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An industrial pollution emission control model
incluo~rlal mnginolring PERGAMON
Computers & Industrial Engineering 37 (1999) 371-374
AN INDUSTRIAL POLLUTION EMISSION CONTROL MODEL
*Aniekan A. Ebiefung 1, **Godwin Udo *Department of Mathematics University of Tennessee, Chattanooga, TN 37403 **Center of Excellence for Information Systems Tennessee State University, Nashville, TN 37203 ABSTRACT A mathematical model for controlling the generation of industrial pollution in a given economic system is presented. For each sector of the economy, the model detem-hnes the appropriate technologies that provide for pollutant levels not exceeding some specified range. Given specific pollution emission information, the model provides an information system for analyzing industrial pollution problems. An algorithm for solving the problem is also provided. © 1999 Elsevier Science Ltd. All rights reserved. KEYWORDS Pollution Control, Input-output, Mathematical Model, Pollutants, Mathematical Programming, Economic Sectors, Choice of Technology, Leontief Models, Pollution Emission. 1. INTRODUCTION The need to have a safe and pollutants free environment is a widely discussed issue in our society today. Manufacturing companies are looking for ways to reduce the amount of pollutants they emit into the atmosphere. For companies to come to grip with the pollutant's emission problem, they must work to eliminate from their production processes those factors that cause high pollution emission. In most industrial set up, a major factor that influences the amount of pollutants emitted into the atmosphere is technology. For instance, two sets of machines for producing an item emit different levels of pollutants into the atmosphere. In this paper, we provide a model which manufacturing companies can use to choose the set of technologies that reduces industrial pollutants. The structure of the paper is as follows. In section 2, the Leontief input-output model is explained. Terminology associated with this model shall be used extensively in the paper. We formulate the model in Section 3 and provide solution technique in Section 4. In Section 5, we summarize our results. 2. LEONTIEF PRODUCTION MODEL The Leontief input-output production model describes the interrelationship among prices, production levels, and demands in a given economic system [7]. For a fixed period of activities, the Leontief input-output production model is described by the equation IThis projectis partially supportedby CECA Scholar'sProgramat U.T. Chattanooga,Tennessee, USA. 0360-8352/99 - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-8352(99)00096-0
372
Proceedings of the 24th International Conference on Computers and Industrial Engineering
yj= ~-~ ajky k +bj
(1)
k=l
where n = number of industries / sectors in the system; yj -- total outputs from industry j; bj = units available/needed at industry j to satisfy demand; ajk= technical coefficient representing units of production of industry j required by industry k. The vectors y= [Yl . . . . .
yn]t and b = [b 1. . . . .
bn ]t are called production vector and demand
vector, respectively. The associated matrix [a,~k is called the coefficient matrix or technology matrix. In the next section, we extend this model to address the pollution control problem. 3. MODEL F O R M U L A T I O N Consider an economy with n industries, each of which must produce an item. Let mj > 1 be the number of different technologies available for the production of output j by industryj. Let xj.. = the amount of pollutants produced by industryj. b I = the amount of pollutants required to satisfy external emission limit for industryj aJik= units of output of pollutants by technology i in industry j for industry k. The condition that the amount of pollutants produced by industry j meets pollutants limits for both internal and external production demands, is equivalent to j
n
x.=b
.
J xk
J
k = 1 ik
,.]=l,...,n,l
(2)
Observe that the sum £ a~xk represents the total amount of pollutants produced due to internal k=l
or inter=sector demands. 4. SOLVING THE PROBLEM
To solve equation (2), we need to combine technologies from different sectors of the economy. If there are n sectors each with two technologies to choose from, then there are 2 n equations to solve, which is exponential. The algorithm given below reduces the number of equations solved in order to obtain the optimal set of technologies. First, we formulate the problem using matrix notation. Let m.
--I
m., n
1,J =1
Proceedings of the 24th International Conference on Computers and Industrial Engineering
373
where dJ is an mjxl column vector, and bJi : b j, i : 1. . . . , mj. Moreover, define
re,
-1.o
=Io
~t where e, is a mjxl column vector with each entry one. Set m= ~ m j .
o...
ol
..... Then E and A are mxn
j=l vertical block matrices. The jth block in A, Aj, corresponds to the pollution technology matrix of the jth industry. Observe that the partition of d conforms to the blocks in A. In matrix notation, equation (2) is equivalent to Nx = d, x >_.0, where N = E-A, and E, A, and d are as defined above. Below is an algorithm for solving problem (2). ALGORITHM Step 1: For each j, solve g,(x) =
min(Nrx- d r ) ,
subject to x > O, N x - d = O .
l~il
Step 2: For each j, let 1 < ir < mr , be the index at which min{gt(x);i = 1,...,m r occurred. Define a matrix M and a vector q e R" by M r. = N!',' qr = q ' ,J' where Mr. is the jth row of M. The equation Mx = q,
x >0
(4)
solves equation (2). T H E O R E M . A solution of equation (4), if one exists, solves equation (2)• PROOF. Let x be a solution of Mx = q, x > 0. Then by the definition of M, we have for each j, 0 = (Mr.i = q r ) < ( N r y c - d s ) = 0 , x > 0 , i = 1 , . . . , mj. m
Thus/v'2 = d. That is, x is a solution o f N x =d, x >_O. This result can be interpreted as follows. The technology corresponding to the matrix vector Mr. is the one chosen for industry j. The pollution generated by this technology meets exactly pollution limit for the sector, which is divided into pollution limits for internal and external demands. The above Algorithm can be easily implemented on a computer using existing linear and nonlinear problem solvers such as Lindo and Gino.
374
Proceedingsof the 24th International Conference on Computersand Industrial Engineering
5. CONCLUSION A mathematical model for controlling the generation of industrial pollution is presented. The model is based on the Leontief input-output production and pollution coefficients, and mathematical programming. Information used for the model construction is available for the US and other advanced economies. Nevertheless, many industrial nations have undertaken similar studies and the concept of this paper applies to their economies as well. 6. REFERENCES
1. Baker, T. S. (1972). Foreign Trade in Multisectional Models, Input-Output Techniques, Edited by Brody, A. and Carter, A. P., North-Holland Publishing Company. 2. Dantzig, G. B. (1955). Optimal Solution of Leontief Model with Substitution, Econometrica, vol. 23, pp. 295-302. 3. Ebiefung, A. A., and Kostreva, M (1993). The Generalized Leontief Input-Output Model and its Application to the Choice of New Technology, Annals of Operations Research 44, 161172. 4. Lemke, C. E.(1965). Bimatrix Equilibrium Points and Mathematical Programming, Management Science, vol. 11, pp. 681-689. 5. Leontief, W. W. (1985). The Choice of Technology, Scientific American, pp. 37-45. 6. Leontief, W. W. (1949).The Structure of the American Economy, 1919-1935, Oxford University Press, London & New York. 7. Leontief, W. W. and Daniel, F. (1972). "Air Pollution and the Economic Structure: Empirical Results of Input-Output Computations." Input-Output Techniques. Brody, A. and Carter, A. P. editors. North-Holland Publishing Company. 8. Saigal, R. (1970). On a Generalization of Leontief Systems, University of California, Berkeley. 9. Veinott, A. F. (1968). Extreme points of LeontiefSubstitution Systems, Linear Algebra and its Applications, vol. 1, pp. 181-194. 10.Wu, R., and Chen, C.(1990). On the Application of Input-Output to Energy Issues, Energy Economics, pp. 71-76.
Computers & Industrial Engineering 37 (1999) 371-374
AN INDUSTRIAL POLLUTION EMISSION CONTROL MODEL
*Aniekan A. Ebiefung 1, **Godwin Udo *Department of Mathematics University of Tennessee, Chattanooga, TN 37403 **Center of Excellence for Information Systems Tennessee State University, Nashville, TN 37203 ABSTRACT A mathematical model for controlling the generation of industrial pollution in a given economic system is presented. For each sector of the economy, the model detem-hnes the appropriate technologies that provide for pollutant levels not exceeding some specified range. Given specific pollution emission information, the model provides an information system for analyzing industrial pollution problems. An algorithm for solving the problem is also provided. © 1999 Elsevier Science Ltd. All rights reserved. KEYWORDS Pollution Control, Input-output, Mathematical Model, Pollutants, Mathematical Programming, Economic Sectors, Choice of Technology, Leontief Models, Pollution Emission. 1. INTRODUCTION The need to have a safe and pollutants free environment is a widely discussed issue in our society today. Manufacturing companies are looking for ways to reduce the amount of pollutants they emit into the atmosphere. For companies to come to grip with the pollutant's emission problem, they must work to eliminate from their production processes those factors that cause high pollution emission. In most industrial set up, a major factor that influences the amount of pollutants emitted into the atmosphere is technology. For instance, two sets of machines for producing an item emit different levels of pollutants into the atmosphere. In this paper, we provide a model which manufacturing companies can use to choose the set of technologies that reduces industrial pollutants. The structure of the paper is as follows. In section 2, the Leontief input-output model is explained. Terminology associated with this model shall be used extensively in the paper. We formulate the model in Section 3 and provide solution technique in Section 4. In Section 5, we summarize our results. 2. LEONTIEF PRODUCTION MODEL The Leontief input-output production model describes the interrelationship among prices, production levels, and demands in a given economic system [7]. For a fixed period of activities, the Leontief input-output production model is described by the equation IThis projectis partially supportedby CECA Scholar'sProgramat U.T. Chattanooga,Tennessee, USA. 0360-8352/99 - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-8352(99)00096-0
372
Proceedings of the 24th International Conference on Computers and Industrial Engineering
yj= ~-~ ajky k +bj
(1)
k=l
where n = number of industries / sectors in the system; yj -- total outputs from industry j; bj = units available/needed at industry j to satisfy demand; ajk= technical coefficient representing units of production of industry j required by industry k. The vectors y= [Yl . . . . .
yn]t and b = [b 1. . . . .
bn ]t are called production vector and demand
vector, respectively. The associated matrix [a,~k is called the coefficient matrix or technology matrix. In the next section, we extend this model to address the pollution control problem. 3. MODEL F O R M U L A T I O N Consider an economy with n industries, each of which must produce an item. Let mj > 1 be the number of different technologies available for the production of output j by industryj. Let xj.. = the amount of pollutants produced by industryj. b I = the amount of pollutants required to satisfy external emission limit for industryj aJik= units of output of pollutants by technology i in industry j for industry k. The condition that the amount of pollutants produced by industry j meets pollutants limits for both internal and external production demands, is equivalent to j
n
x.=b
.
J xk
J
k = 1 ik
,.]=l,...,n,l
(2)
Observe that the sum £ a~xk represents the total amount of pollutants produced due to internal k=l
or inter=sector demands. 4. SOLVING THE PROBLEM
To solve equation (2), we need to combine technologies from different sectors of the economy. If there are n sectors each with two technologies to choose from, then there are 2 n equations to solve, which is exponential. The algorithm given below reduces the number of equations solved in order to obtain the optimal set of technologies. First, we formulate the problem using matrix notation. Let m.
--I
m., n
1,J =1
Proceedings of the 24th International Conference on Computers and Industrial Engineering
373
where dJ is an mjxl column vector, and bJi : b j, i : 1. . . . , mj. Moreover, define
re,
-1.o
=Io
~t where e, is a mjxl column vector with each entry one. Set m= ~ m j .
o...
ol
..... Then E and A are mxn
j=l vertical block matrices. The jth block in A, Aj, corresponds to the pollution technology matrix of the jth industry. Observe that the partition of d conforms to the blocks in A. In matrix notation, equation (2) is equivalent to Nx = d, x >_.0, where N = E-A, and E, A, and d are as defined above. Below is an algorithm for solving problem (2). ALGORITHM Step 1: For each j, solve g,(x) =
min(Nrx- d r ) ,
subject to x > O, N x - d = O .
l~il
Step 2: For each j, let 1 < ir < mr , be the index at which min{gt(x);i = 1,...,m r occurred. Define a matrix M and a vector q e R" by M r. = N!',' qr = q ' ,J' where Mr. is the jth row of M. The equation Mx = q,
x >0
(4)
solves equation (2). T H E O R E M . A solution of equation (4), if one exists, solves equation (2)• PROOF. Let x be a solution of Mx = q, x > 0. Then by the definition of M, we have for each j, 0 = (Mr.i = q r ) < ( N r y c - d s ) = 0 , x > 0 , i = 1 , . . . , mj. m
Thus/v'2 = d. That is, x is a solution o f N x =d, x >_O. This result can be interpreted as follows. The technology corresponding to the matrix vector Mr. is the one chosen for industry j. The pollution generated by this technology meets exactly pollution limit for the sector, which is divided into pollution limits for internal and external demands. The above Algorithm can be easily implemented on a computer using existing linear and nonlinear problem solvers such as Lindo and Gino.
374
Proceedingsof the 24th International Conference on Computersand Industrial Engineering
5. CONCLUSION A mathematical model for controlling the generation of industrial pollution is presented. The model is based on the Leontief input-output production and pollution coefficients, and mathematical programming. Information used for the model construction is available for the US and other advanced economies. Nevertheless, many industrial nations have undertaken similar studies and the concept of this paper applies to their economies as well. 6. REFERENCES
1. Baker, T. S. (1972). Foreign Trade in Multisectional Models, Input-Output Techniques, Edited by Brody, A. and Carter, A. P., North-Holland Publishing Company. 2. Dantzig, G. B. (1955). Optimal Solution of Leontief Model with Substitution, Econometrica, vol. 23, pp. 295-302. 3. Ebiefung, A. A., and Kostreva, M (1993). The Generalized Leontief Input-Output Model and its Application to the Choice of New Technology, Annals of Operations Research 44, 161172. 4. Lemke, C. E.(1965). Bimatrix Equilibrium Points and Mathematical Programming, Management Science, vol. 11, pp. 681-689. 5. Leontief, W. W. (1985). The Choice of Technology, Scientific American, pp. 37-45. 6. Leontief, W. W. (1949).The Structure of the American Economy, 1919-1935, Oxford University Press, London & New York. 7. Leontief, W. W. and Daniel, F. (1972). "Air Pollution and the Economic Structure: Empirical Results of Input-Output Computations." Input-Output Techniques. Brody, A. and Carter, A. P. editors. North-Holland Publishing Company. 8. Saigal, R. (1970). On a Generalization of Leontief Systems, University of California, Berkeley. 9. Veinott, A. F. (1968). Extreme points of LeontiefSubstitution Systems, Linear Algebra and its Applications, vol. 1, pp. 181-194. 10.Wu, R., and Chen, C.(1990). On the Application of Input-Output to Energy Issues, Energy Economics, pp. 71-76.