OPTIMIZATION
OF A M B I E N T A I R Q U A L I T Y M O N I T O R I N G NETWORKS
(Part II) P R A S A D M. M O D A K
Environmental Science and Engineering Group, Indian Institute of Technology, Powai, Bombay 400 076, India and B. N. L O H A N I
Environmental Engg Division, Asian Institute of Technology, P.O. Box 2754, Bangkok, Thailand, 10501
(Received 9 July, 1984) Abstract. Minimum Spanning Tree (MST) algorithm developed by Modak and Lohani (1984a) has been extended to consider multiple objectives for the optimum siting of ambient air monitors. Two approaches have been proposed, namely one based on the utility function and another based on the principles of sequential interactive compromise. The sequential interactive approach is heuristic but perhaps best suited to consider several objectives at a time, and particularly when professional judgements are also involved. The utility function approach may be normally restricted to two objectives at a time, but could be extended to consider a number of pollutants in the optimum design. For the purpose of illustration, the case of Taipei City, Taiwan has been considered.
1. Introduction
In part I of the paper (Modak and Lohani, 1984a) a methodology was developed to find a joint solution to the problem of the optimum number of monitors and optimum monitor configuration. The objective of this method was to represent regionwide air quality patterns (in terms of the spatial variance) for a stipulated value of cut-off correlation coefficient or specified monitoring budget. In real situations however, representation of the regionwide air quality patterns is just one of the several objectives of the Air Quality Monitoring Network (herein referred as AQMN). An objective such as compliance with ambient air quality standards for instance, is yet another important objective of the AQMN. It is then quite logical that the AQMN must be optimized for multiple objectives. Little work has been done multiple objectives in the AQMN design. But the recent literature shows a growing interest for attempting this problem. Naito and Ochiai (1981) and Munn (1981) consider that the incorporation of multi-objectives is extremely important to lay the foundation of the future practices towards AQMN optimization. Smith and Egan (1979) suggest that optimization should be undertaken for each objective. Munn (1981) suggests an overlay of these designs since there could be a subset of these designs which could probably meet the interest of more than one objective. Environmental Monitoring and Assessment 5 (1985) 21-38. 9 1985 by D. Reidel Publishing Company.
0167-6369/85.15.
22
P. M, M O D A K
A N D B. N. L O H A N I
Once a multi-objective framework is decided, it may be rather tempting to recommend several sub-configurations for each objective. In such instances however, the number of monitors is normally expected to increase, and this could be particularly true when the objectives imposed are in conflict with each other. Such situations would only be ideal, and particularly not feasible in the case of developing countries where funds for monitoring activities are severely limited. It is quite evident therefore that when the resources on monitoring are limited or constrained, it may not be possible to maximize the interests of each objective. While it may not be logical to neglect the role of some of the objectives altogether, due to limitations on resources available. The multi-objective methodologies are essentially a compromise philosophy to the AQMN design. The interest of the Decision Maker (herein refered as DM) is then to investigate how far the selected AQMN configuration could do justice to more than one objective - all under a common economic constraint. In this paper, two different approaches have been proposed for the multi-objective optimization of the AQMN. These methodologies are essentially an extension of the basic Minimum Spanning Tree (MST) algorithm developed in Part I of the paper (Modak and Lohani, 1984a). The first method makes use of the Utility Function (herein referred as UF), (Cohon and Marks, 1975) and the second methods makes use of a Sequential Interactive Compromise, commonly referred as the 'An Interactive Method of Progressive Articulation of Information' (Hwang etal., 1980). The first method will be hereafter referred to as the U-approach and the later as the S-approach. In this study, for the purpose of demonstration, two objectives have been considered as of interest; namely representation of spatial-temporal patterns and the detection of violations of ambient air quality standards. In addition, estimation of population dosage product has been treated as an auxiliary monitoring objective. Since pattern score (Modak and Lohani, 1984a), has been identified as the decision variable for representing air quality patterns, it is necessary that a decision variable be identified for representing the objective of compliance. 2. Detection of Violations over Ambient Standards
The potential of a monitoring site for the detection of violations is considered herein in terms of violation scores. A location having a high violation score is then considered to have a high potential for detection of violations. The computation of violation scores is essentially a weighted scoring of the concentrations above the prescribed thresholds. These scores are therefore dependent on, (1) The threshold levels. (2) The weighing factors between each threshold range, or the weighing function. A decision on thresholds and weighing factors (i.e. severity) is indeed pollutantspecific and further dependent upon the averaging time and populations concerned. Several weighing functions have been reported such as linear functions, segmented linear functions, non-linear function, segmented non-linear functions etc. (Ott, 1978; Jain et aL, 1977). In this case, a segmented nonlinear weighing function is proposed as,
23
O P T I M I Z A T I O N OF AMBI ENT AIR QUALITY M O N I T O R I N G NETWORKS
T N[ = Z i=1
N~ Z
(v~ + I - v~)(xi - X k ) X / ( x k + I - xk)
(1)
k=l
where vk = the weighing factor corresponding to threshold x~, x k = the k-th threshold, X =0if(x i-xk) 0. The U-approach need not be restricted to consideration of pattern and violations but could be extended to the question of placing monitors in populated areas of probable concern. In this case, the U F could be expressed as, VV = Np x (N L, x K) b
(6)
where K is population activity around the location. It is to be noted that a product of K and N o yields the Population Dosage Product (herein referred to as PDP).
5. Basic Mechanism in the Utility Approach Table I shows the effect of U F on the selection of the locations in terms of a decision process. It is clear from Table I that the U F shifts the monitor configuration towards TABLE I Effect of the structure of utility function on the selection of the monitoring locations Location
~
~
UF b=0
UF b=l
UF b=2
A B C
4 9 6
10 5 3
4 9 6
40 45 18
400 225 54
B..C..A
B..A..C
A..B..C
Decision
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
25
meeting the associated objective. In Table I for instance, on the imposition of the utility, station A was preferred to station B, despite its low pattern score. The violation scores were subsequently improved, i.e., from 5 to 10, by shifting the decision. The utility approach could therefore be considered as a method of weighted MST. Since the termination criterion of the M S T algorithm is to ensure total coverage, such a shift in the priority is expected to lead to an increase in the number of monitors, i.e., the cost of the A Q M N design. It is not necessary that the number of monitors in the utility case could always be higher than obtained due to the sole consideration of patterns. It is possible that on the imposition of utility, the number of monitors in the A Q M N remains the same but only the configuration is altered. Such situations are expected for lower values of C O and especially when the pattern scores and the violation scores are not in conflict with each other. This elasticity of the solution is ideal since in such instances, the cost of the A Q M N would remain the same while some improvement is possible in meeting the associated objective. As the magnitude of coefficient b is increased, the solution is no more elastic and the optimal number of monitors starts increasing, in addition to relocation within the configuration. After a certain value of b is reached, the solution offers no more changes or alterations. This is because all of the locations which help meet the associated objective have been selected. In the example case illustrated in Table I, the decision on A as the best location is unaltered after b = 2 for any positive value of b. It should be noted that improvement in the associated objective (in this case the compliance) may not be necessarily that of a monotone increasing nature with that of b. In other words, though the solution obtained for b > 0 is more effective than that of b = 0, this effectiveness does not necessarily increase as the value ofb is increased. This behaviour is due to the use of the utility function U F as well as due to the sequential policy of selecting optimal locations.
6. Formulation and Computational Algorithm for th~ Utility Approach Normally, the problem of A Q M N optimization is financially constrained, and therefore it is not possible to maximize the interest of both objectives. For a given budget, it is then necessary to identify the optimum value of C O and that of b which would lead to a possible maximization of program objectives. This compromise can be identified by increasing the coefficient b at a fixed C Otill the cost of the system increases beyond the prescribed budget. Once the budgetary constraints are violated, the value of C Ocould be lowered, since the cost of the system has direct relationship with C O (refer to Figure 4 in Modak and Lohani, 1984a). Once again, the sequence of increasing the level of b could be reinitiated. A procedure of compromising over C ~ would continue till a minimum prescribed level of C Ois reached. Since the effectiveness of the associated objective does not have a monotone increasing relationship, all intermediate solutions need to be printed out with summary statistics
P. M. MODAKAND B. N. LOHANI
26
TABLE 11 A comparision between the Utility and the Sequential Interactive approaches No.
Utility approach
Sequential Interactive approach
1.
Normally applicable up to two joint objectives
Can consider up to 7 objectives at a time
2.
Necessary to formulate a Utility Function, difficultto accomodate interests of heterogeneous objectives
No need to formulate a Utility Function, heterogenious objectives could be considered
3.
No heuristics is involved once the Utility Function is specified
Heuristics is the idea
4.
For a specified Utility Function, an optimum could be identified
Optimum may not be achieved
5.
Could be extended to multiple pollutants
Restricted only for a single pollutant
at the end of the calculations. M o d a k (1984) should be referred for computational details. T h e optimization problem over the utility could therefore be stated as (in set notations), M a x ( M 1 w m 2 t,.) . . . w M m )
(9)
while, Max
~
(10)
NIl.
i=1
Such that (M 1 u M e w ... w M m) = (M l w Me...
w M u)
(11)
and m = m~
(12)
where, M i = the set of Np monitors correlated with location i above C ~ rn = the n u m b e r of m o n i t o r s ; m ~ : the allowable n u m b e r of m o n i t o r s ; N = the n u m b e r of candidate locations. The c o m p u t a t i o n a l algorithm for the U-approach is quite similar to that of the original M S T algorithm presented in M o d a k and Lohani (1984a). I n this case however, instead of the pattern scores alone, violation scores are also considered in the form of a multiplicative utility function. The attribute for the selection o f t h e best and the next best m o n i t o r s is thus o n a utility basis rather than solely on the consideration of the pattern scores.
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
27
7. Application of U-Approach to the Taipei City Case As an illustration of the utility approach, the example of Taipei City (refer to Modak and Lohani, 1984a) is now reconsidered. The concentrations of sulfur dioxide in Taipei City are fairly low, and violations of the standards of 0.10 p p m (monthly averaged hourly concentrations) are almost nonexistent. For the purpose of illustration therefore, the concentrations of sulfur dioxide were scaled up by a factor of 5, to justify the objective of compliance. It is to be noted that a constant multiplier such as 5 does not change the correlation matrix, so the analysis is not affected as far as the patterns are concerned. It is rather difficult to prescribe a weighing function in this example, since data on the hazardous effects of monthly averaged hourly concentrations has seldom been reported in the literature. The normal practice has been to consider hourly or daily averages in the development of the weighing functions (Ott, 1978). For the purpose of illustration therefore, a non-linear weighing function of the form such as in Table III is TABLE III Non-linear segmented function for computing violation scores for sulphur dioxide Concentration SO2 in ppm
Score
0.50 < 1.00 > 1.00 < 2.00 > 2.00
0.5 1.5 3.0 5.0
considered. This weighing function bears a close relationship to the one reported by Jain
et aL (1977) for assessing the impacts due to annually averaged daily sulfur dioxide concentrations. More appropriate weighing functions could be considered later when relevant effects data are available. For the purpose of illustration, a budgetary constraint of 800000 US$ was considered. Maximum and minimum level of C O was considered as 0.95 and 0.85, respectively. Table IV shows a summary of the results. Costs are calculated in the way described in Modak and Lohani (1984a). Percentage Exposure has been defined as a ratio of the P D P detected by the A Q M N to that of the total P D P calculated on summing over all the candidate locations. Percentage Compliance denotes a ratio of the violation score detected by the A Q M N to that of the total violation score calculated by summing over the violation scores at all candidate locations. Percentage Variance is the square of the population correlation coefficient C O. At the outset, the results of Table IV show that consideration of multiple objectives is rather important in the A Q M N design. In this case, the solutions based on the sole consideration of patterns are observed to render a poor performance as far as compliance is concerned.
28
P.M. MODAK AND B. N. LOHAN1 TABLE
IV
A Summary of multi-objective designs based on utility approach No.
Cost 1000 US$
Percentage exposure
Percentage compliance
Percentage variance
b
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
914.8 711.5 914.8 813.2 711.5 711.5 813.2 609.9 609.9 609.9
38.8 27.1 34.8 27.8 27.0 28.2 38.1 17.7 23.8 24.3
33.6 27.8 32.9 30.2 27.8 29.9 34.5 23.4 24.0 25.4
90.2 88.4 88.4 86.5 84.6 84.6 84.6 82.8 82.8 82.8
0.0 0.0 2.0 0.0 0.0 2.0 4.0 0.0 2.0 4.0
24 23 22 21 2C 19 18 17 16 15
t~ =3 II 9 8
2 3 4 5 6 7 8 9 I0 II 121314 1516171819 KILOMETERS
I 2 3 4 ,5 6 7 8 9 I0 II 121314 15 16 17 1819 KILOMETERS
Note : Numbers indicate the order of station selection
Solution Number 4 Objective
Solution Number 7
Effectiveness
Objective
Compliance
2Z 80 %
Exposure
30. 20 %
Compliance Exposure
38. I0 ~ 34.50 ~
Variance
86.50 %
Variance
84.60 %
Fig. 1.
Effectiveness
Configurations of optimal A Q M N designs based on Utility Approach.
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORINGNETWORKS
29
It is observed that as coefficient b in the utility function is progressively increased, accomplishment of the associated objective is also enhanced. As discussed earlier, this improvement is not necessarily of a monotone increasing nature. It is to be noticed that in some cases, the costs of AQMN remain unaltered. Solution numbers 5 and 6 in Table III could be considered as examples of this situation. Such a possibility has already been discussed. When the costs of the AQMN cross beyond a prespecified budgetory constraint, (in this case, 800000 US$) the level of C O is lowered as a compromise. It appears that the objectives of compliance and estimation of PDP are commensurate with each other. It could be seen that as the coefficient b is increased in case I, the estimation of the PDP is also improved along with that of compfiance. A commensurate relationship is rather evident because most of the contribution to the sulfur dioxide in Taipei city is primarily due to transportation sources, i.e., mobile pollution, sprawled in the densely populated areas. Figure 1 shows the configuration of the AQMN corresponding to solution numbers 4 and 7 of Table III. Table V (extracted from Table IV) shows how the utility approach
TABLE V A Summary of the results for utility approach - extracted from Table IV No.
Cost 1000 US$
Percentage exposure
Percentage compliance
Percentage variance
b
4. 7.
813.2 813.2
27.8 38.1
30.2 34.5
86.5 84.6
0.0 4.0
optimizes the possibility of meeting the associated objectives (compliance and estimation of the PDP) in the same budget by carrying out a compromise over the reliability of pattern representation.
8. The Sequential Interactive (S) Approach In certain situations, it may not be possible to identify the form of the UF. A choice of the structure of the utility function is sometimes difficult when several heterogeneous objectives are to be considered. This may be particularly true when professional judgements also come into play. In such instances, the S-approach could be considered as a potential method for applications. In this method, the priority on objectives is successively shifted, with the help of the Decision Maker (DM), until an acceptable compromise is reached. This procedure is thus highly amenable to interactive versions. The S-approach is a style of compromise programming and bears a close relationship with the Interactive Decision Evaluation Aid (IDEA) as discussed by Zeleny (1982).
30
P. M . M O D A K
A N D B. N . L O H A N I
9. Computational Algorithm for Sequential Interactive Approach The various steps of this algorithm are as follows. (1) Specify: (a) The maximum allowable monitoring budget for the AQMN. If the desired coverage effectiveness is not specified, then the design would be reported for the total coverage. (b) The minimum and maximum acceptable level o f C ~ i.e., the minimum and maximum reliability of the representation of the spatial-temporal pattern. (2) Arrange the grid points in descending order as per the scores of violations. Alternatively or in addition, a sorting of sites according to the population dosage product could be considered. In interactive methods, it has been reported that the DM could apply it to 7 objectives of interest and therefore a simultanious consideration of compliance and estimation of population dosage product should not be a problem. A policy of ranking the locations in relation to the associated objectives assists the DM in the decision-making process. (3) For the maximum value of C o, identify the locations of monitors based on the M S T algorithm. (4) If the cost of the design in step 3 exceeds the budgetary constraint prescribed in step 1 (a), then the D M is asked to lower the C o value. This step is evident due to the relationship between C o and m. If the budgetary constraint is not violated, then move to step 5. Otherwise the DM is asked to lower the C o value and move to step 4 again. (5) It is possible that locations with a high number of violations and/or that are needed to estimate P D P are not included in the array of monitors selected by the coverage criterion of step 4. The DM is now asked whether he intends to include any station missed in the final design. The S-approach would respond to the DM's request of inclusion, for one station at a time. If constraint on the budget specified in 1 (a) is not violated, then the DM is questioned again whether any additional station is to be included in the design. A ranking of the associated objectives provided on the screen enables the D M to select such potential monitors. (6) Once the constraint as in 1 (a) is violated, then the program would once again request a further reduction in the C ~ Step 4 is executed once again, to get the new (possibly reduced) number and location of the monitors, and the procedure is iterated till the monitors indicated by the DM, based on the other objectives are accommodated within the budgetary constraint. It should be noted however that the monitors introduced by the DM are now used to initialize the status matrix (refer Modak and Lohani, 1984a), before any selection of additional monitors is made. (7) Steps 4, 5, and 6 are now iterated till the value of C ~ becomes less than the minimum level, as specified earlier in step 1 (b). At each stage of iteration, the locations of interest of the associated objective are introduced by compromising on C o. Once the program is terminated, all the solutions with the DM's response are printed out which could be consulted later to pick up the best A Q M N design. It is to be noticed that step 7 in the algorithm adds the monitors to the A Q M N configuration. Step 8 on the other hand reduces the total number of monitors while retaining those suggested in step 7. At each addition the DM meets his objectives better
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
31
(other than that of the pattern representation) and at each such addition - deletion process his emphasis on the objective of pattern representation is reduced. In fact such a shift in emphasis is envisaged to amplify his other objectives of interest - as in the case of ambient air quality compliance and of estimating the population dosage product. This process of addition-deletion moves sequentially at a series of DM's responses till a situation such at as step 9 is arrived as an acceptable compromise. In the S-approach therefore, the D M has a good opportunity to learn and familiarize himself with the various implications of the A Q M N design.
10. Application of S-Approach to Taipei City The interactive methodology as described in the previous section has been programmed as ' S U S E R ' (for details refer M o d a l 1984). For the purpose of illustration, the data used for U-Approach: i.e., Taipei City sulfur dioxide AQMN, was once again considered. The salient results of the ' S U S E R ' output are extracted in the Appendix in the form of the first three interaction screens. In this case, similar to the U-approach, a budgetary constraint of 800 00 US$ was imposed and a maximum and minimum level of C o was considered as 0.95 and 0.85 respectively. Table VI shows a summary of the solutions. T A B L E VI A S u m m a r y of multi-objective designs based on sequential interactive approach Step
Step Cost 1000 US$
Percentage compliance
Percentage exposure
Percentage variance
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
914.8 711.5 711.5 711.5 813.2 813.2 813.2 711.5 813.2 711.5 813.2 813.2 813.2 711.5 813.2
33.6 27.8 29.1 28.1 33.8 33.7 33.7 28.8 34.0 34.8 32.6 32.6 32.6 27.7 32.0
33.8 23.5 23.9 24.9 32.6 32.6 32.6 23.9 31.1 36.3 35.7 35.7 35.7 34.4 40.4
90.25 88.36 88.36 88.36 88.36 86.49 86.49 82.81 82.81 81.00 79.21 77.54 75.69 73.96 73.96
Table VI presents solution number 9, where C Ois reduced to 0.91 (variance 82.36 Yo) at a benefit of increasing the interests of the other associated objectives. Solution number 15 reduces C O even further (C O = 0.86 or variance 7 3 . 6 ~ ) as a solution of maximum allowable compromise. This solution provides maximum attainment to the objective of estimating the PDP. An examination of solutions numbers 9 and 15
32
P. M. MODAK AND B. N. LOHANI 2A
-)4 ->3 _:22 21 _:?C 19 18 17 16 15 14
I I I I I
12 II I0 9 8 7
B
2 3 4 5 6 7 8 9 10 II 1213141516171819 KILOMETERS
r 2 3 4 5 6 7 8 9 I011121314 1516171819 KILOMETERS
Note : Numbers indicate the order of station selection
Solution Number 9
Objective
Fig. 2.
Solution Number 15
Effectiveness
Objective
Compliance
34. O0 ~
Compliance
32.00 %
Exposure
31. IO ~
Variance
82.81 ~
Exposu re Variance
40.40 ~ 73.56 ~
Effectiveness
Configurations of optimal A Q M N designs based on Sequential Interactive Approach.
however indicates that solution number 9 could be considered as that of a good compromise from overall considerations. Figure 2 shows the configuration of the AQMN obtained for solution 9 and solution 15.
11. A Comparision between the Sequential Interactive and the Utility Function Approaches The U-approach developed in this paper is applicable for joint consideration of pattern representation and detection of violations or with that of the estimation of the PDP. Normally, the utility approach is applicable for not more than two objectives at a time: the primary objective in every case being that of the pattern representation. When the number of objectives increases, several runs of palrwise combinations of the problem are required: and in this case the set of non-dominated solutions could become quite
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
33
large. The S-approach on the other hand can handle all kinds of heterogeneous objectives with which the DM is familiar, all at the same time. This could be considered as an important advantage of the S-approach over the utility method. In the S-approach, the optimal monitoring stations are selected with an explicit consideration to the objectives. The DM therefore knows why a location is selected. This is indeed very important in the operation and the organization of the monitoring network. The procedure for selection of sites in other words, is entirely objective specific. One of the advantages of the S-approach is its flexibility. In many cases, a part of the AQMN is already in operation, or the DM is interested in retaining a few of the existing monitors. In the case of the S-approach, these monitors could be readily specified by user interaction. In the case of the U-approach, though such an inclusion is possible, it could only be off-line. In the case of the U-approach, the DM has absolutely no control in station selection, once the structure of the Utility Function (UF) is specified. However, a shift towards the associated objective through a progressive increase in b, assures an identification of the best compromise. The S-approach though convenient, depends on the response of the DM. The procedure is therefore rather heuristic, and there is a good possibility that the optimum is missed. Table VII shows a summary of comparision between the U and the S approaches.
T A B L E VII A comparision between the optimal designs obtained by the S and the U approach Method
Cost 1000 US$
Percentage compliance
Percentage exposure
Percentage variance
S S U
813.2 813.2 813.2
34.0 32.0 38.1
31.1 40.4 34.5
82.8 73.9 84.6
It is difficult to comment on which of the two methods is effective. Due to the differences in the philosophy of design, the solutions of S and the U approach are expected to be different for the same level of budgetary constraint. Table VII shows a comparision between the solutions obtained by the S and the U approach for Taipei City. In this case, it appears that the U approach is more effective than the S approach. The attractiveness of the utility method is expected to increase when multiple pollutants are under consideration. In this case, the interactive methods would have to provide multiple screens to the DM and the DM would have to respond specifically for each pollutant. Decision-making in such situations is expected to be rather difficult. The S-approach could therefore be considered to be applicable for only the case of single pollutant AQMN. A case of multiple pollutants under multiple objectives is discussed in Part III (Modak and Lohani, 1984c) of this paper.
34
P . M . MODAK AND B. N. LOHANI
12. Source-Interpretation in Multi-Objective Design It is clear that the role of the multi-objective philosophy is to shift the decision sequence of the MST algorithm (pattern solution) in favour of the associated objectives. It is important to note that a shift from the pattem solution in this approach depends upon the relationships between the pattern scores Np and the associated objective (such as violation scores Nv). An investigation of such relationships is important for the interpretation of the multi-objective design. If Np and Nv for instance, are in conflict with each other, then the number of monitors required would be significantly more than those required for the sole consideration of pattern. On the other hand, if Np and N~ have a commensurate relationships, then the number of monitors in the case of multi-objective design may not be significantly different. A case of conflict between the pattern and the violation scores could be envisaged as a situation having a large number of point sources scattered in the region. A case of commensurate relationships is the situation where a significant amount of pollution is contributed by area and mobile souces sprawled over the region. A design of the AQMN with multiple objectives has therefore a direct bearing on the source distribution in the urban areas. Conversely, it could also be said that a multi-objective approach to AQMN design could serve as a cursory tool for regional source dignosis. A cluster analysis between the pattern and the violation scores could be a potential method for further investigations. These deductions, in the case of the example of Taipei City, Taiwan, were found to be quite true. The results showed a commensurate relationship between the pattern scores N~, and that of the estimating the PDP. These relationships therefore imply that the sources of the mobile and population-related activity play a dominant role in the overall air quality. Further, it could be inferred that the populated regions of concern are concentrated in the busiest areas of traffic or mobile pollution. The land-use map of Taipei City (Taipei City, Summary of Statistics, 1981) confirms such deductions.
13. Summary and Conclusions (1) A Minimum Spanning Tree (MST) algorithm developed by Modak and Lohani (1984a) is extended to consider multiple objectives in the optimum AQMN design. (2) This extension is possible via two approaches, namely one based on the utility function and another based on the principles of sequential interactive compromise. The sequential approach is heuristic but perhaps best suited to consider several objectives at a time, and particularly when professional judgements are also involved. The utility function approach which may be restricted to two objectives at a time, is however better suited when a number of pollutants are to be considered. (3) One of the important outcomes of any multi-objective study is the experience and the insight the DM could gain about the system in total. A preliminary source diagnosis could be made based on examination of the relationship between objectives - i.e.
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
35
whether these are in friendly competition or in conflict. A multi-objective optimization of the A Q M N has thus several other useful implications besides mere optimization of network density a n d configuration. Optimization is then only a beginning for seeking policies in search of effective air quality m a n a g e m e n t .
Acknowledgements The various c o n t r i b u t i o n s m a d e in this paper are basically a part of the doctoral research carried out by the first author while at A s i a n Institute of Technology, Bangkok, Thailand. All c o r r e s p o n d e n c e regarding this paper should be therefore addressed to the first author. T h e support of the G o v e r n m e n t of J a p a n is gratefully acknowledged for providing financial assistance to complete the above doctoral research.
References Cohon, J. L. and Marks, D. H.: 1975, 'A Review and Evaluation of Multi-objective Programming Techniques', Water Resources Research II, 208-220. Godfrey, S. M., Novak, J. H., and Turner, D. B.: 1975, 'A Modelling Study to Examine Detection of Violationsof Air Quality Standards Using Various Sampling Station Networks in the Vicinityof a Single Plant', Report 76-23.3, Pittsburgh; Air Pollution Control Association. Hwang, C. L., Paidy, S. R., Yoon, K., and Masud, A. S. M.: 1980,'Mathematical Programmingwith Multiple Objectives: A Tutorial', Computers and Operations Research 7, 5-31. Jain, R. K., Urban, L. V., and Stacey, G. S.: 1977,Environmental ImpactAnalysis:A New Dimension in Decision Making, Van Nostrand Reihold, New York, U.S.A., Appendix B, pp. 178-209. Modak, P. M.: 1984, 'Optimum Siting of Ambient Air Monitors', A Dissertation submitted as the partial fulfillmentfor a degree of Doctor of Engineering, Asian Institute of Technology, Bangkok, Thailand. Modak, P. M. and Lohani, B. N.: 1984a, 'Optimization of Ambient Air Quality Monitoring Networks Part I', Environmental Monitoring and Assessment 5, 1-19. Modak, P. M. and Lohani, B. N.: 1984c, 'Optimization of Ambient Air Quality Monitoring Networks Part III', Environmental Monitoring and Assessment 5, 39-53. Munn, R. E.: 1981,Design of Air Quality Monitoring Networks, Macmillan Press Ltd, Basingstoke, Hampshire, U.K. Naito, M. and Ochiai, M.: 1981,'On Optimal Allocation of Air Monitoring Stations', International Seminar on Air Quality Management and Related Energy Policies, Tokyo, Japan, pp. 247-265. Ott, W. R.: 1978, Environmental Indices: Theory and Practice, Ann Arbour Science. Smith, D. G. and Egan, B. A.: 1979, "Design of Monitoring Networks to Meet Multiple Criteria', Journal of Air Pollution Control Association 29, 710-714. Taipei City Summary of Statistics: 1981, Taipei, Taiwan. World Health Organization (WHO): 1977, 'Air Monitoring Programme Design for Urban and Industrial Areas', Global Environmental Monitoring System, WHO Offset Publication No. 38. Zeleny, M.: 1982, Multiple Criteria Decision Mala'ng, McGraw Hill, New York.
36
P. M. MODAK AND B. N. LOHANI
Appendix
SCREEN
NUMBER
1
COST OF THE A Q M N IS.. IN 1000 US$ A N D CMAX IS 800.00 C U R R E N T C U T - O F F IS ... NO OF M O N I T O R S
914.86
0.95
9
PATTERN MST
COMPLIANCE
PDP
10 31 2 28 32 11 18 21 8
11.17 6.42 15.56 6.53 1.28 7.20 14.74 12.59 10.76
55.84 6.42 46.68 6.53 1.28 21.61 88.42 12.59 75.33
TOTAL C O M P L I A N C E POTENTIAL ,.. TOTAL PDP POTENTIAL ... VIOL RANKS ... 1 15.75 2 4 13.27 19
15.56 13.08
PDP RANKS ... 18 88.4 8 10 55.8 13
75.3 52.9
314.69
18 9
5 2
86.24
14.74 12.74
72.4 46.7
6 19
5 21
14.49 12.59
71.5 39.2
4
R E P O R T I N G THE STATUS OF THE COVERAGE 10 17.00 31 22.00 2 25.00 28 28.00 II 31.00 18 32.00 21 33.00 8 34,00
6
14.31
66.4
32
30.00
I N D I C A T E YOUR CHOICE OF R E D U C I N G CRX . . .
0.01
THIS STEP IS ESSENTIALLY THAT OF C O M P R O M I S I N G ON C o TO MEET THE PRESCRIBED B U D G E T O R Y CONSTRAINTS
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
SCREEN
NUMBER
37
2
711.55
COST OF THE A Q M N IS.. IN 1000 US$ A N D CMAX IS 800.00 C U R R E N T C U T - O F F IS ... NO OF M O N I T O R S
0.94
7
PATTERN MST
COMPLIANCE
PDP
18 15 3 28 32 4 21
14.74 10.65 12.47 6.53 1.28 13.27 12.59
88.42 31.96 12.47 6.53 1.28 66.37 12.59
TOTAL C O M P L I A N C E POTENTIAL ... TOTAL PDP POTENTIAL ... VIOL RANKS ... 1 15.75 2 4 13.27 19
15.56 13.08
PDP RANKS ,., 18 88.4 8 10 55.8 13
75.3 52.9
219.62
18
5
71.53
14.74
72.4
5
6
14.49
71.5
4
6
14.31
66.4
R E P O R T I N G THE STATUS OF THE COVERAGE 18 19.00 15 24.00 3 27.00 28 30.00 4 33.00 21 34.00
32
32.00
SO YOU W A N T TO INTERACT ?... TYPE YES OR NO yes
? 1
L O C A T I O N 1 IS F O R C E D TO ENTER THE MST SOLUTION BY E X A M I N I N G ITS PERF O R M A N C E WITH RESPECT TO THE O T H E R OBJECTIVES OF INTEREST
38
P. M. M O D A K A N D B. N. LOHANI
SCREEN
NUMBER
3
COST OF THE A Q M N IS.. IN 1000 US$ A N D CMAX IS 800.00 C U R R E N T CUT-OFF IS ... NO OF M O N I T O R S
711.55
0.94
7
PATTERN MST
COMPLIANCE
PDP
1 18 15 28 32 4 21
15.75 14.74 10.65 6.35 1.28 13.27 12.59
15.75 88.42 31.96 6.53 1.28 66.37 12.59
TOTAL COMPLIANCE POTENTIAL ... TOTAL PDP POTENTIAL ... VIOL RANKS ... l 15.75 2 4 13.27 19 18 10
88.4 55.8
8 13
15.56 13.08 75.3 52.9
74.80
222.89
18
5
14.74
72.4
6
5
14.49
71.5
4
R E P O R T I N G T H E STATUS OF THE COVERAGE 18 22.00 15 27.00 28 30.00 32 32.00 21 34.00
6
14.31
66.4
4
33.00
N O T E THAT LOCATION 1 BECOMES A PART OF THE MST SOLUTION DUE TO THE D E C I S I O N T A K E N ON SCREEN 2. I N T E R A C T I O N C O N T I N U E S ...
OF A M B I E N T A I R Q U A L I T Y M O N I T O R I N G NETWORKS
(Part II) P R A S A D M. M O D A K
Environmental Science and Engineering Group, Indian Institute of Technology, Powai, Bombay 400 076, India and B. N. L O H A N I
Environmental Engg Division, Asian Institute of Technology, P.O. Box 2754, Bangkok, Thailand, 10501
(Received 9 July, 1984) Abstract. Minimum Spanning Tree (MST) algorithm developed by Modak and Lohani (1984a) has been extended to consider multiple objectives for the optimum siting of ambient air monitors. Two approaches have been proposed, namely one based on the utility function and another based on the principles of sequential interactive compromise. The sequential interactive approach is heuristic but perhaps best suited to consider several objectives at a time, and particularly when professional judgements are also involved. The utility function approach may be normally restricted to two objectives at a time, but could be extended to consider a number of pollutants in the optimum design. For the purpose of illustration, the case of Taipei City, Taiwan has been considered.
1. Introduction
In part I of the paper (Modak and Lohani, 1984a) a methodology was developed to find a joint solution to the problem of the optimum number of monitors and optimum monitor configuration. The objective of this method was to represent regionwide air quality patterns (in terms of the spatial variance) for a stipulated value of cut-off correlation coefficient or specified monitoring budget. In real situations however, representation of the regionwide air quality patterns is just one of the several objectives of the Air Quality Monitoring Network (herein referred as AQMN). An objective such as compliance with ambient air quality standards for instance, is yet another important objective of the AQMN. It is then quite logical that the AQMN must be optimized for multiple objectives. Little work has been done multiple objectives in the AQMN design. But the recent literature shows a growing interest for attempting this problem. Naito and Ochiai (1981) and Munn (1981) consider that the incorporation of multi-objectives is extremely important to lay the foundation of the future practices towards AQMN optimization. Smith and Egan (1979) suggest that optimization should be undertaken for each objective. Munn (1981) suggests an overlay of these designs since there could be a subset of these designs which could probably meet the interest of more than one objective. Environmental Monitoring and Assessment 5 (1985) 21-38. 9 1985 by D. Reidel Publishing Company.
0167-6369/85.15.
22
P. M, M O D A K
A N D B. N. L O H A N I
Once a multi-objective framework is decided, it may be rather tempting to recommend several sub-configurations for each objective. In such instances however, the number of monitors is normally expected to increase, and this could be particularly true when the objectives imposed are in conflict with each other. Such situations would only be ideal, and particularly not feasible in the case of developing countries where funds for monitoring activities are severely limited. It is quite evident therefore that when the resources on monitoring are limited or constrained, it may not be possible to maximize the interests of each objective. While it may not be logical to neglect the role of some of the objectives altogether, due to limitations on resources available. The multi-objective methodologies are essentially a compromise philosophy to the AQMN design. The interest of the Decision Maker (herein refered as DM) is then to investigate how far the selected AQMN configuration could do justice to more than one objective - all under a common economic constraint. In this paper, two different approaches have been proposed for the multi-objective optimization of the AQMN. These methodologies are essentially an extension of the basic Minimum Spanning Tree (MST) algorithm developed in Part I of the paper (Modak and Lohani, 1984a). The first method makes use of the Utility Function (herein referred as UF), (Cohon and Marks, 1975) and the second methods makes use of a Sequential Interactive Compromise, commonly referred as the 'An Interactive Method of Progressive Articulation of Information' (Hwang etal., 1980). The first method will be hereafter referred to as the U-approach and the later as the S-approach. In this study, for the purpose of demonstration, two objectives have been considered as of interest; namely representation of spatial-temporal patterns and the detection of violations of ambient air quality standards. In addition, estimation of population dosage product has been treated as an auxiliary monitoring objective. Since pattern score (Modak and Lohani, 1984a), has been identified as the decision variable for representing air quality patterns, it is necessary that a decision variable be identified for representing the objective of compliance. 2. Detection of Violations over Ambient Standards
The potential of a monitoring site for the detection of violations is considered herein in terms of violation scores. A location having a high violation score is then considered to have a high potential for detection of violations. The computation of violation scores is essentially a weighted scoring of the concentrations above the prescribed thresholds. These scores are therefore dependent on, (1) The threshold levels. (2) The weighing factors between each threshold range, or the weighing function. A decision on thresholds and weighing factors (i.e. severity) is indeed pollutantspecific and further dependent upon the averaging time and populations concerned. Several weighing functions have been reported such as linear functions, segmented linear functions, non-linear function, segmented non-linear functions etc. (Ott, 1978; Jain et aL, 1977). In this case, a segmented nonlinear weighing function is proposed as,
23
O P T I M I Z A T I O N OF AMBI ENT AIR QUALITY M O N I T O R I N G NETWORKS
T N[ = Z i=1
N~ Z
(v~ + I - v~)(xi - X k ) X / ( x k + I - xk)
(1)
k=l
where vk = the weighing factor corresponding to threshold x~, x k = the k-th threshold, X =0if(x i-xk) 0. The U-approach need not be restricted to consideration of pattern and violations but could be extended to the question of placing monitors in populated areas of probable concern. In this case, the U F could be expressed as, VV = Np x (N L, x K) b
(6)
where K is population activity around the location. It is to be noted that a product of K and N o yields the Population Dosage Product (herein referred to as PDP).
5. Basic Mechanism in the Utility Approach Table I shows the effect of U F on the selection of the locations in terms of a decision process. It is clear from Table I that the U F shifts the monitor configuration towards TABLE I Effect of the structure of utility function on the selection of the monitoring locations Location
~
~
UF b=0
UF b=l
UF b=2
A B C
4 9 6
10 5 3
4 9 6
40 45 18
400 225 54
B..C..A
B..A..C
A..B..C
Decision
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
25
meeting the associated objective. In Table I for instance, on the imposition of the utility, station A was preferred to station B, despite its low pattern score. The violation scores were subsequently improved, i.e., from 5 to 10, by shifting the decision. The utility approach could therefore be considered as a method of weighted MST. Since the termination criterion of the M S T algorithm is to ensure total coverage, such a shift in the priority is expected to lead to an increase in the number of monitors, i.e., the cost of the A Q M N design. It is not necessary that the number of monitors in the utility case could always be higher than obtained due to the sole consideration of patterns. It is possible that on the imposition of utility, the number of monitors in the A Q M N remains the same but only the configuration is altered. Such situations are expected for lower values of C O and especially when the pattern scores and the violation scores are not in conflict with each other. This elasticity of the solution is ideal since in such instances, the cost of the A Q M N would remain the same while some improvement is possible in meeting the associated objective. As the magnitude of coefficient b is increased, the solution is no more elastic and the optimal number of monitors starts increasing, in addition to relocation within the configuration. After a certain value of b is reached, the solution offers no more changes or alterations. This is because all of the locations which help meet the associated objective have been selected. In the example case illustrated in Table I, the decision on A as the best location is unaltered after b = 2 for any positive value of b. It should be noted that improvement in the associated objective (in this case the compliance) may not be necessarily that of a monotone increasing nature with that of b. In other words, though the solution obtained for b > 0 is more effective than that of b = 0, this effectiveness does not necessarily increase as the value ofb is increased. This behaviour is due to the use of the utility function U F as well as due to the sequential policy of selecting optimal locations.
6. Formulation and Computational Algorithm for th~ Utility Approach Normally, the problem of A Q M N optimization is financially constrained, and therefore it is not possible to maximize the interest of both objectives. For a given budget, it is then necessary to identify the optimum value of C O and that of b which would lead to a possible maximization of program objectives. This compromise can be identified by increasing the coefficient b at a fixed C Otill the cost of the system increases beyond the prescribed budget. Once the budgetary constraints are violated, the value of C Ocould be lowered, since the cost of the system has direct relationship with C O (refer to Figure 4 in Modak and Lohani, 1984a). Once again, the sequence of increasing the level of b could be reinitiated. A procedure of compromising over C ~ would continue till a minimum prescribed level of C Ois reached. Since the effectiveness of the associated objective does not have a monotone increasing relationship, all intermediate solutions need to be printed out with summary statistics
P. M. MODAKAND B. N. LOHANI
26
TABLE 11 A comparision between the Utility and the Sequential Interactive approaches No.
Utility approach
Sequential Interactive approach
1.
Normally applicable up to two joint objectives
Can consider up to 7 objectives at a time
2.
Necessary to formulate a Utility Function, difficultto accomodate interests of heterogeneous objectives
No need to formulate a Utility Function, heterogenious objectives could be considered
3.
No heuristics is involved once the Utility Function is specified
Heuristics is the idea
4.
For a specified Utility Function, an optimum could be identified
Optimum may not be achieved
5.
Could be extended to multiple pollutants
Restricted only for a single pollutant
at the end of the calculations. M o d a k (1984) should be referred for computational details. T h e optimization problem over the utility could therefore be stated as (in set notations), M a x ( M 1 w m 2 t,.) . . . w M m )
(9)
while, Max
~
(10)
NIl.
i=1
Such that (M 1 u M e w ... w M m) = (M l w Me...
w M u)
(11)
and m = m~
(12)
where, M i = the set of Np monitors correlated with location i above C ~ rn = the n u m b e r of m o n i t o r s ; m ~ : the allowable n u m b e r of m o n i t o r s ; N = the n u m b e r of candidate locations. The c o m p u t a t i o n a l algorithm for the U-approach is quite similar to that of the original M S T algorithm presented in M o d a k and Lohani (1984a). I n this case however, instead of the pattern scores alone, violation scores are also considered in the form of a multiplicative utility function. The attribute for the selection o f t h e best and the next best m o n i t o r s is thus o n a utility basis rather than solely on the consideration of the pattern scores.
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
27
7. Application of U-Approach to the Taipei City Case As an illustration of the utility approach, the example of Taipei City (refer to Modak and Lohani, 1984a) is now reconsidered. The concentrations of sulfur dioxide in Taipei City are fairly low, and violations of the standards of 0.10 p p m (monthly averaged hourly concentrations) are almost nonexistent. For the purpose of illustration therefore, the concentrations of sulfur dioxide were scaled up by a factor of 5, to justify the objective of compliance. It is to be noted that a constant multiplier such as 5 does not change the correlation matrix, so the analysis is not affected as far as the patterns are concerned. It is rather difficult to prescribe a weighing function in this example, since data on the hazardous effects of monthly averaged hourly concentrations has seldom been reported in the literature. The normal practice has been to consider hourly or daily averages in the development of the weighing functions (Ott, 1978). For the purpose of illustration therefore, a non-linear weighing function of the form such as in Table III is TABLE III Non-linear segmented function for computing violation scores for sulphur dioxide Concentration SO2 in ppm
Score
0.50 < 1.00 > 1.00 < 2.00 > 2.00
0.5 1.5 3.0 5.0
considered. This weighing function bears a close relationship to the one reported by Jain
et aL (1977) for assessing the impacts due to annually averaged daily sulfur dioxide concentrations. More appropriate weighing functions could be considered later when relevant effects data are available. For the purpose of illustration, a budgetary constraint of 800000 US$ was considered. Maximum and minimum level of C O was considered as 0.95 and 0.85, respectively. Table IV shows a summary of the results. Costs are calculated in the way described in Modak and Lohani (1984a). Percentage Exposure has been defined as a ratio of the P D P detected by the A Q M N to that of the total P D P calculated on summing over all the candidate locations. Percentage Compliance denotes a ratio of the violation score detected by the A Q M N to that of the total violation score calculated by summing over the violation scores at all candidate locations. Percentage Variance is the square of the population correlation coefficient C O. At the outset, the results of Table IV show that consideration of multiple objectives is rather important in the A Q M N design. In this case, the solutions based on the sole consideration of patterns are observed to render a poor performance as far as compliance is concerned.
28
P.M. MODAK AND B. N. LOHAN1 TABLE
IV
A Summary of multi-objective designs based on utility approach No.
Cost 1000 US$
Percentage exposure
Percentage compliance
Percentage variance
b
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
914.8 711.5 914.8 813.2 711.5 711.5 813.2 609.9 609.9 609.9
38.8 27.1 34.8 27.8 27.0 28.2 38.1 17.7 23.8 24.3
33.6 27.8 32.9 30.2 27.8 29.9 34.5 23.4 24.0 25.4
90.2 88.4 88.4 86.5 84.6 84.6 84.6 82.8 82.8 82.8
0.0 0.0 2.0 0.0 0.0 2.0 4.0 0.0 2.0 4.0
24 23 22 21 2C 19 18 17 16 15
t~ =3 II 9 8
2 3 4 5 6 7 8 9 I0 II 121314 1516171819 KILOMETERS
I 2 3 4 ,5 6 7 8 9 I0 II 121314 15 16 17 1819 KILOMETERS
Note : Numbers indicate the order of station selection
Solution Number 4 Objective
Solution Number 7
Effectiveness
Objective
Compliance
2Z 80 %
Exposure
30. 20 %
Compliance Exposure
38. I0 ~ 34.50 ~
Variance
86.50 %
Variance
84.60 %
Fig. 1.
Effectiveness
Configurations of optimal A Q M N designs based on Utility Approach.
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORINGNETWORKS
29
It is observed that as coefficient b in the utility function is progressively increased, accomplishment of the associated objective is also enhanced. As discussed earlier, this improvement is not necessarily of a monotone increasing nature. It is to be noticed that in some cases, the costs of AQMN remain unaltered. Solution numbers 5 and 6 in Table III could be considered as examples of this situation. Such a possibility has already been discussed. When the costs of the AQMN cross beyond a prespecified budgetory constraint, (in this case, 800000 US$) the level of C O is lowered as a compromise. It appears that the objectives of compliance and estimation of PDP are commensurate with each other. It could be seen that as the coefficient b is increased in case I, the estimation of the PDP is also improved along with that of compfiance. A commensurate relationship is rather evident because most of the contribution to the sulfur dioxide in Taipei city is primarily due to transportation sources, i.e., mobile pollution, sprawled in the densely populated areas. Figure 1 shows the configuration of the AQMN corresponding to solution numbers 4 and 7 of Table III. Table V (extracted from Table IV) shows how the utility approach
TABLE V A Summary of the results for utility approach - extracted from Table IV No.
Cost 1000 US$
Percentage exposure
Percentage compliance
Percentage variance
b
4. 7.
813.2 813.2
27.8 38.1
30.2 34.5
86.5 84.6
0.0 4.0
optimizes the possibility of meeting the associated objectives (compliance and estimation of the PDP) in the same budget by carrying out a compromise over the reliability of pattern representation.
8. The Sequential Interactive (S) Approach In certain situations, it may not be possible to identify the form of the UF. A choice of the structure of the utility function is sometimes difficult when several heterogeneous objectives are to be considered. This may be particularly true when professional judgements also come into play. In such instances, the S-approach could be considered as a potential method for applications. In this method, the priority on objectives is successively shifted, with the help of the Decision Maker (DM), until an acceptable compromise is reached. This procedure is thus highly amenable to interactive versions. The S-approach is a style of compromise programming and bears a close relationship with the Interactive Decision Evaluation Aid (IDEA) as discussed by Zeleny (1982).
30
P. M . M O D A K
A N D B. N . L O H A N I
9. Computational Algorithm for Sequential Interactive Approach The various steps of this algorithm are as follows. (1) Specify: (a) The maximum allowable monitoring budget for the AQMN. If the desired coverage effectiveness is not specified, then the design would be reported for the total coverage. (b) The minimum and maximum acceptable level o f C ~ i.e., the minimum and maximum reliability of the representation of the spatial-temporal pattern. (2) Arrange the grid points in descending order as per the scores of violations. Alternatively or in addition, a sorting of sites according to the population dosage product could be considered. In interactive methods, it has been reported that the DM could apply it to 7 objectives of interest and therefore a simultanious consideration of compliance and estimation of population dosage product should not be a problem. A policy of ranking the locations in relation to the associated objectives assists the DM in the decision-making process. (3) For the maximum value of C o, identify the locations of monitors based on the M S T algorithm. (4) If the cost of the design in step 3 exceeds the budgetary constraint prescribed in step 1 (a), then the D M is asked to lower the C o value. This step is evident due to the relationship between C o and m. If the budgetary constraint is not violated, then move to step 5. Otherwise the DM is asked to lower the C o value and move to step 4 again. (5) It is possible that locations with a high number of violations and/or that are needed to estimate P D P are not included in the array of monitors selected by the coverage criterion of step 4. The DM is now asked whether he intends to include any station missed in the final design. The S-approach would respond to the DM's request of inclusion, for one station at a time. If constraint on the budget specified in 1 (a) is not violated, then the DM is questioned again whether any additional station is to be included in the design. A ranking of the associated objectives provided on the screen enables the D M to select such potential monitors. (6) Once the constraint as in 1 (a) is violated, then the program would once again request a further reduction in the C ~ Step 4 is executed once again, to get the new (possibly reduced) number and location of the monitors, and the procedure is iterated till the monitors indicated by the DM, based on the other objectives are accommodated within the budgetary constraint. It should be noted however that the monitors introduced by the DM are now used to initialize the status matrix (refer Modak and Lohani, 1984a), before any selection of additional monitors is made. (7) Steps 4, 5, and 6 are now iterated till the value of C ~ becomes less than the minimum level, as specified earlier in step 1 (b). At each stage of iteration, the locations of interest of the associated objective are introduced by compromising on C o. Once the program is terminated, all the solutions with the DM's response are printed out which could be consulted later to pick up the best A Q M N design. It is to be noticed that step 7 in the algorithm adds the monitors to the A Q M N configuration. Step 8 on the other hand reduces the total number of monitors while retaining those suggested in step 7. At each addition the DM meets his objectives better
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
31
(other than that of the pattern representation) and at each such addition - deletion process his emphasis on the objective of pattern representation is reduced. In fact such a shift in emphasis is envisaged to amplify his other objectives of interest - as in the case of ambient air quality compliance and of estimating the population dosage product. This process of addition-deletion moves sequentially at a series of DM's responses till a situation such at as step 9 is arrived as an acceptable compromise. In the S-approach therefore, the D M has a good opportunity to learn and familiarize himself with the various implications of the A Q M N design.
10. Application of S-Approach to Taipei City The interactive methodology as described in the previous section has been programmed as ' S U S E R ' (for details refer M o d a l 1984). For the purpose of illustration, the data used for U-Approach: i.e., Taipei City sulfur dioxide AQMN, was once again considered. The salient results of the ' S U S E R ' output are extracted in the Appendix in the form of the first three interaction screens. In this case, similar to the U-approach, a budgetary constraint of 800 00 US$ was imposed and a maximum and minimum level of C o was considered as 0.95 and 0.85 respectively. Table VI shows a summary of the solutions. T A B L E VI A S u m m a r y of multi-objective designs based on sequential interactive approach Step
Step Cost 1000 US$
Percentage compliance
Percentage exposure
Percentage variance
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
914.8 711.5 711.5 711.5 813.2 813.2 813.2 711.5 813.2 711.5 813.2 813.2 813.2 711.5 813.2
33.6 27.8 29.1 28.1 33.8 33.7 33.7 28.8 34.0 34.8 32.6 32.6 32.6 27.7 32.0
33.8 23.5 23.9 24.9 32.6 32.6 32.6 23.9 31.1 36.3 35.7 35.7 35.7 34.4 40.4
90.25 88.36 88.36 88.36 88.36 86.49 86.49 82.81 82.81 81.00 79.21 77.54 75.69 73.96 73.96
Table VI presents solution number 9, where C Ois reduced to 0.91 (variance 82.36 Yo) at a benefit of increasing the interests of the other associated objectives. Solution number 15 reduces C O even further (C O = 0.86 or variance 7 3 . 6 ~ ) as a solution of maximum allowable compromise. This solution provides maximum attainment to the objective of estimating the PDP. An examination of solutions numbers 9 and 15
32
P. M. MODAK AND B. N. LOHANI 2A
-)4 ->3 _:22 21 _:?C 19 18 17 16 15 14
I I I I I
12 II I0 9 8 7
B
2 3 4 5 6 7 8 9 10 II 1213141516171819 KILOMETERS
r 2 3 4 5 6 7 8 9 I011121314 1516171819 KILOMETERS
Note : Numbers indicate the order of station selection
Solution Number 9
Objective
Fig. 2.
Solution Number 15
Effectiveness
Objective
Compliance
34. O0 ~
Compliance
32.00 %
Exposure
31. IO ~
Variance
82.81 ~
Exposu re Variance
40.40 ~ 73.56 ~
Effectiveness
Configurations of optimal A Q M N designs based on Sequential Interactive Approach.
however indicates that solution number 9 could be considered as that of a good compromise from overall considerations. Figure 2 shows the configuration of the AQMN obtained for solution 9 and solution 15.
11. A Comparision between the Sequential Interactive and the Utility Function Approaches The U-approach developed in this paper is applicable for joint consideration of pattern representation and detection of violations or with that of the estimation of the PDP. Normally, the utility approach is applicable for not more than two objectives at a time: the primary objective in every case being that of the pattern representation. When the number of objectives increases, several runs of palrwise combinations of the problem are required: and in this case the set of non-dominated solutions could become quite
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
33
large. The S-approach on the other hand can handle all kinds of heterogeneous objectives with which the DM is familiar, all at the same time. This could be considered as an important advantage of the S-approach over the utility method. In the S-approach, the optimal monitoring stations are selected with an explicit consideration to the objectives. The DM therefore knows why a location is selected. This is indeed very important in the operation and the organization of the monitoring network. The procedure for selection of sites in other words, is entirely objective specific. One of the advantages of the S-approach is its flexibility. In many cases, a part of the AQMN is already in operation, or the DM is interested in retaining a few of the existing monitors. In the case of the S-approach, these monitors could be readily specified by user interaction. In the case of the U-approach, though such an inclusion is possible, it could only be off-line. In the case of the U-approach, the DM has absolutely no control in station selection, once the structure of the Utility Function (UF) is specified. However, a shift towards the associated objective through a progressive increase in b, assures an identification of the best compromise. The S-approach though convenient, depends on the response of the DM. The procedure is therefore rather heuristic, and there is a good possibility that the optimum is missed. Table VII shows a summary of comparision between the U and the S approaches.
T A B L E VII A comparision between the optimal designs obtained by the S and the U approach Method
Cost 1000 US$
Percentage compliance
Percentage exposure
Percentage variance
S S U
813.2 813.2 813.2
34.0 32.0 38.1
31.1 40.4 34.5
82.8 73.9 84.6
It is difficult to comment on which of the two methods is effective. Due to the differences in the philosophy of design, the solutions of S and the U approach are expected to be different for the same level of budgetary constraint. Table VII shows a comparision between the solutions obtained by the S and the U approach for Taipei City. In this case, it appears that the U approach is more effective than the S approach. The attractiveness of the utility method is expected to increase when multiple pollutants are under consideration. In this case, the interactive methods would have to provide multiple screens to the DM and the DM would have to respond specifically for each pollutant. Decision-making in such situations is expected to be rather difficult. The S-approach could therefore be considered to be applicable for only the case of single pollutant AQMN. A case of multiple pollutants under multiple objectives is discussed in Part III (Modak and Lohani, 1984c) of this paper.
34
P . M . MODAK AND B. N. LOHANI
12. Source-Interpretation in Multi-Objective Design It is clear that the role of the multi-objective philosophy is to shift the decision sequence of the MST algorithm (pattern solution) in favour of the associated objectives. It is important to note that a shift from the pattem solution in this approach depends upon the relationships between the pattern scores Np and the associated objective (such as violation scores Nv). An investigation of such relationships is important for the interpretation of the multi-objective design. If Np and Nv for instance, are in conflict with each other, then the number of monitors required would be significantly more than those required for the sole consideration of pattern. On the other hand, if Np and N~ have a commensurate relationships, then the number of monitors in the case of multi-objective design may not be significantly different. A case of conflict between the pattern and the violation scores could be envisaged as a situation having a large number of point sources scattered in the region. A case of commensurate relationships is the situation where a significant amount of pollution is contributed by area and mobile souces sprawled over the region. A design of the AQMN with multiple objectives has therefore a direct bearing on the source distribution in the urban areas. Conversely, it could also be said that a multi-objective approach to AQMN design could serve as a cursory tool for regional source dignosis. A cluster analysis between the pattern and the violation scores could be a potential method for further investigations. These deductions, in the case of the example of Taipei City, Taiwan, were found to be quite true. The results showed a commensurate relationship between the pattern scores N~, and that of the estimating the PDP. These relationships therefore imply that the sources of the mobile and population-related activity play a dominant role in the overall air quality. Further, it could be inferred that the populated regions of concern are concentrated in the busiest areas of traffic or mobile pollution. The land-use map of Taipei City (Taipei City, Summary of Statistics, 1981) confirms such deductions.
13. Summary and Conclusions (1) A Minimum Spanning Tree (MST) algorithm developed by Modak and Lohani (1984a) is extended to consider multiple objectives in the optimum AQMN design. (2) This extension is possible via two approaches, namely one based on the utility function and another based on the principles of sequential interactive compromise. The sequential approach is heuristic but perhaps best suited to consider several objectives at a time, and particularly when professional judgements are also involved. The utility function approach which may be restricted to two objectives at a time, is however better suited when a number of pollutants are to be considered. (3) One of the important outcomes of any multi-objective study is the experience and the insight the DM could gain about the system in total. A preliminary source diagnosis could be made based on examination of the relationship between objectives - i.e.
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
35
whether these are in friendly competition or in conflict. A multi-objective optimization of the A Q M N has thus several other useful implications besides mere optimization of network density a n d configuration. Optimization is then only a beginning for seeking policies in search of effective air quality m a n a g e m e n t .
Acknowledgements The various c o n t r i b u t i o n s m a d e in this paper are basically a part of the doctoral research carried out by the first author while at A s i a n Institute of Technology, Bangkok, Thailand. All c o r r e s p o n d e n c e regarding this paper should be therefore addressed to the first author. T h e support of the G o v e r n m e n t of J a p a n is gratefully acknowledged for providing financial assistance to complete the above doctoral research.
References Cohon, J. L. and Marks, D. H.: 1975, 'A Review and Evaluation of Multi-objective Programming Techniques', Water Resources Research II, 208-220. Godfrey, S. M., Novak, J. H., and Turner, D. B.: 1975, 'A Modelling Study to Examine Detection of Violationsof Air Quality Standards Using Various Sampling Station Networks in the Vicinityof a Single Plant', Report 76-23.3, Pittsburgh; Air Pollution Control Association. Hwang, C. L., Paidy, S. R., Yoon, K., and Masud, A. S. M.: 1980,'Mathematical Programmingwith Multiple Objectives: A Tutorial', Computers and Operations Research 7, 5-31. Jain, R. K., Urban, L. V., and Stacey, G. S.: 1977,Environmental ImpactAnalysis:A New Dimension in Decision Making, Van Nostrand Reihold, New York, U.S.A., Appendix B, pp. 178-209. Modak, P. M.: 1984, 'Optimum Siting of Ambient Air Monitors', A Dissertation submitted as the partial fulfillmentfor a degree of Doctor of Engineering, Asian Institute of Technology, Bangkok, Thailand. Modak, P. M. and Lohani, B. N.: 1984a, 'Optimization of Ambient Air Quality Monitoring Networks Part I', Environmental Monitoring and Assessment 5, 1-19. Modak, P. M. and Lohani, B. N.: 1984c, 'Optimization of Ambient Air Quality Monitoring Networks Part III', Environmental Monitoring and Assessment 5, 39-53. Munn, R. E.: 1981,Design of Air Quality Monitoring Networks, Macmillan Press Ltd, Basingstoke, Hampshire, U.K. Naito, M. and Ochiai, M.: 1981,'On Optimal Allocation of Air Monitoring Stations', International Seminar on Air Quality Management and Related Energy Policies, Tokyo, Japan, pp. 247-265. Ott, W. R.: 1978, Environmental Indices: Theory and Practice, Ann Arbour Science. Smith, D. G. and Egan, B. A.: 1979, "Design of Monitoring Networks to Meet Multiple Criteria', Journal of Air Pollution Control Association 29, 710-714. Taipei City Summary of Statistics: 1981, Taipei, Taiwan. World Health Organization (WHO): 1977, 'Air Monitoring Programme Design for Urban and Industrial Areas', Global Environmental Monitoring System, WHO Offset Publication No. 38. Zeleny, M.: 1982, Multiple Criteria Decision Mala'ng, McGraw Hill, New York.
36
P. M. MODAK AND B. N. LOHANI
Appendix
SCREEN
NUMBER
1
COST OF THE A Q M N IS.. IN 1000 US$ A N D CMAX IS 800.00 C U R R E N T C U T - O F F IS ... NO OF M O N I T O R S
914.86
0.95
9
PATTERN MST
COMPLIANCE
PDP
10 31 2 28 32 11 18 21 8
11.17 6.42 15.56 6.53 1.28 7.20 14.74 12.59 10.76
55.84 6.42 46.68 6.53 1.28 21.61 88.42 12.59 75.33
TOTAL C O M P L I A N C E POTENTIAL ,.. TOTAL PDP POTENTIAL ... VIOL RANKS ... 1 15.75 2 4 13.27 19
15.56 13.08
PDP RANKS ... 18 88.4 8 10 55.8 13
75.3 52.9
314.69
18 9
5 2
86.24
14.74 12.74
72.4 46.7
6 19
5 21
14.49 12.59
71.5 39.2
4
R E P O R T I N G THE STATUS OF THE COVERAGE 10 17.00 31 22.00 2 25.00 28 28.00 II 31.00 18 32.00 21 33.00 8 34,00
6
14.31
66.4
32
30.00
I N D I C A T E YOUR CHOICE OF R E D U C I N G CRX . . .
0.01
THIS STEP IS ESSENTIALLY THAT OF C O M P R O M I S I N G ON C o TO MEET THE PRESCRIBED B U D G E T O R Y CONSTRAINTS
OPTIMIZATION OF AMBIENT AIR QUALITY MONITORING NETWORKS
SCREEN
NUMBER
37
2
711.55
COST OF THE A Q M N IS.. IN 1000 US$ A N D CMAX IS 800.00 C U R R E N T C U T - O F F IS ... NO OF M O N I T O R S
0.94
7
PATTERN MST
COMPLIANCE
PDP
18 15 3 28 32 4 21
14.74 10.65 12.47 6.53 1.28 13.27 12.59
88.42 31.96 12.47 6.53 1.28 66.37 12.59
TOTAL C O M P L I A N C E POTENTIAL ... TOTAL PDP POTENTIAL ... VIOL RANKS ... 1 15.75 2 4 13.27 19
15.56 13.08
PDP RANKS ,., 18 88.4 8 10 55.8 13
75.3 52.9
219.62
18
5
71.53
14.74
72.4
5
6
14.49
71.5
4
6
14.31
66.4
R E P O R T I N G THE STATUS OF THE COVERAGE 18 19.00 15 24.00 3 27.00 28 30.00 4 33.00 21 34.00
32
32.00
SO YOU W A N T TO INTERACT ?... TYPE YES OR NO yes
? 1
L O C A T I O N 1 IS F O R C E D TO ENTER THE MST SOLUTION BY E X A M I N I N G ITS PERF O R M A N C E WITH RESPECT TO THE O T H E R OBJECTIVES OF INTEREST
38
P. M. M O D A K A N D B. N. LOHANI
SCREEN
NUMBER
3
COST OF THE A Q M N IS.. IN 1000 US$ A N D CMAX IS 800.00 C U R R E N T CUT-OFF IS ... NO OF M O N I T O R S
711.55
0.94
7
PATTERN MST
COMPLIANCE
PDP
1 18 15 28 32 4 21
15.75 14.74 10.65 6.35 1.28 13.27 12.59
15.75 88.42 31.96 6.53 1.28 66.37 12.59
TOTAL COMPLIANCE POTENTIAL ... TOTAL PDP POTENTIAL ... VIOL RANKS ... l 15.75 2 4 13.27 19 18 10
88.4 55.8
8 13
15.56 13.08 75.3 52.9
74.80
222.89
18
5
14.74
72.4
6
5
14.49
71.5
4
R E P O R T I N G T H E STATUS OF THE COVERAGE 18 22.00 15 27.00 28 30.00 32 32.00 21 34.00
6
14.31
66.4
4
33.00
N O T E THAT LOCATION 1 BECOMES A PART OF THE MST SOLUTION DUE TO THE D E C I S I O N T A K E N ON SCREEN 2. I N T E R A C T I O N C O N T I N U E S ...
