EPA-R 5 - 72-0 04
September 1972
Socioeconomic Environmental Studies Series
An
Investment Decision Model
for
Control Technology
National Environmental Research Center
Office of Research and Monitoring
U.S. Environmental Protection Agency
Cincinnati, Ohio 45268
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EPAR5 72-004
September 1972
AN INVESTMENT DECISION MODEL
FOR CONTROL TECHNOLOGY
Robert M. Clark
Office of Program Coordination
National Environmental Research Center, Cincinnati
Program Element 1D1312
US EPA WGIOfi 4 LIBRARY
AFC-TOWER 9th FLOOR
61 FORSYTH STREET SW
ATLANTA, GA. 30303
NATIONAL ENVIRONMENTAL RESEARCH CENTER
OFFICE OF RESEARCH AND MONITORING
U.S. ENVIRONMENTAL PROTECTION AGENCY
CINCINNATI, OHIO 45268
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REVIEW NOTICE
The National Environmental Research Center, Cincinnati/
U. S. Environmental Protection Agency, has reviewed
this report and approved its publication. Mention
of trade names or commercial products does not con-
stitute endorsement or recommendation for use.
ii
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FOREWORD
Man and his environment must be protected from the
adverse effects of pesticides, radiation, noise and
other forms of pollution, and the unwise management
of solid waste. Efforts to protect the environment
require a focus that recognizes the interplay between
the components of our physical environment—air, water
and land. The multidisciplinary programs of the
National Environmental Research Centers provide this
focus as they engage in studies of the effects of
environmental contaminants on man and the biosphere
and in a search for ways to prevent contamination
and recycle valuable resources.
ANDREW W. BREIDENBACH, Ph.D.
Director, National Environmental
Research Center, Cincinnati
ill
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ABSTRACT
Investment decisions in control technology for en-
vironmental management are similar to those in other
areas of public finance. These decisions, which may
include the decision to construct a water or waste-
water treatment system or an incinerator, depend on
adequate financial support, which means not only
availability of money in sufficient quantity, but
also at the time when needed. A mathematical model,
incorporating borrowing and lending variables, has
been structured to provide an efficient method of
studying the problem. The model formulation assumes
that investment decisions for control technology can
be separated into a total operating and capital cost
decision and an investment cost decision. These
costs are minimized in two stages. The first stage
utilizes a fixed-charge algorithm and the second
stage, a linear programming algorithm. A problem
is solved utilizing the construction, operation and
financing of an incinerator subject to capacity and
monetary constraints for an example.
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AN INVESTMENT DECISION MODEL
FOR CONTROL TECHNOLOGY
Investment decisions in control technology for en-
vironmental management are similar to those in
other areas of public finance. These decisions,
which may include the decision to construct a water
or wastewater treatment system or an incinerator,
depend on adequate financial support, which means
not only availability of money in sufficient quan-
tity but also at the time when needed. To illustrate
the interdependence of engineering design (and oper-
ation) of control technology systems and the timing
of sufficient quantities of money, an investigation
has been undertaken to isolate the critical variables
involved. The problem has been structured in the
form of a mathematical model to provide a more effi-
cient method of defining the important variables.
Basically, the model says nothing more than money
comes in from taxes or user charges and money goes
out to pay for control technology management, but
the money coming in may not match the money going
out in quantity or timing. For this reason, a method,
such as a borrowing or lending device, is needed to
bring the expenditures into phase with incoming
funds. In addition, the model must be consistent
with reality by conforming to reasonable limitations
on such things as treatment capacities and user
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charges or tax burdens. The study might be charac-
terized as one that studies the interface between
engineering design and adequate financial support.
The manager of control technology facilities must
plan a system that will meet the desired operational
or functional requirements of the community at mini-
mum cost. In such a system, development and enlarge-
ment of engineering facilities over time must be
considered, as well as long-range financial planning.
The manager is concerned with optimal financing and
with timing the enlargement of facilities in accord-
ance with growth requirements, borrowing and payback
opportunities, service charge possibilities, etc.
In view of the uncertainty associated with informa-
tion required for an analysis of this problem (e.g.,
growth rates and market opportunities), it seems un-
likely that any analytical solution would result in
a conclusive optimal design in the sense that the
answer is the best of all possible designs. Standard
approaches to the evaluation of capital investment
decisions are based on the present value or present
worth concept [1]. These methods do not allow the
consideration of limitations on fund availability.
What is needed, however, is the ability to study a
particular situation in detail and to assess a variety
of possible conditions for which suboptima can be gen-
erate [2]. Techniques from mathematical programming
are used here to suggest an approach to such a study.
The model formulation will assume that investment
decisions for control technology can be separated
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into a total operating and capital cost decision and
an investment cost decision.
A model will be developed that performs a two-stage
minimization of these costs. The first stage mini-
mizes the total operating and capital cost of a facil-
ity subject to operational constraints, and the second
stage minimizes the cost of financing its construc-
tion. The facility will be assumed to have a char-
acteristic capital and operating cost function and
an investment cost dependent on the rate of return
associated with the bonds used to finance the capital
expenditures.
The operating and capital cost function will take the
form C = MK + Cq» where:
C » total construction and operation costs;
M = slope;
K = capacity supplied; and,
Co = set-up or start-up cost.
The total operating and capital cost function may be
composed of one or of many of these types of rela<-
tionships and will be minimized subject to operational
requirements. The solution to this problem will in-
dicate the amount of capital investment that must be
undertaken in each period to supply the required
level of capacity. To meet requirements, for example,
one facility might have to be constructed in period
one and another in period three. We must, therefore,
invest in capital equipment in each of these periods,
equal to the fixed cost or set-up charge. The total
cost of required capital investment in each period
will be the sum of the construction costs required
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for that period. This problem is essentially a fixed
charge problem and requires appropriate solution tech-
niques .
After finding the "best" solution to our operational
problem/ we will determine the most efficient alloca-
tion of funds for capital investment. To examine the
inflow and outflow of funds for timing and investment
requirements and for investment alternatives, a linear
programming approach will be applied to this problem.
It should be emphasized that this approach leads to
a suboptimal solution. The "best" approach would
be to minimize a total cost function including oper-
ating, capital, and financing costs. Unfortunately,
the state-of-the-art in mathematical programming is
such that there are no techniques available to solve
a problem formulated in this manner. The approach
suggested in this paper does, however, provide a sys-
tematic method for evaluating various alternatives
for minimizing the cost of control technology facilities.
To rationally apply this approach to control tech-
nology investment decisions, it is important to under-
stand something about control technology systems.
CONTROL TECHNOLOGY SYSTEMS
Control technology systems are those systems that
assist in environmental management by abating and
controlling pollution and/or protecting the public's
health. These systems may be designed for water and
wastewater treatment, air pollution control, solid
waste management, and noise abatement and control.
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Although the example that will be developed here is
for investment, operating, and capital cost decisiohs
in solid waste management, the general techniques
apply equally well to other investment decision prob-
lems in control technology.
The most common methods presently used for treatment
and disposal of solid waste are the sanitary land-
fill, the central incinerator, or combinations of
both. Incineration is a volume reduction method and
requires investment in an expensive fixed facility.
Sanitary landfill requires acquisition of land area,
which may be excavated; waste material is deposited
in the excavation and covered with earth at the end
of each day's operation.
Central incineration will be considered in this an-
alysis as an example for investment in control tech-
nology. An incinerator may have a high initial cost
and a low operating cost or a low initial cost and a
high operating cost. The selection of an incinerator
to be used in a specific community strongly depends
on the availability of funds for investment and oper-
ation and on the amount of wastes to be handled.
Decisions such as this must be made within a time
frame or planning horizon.
FACILITY COST MINIMIZATION MODEL
A model for this problem must reflect the require-
ments of the situation, which include the planning
period, the level and type of operation, and quan-
tity of waste to be treated. The objective is to
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minimize the total cost of operation and equipment
acquisition for the time periods under consideration,
while respecting the requirements or restrictions
imposed on the problem. An optimal solution to the
problem will indicate: (a) the types of facilities
to be constructed in each time period; and, (b) the
types of facilities to be constructed for each in-
crement of capacity.
To provide the required capacity for the period under
consideration, we can choose from among any of sev-
eral available incinerators. We can assume n dif-
ferent incinerators and j periods in which treatment
capacity must be supplied. Thus, the capacity of
treatment available must be sufficient to handle the
quantity of waste Qj. For the first period, this
can be:
(1)
n
z
i=l
x • i > Q,
ll — 1
in which x^ represents the increment of capacity
of incinerator type 1 required in period one; X21'
the increment of capacity of incinerator type 2 re-
quired in period one, etc. Incremental additions to
incinerator capacity in period j can be obtained from
among the n types, x, ., x-. . . . x ., such that for
i j zj nj
any period j, (j = 2, 3, . . . m) the capacity is
given by:
(2)
n
E X. . > Q.
i=l ~ J
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where (j = 2 . . . m) is the incremental waste
quantity handled in period j. This formula assumes
that each increment directly augments capacity. To
describe the situation in a realistic manner, we
can introduce the idea of a fixed cost, f^, which
is associated with the use of activity x. and is
incurred with the first increment of x^ constructed.
The value for f^ represents the capital cost or ac-
quisition cost of incinerator i; thus, if in period
one, treatment capacity is required from incinera-
tor 1, we would designate this capacity as This
implies that a capital investment cost of f^ will be
incurred. If more capacity is needed, for example
x12 in period two, the only additional cost will be
the variable cost C]^XJ2' where x^ is the level of
capacity for incinerator 1 constructed in period
two [4].
In addition to the fixed cost of the initial facility
construction, we can also allow for a fixed cost of
facility expansion. For example, in incinerator 1,
the expansion capacity is anc* fixe<* cost
associated with it would be f^.2' We can f°rmula'te
the model as follows:
n n m
(3a) Minimize; z = E (f. + c. x.) + I E (f.. + 0 x..)
i=l 1 11 i=l j=2 ^
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Subject to;
xll+x12+***+Xlm ~ X!
+X21+X22+*'*x2m = X2
(3b) +xnl+xn2+"* *+Xnm = Xn
X11 +x21+ +xnl - Q1
x12 +x22 +xn2 ~ °2
(3c) xlm +x2m +xnm - Qm
xij ^ °
Equation (3a)f composed of the concave cost functions
for the various investment alternatives, represents
the function to be minimized. For example, consider
the cost function for incinerator 1 represented by
fl + clxl; ^ere ^l represents the purchase or fixed
cost for acquiring the treatment facility, and c^ is
the variable cost associated with the level of activ-
ity, x^, at which it is operated. The fixed cost of
adding the incremental capacity, x^/ wou1^ be f]_2*
The (3b) group of equations gives the total capacity
at which the facility is operated. Again, when dis-
cussing incinerator 1, x^ is the total capacity needed
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by incinerator 1; is the increment of capacity
needed in period one; x12 is the increment of capa-
city needed in period two? etc., which can be written
as:
X1 X11 + x12 + * * * + xlm
Equations (3c) are the constraints on capacity re-
quired in period one . . . m. In period one, the
required capacity must be greater than . The op-
tions for capacity include the available capacity
of incinerator 1 in period one, x^i' or incinerator 2,
in period one, *21' etc* This holds true for each
period of interest.
If incineration capacity is not available in certain
time periods, the capacity equals zero. Thus, if
there will be no capacity available for incinerator x2
in time period one, x2^ 83 0, and,
x2 = X22 + • • • + x2m
and, x12 + x32 • • • + xn2 - ^2
This allows us to specify that some investment al-
ternatives will not be available until later in the
planning period. Any constraints on the size of
capacity for each incinerator are given by the form:
xij i Ks
where x^ is the ith type treatment in the jth period,
and K is its constraint,
s
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This model demonstrates the capital cost that must
be incurred each period to maintain capacity subject
to operational requirements. If the results of the
optimization indicate that the first increment of
capacity for incinerator 1 is to be constructed in
period one, the first increment of capacity for in-
cinerator 2 is to be constructed in period three,
and no other incinerators are to be constructed, then
the capital investment costs will be f^ in period one
and f2 in period three. The following notation is
used to designate the capital investment required in
any period j; is the capital investment required
in period one, the capital investment required in
period two, etc. In our example, = f^ and F^ = f2/
with all other capital investment being zero.
BORROWING MODEL
The construction of solid waste facilities or of any
types of control technology involve sizable sums of
money obtained through some financing procedure, nor-
mally through general tax funds. We will assume,
however, for this analysis that revenue bonds will
be used to finance the project. If this financing
method is unavailable to a particular community, the
revenue and individual taxpayer cost must be analyzed
to assess the economic validity of the level of ser-
vice to be rendered. Where financing of control tech-
nology systems is entirely through taxation, a mean-
ingful analysis requires the budget and the tax resources
available to the particular project to be partitioned [2].
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For any period, the funds required for construction
of control technology facilities must be sufficient
to cover the cost of construction of plant capacity
in that period, and they must be available to pay
interest and principal payments due in that period.
At the beginning of a period, the funds available
must represent the accumulated revenue collected in
all preceding periods. In this analysis, the time
value of money is considered, as well as differences
between borrowing and lending rates, dependence of
funds usages, capital rationing, and "imperfect mar-
kets."
To formulate equations for these concepts, we must
be able to write explicit constraints for: usage
opportunities of funds; requirements and timing of
funds, e.g., interest payments, construction charges,
etc.; and the interdependence of fund usages.
The variables in the capital allocation model include
borrowings and amounts set aside for reserves. These
are related, through expressions, to the funds avail-
able for the uses in each period and to the needs that
the available funds must cover in each period [3].
The funds available during period one for construc-
tion of the facility must be greater than the obli-
gations for that period, or:
(4) Fq + Z Lkl > £ Lklpkl + Z Lklrk + Fx - E Akl
In equality (4), the left side of the inequality con-
sists of: cash on hand at the beginning of period
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one, F0; plus the loan (or funds borrowed) of bor-
rowing type k at the beginning of period one. The
right side of the inequality consists of the funds
employed to retire a portion, Pj^' °f the outstanding
debt, or E Lklpkl? Plus the money required for
interest payments, jE Lkirk; plus the capital invest-
ment required in period one, F^; minus additional
funds used to develop cash reserves (or sinking funds)
to decrease the outstanding debt or any other special
purposes, E A. ,.
k KX
We can now define a variable that is equal to:
(5) Wx = Fo + £ Lkl - E Lklpkl - E Lklrk - F1 + E AR1
Equation (5) represents the accumulated funds remain-
ing at the end of the first period.
During period two, the funds available will consist
of the revenue collected at the end of period one,
cjlPl' where g^ is the service charge, and is the
population served during period one; plus the funds
remaining from period one, W^; plus funds borrowed
at the beginning of period two, E Lk2* The sum of
the funds available in period two must be greater
than or equal to the funds employed to retire a por-
tion, Pk2' of the outstanding debt, or E L^P]^'
minus the interest due on the outstanding de^t,
or E (1 " Pfcj.)» and, minus additional funds used
to develop cash reserves, jE Ak2; plus funds employed
to retire a portion, Pk2' the outstanding debt,
Lk2' or 1 Lk2pk2; P*us interest due on the outstanding
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debt, I Lk2rk' Plus the capital investment required
Jv
in period two, F2> In summary, funds available at
the beginning of period two can be stated as:
glPl + W1 + £ Lk2 ^ k Llkpk2 + £ Lklrk (1 " pkl) ~ £ Ak2
+ £ Lk2pk2 + £ Lk2rk + F2
VI2 is defined as the funds left over at the end of
period two, or:
W2 " glPl + W1 + £ L12 Lklpk2 + £ Lklrk (1 " pkl)
" £ ^Sc2 + £ Lk2pk2 + J Lk2rk + F2}*
For all succeeding periods, j, (j - 3, 4, 5 . . . n),
the funds available at the beginning of period will be
given by:
9j-lpj-l + Wj-1 + £ Lkj i £ ^ Lkbpkj + J ^ ^b'k
(1 " pks> " * *kj + £ ^j'kj + J Lkjrk + Fj
We can categorize the funds in three different ways:
E £, Wk {(\ + x) - * "k <«k -s + x>>
=1 s=i
1. Interest payments:
3 «k
k b
2. Funds recovered for investment:
- I I (1 + iK)3"15
k b-1 kb b
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3. Revenue from service charges:
j
b-i b b
In these terms, the objective may be stated as:
Minimize total cost, C:
M,
3 k
(6) C = E E Lkbrk {(Mk + 1) " E pks (Mk ~ s + 1)}
k b=l s=l
- J J, A*b + vj"b + j, Vb
where is the life of issue and ifa is the interest
rate in period b. The objective is to be attained
subject to the constraints detailed above on the
variables involved. By combining this model with
the facility selection model, we can achieve a more
efficient allocation of funds for control technology
investment decisions.
EXAMPLE
This technique is used to solve the following prob-
lem. Assume three basic incinerators designated
by x1# x2, and x3, respectively. The first type,
has a capital investment cost, f^, of $3 million
and a variable operating cost of $2 per ton of solid
waste processed; this means that $3 million is needed
to acquire the facility represented by treatment type
x^. We will also assume a zero fixed cost for capa-
city addition to the incinerators. After the facility
is built, the operating cost will be $2 for every
ton of solid waste processed. Incinerator x2 has
a capital cost of $2 million and an operating cost
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of $3 per ton, and incinerator x^ has a capital cost
of $1 million and an operating cost of $4 per ton.
Combinations of these incinerators can be used to
satisfy the demand over four planning periods.
Assume that a municipality has ah increasing produc-
tion of solid waste over time; 5 million tons must be
treated in period one, and the incremental capacity
requirements in the following periods are 7 million
in period two, 8 million in period three, and 9 mil-
lion in period four. The constraint set for this
problem is:
xll+x12+x13+x14
= x.
(7a)
+x21+X2 2+X2 3+X2 4
= x.
+x31+x32+x33+x34 " x3
l X31 — 5 ^
+x, 0 +X27 > 7 X 10?
(7b) 12+x1- +x23 +x^ 7 8 X 10£
+x14 24 +x34 E 9 x 1°
Equations (7a), the sum of which equals the total ca-
pacity for the treatment type, represent the capacity
available over the four periods. That is, is the
capacity of incinerator 1 used in period one, x12 the
capacity of incinerator 2 used in period two, etc.
Equations (7b) represent the capacity of the various
treatment types available to fulfill the demand in
each period, e.g., in period' one: we have the
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capacity of incinerator 1 in period one; plus x21'
the capacity of incinerator 2 in period one; plus
x^/ the capacity of incinerator 3 in period one.
The following system could then be solved:
Minimize:
z = (3 x 106 + 2x,) + (2 x 106 + 3x0) + (1 x 106 + 4x,)
Subject to:
X1 ~xll~x12~x13~x14 = °
x2 "X21~X22~X23~X24 = 0
x3 ~x31~x32~x33~x34 = 0
XH +x21 +x31 15x10
x12 +x22 +X32 > 7 x 10
x13 +x23 +X33 - 8 x 10
x14 +x24 +x34 - 9 x 10
After solving the model with the use of Walker's
algorithm, we find incinerator 1 provides the least
expensive solution to handle capacity for all four
periods [5]. Incinerator x^ will be built in per-
iod one and will require a capital investment of $3
million; therefore, F.^ = 0, P3 = 0, F4 = 0.
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The conditions for the bond type, assuming one type
of borrowing available only at the beginning of per-
iod one, are given as:
Bond
Type
(k)
Life of
Issue
(Mk)
Periods
pks
% of Original Loan to be
Retired in Each Successive
5-Year Period
Borrowing
Interest
Rate Per
Period
%
(rk)
S
1
2
3
1
3
21
32
47
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Assume, for simplicity, an Fc = 0, with no sinking
funds, and at the beginning of each period, a popu-
lation of 2,000, 3,000, and 4,000, respectively. A
constraint placed on the user charges requires that
they be equal in each period. The constraint set
will be as follows:
(8a) 0.61LX1 > 3 x 106
l,000g1 + 0.15L11 > 3 x 106
2,000g1 + 3,000g2 - 0.40L1;l > 3 x 106
2,000g1 + 3,000g2 + 4,000g3 > 3 x 106
(8b) g-L = g2 = g3
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Constraints (8a) are of the same form as shown in
constraint (5), and (8b) reflects the equality con-
ditions on the user charges. The objective function
is obtained from equation (6) and has the form:
Minimize:
z = 0.41L11 + 2,000g1 + 3,000g2 + 4,000g3
The solution to the above system is obtained with the
use of a standard linear programming algorithm [3].
The values are:
Ln = 5.80 x 106 dollars
g^ = 1,060 dollars/period
gj = 1/060 dollars/period
g3 = 0,060 dollars/period
Total cost of the system is:
C = 11.90 x 10^ dollars.
SUMMARY AND CONCLUSION
With this approach, the most efficient allocation
of funds for investment in control technology can
be analyzed. Obviously, there is a great deal of
flexibility in this method. The various demand
patterns can be examined, as well as many different
alternatives for investment. By using sensitivity
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analysis in conjunction with the linear programming
portion of the model, the effects of changing the
investment pattern on the borrowing costs can also
be examined. The writer believes this technique
can be beneficially applied to financing control
technology systems.
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008902
REFERENCES
Barish, N. N. , Economic Analysis for Engineering
and Managerial Decision-Making, McGraw-Hill
Book Company, New York, 1962, p. 51.
Lynn, W. R., "Stage Development of Wastewater
Treatment Works," Journal Water Pollution
Control Federation, Vol. 36, (June 1964) ,
pp. 722-751.
Dean, J. , Capital Budgeting; Top-Management Policy
on Plant Equipment, and Product Development,
Columbia University Press, New York, 1951,
p. 210.
Hillier, F. S., and Lieberman, G. J., Introduction
to Operations Research, Holden-Day, Inc., San
Francisco, 1967, pp. ?64-565.
Walker, W., "Adjacent Extreme Point Algorithms
for the Fixed Charge Problem," Technical
Report No. 40, Cornell University, Ithaca,
1968, pp. 4-6.
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759-549/1034
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