500
500
-
"00
500
-
-
500
"00
-
',00
.;>5 x M.V.
tonc/yr
378
-
-
690
-
378
-
378
378
-
.78
378
-
:78
378
-
-
573
Gross Heavy Metals
Volume
ton.i/yr
5,500
100
"
780
Il.teo
5,300
115
5,lB5
5,500
I'O
~
5,170
5,500
I'uh
-
5,156
5,300
156
-
5,1:1.
1.25 x M.V.
tons/yr
6,600
IOC
780
5,720
6,600
115
6;85
6,600
150
~
6,470
6,600
11,!,
6,1,56
6/.00
156
"
•o,H*
Effluent
Volume
= -f t/yr
lU5 , 000
512,000
88,000
71,5,000
165,000
580,000
71.5,000
185,000
560,000
71*5,000
205,000
51.0,000
71,5, ooo
250,000
1.95,000
1.25 X M.V.
acre -f t/yr
11*5,000
512,000
273,000
950,000
165,000
765,000
9 JO, 000
185,000
7^5,000
930,000
205,000
725,000
930,000
250,000
685,000
-------
TABLE XXX
PROJECTION OF DOMESTIC WASTE LOAD COEFFICIENTS6
Year
1965
1980-
1990
2000
Domestic Waste Load
Ib/capita/day
ton/103 pop./yr
Domestic Waste Load
Ib/eapita/day
ton/103 pop./yr
Domestic Waste Load
Ib/capita/day
ton/103 pop./yr
BOD
0.17
28.07
0.21
34.67
0.24
39.63
Nitrogen
0.043
7-10
0.051
8.42
0.059
9-74
Phosphate
0.008
0.13
0.010
0.16
0.012
0.20
TSS
0.10
16.5
0.12
19.81
0.14
23.12
Oil and
Grease
0.025
4.13
0.025
4.13
0.025
4.13
Gross
Heavy
Metals
0.004
0.06
0.004
0.06
o.oo4
0.06
See Reference [10].
-------
TABLE XXXI
DOMESTIC EFFLUENT DISCHARGE INTO THE SAN FRANCISCO BAY UNDER VARIOUS
COMBINATIONS OF LEVELS OF PRIMARY AND SECONDARY TREATMENT51
Constituent
BOD
TSS
Nitrogen
Gross Heavy
Metals
Level of Treatment
% of Total Effluent
Primary
60
75
90
60
75
90
60
75
90
60
75
90
60
75
90
Secondary
35
50
65
35
50
65
35
50
65
35
50
65
35
50
- 65
Domestic Pollutants Discharged (tons/yr)
1965
59,638
51,118
40,164
21,463
17,886
15,739
27,552
26,167
24,462
101
91
83
503
478
450
1970
67,987
58,275
45,787
24,467
20,389
17,930
31,410
29,838
28,076
115
103
95
574
545
513
1975
76,330
65,430
51,410
27,472
22,893
20,130
35,267
33,494
31,524
130
116
106
645
612
577
1980
104,613
89,668
70,454
36,591
30,497
26,840
46,405
44,072
41, 480
144
129
118
88l
837
788
1990
108,787
85,476
37,000
32,560
56,299
53,469
50,324
175
156
143
1,069
1,015
956
aTreatment Efficiencies -Percent Removal (see Reference [10]).
Treatment I BOD I Nitrogen I Phosphates
TSS
Gross Heavy
Metals
Primary
Secondary
25-35
85-95
30-40
JO-40
60-70
85-95
50-60
75-85
-------
Rl
R2
R3
R4
P5
R6
R7
R8
R9
RIO
Rll
R12
R13
R14
R15
R16
R17
R18
R19
R20
R21
R22
R23
FORMAT OF UNTREATED EFFLUENT LINEAR FRO
-------
TABLE XXXII
RESULTS OF UWTREA1ED EFFLUENT PROGRAMMING MODEL
Year
1963
1970
1975
1980
Contribution
of Delta
+
+
Gross Regional
Product
15,1196.57
22,576.06
22,658.99
29,712.05
29,01 14
33,1814.1.8
" c fit: j,o
'J?3 jfO- '
Constituents
b
Allowable Level
. /
tons/yr
Actual Level of
Discharge
tons/yr
Shadow Price
$103/ton
Allowable Levelb
, /
tons/yr
Actual Level of
Discharge
tons/yr
Shadow Price
$103/ton
Allowable Levelb
+ /
tons/yr
Actual Level of
Discharge
tons/yr
Shadow Price
$103/ton
Allowable Levelb
tons/yr
Actual Level of
Discharge
tons/yr
Shadow Price
$103/ton
Allowable Levelb
tons/yr
Actual Level of
Discharge
tons/yr
Shadow Price
$103/ton
Allowable Levelb
tons/yr
Actual Level of
Discharge
tons/yr
Shadow Price
$103/ton
Allowable Levelb
tons/yr
Actual Level of
Discharge
tcns/yr
Shadow Price
$103/ton
BOD
59,360.00
36,127-20
51,030.00
51,050.00
70.20
814,050.00
52,575-35
60,000.00
60,000.00
51-13
75,660.00
63,006.50
65,000.00
65,000.00
861 . 51
70,000.00
63,132.29
Nitrogen
36,595-00
18, -55. 80
53,595-00
26,001-38
149,095.00
26,739-73
50,595.00
30,595-00
25.02
1.5,095.00
32,326.19
32,1471.1.3
32,1.71.113
6.27
32,3'iO.OO
31,899.26
Phosphate;;
597.00
5714.00
918.00
822.95
-
828.00
828 . oo
,2
1,039.00
1,009.70
-
1,000.00
1,000.00
0 . ,
o, 64fo • 52
1,235-00
1,065.9''
-
1,275.00
1,028.70
Oil und
Grea se
26,050.00
5,738.55
25,1.00.00
8,165-08
-
33, ^OO. 00
8,352.1.1
21., 500. 00
9,1.22.82
-
52,500.00
9,6214.55
25,700.00
9,722.61
51,700.00
9,919-17
TSS
536,81.0.00
186,871.145
-.29,600.00
272,5146.00
-
1.11,000.00
268,081.00
323,800.00
323,800.00
3.26
14 Oh, 000. 00
2914,778.514
327,100.00
326,9148.1.3
-
317,000.00
317,000.00
31.8.17
Pheno.
350.00
2147.95
'62.28
362.28
297.62
1400.00
361.83
1175.09
1.75.09
155.20
500.00
1.69 .88
1,75.1.8
1.75.1.8
3li, 792. 00
1.80.00
1.1.6.65
Gror;c Heavy
Metals
5,200.00
I,9li7.76
5,185.00
2,855.87
-
6,1.85.00
2,769.21.
5,170.00
3,1189.02
-
6,lt70.00
2,950.1.8
5,156.00
3,307.80
-
6,1.56.00
3,1.19.1.1
Total
Effluent
(acre-ft)
600,000.00
178,989.1.6
580,000.00
257,'*92-72
-
765,000.00
258,509.08
560,000.00
312,238.32
-
71.5,000.00
307,11.3.28
51.0,000.00
3111,1.52.29
. -
725,000.00
326,287.20
Plus ( + ) indicates Alternative B —with flushing from Delta.
So-rie figures are adjusted upvards tc arrive at a feasible solution.
-------
TABLE XXXIII
OPTIMAL ACTIVITY LEVELS - PRIMAL VARIABLES SOLUTION TO THE
UNTREATED EFFLUENT PROGRAMMING MODEL
Year
Gross
Regional
Product
$109
Output
$106
Sectors
1
2
3
If
5
6
7
8
9
10
11
12
13
1U
1963
15A96.57
5if8.3if
82.73
3,553-10
3,138.29
335-29
581.85
1,635-53
303.08
679-89
5,227-9^
2,897.67
3,620.68
lf,2lf5-52
5,0if0.20
1970
22,576.06
Without
Flushing
771-23
122.02
5,202.13
^,363-85
if76Ao
81*9-97
2^23.99
iflfO.73
985 A9
7,651-19
if, 229. 05
5,293-08
6, 206 A3
7,350.19
22,658.00
With
Flushing
799-3^
118. 2if
5,20if.20
if, 606. 88
if 80- 93
8149.38
2,32^.00
if If 2. If 8
990 . 10
7,659-56
if, 237. 18
5,303.07
6,210.92
7,366.72
1975
29,712.05
Without
Flushing
895-13
150.65
6,989A^
lt,76if.l3
606-73
1,115-86
2,925-30
582.79
1,291.22
10,2lflA5
5,583.36
7,065.98
8,311-71
9,782.06
29, 8lifA if
With
Flushing
971-65
132.77
6, 992- if 2
5,303 A9
6lif-91
l,105.6if
2,46lf.05
585-91
1,297.1^
10,253-83
5,579-89
7,o81f.l2
8,315-23
9,807A9
1980
33,l8ifA8
Without
Flushing
95^-23
liflf.08
6,126.19
5,232.58
638.15
I,0lf8.97
2,773-91
535-63
1,269-85
9,^78.83
5,36^.28
8,831.06
10,750.67
12,27if.7l
35,676A2
With
Flushing
963-06
153-57
7,966.98
5,250.67
669-58
1,116.92
2,865-55
6U6.91
1,360.29
9,855- if5
7,013-36
9,o6lf-79
10,889.89
12,605-05
-------
72
It is clear from these results that while a gradual increase in the gross
regional product is expected to take place throughout the planning horizon of the
model, the effect of Delta inflow to the Bay remains small, at least until 1980
when the difference in the values of GEP of the two alternatives starts to increase.
The small effect of the flushing of the Bay is probably due to the fact that a low
growth rate of 6 percent was specified as the upper bound on the growth of final
demand, thus having for most sectors in most of the analyzed time periods in both
alternatives an optimal solution which satisfies these upper bounds. With a higher
expected growth rate, the differences between the two alternatives probably would
be greater.
This conclusion is also valid for the values of the primal variables — the
optimum sectoral outputs. Yet due to flushing effect, a slight variation in the
distribution of sectoral outputs is realized and thus for a few sectors the optimal
output without flushing from the Delta may be slightly higher at the expense of
those heavy polluting sectors whose output drops to a considerable extent (especially
sectors 6 and 7)•
While the critical time periods as presented in Table XXXIV can be observed
for those constituents which result in shadow prices at a certain time in the future,
i.e., the allowable amount to be discharged is exhausted, it is appropriate to remark
that for some constituents at certain future years, an infeasible solution resulted
from the specified right-hand side levels. To overcome these problems an upward
adjustment was made so as to arrive at optimal feasible solution for the whole
planning horizon. Thus the results as presented incorporate some violations of the
quality standards set up in the preceding chapters. The extent of these violations
can be evaluated by comparing the actual amounts of each type of pollutant discharged,
as given in Table XXXII, with the original allowable levels of discharge as derived
in this chapter and presented in Table XXIX.
Thus the primal solution to the untreated effluent programming model points
out the necessity of immediately undertaking treatment of the industrial waste in
order to maintain the quality standards set up for the Bay. The argument for
extensive treatment will be strengthened by the analysis of the dual problem of
the model which reveals the pricing aspects of potential treatment procedure.
The Dual Problem — Shadow Prices . Theoretically there may be three sets of
dual variables in an optimal solution to the programming problem, as presented in
a preceding section — it, y, p. Yet, while the dual feasibility of the linear
programming problems has to be satisified by the nonnegativity of these variables,
usually only one of the resources contraints in the primal problem will be tight
and thus only one dual variable will be at positive level. This positive level is
a result of all the interrelationships of the specified problem and its related
data and thus its economic, interpretation has to be considered within the frame of
the problem solved. Thus a shadow price for BOD in 1975 °f 51-13 $/ton as given in
Table XXXII has to be considered with all other data and results of the problem and
is meaningful only with all the specified constraints.
With this remark in mind it is possible to interpret the dual variable y as
that marginal change in the objective function with a change of unit in the resource.
This can be done by partially differentiating the objective function of the dual with
respect to the allowable levels of discharge:
-— = y $/tons of pollutant
and
*r- = p $/acre-ft of effluent
-------
TABLE XXXIV
CRITICAL TIME PERIODS RESULTING FROM UNTREATED
EFFLUENT - PROGRAMMING MODEL
^^~~^Cons t itue nt s
Alternative ^^^^
A - Without Flushing
B - With Flushing
BOD
1973
-
Nitrogen
1975
-
Phosphates
1965-67
1967-70
Oil
and
Grease
-
-
TSS
1980
-
Phenol
1968
1970
Gross
Heavy
Metals
-
-
Effluent
-
-
-------
The resulting shadow prices can "be interpreted as the accounting price of
treatment. Thus comparing the shadow prices given in Table XXXV for some of the
limiting constituents with current costs of treatment processes, it is obvious that
the value of treatment to the regional economy is much higher than the actual costs
as shown in Table XXXV.
MULTIPHASE TREATMENT PROGRAMMING MODELS
Formulation of the Programming Models
Because the results of the preceding programming model lead to the conclusion
that treatment of industrial and domestic waste is essential to satisfy the water
quality standards of the Bay, an extension of the basic programming model was
formulated which incorporated the various treatment aspects of industrial waste.
Since each industry may have available to it several treatment processes,
each with different unit costs, or at least the various processes can be aggregated
into three major categories — primary, secondary, and tertiary treatment - the
modified treatment model has to account for the possibility of adoption of several
treatment phases.
Thus assuming there are h* processes available to industry j, the basic
model may be restated as one whose objective is still the same as before, i.e., to
maximize gross regional product net of treatment costs :
Maximize
subject to the economic growth constraints
(i = 1, ..., n)
and the water quality constraints
h=i
(p = 1, ..., m)
Total Amount of
Pollutant p
from all
Sectors j
tons
$10
Total Amount of
Pollutant p
Removed by all
Treatment
Process h* of
all Sectors j
tons
acre-ft
x acre-ft s
Allowable Amount
of Pollutant
p to be
Discharged
into Bay
tons
-------
TABLE XXXV
COMPARISON OF SHADOW PRICES FOR TREATMENT OF POLLUTANT WITH ACTUAL TREATMENT COSTS
Constituent ->
Shadow Price, 10a x $/ton ->
Sector
Agriculture 1
Food 4
Paper 5
Petroleum 6
Stone 7
Chemicals 8
Fabricated Metals 9
All Others 10
Domestic
BOD
50 -70
Nitrogen
25
Phosphates
4,300 - 8,600
Phenol
300 - 34,800
TSS
1* - 348
Unit Cost of Treatment (dollars per ton)
Pri-
mary
-
37
100
100
260
820
2,020
-
77
Secon-
dary
-
75
20k
20k
530
1,680
5,8Uo
-
158
Terti-
ary
-
119
321.
32l+
84o
2,630
6,100
-
251
Pri-
mary
-
-
-
-
-
-
-
-
-
Secon-
dary
2JO
565
8,000
360
49,500
2,900
3,4oo
1,310
636
Terti-
ary
364
900
12,700
575
79,000
U, 600
5,1*00
2,080
1,010
Pri-
mary
-
-
-
-
-
-
-
-
-
Secon-
dary
-
12,000
8,100
31,000
-
87,200
-
6,600
34,1*86
Terti-
ary
-
19, ooo
13,000
1*9,200
-
1*3,000
-
10,1*00
54,772
Pri-
mary
-
-
-
-
-
-
-
-
-
Secon-
dary
-
-
-
3,iOOa
1-5 x 106
1.1 x 10s
-
-
-
Terti-
ary
-
-
-
4,900a
2.56 x 103
1.74 x 10s
-
-
-
Pri-
mary
-
16
640
118
48
14
-
52
132
Secon-
dary
-
3^
1,300
240
99
28
-
107
270
Terti-
ary
-
53
2,070
380
157
45
-
169
428
These figures may "be overly conservative since special treatment processes for phenol removal were not considered.
-------
16
A third set of constraints is added by the logic . that the maximum amount of
effluent from sector j that will be treated cannot exceed the amount of effluent
generated "by this sector. It is assumed that this maximum amount of effluent
generated "by sector j will first be treated by the least -cost treatment process
available to the industry which, of course, depends on the pollutant composition of
its effluent . The amount of sectoral effluent treated by an advanced phase of
treatment with higher unit costs cannot exceed the amount generated by this sector
and treated by a lower ranking treatment process; thus
Vjhf Sej Xj
and (h_ = the least treatment process
available to sector j)
(h-i = the treatment process
next to process h)
(h = i, ..., h )
J
where
W = amount of effluent from industry j to be treated by process
J h (effluent volume in acre-ft).
C., = unit cost of treatment process h to industry j ($ value/effluent
^ volume in $/acre-ft).
g = amount of pollutant p removed by treatment process h from one unit
JP of industry j's effluent (pollutant weight/effluent volume in
tons/acre-ft).
The number of sectors in the above formulation in which two or more treatment
processes are adopted will not exceed m [kQ]. As Carew and Van Slyke have shown
the dual of such a multistage model is useful in the evaluation from the regional
viewpoint of the adoption of new treatment techniques.
A Model With natural Runoff and
Domestic Waste Water Considerations
Thus far only waste resulting directly from economic activities has been
explicitly considered. Additionally, however, estuarine waters customarily receive
pollutant flows from natural runoff waters and from discharge of domestic wastes.
The entry into the estuary of a major portion of runoff is through rivers
and streams thus making control of this effluent source quite difficult. For these
reasons natural runoff is not incorporated into the regional control system
explicitly. It is accounted for in the models through reduction of the maximum
allowable amount of the pollutant p, Sp, by the amount of the pollutant introduced
into the system via this particular source.
Control of natural runoff, as well as other relatively minor sources of
pollutants such as sanitary landfill, is likely to be most effectively undertaken
through regulations on a local basis.* In the case of the one pollutant from which
runoff say be a major source, pesticides, control may have to be instituted on a
federal level through restrictions on production and use -
*
Por examples of estimations of recreational benefits derived from improved water
quality see the articles t>y J. E. Stevens and by P. Davidson et al_., see Reference [39]
-------
77
Domestic waste water flows emanating from sanitary fixtures, garbage
disposals, washers, air conditioners, etc. constitute a prime source of some
pollutant measures such as BOD ("biological oxygen demand) and nitrogen. The
inclusion of domestic waste flows within the model can "be undertaken through the
addition of a domestic sector to the constraints. Letting M_ equal the predicted
flow pollutant p from domestic waste M "based primarily upon population estimates,
dp^, the amount of pollutant p removed through the treatment process h of one unit
of domestic waste, and M^ the amount of domestic effluent to "be treated "by process
h, the water quality constraint "becomes
h,
n m
V V 1
L L 1
j=l p=l
f
i ejp xj - E SJPI
\ h=l
W.,
i Jh
-d,M.I«S+M (p = 1, ..., m)
ph In. p p
W., ^ e. X.
Jh J J
M s M (for h = the least cost
treatment process avail-
able for domestic treatment)
and
and the objective function of this model will thus "be
where Cdh is the cost of treating one unit of domestic waste water by process h; and
M is the total domestic untreated effluent.
Treatment Cost Minimization Models
It has been demonstrated elsewhere [59], in a case study of the Delaware
Estuary, that direct and indirect effects on the region's economy resulting from
output limitations upon the production of the Paper, Chemical, and Petroleum sectors
to maintain the established water quality standards are relatively quite large
compared to the additional treatment costs necessary to accomplish the same objectives,
In light of these results a model which focuses not upon output restrictions but
only upon treatment costs and operations, may be formulated as another version of
the two treatment models which will differ only in the objective function which will
be the minimization of treatment costs.
-------
78
The program is thus to
h
n j
V V
Minimize Z = / / C., W.,
Li Li ah jh
j=i h=i
subject to the same set of constraints.
With the introduction of domestic effluent this version will be formulated
tc
h h
J 3
V V n W , V
) C., W.
Li Jh j
Minimize ) >C., W.+ /C M, ,
Li L jh jh Lt dh h
j=i h=x h=x
again subject to the same set of constraints.
Summary of the Four Versions of
Multiphase Treatment Models
In light of the foregoing discussion, the multiphase treatment programming
model will be employed in the following four versions :
A —Maximization of gross regional product models
(A-l) Without explicit consideration of domestic effluent
Maximize V X - C W
Subject to YU g (I-A) X g Y1
(epX -gpWs Sp (p= 1, ..., m)
W.h,e. X. (j =1, ..., n)
(A-2) With explicit consideration of domestic effluent
Maximize VX-CW-CM
d
Subject to YU s (I-A) X s
-------
79
epX -gpW -dpMg Sp+Mp (p = 1, ..., m)
X. (
(h = least cost treat-
ment process)
"h*
M
Vs
(h = 1, ..., h.)
B — Minimization of multiphase treatment costs
Minimize C W (B-l)
Subject to
same constraints as for (A-l) model
Minimize C W + C, M (B-2)
d
Subject to
same constraints as for (A-2) model
Two examples of the formulations of the multiphase treatment programming
models are shown on the following pages.
Input Data Aspects
Phases of Treatment - Primary, Secondary, and Tertiary Processes. Treatment
techniques for domestic as well as industrial sewage have been largely perfected,
and treatment through primary, secondary, and tertiary stages of domestic wastes can
render waste water of almost pure quality though each succeeding stage adds to the
costs of treatment.
Primary treatment consist of settling raw sewage until the solid material
(sludge) is precipitated and the floating particles surface. At this point grease
and floating particles are skimmed off and the waste flow that emerges from this
process had only one-third of its oxygen-consuming degradable compounds removed.
In secondary treatment, various processes such as chlorination with filtration
or activated sludge are used to further decompose wastes. This treatment removes
8O to 90 percent of the BOD in wastes.
-------
FORMAT OP MULTIPHASE TREATMENT PROGRAMMING MODEL -MAXIMIZATION OF GROG:; REGIONAL PRODUCT (A-2)
XXX
» X X X X
X V X 1 ! 1 L I l» I'
i M s s «; 5
n P i <» s fr
i?
6 f f F
SO T N l> r> n f- F F
I M r T T T T T T 1 F r. 1 7 f II :.1 1
6 7 8 <) 0 f) ' E I 7
1
PI
P.2, ....
Pi
"5
Q7
ax
"1 1
P1?
PI 1
P I A
''I7
P 1 0
"p?l
"??
R26
R?Q
P^l
R13
P17
PA?
R4i-
R47
IIPPF^
r # - * - * - * - 1 -*-*-*-*-* *
^ »*-*-*-*-#-*- »- *-**-*-#-*-* *
P, _£_ A -*_<:-. ^— *— *- X'- #-*-*-*-*# *
j **« _*_*-* _*_*_##
0 * -1
C * -1
r, * -i
r, » -1
C * -1
o * -1
r. * -I
r, * -1
._a i -i
n i -i
r, l -I
r, I -1
n 1-1
r, 1 -1
.-C .-- 1 -1
<-. 1 -1
r, 1 -l
r- i -i
r, I -1
r i-l
G 1-1
r 1 -V
~, 1-1
rtfiJMO * *
1 ***** *
* *
-------
FORMAT OF MULTIPHASE TREATMENT PROGRAMMING MODEL -MINIMIZATION OF TREATMENT COSTS (B-2)
&
A F P F
R2
PS
Rt) .
"1 1
P 1 T
C 18
R2S
R? »
R11
P )4
Olf,
P41
> Y x y x P ') S n i
'
r; -* -* *-*- »- «-•*-*-*-*-*-*_* *
r, ************** *
G
, *
,
"
'
1 —
'
L ^ * »
I****14'*
(_ 44* _*_*_<, _*-*_* *
L * * -*-* -* -*-* -* -* *
r, * -i
r, * -1
G * -I
r, * -l
G * - 1
C * -1
r * -i
C, * -I
1 1 -1
r, l -1
r I -1
r i -i
n i -i
", 1-1
. G . 1 -I
r 1-1
r 1-1
r 1 -'
1-1
r 1 ? ** n ) •)
F 1 ? 7 I 2 1
* ft # * & $ *
£ * * V * * *
'
*
#£#$#$
-1
-1
-1 CO
-I H
-------
o2
Tertiary treatment is the last step in achieving almost totally pure water,
"but it is also an expensive process. At this stage water may be, for example,
passed over sand filters and then over activated carbon. Other methods are also
available to remove the nondegradable chemicals in the waste waters.
While some industrial wastes differ in their composition from domestic wastes,
many can be controlled by the same treatment processes used for domestic wastes as
described above. Other vastes (from refineries, for example) may receive a degree
of treatment roughly equivalent to secondary treatment.
Thus for the purposes of this model the various treatment processes used
for domestic and industrial wastes are defined in terms of these three stages with
their corresponding removal efficiencies and unit costs.
Unit Costs and Removal Efficiency. In order to apply the models, data of
units costs of each type of treatment process ($/acre-ft) and of removal efficiencies
in terms of percentage removal were collected from various sources. Average figures
are presented in Tables XXXVI and XXXVII-
TABLE XXXVI
TOUT COSTS OP WASTE WATER TREATMENT61
Treatment Process
Primary Treatment
Conventional
Advanced
Secondary Treatment
Activated Sludge,
Chlorination with
Filtration
Tertiary Treatment
Chemical,
Filtration,
Chlorination,
Additional Chemical
and Activated Carbon
Activated Sludge,
Filtration,
Chlorination,
Tertiary Chemical
and Activated Carbon
Domestic Wastes
Primary
Secondary
Size of Plant
10T3
ragd
100
mgd
1000
mgd
Costs
$/mil gal
92
95
201
308
336
$/acre -ft
31.12
65.53
100.42
12
20
$/mil gal
i+9
51
104
175
189
$/mil gal
38
38
77
124
142
See Keference [10].
Costs of this plant size were used for calculation in the programming
model, thus results can be judged with respect to these conservative estimates.
'See Reference
-------
TABLE XXXVII
WASTE WATER TREATMENT PROCESS PERFOBMAWCE
Percent Removal
Constituents
Treatment Process
§
O
cu
CO
T
tr\
O
-P
PH
as
ra
-ci
O
•d
ft
CO
CO
•P
03
05
O
*H
O
a
CD
CM
O
CD
to
CO
CD
T3
a
C8
,-t
•H
O
Primary Treatment
Conventional
Advanced
Secondary Treatment
Activated Sludge and
Chlorination
With Filtration
Tertiary Treatment
Chemical, Filtration,
Chlorination, Additional
Chemical and Activated
Carbon
Activated Sludge, Filtration
Chlorination, Tertiary
Chemical and Activated
Carbon
25-35
40-50
85-95
90-97
99
90-95
30
30
60-70
55-65
80-90
85-95
99
99
50-60
60-70
75-85
85-95
99
99
50-65
50
iSee Reference [10].
3See Reference [27], Table V-J.
DSee Reference [27], Table IV-12.
-------
84
Results of the Multiphase Treatment
Mode1 for the San Francisco Bay
Primal Problems. In addition to the various aspects of the programming
model to be analyzed, as mentioned in the preceding section, a comparison "between
the results of the four versions of the multiphase programming model is made as
well as a comparison with the results of the untreated effluent programming model.
1. A comparison of the optimal values of the objective functions of the
tvo model versions which maximize gross regional product net of treat-
ment cost with the preceding model show very clearly that treatment of
industrial and domestic waste to a certain extent is economically
efficient and results in higher gross product for the region. The
periodical increments to the GRP which result from the various industries
operating to their upper limit (satisfying the upper limit of final
demand) are larger for the later part of the planning horizon which
follows the logic that the limits put on the sectoral growth are more
significant as time passes-
It should also be realized that the actual difference would even be
greater as the results of the untreated effluent model incorporate
existing violations of the quality standards for the Bay in order to
achieve a feasible solution- Thus in order to satisfy even the
projected lower growth rate of deliveries to final demand, treatment
processes have to be undertaken immediately-
A summary of the results and comparison is given in Table XXXVIII.
TABLE XXXVIII
C&OSS REGIONAL PRODUCT OF THE VARIOUS PROGRAMMING MODELS
Untreated Effluent
Model
Multiphase
Treatment
Model
Without Domestic
Treatment
With Domestic
Treatment
1963
15,^96
15,496
15,451
1970
With
Flush-
ing
22,659
22,699
22,629
With-
out
Flush-
ing
22,576
22,631
25,569
1975
With
Flush-
ing
29,8ll+
30,485
30,418
With-
out
Flush-
ing
29,712
30,398
29,698
1980
With
Flush-
ing
35,676
40,595
39,773
With-
out
Flush-
ing
35,184
40,362
33,956
The results of the minimum treatment costs models are given in Tables
XXXIX and XL. The clear increase in treatment costs when domestic
effluent is explicitly considered is not only logic Tout, as the results
of the model indicate, this difference in treatment costs is implicitly
absorbed by the domestic "sector" by already assuming a certain level of
treatment within this sector in version (A-l).
-------
TABLE XXXIX
MULTIPHASE TREATMENT MODEL - MINIMIZATION OF TREATMENT COSTS
RESULTS OF INDUSTRIAL TREATMENT ONLY
Year
1963
1970
1975
1980
Delta
Contribution
into Bay
With
Flushing
Without
Flushing
With
Flushing
Without
Flushing
With
Flushing
Without
Flushing
Treatment
Costs
$10 6
0
0
55-68
10.12
77-78
9!*. 88
220.11
^"\^ Sectors
Type of ^»s^
Treatment ^\-
Wo Treatment
"No Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
Ho Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
Effluent to be Treated by Sectors
1
acre -ft
1*0,000
51,712
1*9,200
56,998
56,998
66,000
66,000
1*
acre -ft
22,299
27,679
27,428
31,776
31,776
36,794
36,794
36,794
36,794
36,794
5
acre -ft
29,021
38,055
35,695
12, 111*
12, Hit
12, Hit
41,354
41,354
31,417
31,417
31,417
1*7, 884
38,954
38,95!*
38,951*
1*7,881*
1*7,881*
1*7,881*
1*7,884
6
acre -ft
10,796
ll*,76l
13,279
15,385
15,385
1,1*72
1,472
1,472
17,813
1,620
1,620
1,620
17,813
3,l89
3,189
3,189
7
acre -ft
54,132
79,063
66, 582
77,138
77,138
89,301
89,301
17,1*89
17,489
17,489
8
a ere -ft
6,719
8,1*81*
8,265
9,575
9,575
11,087
11,087
9
acre -ft
7,169
9,270
8,818
10,216
10,216
11,829
11,829
10
acre -ft
5,159
6,529
6,3^6
6,346
6,346
6,346
7,353
5,237
5,237
5,237
7,352
7,352
7,352
7,352
8,513
8,513
8,513
8,513
8,513
8,513
8,513
8,513
CO
VI
-------
TABLE XL
CD
ON
MULTIPHASE TREATMENT MODEL - MINIMIZATION OF TREATMENT COSTS
RESULTS OF INDUSTRIAL AND DOMESTIC TREATMENT
Year
196?
1970
1975
1980
Delta
Contribution
into Bay
With
Flushing
Without
Flushing
With
Flushing
Without
Flushing
With
Flushing
Without
Flushing
Treatment
Costo
$106
1*2.7
59-9
97-6
81.3
168.8
255-5
528.6
,_
^^^^
^v^ Sectors
Type of^^v.
Treatment ^\^^
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
Mo Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
Effluent to be Treated, by Sectors
Domestic
a ere -ft
291 ,-379
352,169
11,085
11,085
332,169
572,960
372,960
533,817
166,38*1
166,58!*
513,817
1
acre -ft
1*0,000
1*9,200
49,200
56,998
56,996
2,882
2,882
66,000
23,255
25,233
66,000
1^,105
lit, 105
h
a ere -ft
22,299
18,507
18, 507
18,507
27,1*28
27,1*28
27,1*28
27,1*28
27,1* 28
22,633
22,633
22,633
31,776
28,585
28,583
28,383
31,776
31,776
31,776
31,776
56,79^
36,791*
56,791+
56,79lt
36,79*+
36,79*1
36,791+
36,79!+
5
a ere -ft
29,021
3,608
5,608
3,608
35,695
'.5,695
27,860
27,860
27,860
1+1,351+
13,675
13,675
15,675
1*1,35^
1*1,551*
1*1,351+
1+1,351+
1*7,881*
1+7,881*
1*7,881*
1*7,881*
1*7,881+
l*7,88)t
1*7,881+
1*7,881*
6
acre -ft
10,796
13,279
13,279
15,385
15,385
9,531+
9,531+
9,531+
17,811*
17,811*
I7,8ll*
17,8ll*
17, 8] U
16, 502
16,502
16,502
7
acre -ft
51* , 132
66,582
66,582
77,138
77,138
89,301
89,501
89,301
89,501
89,301
8
acre -ft
6,719
8,265
8,265
9,575
9,575
11,087
11,087
9
acre -ft
7, 169
8,818
8,818
10,216
10,216
11,829
11,829
10
a ere -ft
5A59
6,31+6
6,31*6
7,552
7,352
2,208
2,208
2,208
8,513
8,515
8,515
8,515
8,513
8,513
8,513
8,513
-------
The value of the flushing water from the Delta at a rate of 2,000 cfs,
which is assumed to increase the assimilative capacity of the Bay by
25 percent, can be directly obtained from the difference between the
values of minimum treatment costs of the two alternatives as follows:
TABLE XLI
VALUE OF FLUSHING FROM DELTA
Minimum Treatment
Costs Without
Domestic Effluent
Without Flushing ($10e)
With Flushing ($10e)
Value of Flushing ($106)
Minimum Treatment
Costs With
Domestic Effluent
Without Flushing ($106)
With Flushing ($10e)
Value of Flushing ($106)
Year
1970
35-68
0
35-68
97-60
59-90
37-70
1975
77-78
10.12
67-66
168.80
81.30
87-50
1980
220.11
94.88
125-23
528.60
235-30
293-30
In contrast to the first programming model, the primal variables of
main interest are those which express the recommended level of treat-
ment processes required by each sector as presented in Tables XLII
and XLIII.
The main features of these results are as follows:
a. Only a few industries are required to treat their effluent.
b. The number of industries increases for the later part of the
planning horizon.
c. Some of the industries in some of the time periods which are
recommended to treat their effluent are different in the two
models (B-l) and (B-2).
d. Once an industry is expected to treat its effluent the optimal
solutions of the models advocates full treatment, i.e., primary,
secondary, and tertiary. These types of optimal solutions may
be the result of too low unit costs assigned advanced treatment
processes although these costs are the average ones given in
the literature. Different results may be obtained with a greater
range between the various unit costs.
e- The results of the models which incorporate domestic effluent
point out decisively that treatment of industrial waste rather
than domestic should be preferred. These conclusions result
from both types of models.
-------
TABLE XLII
MULTIPHASE TREATMENT MODEL - MAXIMIZATION OF (BOSS REGIONAL PRODUCT
RESULTS OF INDUSTRIAL TREATMENT ONLY
Year
1963
1970
1975
1980
,
Delta
Contribution
into Boy
With
Fluching
Without
Flushing
With
Flushing
Without
Flushing
With
Flushing
Without
Flushing
Gross Regional
Product
$10°
15,^96
22,699
22,631
30A85
30,398
^0,595
1+0,362
^—
^*^^^
^x. Sectors
TV'W of ^"N"x«^
*-J r"-" ^ ^*^*^^
Treatment ^^'^s.
No Treatment
Mo Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
Effluent to be Treated by Sectors
1
a ere -ft
;il,153
60,000
60,000
80,800
80,800
6,l+6l
6,1*61
108,000
5, UUl
5,1.1+1
108,000
26,876
26,876
1+
a ere -ft
22,78'.
33,1+1+9
33,1+1+9
1+5,01+5
i* 5, Ql+5
9,1+05
9,1+05
9,1+05
r~0,208
32,26?
32,262
32,262
60,208
60,208
60,208
60,208
5
acre -ft
29/f+O
^5,53?
^3,532
26,81?
26,817
26,817
58,622
30,079
30,079
30,079
58,622
58,622
58,622
58,6,22
78,355
78,355
78,355
78,355
78,355
78,355
78,355
78,355
6
a ere -ft
11,037
16,195
16,195
2,059
2,059
2,059
21,808
^,515
^515
'.,515
21,808
6,128
6,128
6,128
29,1 50
9,836
9,836
9,836
29,150
29,150
29,150
29, 150
7
acre -ft
5'i,725
8l,197
81,197
109,31+6
109,31(6
iU6,i39
1^6,139
55,311
55,311
55,511
8
a ere -ft
6,898
10,079
10,079
13,57>»
13,57^
l8,ii+3
18, ll+3
9
acre -ft
7,370
10,7s1*
10, 75^
ll+,l+82
lk,k82
19,358
19,358
10
a ere -ft
5,280
7,71+0
1,555
1 R^C
1,??5
1,555
7,7^0
7,71+0
7,7^0
7,7^0
10,1+22
10,1+22
10,1+22
10,l+22
10, 1+22
10,1+22
10,1+22
10, 1+22
13,931
13,931
13,931
13,931
13,931
13,931
13,931
13,931
-------
TABLE XLIII
MULTIPHASE TREATMENT MODEL - MAXIMIZATION OF GROSS REGIONAL PRODUCT
RESULTS OF INDUSTRIAL AND DOMESTIC TREATMENT
Year
1963
1970
1975
1980
Delta
Contribution
into Bay
With
Flushing
Without
Flushing
With
Flushing
Without
Flushing
With
Flushing
Without
Flushing
Gross Regional
Product
$10 s
15,451
22,629
22,569
30, 4 18
29,698
39,775
33,956
^^^-^s^ Sectors
Type of^~\^^
Treatment ^^\^
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
No Treatment
Primary Treatment
Secondary Treatment
Tertiary Treatment
Effluent to be Treated by Sectors
Domestic
a ere -ft
291,379
332,169
332,169
372,960
372,960
513,817
513,817
145,469
145,469
1
a ere -ft
41,153
60,000
60,000
402
402
80,800
76, 129
27,190
27, 190
107,315
50,570
50,570
72,159
30,716
30,716
4
a ere -ft
22,783
18,421
18,421
18,421
33,448
32,738
32,738
32,738
32,449
24,321
24,321
24,321
45,045
23,365
23,365
23,365
41,088
36,148
36,148
36,148
59,712
59,712
59,712
59,712
37,659
37,659
37,659
37,659
5
a ere -ft
29,740
4,934
4,934
4,934
43,532
4,930
4,930
4,930
43,532
34,662
34,662
34,662
58,622
52,211
52,211
52,211
56,727
56,165
56,165
56,165
76,705
76,705
76,705
76,705
60,764
60,764
60,764
60,764
6
acre -ft
11,037
16,195
16,195
2,059
2,059
2,059
21,808
4,515
4,515
4,515
19, 164
4,188
4,188
4,188
2)4,713
24,713
24,713
24,713
21,067
21,067
21,067
21,067
7
a ere -ft
54,725
81,197
81,197
109,346
82,457
98,996
2,009
2,009
2,009
94,336
94,336
94,336
94,336
8
acre -ft
6,898
10,079
10,079
13,574
13,387
17,984
12,629
9
a ere -ft
7,570
10,754
10,754
14,482
l4,l49
19,021
14,634
10
a ere -ft
5,280
7,740
7,740
7,740
7,740
10,422
10,368
10,368
10,368
10,368
13,875
13,875
13,875
13,875
12,589
12,589
12,589
12,589
-------
90
In the multiphase treatment models it was not required to make any
adjustment to the right-hand side values in order to obtain feasible
solutions, thus the levels of untreated effluent to be discharged
into the Bay are those -which will satisfy the specified quality
standards for the various pollutants. These levels are given in
Tables XXXIX, XL, XLII, and XLIII as the difference between the
amounts of total effluent and the amounts to be treated.
Dual Problems . As in the previous model, dual problems for the primal
problems may be formulated and an economic interpretation may be applied to the
dual variables, i.e., shadov prices. There are thus n accounting or shadow prices,
a , .-., a^t associated with the initial set of constraints. Variables a^ = 5z/9y^
represent the marginal increase in mini mum total treatment costs per unit increase
in the final demand of industry i- From the nature of the formulation, o.^ § 0 since
an increase in the maximum production level of a particular industry which satisfies
the upper bounds of final demand will result in an increase of effluent production
and may therefore result in greater minimum treatment costs. The o^ are useful in
ascertaining the effects upon the minimum treatment cost required to maintain a
given quality standard in the face of relatively snail changes in the final demands
for the outputs of the various regional industries .
There is secondly a set of m accounting prices, \ , ..., X^ X_ = Sz/dSp or
+ Mp) which indicates the marginal increases in minimum cost resulting from
a unit increase in the amount of effluent component p allowed in the estuary.
\^ 5 0 since the greater the amount of pollutant p allowed in the estuarine waters,
the less the amount that must be subjected to treatment .
The use of the X- dual variables may be exemplified through the assignment
of values to various levels of augmentation of the estuarine waters such as the
flushing from the Delta. Releases into the estuary of large quantities of fresh
water from reservoirs are sometimes desirable, not only to remove pollutants
discharged into the estuary, but to counteract sea water intrusion through the ocean
inlet. If at mean volume v of the waters Sp is the quality standard for pollutant
p, then with an augmentation of volume v' the new standard may be represented as
+ A v' where the value of Ap is such as to maintain the maximum allowable
S + A v1 S
P P . = _£.
v + v' v
The value of augmentation v' to the regional economy in terms of reduced treatment
cost is thus - Ap v' X_. This shadow price of the flushing from the Delta is the
value of flushing obtained from the dual problem.
concentration ratio.
There is lastly a set of n dual variables 3^ = Sz/olcj Xj g 0 which indicate
the increase in treatment costs per unit of additional effluent produced by sector
j . However, as earlier discussed, the set of Y^ prescribed from the first set of
constraints will precisely determine the set of X.^ forthcoming from the regional
economy. Thus under the assumption of fixed technical and effluent coefficients,
the 3i yield no new information since they can be determined from the values of
the a.
ai
k (I-A)"1 oY. k. (l-A)-Z
-------
91
The shadow prices for the different models for the various time periods are
given in Tables XLIV and XLV.
EXTENSIONS
Multiple Quality Standards
If the pattern of withdrawers of, and dischargers into, the estuarine waters
and the physical configuration of the waters is such that differing beneficial uses
of the vaters can "be ascribed to various portions of the estuary, it would of course
be highly inefficient to prescribe a single quality standard for any pollutant to
the entire estuary. Quality intake standards have been shown to vary significantly
between industries .[^3] and as a result widely differing quality standards may be
established for various segments of the estuary. This potential variation of
standards is accentuated by the possibility that portions or segments of the estuary
may be reserved primarily for the promotion of fish life and for recreational uses.
In recommending standards for the San Francisco Bay, the San Francisco
Bay-Delta Water Quality Control Program has stated that
"The uses and the water quality necessary for those uses vary
from area to area. Both the uses and the water quality in
the Delta are very different from the water quality and
associated uses in the central portions of San Francisco Bay.
To handle this problem, ten Water Quality Zones were established
over the Bay-Delta system to permit differentiation of uses and
water quality in different parts of the system. These Zones
were determined by the physical configuration of the Bay, or by
the nature of the physical, chemical, and biological data
available for this study." [10]
Once an estuary has been so apportioned into various segments on the basis
of the above criteria, the overall objective of the regional water quality program
is to minimize the sum of the treatment costs incurred in each of the estuarine
segments. Assuming an estuary divided into L zones or segments and ignoring, for
purposes of exposition, domestic waste waters and the possibility of multistage
treatments, the objective in terms of our programming model is to
Minimize ,C. ,W.
The water quality constraints of the program must now take into consideration
the fact that effluent discharge into one segment of the estuary may have, through
dispersion, a damaging effect upon the quality of the waters in adjacent segments.
We thus introduce into the model the coefficient ^z which represents the amount of
pollutant p which appears through dispersive action in segment or zone i. In keeping
with the general limitation of the program to linear relationships, the appearance
of a pollutant in any one segment through dispersive action is assumed to be a linear
function (or a function that can be readily linearly approximated) of the amount
discharged into an adjacent segment. The effect upon z.one k in terms of a particular
pollutant p resulting from economic activity in segment & may now be calculated by
applying the new coefficient to the net amount of the pollutant entering the waters
of segment SL to yield
-------
TABLE XLIV
DUAL VARIABLES (SHADOW PRICES) FOR TREATMENT MODELS WITHOUT FLUSHING ($10S)
Shadow
Price
a
X
Year
Model
Rowa
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
23
1970
A-l
- 0.673
- 0.795
- 0.808
- 0.595
- 0.538
- 0.738
- 0.386
- 0.660
- 0.787
- 0.765
- 0.891
- 0.951*
- 0.938
- 0.872
- 0.212
- 0.048
A -2
- 0.51+5
- 0.791*
- 0.807
- 0.573
- 0.543
- 0.733
- 0.383
- 0.659
- 0.785
- 0.764
- 0.891
- 0.953
- 0.936
- 0.870
- 0.0001
- 0.005
- 0.194
- 0.012
B-l
- 0.002
- 0.001
- 0.003
- 0.015
- 0.198
- 0.018
- 0.020
- 0.008
- 0.004
- 0.007
- 0.001
- 0.002
- 0.001
- 0.003
•f 0.212
B-2
- 0.002
- 0.001
- 0.003
- 0.017
- 0.177
- 0.020
- 0.017
- 0.009
- 0.004
- 0.007
- 0.001
- 0.002
- 0.001
- 0.003
+ 0.0005
+ 0.167
1975
A-l
- 0.545
- 0.791*
- 0.807
- 0.571
- 0.5^2
- 0.731
- 0.378
- 0.660
- 0.785
- 0.764
- 0.890
- 0.953
- 0.936
- 0.870
- 0.005
- 0.255
- 0.012
A -2
- 0.475
- 0.779
- 0.774
-
- 0.500
4- o.o6ob
+ 0.0l4b
- 0.621
- 0.764
- 0.735
- 0.871
- 0.939
- 0.926
- 0.837
- 0.0001
- 0.005
- 0.194
- 0.408
- 0.012
B-l
- 0.003
- 0.001
- 0.004
- 0.015
- 0.199
- 0.044
- 0.021
- 0.009
- 0.005
- 0.008
- 0.001
- 0.002
- 0.001
- 0.003
+ 0.212
4- 0.048
B-2
- 0.129
- 0.001
- 0.004
- 0.030
- 0.168
- 0.045
- 0.022
- 0.011
- 0.006
- 0.009
- 0.002
- 0.002
- 0.002
- 0.004
4- 0.0008
4- 0.005
+ 0.164
1980
A-l
- 0.544
- 0.794
- 0.807
- 0.570
- 0.597
- 0.727
- 0.33^
- 0.661
- 0.786
- 0.765
- 0.889
- 0.953
- 0.936
- 0.870
- 0.005
- 0.758
A -2
•L
+ 1.312*
- 0.403
V.
4- 0.062to
•\_
4- i4.562D
. "h
+ o.524b
•u
+ 20.211
•u
+ 10.166°
•v
4- 0.344b
- 0.223
- 0.004
- 0.368
- 0.576
- 0.667
-
- 0.005
- 0.803
- 10.761
B-l
- 0.005
- 0.002
- 0.005
- 0.025
- 0.135
- 0.055
- 0.072
- 0.008
- 0.004
- 0.007
- 0.003
- 0.002
- 0.001
- 0.004
4- 0.811
4- 0.010
B-2
- 0.132
- 0.002
- 0.005
- o.o4o
- 0.135
- 0.055
- 0.072
- 0.008
- 0.006
- 0.008
- 0.003
- 0.003
- 0.003
- 0.005
4- 0.005
H- 0.803
^ow 2 corresponds to sector 1, ..., row 15 corresponds to sector l4; rows 16-23 correspond to constituent constraints (BOD, nitrogen,
,.., effluent).
^Final demand at lower limit.
-------
DUAL VARIABLES
TABIE XLV
3HADOW PRICES) FOR TREATMENT MODELS WITH FLUSHING ($10°)
Shadow
Price
a
X
Year
Model
Row
2
*>
k
5
6
7
8
9
10
11
12
15
111
15
16
17
18
19
20
21
22
23
1970
A-l
- 0.675
- 0.795
- 0.809
- 0.597
- 0.557
- 0.766
- 0.589
- 0.662
- 0.788
- 0.766
- 0.891
- 0.95^
- 0.958
- 0.872
- 0.195
- 0.195
A -2
- 0.673
- 0.795
- 0.809
- 0.591*
- 0.560
- 0.763
- 0.390
- 0.660
- 0.788
- 0.767
- 0.891
- 0.951*
- 0.938
- 0.872
- 0.0005
- 0.167
B-l
-
-
-
-
-
-
-
-
-
-
-
B-2
- 0.0008
- 0.0004
- 0.0015
- 0.0053
- 0.061
- 0.015
- 0.003
- 0.008
- 0.001
- 0.002
- 0.0004
- 0.0006
- 0.0003
- 0.001
+ 0.0016
1975
A-l
- 0.673
- 0.795
- 0.808
- 0-595
- 0.538
- 0.758
- 0.386
- 0.660
- 0.787
- 0.765
- 0.891
- 0.951*
- 0.958
- 0.872
- 0.212
- 0.048
A -2
- 0.673
- 0.795
- 0.809
- 0.595
- 0.559
- 0.742
- 0.389
- 0.659
- 0.787
- 0.766
- 0.891
- 0.95^
- 0.938
- 0.872
- 0.0005
- 0.167
- 0.038
B-l
- 0.002
- 0.001
- 0.003
- 0.015
- 0.180
- 0.017
- 0.018
- 0.007
- 0.004
- 0.007
- 0.001
- 0.002
- 0.001
- 0.003
+ 0.195
B-2
- 0.002
- 0.001
- 0.003
- 0.016
- 0.177
- 0.020
- 0.017
- 0.009
- o.oo4
- 0.007
- 0.001
- 0.002
- 0.001
- 0.003
+ 0.0005
+ 0.167
1980
A-l
- 0.545
- 0.79^
- 0.807
- 0.571
- 0.542
- 0.731
- 0.378
- 0.660
- 0.785
- 0.764
- 0.890
- 0.953
- 0.936
- 0.870
- 0.005
- 0.255
- 0.012
A -2
- 0.474
- 0.779
- o.nk
-
- 0.554
+ o.o6ib
+ o.o6ob
- 0.623
- 0.764
- 0.736
- 0.869
- 0-939
- 0.926
- 0.838
- 0.005
- 0.758
- o.4o5
B-l
- 0.003
- 0.001
- o.oo4
- 0.015
- 0.199
- 0.449
- 0.021
- 0.009
- 0.005
- 0.008
- 0.001
- 0.002
- 0.001
- 0.003
+ 0.212
+ o.o48
B-2
- 0.131
- 0.002
- 0.005
- 0,039
- 0.174
- 0.054
- 0.045
- 0.009
- 0.006
- 0.009
- 0.003
- 0.003
- 0.003
- 0.005
+ 0.005
H- 0.446
^ow 2 corresponds to sector 1, ..., row 15 corresponds to sector 14; rows 16-23 correspond to constituent constraints (BOD,
nitrogen, ..., effluent).
Final demand at lower limit.
VJ4
-------
0=1
where the variables X. appearing in the expression are as previously defined but now
set in a spatial context toy the subscript £.
The water quality constraints for each zone in the program indicate as "before
that the amount of pollutant p released after treatment cannot exceed the established
standard, and secondly that the amount of effluent treatment in each spatial sector
cannot exceed the amount of effluent produced by that sector,
ksp
n
.1=1
n
\
+ , .z / (e. JC, - g, .W,)] «= 0
X B '
, .z / e. , - g, .,
kl P L X JP * j BJP •« j
U = *)
(p = 1, ..., m)
(j = 1, ..., n)
(J = 1, ..-, L)
Once the outputs of each sector have been spatially identified, it is desirable to
develop an economic model that is structured in such a manner as to yield information
as to the effects throughout the system which will result from changes in the economic
demands upon any given industry. The increased production of industry i in zone I,
may result in a greater amount of effluent in zone k, not only through hydro logic
dispersive action but also through economic linkages as well by stimulating or
inducing increased output from industry j located in zone k. Multiplier effects
such as these can be estimated through the extension of the regional input-output
model into an interregional formulation.
Interregional input-output analysis differs from the regional model in that
the former attempts to incorporate spatial location. Regional interdependence
analysis disaggregates production and consumption totals in terms of commodities or
industries; interregional theory likewise breaks down these totals by commodities
but also in terms of geographic location. The interregional model concentrates
upon the economic relationships not only within each region but among the regions
as well. If an industry i in region I requires as an input the product of industry
j in region m, a direct economic "tie" or interdependence exists between the two
industries and thus the two regions. If industry k in region n then purchases for
its production process the product of industry i in region i, in addition to the
direct link formed between regions n and /, an indirect tie is established between
regions n and m. Interregional input-output theory is designed to reveal both
these direct and indirect interdependencies among regions [hk].
Identifying each region with a particular estuarine zone, the interregional
model enables us to provide the type of analysis needed for the new segmented system.
-------
95
In our programming model the economic conditions may toe restated as
kxj = kYi (i = 1' •••>n)
(k = 1, ..., L)
where j^a.ji is the amount of production of industry i in region k required by industry
j in region 4 for the latter industry to produce one dollar of gross output.
Interzonal Effluent Shipments
Due to the tidal characteristics of estuary waters, almost all pollutants
vill ultimately be deposited in the ocean. The dilution capabilities due to tidal
action will vary significantly, however, between the different estuarine segments.
The nearer the disposal site is to the ocean inlet, the more effective will be the
tidal flushing action. In the San Francisco estuary, for example, the dilution
capabilities of the central portions of the Bay which are adjacent to the connecting
passage with the ocean at the Golden Gate, are up to fifty times greater than those
of the dispersive actions of the extreme northern estuarine waters.
As an additional extension of the model we might therefore consider the
possibility of transporting effluent between zones, now that the model of the estuary
has been spatially segmented. As economic activity expands and higher cost treatment
processes are necessitated, the possibility of shipping effluent between estuarine
segments via pipelines becomes an increasingly viable economic alternative to on-site
treatment. Such shipments may be made from zones where quality standards are being
approached to other segments where conditions are more favorable to waste disposal.
Alternatively, the destination of the waste shipments might be to a regional treat-
ment plant which is designed to take advantage of certain economies of scale by
treating the wastes from two or more zones.
Assuming the cost of the effluent shipment to be linearly dependent upon
the volume transported, we may let ^t-i represent the total volume of effluent from
industry j in zone £ to be conveyed to zone k for disposal and j^c I the cost of
transporting one unit of this effluent. To simplify the exposition let us assume
that region k represents the only logical zone to which shipments may be made. The
objective of the program then is to minimize the total regional cost of a system of
treatment operations and interzonal shipments, i.e., to:
L n
Minimize Z = £
4=1 j=i h=i 4=i j=
and the water constraints may be restated as
,-E
e . «A . -
JP 4 j
11=1 (4 = i, ..., L)
(i = 1, ..., n)
(p = 1, ..., m)
-------
96
»,t., .W., g 0 for all subscripts £, j, and h.
-KK j * jn
JJ
-z
/=!.
= JP ^kAj
h=i
t.k p
3=1
a o
(p = 1, ..., m)
Assuming an estuary apportioned into two zones, the above constraint is presented
diagramatically for p = 1, ..., m.
Region k
k%(L-T) -.
*K !
Estuary Zone k I Estuary Zone I
(L-T)
z (K+T)
Region
where
JP *XJ - Z
-------
97
11 / "n
- 7 f. x . y
Li \ JP i j L>
jph
J=l
h=l
T = ) e
^Li OP
J-i
-------
SECTION III
Summary And Conclusions
Phases 1-5
by
P. H. McGauhey
-------
XI. SUMMARY AND CONCLUSIONS
SUMMARY
Studies of the Economic Evaluation of Water Quality we.re initiated in 1962
in recognition of the growing percentage of the nation's water resource being
degraded in quality through beneficial use; and the ever-increasing investment in
waste water treatment works necessary to repair the damage to receiving waters. At
that time the question of how water quality criteria established by regulatory
agencies for any given sector of the water resource might "be achieved at a minimum
cost was being asked in many quarters, but the concepts of "benefit-cost and pollution
control current during that period did not lead to satisfactory answers.
The economic evaluation study was proposed on the rationale that modern
methods of mathematical modeling, systems analysis, and operations research could be
made to predict the consequences of alternative decisions, if not indeed to reveal
optimum economic solutions.
Exploration of the possibility of developing an optimizing model was begun
by Mr. Richard Frankel, then an advanced student in water resources engineering,
under a one-year predoctoral fellowship from Resources for the Future. Early results
indicated that the subject of economic evaluation of water quality might be a fruitful
line of research and in 1963 Grant No. WP-00597 was made by the National Institutes
of Health in response to a proposal by Professor P. H. McGauhey of the University of
California. Subsequently, when the national water resources program was transferred
from the Public Health Service to the Department of the Interior, the project was
continued under the Federal Water Pollution Control Administration as Grant No.
USDI WP-597 until the close of the project in December 19^9- During its tenure it
produced annual reports on four phases of the study, and a Final Report (herein
presented) in which these four reports are summarized and a fifth and final phase
of the study presented in detail-
A number of authors are credited with the various reports and the project
produced doctoral dissertations in the fields of water resources engineering,
economics, and operations research.
The initial phase of the study was an engineering economic model for water
quality management. It involved the development of a computer model and its
application to a study of the combination of waste water treatment, natural self-
purification, and water treatment processes which would involve the least cost to a
group of communities in achieving water resource quality goals for a river subject
to various water management schemes and multiple "beneficial uses. For purpose of
the study, the oxygen resource of the stream in terms of dissolved oxygen (DO) was
taken as the critical factor governing waste treatment processes. The model, however,
was flexible enough to accommodate any combination of water quality criteria and of
waste inputs to the river. During the course of phase 1 of the study there was
evidence that the normal criteria of average DO concentration of 5 or 6 mg/£ might
prove inadequate because of the diurnal changes in oxygen concentration and variations
in oxygen tension might prove critical and so lead to a whole new concept on which to
base the DO criterion of water quality. The second phase of the study was therefore
directed to the development and testing of a mathematical model expressing the
relationship between the oxygen-demanding fractions of a waste water and the responses
of a living stream.
Although the results of the first two phases of the study bore out the
rationale of the project, it was evident that with respect to one single criterion
or parameter the problem of "indivisibility" or scale factors of treatment processes
made for insensitivity of the model in the case of a single stream receiving wastes
from but one city and serving another as a source of water supply. That is, the
99
-------
100
model was best suited to a regional network of resources and uses. However, as
domestic, industrial, and agricultural wastes may contain various toxic as well as
stable compounds which may adversely affect human health or marine ecosystems, DO
can no longer be considered the critical standard of water quality. Instead, a
multicomponent water quality approach is needed in which limiting concentrations of
several important water quality components express the standard.
Phase 3 of the study was designed to reflect a growing emphasis in public
policy on "quality of the resource" as contrasted with control of pollutants. It
was conducted to develop, and to illustrate by application to an example, a linear
programming multicomponent water quality control model capable of minimizing the
cost of achieving any given water quality goal when the quantities of various wastes
are reduced to levels imposed by presently-known technology. The model made possible
an estimate of the minimum cost of achieving quality standards in terms of the
maximum allowable quantities of each waste constituent discharged to the receiving
water; in this case, one section of San Francisco Bay. Its principal limitation was
that instant mixing and dispersion was an implicit assumption. That is, it was
assumed that the maximum permissible concentration of a single pollutant resulted
from discharging into the estuary the amount of pollutant necessary to produce such
a concentration in a dispersed system.
Phase k of the study was designed to eliminate to the extent possible the
limitation of phase 3 "by "the development of dispersion models for the optimal
allocation of water quality and integration of a dispersion model with the water
quality control model of phase 3. The result was a waste treatment model which
generates an optimal waste treatment plan for all waste-producing agencies in the
study region by minimizing the total waste treatment costs. Finally, a complete
model of estuarine water quality management was developed and its use illustrated
for Suisun Bay of the San Francisco Bay system.
The final phase of the study develops a multisectional programming analysis
for the management of the waste water economy of the San Francisco Bay region,
drawing upon a nine-county multisector model of the Bay area economy constructed
(by the authors) for the specific purposes of this phase of the study; waste water
discharge data and water quality standards for San Francisco Bay; and the steady
state linear optimal dispersion model of phase 4. This model included economic and
waste water multiplier analyses as well as waste constituent interaction tables and
a critical time period analysis. Finally the Bay area waste load data and final
denand projections were incorporated into linear programming water quality management
models for both untreated effluents and multiphase waste treatment. Thus the final
phase of the study presents a method for predicting the discharges which may be
permitted in any estuary having known dispersion and advection characteristics.
The model was applied under the constraints of the water quality requirements
presently imposed upon San Francisco Bay- It also predicts the time when stricter
levels of waste treatment must be imposed.
CONCLUSIONS
From the work completed under phases 1 and 2 of the study it was generally
concluded that the engineer ing-economic model has wide applicability to surface
water pollution problems. The more than 20 specific conclusions reported include
the following:
1. Conventional water treatment plants can handle the pollution load caused
by domestic waste disposal at small increases in average annual operating
costs.
2- Since large additional downstream treatment costs are incurred only at
relatively infrequent intervals, the costs Imposed by flows of low
probabilities of occurrence are relatively small as compared to the
costs imposed by flows occurring more frequently.
-------
101
3. The costs imposed dovnstream decrease as the performance of upstream
treatment increases, "but primary effluent disposal offers little
advantage over raw waste disposal in terms of savings to municipal
water treatment downstream.
k. Distance is an important factor in imposed costs downstream when little
or no treatment is provided upstream "but is of minor importance when
secondary or higher treatment is provided upstream.
5- When sewage loads are discharged into initially clean streams, the
increase in cost to downstream municipal treatment plants is relatively
large for the initial pollutional load. Similar increases in pollution
loads in already polluted streams result in lower incremental costs to
the water treatment plant.
6. The savings to municipal water treatment plants downstream is a direct
"benefit of improved upstream treatment. The ratio of cost savings
dovnstream to the costs of additional treatment upstream is quite small
and varies from negligible to about 0.10. Maximum return per additional
investment dollar is realized when secondary treatment is added to
primary•
7. The minimum cost conventional treatment alternative to meet coliform
standards in the receiving waters is primary waste treatment with
heavy chlorination. Average annual costs of maintaining a coliform
standard of g 2kQQ MPN/100 ud for bathing and recreation uses imposes
an added cost to the waste-stream-water system of approximately
$51,600 per mgd of domestic sewage discharged for plants of 2.5-mgd
capacity and $22,300 per mgd for plants of 10-0-mgd capacity.
8. The costs of maintaining minimum dissolved oxygen levels in the
receiving waters for fish and wildlife is very dependent upon stream
quality and stream characteristics. Average annual cost estimates
based on conventional treatment methods vary between $21,^00 and
$45,000 per mgd of sewage discharged depending upon the specified
dissolved oxygen level, the design flow, and the size of the treatment
plant.
9. Variants to conventional treatment operations such as reducing treat-
ment as flow conditions improve, lagooning during low flow periods,
or using more operating intensive treatment processes in preference to
fixed capital intensive processes, show considerable savings in average
annual costs of sewage treatment.
From phase 3 of the report it was evident that a linear water quality control
lodel is an exceedingly useful tool for optimizing the cost of waste treatment.
Specifically it was concluded that:
1. Although in the illustrative example the model was applied to secondary
treatment processes only and to ten common quality factors, the model
could quite as readily be applied to wastes restricted to different
quality standards for different quality factors.
2. Only in the case of cities is it necessary to treat the entire waste
stream. Industry, by keeping its waste streams separate and applying
different treatments as appropriate, has greater latitude in optimizing
its costs by use of mathematical models.
3. A considerable reduction in overall cost of achieving imposed standards
for the 10 quality factors from the 12 sources examined in the study
would result from requiring different levels of treatment by different
dischargers of the same waste factor, rather than from imposing the
same limit of concentration on each discharger.
-------
102
In the more sophisticated model utilized in phase k, the physical system of
waste transport and degradation in an estuary was described by a one-dimensional
diffusion equation and an estuarine water quality management model developed from
a synthesis of dispersion and waste treatment models. From the illustrative examples
it was concluded that:
1. The limited boundaries of knowledge about water pollution, waste
treatment, and mixing processes in estuaries leave much to be
desired in terms of understanding of these processes so that they
can be described in precise mathematical terms.
2. A reliable theoretical method is available for most estuarine
situations. Additional research is needed to develop a three-
dimensional description of the dispersion process along with a
method of estimating directional dispersion coefficients.
5- Movement and resulting concentration of conservative substances
can be described adequately if a proper description of the
dispersive and advective processes is available. However, for
nonconservative constituents the accuracy of a model is dependent
to a great extent on the reliability of the estimates of the
reaction rate constants. Usually these estimates are arrived at
by subjective judgments based on the experience of the investigator
with the particular environment and waste discharges. There is a
need to seek better mathematical descriptions of nonconservative
processes, particularly of the biological reactions.
h. The solutions of the water quality control model proposed is
sensitive to the water quality standards used. Multicomponent
water quality standards are still not available for many estuarine
basins. The harmful effects of many constituents by themselves or
in conjunction with other constituents on human health and marine
ecology are not yet completely understood in quantitative terms.
Further research and legislation are needed for establishing proper
water quality standards.
5. Quantitative information regarding harmful effects and social costs
of pollution and benefits from quality improvement projects are
lacking today so that these costs and benefits cannot be included
in optimization models. In the report the author has developed
deterministic models and considered concentrations averaged over
one or more complete tidal cycles. In unsteady state models, only-
long -term or seasonal variations are considered. Stochastic models
may be developed to study variations of water quality during a tidal
cycle and effects of random variations of volume of flow in the
estuary. Biere has been some progress in this direction.
Among the most important conclusions from the final phase of the study
(Section II of report) are the following:
1. A multisector linear programming water quality model can be formulated
and applied to set up optional guidelines for water quality manage-
ment either to achieve:
a. minimum costs, or
b. maximum regional economic development.
2. The use of optimizing techniques for achieving specified water quality
levels in receiving waters at minimum cost is a relatively new develop-
mant; and its extensions indicated in this study are a completely
novel development.
-------
103
3 • A pollution multiplier model can be used to forecast the critical time
periods under assumption of a growth pattern of the regional economy
which is not controlled by any consideration of economic efficiency.
4. The primal-dual relationships determined in the study provide broad
guidelines for evaluating the damage costs of effluent discharges and
the benefits of maintaining desired water quality levels.
5- An untreated effluent programming model will result in a growth
pattern which maximizes the gross regional product while satisfying
the required water quality constraints.
6. Treatment of domestic and industrial wastes to a certain extent is
economically efficient and permits a higher gross product output
for the region while meeting receiving water objectives.
7. The periodic increments to the GRP which result from operating
various industries to their upper limit of output are larger for
the latter part of a planning horizon.
8. In order to satisfy even the projected lower growth rate of deliveries
to final demand, waste treatment processes have to be undertaken
immediately in the San Francisco Bay area.
9- The value of fresh water for flushing and dilution purposes can be
evaluated within the framework of the technique herein developed.
10. The clearly established pattern, from an evaluation of the study
results, is that industrial wastes pose the most difficult problems
in maintaining water quality; and the most efficient way of meeting
set water quality standards is for industrial dischargers to treat
their waste to high levels.
11. The results of the linear programming models show that once it
becomes necessary for some level of treatment of an industrial
waste to be implemented, complete treatment of total amounts
discharged is necessary to meet regional water quality objectives.
-------
APPENDIX
Effluent "Interactions" Tables
-------
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
5
6
7
8
9
10
11
12
13
14
ROW
1
2
3
5
6
7
0
9
10
11
12
13
1
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6.0997129
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.0311219
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SUM
.995C154
.3119313
1.1846663
8.7491118
36. 838167C
3.5199797
6.9509328
7.7211946
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-------
1
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6
7 "
9
10
11
12
13
14
1
2
3
5
6
7
8
9
10
11
12
13
ROW
1
2
3
4
5
6
7
8
9
10
11
12
13
14
I
24.2934205
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SUM
24.4931909
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4.1755264
1.1245457
1.3280027
3.7989895
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-------
1
2
3
4
5
6
7
8
9
10
11
12
13
2
3
4
5
6
7
8
9
10
11
12
13
14
ROW
1
' 2
3
4
5
6
7
9
10
11
i 12
13
14
1
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C.f.'COOCr.Q .GCC3'J26 O.OUCO^OQ
0.0_COOOs'0 .0017342 C.oOCOOOu
o.oooocoo
0.0000000
c.cuoooco
O.OCOOOoO
O.OCOOC'OO
O.OCOOCCJ
.CC47834
.C142536
.OtG7499
.0004823
.CO'19132
C.OKOOoOO
O.OCOOCJO
O.COOOCO-;
c.oopc'poo._
C.COCCO'JU
C.C'C-OOC-UO
5
.0042583
f 0_C 3 6 760
.0119094
.0193649
.9250364
.C311372
.CC62716
.C353415
.0153645
.0216055
.0036334
.OC83051
.OC36089
.0110195
12
o.ooooonn
o.ocoooc-o
o.ooooouo
c.oooccco
O.C'COCOUO
o.ccocooc
C.CtOC'OOO
O.UCOOQOO
0.0000000
c.cooocoo
O.CCOC-CCw
c.ocooooo
C. 0000000
6
.0024868
.00 12340
.0028485
.0013577
.0041273
.1023220
.0033572
.0039429
.0020968
.0030943
.OG06439
.CCC5859
.0005669
.0016183
13
0.0000000
O.OCCQOQG..
C.OOCOCCO
G.OOCOCOO ._
O.OfcOOOOO
O.OOCOOOO
O.OCOOCOO
0,OOGOCOO
O.OCOOCOO
C. 0000000
o.oooooco
c.ooc-cooo
o.ooooc-co
O.OCrCOCOO
7
.0010360
.0005140
.0010735
.0005196
.C007813
.0019933
.0433382
.0007079
.0004534
.0003828
.0013790
.C004984
.0004289
.0004876
_L4-
0.0000000
0.0000000
0.0000000
0.0000000
o.oocoooo
o.cocoooo
0.0000030
0.0000000
o.coooooo
0.0000000
. - .jp_.cj&pec_oo
0.0000000
0.0000000
PHOSPHATES
-------
1
2
3
4
5
6
7
9
9
10
11
12
13
}*
1
3
5
6
7
a
9
10
11
i?
13
1
2
3
4
5
6
7
8
9
10
11
12
13
14
j^
O.OOOoCCO
U.OOOOOCu
u.ooooouo
o.ooooooa
C. 000000 1)
e.coooocc
i). 00000 oc
c.cooccoc
O.tOOOCCC
Q.OOCOCOC
0.0000000
C.OOO&GOi,
8.
.0060034
.0000610
.0009105
.0000072
.0000128
.0000036
.DO 1C 892
.0090111
.0000 Lit
.0000040
.0000041
.OOOJC56
.0000061
ROW SUM
.1692861
.0347593
.0753468
1.4036344
.0847906
1.1835791
1.6215667
.0740887
.0430424
.0526843
.0647775
.0382128
.0272647
.0759408
£
0.0000000
Ojocj&ooeo
O.OtOOOOC
u.ot ooooo
O.vCOOCOO
y.ot-oogoo
O.OuOOOOO
0.0000000
o.ocooooo
v. 0000000
O.UbOQGOO
0*0000000
9
0.0000000
O.QUCtJOCO
o.ooooccc
o.ooocouo
O.OOOOC/UU
olooooooo
O.OtkJyCOv/
C. 00000 "JO
OiCt 00000
O.OLCOOOO
0.0000000
3
c.oocoooo
C.OfaCwOOC
o.oooocoo
O.OOCtCOO
o.coocooc
o.ocoooco
C.OOCOivOC
g.ctiucooo
o.ccooooo
0.1)000000
Ift
O.CCOO'JOO
O.bCOOJUO
o.cocouoo
C.OCOOOOO
O.CCCGwOC
.._.... o.cogcvoc
r.ooocooo
C. 0000000
o.cowocco
4
.1053566
.0930345
.0061947
1.3704C44
.01293C1
.0364279
.OC 44668
.0057586
.uC 44661
,0962120
.0076779
.013864U
.0056497
.0411441
11
O.OOCOOOO
0,900^ 3,0
0.0000000
C.C0COOOO
O.OoC'OOOO
O.Of Cu«'OJ
O.yOCOOUO
o.uooocoo
0. OOOOOOC
o.oooocno
o.ooocooc
5
o.ocococo
o.tcoooco
....W.OCQQOCO
o.ococooo
c.ccooooc
c.ooocooo
O.OOOi.OC'0
c.ooocooo
o.coooooo
O.OCOOf'CO
t.OCOOOuO
O.CCOCOCO
12
O.OLOOC'OO
c.ooocooo
O.CCOfOOO
o.rcoccco
O.OCOtOOO
o.ococcoo
o.ovooouo
C.OCOPO(.0
O.OOCCCrCC
o.otcooio
o.ccooooo
6
.0261118
.0299089
.0142557
.0433362
^1.0743815
.0352504
.0414009
.0220163
,C3249Qi
.OC67608
.OC6112J
.0059529
.0169924
13 _
O.COOOOOO
O.QCOQSQJD
O.COCCOOC
0*C£O.QJ2QC
C. 0000000
0.0000000
C. 0000000
O.OCODOOO
O.OOOOOC'O
._ O.OCQOOOO__
o.ocooooo
0.0000000
7
.0378143
.0187AL1.
,0391822
,0189638
.0285171
1.5818459
.0258399
.0165489
a0119JDS.
.0503347
.0156565
.0177982
14
0.0000000
O.OOCOOOO
0.0000000
0.0000000
0.0000000
O.OOCOOOO
0.0000000
0.0000000
O.OOOOQOO
0.0000000
o.ooocooo
0.0000000
0.0000000
OIL a GREASE
-------
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
9
10
11
12
'5J
RQtt
1
2
3
6
7
8
9
10
11
12
13
14
u. Jt JC.C/C/U
O.oOOOOOC
O.COOCCC'v
C.OGGjGCK'
C.f'l/CLOGO
O.OGOOC'ltU
Ji.wiLOJiCjLk
C.&OOOCCO
OVX»9COi£
a.OvKiOOOC
U.k/GCbuilCi
8
.0513467
.1077332
.9188724
.1585578
.1090755
.1929500
.0546680
16.4147631
.1667386
.1664460
.0607844
.0612734
.0837513
.0925156
1UM
4.042C962
1.5081776
4.1990847
. ~ "11 "\ ft SS Q
102.9232559
9.0981889
2d. 7650 809
2.7589033
4.3895231
1.0451534
.9663264
.8266513
2.4719871
t'.GC 00000
O.OOOOOOC
O.OOCOOOO
O.OOOOCOO
tlouoocoo
G.CCGGCut)
c. ocx 000
0.0000000
9
o.ocooooo
c.ctoooco
c.coccooo
C'.'X 00000
O.OoOOCOO
?lSicctSo
J.OCOCOOO
o.o&ooooo
O.ccoyuot.
o.oooocoo
0.0(00000
o.wooooc
o.oocoooo
O.OOOfJOOO
Of OCCJQOQ
o.cccocoo
O.OOl/OOOO
C.wCOOOOC
c.ooccooo
o.ocooooo
C. 0000500
C.COOOtuC
c.ocooo^o
0. CO 00000
o.oooocco
10
.0254997
.0457812
.1566727
.C349996
.0928277
.0600255
.0187621
.0455192
.2965694
.8837259
.0464966
.0311995
.0 2990 56
.1185583
1.3320078
.0383641
.0783181
17.3258270
.1634735
.46C5525
.0564729
.0728055
.0564642
.0785379
.0970708
.1752810
.0714287
.5201784
11
O.OOOOOOC
0.0000000
o.oooooou
c.oooooc-o
0.00000'JO
0.0000000
O. 0000000
0.0000000
O.OUOCOOC
o.ocoooco
o.oooccoo
0.0000 COO
C.OOCOC30
o.coaoooo
TSS
.0264368
.0228219
.0739375
^,1202237
5.7429343
.1933101
.0389364
.2194120
.0953878
.0225576
.0515609
.C224051
.0684128
12
O.OCOCOOP
c.orocooo
0.0000000
O.OPOCOCO
o.soooooo
O.OOCCOOO
o.coooooo
O.tJCOOOCO
0.0000000
o.oooeooo
o.oocoooo
O.OOOGOtO
2.47316CO
U2271709
2.8328C45
UMC_222,3
4.1045586
1C1. 7592768
3.3387159
3.9212564
2.0852553
3.0773029
.6403489
.5827153
.5638267
1. 60 941 89
.13
C*OCOOOOO
... AiUOCCOGCL.
O.OCOGOOO
0. 0000000
o.ocoocoo
o.ooooocc
o.cocoooo
c.ccoocoo
o.ocooooo
o.ooocoon
o.ocoocoo
0.0000000
.1336452
.0663064
.1384795
.0670227
.1007864
.2571410
5.5906336
.0913247
.0584880
.0493766
.1778951
.0642962
.0553339
.0629033
0.0000000
O.COOCOOQ
O.OOOOOOC
0.0000000
0.0000000
o.ocooooo
o.ccooooo
o.oooocoo
o.ocooooo
0.0000000
__ o.pooooop
o.ocooooo
o.oocoooo
-------
i
1
2
3
4
5
6
7
a
9
10
11
12
13
14
1
2
a
4
5
6
7
a
9
10
11
12
13
14
ROM
1
2
3
4
5
6
7
0
9
10
11
12
13
14
1
G.SOOPOOv
O.OGC'fcSpv
O.QOOOf 00
OjiflfifliMfWV
O.COC'O't-tO
C.QOOwCCC
O.OOOCGCU
C.COUCOl'4
O.COOOOCu
Cioooocoo
O.OOOOOi'l!
0.0000000
o.&oooooo
C. 0000000
_ 8
.0000034
.0000071
.C000610
.OOOC105
.0000072
.0000128
.OCOOC36
.0010692
.0000111
.0000 lib
•OOOOC4U
.C000041
.OOUOC56
.0000061
SUM
.0109437
.OOS435H
.0114038
.0054998
.0083140
.0235009
.4551391
.0086212
.0048241
.0041074
.0145000
.0052521
.0045237
.0051666
i
0.0000003
O.OCOOOCO
o.ocuocoo
o.ocooovo
O.OuOOCOO
O.OOUOGOO
o.ocoooco
o.Qooopiw)
c.ocoocoo
U.OOOOCO'J
U.OCOCCOi1
O.OtOOOOO
u.cooocou
0.0000000
9
o.orcoooo
O.OCOOOOQ
C. 0000000
0.0000000
o.ccooooo
C.OCOOOOO
C.OCOOtUu
O.OOOOQOQ
o.ouooooo
O.OCuCOOO
C.OPOOOOC
O.CCOOCC'J
o.ocootoc
v.OuOOCOO
.3..
o.coooooo
5jLCU.OCLOCC__ ..
o.ccooocu
O.CCOOC'CC
C.tCOOOuO
O.Ot 00000
U.CCCOCOC
ItPilfiflCO*) .. .
o.coooooo
r.ccoooco
C..CQCOCOO
O.COCUCUC
C.OI.OOCOO
fc.OuOOOOC
10
o.cocoocc
u*OOU^OCO
O.OOOOOCC
O.CCOJOCO
c.ocooooc
(...Ct'CJOOC
0.0( COCOC
Vjtpocottc
C..COC90C-C
O.OCOIf'CX,
O.OOC030C
o.uocooor
O.OfOO'JCC
C'.OOGOOOO
4 .
O.OOC0003
C.OCOOCuO
o.coooooo
0. 00.00090- .. _
0.0000000
O.COWOOP
o.ococooo
o.ooooooc
O.OCOUCOC
o.ro^ocoo
O.OCCOCoO
O.OOOuCOO
0.0000000
0.0000000
11 1
o.oocoooo
0.0000000
o.oocoooo
o.oooooco
O.OwCOOOU
o.oooc-coo
o.coootoo
O.OOOCOvO
O.OCOOCO"
O.OOCOCOJ
C.OC'OOOufi
o.teoofuo
o.ocoocoo
o.ocoooo
PHENOL
5
o.ooooooo
0*JD00001)0.
C. 0000000
O..CCOOQQO
o.ooocooo
o.ooooooo
o.cccoooo
fl*J'fi.CttSLQP ...
O.OCOCQOO
o.ocooooo
c.ooococo
o.oooooco
o.orooooo
O.COOOOGO
2
o.ooooooc
o,ccococo_ .
c.ococooo
G.OCOCOOO
o.ooocovo
o.oooooco
0.0000000
O.uOOOOCO
O.CCCOuOO
O.OCOOvOO
o.ooocooo
o.ocooooo
o.ococcco
o.ttoooco
ft
.0000622
.0000308
.0000712
.0000339
.0001032
,002558L_
.C000839
.0000986
.0000524
.otfoom
.0000161
.0000146
.0000142
.0000405
13
c.ooooooc
o.ocoocoo
o.oocoooo
o.oocoooc
0.0000000
o.oooococ
o.ocooooc
o.occocoo
O.OOCOCGO
o.oooccoo
O.OOOOOCO
0.0000000
O.OCOODOO
O.OOOliCOC
7
.0108781
.0053970
.0112716
.0054553-
.0082035
.0209301
.4550516
,OQ 74334
.0047606
_.00 40190,
.0144798
1.0052334-
.0045C39
.0051200
14
0.0000000
0.0000000
0.0000000
o.oococoo
o.ooooooo
o.coooooo
o.ooocooo
0,0000000
o.ooocooo
o.oocoooo
o.oocoooo
0.0000000
0.0000000
0.0000000
-------
1
3
4
5
6
7
8
9
10
11
12
13
14
1
2
3
4
I
7
8
9
10
11
12
13
1
3
5
6
7
ft
9
10
11
12
13
1 14
1
i^SSc
O.C'OO'IWC
c.ccoccoo
^.1010'C'JO
0..-OOOCC'.'
<.. .0000000
O.UODCCO
O.OOWliCOO
C.GOCOGCO
8
Q.OOCQCOa
y.oooaoco
cloouucec
O.OCOQOC'O
O.GOOCOCH
o.ooocooa
0.0000000
o.cxioococ
O.GOGOf 00
O.COCDCC-C
o.ooooocc
o.ooooooc
0.0000000
ROW SUM
.0378722
.0187919
.0431814
.0612125
1.4871580
.1245210
.0584116
.0311969
.0455375
.0117497
.00936,83
.0089714
.0243190
2 3
•J.UC COCOO O.OCOCl/CC
O.vJCjiJCCC C.OuOCOf/t1
O.iCOOCCO O.CCittOOor-
J.CVCCCOO u.OCCtt<>0*>
Otccooo^c o. ttor-c-or
cSocco uctSoco
Il'toSS? C.'tC?Wf'
j. «,(, JOC U'J O.COt OCCt
U.'XOCCiC'. C.tCCCO^v
v.OwOwOCO C.OOCJCf
o.ocooooo c.orooooc>
9 1C
C.UCOUOLC f;.OGoC'»rO
c.otoacoa o.oooooou
0.0000009 O.OO'.Jv-C-*)
O.OOOwCOO 0. liC-UoOO'.1
O.OOOCOOO O.OC-WvOw
O.COOOOOO O.CCC«>rOf'
a. 0000000 G.Pt'OvJOX
u.C'OOOoOO C.oC<.OOOr
O.OCOOCOO O.OOC.OwOC
O.OCOOOOO C.CvCOOOC
o.ocoooo'j a.oocoGoc-
O.OCOOOOO O.OOOOOOC
O.CPO'JOO'J 0.0000300
4
C.OSOOCCO
o.rocococ
O.OOCOOOu
O.OOO&CiO
0. 000000 J
C'loooocoo
cloooc"^
il/. 0000000
11
c.or-oocoo
0.0'KCCf'i
C.&COOOJO
O.OCOOOJO
O.OPOOOOO
o.oooooao
o.cooocco
O.OOOOCOO
GROSS HEAVY
5
c.cooroco
o.oocoooc
o.ocotocc
o.occocoo
o.coco^oo
o.ooocooo
o.ctooooc
0.0000000
O.OCOOOOO
o.cooocoo
0.0000000
0.4fOOCOCO
O.OCCOOCD
12
G.COOOOCO
o.ooococo
o.ccoooco
C.C'OOf-000
o.coocooo
o.f-cococo
O.OOOCOOQ
2.CCCCCOO
o.ococooo
D.Ct 00000
0.0000000
O.COOOOOt)
METALS
6
.0360591
.C178924
.0413028
.0196865
.0598452
1.4836697
.C486791
.0571727
.C304C34
.0448677
.0093364
.OC84961
.C082207
.0234656
13
O.OOOOCOO
O.C1C-QOCQ.
e.cocococ
c.oocoooo .
o.oooocoo
C.CCC'COOO
o.ooooooc
0.0000000
o.occoooo
c.ocooooo
c.oooococ
(.'.0000000
o.ooooooc
7
.0018130
.0008995
.0018786
.0009092
.0013673
.00 34883
.0758419
.0012389
.0007934
.0006698
.0024133
.0008722
.0007507
.0008533
1,4
0.0000000
o.ooooooo
O.OCOOOOO
. 9..QJ3000QQL
C. 0000000
O.OOOOQOOL
0.0000000
0.0000000
o.oooocoo
0.0000000
O.OCOOOOO
0.0000000
0.0000000
0.0000000
-------
1
2
3
_*__
5
6
7
B
9
lfi_
11
12
13
1
3
4
5
6
7
fl
9
IP
11
12
13
14.
1
3
4
5
6
7
8
9
10
11
12
13
84.4866177
.1532122
.2888381
.2065699
.4440*528
.0866131
.13157(6
•2806C58
.6953553
.1787150
.266021/5
. 892686C
.6968198
a
.0775481
.1627C78
1.3877595
.2394676
.1647351
.2914096
.0825643
24. 79C976D
.25 IB 229
.2513810
.0918018
.0925403
.1264883
.1397249
ROW SUM
87.4410076
1.7169030
6.2021250
23.3780221
116.8678447
48.9311164
23.1749786
31.5529788
15.6663687
6.9892135
1.8756191
2.0704459
2.5908566
3.6941694
Vt'/(&CCk>4
O.OOOOl'jg
O.OUoOOOO
C.OCOCCOO
O.OCOOCOu
C. Oi 00000
O.wCOwCOO
o.OOOOOOO
U. 00 00000
o.CCGOOOO
b.ocoocoo
....a....... -
.0963233
.Q940005
1.0625042
,3541816
.2C 54081
,2842711 .
.2163493
. .1B01941
11.6370475
__ _, 4251940
.1C86554
.U11979
.1321282
3
O.OOOOCOO
^ tJCWf CL'?^^
tj» C(?C 0*3l>0
c.oooooco
o.cococoo
C.O'JCOOQO
o.cctoooo
O.OOl'OOOO
u.oooooon
o.coooooc
o.oocooro
0.0006000
10
.C415399
.£1745791. .
.2552249
.0570155
.1512192
.0977835
.03C5641
-. .074L522.._
.4831211
1.439618Q
.0757445
.0508250
.0487172
.1931352
4
.6829362
.0196697
.0401546
8(8831571
.0838148
.2361307
.0289543
.0373282
.0289499
.0402673
.C 497 69 3
.0898687
.0366223
.2667016
11
o.oocoooo
C.COOL'OJy
olocc'co-jj
c.ooceooo
O.OCCCCv'J
O.OCOOuOt
_Q.OO_QOr.OC .
O.OOCOo.J'j
"O.OCOC'tvt
c.oocooc
5
.5245952
.4528649
1.4671705
... 2.3856475
113.9593449
3.8359302
.7726300
4.3538795
1.8928173
2,6616720
.4476203
1.0231440
.4445929
c.ccfccocb
u.ococoro
c.oocooco
t.OUOJO'JO
C.OC 0(000
OtVVGOOQO
c.rcococo
O.ilOCCK'Q
O.OC-wuOC-0
TOTAL EFFLUENT
1.0401198
,5161027_._.
1.1913731
.5678537
1.7262259
42.7961966
1.4C41407
._it.64.9J.357_. _
.8769814
1.2942000
.2693C71
.245P686
.2371247
j, 676 8 6.22 ___
13
o.cocotco
...0,.CCCJCJLCO __
C.OOJCCOO
O.OCO.ODQO... .......
O.CCGOCOO
o.ococooo
C.OCOO&CO
.4913274
.5091000
,2463992
.3705268
.9453419
20.5531626
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