A Mathematical Programming Modelfor Air Pollution Control*

Robert E. Kohn1Assistant Professor of Economics at Southern Illinois University�

Edwardsville Campus, Edwardsville, IllinoisCommittee for Environmental Information

In this paper, a mathematical programming model is proposed,which can be useful in determining what air pollution controls shouldbe adopted in an airshed. The methodology is based on the premisethat air quality goals should be achieved at the least possible cost.Advantages of the model are its simplicity, its emphasis on economicefficiency, and its appropriateness for the kind of data that is alreadyavailable.To illustrate the model, consider a hypothetical airshed with a sin-

gle industry, cement manufacturing. Annual production is 2,500,000barrels of cement. Although the kilns are equipped with mechanicalcollectors for air pollution control, they are still emitting two poundsof dust for every barrel of cement produced. The industry can be re-quired to replace the mechanical collectors with four-field electro-static precipitators which would reduce emissions to .5 pound of dustper barrel of cement or with five-field electrostatic precipitators whichwould reduce emissions to .2 pound per barrel. If the capital and oper-ating cost of the four-field precipitator are 14^ per barrel of cementproduced and the five-field precipitator 18^ per barrel, what controlmethods should be required of this industry? Assume that, for thishypothetical airshed, it has been determined that particulate emis-sions (which now total 5,000,000 pounds per year) should be reducedby 4,200,000 pounds.The model can be stated as follows:

Minimize C = $.14Zi + tiSXs

Subject to Zi + X2 < 2,500,000

1.5Zi + 1.8Z2 ^ 4,200,000

Xi, X2 ^ 0

where

C==cost of control

* Paper presented at the 1968 Annual Convention of the Central Association of Science and MathematicsTeachers 9:00 A.M., November 30, 1968, Granada Room, Sheraton Jefferson Hotel, Saint Louis, Missouri.

1 This investigation is supported by Public Health Service Fellowship 37, 378-012 from the National Centerfor Air Pollution Control. It is part of a doctoral dissertation written in the Department of Economics, Washing-ton University, Saint Louis, Missouri.

487

488 School Science and Mathematics

X-i== barrels of annual cement production subject to the four-fieldelectrostatic precipitator control method, whose cost is $.14 abarrel of cement produced, and with which the pollutant re-duction is (2.�.5) or 1.5 pounds of particulates per barrel ofcement produced.

Xa= barrels of annual cement production subject to the five-fieldelectrostatic precipitator control method, whose cost is $.18and with which the particulate reduction is (2.�.2) or 1.8pounds per barrel of cement produced.

The first equation states that our objective is to minimize air pol-lution control costs; the second that barrels of cement productionsubject to the two control methods cannot exceed the annual produc-tion; the third that the particulate reduction from the two methodsmust be greater than or equal to the particulate reduction target;while the final expression precludes a solution of the model with nega-tive quantities of cement.

Figure 1 illustrates a graphic solution to the problem. The two in-equalities are graphed on a coordinate system whose axes are thenumbers of barrels of cement production subject to control methods1 and 2. Note that the requirements of these two inequalities are ful-filled in the crosshatched region. The dotted lines illustrate the cost ofcontrol equation for several different costs. To minimize cost of con-trol we want that iso-cost line which is closest to the origin but whichtouches the crosshatched region. By inspection it can be seen that theiso-cost line of smallest value, which touches the required region, goesthrough point A where the two constraint boundaries intersect. Theleast cost solution would be to install the four-field precipitator onkilns producing 1,000,000 barrels (Zi= 1,000,000) and the five-field precipitator on kilns producing 1,500,000 barrels of cement(Z2== 1,500,000).Now consider a more realistic model. There are many pollution

sources and not one but five major pollutants. In the larger model onwhich I am working, the required pollutant reductions in the SaintLouis airshed for the year 1970 are as follows:

sulfur dioxide: 485,000,000 poundscarbon monoxide: 1,300,000,000 poundshydrocarbons: 280,000,000 poundsnitrogen oxides: 75,000,000 poundsparticulate matter: 180,000,000 pounds

The model includes a wide variety of possible control methods.Among them are the installation of exhaust and crankcase deviceson used as well as new automobiles, the substitution of natural gas for

A Model for Air Pollution Control 489

BARRELS OF

CEMENT PRODUCED

USING CONTROL:METHOD 2

FIG. 1

coal; the catalytic oxidation system to convert sulfur dioxide in thestacks of power plants to saleable sulfuric acid, even the municipalcollection of leaves as an alternative to burning.The most contested control method in the Saint Louis airshed has

been a restriction on the sulfur content of coal. Consider a particularcategory of traveling grate stoker which burns 3.1% sulfur coal. Letcontrol method 3 be the substitution of 1.8% sulfur coal for the highsulfur coal in these stokers. The variable, Xs, represents the numberof tons of 3.1% sulfur coal replaced with low sulfur coal.

Total cost of this control method is

C = ($2.50)X3

where

$2.50 is an estimate of the incremental cost of the low sulfur coal.Just as the barrels of cement controlled by any process wereconstrained in our simple example, so

Xz<, 200,000

490 School Science and Mathematics

where

200,000 tons is the estimate of the quantity of coal that will beburned in this category of traveling grate stokers in1970.

For every ton of 3.1% sulfur coal replaced by 1.8% sulful coal, sul-fur dioxide emissions are reduced

[(.031 sulfur content) (2,000 pounds per ton of coal) (.95 complete burning) (2)]- [(.944) (.018 sulfur content) (2,000 pounds) (.95) (2)] = 53.2 pounds

where the factor (2) doubles the weight of sulfur burned to get theweight of sulfur dioxide; where the factor (.944) accounts for thehigher BTU content of the low sulfur coal which permits .944 ton of itto replace one ton of the high sulfur coal; and where (.95) incorporatesan assumption of 95% complete burning. The two square bracketedterms represent emission of sulfur dioxide from 3.1% and 1.8% sulfurcoal respectively. Thus we have

(53.2)Zs = pounds of sulfur dioxide reduced.

The remaining pollutant reductions are

(.2) -X’3== pounds of carbon monoxide reduced,(.1) Xs== pounds of hydrocarbons reduced,

(1.1) Xs== pounds of nitrogen oxide reduced,(12.2) ^3==pou,nds of particulates reduced.

The relatively high reduction in particulates reflects not only thefact that .944 ton of the 1.8% sulfur coal is burned in place of onewhole ton but also the lower ash content of the substituted coal.(Reduction coefficients are not always positive; low sulfur coal in apulverized coal boiler, having a high efficiency electrostatic precipi-tator, can cause an increase in particulate emissions. The presence ofless sulfur dioxide in the flue gas reduces the chargeability of theparticles so that the benefits of the lower ash and higher BTU con-tent may be offset by the reduced efficiency of the electrostatic pre-cipitator.)

Figure 2 illustrates the mathematical programming model for1970. Note that control methods X\ and X^ for the cement industryare included, as well as control method X^ The dots represent theremaining two to three hundred control methods.The pollutant reduction requirements mentioned above appear in

the model on the righthand side. In summing the pollutant reductionscontributed by the various control methods, we are assuming that allpounds of any pollutant are homogeneous, regardless of where or

A Model for Air Pollution Control 491

Minimize C==4 .14 Xi+f .18 ^4-$2.50 Zs+ �������

Subject to Xi+ Xz :< 2,500,000Zs+ �������< 200,000

....... ^.......<

....... ^.......<53.2 Zg+ ������- ^485,000,000 pounds of

sulfur dioxide.2 Zs4- ....... ^l,300,000,000poundsof

carbon monoxide.1 Xs+ ...�.�. ^280,000,000 pounds of

hydrocarbons1.1 Z3+ .......>: 75,000,000 pounds of

nitrogen oxides1.5 Xi+ 1.8 ^2+12.2 Xs-}- .......> 180,000,000 pounds of

particulatesZi, X^ Xs, ....... >0

FIG. 2. The linear programming model.

when they are emitted. This is a limitation of the model, for it isdependent on a close correspondence between a pollutant reductionand a specific concentration measured in parts per million or micro-grams per cubic meter of that pollutant in the ambient air.However, where necessary, meteorological sophistication can be

incorporated in the model by selective weighting of those sourceswhich seem to have a greater or lesser proportional effect on airquality than others.The solution of this model, which is obtained with a computer,

gives the cost of achieving the required air polution reduction. It alsoindicates which of the control methods should be used and which onesare inefficient. This should provide a useful guide for a community inselecting regulations that have sound economic justification.

APPENDIX: The Dual Model

The model has what is called its dual. Our simple problem can berewritten as

Minimize C == $.14Zi + $.18^2

Subject to �Xz � X^ > � 2,500,000

1.5Zi + 1.8Z2 > 4,200,000

^i, X^ 0

The second equation of the original problem has been multipliedthrough by (�1) and the inequality reversed. In matrix notation,the above can be written

492School Science and Mathematics

Minimize C = [.14 .18] 1

LX^Jr-i -nrzr) r-^500^00!Subject to >Ll.5 I^JIAJL 4,200,000j

K> 0L^2

The symmetric dual is a maximization problem, in terms of two newvariables, Wi and ^2.

[Wi~jMaximize C7 = [-2,500,000 4,200,000]W2J

r-i. 1.5-1 i-wii r$.i4-isubiectto L-i. JLJ4..J

p.-iWl h o-W2JLW2J

Note that the righthand side constraints have been interchanged withthe control method prices, the matrix

r-i. -i.-iL 1.5 1.8J

has been transposed, the inequality reversed, and the system ex-pressed in terms of Wi and W2. Figure 3 illustrates the solution of thedual problem. In this case, C’ is maximized when the dotted line isfarthest to the left, while still touching the crosshatched area. Thisoccurs at point A’, where wi=$3/50 and W2==$4/30. Note that thevalue of Cf in the dual problem,

[C1 = (-2,500,000) ($3/50) + (4,200,000) ($4/30) == $410,000.],

equals the value of "C" in the primal problem. Thus wi can be in-terpreted as a ^shadow price55 of the constraint --2,500,000 and w^as a ^shadow price77 of the pollutant reduction requirement, 4,200,000.The ^shadow price77 of the pollutant constraint is the price thatsociety would pay, if there were a market for such a good, for theelimination of one additional pound of particulate matter. This canbe illustrated as follows.

Suppose, in our original problem, the particulate reduction require-ment were 4,200,001 pounds instead of 4,200,000. The optimal solu-tion would still be at the intersection of the two constraint boundarylines, which is the simultaneous solution of

A Model for Air Pollution Control

( Xi+ Za == 1,500,000^ll.SXi + l.SZs == 4,200,OOlJ

which is

Zi == 999,996| barrels of cement,

and

X^ = l,500,003i barrels of cement.

.01.02 .03«04.05 .06 .07 .08 .09 .10

FIG. 3.

For this combination of control methods in the primal problem,

C == $.14(999,9961) + $.18(1,500,0031)

== $.14(999,9961) + ($.14 + $.04) (1,500,0031).

494 School Science and Mathematics

= $.14(999,996§ + 1,500,0031) + $.04(1,500,000 + 3i)= $.14(2,500,000) + $.04(1,500,000) + $.04(3^)

== $350,000. + $60,000. + $.04(10/3)== $410,000. + $4/30.

Thus $4/30 represents the increase in total cost over the originalproblem which has resulted by raising the pollutant reduction re-quirement by one pound.This may prove a valuable ^bonus^ of the mathematical pro-

graming model. The shadow prices thus generated permit us to seejust how much more it would cost to improve air quality with re-spect to any of the pollutants. Scientists, who evaluate the effects ofair pollution on humans, vegetation, and materials, could advisewhether the relative damaging effects of the various pollutants areproportional to their control costs. If they were not, this would indi-cate that resources being spent to attain a specific set of air qualitygoals could be reallocated to purchase a higher level of air quality.

PREVENTING MENTAL ILLNESS

Just as flu or polio vaccine prevents physical illness, so certain measures canprevent mental illness.As mental illness increasingly becomes a problem in our society, prevention be-

comes more and more vital, a University of Wisconsin psychiatrist believes."Preventive psychiatry is most effectively applied at the family level," accord-

ing to Dr. William Bolman, associate professor of psychiatry at the WisconsinMedical School.Family treatment can be approached in one of three ways, Dr. Bolman ex-

plains:First is the community-wide approach which affects all families in a town or

neighborhood.Many such measures are known, but have not been fully exploited; for ex-

ample, assisting community development in needy areas or providing essentialfamily needs such as employment and education. These needs are met in mostmiddleclass communities, but are not being met in lower class ones.Another emerging community-wide approach is genetic counseling. Parents

with a family history in inheritable disease can be advised of their chances ofhaving normal children.A second, "high risk" approach in preventive psychiatry is directed toward

families showing a greater than average likelihood of mental disorders. Generallypoor and undereducated, these families merit particular attention since they pro-duce an estimated one-fifth of America’s children.Head Start, the Job Corps, and other anti-poverty programs are examples of

the high risk approach. They are aimed almost exclusively at one group of fami-lies�those caught in the poverty cycle.The third approach in treating the family unit, the "milestone approach" offers

services to family members at milestones in their lives such as birth, marriage,pregnancy or retirement. School entry is an especially important milestone foridentifying a wide range of correctable problems in children.