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.

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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

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    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.

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         "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.

-------
                     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

-------
                                                                                  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.

-------
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.

-------
                                                                                 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.

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                 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.

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                                                                                  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

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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].

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                                   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.

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                                         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

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                    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.

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                                     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

*
*
*
*
*
*
*
*
*
*
*

I


*
*
*
*
*
*
*
*
*
*

1 ]



*
*
*
*
*
*
* *
* <
* *

?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
*
*
*
*
*
*
*
* 1
* 4
* 4
* si
* ^

-*
*-
*
*
*
*
*
It *
It *
i *
c ^

+
-*
*-
*
*
*
*
*
*
*
£

^

-*
*-
*
*
*
*
*
*

^


*
*-
*
*
*
*
*

+



-*
*-
*
*
*
*

^




-*
*-*
* *-*
* * *-*
* * * * *-*


                                              -*-*-*-*-*-*
                                              -*-*-*-*-*-*-*
                                              -*-*-*-
     bound
                            ********
  -*-*-*-<
  -*-*-*-*-*_*-*_<
  -*-*-*—*-*-*-*-*-*-*-*_*_*
•*-*-*-*-*-*-*-*-*-*-*—*-*_*_#
 *-*-*-*-*-*-*-*-*-*-*-*_*_*_*_«
 *

-------
                  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.

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                                             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.

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                       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

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                                                                                   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

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                                                                                             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

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                                                                                  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).

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                                                                                 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.,

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        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•

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                               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.

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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
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2
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SUM
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1.1846663
8.7491118
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3.5199797
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7.7211946
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-------
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8
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11
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2
3
4
5
6
7
8
9
10
11
12
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1
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6
.0261118
.0299089
.0142557
.0433362
^1.0743815
.0352504
.0414009
.0220163
,C3249Qi
.OC67608
.OC6112J
.0059529
.0169924
13 	 _
O.COOOOOO
O.QCOQSQJD
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0*C£O.QJ2QC
C. 0000000
0.0000000
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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
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16.4147631
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102.9232559
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.0928277
.0600255
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.0464966
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.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
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0.0000000
O.OUOCOOC
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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
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0.0000000
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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
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C*OCOOOOO
... AiUOCCOGCL.
O.OCOGOOO
0. 0000000
o.ocoocoo
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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
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O.OOOOOi'l!
0.0000000
o.&oooooo
C. 0000000
_ 8 	
.0000034
.0000071
.C000610
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.0000072
.0000128
.OCOOC36
.0010692
.0000111
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SUM
.0109437
.OOS435H
.0114038
.0054998
.0083140
.0235009
.4551391
.0086212
.0048241
.0041074
.0145000
.0052521
.0045237
.0051666
i
0.0000003
O.OCOOOCO
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U.OOOOCO'J
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u.cooocou
0.0000000
9
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C. 0000000
0.0000000
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C.OCOOOOO
C.OCOOtUu
O.OOOOQOQ
o.ouooooo
O.OCuCOOO
C.OPOOOOC
O.CCOOCC'J
o.ocootoc
v.OuOOCOO








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o.coooooo
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C..CQCOCOO
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10
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0.0( COCOC
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O.OfOO'JCC
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4 .
O.OOC0003
C.OCOOCuO
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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
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O.OOOCOvO
O.OCOOCO"
O.OOCOCOJ
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PHENOL
5
o.ooooooo
0*JD00001)0. 	
C. 0000000
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O.OCOCQOO
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o.ooooooc
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G.OCOCOOO
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0.0000000
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o.ooocooo
o.ocooooo
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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
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O.OOCOCGO
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O.OOOOOCO
0.0000000
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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
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C.GOCOGCO
8
Q.OOCQCOa
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cloouucec
O.OCOQOC'O
O.GOOCOCH
o.ooocooa
0.0000000
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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
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2 3
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a. 0000000 G.Pt'OvJOX
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4
C.OSOOCCO
o.rocococ
O.OOCOOOu
O.OOO&CiO
0. 000000 J
C'loooocoo
cloooc"^
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11
c.or-oocoo
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C.&COOOJO
O.OCOOOJO
O.OPOOOOO
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O.OOOOCOO


GROSS HEAVY






5
c.cooroco
o.oocoooc
o.ocotocc
o.occocoo
o.coco^oo
o.ooocooo
o.ctooooc
0.0000000
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0.0000000
0.4fOOCOCO
O.OCCOOCD
12
G.COOOOCO
o.ooococo
o.ccoooco
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0.0000000
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METALS






6
.0360591
.C178924
.0413028
.0196865
.0598452
1.4836697
.C486791
.0571727
.C304C34
.0448677
.0093364
.OC84961
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.0234656
13
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0.0000000
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c.oooococ
(.'.0000000
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7
.0018130
.0008995
.0018786
.0009092
.0013673
.00 34883
.0758419
.0012389
.0007934
.0006698
.0024133
.0008722
.0007507
.0008533
1,4
0.0000000
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. 9..QJ3000QQL
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0.0000000
0.0000000
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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
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....a....... -
.0963233
.Q940005
1.0625042
	 ,3541816
.2C 54081
	 ,2842711 .
.2163493
. .1B01941
11.6370475
__ _, 4251940
.1C86554
.U11979
.1321282

3
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0.0006000
10
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.£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
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"O.OCOC'tvt
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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
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...0,.CCCJCJLCO __
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  .4913274
  .5091000
  ,2463992
  .3705268
  .9453419
20.5531626
  .2150227
  .1815259
  .6540058
  ,2363759
  .2034271
14
 O.OOOCOOO
 Q.OQOOQQO
 0.0000000
 IL.MQQO.Q.O..
 0.0000000
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  o.ocooooc
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  C.C-C>Oi/COO
 O.OOOOODO
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 0.0000000
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 0.0000000
...0»fiOjCO..OJlO_
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 0.0000000

-------
                                    REFERENCES


  1.   Frankel, R. J.  Economic Evaluation  of Water quality:  An Engineering-Economic
             Model for Water Quality Management^First Annual Report.SERL Report
             No. 65-3-Berkeley:Sanit. Eng. Research Lab., Univ. of Calif.,
             January 1965.

•  2.   Hansen, W. W. and R. J. Frankel.  Economic Evaluation of Water Quality;  A
             Mathematical Model of  Dissolved  Oxygen Concentration in Freshwater
             Streams.Second Annual Report.SERL Report No. 65-11.Berkeley:
             Sanit.  Eng. Research Lab., Univ. of Calif., August 1965.

  3-   Carew, J. P. and  P. H. McGauhey.  Economic Evaluation of Water Quality;  A
             Linear  Programming Water Quality Control Model.  Third Annual Report.
             SERL Report No. 68-2.Berkeley:Sanit. Eng. Research Lab., Univ. of
             Calif., February 1968.

  k.   Mukherjee, S. K.  Economic  Evaluation of Water Q-uallty:  A Multicomponent Model
             of Optimal  Quality Control in Estuarine Waters .  Fourth Annual Report.
             SERL Report No. 69-2.Berkeley:Sanit. Eng. Research Lab., Univ. of
             Calif., January 1969.

  5.   Ellis, M. M.  "Detection and Measurement of Stream Pollution," Bulletin of the
             U. S. Bureau of Fisheries, k8_(22), 1937-

  6.   Storrs, P. N. et_  al.  A Comprehensive Study of San Francisco Bay, 1961-62 -
             South San Francisco Bay Area, Suisun Bay-Lower San Joaquin River Area,
             San Pablo Bay Area.  SERL Report No- 63-3.  Berkeley:  Sanit. Eng.
             Research  Lab., Univ. of Calif., April 1963.  (Appendices SERL Report
             No. 63-4, same date.)

  7-   The  Nation's Water  Resources,  Summary Report.  United States Water Resources
             Council,  Wash., D.  C., 1968.

  8.   Powell, S. T-   "Water-Quality  Characteristics," in Handbook of Applied Hydrology,
             Ven Te  Chow (Ed.).  New York:  McGraw-Hill Book Co., 1964.

  9-   Stann, E. J. and  R. J. Ringwood.  "A Systems Engineering Approach to Water
             Quality Management," Civil Engineering, ASCE, 39_(6):74, June 19&9-

 10.   San  Francisco Bay — Delta Water Quality  Control Program, Final Report.
             Sacramento:  State  Water Resources Control Board, March 1969-

 11.   San  Francisco Bay Plan Supplement.   San  Francisco Bay Conservation and Develop-
             ment Commission, San Francisco,  15 August 1968.

 12.   Pearson, E. A.  A Comprehensive Study of San Francisco Bay.  (Project includes
             numerous  volumes yet unfinished, dating back to 1962.)  Berkeley:
             Sanit.  Eng. Research Lab., Univ. of Calif., 1962-1970.

 13-   Economic Developnent Agency of California.  California Statistical Abstract,
             1968.   Sacramento:  State Printing Office, 1968.
 14.   Bargur, J.  S.,  H.  C- Davis, and E. M. Lofting.  An Eighty-One Sector Input-Output
             Table for  The  California Economy for  1963.  Berkeley:  Sanit. Eng.
             Research Lab., Univ.  of Calif., May 1969  (mimeo.).
                                         115

-------
116
REFERENCES (continued)

15.  U. S. Department of Commerce, Office of Business Economics.   The National Income
             and Product Accounts of the United States,  1929-1963 :  Statistical
             febles .  A supplement to the Survey of Current Business.  Wash.,  D.  C.,


16.  Economic Development Agency of California.  California Statistical Abstract,
             1964.  Sacramento:  State Printing Office,  1965-

17.  U. S. Department of Commerce, Office of Business Economics.   "The  Transactions
             Table of the 1958 Input -Output Study, and Revised Direct and Total
             Requirements Data," Survey of Current Business, Vol. k$, No. 9,
             September
 18.  Walras, L. (tranl. by W. Jaffe).  Elements of Pure Economics.  Homewood,  111.:
             Richard D. Irwin, Inc.,
 19.  Leontief, W.  "Quantitative Input -Output Relations in the Economic System of
             the U. S.," Rev. EC on. and Stat., l8(3) :105-25, August 1936.

 20.  Baumol, W. J.  Economic Theory and Operations Research.  Englevood Cliffs,
             N. J.:  Prentice -Hall, Inc., p. 309, 19&1-

 21.  Carasso, M.  A FORTRAN IV Computer Program for Preparing a Regional Inter-
             industry Transactions Table from Secondary Data Sources, program
             developed by Bargur, Davis, and Lofting, Univ. of Calif., Berkeley,
             1968.

 22.  Moore, F- T- and J. W. Petersen.  "Regional Analysis:  An Interindustry Model
             of Utah," Rev. EC on. and Stat., 51:368-83, November 1955-

 23.  Fortune Iferketing Service.   1966 Input -Output Matrix, Fortune's Input/Output
             Portfolio.  New York:  Time Inc., 1968 •

 2^.  Isard, W. and E. Romanoff.   Water Use and Water Pollution Coefficients,
             Tech. Paper 6, Regional Inst., Cambridge, Mass., November 1967 (mimeo.).

 25.  Final Report, Task II -U, Determination of Present Water Use and Waste Loads.
             Kaiser Engineers Report to the California State Water Quality Control
             Board for the  project San Francisco Bay-Delta Water Quality Control
              Program,  15 December 1967-

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