Integrated approach to urban wastewater management


Material Information

Integrated approach to urban wastewater management
Physical Description:
xxi, 230 leaves : ill. ; 28 cm.
Hasan, Sheikh Mohammad, 1933-
Publication Date:


Subjects / Keywords:
Urban runoff -- United States   ( lcsh )
Sewage -- Environmental aspects   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis--University of Florida.
Includes bibliographical references (leaves 221-229).
Statement of Responsibility:
by Sheikh Mohammad Hasan.
General Note:
General Note:

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 000171114
notis - AAT7536
oclc - 02948793
System ID:

Full Text







To A{f Farniey



I would like to thank the members of my graduate

committee for their encouragement and advice throughout

the research effort. The primary influence for the overall

research has come from Dr. J. P. Heaney and Dr. W. C. Huber.

I am deeply grateful for their guidance. I would especially

like to thank Dr. J. P. Heaney for his undivided attention

and assistance from the early conception of the research

to the final days of writing and review.

I wish to acknowledge the assistance of many

people who have directly or indirectly helped me in this

research. Special thanks are due to Mr. Michael P. Murphy,

who generated the stormwater isoquants, and Mr. Henry Malec,

who developed the mathematical equations for these iso-

quants. Special thanks are also due to Ms. Mary Polinski

for typing the rough draft of the manuscript. The excellent

typing of Pat Whitehurst on the final draft is sincerely







NOTATION .......

ABSTRACT .......


1 INTRODUCTION ..................................


2.1 Introduction .............................
2.2 Wastewater Characteristics ...............
2.3 Wastewater Control Devices ...............
2.4 Wastewater Treatment Costs ...............
2.5 Control Criteria ........................
2.6 Hypothetical Planning Area ...............

PLA NN ING ......................................

3.1 General Statement of the Problem .........
3.2 Economic Analysis ..................
3.3 Multiobjective Planning ..................
3.4 Efficiency and Equity ....................
3.5 Multipurpose-Multigroup Planning .........
3.6 Summary ..................................


4.1 Problem Definition .......................
4.2 Strategies for Dry-Weather Quality
Co n tro l ... .. .. .... .. ... ... .. ... ... .
















. . . . v




4.3 Optimization Procedure ............... 96
4.4 Illustrative Example ................. 99
4.5 Summary .............................. 103

5 STORMWATER MANAGEMENT ..................... 105

5.1 Problem Definition .................... 105
5.2 Strategies for Wet-Weather Quantity
Control ............................. 106
5.3 Wet-Weather Quantity Control
Optimization ....... ........ ........ 110
5.4 Strategies for Wet-Weather Quality
Control ............................ 116
5.5 Wet-Weather Quality Control
Optimization ......................... 135
5.6 Summary .............................. 142


6.1 Problem Definition ....... ............ 143
6.2 Strategies for Multipurpose Wastewater
Management ............... ............ 145
6.3 Optima1 Sequence of Dry-Weather and
Wet-Weather Quality Control .......... 156
6.4 Optimal Nultipurpose Wastewater
Management ........................... 160
6.5 Summary ............................... 165


7.1 Proolem Definition ................... 167
7.2 Efficiency/Equity Criteria ........... 170
7.3 Multiobjective Solution Techniques ... 178
7.4 Conventional Cost-Sharing/Cost-
Allocetion Techniques ................ 184
7.5 Cooperative N-person Same Theory ..... 189
7.6 Application of Game Theory to Cost
Sharing/Cost Allocation ....... .... 201
7.7 Relationship Between Game Theoretic
Approach and Conventional Cost-Sharing/
Cost-Allocation Procedures ........... 210
7.8 Summary .............................. 214

8 SUMMARY .................. ................. 218

REFERENCES .. ....................................... 221

BIOGRAPHICAL SKETCH ............................... 230


Table Page

2-1 Typical Values of Parameters for Domestic
Sewage 17

2-2 Wet-Weather Pollutant Loading Factors 21

2-3 Domestic Wastewater Treatment Performance
Data 27

2-4 Wet-Weather Treatment Plant Performance
Data 29

2-5 Cost Functions for Domestic Wastewater
Control 30

2-6 Installed Cost for Wet-Weather Treatment
Devices 32

2-7 Cost Functions for Wet-Weather Control
Devices 33

2-8 Capital Cost of Storage Facilities 35

2-9 Characteristics of Hypothetical Planning
Area 40

4-1 Annual Costs for Various Strategies for
Domestic Wastewater Management-Hypotheti-
cal Planning Area 104

5-i Data for Wet-Weather Quantity Control-
Hypothetical Planning Area 112

5-2 Computations of Required Storage Volume for
Wet-Weather Quantity Control -Hypothetical
Planning Area 113

5-3 Annual Costs for Wet-Weather Quantity
Control-Hypothetical Planning Area 117


Table Page

5-4 Values of Parameters and Ccrrelation Co-
efficients for Isoquant Equations--Percent
Runoff Control or Percent BOD Control
Without First Flush 128

5-5 Values of Parameters and Correlation
Coefficients for Isoquant Equations-
Percent BOD Control with First Flush 133

5-6 Design Data for Wet-Weather Quality
Control-Hypothetical Planning Area 140

6-1 Annual Costs for Various Multipurpose
Wastewater Management Strategies. City
6-Hypothetical Planning Area 166

7-1 Apportionment of Annual Costs Using Game
Theoretic Approach: Domestic Wastewater
Management-Hypothetical Planning Area 211

7-2 Comparison of Characteristic Functions
Used by Alternative Cost Method and
Shapley Value Method 213

7-3 Comparison of Cost Apportionment Using
Game Theoretic Approach vs. Alternative
Cost Approach: Domestic Wastewater
Management-Hypothetical Planning Area 215



Figure Page

2-1 Generalized Performance Curve of a Given
Size Dry-Weather Plant 26

2-2 Hypothetical Planning Area 39

3-1 Determination of Optimal Level of Input
and Output: One Input, One Output 46

3-2 Determination of Optimal Combination of
Inputs: Two Inputs, One Output 48

3-3 Determination of Optimal Combination of
Outputs: One Input, TWo Ouputs 51

3-4 Generalized Isoquants for Dry-Weather
Control 54

3-5 Generalized Isoquants for Wet-Weather
Control 55

3-6 Edgeworth Box: Two Inputs, Two Outputs,
Two Purposes 57

3-7 Noninferior Set for Two Outputs 59

3-8 Determination of Best Compromise Solution 64

3-9 Cost Function for On-Site Stormwater
Control 69

3-10 Network Representation of Example Problem 72

3-11 Shadow Price for Area 1 for Assumed Value
of Q1 75

3-12 Demand for Off-Site Discharge 80

3-13 Relationship Betwee.i Tncuts and Ououts-
Urban Wastewater Management 85


Figure Page

4-1 Permitted Piping-Treatment Combinations
for Domestic Wastewater Management with
Upstream Pumping-Hypothetical Planning
Area 100

4-2 Piping-Treatment Combinations for Domes-
tic Wastewater Management Without Up-
stream Pumping-Hypothetical Planning
Area 102

5-1 Effect of Urbanization on Peak Discharges 107

5-2 Network Representation of Wet-Weather
Quantity Control Problem-Hypothetical
Planning Area 115

5-3 Storage Versus Treatment for Wet-Weather
Quality Control 119

5-4 Optimal Storage-Treatment for Wet-Weather
Quality Control 120

5-5 STORM Model Simulation of Storage and
Treatment for Wet-Weather Quality Control 121

5-6 Regional Boundaries for Nation-Wide
Assessment 123

5-7 Storage-Treatment Isoquants for Various
BOD Control Levels--Region IV: Atlanta 124

5-8 Storage-Treatment Isoquants for Various
BOD Control Levels--Region IV: Atlanta 131

5-9 Comparison of a Storage-Treatment Isoquant
with and without First Flush--Region IV:
Atlanta 132

5-10 Control Costs for Primary and Secondary
Devices as a Function of Percent BOD
Removal-Region IV: Atlanta 137

6-1 Pollutant Removal in a Primary Tank as a
Function of Detention Time 149

Fi ure Page

7-1 Illustration of Equity-Criteria for Al-
location of Capacity Example 171

7-2 Illustration of Equity Criteria for
Cost-Sharing/Cost-Allocation Example 173

7-3 Illustration of "Ideal" Solution by
STEM for Resource Allocation Problems 181

7-4 Illustration of "Ideal" Solution Sys-
tem for Cost-Sharing Problem 182

7-5 Illustration of Conventional Cost-Shar-
ing/Cost-Allocation Techniques 188

7-6 Relationship Between Edgeworth Box and
Core 197

7-7 Illustrations of Game Theoretic Approach
for Cost Sharing/Cost Allocation 203


The following symbols have been adopted for use

in this study:

a = coefficient (inches per hour)

a.. = coefficient for city i in region j (inches per hour)

A = area (acres)

AR = annual runoff (inches per year)

ARi = annual runoff in city i (inches per year)

AR. = annual runoff in test city for region j (inches
per year)

b = coefficient (inches per hour)

b. = coefficient for test city in region j (inches per
J hour)

b.i = coefficient for city i in region j (inches per hour)

B = runoff coefficient

BD = runoff coefficient after development

B = runoff coefficient prior to development
BOD = biochemical oxygen demand (mg/1)

BOD = biochemical oxygen demand (mg/l)at 5 days (mg
BOD5 = biochemical oxygen demand at 5 days (mg!l)

c = unit cost of control ($)

c. = unit cost of resource i ($)

c = initial concentration (mg/1)

c = concentration at time t (mg/1)

c = unit cost of control i in area j ($)

cE = unit treatment cost for excess capacity (annual
collars per inch per hour

cS = unit cost of storage (annual dollars per acre-inch)

cT = unit cost of treatment (annual dollars per acre-inch
per hour)

c(i) = characteristic function representing individual
cost ($/year)

c(ij) = characteristic function representing joint cost

cD = unit cost of dry-weather treatment (dollars per

c(S) = characteristic function representing cost to caoli-
tion S ($/year)

c(N) = characteristic function representing total cost
C = annual control cost to city i for treatment level

C. = annual control cost to cities i and j for treatment
level (5/year)

CA = amortized capital costs ($/year)

CA = amortized capital costs for treatment level y

CA = amortized capital costs to city i for treatment
I level 1 ($/year)

d = coefficient (inch- )

d.. = coefficient for city i in region j (inch-l)
D = annual average dry-weather flow (mgd)

D = design capacity of dry-weather plant (mgd)

Di = annual average dry-weather flow from city i (mgd)












F (y)










= design capacity of dry-weather plant (mgd)(city i)

Annual quantity of dry-weather BOD (pounds per

= annual quantity of dry-weather flow (inches per

= dissolved oxygen (mg/l)

= annual cost of wet-weather pollution control,
in eastern U. S., usinr primary devices ($/acre)

= annual cost of wet-weather pollution control,
in eastern U. S., using secondary devices

= maximum weighted difference between an objective
and its maximum value

= coefficient (percent R)

=coefficient for city i in region j (percent R)-

= total annual cost ($.)

= sum of resource costs ($)

= sum of damages ($)

= resource costs to purpose or group j ($)

= production function

= constraint set

= constraint set for purpose or group j

= constraint set

= production function relating percent control
(R) to storage (S) and treatment (T)

= conversion factor

= coefficient (percent R)1

= coefficient for test city in region j (percent R)-

= coefficient for city i in region j (percent R)-1

= imperviousness, a fraction of percent


i = area or input

I = intensity of rainfall (inches per hour)

IC, = incremental cost of dry-weather treatment ($)

j = control option or output

J = constant
k = BOD removal rate constant, time-

K = constant

KA = constant

1 = constant

1= constant for treatment level p

L = pipeline distance (miles)

L. = pipeline distance from i to j (miles)
L*. = break-even pipeline distance from i to j (miles)
LA = Lagrangian function

m = exponent (less than 1)

m, = exponent for treatment level y (less than 1)

M = pollutant loading averaged over different land
uses (Ibs/year)

M1 = pollutants removed from wet weather (Ibs/year)

M* = optimal amount of pollutants removed from wet
weather prior to initiating tertiary treatment

Mc = pollutant loading in combined sewered areas

M. = maximum value of objective j
Ms = pollutant loading in separate sewered areas

Me. = increased dry-weather BOD removal through ter-
ert tiary treatment (lbs/acre)

MB. = marginal benefit of output j ($)

MC. = marginal cost of i ($)

MRS = marginal rate of substitution

MRT = marginal rate of transformation

N = nitrogen content of wastewater (mg/l)

NSC = nonseparable cost ($/year)

OM = operation and maintenance cost ($/year)

OM = operation and maintenance cost for treatment level
I ($/year)

OM. = operation and maintenance to city i for treatment
1 level ($/year)

P = constant

P = constant for treatment level P

P. = unit price for output j ($)

P = precipitation rate (inches per year)

PD = population density (persons/area)

q = exponent (less than 1)

q = exponent for treatment level 4

Q. = release for city or area i

r1 = constant

r = constant

R = percent runoff control

R = percent pollutant control

RI = maximum percent pollutant control

RO = runoff rate (cubic feet per sec)

s = number of players

S = coalition with s players









= storage volume (mg or inches)

= optimal storage volume (inches)

= separable cost to city in purpose i

= detention time, time

= unit cost of transmission for area i ($)

= treatment rate (inches per hour)

= optimal treatment rate (inches per hour)

= coalition T

= treatment rate at which isoquant is parallel
to the ordinate (inches per hour)

= treatment rate at which isoquant intersects
the abcissa (inches per hour)

= total annual cost ($/year)

= total annual cost for treatment level i ($/year)

= total annual cost for secondary treatment

= total annual cost for tertiary treatment ($/year)

= constant

= constant for treatment level p

= constant

= constant for treatment level 4

= characteristic function for player i

= characteristic function for coalition S

= characteristic function for all players

= volume of storage required for wet-weather quan-
tity control (acre-feet)

= maximum allowable release (acre-feet)

= wet-weather quantity control volume required by
city i(acre-feet)











w = constant

wI = constant

w2 = constant

wzp = cost of wet-weather control using primary device
for the western U. S. ($/acre)

wzs = cost of wet-weather control using secondary device
for the western U. S. ($/acre)

w allowable release

x = input

xi = ith input

x(i) = cost assigned to group or purpose i

x(ij) = quantity controlled by th control option in
area i

x.. = maximum available control by jth control option
1J in area i

X = input vector

y = outputs

y = jth output

y = output vector

z = constant

z, = constant

z2 = constant

Z = total annual cost ($/acre)

Z* = optimal total annual cost (dollars per acre)

Zk(X) = kth objective

Z = annual cost for primary control unit (dollars per
P aacre)

Z = annual cost for secondary control unit (dollars
per acre)


Z = annual cost of dry-weather quality control (dollars
per acre)

Z2 = annual cost of wet-weather quality control (dollars
per acre)

Z = annual cost of wet-weather quantity control
3 (dollars per acre)

Z = annual cost of dry-weather quality control and
I2 wet-weather quality control (dollars/acre)

Z = annual cost of wet-weather quality control and
23 wet-weather quantity control (dollars/acre)

ZI3 = annual cost of dry-weather quality control and
wet-weather quantity control

Z = annual cost of dry-weather quality, wet-weather
123 quality, and wet-weather quantity control (dollars
per acre)

c(i,j) = pollutant load j from separate area land use i

B(i,j) = pollutant load j from combined area land use i

Si = allocation vehicle

Y = imputation vector

Yi = imputation i
6 = imputation vector

64 = imputation i

S= treatment level

= treatment level

n = efficiency of treatment

T = shadow price

.i = cost sharing and/or cost allocation

X = Lagrange multiplier

n = efficiency of secondary treatment

n = efficiency of tertiary treatment


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Sheikh Mohammad Hasan

March, 1976

Chairman: J. P. Heaney
Major Department: Environmental Engineering Sciences

Traditionally, the problem of urban wastewater

management has been viewed as the control of urban storm

drainage and dry-weather sewage treatment. Each of these

purposes has been accomplished independently in a least-

cost manner. With the enactment of Public Law 92-500,

communities within the designated urban areas are required

to plan for coordinated wastewater management. A third

facet of urban wastewater management. i.e., wet-weather

quality control, has also become important as the nation

strives to achieve the 1983 goal of swimmablee and fishable"

waters. Further, the emphasis of urban wastewater manage-

ment has shifted from a "least-cost" solution to a "cost-

effective" solution.

In this work, the urban wastewater management prob-

lem is viewed as a multigroup, multipurpose and multiob-

jective problem. Procedures have been developed for
x i x

formulation of control strategies and for evaluating their

costs for accomplishing each purpose independently. If

the three purposes are viewed as a part of the overall

management plan, then it is possible to take advantage

of the complementarities that exist among these purposes

in order to reduce the cost of urban wastewater management.

Procedures for evaluating multipurpose strategies are also

presented. The identification of the cost-effective solu-

tion then involves identifying tradeoffs between monetary

and nonmonetary costs of various strategies and the selection

of a best compromise solution.

The equity question deals with apportioning of

the project costs among groups/purposes in a fair manner.

Economic analysis indicates that users should be charged

the marginal cost of adding them to the plan. However,

in single-purpose problems involving several groups and

in multipurpose problems, if purposes) are accomplished

simultaneously, charges equalling marginal costs do not

raise enough revenues to cover total project cost. For

this reason, it is customary to apportion the cost of a

multipurpose or a multigroup plan so that each user pays

its marginal or separable cost plus a portion of the joint

or nonseparable cost. The conventional techniques for

accomplishing this do not necessarily result in an equitable

arrangement. Further, they are difficult to use in multi-

purpose wastewater management. The equity problem has

been viewed in the context of N-person cooperative game

theory. It is shown that these concepts when applied to

cost sharing/cost allocation have desirable equity proper-





Over the last century, the United States has under-

gone a transformation from an essentially rural to an ur-

ban society. In the process, two very definite trends

have been established. First, people have concentrated

in urban areas. In 1970, two-thirds of the total popula-

tion was living in Standard Metropolitan Statistical Areas

(SMSA's) (Bureau of Census, 1971). The second noticeable

trend of concern has been a steady decrease in average

urban population density per unit area (Pickard, 1967).

Taken together, these two trends result in massive urban


During the transition period, streets that were

largely unpaved with side sales or ditches that stored

appreciable amounts of rain before any runoff occurred

were replaced by smooth pavement and curbs. Natural

tributary channels and water courses were replaced by

smooth, lined underground conduits. Large parcels of

undisturbed land were converted into parking lots and

shopping centers. These factors combined to create an

urban drainage problem of tremendous magnitude. Engineering

response to these drainage needs consisted of intercepting

the surface runoff (also called wet-weather flow) as

quickly as possible and then conveying this stormwater

by means of storm sewers to the nearest water course.

These practices caused tremendous increases in the peak

stormwater runoff discharged to the receiving water

resulting in stream bank overflows, increases in frequency

of flooding, and instream erosion due to high velocities.

In addition to stornwater runoff, cities have also

been traditionally concerned with disposal of domestic

wastes (also called dry-weather sewage flow). Continuous

discharge of relatively small concentrated volumes of

dry-weather sewage is of concern to public health offi-

cials as it contains infectious bacteria. The strategy

has been to dispose of this wastewater in the least expen-

sive manner. At the outset, sewers designed for storm-

water runoff were utilized to carry dry-weather sewage

flow and reliance was placed on the capability of surface

waters to provide enough dilution to this water to pre-

vent any health hazard. As the areas increased in size,

the dilution capability of surface waters was overtaxed.

Objections from citizens and public health officials

forced cities to install treatment facilities at the end

of the sewer network to provide a minimum level of treat-

ment to the dry-weather sewage flow. These facilities

were sized to accommodate two to five times the average

dry-weather flow. This excess capacity was provided pri-

marily to accommodate the normal daily or seasonal varia-

tions in the dry-weather flow. If, in the event of a

rainfall, the rate of flow was larger than the treatment

capacity, the excess flow containing a mixture of untreated

sewage and stormwater was bypassed directly to the receiving

water. This phenomenon is frequently referred to as a

combined sewer overflow.

As public awareness of water pollution grew, the

Water Quality Act of 1965 was enacted. Under the Act,

the first step was to establish the "beneficial use" to

which a particular reach of water was to be put. Then

water quality standards that would protect this use were

established. The pollution budget was then allocated,

ignoring non-point sources (wet-weather flows) as well

as future growth patterns, among various polluters in

the area. Cities and industries were then issued waste

discharge permits, allowing them to discharge pollution

up to a certain specific amount. Since this Act, in general,

required a higher level of treatment to dry-weather flow,

communities began to investigate coordinated or regional

wastewater treatment strategies. The objective is to

take advantage of the economies of scale in wastewater

treatment and/or less restrictive discharge requirements

of some locations. However, there has been little success

in implementing such proposals due partially to the

nonexistence of a real-world regional authority with neces-

sary power to shift decisions in this direction.

During the late 1960's, concern was expressed

regarding the discharge of combined sewer overflows since,

during storms, untreated combined sewage (mixture of dry-

and wet-weather flows) is directed to the receiving water

(Burm and Vaughan, 1966). Cities were installing separate

sewer systems in newly developing areas to carry polluted

domestic wastewater and supposedly clean stormwater in

separate pipes. Also, programs were initiated to separate

the existing combined sewer systems. However, preliminary

cost estimates indicated that this separation program

would be very expensive (American Public Works Association,

1967). In addition, concurrent stormwater sampling pro-

grams indicated that stormwater from urban areas contains

significant quantities of pollutants picked up from street

surfaces, parking lots and drainage structures (Benzie and

Courchaine, 1966). Uncontrolled discharge of these flows

may have substantial impact on receiving water quality. It

has recently been shown that, approximately 20 percent of

the time, receiving water quality in an urban area is con-

trolled by stormwater runoff (Colston, 1974).' Thus, sewer

separation may not be very cost effective. Recognition of

this situation has led many communities to institute con-

trols on stormwater quantity and quality. As an example,

regulations of Orange County, Florida state that treatment

is required for stormwater in all drainage systems to

reduce pollutants and other objectionable material con-

tained in the stormwater runoff (Orange County, 1972).

A great deal of controversy presently exists on

the question of treatment of stormwater and combined sewer

overflows versus treatment of dry-weather sewage flow

beyond secondary treatment. Studies have shown that peak

stormwater runoff after development ranges from three to

eight times the redevelopment peak runoff (Carter, 1961;

Waananen, 1961). Other recent studies (Field and Lager,

1974) indicate that

(1) Biochemical oxygen demand (BOD5) con-
centrations of stormwater approximately
equal the strength bf domestic sewage
after secondary treatment from the same
land use. For combined sewer overflow,
the BOD5 loadings average approximately
one-half the strength of untreated domes-
tic sewage.

(2) The total suspended solids (TSS) content
of the stormwater runoff is generally
about three times that of untreated sew-
age but consists mostly of inorganic

(3) Bacterial coliform content of runoff is
about two or four orders of magnitude
smaller than untreated sewage. However,
it is two to five orders of magnitude
higher than is considered safe for water
contact recreation.

Research has also revealed that a "first flush" phenomenon

is exhibited during the initial periods of a storm event.

Many of the pollutants in stormwater runoff are carried off

during this period (Bryan, 1972). This is also true for

combined sewer overflow (Lager and Smith, 1974). Due to

these factors some people nave argued that it may be more

cost-effective to treat combined sewer overflow and/or

urban stormwater runoff (Bryan, 1972). This question re-

mains unresolved. The management of water quality in

urban areas is further complicated hy the fact that these

areas usually encompass more than one city and the waste-

water problems and management goals of each city within

the area differ. As a result of all .ne above factors a

general degradation in the water quality in urban areas


Because of the public health aspects of water

quality, states and the federal government have played

the role of regulating agencies. Subsidies have been

provided to the local governments as an incentive to

control dry-weather flows. By enacting Public Law 92-500,

Congress initiated the most comprehensive program against

water pollution the nation has ever experienced. Known

as the Federal Water Pollution Control Act Amendments of

1972, the Act has two general goals (1) to achieve

where possible, by July 1, 1983, water that is clean

enough for swimming and other recreational uses, and

clean enough for the protection and propagation of fish,

shellfish and wildlife; and (2) by 1985, to have no dis-

charge of pollutants into the nation's waters. Attainment

of these goals is expected to require commitments of

national resources far in excess of those ever committed

before to water pollution control. In a recent report to

Congress, the costs of wastewater treatment plants required

between now and 1983 are estimated at $33 billion. Another

$235 billion is estimated for controlling stormwater sys-

tems (Environmental Protection Agency, 1974). Public Law

92-500 provides for 75 percent construction grants for

water pollution control works.

To encourage efficient use of national resources in

accomplishing the above goals, Public Law 92-500 requires

preparation of Basin Plans (Section 303), Area-wide Waste

Treatment Management Plans (Section 208), and Facility

Plans (Section 201). The Basin Plan is a management docu-

ment, which identifies the water quality problems of a

particular basin and sets forth an effective remedial

program to alleviate those problems. It is neither a

broad water and related land resources plan nor a basin-

wide facilities plan. The value of the basin plan lies

in its utility in making water quality management decisions

on a basin-wide scale. The Area-wide Waste Treatment Plan

is a document designed for water quality management in

those urban areas, within the basin, having a substantial

water quality problem. The plan is directed towards meet-

ing the 1983 goal of the Act. It identifies anticipated

municipal and industrial treatment works construction over

a 20-year period. It also identifies urban runoff and

other non-point sources of pollution and methods to control

these sources to the extent feasible. The plan also sets

out construction priorities over the next five-year and

20-year periods and designates agencies necessary to con-

struct, operate and maintain the control facilities. It

is also a mechanism which will be used for regulatory

purposes and for issuing waste discharge permits. The

Facilities Plan is a document similar to the area-wide

plan except that it is much narrower in scope and detailed

consideration of non-point sources is not required.

This work is motivated by the ongoing research at

the University of Florida on management of stormwater and

combined sewer overflows. The research problem is to devel-

op (1) a decision making model for formulation and evalu-

ation of a set of strategies for controlling dry-weather

and wet-weather flows in order to identify a cost-effective

solution to urban wastewater management problems in the con-

text of Section 208; and (2) procedures for apportioning the

costs of the selected strategy amongst various project pur-

poses and groups in a fair and equitable manner in order to

enhance the implementation feasibility of that strategy.

In this work, the problem of urban wastewater management

is viewed as a rmultiobjective, multipurpose as well as a

multigroup problem as opposed to single purpose, single

group problems considered ir the past. iWh le the single

purpose, single group problems have traditionally been

solved for least-cost, multipurpose/multigroup problems

need to evaluate equity as well as efficiency. These

objectives are demonstrated by means of examples presented

in Sections 3.1 and 3.4. Mathematical formulations of the

urban wastewater management problems incorporating effi-

ciency/equity are presented in Section 3.5. New procedures

for attaining efficiency as well as equity are presented

in Section 7.4 and demonstrated by means of examples in Sec-

tion 7.5. Single purpose, single group analysis for deter-

mining a least-cost solution have been generalized and ex-

tended so that alternative strategies can also be formulated

and evaluated. The procedures for accomplishing this for

domestic wastewater management are discussed in Section 4.1

and for wet-weather quantity control in Section 5.1. For wet-

weather quality control, due to its recent importance and

lack of sufficient data, it is necessary to develop perfor-

mance and cost data for various control devices in addition

to the procedures for formulation and evaluation of strate-

gies. The performance data are developed in Section 2.3 and

the cost data in Section 2.4. Procedures for formulation

and evaluation of wet-weather quality control strategies are

developed in Section 5.4. In Section 6.1, engineering con-

cepts that may be utilized to increase the treatment capabil-

ities of the dry-weather facilities are presented in order

that these facilities may be used for wet-weather quality

control. Procedures for evaluation of various multipurpose

strategies are discussed in Sections 6.3 and 6.4.

Chapter 2 presents an overview of Section 208 of

the 1972 Act Arendments, characteristics of wastewater and

procedures for estimating these parameters. A review of

various control devices is also included and cost functions

for these devices are developed. A brief description of

Basin Plans establishing control criteria is presented and

data on a hypothetical planning area are outlined.

Chapter 3 presents an overview of the planning

objectives discussed in this work. Economic analysis is

presented first. Subsequently, multiobjective planning

is discussed and the need for efficiency and equity in

wastewater management is outlined, and mathematical formu-

lations are developed.

Chapter 4 presents optimization procedures for

determining the resource costs associated with various

strategies for domestic wastewater management. The use

of these procedures is illustrated by solving the domestic

wastewater management problem of the hypothetical planning


Chapter 5 presents procedures for formulation and

evaluation of strategies for wet-weather control. The

wet-weather quantity control problem is discussed first.

Techniques for determining resource costs associated with

various wet-weather quantity control strategies are pre-

sented. Subsequently the wet-weather quality problem is

discussed and optimization and evaluation procedures for

wet-weather quality control strategies are described.

The wet-weather control problem of the hypothetical plan-

ning area is solved to illustrate the various concepts.

Chapter 6 discusses procedures for multipurpose

planning for water quality control. The concept of joint

dry-wet weather treatment is presented and optimization

procedures for evaluating the resource costs for various

multipurpose strategies are outlined.

Integrated efficiency and equity analysis is

presented in Chapter 7. Multiobjective solution tech-

niques as well as existing cost sharing/cost allocation

techniques are discussed. A brief review of cooperative

N-person game theory is presented. Subsequently, appli-

cation of these concepts for resolving the equity ques-

tions is outlined and its relationship to existing cost

sharing/cost allocation procedures is highlighted.

Lastly, Chapter 8 summarizes the various concepts

discussed in this work.



2.1 Introduction

Section 208 of the Federal Water Pollution Control

Act (FWPCA) Amendments of 1972 emphasizes the development

of sophisticated area-wide waste management systems which

will result in the successful management of water quality

at the substate level in highly complex areas of urban-

industrial concentrations. An urban-industrial concentra-

tion is that portion of a standard metropolitan statis-

tical area (SMSA),or those portions of SMSAs, having sub-

stantial concentrations of population and manufacturing

production or other factors which result in substantial

water quality control problems. In such areas, water

pollution sources can be categorized as point and non-

point sources. A point source discharges its effluent

into the water body through a pipe or conduit, e.g.,

municipal wastewater treatment plant or industrial waste

effluents. A non-point source is diffuse; it either seeps

into the ground through the soil or is carried over the

surface of the land by rainwater. Runoff from urban areas,

construction sites, farms, forest lands, and strip mines

produces non-point source pollution. The stated objectives

of area-wide waste treatment management are to provide

(Federal Reaister, 1973)

(1) cost-effective point source treatment
and control;

(2) control of non-point sources; and

(3) coordinated wastewater management.

Pollution emanating from industrial wastes, con-

struction sites, farms, forest lands and strip mines is

highly site specific. Thus, control of those sources

would depend upon local conditions. Dry-weather sewage

flow and urban stormwater are the only two sources for

which generalized data can be used for estimating volumes

and pollutant loadings. Further, a generalized approach

can be used to formulate and evaluate control strategies

for these sources. Thus, in this work only these two

wastewater sources will be considered.

Designation of area-wide planning boundaries, the

key first step in implementing Section 208, is based on

the following criteria (Federal Register, 1973):

(1) The area must have substantial water
quality control problems, i.e.,.when
water quality has been degraded to the
extent that desired uses are impaired
or precluded.

(2) The area has in operation a coordinated
waste treaLtent: management system or
the local governments within the area
show their intent to develop and imple-
ment a plan which will result in coor-
ainated waste treatment management.

Following the designation of the area, it is neces-

sary to inventory existing wastewater sources, estimate

present and future wastewater volumes and pollutant load-

ings, develop cost data on various wastewater control

devices and determine the extent of controls that would

be required to alleviate or prevent water quality problems.

Based on this information various control strategies can

be formulated and evaluated to determine a cost-effective


In this chapter various pollutants that are of

significance will be described and cost-data on various

control devices will be presented. Criteria for deter-

mining the extent of controls required will be outlined.

Data on a hypothetical planning area will then be pre-

sented in order that the concepts outlined in later chap-

ters may be demonstrated in the context of this planning


2.2 Wastewater Characteristics

Several parameters are used to define the charac-

teristics of domestic wastes and stormwater. The four

most widely used parameters are wastewater volume, bio-

chemical oxygen demand (BOD), suspended solids (SS) and

coliform organisms. Other parameters of importance in-

clude dissolved oxygen (DO), temperature, pH, nitrogen (N)

and phosphorous (P).

The volume of wastewater is important for three

reasons. First, it determines the size of the conveyance

and treatment facilities. Secondly, it is needed to de-

termine the mass loadings of pollutants. Finally, controls

on quantity may be required with or without controls on

quality. The latter is especially true for wastewater from

urban stormwater runoff where control of flooding may be

one of the objectives of planning.

The BOD of wastewater is a measure of its strength

in terms of quantities of oxygen required to biochemically

oxidize the organic matter contained in wastewater. When

a wastewater is released, the dissolved oxygen content of

the receiving water is utilized to satisfy the biochemical

oxygen demand of wastewater thereby depressing or even

depleting the DO of the receiving water. The DO of the

receiving water is a major parameter which determines its

beneficial use. Wastewater BOD is usually expressed in

terms of the amount of oxygen utilized during a five-day

period (BOD5). The higher the BOD5, the more damaging

the waste is to the receiving water. To date, BOD5 is

perhaps the most important parameter used for evaluating

wastewater control strategies.

Suspended solids refer to the solids which can be

mechanically filtered from wastewater. The suspended

solids render the wastewater unsightly and unless removed,

tend to settle in the receiving water.

Coliform organisms are organisms of intestinal

origin and are present in large amounts in municipal

sewage. Their presence in a water supply source is an

indication of pollution from human sources. Coliform

levels are commonly reported as the most probable number

of organisms per 100 milliliters of sample (MPN). Sewage

requires disinfection before being discharged into the

receiving water.

Nitrogen and phosphorous content of the wastewater

is important when the receiving water body is a lake where

continuous discharge of nitrogen and phosphorous results

in its fertilization. This is characterized by explosive

"blooms" of algae of such intensity that clear, sparkling

lakes are transformed to turbid colored bodies of water.

Decomposition of the algae then produces obnoxious odors

and floating decomposing mats of organic matter.

The volume of domestic wastes and the quantity of

pollutants originating within an area is a function of

land use, population density, market value of the dwelling

units, use of garbage grinders and characteristics of

water supply. Typical values of various parameters of

dry-weather sewage flow are listed in Table 2-1. Assuming

an annual average domestic wastewater flow of 100 gallons/

person-day and an annual average domestic wastewater BOD5

of 0.17 pounds/person-day, the following equations result:

Table 2-1

Typical Values of Parameters for Domestic Sewage


Typical Values

Volume 100 gallons per capital per day

BOD5a 200 mg/l

SSa 200 mg/l

Pa 10 mg/1

Na 40 mg/i

Coliformsa 5 x 107 MPN/100 ml

aLager and Smith, 1974.

DWF = 1.34 (PD)

DWB = 62.1 (PD)



where DWF = annual quantity of dry-weather flow,

DWB = annual quantity of dry-weather BOD,,
lbs/ac-yr; and

PD = population density, persons/acre.

The volume and pollutant loadings resulting from

wet-weather flows are a function of the total area, type

of land use, population density, area under each land use,

type of conveyance system such as separate, combined or

surface, antecedent storm conditions and amount and dura-

tion of rainfall. For wet-weather flows, one is concerned

not only with the peak flow and the pollutant loadings

but also with their temporal variations. With present-

day technology, characteristics of wet-weather flows can

only be estimated roughly. The procedure involves the

use of simulation models which describe the rainfall-

runoff-quality process. Brandstetter (1975) and Brown

et al. (1974) assess available models. Two of these

models, the U.S. Environmental Protection Agency (EPA)

Storm Water Management Model (SWMM) and the Corps of

Engineers model, STORM, will be -mentioned briefly because

of their general acceptance within the engineering com-


The SWMM, developed in 1969-1970 (Environmental

Protection Agency, 1971), was formulated as a design

model in that it simulates a single storm event. This is

characterized by short time steps and total simulation

times (i.e., minutes and hours, respectively) and rela-

tively high degree of detail in catchment schematization.

The RUNOFF block of the SWMM predicts surface runoff quan-

tity and quality from a given storm event. The TRANSPORT

block routes the flow and pollutants through the convey-

ance network. The output from the TRANSPORT block

provides data on the quantity and quality of wet-weather

flows generated from tne planning area during each time

step. The SWMM also has the capabilityy of simulating

areas with combined sewer systems. In addition, various

wet-weather control devices can also be simulated using


Within the RUNOFF block of the SWMM, quantity and

quality routing occurs in two distinct segments, initially

as overland flow and then, if designated, as gutter/pipe

routing. The input data required to generate the hydro-

graph and pollutograph include the area, fraction imper-

viousness, ground slope, roughness factors, surface de-

pression storage, overland flow width, hyetograph of

precipitation and the infiltration coefficients of Horton's

exponential function. Pollutant loads are introduced

into the stormwater outflow of the RUNOFF block through

two mechanisms, the surface pollutant load and catchbasin

BOD5 load. Eight pollutants are modeled by the RUNOFF

block: BOD5, SS, total coliform, COD, nitrogen, phos-

phorous, settleable solids and grease. For simulating

sewer systems and treatment devices, additional appropri-

ate data are included.

In contrast to SWMM, STORM (Hydrologic Engineer-

ing Center, 1975) is a planning model capable of continuous

simulation in hourly time steps for simulation periods of

many years. Hourly runoff volumes are computed as the

difference between rainfall and surface storage, multiplied

by the runoff coefficients. There is no pipe or gutter

routing in the STORM program. Modeling of runoff quality

is based on a percentage washoff of available pollutants

as a function of runoff rate. The treatment system is

simulated as a black box with its performance specified

by the user.

The quality prediction techniques found in most

simulation models (e.g., SWMM, STORM) rely upon genera-

tion of an initial surface load of pollutants. This load

is usually expressed in units of lbs, Ibs/acre, Ibs/curb-

mile, lbs/day-acre, or Ibs/day-curb-mile. Normalized

loads are, of course, multiplied by a unit of area, dry

days, etc., to produce an initial mass of pollutants at

the start of the storm. Pollutants are then "washed off"

during the storm in an exponential fashion, in which the

amount removed per time step is proportional to the amount

present, the runoff rate and other factors. Based on

thorough review of stormwater sampling studies done during

the past 20 years, Heaney et al. (1976) have updated the

pollutant loading estimates. The results simplify the

original loading factors by eliminating the use of dust

and dirt and curb miles. They propose the equations

listed in Table 2-2 for predicting annual average loading

factors as a function of land use, precipitation and popu-

lation density. Based on their findings that the fraction

of the developed land under residential, commercial,


Table 2-2
Wet-Weather Pollutant Loading Factors
(Heaney et al., 1976)

Separate Areas: Ms = a(i,j) P f(PD) acre-yr

Combined Areas: M = 3(i,j) P fI(PD) l
c acre-yr
where M = lb of pollutant j generated per acre
of land use i per year;
P = annual precipitation, inches;
PD = population density, persons per acre; and
c,6 = factors given in table below

Land uses:


= 1
= 2
= 3
= 4

= 1
= 2
= 3
= 4
= 5

Population Function

Other (assume PD = 0)

BOD5, Total
Suspended Solids (SS)
Volatile Solids, Total (VS)
Total P04 (as P04)
Total N

: i = i f (PD) = 0.142 + 0.533 PD0 145
i = 2,3 f (PD) = 1.0
i = 4 f (PD) = 0.142

Factors a and 8 for Equations: Separate factors, a, and
combined factors, 3, have units Ib/acre-yr-in. To
convert to kg/ha-yr-cm, multiply by 0.442.

Pollutant, j

Land Use, i
1. Residential
Separate 2. Commercial
Areas, ax
3. Industrial
4. Other

1. Residential
Separate 2. Commercial
Areas, 3. Industrial
4. Other

1. BOD5


2. SS


3. VS


4. POD 5. N
0.0336 0.540
0.0757 0.296
0.0705 0.276
0.00994 0.0605



industrial and open space uses is fairly constant (0.534,

0.086, 0.148, and 0.182, respectively), an average pollu-

tant loading factor over all lands uses can be derived

by using these fractions as weighting factors. The result-

ing equation for predicting BOD over all land uses is as

fo lows:

M = KA(0.467 P (0.142 + 0.533PD0'145) + 0.459P) (2.3)

where M = average annual BOD loading over four land uses,

P = annual precipitation, in;

PD = population density; and

A = 1 storm or unsewered area, or
KA {
= 4.12 combined sewer area.

2.3 Wastewater Control Devices

The primary concept upon which the control of

domestic wastes has been predicated is the matter of BOD5

reduction. Therefore, dry-weather flow control devices

have been built around satisfying the oxygen demand of

degradable organic matter before releasing it to the

receiving water. In general, the efficiency of BOD re-

moval of a dry-weather control device is a function of

the detention time. The performance of dry-weather control

devices can be characterized by the following simplified


ct/c = 1 e (2.4)

where ct = outlet concentration of BOD5, mg/l;

c = inlet concentration of 80D mg/l;

k = BOD5 removal rate constant, time 1 ; and

t = detention time in control device, time.

A dry-weather control device is usually comprised

of a combination of treatment processes or levels placed

in series. A combination of storage and treatment is

considered cost-effective only when the dry-weather flow

contains highly variable quantities of industrial wastes.

In actuality, dry-weather flow, in itself, undergoes daily

fluctuations. The treatment processes are designed on

the basis of average daily flow with provision to accommo-

date daily mass. Thus, the treatment process is under-

utilized a part of the time and is overloaded at other

times. The performance of the facility fluctuates in

accordance with the above. Recently it has been argued

that a combination of storage and treatment may be more

cost-effective (American Society of Civil Engineers, 1975).

The four different levels of treatment most com-

monly utilized for dry-weather treatment are called

preliminary, primary, secondary and tertiary. Initially,

wastewater receives oreiiminary treatment consisting of

bar racks and grit tanks. This is followed by primary

treatment consisting of a gravity sedimentation tank.

Primary treatment provides 30-35 percent BOD5 removal

and 55-60 percent SS removal.

When BOD5 and SS removals in excess of those attain-
able from primary treatment are required, then it is cus-

tomary to subject the wastewater to secondary or biological

level of treatment. As a general rule secondary treatment

includes primary and secondary levels. However, in some

cases, primary treatment may be omitted. Several treatment

processes are available for accomplishing the secondary

level. All of these processes rely on the ability of the

microorganisms to utilize as their food waste matter present

in sewage. The overall BOD5 and SS removals from secondary

treatment are approximately 85-90 percent and 90-95 percent,


Tertiary treatment is necessary when reduction

in BOD, and SS in excess of that accomplished by secon-

dary treatment is required or when a substantial reduction

in the nitrogen and/or phosphorous content of the waste-

water is desired. Tertiary treatment can be accomplished

either by biological-physical, biological-chemical or

physical-chemical processes. In the biological-physical

and biological-checmical processes, a third level of treat-

ment (either physical or chemical) follows secondary treat-

ment. In the case of physical-chemical treatment, the

biological or secondary level is omitted.

Assuming that conventional sedimentation, conven-

tional activated sludge and biological-physical processes

are representative of primary, secondary, and tertiary

levels, performance data for each of these levels are pre-

sented in Table 2-3. Sewage treatment processes are nor-

mally designed on the basis of annual average flow and

BOD5 expected at the end of a design period (15 to 20

years). Therefore, provisions are made to hydraulically

pass peak flows ranging from two to three times the average

flow. A generalized relationship between the flow and

the efficiency, for a given design flow, is shown in Figure

2-1. During the initial years of plant operation, the

flow and the BODO are less than the design figure and the

plant will usually operate at higher efficiency than the

design. Additional capacity is frequently added before

the actual flow approaches the design flow. Thus, these

plants are seldom operated at flow in excess of the design.

A wide variety of control alternatives is available

for wet-weather quality control and for improving the

quality of wet-weather flows (Field and Struzeski, 1972;

Lager and Smith, 1974; Becker et al., 1973). Rooftop and

parking lot storage, surface and underground tanks and

storage in treatment units are the flow attenuation control

alternatives. Wet-weather quality control alternatives

can be subdivided into two categories: primary devices

and secondary devices. Primary devices take advantage


Figure 2-1.

Generalized Performance Curve of a Given
Size Dry-Weather Plant







Table 2-3

Domestic Wastewater Treatment Performance Data

Overall Performancea Incremental Performance
Control BOD Removal Detention Time BOD Removal Detention Time
Alternative Efficiency Hours Efficiency Hours

Primary level 35 2 35 2

Secondary level 86 8 78 6

Tertiary level 95 10 71 2

aOverall performance for secondary level includes primary
level. Overall performance for tertiary level includes
primary and secondary level.

Including secondary clarifier.

of physical processes such as screening (microstrainer),

settling and flotation (sedimentation, swirl concentrator

and dissolved air flotation). Secondary devices take ad-

vantage of biological processes (contact stabilization)

and physical-chemical processes (physical-chemical treat-

ment). These control devices are suitable for treating

stormwater runoff as well as combined sewer overflows.

However, the contact stabilization process is feasible

only if the domestic wastewater facility is of an acti-

vated sludge type. The quantities of wet-weather flows

that can be treated by this process are limited by the

amount of excess activated sludge available from the

dry-weather plant. At the present time, there are several

installations throughout the country designed to evaluate

the effectiveness of various primary and secondary devices.

A summary of the design criteria and performance of these

devices is presented in Table 2-4. Based on these data,

the representative performance of primary devices is as-

sumed to be 40 percent BOD5 removal efficiency and that

of secondary devices to be 85 percent BOD, removal effi-


2.4 Wastewater Treatment Costs

Costs for dry-weather flow treatment devices have

been reported by many investigators. The major references

are Smith (1971) and Battelle-Northwest (1974). Cost

functions for various levels of domestic waste treatment

are presented in Table 2-5. Also included in this table

is the cost function for wastewater transmission. All

of these functions are of the form C = 1Dm, with m less

than one, reflecting the economies of scale in wastewater

treatment and transmission. Since the control of wet-

weather flow has gained attention only recently, well-

defined cost data for wet-weather control devices are not

available. The costs of various storage and treatment units

built around the country for handling wet-weather flows

were examined in order to develop generalized cost func-


Table 2-4

Wet-Weather Treatment Plant Performance Data

BOD5 Removal
Device Control Alternatives Design Criteria Efficiency, n

Primary Swirl Concentratorab 60.0 gpm/sq ft 0.25 0.50

Microstrainerc 20.0 gpm/sq ft 0.40 0.60

Dissolved Air Flotatin 2.5 gpm/sq ft 0.50 0.60
w/ Chemical Addition

Sedimentatione 0.5 gpm/sq ft 0.25 0.40

Representative Performance 0.40

Secondary Contact Stabilizationf Cont. 0.25 hrs 0.75 0.88
Stab. 3.0 hrs
Physical-Chemical9 3.0 0.85 0.95

Representative Performance 0.85

aield and Moffa,(1975)

cMaher, (1974)

dLager and Smith, (1974)

ePerformance data based on domestic wastewater treatment

Agnew et al.,(1975)

9Estimate based on performance of these units for domestic wastewater

V3 Ot

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Cost data for installed wet-weather treatment de-

vices are listed in Table 2-6. Since wet-weather control

facilities operate intermittently, annual operation and

maintenance costs are greatly affected by the number of

hours the facility is utilized. As a general rule, a

facility will operate a greater amount of the time if it

incorporates storage. An examination of Table 2-6 reveals

that annual operation and maintenance costs are 16.7 per-

cent of the total annual costs for the contact stabiliza-

tion unit. In the case of the swirl concentrator, the

percentage is 27.3. Annual operation and maintenance

costs for other units fall in between these two values.

Based on this analysis, it was decided to assume annual

operation and maintenance costs as 20 percent of the total

annual costs for all treatment devices. Cost functions

developed for various wet-weather quality control devices

are presented in Table 2-7. These costs include provisions

for sludge handling, engineering, contingencies and land


All treatment units exhibit economies of scale,

i.e., z < 1. Thus, there is an incentive to build larger

units. The optimal size treatment unit can be found by com-

paring the savings in treatment cost of going to a larger

unit with the increased piping costs. Unfortunately, suffi-

cient data on the number and flow rate of stormwater discharges

in urban areas could not be found. Thus, it is not possible

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to determine the optimal mix of treatment plants and pipe-

lines. Therefore, "representative" treatment costs were

developed as shown in Table 2-7.

Cost data on detention basins built in the Chicago

area for temporary storage of runoff are listed in Table

2-8. Costs of storage tanks built for the purpose of

wet-weather quantity and quality control as well as for

dry-weather quantity control are also included in this

table. Due to the wide variations in these figures, an

attempt was made to verify these costs using excavation

costs as the basis. Storage costs based on unit excava-

tion costs are listed in Table 2-8. The unit cost of

equalization storage basins for sewage treatment plants

and the estimated cost of rooftops and parking lot storage

are also shown in Table 2-8. Lastly, analysis of recent

estimates of storage costs developed by Culp et al. (1976)

indicate the following unamortized capital cost C ($ x 10 )

as a function of storage volume, S (mg)

Unit Cost @ S = 10 mg
Type Equation $/gal

Earthen C = 0.025 S0.73 $0.013

Concrete w/o. Cover C = 0.350 S0.58 $0.133

Concrete w. Cover C = 0.400 S 079 $0.250

The data indicate wide variation in the costs of storage.

Thus, the relatively simple relationship shown in Table 2-7

Table 2-8
Capital Cost of Storage Facilitiesa

Storage Reservoirs0 Capacity Capital Cost
mil gal $/aal
Hillside Park 11.4 0.01 Earthen Basin
Heritage Park 36.5 0.01 Earthen Basin
Oak Lawn 7.8 0.02 Earthen Basin
Middle Fork North Branch 195.5 0.02 Earthen Basin
Wilke-Kirchoff 32.6 0.03 Earthen Basin
Melvina Dutch 53.8 0.03 Earthen Basin
Oak Hill Park 25.1 0.02 Earthen Basin
Dolphin Park 53.8 0.01 Earthen Basin
Average 52.1 0.019

Storage Tankse
Cottage Farm, Bostonc 1.3 5.21d Covered Conc. Tanks
Spring Creek, New Yorkc 10.0 2.33 Covered Conc. Tanks
Chippewa Falls, Wisconsinc 2.8 0.29 Asphalt Paved Basin
Humboldt Avenue, Milwaukeec 4.0 0.55 Covered Conc. Tanks
Seattle, Washington 32.0 0.25 In-line
Whittier Narrow, Columbusc 4.0 1.70 Open Concrete Tanks
Average 9.0 1.72
Based on Excavation Costs
$2/cu yd 0.01 Earthen Basin
$5/cu yd 0.025 Earthen Basin in Rock
Equalization Basins for Dry
Weather Sewage Treatment
Plants9 1.0 0.22 Earthen Basin
3.0 0.10 Earthen Basin
10.0 0.06 Earthen Basin
1.0 0.39 Concrete Basin
3.0 0.2S Concrete Basin
10.0 0.25 Concrete Basin
Parking Lots 0.10
Rooftops 0.05

aBased on ENR = 2200.
Source: metropolitann Sanitary District of Greater Chicago.
cAlso used for stor;;,.ater treatment.
Includes pumping station, chlorination and outfall facilities.
fSource: Lager and Smith, (1974)
Soil Conservation Service, Gainesville, Florida
gF!ow Equalizatcon Plus for Wastewater Treatment Plants, American Society of
of Civil Engineers, (./;,975)
hSource: lWJiswall and Robbins, (1975).

was used. Annual storage costs are estimated as a function

of population density. The curve was derived using an

unamortized capital cost of $0.10 per gallon for PD = 5

persons per acre and $0.50 per gallon for PD = 15 persons

per acre.

Since most of the operation and maintenance costs

attributable to storage are for solids handling and because

these solids are usually handled at the treatment facility,

operation and maintenance costs of storage facilities have

been assumed at zero.

2.5 Control Criteria

Basin plans proposed under Section 303(e) constitute

the overall framework within which 208 plans are developed

for specific portions of a basin with complex pollution con-

trol problems. Basin plans define (1) water quality stand-

ards and goals; (2) critical water quality conditions; and

(3) waste load constraints.

In the basin plan, the basin is divided into water

quality and effluent limited segments. Water quality

limited segments occur when application of best practicable

treatment for industrial waste discharges and secondary

treatment of domestic discharges would be insufficient

to achieve water quality standards. In this case, some

significant point sources must be subjected to further

control or some non-point sources must be controlled or

some combination of point and non-poirt source treatment

must be implemented. The effluent limited segments are

those in which water quality is and will continue to be

at least equal to the applicable water quality standards,

or if water quality does not meet standards, it will do

so after the application of best practicable control

technology by industrial sources and secondary treatment

by domestic sources. Most of the urban areas requiring

208 planning are expected to contain water quality limited


The basin plans would, in general, specify either

the waste load allocation, i.e., amount of pollutants

(# of BOD5 per year, etc.) that can be released from point

sources within the area or the minimum degree of treatment

that must be provided to these point sources. For non-

point sources, such waste load allocation is not expected

to be available in basin plans and must be developed during

the 208 planning process. This requires water quality

modeling studies to determine the extent to which non-

point sources must be controlled to meet the water quality

standards. Based on these studies control criteria such

as the pounds of BOD5 that can be released on an annual

basis can be established.

In addition, stormwater quantity control criteria

may be established as a part of the planning process or

existing criteria within the region may be assumed to be

followed for future planning. These criteria would be

in terms of an allowable rate of release of wet-weather


2.6 Hypothetical Planning Area

The layout of a hypothetical planning area is pre-

sented in Figure 2-2. The planning area is assumed to be

located in the southeastern part of +he U. S. Data on

the planning area are listed in Table 2-9. Seven cities

are located within the planning area. Existing institu-

tional arrangements are such that cities 1, 2, and 3

belong to one polity; cities 4 and 5 to the second one,

and cities 6 and 7 to the third. City 3 has separate

sanitary and storm sewer systems while city 6 has a com-

bined sewer system. Both of these cities have existing

sewage treatment facilities which neither have any excess

capacity nor can be expanded. None of the other cities

has a sewer collection or treatment system. The control

criteria assumed to be met by each of the cities within

the planning area are also listed in Table 2-9.

Methodology for conducting area-wide planning pre-

sented in the later chapters will be illustrated by using

this hypothetical planning area. Planning objectives in-

volved in area-wide planning are discussed in the next


/ I -

I ,


z '-

z o

a. \L ---- a -

Si in |

Z .r-
I *.1

2 2a

wI I /
M 3I

z I
, I--

St 0

Co- 'N -------------------- 1 6
7- U-

r J
o 0 rj
1- 4 Z L
a cr,
0, / '*
c- wn j
3 z 0
--- GM

Table 2-9
Characteristics of Hypothetical Planning


1 2 Ci4 5 6 ty
1 2 3 4 5 6 7

1. Total Area, Acres
2. Residential Area and
Open Space, Acres
3. Industrial and Com-
mercial Area, Acres
4. Undeveloped Area, Acres
5. Population
6. Exist. Sewerage System
7. Exist. Treat. Plant
Year Built
Design Flow, MGD
1975 Flow, MGD
Design BOD5, #/day
1975 BOD5, #/day
% BOD5 Removal
Type of Facility
8. Projected Waste Loads
1990 Volume MGD
1990 BOD5 #/day
9. Waste Treatment
Volume, MGD
BOD5 Removal, m/day
10. Control Criteria
Dry-Weather Flow
% BOD Removal

1000 660 4000 2000 400 15000 2000

400 200 1500 800 100
550 330 2000 1200 200


300 200 15
400 300 25

8000 800
10000 1200

2000 200
3000 300

500 430 2200 1000 285 5000 1000
300 270 1600 500 175 2000 500

2000 1000 15000 5000 500 100000 1000
3000 2000 21000 7000 1000 125000 7000

- separate

(1 )




- combined


0.2 2.1 0.7 0.1 12.5 0.7
400 4200 1400 200 25000 1400

0.2 2.1 0.7 0.1 12.5 0.7
360 3780 1260 180 22500 1260

(2) (2)
(5) (5)

(2) (2)

(1) = Primary Treatment
(2) = Secondary Treatment
(5) = StormwTater quantity not to exceed the volume for undeveloped area
under 10-year 24-hour rain-all .



3.1 General Statement of the Problem

Urban wastewater planning, under Public Law 92-500,

seeks a cost-effective solution for point and non-point

sources of water pollution, on an area-wide basis that is

politically acceptable and would achieve the 1983 goal of

swimmablee and fishable" waters. As stated previously,

two main sources of water pollution in urban areas are

dry-weather sewage and stormwater runoff. Dry-weather

sewage flow is of concern primarily from a quality stand-

point. For stormwater runoff, however, both quantity and

quality are relevant. Thus, the three facets of urban

water management that must be examined in order to achieve

the goal of swimmablee and fishable" waters are control of

(1) dry-weather sewage flow quality,

(2) stormwater quantity, and

(3) stormwater quality.

Since the federal government provides construction grants

for required facilities, it has set up guidelines for con-

ducting area-wide planning (Environmental Protection Agency,

1975). These guidelines require an integrated approach for

evaluating the above three purposes in a cost-effective

manner. Cost-effectiveness criteria involve identifying

a plan that minimizes "total cost to society," i.e.,

resource, environmental, and social costs. Resource costs

are the values of goods and services representing primary

project inputs and include capital costs plus operation,

maintenance and replacement costs of water pollution con-

trol works required to accomplish the above goal. Environ-

mental and social costs may be viewed primarily as damages

associated with the project outputs. In general, there

may be numerous strategies for accomplishing all three

purposes. Each strategy may have different resource costs

as well as environmental and social costs. The cost-

effective solution is one which minimizes the total costs.

An urban area will usually embody several separate

political entities. Since these groups must bear a portion

of resource costs as well as the damages, the selected

plan must be acceptable for it to be implementable. Thus,

preparation of the area-wide plan involves not only engineer-

ing considerations but also economic, financial, social and

environmental analysis.

This chapter first reviews basic concepts from

economic theory and presents optimality criteria for simple

cases of water quality management. Difficulties in apply-

ing these criteria are discussed and the need for multiobjective

planning is outlined. Various concepts involved in multi-

objective, multipurpose and multigroup planning, in the

context of the area-wide plan, are then presented.

3.2 Economic Analysis

Water quality management may be viewed as a produc-

tion process whose basic purpose is to convert resources

from a given form (input) into a more useful form (output).

A water quality management project is constructed to pro-

duce such desired final outputs as the protection of

health, esthetic and other beneficial uses of the water.

The immediate output can be assumed to be the amount of

pollutant removal. System inputs can be thought of as

the physical structures, e.g., treatment plant, storage

device, that define system design. These inputs can be

directly translated into primary resources such as land,

labor and capital.

The relationship between inputs X = (x1,...,x )

and outputs Y = (yl,...,y ) is expressed by means of a

production function

g(X,Y) = 0 (3.1)

which identifies those combinations of outputs and inputs

by which it is impossible to produce more of one output

without either producing less of some other output or re-

quiring more of some input.

Associated with each input and output are the

resource costs and the pollutant damage costs.

The economic optimization problem for water quality

management can be stated as follows:

minimize Z = F1(X) + F2(Y)

subject to g(X,Y) = 0 (3.2)

X,Y > 0

where F1(X) = sum of resource costs;

F2(Y) = sum of damage costs; and

Z = objective function.

Thus, the objective is to find an alternative having

minimum value of Z with the constraint that only alternatives

contained on the production function, g(X,Y) = 0, need be

considered. Relevant simple cases are examined using a

geometrical approach.

1. Single Facility, Single Purpose (one input,
one output)

Assume a single treatment plant with its input

represented by n:e BC 0 removed. This case is

typical of dry-weather flow control. The output

vector Y is a compilation of the quantities of all

goods and services, including labor, required to

construct a treatment facility of specified size

(flow rate). The output can be represented by a

single component, y, the cost in dollars of primary

input resources. The input vector is the amount

of various pollutants removed and may be repre-

sented by a single component, x, representing BOD5

removed. The simplified production function be-


y = g(x) (3.3)

where x = pounds of BOD removed per period; and

y = cost of treatment plant of specified
capacity in dollars per period.

The resource costs are illustrated in Figure 3-1.

Since the damages to society are incurred as

a result of the BOD released, these costs are

inversely proportional to the input x (BOD removed

or withheld). A generalized curve of these costs

vs. the input is shown in Figure 3-1. The objec-

tive function is derived by the vertical summation

of the two curves. From this curve, the optimal

output y and input x0 is at the point where the

resource costs plus the societal damages are at

their minimum as shown in Figure 3-1.

2. Dual Facilities, Single Purpose (two inputs,
one output)

Assume that we nave two facilities such as

a storage tank and a treatment plant to accomplish





Figure 3-1.

Determination of Optimal Level of Input and
Output: One Input, One Output



BOD removal. This case is typical for control

of wet-weather flows. In this case, the two in-

puts are the treatment rate and the storage volume.

The output is the amount of BOD removed. The pro-

duction function can be represented by

y = g(x x2) (3.4)

where y = pounds of BOD removed;

xI = treatment rate; and

x2 = storage volume.

The above formulation allows isoquants (lines of

equal output) to be generated showing all possible com-

binations of x1 and x2 for a specified output level y.

Several hypothetical isoquants as a function of

treatment rate and storage capacity are shown in

Figure 3-2. For a given isoquant, the same output

y can be produced by a mix of x1 and x2 which yields

a given output at the lowest cost. If the costs

are linear, then

F1(X) = c1X1 + c2x2 (3.5)

where F1(X) = total cost;

cI = unit cost of resource, xl; and

c2 = unit cost of resource, x2.

Graphically, this shows up on Figure 3-2 as a system



-Y2 y= g(x,,x2)


Figure 3-2. Determination of Optimal Combination of
Inputs: Two Inputs, One Output


of parallel iso-cost lines. Production of a given

level of output, y, with least-cost combination

of resources occurs where an isoquant is tangent

to the iso-cost line. At this point

c1 MC1
MRS2 1 c2 MC (3.6)
1 c 2 2

where MRS2,1 = marginal rate of substitution of
'2 for xl; and

MCi = marginal cost of input i.

The expansion path, shown in Figure 3-2, traces out

the locus of the optimal combination of inputs for

various output levels. The expansion path then can

be used to derive the curve shown in Figure 3-1,

from which the optimal amount of BOD5 removal can

be determined. Lastly, the optimal mix of inputs,

xI0 and x2, can be determined from Figure 3-2 as


3. Single Facility, Dual Purpose (one input, two
ou puts)

Assume a treatment facility designed to accom-

plish BOD removals from dry-weather flow as well as

wet-weather flow. The input x can be taken as the

total resource costs. The two outputs are the amount

of dry-weather BOD removed, yl, and the amount of

wet-weather BOD removed, y,. The production func-

tion becomes

x = g(y ', y2) (3.7)

A family of curves can be obtained showing outputs

yl and Y2 that can be produced with a given input,

x. These product transformation curves, shown in

Figure 3-3, identify the locus of all possible com-

binations of outputs that can be produced with a

given fixed input. A concave curve indicates that

the two outputs are complementary. The slope of

the product transformation curve is called the

marginal rate of transformation. A family of parallel

lines called iso-revenue lines or iso-benefit lines

may also be drawn in Figure 3-3. Each of these

lines shows the different combination of outputs

that could be obtained for the same amount of gross

revenue or benefits. The optimal mix of the outputs

is achieved where

pl MB1
MRT (3.8)
1,2 p2 MB 2

where MRT1 2 = marginal rate of transformation
of Y for Y2;

pj = price for output yj; and

MB. = marginal benefits from output, yj.













Figure 3-3. Determination of Optimal Combination of
Outputs: One Input, Two Outputs

x = 9(y,2)

An expansion path showing the ootiral combination

of yl and Y2 for given levels of benefits can be

generated only if the output prices or their mar-

ginal benefits are known. The expansion path then

can be used to construct Figure 3-1, yielding the

optimal amount of total BOD5 removal. Tte opTimal

combination of outputs is determined using Figure


4. Single Facility vs. Dual Facility

Assume two sources of dry-weather flow are lo-

cated at some distance from each other. A typical

example may be two cities along a river. The

problem in this case is to determine whether each

city should build its own treatment facility or

build one joint facility. Economic analysis re-

quired to determine the optimal amount of output

(BOD removed) for two single facilities is similar

to the single facility, single purpose case shown

in Figure 3-1, except that additional environmen-

tal and social costs (externalities) may be incurred

by one or both cities as a result of the facility

of the other community. The optimal joint output,

y for two facilities is equal to

y i, + Y2

where yi = optimal output (BOD removed) by plant i.

The total resource costs are

1 1 1
x = x1 + x2

where xi = resource cost of plant i.

The economic analysis for the joint facility is simi-

lar to the single facility-dual purpose case illus-

trated in Figure 3-3. The optimal joint output is

y" and the total resource cost is x". Then the

joint facility is cost effective if and only if

1 1 1
x" < x1 + x = x (3.9)
X1 2

5. Allocation of Capacities Between Purposes

Assume a treatment facility used primarily for

dry-weather BOD control and a storage facility used

primarily for wet-weather BOD control. The objec-

tive in this case is to determine if the total out-

put can be increased by a reallocation of treat-

ment and storage. The storage-treatment isoquants

for dry-weather BOD and wet-weather BOD are shown

in Figures 3-4 and 3-5. Let T1 be the available

capacity of the treatment facility and S2 be the

available storage capacity.

Conditions that must be fulfilled to guarantee

efficient allocation of resources between two consumers

S 3mn1OA 301dOIS






Figure 3-5. Generalized Isoquants for Wet-Weather Control

are usually demonstrated in economic theory by an

Edgeworth Box diagram (Leftwich, 1965; Cohen and

Cyert, 1965; Beli and Todaro, 1969). Using this

procedure, the Edgeworth Box diagram for this prob-

lem is presented in Figure 3-6. The dry-weather

BOD removal, y, increases from bottom left to top

right and the wet-weather BOD removal, y2' increases

from top right to bottom left. As can be seen from

Figure 3-6, movement along '3oquant y1 reallocates

storage and treatment such that dry-weather BOD

removal remains y1 while wet-weather BOD removal

increases. At point B, wet-weather BOD removal is
0 3
increased from y2 to y2 while dry-weather BOD
removal remains constant at y, At this point,

T1 of the treatment capacity and S1 of the storage

volume have been allocated to dry-weather BOD

removal and the remaining treatment and storage to

wet-weather BOD removal. Similarly, if a move is

made along isoquant y20, wet-weather BOD removal

remains constant while dry-weather BOD removal in-

creases. At point A, dry-weather BOD removal has
0 3
increased from y1 to yl while wet-weather BOD

removal remains the same. This point corresponds
2 2
to reallocation of T1 of treatment and S1 of

storage volume to dry-weather control and the re-

mainder to wet-weather control. Curve AB is called





2 So


Figure 3-6. Edgeworth Box: Two Inputs, Two Outputs,
Two Purposes

the contract curve. It will be designated as the

noninferior set because dry-weather BOD removal

can only be increased by decreasing the wet-weather

BOD removal or vice versa. This is shown in Figure

3-7, which is derived from the contract curve AB.

As in case three above, the optimal mix of the two

outputs can be determined only if the product price

or marginal benefits are known. Knowing the optimal

mix of the two outputs, optimal allocation of treat-

ment and storage to the two purposes can be deter-

mined from the Edgeworth Box. The above analysis

would also apply if the two purposes were to be

replaced by two groups such as city 1 and city 2.

The results yield the optimal allocation of treat-

ment and storage between these cities.

Application of these economic concepts to water

quality management is usually not practical for

several reasons. First, it requires the evaluation

of environmental and social costs or the "societal

damage cost function" which relates project outputs

and damages in order to determine the optimal out-

put. These costs are not identifiable in monetary

terms. Rather, they are usually described using

Qualitative and quantitative terms as the adverse

environmental and social impacts. Thus, the eco-

nomic optimization problem of equation 3.1 needs




Figure 3-7. Noninferior Set for Two Outputs






LL 2


to be extended to the following.

minimize Z = (resource cost; environmental
and social costs)

= F1(x); F2(y) (3.10)

subject to g(x, y) = 0

x, y > 0

The economic analysis above also requires specifi-

cation of the product prices or marginal benefits

in order to determine the optimal mix of outputs.

If it is assumed that the marginal benefits from

removal of one pound of dry-weather BOD are the

same as that obtained from removal of one pound of

wet-weather BOD, the marginal benefit curve would

be a 450 line and the optimal mix of outputs can

be determined. If the marginal benefit curve can-

not be specified, then the solution would be the

entire set such as shown in Figure 3-7. Further,

even where the optimal mix of outputs can be ob-

tained by using the marginal benefit curve, the

economic theory derives this solution based on

efficiency alone and does not concern itself with

whether the allocation is equitable. Thus, the

economic optimization problem in this case needs to

be extended as follows:

maximize Z = (Dry-weatner BOD; Wet-weather BOD)

= (yI ', 2) (3.11)

subject to g(S ,T1,y1) = 0

g(S2,T2 2) = 0

S1,T1' S2' ,T22',y 2 0

The problem may be viewed as an efficiency and equity


3.3 Multiobjective Planning

Problems involving objective functions of the form

given by equations (3.10) and (3.11) are usually referred

to as multiobjective problems as they involve optimization

of a vector of objectives. During the last few years,

multiobjective planning has been promulgated in the general

area of water resources development. Traditionally, projects

for water resources development have been evaluated on the

basis of a single criterion, national economic efficiency.

The procedure involves an evaluation of the total benefits

and costs of the project to examine its economic impact.

Strong objections have been levied against benefit-cost

analysis (Maass, 1970; Prest and Turvey, 1965; Whipple,

1971; Cohon, 1973). The criticism has led to the promul-

gation of multiobjective planning (Water Resources Council,

1970) using the following four objectives:

(1) national economic development (i.e., national
economic efficiency),

(2) environmental quality,

(3) social well being, and

(4) regional development.

Howe (1971) argues that social well-being and regional

development can be classified as distributional problems,

i.e., that these objectives address the question of how the

benefits and costs are distributed. The same is also par-

tially true for environmental quality. These problems are

generally referred to as equity questions. Thus, the above

four objectives can be narrowed down to two objectives,

i.e., efficiency and equity. The objectives to be accom-

plished in urban wastewater management are quite similar

to those for water resources planning. Thus, the multi-

objective planning in urban wastewater management may be

viewed as an extension of the policy to water quality


The general theory of multiobjective planning is

outlined by Major (1969) and Howe (1971). While benefit-

cost analysis maximizes economic efficiency, multiobjec-

tive analysis maximizes a vector quantity, the elements

of which are the net benefits associated with each objec-

tive. If, for example, the objectives are to minimize the

cost of treatment (dollars) and pollution load (estimated

as # of BOD, discharged) subject to appropriate constraints,

a transformation curve can be generated by successive solu-

tion of this vector minimization problem. The transforma-

tion curve defines the set of noninferior solutions. This

set is comprised of those solutions wherein it is impossible

to increase the value of one objective without decreasing

the value of the other objective. The transformation curve

and the noninferior set are illustrated in Figure 3-8. If

one knows the indifference curve (rate of tradeoff between

or among objectives), an appropriate mix of these objectives,

known as the best compromise solution, can be found as illus-

trated in Figure 3-8. This solution then yields a best

possible distribution of treatment cost and quantity of

BOD discharged. Unfortunately it is no trivial matter to

determine both curves in actual practice.

A general formulation of the multiobjective prob-

lem is as follows:

minimize Z(X) = Z (X),.......,Z (X)

subject to gi(X) < 0 -i (3.12)

X > 0

where X = a vector of decision variables;

Zi(X) = ith objective; and

gi(X) = constraints.


i I


't i






Figure 3-8. Determination of Best Compromise Solution



Mathematical programming techniques offer a promis-

ing way of analyzing the above multiobjectives (Cohon, 1973;

Cohon and Marks, 1973). By using either weighting tech-

niques or constraint techniques, the transformation curve

is generated. In the former case, this is accomplished by

varying the weights on each of the objectives. It is easy

to do this computationally using parametric programming.

The alternative approach optimizes one of the objectives

subject to the usual constraint set and an additional

constraint which indicates a prespecified level of attain-

ment of the second objective. Thus, from a computational

point of view, the noninferior set of solutions can be

determined. However, to solve the problem, the rate of

tradeoff between objectives needs to be articulated. In

actual practice, the set of noninferior solutions is

submitted to the decision makers) who then selects the

best compromise solution. Since the indifference curve

is usually not available, selection of the best compromise

solution implies value judgements. Such a situation does

not ensure that the solution attains the equity objective.

Some multiobjective solution techniques for deriving the

best compromise solution are discussed in Chapter 7. The

next two sections discuss the efficiency and equity ques-

tions that arise in the context of urban wastewater manage-


3.4 Efficiency and Equity

There has been sustained interest over the past

decade in determining optimal regional environmental quality

management strategies. Numerous investigators have demon-

strated that coordinated wastewater treatment strategies

are more efficient, in an economic sense, than decentral-

ized treatment plants, e.g., Heaney et al. (1971). Similar

results have been obtained for air pollution, e.g., Teller

(1968) and Seinfield and Kyan (1971) and will also be

demonstrated to occur in urban waste management. However,

there has been little success in implementing such pro-

posals due partially to the nonexistence of a real-world

regional authority with necessary power to shift decisions

in this direction. Lacking such a regional authority, Hass

(1970) investigated the possibility of setting up a system

wherein price guides could be used to direct the activi-

ties of the individual waste dischargers toward the regional

optimum. He structured the problem using the decomposition

principle (Dantzig and 'Wolfe, 1960). Briefly, the decom-

position principle partitions the total regional problem

into a series of subproblems--one for each waste discharger,

and a regional master problem. Each w, aste discharger sub-

mits a provisional control plan to the regional authority

who runs tne master problem to see if a regional optimum

has been aciie/ed. If n)t, he trans3,its a revised set of

criterion elements, cost coefficients in this case, to the


individual waste dischargers. They resolve their problem

and may decide to submit an additional solution for con-

sideration. Each such solution represents an extreme

point from their feasible region. Since there are only a

finite number of such extreme points, the algorithm even-

tually converges. The resultant optimal regional solution

is actually a weighted average of the extreme points of

the solutions submitted by the individual waste dischargers.

However, it may occur that the optimal solution for an

individual is a convex combination of two adjacent extreme

points so that the notion of decentralization by price

guides alone breaks down (Baumol and Fabian, 1964). The

reason is that the individual is now indifferent among

solutions along this edge connecting the adjacent extreme

points while the regional authority knows precisely where

along the edge the individual should act in the interest

of regional efficiency. Thus, in general, more than price

guides are necessary to achieve the regional optimum.

Charnes, Ciower and Kortanek (1967) suggest the inclusion

of preemptive goals as z device ecr providing the requisite

amount of information to attain a stable condition.

Dorfman and Jacoby (i1970) have argued that the

optimization models might be used to screen the number of

alternatives down to a reasonable number (say five to

ten) and their submit tnese Pa-eto-admissible solutions

to further scrutiny by emp:loying a simple political

simulation model. The essence of this approach is to assign

weights based on the relative importance of each decision-

making group. One can then generate various weighting

schemes and examine how sensitive the solution is to the

assumed weights. Burke and Heaney (1975) have devised a

more formalized political simulation model based on work

by Bulkley and McLaughlin (1966) and applied it to the

Dorfman-Jacoby example. Their effort describes the rela-

tive power of several interest groups and simulates the

bargaining process and coalition formation among these


An essential component of any workable program is

the notion that the resultant solution not only is effi-

cient but also is fair to every one of the participants.

Thus, if the performance standards or control criteria

are specified, the efficient solution is the least-cost

solution subject to meeting this control criterion. Given

this solution, the question of equity must then be resolved.

Some of the concepts involved in efficiency and equity

are outlined below.

There are numerous ways to control pollution using

on-site control and/or off-site control. Thus, a general

framework is needed for addressing the problem. The

selected approach is based on the planning theory of zoning

(Herzog, 1969). Using the planning theory of zoning, each

source of pollution calculates the cost of handling the

problem on-site as a function of the allowable rate of

release from his area. The cost function would look like

the curve shown in Fiaure 3-9. Costs decrease as the allow-

able release increases. In fact, the cost falls to zero

as it is allowed to release more and more pollution.

Assume that such a curve exists for each area. Note that

the abscissa could be air pollutants, water pollutants,

noise or any other "nuisance" which is normally covered

by zoning regulations.


$/ Q


Figure 3-9. Generalized Cost Function for On-site Storm-
water Control

In this case, the policy question is to determine

how much pollution can be released from each of these

areas. Using the planning theory of zoning, the answer

depends on a determination of "assimilative capacity."

Assume that the required level of control can be specified.


The following example is presented in order to

illustrate many of the concepts to be discussed. Assume

the region under consideration has been partitioned into

three study areas. Each study area has two options: (1)

on-site control, and/or (2) off-site control at a central

control facility. The following'notation is used. Let

x = decision variable: number of units of
control j selected for area i;

xij = upper bound on x..

c = unit cost for control j in area i;

D. = quantity of commodity originating in area i;

Q = maximum allowable release of pollutant from
area i;

Qi = quantity of pollutant released from area i;

Zi = total control cost to area i;

T. = reduction in control cost to area i if Qi
is increased by one unit;

t = unit cost of transporting pollutant from
area i to the central control location;

c = unit cost of central control; and

i = maximum control at central facility.


The example problem is shown in Figure 3-10 using a net-

work representation. This problem is deliberately over-

simplified to permit us to understand the concepts without

getting enmeshed in computational difficulties. The results

can be extended easily to more realistic cases where mul-

tiple central control facilities exist.

The overall objective function seeks to minimize

the total cost of on-site and off-site control. The problem

facing each study area is to


subject to

Zi = L cij x. +
i = D

Sx. +Q =D.
13 1 1

Qi < Q.
1l- 1l

x. < x for all j,
IJ -- l3

Xij 1 0

for all j, and

Qi > 0

For area 1, the problem is to select x11, x12, and x13

so as to minimize Z1 = lOx11 + 5x12 + Ox13 +4Q1

subject to x1l + x12 + x13 + Q1 = 500

xli < 100
X < 100
X12 < 200
x13 < 200

x11,X12,X13,Q1 > 0





2= 400

Figure 3-10. Network Representation of Example Problem

This is a very simple problem to solve if Q1 is

known. As it turns out this is a question of critical

importance in some cases. In the context of environmental

quality management Q1 represents a judgement on the part

of the administrator as to the "assimilative capacity" or

availability of off-site controls to area 1. Traditionally,

the natural system has provided these off-site controls

free of charge. For example, the off-site area in this

example could be a swamp, river, or the atmosphere.

Willingness to Pay for Off-Site Control

In order to find Qi, we need to know how much each

area is willing to pay for off-site control based on its

alternative on-site control costs. This can be done by

solving the above linear program for various values of

Qi, assuming on-site control is required. Thus, for the

moment, Qi is analogous to an effluent standard imposed

on area i. The problem can be solved by deleting the

last term, (t. + c)Q from the objective function and

finding the optimal solution for assumed levels of Qi.

Computationally this can be done quite easily using para-

metric programming as explained below. Initially, set

Qi = 0 and solve the linear programming problem. Then,

as a postoptimal procedure, one can vary Qi continuously

from 0 up to any prescribed upper bound using as the right

hand side 0. = 0 + 6r where & equals a parameter which

will increase continuously from 0 to 1 and r is a scalar,

say 1OC0, in this case. This solution to this problem

tells the total cost to area i for any value of Q. which

is of interest. This is a very attractive feature of

linear programming. Another aspect of linear programming

which is of interest is duality theory. The solution to

the dual problem is obtained when the above problem (primal

problem) is solved. Among other things, it tells, for a

given Qi, the reduction in cost to area i if Qi is increased

by one unit. This unit cost is called the "shadow price"

with respect to Qi and will be denoted as 7..

Solve the example problem for area 1 assum-

ing Ql = 0. The answer is simply that on-site control is

used. As the constraint on 01 is relaxed, area 1 will sub-

stitute off-site control for its most expensive on-site

control, x11, in this case. Thus, it is saving $6 per unit

change in Q, in this range. The analysis continues in

this manner until all solution possibilities have been

identified. The results are shown in Finure 3-11.

Assume that a similar analysis was done for areas

2 and 3. Then the willingness to pay for each of the

three areas would be known. Next assume each area was

offered as much discharge as it wanted at its cost of

(ti c) dollars per unit. In this case, the aggregate

demand would be Q = 300, Q2 = 300, Q = 250, or a total

demand of 85C units. But suppose only 420 units are




Figure 3-11.

Shadow Price for Area 1 for Assumed Value
of Q1


$ i,



available. How should the available capacity be allocated?

It can be assumed that the central agency or planning auth-

ority will attempt to maximize the aggregate savings to

areas 1, 2, and 3 from using the regional facility.

Alternatively preference will be given to those areas

whose on-site control costs are the highest.

Solution of the Regional Problem

The answer to the above question can be obtained

by solving one larger optimization model which is formu-

lated on the following page. The constraints for this

problem can be divided into two categories: (1) three sets

of study area constraints, and (2) a coupling constraint.

The coupling constraint is the only linkage among the

three study areas. If the values of Q1, Q2, and Q3 are

prespecified, then the larger problem can be completely

decomposed into three independent subproblems. There are

many real-world situations in which an a priori apportion-

ment is used, e.g., allocation to each area based on its

size, population, etc. For a oollutant, W might be the

assimilative capacity of che receiving water, which has

been apportioned. In the case of a pollutant', the appro-

priation is equivalent to prescribing effluent standards.

Assume an apportionment is established such that

Q, = Q2 = 3 = 420/3. Given this apportionment, find Z,

Z2, and Z3, the least-cost solutions for the three areas.


o 0

I: v I A



o 0

Ix c c

VI ^A Al

"I" C'


n n

n vi Al
ii vl | A

*-* cr c> =

cn m C-1 m


+ I + +

1- -i )

C"j C" (j
x x x

-, C

Is Cf (


+ 0 4++

x X X 0
<-> rrw

U --

+ ..


X x x


'E .


re -
11 o

- E

i l'
= 0.


C 00




S 0c


The results are

Z1 = $1360 : 71 $1

Z2 = 1540 : T2 = 4

Z3 = 2560 : 3

Z. = $5460

Q1 = 140

Q2 = 140

Q3 = 140

Q = 420

Apportioning the capacity among the three areas results in

a combined cost of $5,460. Next examine whether,

from a least-cost point of view, it would be possible to

select Q, Q2, and Q3 such that costs are reduced. This

problem can be solved by running the entire linear program

with Q Q2', and Q3 as decision variables. This approach

is equivalent to receiving system standards wherein the

coordinator allocates the assimilative capacity in an op-

timal manner. This is precisely the problem to be addressed

here. The optimal solution is

Z = $1400

7 = 1020

Z = 2650

Zi = $5070 : T = $4

Q1 = 100

Q2 = 270

Q3 = 50

Qi = 420

The solution to the overall optimization problem reduced

total costs from $5,450 to $5,070 by making more effective

use of the available capacity. As can be seen by examining

the solutions, the model took capacity from 1 and 3 and

allotted it to 2 since its net savings (Tr2) were higher.

While this latter solution is the least costly from an

overall point of view, if the costs are assigned as shown

above, areas 1 and 3 are made worse off while area 2 is

made significantly better off. Thus, areas 1 and 3 might

reasonably object to such a solution. This is precisely

the problem that has thwarted implementation of optimal

programs, i.e., while they are the least costly, they do

not seem to be "fair" to everyone. Therefore one needs

a procedure which is not only efficient but also equitable.

Solution Using Market Price Concept

Perhaps one could use demand theory from economics

to determine a better solution. The aggregate demand curve

for the three areas is shown in Figure 3-12. Knowing the

demand curve and the supply curve, then it is possible to

determine the 'market price" for the central facility.

Referring to Figure 3-12, it is $4/Q. Thus, according to

economic theory. to achieve efficient resource allocation,

a market price of $4 per unit of storage should be used.

Let T denote the market price (which is the same as the

shadow price of the central facility from the overall op-

timization model), then the assessment to each study area

per unit is (7 + c + t ).