OPTIMIZATION AND COST/EFFECTIVENESS
MODELS IN WATER QUALITY MANAGEMENT
Ethan T. Smith
Federal Water Pollution Control Administration
Edison, N. J.

-------
CWT 10-19
Optimization and Cost/Effectiveness
Models in Water Quality Management
Ethan T» Smith
Chief, Program Management Section
Hudson-Delaware Basins Office
Federal Water Pollution Control Administration
Edison, N. J.
Mr. Secretary, Ladies and Gentlemen:
During the Second World War a number of techniques were developed
for use in planning strategy and tactics. Typically, these methods
employed various mathematical techniques to determine the best probable
course of action when information is incomplete and resources are
limited. Today these techniques can be applied to non military problems
in government and industry which bear these same characteristics„
There are two basic parts to this approach. The first is systems
analysis: this can be defined as the resolution of a problem into many
separate components which form a system. The relationships between the
components are analyzed to determine exactly what effect a change in
any given component has on each of the other components. This analysis
usually involves mathematical equations, with which real world cause
and effect relationships can be simulated<> Once the system has been
quantified in this way, it is possible to undertake the second part of
the approach, which is usually called operations research. This step
involves the optimization of the system in order to achieve a defined
objective. This means that alternative ways of attaining a given objec-
tive are systematically examined. The best option, or optimal result,
Presented at the Seminar on Systems Analysis, U. S. Department of the
Interior, Federal Water Pollution Control Administration, Arlington, Va.
March 13, 1970.

-------
2
Is selected in accordance with whatever criteria may be desiredo
Various mathematical techniques can be used co achieve this
jptiraization, many of which have been developed in recent years.
There are several advantages to a system approach like this0
First, it provides a mathematical substitute, a model, for an
ictual situation= This model can be readily manipulated to examine
alternatives and options in a way that cannot be done with the real
?orld situation., The results of alternative actions can be assessed
ihead of time, rather than blindly instituting a given policy and wait-
ing for the results to appear, hoping that such results are favorable.
Second, the use of a model forces objectives to be specifically
lefined. Any possible objective that applies to the problem situation
lust be translated into quantitative terms so that the system can re-
ipond to it. In addition, any desired constraints, or policies, must
ie similarly specified if one desires to examine their effect on the
ystem, A related characteristic is that any desired objective or
:onstraint must be possible. If some particular objective cannot be
ittained, the model will provide this information immediately without
he need for waiting until the real world situation reacts„
Finally, the systems approach allows the problem to be examined
is a whole rather than piecemeal. The traditional sciences of engin-
ering, chemistry, biology, and economics cannot achieve the same
esults as systems analysis because each science looks at a problem
rom the viewpoint of that particular specialty. This is a piecemeal

-------
3
approach. Systems analysis concentrates on defining how each part
of a problem relates to each other part, and thereby uses all the
sciences to view the problem as a whole. It is not sufficient to
attach problems piecemeal, and to seek piecemeal solutions to them.
Systems analysis can provide comprehensive solutions to problems
having complex interrelationships, and quite obviously, environmental
management is a problem of this type.
Application to Water
Quality Management
Systems analysis and operations research can be used to structure
intensive water quality management studies. Such intensive studies
focus effort on individual river basins, or on highly urbanized areas
within basins where water pollution is unusually serious. In such an
area, the u-a.jor objective of water quality management is to secure
levels of water quality in the streams suitable for the multiple use
requirements of the regional population. The uses for various streams
and their associated numerical criteria, are provided by the water
quality standards. A comprehensive river basin study is designed for
a particular span of time, and is always future-oriented. This means
that a plan can be designed, for example, for ten or twenty years aheai
Within this period, there will be various alternatives available for
meeting the demands of the regional population. The alternatives are
strongly influenced by both the level of technology and the economic
resources that can be expected for this period. The growth of the
regional population and its associated demands for conflicting water
uses such as recreation on one hand and waste disposal on the other wi]
force a choice to be made between the available alternatives.

-------
h
Figure 1 shows the major steps in a comprehensive river basin
study. This is the program presently being pursued by FWPCA in the
Raritan River Basin. (5) To prepare a program capable of implemen-
tation, it is necessary to combine data on the physical river system
with socioeconomic data from the basin area. Systems analysis of all
information permits meaningful alternatives to be defined, and opera-
tions research methods are then applied to discriminate among various
alternatives and finally to aid in an optimal choice.
Water Quality Simulation Models
A systems analysis of the river itself is the indispensable first
step in water quality management. This consists of representing with
mathematical equations the cause and effect relationship that exists
between waste effluent and the resultant in-stream water quality.
The outcome is a water quality simulation model that permits alterna-
tive combinations of waste treatment to be examined. Such models can
be used to predict the effects of, effluents containing, e.g., organic
waste loads, toxic substances, bacteria, or thermal discharges. Most
work to date has been concerned with the effect of organic waste loads
Water quality models have various characteristics depending on the
method of formulation. Models generally are composed of a number of
separate geographical reaches within the river system. Each reach is
represented by a separate mathematical equation. They can be designed
for simple river systems, or for estuaries with superimposed tidal
fluctuations. It is possible to apply the model approach to bays and
lakes as well. Models can be formulated so as to predict short term

-------
5
changes in water quality, or average effects on the order of days, weeks,
or months. The latter type of model has proved to be especially suitable
for planning efforts that involve substantial time periods. (1) This kin<
of model fits in well with a ten or twenty year planning horizon.
The procedure in using any of' these models is the same:
First, water quality data from the river basin is required for a
base year. The STORET system for data storage is a very useful tool for
this purpose. Of course, the physical characteristics of the river must
be estimated also, such as length, width, depth, and flow.
Next, the magnitude of waste effluents being discharged to the river
must be measured for the same base year, and precisely located.
Finally, both sets of data are incorporated into a water quality
model which is designed specifically for the river system under study.
This means that the predictions of the water quality model are matched
to the observed water quality in the actual river. Once this is done
any change in the model can be used to predict a change in the real
river. For example, we can test ahead of time the effect of increasing
the flow of water in the river, or we can predict exactly how much the
river water quality will increase if the waste effluents are decreased.
It is this connection between the effluents and river water quality
which is most important.
The water quality model is operated by using a high speed digital
computer, which is necessary because of the extreme complexity of the
mathematical equations. One useful.machine for this work is the Interior
Department IBM. 360 when coupled to the IBM 1130's, which are located in
FWPCA laboratories.

-------
6
This approach has been applied co various bodies of water around
the country,. Figure 2 shows the 30 section model used by FWPCA in the
Delaware Estuary Study0 (2) A similar model has been designed for the
New York Harbor complex* (3) Figure 3 shows the model used in FWPCA
work on the Potomac River0 (4) Other applications of this approach
include the Raritan River Basin in New Jersey (5), the San Francisco
Bay-Delta complex (6), and the Columbia River (7)»
Options for Water Quality Management
Once a water quality model has been formulated for a river system,
it is possible to construct alternative water quality management
schemes, and to examine them for optimality0 Within the context of the
water quality model any number of such alternatives can be constructed,
but they all have several characteristics in commons
First, the objectives for any given scheme are supplied by the
water quality standards, which assign numerical levels of desired water
quality to reaches of a stream. This is the same thing as being given
a desired effect, then asked what can be done to bring about such an
effecto For example, the water quality standards may state that a
particular stream should have water quality suitable for water contact
recreationo That level of water quality then becomes our objective,
and our optimization and cost/effectiveness models are used to find
alternative ways to meet that goal0 In the Delaware Estuary Study,
the water use and water quality goals were embodied in five alternatives,
termed Objective Sets„ These range from the low Objective Set 5, which
represents present degraded water quality, to the very high Objective
Set 1, the maximum feasible enhancement of the Delaware River using

-------
7
technology expected to be available through 1980.
When alternatives are being formulated, there are always constraints
that result from policy decisions that must be imposed on the system.
In this case there were two constraints that affect solutions in an
important way. First, in no case would the water quality in any area be
permitted to decrease below its present level, even if present levels
exceed the legal standard in the area. This of course is the anti-
degradation policy. Second, no effluent source could discharge any
load greater than that discharged at present. If a particular treatment
level were chosen, sources currently treating their wastes to a higher
degree could not then lower their degree of treatment.
The most important problem in construction alternatives for optimi-
zation is probably the selection of criteria. I must emphasize that
these are not the same as the criteria associated with the water quality
standards. Rather, optimization criteria are those factors which we
decide are important enough so that we will seek best solutions in terms
of them. That is, we must decide what subsidiary objectives are impor-
tant enough to warrant consideration, in addition to the prime objective
of attaining a given water quality level in the stream. If some group
of waste sources is discharging into the stream, we must decide what
the most important factors are in reducing these discharges to some
acceptable level. Various possibilities for these optimization criteria
can be proposed.
One, for example, would be to look for the solution that would be
least expensive to implement. Another possibility would be to require

-------
8
waste dischargers in the same part of a river to provide the same
amount of waste treatment. Still a third possibility would be to
require large waste dischargers to provide a greater degree of waste
treatment than small dischargers.
The importance of cost as an optimization criterion is well
illustrated by considering the alternatives available in a simple
example with only two waste sources0 Because the tide moves waste
in both directions, waste load removed at any point in an estuary
has an effect both upstream and downstream from the point of dis-
charge0 In this situation the water quality standards could be
equally well attained by very high treatment requirements at either
source, or by more moderate treatment at both sources„ Here we
have a chance to make trade offs between the two waste sources„
First of all, we can give either source credit for whatever degree
of waste removal it is presently achieving. Next, we can mse the
water quality model to determine how much the abatement of either
source would contribute to meeting the standards,, Finally, we can
use the cost of waste removal at each source to compute the cheapest
way of meeting the standards. The least costly approach in general
results in some combination of treatment requirements at both sources
that would not be obvious from a quick appraisal of the situation.
To generalize this conclusion: If each source has an associated cost
function which states how much it would cost to remove additional
waste load from that particular effluent, these functions can be used
to define solutions to the problem of specifying treatment levels at
many sources.

-------
9
This problem was met in the Delaware Estuary Study by constructing
a series of optimization models. (8)(9) Using the water quality stand-
ards as a goal, it is possible to determine approximately how much waste
load can be discharged to the river by all sources acting together. The
optimization models are constructed to allocate this permissible load
and its associated costs among the individual sources located along the
length of the river. For this reason these models are often called cost
allocation models, and their outputs for each waste source are termed
load allocations. Just as in the case of the example with only two
waste sources, we determine the least expensive way of meeting the
standards. Each waste source is also assigned a non-increasing discharge
which it can never exceed : this is its allocated waste load.
There are three major cost allocation models: these are the Uniform
Treatment model, the Cost Minimization model, and the Zoned Optimization
model. It must be emphasized that each of these models will attain the
same water quality goal, although they differ greatly in efficiency.
This means that each model will achieve the water quality demanded by
the standards, but that each will do so in a different way. These models
can be described in terms of the percent removal of untreated waste load
which each waste source must remove rather than discharge to the estuary.
Since these models involve complex mathematical equations, it is necessary
to use a computer to find the solutions of these equations. In fact, it
would be fair to say that the more advanced models could never be solved
without the aid of a computer.

-------
10
I would now like to describe these three models in somewhat more
detail:
1. Uniform Treatment Model (Figure 4).
The uniform treatment model requires that an identical percentage
of the untreated waste load from each source be removed before discharge
to the river. Thus it is necessary to find a single treatment level
that is high enough to satisfy the water quality standards. Uniform
treatment is the commonly used approach in existing management situa-
tions. Its primary advantage is administrative simplicity; however,
it is economically inefficient in accomplishing the goal. By economi-
cally inefficient I mean that many waste sources are required to in-
crease treatment in high quality reaches of the river because the
standards are not being met in the degraded portions. This is like
requiring a discharger to increase his waste treatment when the water in
the river already meets the standards. Indeed, such dischargers are
penalized by the fact that they must provide the same percent removal
as their neighbors who are located in parts of the river with degraded
water quality. The result is a level of water quality much above the
standards in the reaches already of high quality, thus providing a
measure of the inefficiency of this solution. While each source is
required to remove the same percent of raw load, no allowance is made
for differences in the unit cost of waste removal from source to source.
In spite of apparent equity, the solution is actually quite inequitable,
when dealing with very different waste sources like cities, pulp mills,
and oil refineries. It at best can treat each source identically in

-------
11
terms of treatment level without regard for cost. It does not attempt
to make dischargers pay equal costs or to treat sources similarly because
of similar location or types of activity. Thus, any gain in administra-
tive simplicity is offset by the model's inefficiency and inequity plus
its substantially higher costs to the basin.
2. Cost Minimization Model (Figure 5).
The cost minimization model is designed to reach the water quality
standards at an absolute minimum cost to the basin. This model selects
waste dischargers based on the greatest waste removal per dollar of
treatment cost and simultaneously considers where in the estuary water
quality must be improved. The key here is that each source is consideret
on its own merits; each is separately treated. Because of this, the
solutions show a number of trade-offs between individual sources. This
model systematically seeks out those particular dischargers that are
really preventing the standards from being met : this means those which
are right in the part of the river which is degraded; those waste sources
which have inferior waste treatment now; and those where additional
treatment facilities to improve the river would be cheapest.
This model is obviously very efficient in allocating treatment to
be attained by each source to meet the water quality standards. No
unnecessary treatment is called for at all and only that waste removal
is required which produces an increment in water quality at lowest
possible cost. The cost minimization model is undoubtedly the best
technical way to upgrade the river to meet exactly the water quality
standards. On the other hand, this solution is likely to be extremely

-------
inequitable in the sense of not treating industrial competitors in a
like manner. For example, if waste treatment costs at one firm are low
while at another they are very high, it is likely that the source with
the low-cost treatment capability will be required to treat its waste
to extremely high removals while the other discharger might be required
to do very little. Because of this unequal treatment of some dischargers
which may really be quite similar, this solution could result in some
antipathy on the part of competing firms. One example of this would be
two chemical manufacturers located on the same river.
3. Zoned Optimization Model (Figure 6).
The zoned model is actually formulated as a combination of the
uniform treatment and cost minimization models. This results in uniform
treatment levels for groups of waste sources within zones of the river,
with cost trade-offs possible between zones so that a set of treatment
levels having minimum cost is obtained. The estuary was divided into
four treatment or management zones each between about 10 and 30 miles
long, and all waste sources were grouped according to the zone in which
they are located.
As might be expected, the results of this model are intermediate
between those of the first two. In this case the model produces four
different treatment levels, one for each zone. All the waste dischargers
within a given zone must provide treatment to the extent required for
that zone. The higher waste treatment levels will be required for zones
in which the cost is less, and which are located in degraded parts of
the river.

-------
13
A measure of equity is attained in that sources located near each
other, and adjacent to similar water quality, are treated similarly
because they are in the same zone, and hence subject to the same treat-
ment.
The zone approach is looked on with a certain amount of favor from
the administrative viewpoint. It is nearly as easy to implement as the
uniform treatment method, since it requires only locating waste sources
within the proper management zones. In addition, the equities of similar
treatment for sources located near one another tend to reduce the
objections of individual dischargers regarding their situation as
compared to that of their neighbors.
Cost/Effectiveness Comparison
At the same time that the cost of attaining the standards was being
estimated by means of the three cost models, a benefits analysis was
undertaken to determine what effects could be expected after the en-
hanced water quality is achieved.
Certain monetary benefits can derive from increased use of the
estuary, once its water quality is improved. Numerous benefits are
intrinsic in water quality enhancement programs. These are realized by
such things as a more economic utilization of natural resources, preser-
vation of fish and wildlife, and protection of the health of the
regional population. One of the basic aims of the Delaware Estuary
Study has been to better define and quantify the benefits of enhancing
water quality in the estuary.

-------
14
Quantification of the benefits is an essential part of any feasi-
bility study. However, in the water pollution control field, the state
of the art is new and much methodology is currently being developed.
The Delaware Estuary Study proceeded with an analysis of the anticipated
benefits for several water uses that must have high water quality. It
was not expected that all the benefits could be quantified. The results
included positive benefits for recreation and commercial fisheries based
on the increased use attributable to improved water quality. The recre-
ation included swimming, boating, and fishing by the population living
along the estuary.
Program costs and benefits are shown on Figure 7. On this graph
the water quality standards are indicated by the letters DRBC, which
refers to the standards submitted to the Secretary of the Interior by
the states of New Jersey, Pennsylvania, and Delaware, and by the
Delaware River Basin Commission. For any given objective set the cost
minimization model yields the smallest cost and the uniform treatment
model yields the largest. This is in agreement with the efficiency
concepts discussed earlier for each of these models. To attain the
water quality standards costs between 390 and 440 million dollars,
depending on whether the cost minimization or uniform treatment model
is chosen. The cost of the zoned model is 430 million dollars. The
benefits of attaining the water quality standards are estimated to be
worth 420 million dollars.
Examination of the graph reveals the existence of a benefits
"plateau," Once a particular level of use is reached, no further gains

-------
15
result from increased water quality, at least with respect to that use.
Thus, for example, there are no sport fishing benefits unless a fisherman
is present, no swimming benefits without a swimmer, and no boating bene-
fits without a boater. Therefore, as ever more stringent quality goals
are specified, costs increase tremendously while benefits tend to level
off at a maximum value. To aid in gaining perspective, it should be
noted that objective set 1 represents 92-98% removal of organic waste
load plus in-stream aeration and removal of benthic deposits, at an
estimated cost of $700 million dollars.
Implementation
The Delaware Estuary program is currently being implemented through
the joint efforts of FWPCA, the States, and the Delaware River Basin
Commission. The alternative finally chosen was the zoned approach to
estuary management, which would allow a certain degree of flexibility
to be maintained to meet future conditions. The definition of water
quality standards, and the selection of an allocation model, has made
it possible to derive effluent load allocations for individual firms
and municipalities. These non-increasing loads are the maximum waste
discharge loads that will be permitted under the given conditions.
Similar reasoning must be applied to other river basins. Only if the
waste loads to streams are held to non increasing magnitudes will the
water quality standards of our waterways be maintained through time.
Thank you.

-------
References
1« "Estuarine Water Quality Modeling - State of the Art," 1st Technical
Conference, St. John's College, Annapolis, Md., June 24, 1969; Tracor
Document No. 69-707-U; Tracor, Austin, Texas.
2.	Delaware Estuary Comprehensive Study, "Preliminary Report and Findings,"
U. S. Dept. of the Interior, Federal Water Pollution Control Administra-
tion, Philadelphia, Pa.; 1966.
3.	"Mathematical Models for Water Quality for the Hudson-Champlain and
Metropolitan Coastal Water Pollution Control Project," prepared by
Hydroscience, Inc., Leonia, N. J., for the Federal Water Pollution Control
Administration; 1968.
4.	"Potomac River Water Quality, Washington, D. C. Metropolitan Area,"
U. S. Dept. of the Interior, Federal Water Pollution Control Adminis-
tration, Cincinnati, Ohio; 1969.
5.	Morris, A. R.; Statement for the New Jersey Clean Water Council Hearing
on the Raritan River Basin; Federal Water Pollution Control Administra-
tion, October 21, 1969.
6.	"San Francisco Bay Delta Water Quality Control Program," Final Report
to the State of California, Kaiser Engineers, Oakland, Calif., June 1969.
7.	Yearsley, J. R., "A Mathematical Model for Predicting Temperature in
Rivers and River-Run Reservoirs," Working Paper No. 65 (preliminary),
U. S. Dept. of the Interior, Federal Water Pollution Control Administra-
tion, Portland, Ore.; March 1969.
8.	Smith, E. T. and A. R. Morris, "Systems Analysis for Optimal Water Qua-
lity Management," Jour. Water Pollution Control Federation, Sept., 1969.
9.	Smith, E. T. and E. Mehr, "A Cost Allocation Model for Zoned Optimization
of Waste Treatment Requirements," presented at the 2nd. Mid-Atlantic
Industrial Waste Conference, Drexel Institute of Technology, Phila., Pa.,
Nov. 18-20, 1968.

-------
EXAMPLE OF A NETWORK OF PROJECT ACTIVITIES AND EVENTS

-------
^Schuylkill
Rtver
IDorby
CHESTER creek
PENNSYLVANIA
DELAWARE
A
Siote Lme-
Mile 79 0
WILMINGTON
Raccoon
Creek
OFoni
T 21) \ Creek
'Christina
River
NEW
CASTLE ,
(25b
Che*opeoke
Delaware
Conal
/Soiem
/Cono^
f23)
•vSqlem Creek
7-
.Liston Point-Mile 48 3

5^
Smyrna
River
Pennypack
Creek
/
I Poquessing
Creek
^Frankford
CreeK/
Crostwicks
Creek
Roncocas Creek
'( 41 ,
vPennscuken
. Creek
(15)
N
Vgig Timben
\Creek
\
\ Mantua
Creek
NEW JERSEY
DELAWARE ESTUARY
COMPREHENSIVE STUDY
SECTIONS FOR
MATHEMATICAL MODEL
SCALE MILES

FIGURE 2

-------
N
\ LEGEND
/ =
WEST
VIRGINIA
LOCATION MAP
= MAJOR WASTE
TREATMENT PLANT
= ESTUARY SEGMENTS
GAGING STATIONS
L I T T L E FALLS BRAN C H -
BETHESDA, MD
POTOMAC RIVER -
WASHINGTON, D C
ROCK CR -SHERRILL DRIVE-
WASHINGTON, D C
N E BR ANACOSTIA RIVER -
RIVERDALE , MD
N W BR ANACOSTIA RivER-
HvATE5VlLLE , MO
FOURMluE RUN-
ALEXANDRIA, VA
LIT TLE PIMMIT RUN
ALEXANDRIA, VA
CAMERON RUN-
ALE XANDRI A , VA,
HENSON CREEK -
OXEN HILL , MD
POHICK CREEK -
LORTON, VA
MATTAWOMAN CREEK-
POMONKEY, MD
District of Columbia
ARLINGTON COUNTY
ALEXANDRIA S ANIT AT 10 N
AUTHORITY
FAIRFAX COUNTY - WESTGATE
PLANT
FAIRFAX COUNTY - LITTLE
HUNTING CREEK PLANT
FAIRFAX COUNTY - DOGUE
CREEK PL ANT
POTOMAC RIVER STUDY AREA
FIGURE 3

-------
UNIFORM TREATMENT MODEL
LOCATION OF SOURCE ALONG ESTUARY
FIGURE 4

-------
COST MINI M IZATION MODEL
LOCATION OF SOURCE ALONG ESTUARY
FIGURE 5

-------
ZONED OPTIMIZATION MODEL
LOCATION OF SOURCE ALONG ESTUARY
FIGURE 6

-------
V IY IE DRBC n I
OBJECTIVE SET
FIGURE 7

-------