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A Goal Programming Applied to Regional Water Quality Management
A GOAL PROGRAMMING APPLIED TO REGIONAL WATER QUALITY MANAGEMENT B. N. LohanP and Pakorn Adulbhan 2 Asian Institute of Technology, Bangkok, Thailand
ABSTRACT Out of all the optimization techniques used in regional water quality management studies, linear programming is certainly the simplest. In the case of linear programming, however, only one goal can be optimized. When there are multiple goals with the same or different priorities; goal programming is a useful decision making tool . The paper illustrates the application of goal programming to regional water quality management problem where the following two goals are considered: (1) minimize the total costs, (2) maintain the water quality close to the minimum desired level.
The commonly used optimization techniques include linear programming, dynamic programming, and integer programming. Out of all these techniques used in regional water quality management problems, linear programming is the widely used one. In the case of linear programming; however, it is possible to optimize only one goal and for a regional water quality management problem, the goal has usually been to minimize the total cost of treatment for maintaining dissolved oxygen level greater than or equal to the minimum desired level. Linear programming minimizes the treatment cost but cannot at the same time minimize the deviation of the water quality constraints between the goal and what can be achieved. Maintaining the level of water quality higher than the minimum desired level means higher degrees of removal and thus higher cost of treatment. Goal programming, on the other hand, can optimize multiple objectives and can do so even when priorities of the goals are made. It is obvious that not all goals can be met to the extent desired but priorities can be attached to the achievement of such goals. The intention of this paper is to illustrate the application of goal programming to regional water quality management problems which were previously solved by linear programming.
INTRODUCTION The management of a river basin has received wide attention but a few quantitative studies have been made. The management approach used for maintaining water quality in a stream has been mainly by the use of standards - effluent or stream standards. Models have also been developed to know the quality of the effluent which when discharged will meet a predesired standard in the stream. One of the main objectives of regional water quality authorities is to maintain minimum dissolved oxygen limit in the stream. To achieve a desired dissolved oxygen level, it is necessary to specify the amount of waste removal at each of the polluting sources. There could be a number of combinations giving desired solution but the important consideration is the cost. It is usually desired to find optimal removal policy that will yield a minimum cost solution. The minimum cost objective has led to the need for the development of quantitative decision models. Some of the earlier works worth referring are that of DAY et. al. (1), DEININGER (2), KERRI (3), LIEBMAN and LYNN (5), SOBEL (8), REVELLE et. al. (7) and since then many more models were developed.
PROBLEM STATEMENT A stream on which there are n waste discharges is considered. The stream is then divided into n reaches, where the jth (j=l, 2, ••• , n-l) reach being defined as the stretch of the stream between jth and (j+l) st discharge. The last reach (j=n) is the remainder of the stream from the nth discharge to the mouth of the stream. The quantity and quality (BaD) of the discharges (biochemical oxygen demand - BaD) at each point are assumed to be known. The target
lDoctoral Candidate in Environmental Engineering Division, Asian Institute of Technology, Bangkok 2Chairman of Industrial Engineering and Management Division, Asian Institute of Technology, Bangkok 181
182
B. N. Lohani and P. Adulbhan
considered in the operating policy is the dissolved oxygen concentration at the reaches. The dissolved oxygen is the minimum acceptable stated in a stream standard.
Where a .. is the coefficient of x. and b . 1 is the fight hand side value of tfte constraint. The second set of constraints could be the maximum degree of removal (in percentage) possible at each treatment facility, T . . That is J
Optimal Effluent Water Quality Model
X. In order to enforce the stream quality standards, the amount of dissolved oxygen (DO) at specified locations is usually employed as a key indicator. Organic loading is the major sink for the DO in streams. Therefore, the maximum allowable (BOD) to maintain the desired oxygen level in any reach of a stream must be determined. The DO level downstream from the waste discharge point can be described by Streeter Phelps' an model:
Where
L
DO deficit at the point after a time t from the waste discharge location, mg/l Coefficient of deoxygenation, l/days Coefficient of reaeration, l/days Initial BOD Level, mg/l
D
Initial DO deficit, mg/l
t
time of flow, days
D t Kl K2 0
0
<
J
T . (j=l, 2, .•. , n) J
(5)
Goal Programming Model Goal programming may be considered as a special extension of linear programming which is capable of handling a decision problem with a single goal and multiple subgoals, as well as a problem with multiple goals and multiple subgoals. In goal programming, instead of trying to maximize or minimize the objective criteria directly as in linear programming, the deviations between goals and what can be achieved (within the given set of constraints) are minimized. The deviation may be allowed in two directions, both positive, 6+ (overachievement), and negative, 6- (underachievement). The objective function consists of only these deviational variables. A priority or relative importance may also be attached to the achievement of a goal and will appear in the objective function. The general goal programming model can be expressed as Minimize
The critical time, tc' is given by K2 K2-K l (K -K ) Ln[~(l-~Do)] 2 1 110
Subject to
1
t
c
(2)
From equations (1) and (2), the critical time t and initial BOD, can be determined for a dgsired DO level by iterative method (Ref.6). Linear Programming Model The objective in this model is to minimize the total cost of treatment. The cost is assumed to be a linear function of percentage BOD removal. That is the objective function is of the following form: Minimize
n L
cJ.X . J j=l
(3)
Where C. and X. are respectively the cost per unit BOD r~moval (%) and percentage BOD removal at ith waste treatment facility. The water quality constraints for all the reaches can also be expressed in the following form (Ref. 2): n L
j=l
a . Xj iJ
>
b
i
(i=l, 2, ••• , n) (4)
AX+I 6- -I6+ X, 6-, 6+
>
b 0
(6)
Where n goals are expressed by a n component column vector (b , b , ... , b ), A is l 2 a n x m matrix which expresses thenrelationship between goals and subgoals, X represents variables involved in the subgoals (X , X ' ... , X ), 6- and 0+ are n 2 component vectors formthe variable representing deviations from goals, and I is an identity matrix of size n. The relative priorities should be ranked for the deviational variables, and in the objective function.
0t
°1,
The goal programming model for the regional water quality management problem can now be developed. Two goals with two priorities are hereby considered. The goals are: (i) maintain the total cost of treatment at a specified level; (ii) maintain the water quality constraints in equation (4) close to but not less than the minimum preset quality limit. If the above two goals were ranked with priorities one (PI) and two ( P ) respectively, then the goal programming2 model can be written as follows: Minimize
A goal programming Subject to n L
j=l b.
1.
a .. x. 1.J
J
ter time required for running goal programmings for Case 1, Case 2 and Case 3 were 23.73 seconds, 26.60 seconds and 22.51 seconds respectively.
0:
-
1.
(i=l, 2,
. .. ,
n) (7)
+ °n+j
x. J
T. (j=l, 2, J
. .. ,
n)
n L
j=l
c. x. + 0 J
J
x , o~, 0: j J J
183
+
(2n+l) >
o
(2n+l)
0
0
APPLICATION PROBLEM
Table 3 shows the linear programming and goal programming solutions for pollution .limit of 500 Population Equivalent (PE) or 20 mg/l of BOD. The goal programming solution for Case 2 is the same as Deininger's solution. Change in the priorities (Case 1) changes the optimal removals. As shown in Table 3, the total Cost Units for Case 1 is 1,237.50 compared to 1,231.67 for Case 2. It may be observed that the removal at Plant 3 is 46.2% for Case 1 but for Case 2, there is no removal required at Plant 3. When the water quality and cost were analysed with equal priority (Case 3), the solution is the same as for Case 1. This can be seen from Table 3.
The application of goal programming model is shown to the hypothetical problem solved by DEININGER (Ref. 2). The hypothetical river basin involves a river 110 miles long with 12 cities varying in size from 1,000 to 150,000 people. For convenience, these cities were numbered from 1 to 12, starting with the city farthest upstream. The data used for the study are shown in Table 1. Table 2 shows the linear programming problem formulated by Deininger. Deininger found three sets of solutions for each pollution limit in the following three ways:
The computer programme also gives the output for slack analysis, variable analysis and analysis of the objective. The slack analysis, in Table 4 for Case 1 and Case 2, presents the values of the right hand side (available) and also values of the negative and positive variables for each equation. The slack analysis provides informations to analyze the details of goal attainments when the problem is complex. This analysis often proves to be useful in identifying errors when the model does not represent the desired decision environment.
Solution 1: To require from each polluter the same degree of treatment in terms of per cent removal of pollutant. Solution 2: To allow each polluter to discharges as much as he can without violating the water quality criteria. Solution 3: To devise a system of waste treatment which will maintain the desired water quality and minimize the total costs of waste treatment.
The variable analysis presents the constants of only the basic variables. When the problem under consideration is a very complex one, the variable analysis is especially helpful because it presents only the constants of the basic choice variables as compared to the final simplex solution table. It may be pointed out that the variable number are variable numbers of the choice variable (i.e. Xl' x 2 , •.• , x 12 ).
The goal programming solutions were obtained for the following three cases and compared with the Solution 3 of Deininger: Case 1: Considering minimizing the total cost of treatment as the second priority goal and water quality constraints as the first priority goal. Case 2: Considering water quality constraints as the second priority goal and minimizing the total cost of treatment as the first priority goal. Case 3: Considering both the above goals with equal priority. DISCUSSION OF RESULTS The computer program written for goal programming was run in IBM 370/75 at Regional Computer Centre, Asian Institute of Technology. The computer program may be obtained from Industrial Engineering and Management Division, AIT or from the authors. The compu-
An analysis of the objective presents the value of each goal. These values represent the underattained portion of goals. If the model requires assignment of artifical priority to set up the initial table, the artificial priority will also be printed out. CONCLUSIONS In case of river basin management where there are multiple goals sub goals of different priorities, goal programming appears to be a useful tool because it can handle a decision problem with a single goal and multiple subgoals as well as a problem with mUltiple goals and multiple subgoals. The present paper shows only one of the many applications possible of the approach presented. It may be stated here goal programming approach is not an ultimate solution for all decision problems. The techniques simply gives the best optimal solution for the given constraints and priority goals.
t-'
00
Table 1:
Data for the Application Problem (Pollution Limit is 500 PE/CFS or 20 MG/L BOD) (Ref. 2)
I CITY I CITY I CITY I CITY I 5 I 6 I 7 I 8 I ____________________ I _______ I _______ I _______ I _______ I _______ I _______ 1 _______ 1 ______ 1 ___________________ 1 _______ 1 _______ 1 _______ 1 _______ 1 _______ 1 _______ 1 _______ 1 ______ I I
I I I I POPULATION I -------------------- I I I I WASTEFLOW IN CFS I 1--------------------1 I I I DEOX - COEFF I I -------------------- I I I I RIVER MILEAGE I I -------------------- I I I I TREATMENT COSTS I I ------------------- I I I I TREATMENT LIMIT I I ------- ------------ I I I I I STREAMFLOW IN CFS I ------------------- I I I I REOX - COEFF I I ------------------- I I -------------------- I I I I POLLUTION LIMITS IN I I I I I I POPEQUIVAL/CFS I I I I I B 0 D IN MG/L I -------------------- I I -------------------- I I -------------------- I
CITY I 1
""
I
I 25000 I ------- I I 3.86 I -------1 I .23 I ------- I I 105.00 I ------- I I 200.00 I ------- I I .90 I ------- I I 38.86 I ------_ I I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I ------- I ------- I ------- I
CITY I CITY 2
I
I 1000 I ------- I I .15 I -------1 I .25 I ------- I I 98.00 I ------- I I 18.00 I ------- I I .90 I ------- I I 39.01 I ------- I I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I ------- I ------- I ------- I
3
I CITY
I
I 5000 I ------- I I .77 I -------1 I .20 I ------- I I 86.00 I _______ I I 60.00 I ------- I I .90 I ------- I I 39.78 I ------- I I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I -------1 -------1 -------1
4
I 2000 I _______ I I .31 I -------1 I .20 I _______ I I 76.00 I - ______ I I 30.00 I ------- I I .90 I ------- I I 40.09 I ------- I I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I ------- I ------- I ------- I
I I 1000 I 100000 I _______ I _______ I I I .15 I 15.47 I -------1 -------1 I I .25 I .30 I ------- I - ______ I I I 69.00 I 57.00 I _______ I _______ I I I 18.00 I 600.00 I ------- 1-------- I I I .90 I .90 I ------- I ------- I I I 40.24 I 85.71 I ------- I -------1 I I .55 I .55 I ------- I ------- I ------- I ------- I I I I I I I I I 500.00 I 500.00 I I I 20.00 I 20.00 I ------- I ------- I ------- I ------- I ------- I ------- I
I 2000 I _______ I I .31 I -------1 I .27 I _______ I I 47.00 I _______ I I 30.00 I ------- I I .90 I ------- I I 86.02 I -------1 I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I ------- I ------- I ------- I
I CITY I I CITY I CITY I CITY I 12 I I 11 I 9 I 10 I _______ 1 _______ 1 _______ 1 _______ 1 I _______ 1 _______ 1 _______ 1 _______ 1
I I 1000 I 1000 I ______ I _______ I I I .15 I .15 I I -------1 I I .22 I .20 I _______ I _______ I I I 40.00 I 35.00 I _______ I _______ I I I 18.00 I 18.00 I ------- I ------- I I I .90 I .90 I ------- I ------- I I I 86.17 I 86.32 I -------1 -------1 I I .55 I .55 I ------- I ------- I ------- I ----___ I I I I I I I I I 500.00 I 500.00 I I I 20.00 I 20.00 I ------- I ------- I ------- I ------- I ------- I ------- I
I 10000 I _______ I I 1.55 I -------1 I .22 I _______ I I 27.00 I _______ I I 100.00 I ------- I I .90 I ------- I I 87.87 I -------1 I .55 I ------- I _______ I I I I I 500.00 I I 20.00 I ------- I _______ I ------- I
I 150000 I _______ I I 23.20 I -------1 I .27 I _______ I I 16.00 I _______ I I 825.00 I ------- I I .90 I ------- I I 121.07 I -------1 I .55 I _______ I _______ I I I I I 500.00 I I 20.00 I ------- I _______ I ------- I
I 7000 I _______ 1 I 1.08 I _______1 I .30 I _______ I I .00 I _______ I I 77.00 I - ______ 1 I .90 I ------- I I 122.15 I _______ 1 I .55 I _______ I _______ I I I 0 I 500.00 I I 20.00 I ------- I _______ I _______ I
I;>j
z t""
o ::r III
::l ..... III
::l
0.
'
> 0.
C ..... eT
::r
III
::l
Table 2:
Linear Programming Formulation (Pollution Limit is 500 PElcFS or 20 MG/L BOD) (Ref. 2)
THE BUILT TREATMENT PLANTS MINIMIZE THE TOTAL COSTS THE PROBLEM IS, MATHEMATICALLY SPEAKING MINIMIZE 200.000
18.000
60.000
30.000
18.000
600.000
30.000
18.000
18.000
100.000
825.000
77 .000
SUBJECT TO 643.335
.000
.000
. 000
.000
.000
.000
.000
.000
.000
.000
.OOO.G.E •
143.335
600.891
25.634
.000
.000
.000
• 000
.000
.000
.000
.000
.000
.OOO.GE •
126.525
527.668
.000
.000
.000
• 000
49.888
.000
.000
.000
.000
.000
46.995
24.851
.000
.000
.000
.000
1,166.725
.000
.000
. 000
.000
.000
22.296
125.691
.000
477 .567
20.018
115.130
446.112
18.595
108.455
187.553
7.743
46.258
20.044
10.348
.000
.000
.OOO.GE •
175.655
.000
.000
.OOO.GE.
162.603
.000
.000
.OOO.GE.
145.007
.000
.000
.OOO.GE •
938.670
.000
.000
.OOO.GE.
802.059
.000
.000
.OOO.GE.
711. 752
.000
170.452
6.981
42.547
18.436
9.329
1,031.063
23.250
159.542
6.498
40.160
17.402
8.684
946.341
21.520
11.605
.000
152.105
6.170
38.518
16.691
8.246
889.682
20.353
11.086
11.585
.000
.000
.OOO.GE.
654.436
138.819
5.595
35.493
15.380
7.477
793.987
18.339
10.150
10.675
113.804
.000
.OOO.GE.
649.720
91.055
3.638
23.590
10.222
4.862
504.999
11. 819
6.687
7.095
74.976
1,238.953
.OOO.GE.
1,477.896
77.897
3.073
20.572
8.914
4.106
413.094
9.856
5.758
6.187
64.553
1,033.122
57.307.GE.
1,204.439
INEQUALITY
2 IS REDUNDANT
INEQUALITY
5 IS REDUNDANT
INEQUALITY
7 IS REDUNDANT
INEQUALITY
8 IS REDUNDANT
INEQUALITY
9 IS REDUNDANT
:> 00
0
DJ ,.....
'1:1 '1 0
00
'1
DJ
§ ..... ::l
00
INEQUALITY 12 IS REDUNDANT I-'
00 VI
186
B. N. Loha ni and P. Adulbhan
Table 3:
Linear Programming (LP) and Goal Programming (GP) Solutions
Percentage Plant
GP
LP Solution by Deininger
1 2 3 4 5 6 7 8 9 10
34.0 0.0 0.0 0.0 0.0 75.9 0.0 0.0 0.0 0.0 85.9 0.0
11
12 Total Costs
1,231.67 Cost Units
11
12 13 14 15 16 17 18 19
Solutions Case 2
Case 3
22.3 0.0 46.2 6.0 0.0 74.9 0.0 0.0 0.0 5.7 85.8 0.0
34.0 0.0 0.0 0.0 0.0 75.9 0.0 0.0 0.0 0.0 85.9 0.0
22.3 0.0 46.2 6.0 0.0 74.9 0.0 0.0 0.0 5.7 85.8 0.0
1,231.67 Cost Units
1,237.50 Cost Units
Slack Analysis
Ca se
1 2 3 4 5 6 7 8 9 10
Removal
Case 1
1,237.50 Cost Units
Table 4:
Row
BOD
1
Case
2
Available 143.33501 175.65500 162.60300 938.66992 649.71997 1477. 89600 0.00001 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000
POS-SLK
NEG-SLK
POS-SLK
NEG-SLK
0.0 0.0 0.0 0.0 0.0 0.0 1237.49951 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.67720 0.90000 0.43783 0.84003 0.90000 0.15064 0.90000 0.90000 0.90000 0. 84306 0.04170 0.90000
75.70966 4.00658 0.0 10.46700 0.0 1. 66919 1231.66919 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.55952 0.90000 0.90000 0.90000 0.90000 0.14123 0.90000 0.90000 0.90000 0.90000 0.04144 0.90000
A goal programming
Table 5:
Case
2 Artificial
Case
2
Variable
Amount
Variable
Amount
3
0.46217
6
0.75877
Case
1
Variable Analysis
1
4
0.05997
11
0.85856
6
0.74936
1
0.34048
10
0.05694
-
-
11
0.85830
1
0.22280
-
-
Table 6:
Priority
187
Analysis of the Objective
Case
1 Under-achievement
Priority
2 Under-achievement
2
90.18323
0.0
1
1231.66919
0.0
Artificial
0.0
1237.49951
REFERENCES 1. R.J. DAY, F.T. DOLBEAR, JR. and M. KAMIEN, Regional Water Quality Management - A Pilot Study, Proc. 1st Annual Meeting, AWRA, University of Chicago, 1965. 2. R.A. DEININGER, Water Quality Management The Planning of Economically Optimal Pollution Control System, Proc. 1st Annual Meeting, AWRA, University of Chicago, 1965. 3. K.D. KERRI, An Economic Approach to Water Quality Control, J. WPCF, Vol. 38, 1883 (1966). 4. S.M. LEE, Goal Programming for Decision Analysis of Multiple Objectives, Sloan Management Review (Winter), 11 (1972-73).
5. J.C. LIEBMAN and W.R. LYNN, The Optimal Allocation of Stream Dissolved Oxygen, Water Resources Research, Vol. 2, 1966. ----6. B.N. LORAN I , A Chance - Constrained Approach to Regional Water Quality Management, Doctoral Dissertation (in progress), Environmental Engineering Division, Asian Institute of Technology, Bangkok, 1976. 7. C. REVELLE, D.P. LOUCKS, W.R. LYNN, Linear Programming Applied to Water Quality Management, Water Resources Research, Vo!. 4, No. 1, 1, 1968. 8. M. SOBEL, Water Quality Improvement Programming Problems, Water Resources Research, Vol. I, No. 4, 477 (168).
ABSTRACT Out of all the optimization techniques used in regional water quality management studies, linear programming is certainly the simplest. In the case of linear programming, however, only one goal can be optimized. When there are multiple goals with the same or different priorities; goal programming is a useful decision making tool . The paper illustrates the application of goal programming to regional water quality management problem where the following two goals are considered: (1) minimize the total costs, (2) maintain the water quality close to the minimum desired level.
The commonly used optimization techniques include linear programming, dynamic programming, and integer programming. Out of all these techniques used in regional water quality management problems, linear programming is the widely used one. In the case of linear programming; however, it is possible to optimize only one goal and for a regional water quality management problem, the goal has usually been to minimize the total cost of treatment for maintaining dissolved oxygen level greater than or equal to the minimum desired level. Linear programming minimizes the treatment cost but cannot at the same time minimize the deviation of the water quality constraints between the goal and what can be achieved. Maintaining the level of water quality higher than the minimum desired level means higher degrees of removal and thus higher cost of treatment. Goal programming, on the other hand, can optimize multiple objectives and can do so even when priorities of the goals are made. It is obvious that not all goals can be met to the extent desired but priorities can be attached to the achievement of such goals. The intention of this paper is to illustrate the application of goal programming to regional water quality management problems which were previously solved by linear programming.
INTRODUCTION The management of a river basin has received wide attention but a few quantitative studies have been made. The management approach used for maintaining water quality in a stream has been mainly by the use of standards - effluent or stream standards. Models have also been developed to know the quality of the effluent which when discharged will meet a predesired standard in the stream. One of the main objectives of regional water quality authorities is to maintain minimum dissolved oxygen limit in the stream. To achieve a desired dissolved oxygen level, it is necessary to specify the amount of waste removal at each of the polluting sources. There could be a number of combinations giving desired solution but the important consideration is the cost. It is usually desired to find optimal removal policy that will yield a minimum cost solution. The minimum cost objective has led to the need for the development of quantitative decision models. Some of the earlier works worth referring are that of DAY et. al. (1), DEININGER (2), KERRI (3), LIEBMAN and LYNN (5), SOBEL (8), REVELLE et. al. (7) and since then many more models were developed.
PROBLEM STATEMENT A stream on which there are n waste discharges is considered. The stream is then divided into n reaches, where the jth (j=l, 2, ••• , n-l) reach being defined as the stretch of the stream between jth and (j+l) st discharge. The last reach (j=n) is the remainder of the stream from the nth discharge to the mouth of the stream. The quantity and quality (BaD) of the discharges (biochemical oxygen demand - BaD) at each point are assumed to be known. The target
lDoctoral Candidate in Environmental Engineering Division, Asian Institute of Technology, Bangkok 2Chairman of Industrial Engineering and Management Division, Asian Institute of Technology, Bangkok 181
182
B. N. Lohani and P. Adulbhan
considered in the operating policy is the dissolved oxygen concentration at the reaches. The dissolved oxygen is the minimum acceptable stated in a stream standard.
Where a .. is the coefficient of x. and b . 1 is the fight hand side value of tfte constraint. The second set of constraints could be the maximum degree of removal (in percentage) possible at each treatment facility, T . . That is J
Optimal Effluent Water Quality Model
X. In order to enforce the stream quality standards, the amount of dissolved oxygen (DO) at specified locations is usually employed as a key indicator. Organic loading is the major sink for the DO in streams. Therefore, the maximum allowable (BOD) to maintain the desired oxygen level in any reach of a stream must be determined. The DO level downstream from the waste discharge point can be described by Streeter Phelps' an model:
Where
L
DO deficit at the point after a time t from the waste discharge location, mg/l Coefficient of deoxygenation, l/days Coefficient of reaeration, l/days Initial BOD Level, mg/l
D
Initial DO deficit, mg/l
t
time of flow, days
D t Kl K2 0
0
<
J
T . (j=l, 2, .•. , n) J
(5)
Goal Programming Model Goal programming may be considered as a special extension of linear programming which is capable of handling a decision problem with a single goal and multiple subgoals, as well as a problem with multiple goals and multiple subgoals. In goal programming, instead of trying to maximize or minimize the objective criteria directly as in linear programming, the deviations between goals and what can be achieved (within the given set of constraints) are minimized. The deviation may be allowed in two directions, both positive, 6+ (overachievement), and negative, 6- (underachievement). The objective function consists of only these deviational variables. A priority or relative importance may also be attached to the achievement of a goal and will appear in the objective function. The general goal programming model can be expressed as Minimize
The critical time, tc' is given by K2 K2-K l (K -K ) Ln[~(l-~Do)] 2 1 110
Subject to
1
t
c
(2)
From equations (1) and (2), the critical time t and initial BOD, can be determined for a dgsired DO level by iterative method (Ref.6). Linear Programming Model The objective in this model is to minimize the total cost of treatment. The cost is assumed to be a linear function of percentage BOD removal. That is the objective function is of the following form: Minimize
n L
cJ.X . J j=l
(3)
Where C. and X. are respectively the cost per unit BOD r~moval (%) and percentage BOD removal at ith waste treatment facility. The water quality constraints for all the reaches can also be expressed in the following form (Ref. 2): n L
j=l
a . Xj iJ
>
b
i
(i=l, 2, ••• , n) (4)
AX+I 6- -I6+ X, 6-, 6+
>
b 0
(6)
Where n goals are expressed by a n component column vector (b , b , ... , b ), A is l 2 a n x m matrix which expresses thenrelationship between goals and subgoals, X represents variables involved in the subgoals (X , X ' ... , X ), 6- and 0+ are n 2 component vectors formthe variable representing deviations from goals, and I is an identity matrix of size n. The relative priorities should be ranked for the deviational variables, and in the objective function.
0t
°1,
The goal programming model for the regional water quality management problem can now be developed. Two goals with two priorities are hereby considered. The goals are: (i) maintain the total cost of treatment at a specified level; (ii) maintain the water quality constraints in equation (4) close to but not less than the minimum preset quality limit. If the above two goals were ranked with priorities one (PI) and two ( P ) respectively, then the goal programming2 model can be written as follows: Minimize
A goal programming Subject to n L
j=l b.
1.
a .. x. 1.J
J
ter time required for running goal programmings for Case 1, Case 2 and Case 3 were 23.73 seconds, 26.60 seconds and 22.51 seconds respectively.
0:
-
1.
(i=l, 2,
. .. ,
n) (7)
+ °n+j
x. J
T. (j=l, 2, J
. .. ,
n)
n L
j=l
c. x. + 0 J
J
x , o~, 0: j J J
183
+
(2n+l) >
o
(2n+l)
0
0
APPLICATION PROBLEM
Table 3 shows the linear programming and goal programming solutions for pollution .limit of 500 Population Equivalent (PE) or 20 mg/l of BOD. The goal programming solution for Case 2 is the same as Deininger's solution. Change in the priorities (Case 1) changes the optimal removals. As shown in Table 3, the total Cost Units for Case 1 is 1,237.50 compared to 1,231.67 for Case 2. It may be observed that the removal at Plant 3 is 46.2% for Case 1 but for Case 2, there is no removal required at Plant 3. When the water quality and cost were analysed with equal priority (Case 3), the solution is the same as for Case 1. This can be seen from Table 3.
The application of goal programming model is shown to the hypothetical problem solved by DEININGER (Ref. 2). The hypothetical river basin involves a river 110 miles long with 12 cities varying in size from 1,000 to 150,000 people. For convenience, these cities were numbered from 1 to 12, starting with the city farthest upstream. The data used for the study are shown in Table 1. Table 2 shows the linear programming problem formulated by Deininger. Deininger found three sets of solutions for each pollution limit in the following three ways:
The computer programme also gives the output for slack analysis, variable analysis and analysis of the objective. The slack analysis, in Table 4 for Case 1 and Case 2, presents the values of the right hand side (available) and also values of the negative and positive variables for each equation. The slack analysis provides informations to analyze the details of goal attainments when the problem is complex. This analysis often proves to be useful in identifying errors when the model does not represent the desired decision environment.
Solution 1: To require from each polluter the same degree of treatment in terms of per cent removal of pollutant. Solution 2: To allow each polluter to discharges as much as he can without violating the water quality criteria. Solution 3: To devise a system of waste treatment which will maintain the desired water quality and minimize the total costs of waste treatment.
The variable analysis presents the constants of only the basic variables. When the problem under consideration is a very complex one, the variable analysis is especially helpful because it presents only the constants of the basic choice variables as compared to the final simplex solution table. It may be pointed out that the variable number are variable numbers of the choice variable (i.e. Xl' x 2 , •.• , x 12 ).
The goal programming solutions were obtained for the following three cases and compared with the Solution 3 of Deininger: Case 1: Considering minimizing the total cost of treatment as the second priority goal and water quality constraints as the first priority goal. Case 2: Considering water quality constraints as the second priority goal and minimizing the total cost of treatment as the first priority goal. Case 3: Considering both the above goals with equal priority. DISCUSSION OF RESULTS The computer program written for goal programming was run in IBM 370/75 at Regional Computer Centre, Asian Institute of Technology. The computer program may be obtained from Industrial Engineering and Management Division, AIT or from the authors. The compu-
An analysis of the objective presents the value of each goal. These values represent the underattained portion of goals. If the model requires assignment of artifical priority to set up the initial table, the artificial priority will also be printed out. CONCLUSIONS In case of river basin management where there are multiple goals sub goals of different priorities, goal programming appears to be a useful tool because it can handle a decision problem with a single goal and multiple subgoals as well as a problem with mUltiple goals and multiple subgoals. The present paper shows only one of the many applications possible of the approach presented. It may be stated here goal programming approach is not an ultimate solution for all decision problems. The techniques simply gives the best optimal solution for the given constraints and priority goals.
t-'
00
Table 1:
Data for the Application Problem (Pollution Limit is 500 PE/CFS or 20 MG/L BOD) (Ref. 2)
I CITY I CITY I CITY I CITY I 5 I 6 I 7 I 8 I ____________________ I _______ I _______ I _______ I _______ I _______ I _______ 1 _______ 1 ______ 1 ___________________ 1 _______ 1 _______ 1 _______ 1 _______ 1 _______ 1 _______ 1 _______ 1 ______ I I
I I I I POPULATION I -------------------- I I I I WASTEFLOW IN CFS I 1--------------------1 I I I DEOX - COEFF I I -------------------- I I I I RIVER MILEAGE I I -------------------- I I I I TREATMENT COSTS I I ------------------- I I I I TREATMENT LIMIT I I ------- ------------ I I I I I STREAMFLOW IN CFS I ------------------- I I I I REOX - COEFF I I ------------------- I I -------------------- I I I I POLLUTION LIMITS IN I I I I I I POPEQUIVAL/CFS I I I I I B 0 D IN MG/L I -------------------- I I -------------------- I I -------------------- I
CITY I 1
""
I
I 25000 I ------- I I 3.86 I -------1 I .23 I ------- I I 105.00 I ------- I I 200.00 I ------- I I .90 I ------- I I 38.86 I ------_ I I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I ------- I ------- I ------- I
CITY I CITY 2
I
I 1000 I ------- I I .15 I -------1 I .25 I ------- I I 98.00 I ------- I I 18.00 I ------- I I .90 I ------- I I 39.01 I ------- I I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I ------- I ------- I ------- I
3
I CITY
I
I 5000 I ------- I I .77 I -------1 I .20 I ------- I I 86.00 I _______ I I 60.00 I ------- I I .90 I ------- I I 39.78 I ------- I I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I -------1 -------1 -------1
4
I 2000 I _______ I I .31 I -------1 I .20 I _______ I I 76.00 I - ______ I I 30.00 I ------- I I .90 I ------- I I 40.09 I ------- I I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I ------- I ------- I ------- I
I I 1000 I 100000 I _______ I _______ I I I .15 I 15.47 I -------1 -------1 I I .25 I .30 I ------- I - ______ I I I 69.00 I 57.00 I _______ I _______ I I I 18.00 I 600.00 I ------- 1-------- I I I .90 I .90 I ------- I ------- I I I 40.24 I 85.71 I ------- I -------1 I I .55 I .55 I ------- I ------- I ------- I ------- I I I I I I I I I 500.00 I 500.00 I I I 20.00 I 20.00 I ------- I ------- I ------- I ------- I ------- I ------- I
I 2000 I _______ I I .31 I -------1 I .27 I _______ I I 47.00 I _______ I I 30.00 I ------- I I .90 I ------- I I 86.02 I -------1 I .55 I ------- I ------- I I I I I 500.00 I I 20.00 I ------- I ------- I ------- I
I CITY I I CITY I CITY I CITY I 12 I I 11 I 9 I 10 I _______ 1 _______ 1 _______ 1 _______ 1 I _______ 1 _______ 1 _______ 1 _______ 1
I I 1000 I 1000 I ______ I _______ I I I .15 I .15 I I -------1 I I .22 I .20 I _______ I _______ I I I 40.00 I 35.00 I _______ I _______ I I I 18.00 I 18.00 I ------- I ------- I I I .90 I .90 I ------- I ------- I I I 86.17 I 86.32 I -------1 -------1 I I .55 I .55 I ------- I ------- I ------- I ----___ I I I I I I I I I 500.00 I 500.00 I I I 20.00 I 20.00 I ------- I ------- I ------- I ------- I ------- I ------- I
I 10000 I _______ I I 1.55 I -------1 I .22 I _______ I I 27.00 I _______ I I 100.00 I ------- I I .90 I ------- I I 87.87 I -------1 I .55 I ------- I _______ I I I I I 500.00 I I 20.00 I ------- I _______ I ------- I
I 150000 I _______ I I 23.20 I -------1 I .27 I _______ I I 16.00 I _______ I I 825.00 I ------- I I .90 I ------- I I 121.07 I -------1 I .55 I _______ I _______ I I I I I 500.00 I I 20.00 I ------- I _______ I ------- I
I 7000 I _______ 1 I 1.08 I _______1 I .30 I _______ I I .00 I _______ I I 77.00 I - ______ 1 I .90 I ------- I I 122.15 I _______ 1 I .55 I _______ I _______ I I I 0 I 500.00 I I 20.00 I ------- I _______ I _______ I
I;>j
z t""
o ::r III
::l ..... III
::l
0.
'
> 0.
C ..... eT
::r
III
::l
Table 2:
Linear Programming Formulation (Pollution Limit is 500 PElcFS or 20 MG/L BOD) (Ref. 2)
THE BUILT TREATMENT PLANTS MINIMIZE THE TOTAL COSTS THE PROBLEM IS, MATHEMATICALLY SPEAKING MINIMIZE 200.000
18.000
60.000
30.000
18.000
600.000
30.000
18.000
18.000
100.000
825.000
77 .000
SUBJECT TO 643.335
.000
.000
. 000
.000
.000
.000
.000
.000
.000
.000
.OOO.G.E •
143.335
600.891
25.634
.000
.000
.000
• 000
.000
.000
.000
.000
.000
.OOO.GE •
126.525
527.668
.000
.000
.000
• 000
49.888
.000
.000
.000
.000
.000
46.995
24.851
.000
.000
.000
.000
1,166.725
.000
.000
. 000
.000
.000
22.296
125.691
.000
477 .567
20.018
115.130
446.112
18.595
108.455
187.553
7.743
46.258
20.044
10.348
.000
.000
.OOO.GE •
175.655
.000
.000
.OOO.GE.
162.603
.000
.000
.OOO.GE.
145.007
.000
.000
.OOO.GE •
938.670
.000
.000
.OOO.GE.
802.059
.000
.000
.OOO.GE.
711. 752
.000
170.452
6.981
42.547
18.436
9.329
1,031.063
23.250
159.542
6.498
40.160
17.402
8.684
946.341
21.520
11.605
.000
152.105
6.170
38.518
16.691
8.246
889.682
20.353
11.086
11.585
.000
.000
.OOO.GE.
654.436
138.819
5.595
35.493
15.380
7.477
793.987
18.339
10.150
10.675
113.804
.000
.OOO.GE.
649.720
91.055
3.638
23.590
10.222
4.862
504.999
11. 819
6.687
7.095
74.976
1,238.953
.OOO.GE.
1,477.896
77.897
3.073
20.572
8.914
4.106
413.094
9.856
5.758
6.187
64.553
1,033.122
57.307.GE.
1,204.439
INEQUALITY
2 IS REDUNDANT
INEQUALITY
5 IS REDUNDANT
INEQUALITY
7 IS REDUNDANT
INEQUALITY
8 IS REDUNDANT
INEQUALITY
9 IS REDUNDANT
:> 00
0
DJ ,.....
'1:1 '1 0
00
'1
DJ
§ ..... ::l
00
INEQUALITY 12 IS REDUNDANT I-'
00 VI
186
B. N. Loha ni and P. Adulbhan
Table 3:
Linear Programming (LP) and Goal Programming (GP) Solutions
Percentage Plant
GP
LP Solution by Deininger
1 2 3 4 5 6 7 8 9 10
34.0 0.0 0.0 0.0 0.0 75.9 0.0 0.0 0.0 0.0 85.9 0.0
11
12 Total Costs
1,231.67 Cost Units
11
12 13 14 15 16 17 18 19
Solutions Case 2
Case 3
22.3 0.0 46.2 6.0 0.0 74.9 0.0 0.0 0.0 5.7 85.8 0.0
34.0 0.0 0.0 0.0 0.0 75.9 0.0 0.0 0.0 0.0 85.9 0.0
22.3 0.0 46.2 6.0 0.0 74.9 0.0 0.0 0.0 5.7 85.8 0.0
1,231.67 Cost Units
1,237.50 Cost Units
Slack Analysis
Ca se
1 2 3 4 5 6 7 8 9 10
Removal
Case 1
1,237.50 Cost Units
Table 4:
Row
BOD
1
Case
2
Available 143.33501 175.65500 162.60300 938.66992 649.71997 1477. 89600 0.00001 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000 0.90000
POS-SLK
NEG-SLK
POS-SLK
NEG-SLK
0.0 0.0 0.0 0.0 0.0 0.0 1237.49951 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.67720 0.90000 0.43783 0.84003 0.90000 0.15064 0.90000 0.90000 0.90000 0. 84306 0.04170 0.90000
75.70966 4.00658 0.0 10.46700 0.0 1. 66919 1231.66919 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.55952 0.90000 0.90000 0.90000 0.90000 0.14123 0.90000 0.90000 0.90000 0.90000 0.04144 0.90000
A goal programming
Table 5:
Case
2 Artificial
Case
2
Variable
Amount
Variable
Amount
3
0.46217
6
0.75877
Case
1
Variable Analysis
1
4
0.05997
11
0.85856
6
0.74936
1
0.34048
10
0.05694
-
-
11
0.85830
1
0.22280
-
-
Table 6:
Priority
187
Analysis of the Objective
Case
1 Under-achievement
Priority
2 Under-achievement
2
90.18323
0.0
1
1231.66919
0.0
Artificial
0.0
1237.49951
REFERENCES 1. R.J. DAY, F.T. DOLBEAR, JR. and M. KAMIEN, Regional Water Quality Management - A Pilot Study, Proc. 1st Annual Meeting, AWRA, University of Chicago, 1965. 2. R.A. DEININGER, Water Quality Management The Planning of Economically Optimal Pollution Control System, Proc. 1st Annual Meeting, AWRA, University of Chicago, 1965. 3. K.D. KERRI, An Economic Approach to Water Quality Control, J. WPCF, Vol. 38, 1883 (1966). 4. S.M. LEE, Goal Programming for Decision Analysis of Multiple Objectives, Sloan Management Review (Winter), 11 (1972-73).
5. J.C. LIEBMAN and W.R. LYNN, The Optimal Allocation of Stream Dissolved Oxygen, Water Resources Research, Vol. 2, 1966. ----6. B.N. LORAN I , A Chance - Constrained Approach to Regional Water Quality Management, Doctoral Dissertation (in progress), Environmental Engineering Division, Asian Institute of Technology, Bangkok, 1976. 7. C. REVELLE, D.P. LOUCKS, W.R. LYNN, Linear Programming Applied to Water Quality Management, Water Resources Research, Vo!. 4, No. 1, 1, 1968. 8. M. SOBEL, Water Quality Improvement Programming Problems, Water Resources Research, Vol. I, No. 4, 477 (168).