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Multiobjective mathematical programming models for acid rain control
European Journal of Operational Research 35 (1988) 365-377 North-Holland
365
Theory and Methodology
Multiobjective mathematical programming models for acid rain control J. H u g h E L L I S
Department of Geography and Environmental Engineering, The Johns Hopkins University, Baltimore, AID 21218, U.S.A.
Abstract: Although acid rain control is inherently multiobjective, previous optimization approaches have generally been single-objective, often acting to minimize aggregate abatement cost or emission reductions. Using an updated, least-cost deposition-constrained deterministic model as a basic framework, three multiobjective models are developed that consist of formulations which permit deviations about target deposition levels, the addition of constraints to effect measures of equity and models to enforce restrictions on aggregate emission reduction tonnage. The deposition deviation model shows that large abatement cost savings can be realized if the hard upper bound on maximum allowable deposition limit is preferentially relaxed. The socalled equity model develops strategies that attempt to balance within each state and province, the disparity between fractional emission and fractional deposition reductions. The aggregate emission reduction model shows some of the effects associated with the imposition of a common type of acid rain proposal. Our intent is to demonstrate that the incorporation of multiobjectivity into mathematical programming models for optimizing acid rain control constitutes an important step toward the identification of more representative, more useful and hopefully, scientifically and politically acceptable abatement strategies. Keywords: Multiple criteria programming, optimization, acid rain, environment
Introduction In the short time span of just several years, the desire to control acid rain has seen the development of numerous system analytic methodologies which serve as tools for policy creation and analysis. The approaches themselves are quite diverse in structure and can be categorized in differing, often overlapping ways. They include models which embody optimization procedures, simulation procedures and often, a coupling of both. Moreover, certain approaches focus upon the important ef-
Received April 1987; revised June 1987
fects and interrelationships of coal supply/price equilibria, whereas others stress the linkage between emissions and resulting acidic deposition. Some of these approaches are reviewed briefly below, following a rough chronological order. Following the classification of Atkinson and Lewis (1974), Fortin and McBean (1983) develop an ambient least cost linear programming (LP) model that is deposition constrained. It is applied to a case study comprised of four sources and four receptors. They also address transfer coefficient stochasticity in the context of complete dependency. Streets, Hanson and Carter (1984) document a simulation based approach that has elements of optimization through the ranking of marginal costs
0377-2217/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
366
J.H. Ellis / Multiobjective MP models for acid rain control
of S O 2 removal. Ellis, McBean and Farquhar (1985a) use a linear programming model to identify the least-cost set of controllable source SO 2 removal levels such that maximum wet sulfate deposition rates are not exceeded at prescribed receptor locations. It links emissions to resulting wet sulfate deposition through the use of a long range transport/transformation simulation model, and thus falls into the class of hybrid optimization/simulation approaches. A linear programming framework is also used by Morrison and Rubin (1985) but with the important difference that the emission-deposition linkage is not considered. Their approach more closely fits the 'emission-least-cost' categorization of Atkinson and Lewis (1974). It acts to minimize the total levelized cost of delivered coal plus the cost of abatement equipment subject to satisfying energy demands and sulfur emission restrictions for a 31state region. Similar comments apply to Silverman (1985) in which linear programming is used in a large-scale model of electric utility plant emissions. Young and Shaw (1986) and Shaw (1986) also develop models which can be classified as hybrid, in their use of simulation results in an optimization setting, although they do not use a formal optimization algorithm. Except for the case of a single receptor, their methods do not yield global optima. They conclude that most of the emission reductions should occur in the Ohio River Valley, northern Appalachia, the lower Great Lakes region and the St. Lawrence Valley. Similar results are reported in Ellis (1984). Hordijk (1986) applies Young and Shaw's optimization method to a European setting. Ellis et al. (1985b, 1986) extend the deterministic hybrid approach to incorporate stochasticity of pollutant transport/transformation into the optimization framework. Other stochastic approaches include Fronza and Melli (1984), Guldmann (1986) and Fuessle et al. (1987). Ellis (1987) provides a broad overview of the role of optimization models for acid rain control that brings together and sets in some perspective, the important issues of stochasticity, multiobjectivity and coal supply/price equilibria in the development of acid rain abatement strategies. Certainly all of the aspects of acid rain control that the foregoing studies address are potentially important. Criticisms of any and all of the approaches can (and have) been made, yet the underlying rationale for their development and use
remains steadfastly valid. They share a commonality in their efforts to provide insight into what may be described as some of the fundamentally difficult questions to answer in acid rain control: who pays?; how much?; what level of deposition reduction is likely to occur through specified measures of acid rain control?; and, are reduction estimates reliable? These questions, posed as policy goals, represent some of the conflicting, often noncommensurate objectives of acid rain control. This then, naturally leads to multiobjective formulations for it is precisely these types of problems that the full power of multiobjective programming can be fully brought to bear. We propose that extending the existing single objective approaches to multiobjective forms constitutes a significant advance, both for problem-specific reasons as will be developed in this paper, and for the more general reasons put forth by Cohon (1978). Essentially, Cohon argues that multiobjective programming and planning accomplishes three improvements over single objective approaches, those being, (1) possibilities for establishing more appropriate roles for the participants in the planning and decision-making process, (2) identification of a wider range of alternatives, (3) more realistic problem perception. The usefulness of a multiobjective approach will be demonstrated through the development and execution of three models, each addressing a specific policy objective not found in a typical single-objective (e.g. cost minimization) abatement strategy. These modelling extensions are developed from a deterministic linear programming framework, which, as will be described shortly, has been enlarged to include a configuration of some 138 receptors.
Basic L P framework
The basic cost-minimizing, deposition-constrained LP model is described elsewhere (e.g. see Ellis et al., 1985a, b), hence it is briefly shown here as 235
Minimize
z= ~ j=l
CjRj,
(1)
J.H. Ellis / Multiobjectioe MP models for acid rain control 235
subject to
367
92 CE/(1
- Rj)tij +
j=l
~_~ N E ~ t i k k=l
+ BG i ~< D i (i = 1, 2 . . . . . 20), Rj>/0
(j=1,2
. . . . ,235),
(2) (3)
,t
,I
where cj
= cost-removal function for source j,
= SO 2 removal level (these are the decision variables), C E / = existing emission rate (KT(SO 2)/yr), lij = normalized transfer coefficient ((kg(wet SO4)/ha-yr)/kT(SO2) emitted), for controlled source j and receptor i, N E k = noncontrolled SO 2 emission rate (kT(SO 2 ) / yr), tik =normalized transfer coefficient ((kg(wet SO4)/ha-yr)/kT(SO2) emitted), for noncontrolled source k and receptor i, BGi = b a c k g r o u n d (i.e. nonanthropogenic) wet sulfate deposition rate for receptor i, D t = maximum allowable wet sulfate deposition rate for receptor i.
Rj
Selected characteristics of the model are worthy of mention, for example, the output of the model is a set of 235 optimal values for the R / t h a t minimize (1) subject to the stated constraints ((2) and (3)). In each of the twenty constraints (2), all terms except CEjRjt~j contain only scalars and therefore can be aggregated to the right-hand-sides of the constraint inequalities. In this formulation, there will always be at least one constraint (i.e. receptor) that has a resulting deposition limit equal to the prescribed maximum, D~. Thus, the model will choose values for the R/ as high as the constraints (2) will permit, up to the point where the lefthand-side of (2) equals D i for one or more constraints. Deposition levels for the remaining constraints will be less than D~.
Updated model configuration The model's original configuration of 20 receptors was found to be inadequate, thus we created an enhanced receptor configuration along with the additional transfer coefficients thereby required. The new configuration has a total of 138 receptors (i.e. in (2), i = 1, 2 . . . . . 138) and was motivated by
\ RECEPTOR LOCATIONS
Figure 1. Enhanced receptor configuration the desire to reflect more accurately the breadth and diversity of areas potentially afflicted by acid rain, thus permitting more representative multiobjective analyses. We also attempted to include at least one receptor in every political jurisdiction that could be affected by the imposition of acid rain control measures. The enhanced configuration is shown in Figure 1.
Effects of receptor placement Before proceeding to the multiobjective analyses it is important to see generally, the effects of altering receptor placement. As can be demonstrated by a simple set of analyses, the alteration of receptor configuration has profound ramifications for the results. Figure 2 shows a tradeoff of aggregate abatement cost versus prescribed deposition limit (/9,.) for the two configurations involving 20, then 138 receptors. Translation of the curves to the right for the 138 receptor case clearly shows that instead of the previous value of 18, the minimum attainable deposition limit becomes 24.9 kg (wet SO4)/ha-yr. (The limit of feasibility which yields the minimum attainable deposition level is reached when R j = Rj(Max) for all j = 1, 2 . . . . . 235). A review of deposition isopleths for the 'base case' (i.e. R j = 0, for all j ) shows that the original configuration of twenty receptors completely misses some important locations of high predicted deposition. We note however, that
J.H. Elfis / Multiobjective MP models for acid rain control
368
8C
~ " 70 ¸ 60
I'
= ~ " ~ - -
feasibility limits J
I ~ \
.S. \
20 receptors 138 receptors
\
so.
\ \
00 40 1-
\
\
30
\
¢~ 20'
Can."
"~
18
2'1
\
"i\ ~an.
...
24
27
3'0
~
36
3'9
4"2
4'5
WET SULFATE DEPOSITION LIMIT
Figure 2. Cost-deposition tradeoffs: Original and enhanced receptor configurations
buffering concerns notwithstanding, it is erroneous to presume that receptors with high predicted deposition automatically become the most critical constraints within the LP model (where critical is loosely defined as those locations at which it is most difficult to satisfy the prescribed maxim u m allowable deposition rates; such constraints 'drive' the LP model). This can be demonstrated with a series of constraint relaxation analyses. The relaxation involves running the model with the full complement of 138 receptors, and noting at which receptor (i.e. constraint) a change in D i will result in the greatest change in W.CjRj. Call this constraint index r 1. Next, solve the model again but with only 137 receptors, represented by constraints i = 1, 2 . . . . . 138; i :# r 1. Subsequently, a new most critical constraint (r2) is then identifled, and the entire process repeated. The result, given D i = 24.9, for all i, is shown in Figure 3. The Stilwell Lake, N Y receptor is the location at which it is most difficult to meet the deposition standard, Di, yet it is not the location with the highest predicted base case deposition rate. That distinction goes to the Deep Creek Lake, M D receptor (base case deposition rate = 55.7 kg(wet SO4)/ha-yr) which ranked ninth in the relaxation analyses. The base case deposition rate predicted for Stilwell Lake is 46.2 kg(wet SO4)/ha-yr ). Thus, Figure 3 shows the interesting result that removing the Stilwell Lake receptor
from the model causes an aggregate abatement cost decrease of some 30 billion dollars or about 30%. Furthermore, if the Wooster and Purdue Agricultural F a r m receptors are relaxed, cost drops
~FULL
70
COMPLEMENT OF 138 RECEPTORS
I/
6c
/ /
~" O ~ so "~
, deposition limit = 24.9 kg wet S04(ha-yr)1 ° receptors relaxed in decreasing order of sensitivity
/ /STILWELL LAKE, N.~
~\
/WOOSTER, O.IO \
~4o
\ \
o E ~ 30
/ /PURDUE AG. FARM, INDIANA
jBROOKNAVEN, N.Y. ~B~ 1 X~
PRINCETON, N.J.
~l~f
PIEDMONT STATION, N.C.
"~I~PENN STATE, PA.
2o
FINLEY, N.C.-- / / DEEP CREEK LAKE, MD.--
~
~1~ "11-- -II- --II-
o o
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
no. of 'relaxed' constraints
Figure 3. Constraint relaxation results
J.H. Ellis / Multiobjective MP models for acid rain control
369
Table 1 Deposition results from the relaxation analyses Receptor
Base case deposition a
Stilwell Lake, NY Wooster, OH Purdue Ag. Farm, IN Brookhaven, N.Y. Princeton, NJ Piedmont Stn., NC Penn State, PA Finley, NC Deep Creek Lake, MD
46.2 40.6 35.9 45.0 42.6 33.8 48.6 31.3 55.7
Relaxed deposition rates b (1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
32.5
30.5 25.7
28.8 29.3 28.9
34.5 29.3 28.4 32.3
35.1 29.2 29.3 35.1 27.4
35.4 29.3 29.7 35.0 26.8 27.3
37.2 30.3 29.9 36.3 28.7 27.3 27.5
37.1 30.4 30.0 36.2 28.3 28.7 27.5 26.4
37.0 30.7 30.0 36.1 28.1 28.7 27.4 26.4 25.3
a Deposition units = kg(wet SO4)/ha-yr. b Nominal deposition standard = 24.9. Relaxation order: (1) Stilwell Lake, (2) Stilwell Lake and Wooster, (3) Stilwell Lake, Wooster and Purdue Ag. Farm, etc.
another 20 billion dollars. It must be remembered of course, that when a constraint is relaxed, its resulting after-removal deposition rate from the LP model will be higher than the prescribed maximum. Rates for the nine receptors shown in Figure 3 are provided in Table 1. Note that the LP-derived deposition rates for relaxed receptors need not be substantially larger than the prescribed values of D r Furthermore, the relaxed rates vary as a function of which subset of contraints is relaxed. The need to reduce deposition at other constraints necessarily causes substantial reductions in deposition rates at the relaxed receptors. The only situation where this m a y not occur is if the vast majority of deposition at a critical receptor originates from a highly localized source that does not contribute significantly to deposition at other receptors in the system. Given a westto-east advective flow in the simulation model, this situation is more likely for receptors nearest the east coast or for relatively isolated source-receptor pairs with, respectively, west-east positioning. These observations serve to point up several aspects of the complex and contentious role that receptor placement plays in developing an abatement strategy, and with this new updated modelling framework, we now proceed to a description of the multiobjective analyses.
Deposition deviation model It is clear that the vector of m a x i m u m allowable deposition limits (D~) plays a critically im-
portant role in determining the characteristics of control strategies. Unfortunately, we simply do not know with certainty what the safe, allowable limits should be for protecting the environment from acidification damages. It is reasonable to presume however, that they should exhibit strong site-specificity and perhaps temporal-specificity as well. F r o m a scientific viewpoint for example, limits appropriate for aquatic habitat protection are probably very different from those specific to material degradation. F r o m a policy determination viewpoint, these observations serve to greatly complicate the modelling tasks, especially when one considers that the characteristics of the LP solutions are, to a great extent, driven by the requirement to meet such limits for a small subset of sensitive receptors in the model. We draw some of these important issues together by positing a model based upon a straightforward observation: if the ' h a r d ' upper bounds on m a x i m u m allowable deposition could be preferentially relaxed for certain receptors, then it follows that potentially large cost savings could result. The first multiobjective model was therefore motivated by the desire to replace the hard upper (and currently uncertain) bound on allowable deposition, D i, with a somewhat softer restriction. It permits violation deviations, equivalent to resulting deposition rates in excess of Di, in addition to the over-achievement deviations implicit in the previous model (equations 2). Our interest is in identifying the abatement cost savings thereby achieved. Recall that so-called over-achievement deviations occur frequently in the previous model and correspond to the conditions
J.H, Ellis / MultiobjeetiveMP modelsfor acid rain control
370
4( ..,,~~.~,__'DEVIATION'MODEL-equalweights
Z
_o J
\ \
"30
(comparison with base case)
\
t~ z
25 _o ,~
\
v
¢5 '35
Z
_o
(~20
20 1-
15 ,~ 10 m .J
ffl
0
18
21
24
WET
27 30 33 36 SULFATEDEPOSITION LIMIT
39
42
45
Figure 4. Deviation model results 235
92
E C E j ( 1 - R ~ )lij -~- E N E k t i k + B G i < D i j=l k=l
(4)
where R 7 denotes optimal solution values. Noting the strict inequalities, the over-achievement deviations, denoted V~, are simply 235
92
V/= D i - E CEj(1 - R7 )tij -- E NEgtik - BOi. j=l
k=l
(5) In certain cases, V~>/0 even when R~ = 0, for all j, thus indicating that the deposition standard for constraint i is already met at the existing emission levels of all sources. The resulting multiobjective deviation model can be written: 235
Minimize
where U~= decision variable corresponding to violation deviation, and W,u, W,v = objective function weights. By certain basic principles of linear programruing (e.g. see Hadley, 1963, p. 168), U~ and V~ cannot be simultaneously positive in an optimal solution, hence for each constraint i, U~ will be positive signifying a violation of the nominal deposition standard D i, or V/will be positive signifying a resulting deposition rate less than Di, or both U~ and V~ will be zero, indicating a deposition rate exactly equal to D i. The second objective function acts to minimize the weighted sum of these deviations. In the 'weighting method' (Cohon, 1978) the weights serve a dual purpose in that they permit the generation of tradeoffs between objective one (cost minimization) and objective two, and as well, enable the assignation of constraint-specific deviation tradeoffs.
zl "+ Z2 = E CjRj j=l 138
Application of deviation model
+ ~ (W/uUi + W/vv/),
(6)
i=1 235
subject to
92
E CEj(1 - Rj)tij + ~_, NEktik j=l
k=l
+ B G i - D i - U~+ V / = 0
Rj f> 0 V/>i 0
Vi,
Vj,
(7) (8)
Vi,
(9)
As would be expected, cost differences between the deviation approach and the previous 'hard standard' approach increase dramatically as nominal deposition limit (in fact, now a target) is decreased. As is shown in Figure 4, this is especially true for the lower values of D i. Note that cost comparisons with the base case are not possible below the feasibility limit of the original model (i.e. 24.9 kg(wet SO4)/ha-yr ). Unlike the basic
371
J.H. Ellis / Multiobjective MP models for acid rain control
a p p r o a c h however, the d e v i a t i o n m o d e l can never b e c o m e infeasible. O n c e F . C j R j reaches its maxim u m t h r o u g h the process of progressively decreasing D i, objective n u m b e r one of the d e v i a t i o n m o d e l stays c o n s t a n t a n d the sum of aggregate d e v i a t i o n s ( a n d n u m b e r of violations) increases. This explains w h y d e p o s i t i o n violations are shown for n o m i n a l limits b e l o w 24.9 in F i g u r e 4. T h e difference in total costs shown in F i g u r e 4 p e r t a i n s solely to the U.S. Of special n o t e is the result that U.S. a b a t e m e n t costs (not r e d u c t i o n s ) are equal to total ( i . e . U . S . plus C a n a d a ) costs, b e c a u s e the i n t r o d u c t i o n of d e p o s i t i o n d e v i a t i o n s e l i m i n a t e s C a n a d i a n SO 2 r e m o v a l costs. W i t h o u t deviations, the need to meet d e p o s i t i o n limits at selected U.S. r e c e p t o r s results in large C a n a d i a n a b a t e m e n t costs. W h e n selective violations are p e r m i t t e d , we see that the t r a d e o f f b e t w e e n aggregate cost a n d aggregate d e p o s i t i o n d e v i a t i o n s favors e l i m i n a t i n g C a n a d i a n source removals. This of course, results in higher d e p o s i t i o n rates for C a n a d i a n receptors that receive p r e d o m i n a n t l y C a n a d i a n emissions, b u t n o t to the p o i n t of exc e e d i n g the target limits (i.e. n o d e p o s i t i o n violations occur at C a n a d i a n receptors). These characteristics are s t r o n g l y d e p e n d e n t u p o n the weights chosen, Wi u a n d W, v which, as n o t e d earlier, were i d e n t i c a l for all receptors. If, for e x a m p l e , the weights for certain C a n a d i a n r e c e p t o r s were subs t a n t i a l l y increased, then C a n a d i a n SO 2 a b a t e m e n t w o u l d be forced, especially for those receptors at which d e p o s i t i o n originates p r e d o m i n a n t l y f r o m C a n a d i a n sources. T h e a b a t e m e n t cost red u c t i o n s d o however exact a p e n a l t y in increasing n u m b e r s o f n o m i n a l s t a n d a r d violations as D i decreases, as shown in F i g u r e 4. W h e t h e r the p o t e n t i a l a b a t e m e n t cost r e d u c t i o n s are d e s i r a b l e is clearly a f u n c t i o n of w h e t h e r violations at selected r e c e p t o r l o c a t i o n s (assigned t h r o u g h the weights W, u, Wi v) are p e r m i s s i b l e a n d furthermore, w h e t h e r the m a g n i t u d e s of the a n t i c i p a t e d violations are acceptable. I n o t h e r words, F i g u r e 4 tells o n l y a p a r t of the story. E q u a l l y as i m p o r t a n t are the a s s o c i a t e d e n v i r o n m e n t a l (i.e. d e p o s i t i o n a l ) disbenefits themselves, as d e s c r i b e d in T a b l e 2. W e see for example, the interesting result that at n o m i n a l Di = 24.0, 21 of the 138 limits are exc e e d e d (as o b t a i n e d f r o m F i g u r e 4) such that the m e a n exceedence is 2.5 kg(wet S O 4 ) / h a - y r ; 11 of those 21 receptors violate the s t a n d a r d b y 1.8 kg(wet S O 4 ) / h a - y r or less, a n d finally, the m a x i -
Table 2 Statistics of deposition deviations Nominal/ Statistics of deposition deviations (exceedences) deposition Maximum Median Mean Variance Coefficient limit a of variation 45.0 42.0 39.0 36.0 33.0 30.0 27.0 24.0 b 21.0 b 18.0 b
8.7(19) 11.7(28) 14.1(36) 10.8(30) 12.0(36) 6.9(23) 7.5(28) 7.8(33) 7.8(37) 9.0(50)
2.6(6) 5.6(13) 4.5(12) 3.9(11) 3.2(10) 2.3(8) 2.1(8) 1.8(8) 1.8(9) 1.2(7)
3.5(8) 13.8 6.5(15) 13.8 5.7(15) 19.4 4.9(14) 10.9 4.1(12) 10.9 2.9(10) 4.7 2.9(11) 4.9 2.5(10) 6.1 2.7(13) 7.0 2.6(14) 6.8
1.1 0.6 0.8 0.7 0.8 0.7 0.8 1.0 1.0 1.0
all deposition units = kg(wet(SO4)/ha-yr.) (variance units = (dep. units)2). b infeasible for the original (i.e. non-deviation) model. (.) statistics expressed as a percentage of the nominal limit.
m u m v i o l a t i o n is 7.8 kg(wet S O 4 ) / h a - y r . It occurs at the P u r d u e Ag. F a r m r e c e p t o r in I n d i a n a . T h e l o c a t i o n s of the m a x i m u m violations p r o vide useful i n f o r m a t i o n as well. In using ten different values for D i, the m a x i m u m violations occur at o n l y 4 different locations. T h u s we begin to see p r o n o u n c e d consistency in the results, as these f o u r r e c e p t o r l o c a t i o n s shown in T a b l e 3 regularly seem to p l a y i m p o r t a n t roles. This leads to some p o t e n t i a l l y i m p o r t a n t observations. G i v e n the focal c o n c e r n for these receptors, it w o u l d a p p e a r p r u d e n t to pose, at least, two kinds of questions: (1) are the transfer coefficients for these receptors a c c e p t a b l y representative? (both in a b s o l u t e Table 3 Locations of maximum violations Nominal deposition limit ~
Locations of maximum violations
45.0 42.0 39.0 36.0 33.0 30.0 27.0 24.0 21.0 18.0
Deep Creek Lake, MD (53.7) Deep Creek Lake, MD (53.7) Deep Creek Lake, MD (53.1) Penn State, PA (46.8) Penn State, PA (45.0) Penn State, PA (36.9) Stilwell Lake, NY (34.5) Purdue Ag. Farm, IN (31.8) Purdue Ag. Farm, IN; Stilwell Lake, NY (28.8) Purdue Ag. Farm, IN; Stilwell Lake, NY (27.0)
a deposition units = kg(wet SO4)/ha-yr. (-) maximum deposition rate (nominal limit + violation).
372
J.H. Ellis / Multiobjective M P models for acid rain control
terms and relative to other transfer coefficients in the model); and, (2) if so, can we afford to meet, at these receptors (and perhaps others like them) deposition standards as low as can be achieved for the majority of other receptors in the system? The desire to answer the first question constitutes a strong motivation for explicitly representing pollutant transport and transformation processes as stochastic phenomena in optimization models, as is the case in Fortin and McBean (1983), Ellis (1984), Fronza and Melli (1984), Ellis et al. (1985b, 1986), Guldmann (1986) and Fuessle et al. (1987). Implicit in the second question is one of the extremely sensitive tradeoffs associated with acid rain control, i.e., must we, to some extent, sacrifice environmental quality in certain regions so as to obtain affordable abatement strategies overall? Notwithstanding the ever-present externality issue, this clearly is a difficult argument to sell, for among many objections, it purports to sacrifice precisely those regions that may be least able to assimilate higher wet sulfate loadings. Furthermore, it can (and has) been argued that affordability is a highly ambiguous metric, given our modelling context of global cost minimization. There is merit in the argument that global (i.e. system-wide) costs are meaningless for it is solely the distribution of cost that is important. These issues, at their most fundamental level, involve difficult questions of equity which leads to the following multiobjective model that addresses one such aspect of equity.
Emission and deposition reduction deviations However defined, equity considerations will play a dominant role in designing an abatement strategy which has any realistic chance of political acceptance. Modelling methodologies which do not explicitly attempt, in some sense, to embody equity have severely impaired utility. Complicating the issue of course is the confounding, multiobjective nature of equity, in all its varied and diverse manifestations. We describe here, a simple and admittedly crude representation of one aspect of equity that establishes as metrics, the fractional emission reductions and fractional deposition reductions defined over every state and province in the system. A description of the multiobjective
model and its underlying motivation requires two definitions. Within each state and province, -1
J~Js
" " J~Js
/ -1
Fd=
E
CEjRjtij
~ iE1s j = l
E CEjtij i
,
j=l
(11) where F ~ - - f r a c t i o n a l emission reduction for s t a t e / province S, •Is --subset of controllable sources j in s t a t e / province S, Fd --fractional deposition reduction for s t a t e / province S, I s = subset of receptor locations i in state/province S. The important difference between (10) and (11) is that whereas F~ is determined by emission reductions wholly within each state or province, Fsd is determined by emission reductions at sources both within state/province S and all other sources as well. It might be argued then, that an abatement strategy which results in F~ considerably larger than Fd for state/province S is imposing an inequitable or, at least, impolitic situation: state/province S is, proportionately, removing much more SO2 emission than it is receiving in return, in terms of in-state deposition reductions. Of course, the arguments become much more complicated when one realizes that a relatively large value of F~ benefits downwind receptors in other states, and so on. Furthermore, on an annual average basis, easternmost sources generally do not contribute significantly to deposition at receptors in more western state/provinces hence the easternmost states/provinces inescapably have
Fff > F~. A convenient way to pull some of these observations together involves the use of the following model:
Minimize
zl
+ Z2 = E C j R j
J + ~-'.(W~Us+ W~Vs) S
(12)
J.H. Ellis / Multiobjective MP models for acid rain control
subject to
ECEj(1
-
Rj)tij +
ENE~t,k
+ BG, <~Dg Vi,
(13) -1
-lz z-j..j} i~Is j
z
iEls j
- Us + Vs = 0
vs,
(14)
Rj >10
v j,
(15)
Us , Vs>lO
VS,
(16)
Like the previous model, the second objective (z2) serves to minimize the sum of deviations, with those deviations now corresponding to positive (Us) or negative (Vs) differences between F~ and Fd. The global (but unattainable) minimum of z 2 equal to zero corresponds to the case wherein the fractional emission and deposition fractions are exactly equal for every state and province.
Application of the equity model The costs (in terms of ~CjRj) of attaining a balance of F~ and Fff are shown in Figure 5. As
373
expected, lower values of D i result in larger cost differences between the basic and 'equity' solutions. It is therefore more costly to achieve our version of equity for more stringent maximum allowable deposition rates. This continues only up to a point however. Once below Di = 27 kg(wet SO4)/ha-yr, cost differences drop dramatically. The reason for this behavior is that when approaching the limit of feasibility, the need to satisfy the 'hard' deposition constraints (13) completely dominates the solution. There no longer exists room to move in the sense of removal allocations which will minimize zl, satisfy (13), as well as minimize z 2 (equations (14) are always feasible). If constraints (13) were replaced with their deviation counterparts (7), and a third objective added in the form of the second summation in (6), then the aforementioned dramatic decreases would not occur. This would also permit the generation of useful tradeoffs between abatement cost, deposition deviations and deviations from equity as embodied in F~ and Fd. We note as well that the emission source and receptor subsets Js and I s need not be state or province-specific. Rather, S could represent a national delineation between the U.S. and Canada, or any categorization: politically-motivated, electric utility pool-motivated, or otherwise. Aside from cost, other aspects of the equity formulation are potentially informative. Specifi-
cost comparison -'equity' vs. base case-
20( 18(]
-10
16(] 7
140 12(]
U.S.
lO(]
Can.-----
.5
4
80 604020-
li3
g"
2~
2~, 27 30 33 36 WET SULFATE DEPOSITION LIMIT F i g u r e 5. E q u i t y m o d e l results
39
42
45
m
374
J.H. Ellis / Multiobjective MP models for acid rain control
Table 4 Equity deviation results State/province Arkansas a Georgia Illinois Indiana Kansas a Louisiana Maine a Maryland Michigan Minnesota Missouri Nebraska New Jersey New Mexico a New York North Carolina North Dakota a Ohio Oklahoma ~ Pennsylvania South Dakota a Tennessee Texas Vermont a Virginia West Virginia Manitoba Nova Scotia Ontario Quebec Saskatchewan a
Deposition limit (kg(wet SO4)/ha-yr) 45.0
42.0
39.0
36.0
33.0
30.0
27.0
24.9
V
V
V
V
V
V V
V
V
V V V
V V V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V V V
V V
V V
V V
V V
V V
V
V
V
V
V
V
V
V
V
V
V
V
V
V U
V U
V U
V U V U
V U V U
V U V U
V V V V V
V
V V
V V
V V
V
V
V
V
V
V
V
V
V
V
V V
U V
U V
U V
U V
U V
V
V
V
V
V
V
V
V
V
V
V
V
V
V V V U V V V V V
U
V
V V V U
a Mandatory 'V' entries for all deposition limits (no controllable sources) V -= fractional emission reduction < fractional deposition reduction; U -= fractional emission reduction > fractional deposition reduction.
cally, we c a n e x a m i n e at w h i c h states o r p r o v i n c e s is F~ < F~d, o r F~ > F~a, o r lastly, F• = F~d. T h i s i n f o r m a t i o n is p r o v i d e d in T a b l e 4. R e c a l l t h a t w h e n F~ < Fsa, Us = 0 a n d Vs > 0. S i m i l a r l y w h e n F~ > Fsa, t h e n Vs = 0 a n d Us :~ O. N o t c o u n t i n g states a n d p r o v i n c e s w i t h o u t c o n t r o l l a b l e s o u r c e s (i.e. Us m u s t e q u a l zero), t h e m a j o r i t y o f e n t r i e s s h o w t h a t F~ < Fsd t h u s i n d i c a t i n g t h a t f r a c t i o n a l deposition reductions exceed fractional emission r e d u c t i o n s . T h i s w a s u n e x p e c t e d for s o m e o f t h e states a n d p r o v i n c e s l i s t e d in T a b l e 4. G i v e n r e l a t i v e l y large e x i s t i n g e m i s s i o n levels a n d ' u p w i n d ' l o c a t i o n s ( t h e a n n u a l a v e r a g e a d v e c t i v e w i n d di-
r e c t i o n for e a s t e r n N o r t h A m e r i c a is w e s t to east), o n e m i g h t e x p e c t I l l i n o i s a n d I n d i a n a to h a v e Fff > Fsd f o r t h e l o w e r v a l u e s o f D i, b u t s u c h is n o t t h e case. W e see Fff = F d w h e n D, is 33 k g ( w e t S O 4 ) / h a - y r o r b e l o w . F o r the l a r g e r v a l u e s o f D~, a s s i g n m e n t o f Vs > 0 f o r I l l i n o i s a n d I n d i a n a i n d i cates that their prescribed emission reductions substantially benefit instate receptors. Additional s u p p o r t f o r this o b s e r v a t i o n is p r o v i d e d in T a b l e 5 w h i c h shows, for e a c h level o f D i, t h o s e r e c e p t o r s w h i c h w e r e b i n d i n g w i t h r e s p e c t to d e p o s i t i o n . W h e n b i n d i n g r e c e p t o r l o c a t i o n s are n o t in I n d i a n a , e m i s s i o n r e d u c t i o n s are d r i v e n , in part, b y t h e
J.H. Ellis / Multiobjective MP mode&for acid rain control
375
Table 5 Binding receptor locations for the equity model Receptor
Deposition limit (kg(wet SO 4)/ha-yr) 45.0
Purdue Ag. Farm, IN Rockport, IN Deep Creek Lake, M D Brookhaven, NY Stilwell, NY Wooster, OH Penn State, PA
X
X
42.0
39.0
36.0
33.0
30.0 X X X
X
X
X X X X X X
X X
X
X X X X
X
X X X
X X
27.0
24.9
X
X
X ~- binding receptor.
need to meet the deposition limit at other, out-ofstate receptors. This necessarily results in large in-state deposition reductions to the extent that F~ < Fsd for Illinois and Indiana. Pennsylvania, West Virginia and to a lesser extent, Ohio exhibit expected behavior in that F~ > Fsd. For Pennsylvania and Ohio however, this applies only until D i decreases to the limit of feasibility (24.9). At that limit we then see F~ < Fsd. The switch from Us to Vs is due to relative increases of in-state deposition reductions for Pennsylvania and Ohio that, for the most part, arise from out-of-state emission reductions. Before D~ ---24.9 is reached (e.g. at D i = 27.0) emission reductions for Pennsylvania are already at their respective upper bounds. Those for Ohio are very close to their bounds. Thus, further deposition reductions generally originate elsewhere. Indeed as described above, at the limit of feasibility with respect to deposition, the equity constraints become irrelevant and all source removal levels are set equal to their respective maximums, albeit at very high marginal costs.
Aggregate reduction targets The final multiobjective model uses the basic formulation and an extension based upon the principles of the constraint method of multiobjective programming (see generally, Cohon and Marks, 1973; Cohon, 1978). Appended to the basic model (equations (1)-(3)) is the constraint Y'~ CEjRj = G
(17)
jeJ
where G represents a user-specified aggregate
emission reduction and J a specified subset of sources, in this case, U.S. sources. In effect therefore, three objectives are considered. The purpose of this formulation is to provide a means of analyzing the ramifications and sensitivities of some of the typical acid rain control bills which have been proposed in the U.S. congress in recent years. Such bills prescribe an aggregate emission tonnage reduction, hence the motivation for constraint 17. While these approaches may be politically acceptable, only minimal consideration is given to the environmental improvements they purport to effect, the blanket assumption being that emission reductions will result in proportionate deposition reductions. For five levels of G: 4, 6, 8, 10 and 12 million ton SO 2 reductions, Figure 6 shows the tradeoffs between abatement cost and D i, Where cost remains more-or-less constant over a range of Di, indicates that the solution is being driven by constraint 17. In these plateau-like regions then, the aggregate reduction level G, and hence aggregate cost exceeds what is necessary to meet deposition limits /9,.. The presence of intersecting tradeoff curves reveals yet another informative characteristic of the solutions, and generally, the way in which the model works. For example, at about D~ = 27 kg(wet SO 4)/ha-yr, the 8 million ton and 10 million ton aggregate reduction tradeoff curves intersect, with the associated abatement cost approximately equal to 40 billion dollars. In both solutions, constraint (17) must be satisfied and several deposition constraints (equations (2)) are binding. With respect to total, aggregate cost per ton of sulfur removed, the 8 million ton reduction solution prescribes a relatively inefficient set of removal levels as compared to the 10 million ton
J.H. Ellis / Multiobjective MP models for acid rain control
376
\ 12 M \.
'Aggregate U.S. Emission R e d u c t i o n '
60
\
~-~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
o~ X, 8M \, \
=t
20
r
o
24
r 25'.5
l 27
MAXIMUM
r 28-5
l 30
31.5
3~3
34.5
36
37.5
ALLOWABLE DEPOSITION
Figure 6. Specified emission reduction model results
case, which achieves an additional 2 million tons of emission reduction for the same cost. From a deposition viewpoint however, the 10 million ton reduction solution appears inefficient, in that it satisfies a maximum deposition limitation that could be satisfied with an aggregate emission reduction that is 2 million tons less. This simple comparison serves only to demonstrate certain aspects of the model's behaviour. Obviously, many other questions can be brought to bear, based upon more realistic bases for comparison. For example, there is a certain incompatibility between models which act to prescribe aggregate emission reductions and those which are strictly deposition-constrained. In fact, the former are generally motivated by the desire to avoid the latter. The desire to avoid deposition-constrained approaches is, more precisely, the desire to avoid the use of transfer coefficients and their attendant uncertainties. Thus, a more conventional comparison might involve the 8 million ton reduction solution and its minimum feasib|e deposition limit (i.e. about 27 kg(wet SO4)/ha-yr, as shown in Figure 6) and the 10 million ton reduction and its corresponding minimum limit (25.8). Yet another modelling possibility combines the foregoing model with deviation versions of the deposition constraints (i.e. equations (7)). It is also possible to construct a deviation version of (17) so as to investigate the potential savings associated
with variations about G. It is simply Z CEjRj - Vc + Vc = O.
08)
jET_J
Another objective function component would be constructed in the form of Minimize
W~ U6+ wV vG
(19)
where W~ and Wv are prescribed weights, as before. This might be especially informative for the case where G -- 12 million tons, given that this aggregate reduction level also roughly corresponds to the reduction associated with the minimum feasible deposition limit (24.9) of the basic deposition-constrained model.
Closure
We hopefully have shown that the use of a relatively simple mathematical tool can provide insight into some important aspects of developing acid rain abatement strategies. Certain of these insights are of course, easily anticipated, such as the profound effects of receptor placement on strategy characteristics, similarly, the substantial cost reductions possible through the introduction of allowing deposition limit violations at selected receptors. Other results are much less intuitive,
J.H. Ellis / Multiobjective M P models for acid rain control
such as the d o m i n a n t role p l a y e d b y a subset of four receptors in the system. A s well, the a t t e m p t to b a l a n c e f r a c t i o n a l emission a n d d e p o s i t i o n red u c t i o n s y i e l d e d interesting a n d u n e x p e c t e d shifts in r e m o v a l efforts as a f u n c t i o n of m a x i m u m p r e s c r i b e d d e p o s i t i o n limit. L a s t l y we saw, f r o m a d e p o s i t i o n - c o n s t r a i n e d perspective, the p o t e n t i a l inefficiencies a s s o c i a t e d with strategies b a s e d u p o n aggregate emission r e d u c t i o n stipulations. W e close with the r e a f f i r m a t i o n t h a t the m o d e l s a n d m e t h o d o l o g y d e s c r i b e d herein c a n n o t m a k e policy. T h e y exist to serve as tools for the p o l i c y analyst. T h e i r a d v a n t a g e s lie p r i m a r i l y in the a b i l i t y to pull together, in a c o m p l i c a t e d , highlyi n t e r r e l a t e d setting, d i s p a r a t e a n d o f t e n diverse elements of a system. It is clear that a c i d rain c o n t r o l is precisely such a system. S y s t e m a n a l y t i c a p p r o a c h e s a n d especially m u l t i o b j e c t i v e app r o a c h e s can p r o v e to b e that n e e d e d b r i d g e between, say, h a r d science a n d the d e c i s i o n - m a k i n g process. A t a m i n i m u m , such a p p r o a c h e s s h o u l d possess the a t t r i b u t e s of multiobjectivity, stochasticity of p o l l u t a n t flux processes, the effects of coal m a r k e t / p r i c e e q u i l i b r i a - - a l l w r a p p e d in an interactive p a c k a g e suitable for use b y the decis i o n - m a k i n g c o m m u n i t y itself. R e s e a r c h c u r r e n t l y moves f o r w a r d t o w a r d that goal.
Acknowledgements Sincere t h a n k s to J e r r y C o h o n , Ben H o b b s , N a n c y Kete, C h u c k ReVelle a n d the reviewers for their n u m e r o u s helpful c o m m e n t s a n d criticisms. T h a n k s also to Ms. K e t e a n d F e l i x S e e b a c h e r for p e r f o r m i n g m a n y of the c o m p u t e r r u n s a n d to L a n e E n g l a n d for e x p e r t l y w o r d p r o c e s s i n g the m a n u s c r i p t . T h e s u p p o r t of the N a t i o n a l Science F o u n d a t i o n u n d e r g r a n t N o . ECE-8504582 is gratefully a c k n o w l e d g e d . This research was cond u c t e d using the C o r n e l l N a t i o n a l S u p e r c o m p u t e r Facility, a resource of the C e n t e r for T h e o r y a n d S i m u l a t i o n in Science a n d E n g i n e e r i n g at C o r n e l l University, which is f u n d e d in p a r t b y the N a tional Science F o u n d a t i o n , N e w Y o r k State a n d the I B M C o r p o r a t i o n .
377
References Atkinson, S.E. and Lewis, H.D. (1974), "A cost-effective analysis of alternative air quality control strategies", J. Environ. Econ. and Manage. 1, 237-250. Cohon, J.L. (1978), Multiobjective Programming and Planning, Academic Press, New York. Ellis, J.H. (1984), "Chance-constrained/stochastic linear programming model for development of acid rain abatement strategies", Ph.D. thesis, University of Waterloo, Waterloo, Ontario, Canada. Ellis, J.H., McBean, E.A. and Farquhar, G.J. (1985a), "Deterministic linear programming model for acid rain abatement," J. Env. Eng. Div. Amer. Soc. Civil Eng. 111,119-139. Ellis, J.H., E.A. McBean and Farquhar, G.J. (1985b), "Chance-constrained/stochastic linear programming model for acid rain abatement--I. Complete colinearity and noncolinearity", Atmos. Environ. 19, 925-937. Ellis, J.H., McBean, E.A. and Farquhar, G.J. (1986), "Chance-constrained/stochastic linear programming model for acid rain abatement--II. Limited colinearity", Atmos. Environ. 20, 501-511. Ellis, J.H. (1987), "Optimization models for development of acid rain abatement strategies", Civil Engineering Systems 4, 58-66. Fortin, M. and McBean, E.A. (1983), "A management model for acid rain abatement", Atmos. Environ. 17, 2331-2336. Fronza, G. and Melli, P. (1984), "Assignment of emission abatement levels by stochastic programming", Atmos, Environ. 18, 531-535. Fuessle, R.W., Brill, E.D. and Liebman, J.C. (1987), "Air quality planning: A general chance-constraint model," J. Env. Eng. Div. Amer. Soc. Civil Eng. 113, 106-123. Guldmann, J.-M. (1986), "Interactions between weather stochasticity and the locations of pollution sources and receptors in air quality planning: A chance-constrained approach", Geog. Anal. 18, 198-214. Hadley, G. (1963), Linear Programming, Addison-Wesley, Reading, MA. Hordijk, L. (1986), "Towards a targetted emission reduction in Europe", Atmos. Environ. 20, 2053-2058. Morrison, M.B. and Rubin, E.S. (1985), "A linear programruing model for acid rain policy analysis", J. Air Poll. Contr. Assoc. 35, 1137-1148. Shaw, R.W. (1986), "A proposed strategy for reducing sulphate deposition in North America-II. Methodology for minimizing costs", Atmos. Environ. 20, 201-206. Silverman, B.G., "Heuristics in an air pollution control cost model: The 'Aircost' model of the electric utility industry", Manage. Sci. 31, 1030-1052. Streets, D.G., Hanson, D.A. and Carter, L.D. (1984), "Targeted strategies for control of acidic deposition", J. Air Poll. Contr. Assoc. 34, 1187-1197. Young, J.W.S. and Shaw, R.W. (1986), "A proposed strategy for reducing sulphate deposition in North America - I. Methodology for minimizing sulfur removal", Atmos. Environ. 20, 189-199.
365
Theory and Methodology
Multiobjective mathematical programming models for acid rain control J. H u g h E L L I S
Department of Geography and Environmental Engineering, The Johns Hopkins University, Baltimore, AID 21218, U.S.A.
Abstract: Although acid rain control is inherently multiobjective, previous optimization approaches have generally been single-objective, often acting to minimize aggregate abatement cost or emission reductions. Using an updated, least-cost deposition-constrained deterministic model as a basic framework, three multiobjective models are developed that consist of formulations which permit deviations about target deposition levels, the addition of constraints to effect measures of equity and models to enforce restrictions on aggregate emission reduction tonnage. The deposition deviation model shows that large abatement cost savings can be realized if the hard upper bound on maximum allowable deposition limit is preferentially relaxed. The socalled equity model develops strategies that attempt to balance within each state and province, the disparity between fractional emission and fractional deposition reductions. The aggregate emission reduction model shows some of the effects associated with the imposition of a common type of acid rain proposal. Our intent is to demonstrate that the incorporation of multiobjectivity into mathematical programming models for optimizing acid rain control constitutes an important step toward the identification of more representative, more useful and hopefully, scientifically and politically acceptable abatement strategies. Keywords: Multiple criteria programming, optimization, acid rain, environment
Introduction In the short time span of just several years, the desire to control acid rain has seen the development of numerous system analytic methodologies which serve as tools for policy creation and analysis. The approaches themselves are quite diverse in structure and can be categorized in differing, often overlapping ways. They include models which embody optimization procedures, simulation procedures and often, a coupling of both. Moreover, certain approaches focus upon the important ef-
Received April 1987; revised June 1987
fects and interrelationships of coal supply/price equilibria, whereas others stress the linkage between emissions and resulting acidic deposition. Some of these approaches are reviewed briefly below, following a rough chronological order. Following the classification of Atkinson and Lewis (1974), Fortin and McBean (1983) develop an ambient least cost linear programming (LP) model that is deposition constrained. It is applied to a case study comprised of four sources and four receptors. They also address transfer coefficient stochasticity in the context of complete dependency. Streets, Hanson and Carter (1984) document a simulation based approach that has elements of optimization through the ranking of marginal costs
0377-2217/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
366
J.H. Ellis / Multiobjective MP models for acid rain control
of S O 2 removal. Ellis, McBean and Farquhar (1985a) use a linear programming model to identify the least-cost set of controllable source SO 2 removal levels such that maximum wet sulfate deposition rates are not exceeded at prescribed receptor locations. It links emissions to resulting wet sulfate deposition through the use of a long range transport/transformation simulation model, and thus falls into the class of hybrid optimization/simulation approaches. A linear programming framework is also used by Morrison and Rubin (1985) but with the important difference that the emission-deposition linkage is not considered. Their approach more closely fits the 'emission-least-cost' categorization of Atkinson and Lewis (1974). It acts to minimize the total levelized cost of delivered coal plus the cost of abatement equipment subject to satisfying energy demands and sulfur emission restrictions for a 31state region. Similar comments apply to Silverman (1985) in which linear programming is used in a large-scale model of electric utility plant emissions. Young and Shaw (1986) and Shaw (1986) also develop models which can be classified as hybrid, in their use of simulation results in an optimization setting, although they do not use a formal optimization algorithm. Except for the case of a single receptor, their methods do not yield global optima. They conclude that most of the emission reductions should occur in the Ohio River Valley, northern Appalachia, the lower Great Lakes region and the St. Lawrence Valley. Similar results are reported in Ellis (1984). Hordijk (1986) applies Young and Shaw's optimization method to a European setting. Ellis et al. (1985b, 1986) extend the deterministic hybrid approach to incorporate stochasticity of pollutant transport/transformation into the optimization framework. Other stochastic approaches include Fronza and Melli (1984), Guldmann (1986) and Fuessle et al. (1987). Ellis (1987) provides a broad overview of the role of optimization models for acid rain control that brings together and sets in some perspective, the important issues of stochasticity, multiobjectivity and coal supply/price equilibria in the development of acid rain abatement strategies. Certainly all of the aspects of acid rain control that the foregoing studies address are potentially important. Criticisms of any and all of the approaches can (and have) been made, yet the underlying rationale for their development and use
remains steadfastly valid. They share a commonality in their efforts to provide insight into what may be described as some of the fundamentally difficult questions to answer in acid rain control: who pays?; how much?; what level of deposition reduction is likely to occur through specified measures of acid rain control?; and, are reduction estimates reliable? These questions, posed as policy goals, represent some of the conflicting, often noncommensurate objectives of acid rain control. This then, naturally leads to multiobjective formulations for it is precisely these types of problems that the full power of multiobjective programming can be fully brought to bear. We propose that extending the existing single objective approaches to multiobjective forms constitutes a significant advance, both for problem-specific reasons as will be developed in this paper, and for the more general reasons put forth by Cohon (1978). Essentially, Cohon argues that multiobjective programming and planning accomplishes three improvements over single objective approaches, those being, (1) possibilities for establishing more appropriate roles for the participants in the planning and decision-making process, (2) identification of a wider range of alternatives, (3) more realistic problem perception. The usefulness of a multiobjective approach will be demonstrated through the development and execution of three models, each addressing a specific policy objective not found in a typical single-objective (e.g. cost minimization) abatement strategy. These modelling extensions are developed from a deterministic linear programming framework, which, as will be described shortly, has been enlarged to include a configuration of some 138 receptors.
Basic L P framework
The basic cost-minimizing, deposition-constrained LP model is described elsewhere (e.g. see Ellis et al., 1985a, b), hence it is briefly shown here as 235
Minimize
z= ~ j=l
CjRj,
(1)
J.H. Ellis / Multiobjectioe MP models for acid rain control 235
subject to
367
92 CE/(1
- Rj)tij +
j=l
~_~ N E ~ t i k k=l
+ BG i ~< D i (i = 1, 2 . . . . . 20), Rj>/0
(j=1,2
. . . . ,235),
(2) (3)
,t
,I
where cj
= cost-removal function for source j,
= SO 2 removal level (these are the decision variables), C E / = existing emission rate (KT(SO 2)/yr), lij = normalized transfer coefficient ((kg(wet SO4)/ha-yr)/kT(SO2) emitted), for controlled source j and receptor i, N E k = noncontrolled SO 2 emission rate (kT(SO 2 ) / yr), tik =normalized transfer coefficient ((kg(wet SO4)/ha-yr)/kT(SO2) emitted), for noncontrolled source k and receptor i, BGi = b a c k g r o u n d (i.e. nonanthropogenic) wet sulfate deposition rate for receptor i, D t = maximum allowable wet sulfate deposition rate for receptor i.
Rj
Selected characteristics of the model are worthy of mention, for example, the output of the model is a set of 235 optimal values for the R / t h a t minimize (1) subject to the stated constraints ((2) and (3)). In each of the twenty constraints (2), all terms except CEjRjt~j contain only scalars and therefore can be aggregated to the right-hand-sides of the constraint inequalities. In this formulation, there will always be at least one constraint (i.e. receptor) that has a resulting deposition limit equal to the prescribed maximum, D~. Thus, the model will choose values for the R/ as high as the constraints (2) will permit, up to the point where the lefthand-side of (2) equals D i for one or more constraints. Deposition levels for the remaining constraints will be less than D~.
Updated model configuration The model's original configuration of 20 receptors was found to be inadequate, thus we created an enhanced receptor configuration along with the additional transfer coefficients thereby required. The new configuration has a total of 138 receptors (i.e. in (2), i = 1, 2 . . . . . 138) and was motivated by
\ RECEPTOR LOCATIONS
Figure 1. Enhanced receptor configuration the desire to reflect more accurately the breadth and diversity of areas potentially afflicted by acid rain, thus permitting more representative multiobjective analyses. We also attempted to include at least one receptor in every political jurisdiction that could be affected by the imposition of acid rain control measures. The enhanced configuration is shown in Figure 1.
Effects of receptor placement Before proceeding to the multiobjective analyses it is important to see generally, the effects of altering receptor placement. As can be demonstrated by a simple set of analyses, the alteration of receptor configuration has profound ramifications for the results. Figure 2 shows a tradeoff of aggregate abatement cost versus prescribed deposition limit (/9,.) for the two configurations involving 20, then 138 receptors. Translation of the curves to the right for the 138 receptor case clearly shows that instead of the previous value of 18, the minimum attainable deposition limit becomes 24.9 kg (wet SO4)/ha-yr. (The limit of feasibility which yields the minimum attainable deposition level is reached when R j = Rj(Max) for all j = 1, 2 . . . . . 235). A review of deposition isopleths for the 'base case' (i.e. R j = 0, for all j ) shows that the original configuration of twenty receptors completely misses some important locations of high predicted deposition. We note however, that
J.H. Elfis / Multiobjective MP models for acid rain control
368
8C
~ " 70 ¸ 60
I'
= ~ " ~ - -
feasibility limits J
I ~ \
.S. \
20 receptors 138 receptors
\
so.
\ \
00 40 1-
\
\
30
\
¢~ 20'
Can."
"~
18
2'1
\
"i\ ~an.
...
24
27
3'0
~
36
3'9
4"2
4'5
WET SULFATE DEPOSITION LIMIT
Figure 2. Cost-deposition tradeoffs: Original and enhanced receptor configurations
buffering concerns notwithstanding, it is erroneous to presume that receptors with high predicted deposition automatically become the most critical constraints within the LP model (where critical is loosely defined as those locations at which it is most difficult to satisfy the prescribed maxim u m allowable deposition rates; such constraints 'drive' the LP model). This can be demonstrated with a series of constraint relaxation analyses. The relaxation involves running the model with the full complement of 138 receptors, and noting at which receptor (i.e. constraint) a change in D i will result in the greatest change in W.CjRj. Call this constraint index r 1. Next, solve the model again but with only 137 receptors, represented by constraints i = 1, 2 . . . . . 138; i :# r 1. Subsequently, a new most critical constraint (r2) is then identifled, and the entire process repeated. The result, given D i = 24.9, for all i, is shown in Figure 3. The Stilwell Lake, N Y receptor is the location at which it is most difficult to meet the deposition standard, Di, yet it is not the location with the highest predicted base case deposition rate. That distinction goes to the Deep Creek Lake, M D receptor (base case deposition rate = 55.7 kg(wet SO4)/ha-yr) which ranked ninth in the relaxation analyses. The base case deposition rate predicted for Stilwell Lake is 46.2 kg(wet SO4)/ha-yr ). Thus, Figure 3 shows the interesting result that removing the Stilwell Lake receptor
from the model causes an aggregate abatement cost decrease of some 30 billion dollars or about 30%. Furthermore, if the Wooster and Purdue Agricultural F a r m receptors are relaxed, cost drops
~FULL
70
COMPLEMENT OF 138 RECEPTORS
I/
6c
/ /
~" O ~ so "~
, deposition limit = 24.9 kg wet S04(ha-yr)1 ° receptors relaxed in decreasing order of sensitivity
/ /STILWELL LAKE, N.~
~\
/WOOSTER, O.IO \
~4o
\ \
o E ~ 30
/ /PURDUE AG. FARM, INDIANA
jBROOKNAVEN, N.Y. ~B~ 1 X~
PRINCETON, N.J.
~l~f
PIEDMONT STATION, N.C.
"~I~PENN STATE, PA.
2o
FINLEY, N.C.-- / / DEEP CREEK LAKE, MD.--
~
~1~ "11-- -II- --II-
o o
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
no. of 'relaxed' constraints
Figure 3. Constraint relaxation results
J.H. Ellis / Multiobjective MP models for acid rain control
369
Table 1 Deposition results from the relaxation analyses Receptor
Base case deposition a
Stilwell Lake, NY Wooster, OH Purdue Ag. Farm, IN Brookhaven, N.Y. Princeton, NJ Piedmont Stn., NC Penn State, PA Finley, NC Deep Creek Lake, MD
46.2 40.6 35.9 45.0 42.6 33.8 48.6 31.3 55.7
Relaxed deposition rates b (1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
32.5
30.5 25.7
28.8 29.3 28.9
34.5 29.3 28.4 32.3
35.1 29.2 29.3 35.1 27.4
35.4 29.3 29.7 35.0 26.8 27.3
37.2 30.3 29.9 36.3 28.7 27.3 27.5
37.1 30.4 30.0 36.2 28.3 28.7 27.5 26.4
37.0 30.7 30.0 36.1 28.1 28.7 27.4 26.4 25.3
a Deposition units = kg(wet SO4)/ha-yr. b Nominal deposition standard = 24.9. Relaxation order: (1) Stilwell Lake, (2) Stilwell Lake and Wooster, (3) Stilwell Lake, Wooster and Purdue Ag. Farm, etc.
another 20 billion dollars. It must be remembered of course, that when a constraint is relaxed, its resulting after-removal deposition rate from the LP model will be higher than the prescribed maximum. Rates for the nine receptors shown in Figure 3 are provided in Table 1. Note that the LP-derived deposition rates for relaxed receptors need not be substantially larger than the prescribed values of D r Furthermore, the relaxed rates vary as a function of which subset of contraints is relaxed. The need to reduce deposition at other constraints necessarily causes substantial reductions in deposition rates at the relaxed receptors. The only situation where this m a y not occur is if the vast majority of deposition at a critical receptor originates from a highly localized source that does not contribute significantly to deposition at other receptors in the system. Given a westto-east advective flow in the simulation model, this situation is more likely for receptors nearest the east coast or for relatively isolated source-receptor pairs with, respectively, west-east positioning. These observations serve to point up several aspects of the complex and contentious role that receptor placement plays in developing an abatement strategy, and with this new updated modelling framework, we now proceed to a description of the multiobjective analyses.
Deposition deviation model It is clear that the vector of m a x i m u m allowable deposition limits (D~) plays a critically im-
portant role in determining the characteristics of control strategies. Unfortunately, we simply do not know with certainty what the safe, allowable limits should be for protecting the environment from acidification damages. It is reasonable to presume however, that they should exhibit strong site-specificity and perhaps temporal-specificity as well. F r o m a scientific viewpoint for example, limits appropriate for aquatic habitat protection are probably very different from those specific to material degradation. F r o m a policy determination viewpoint, these observations serve to greatly complicate the modelling tasks, especially when one considers that the characteristics of the LP solutions are, to a great extent, driven by the requirement to meet such limits for a small subset of sensitive receptors in the model. We draw some of these important issues together by positing a model based upon a straightforward observation: if the ' h a r d ' upper bounds on m a x i m u m allowable deposition could be preferentially relaxed for certain receptors, then it follows that potentially large cost savings could result. The first multiobjective model was therefore motivated by the desire to replace the hard upper (and currently uncertain) bound on allowable deposition, D i, with a somewhat softer restriction. It permits violation deviations, equivalent to resulting deposition rates in excess of Di, in addition to the over-achievement deviations implicit in the previous model (equations 2). Our interest is in identifying the abatement cost savings thereby achieved. Recall that so-called over-achievement deviations occur frequently in the previous model and correspond to the conditions
J.H, Ellis / MultiobjeetiveMP modelsfor acid rain control
370
4( ..,,~~.~,__'DEVIATION'MODEL-equalweights
Z
_o J
\ \
"30
(comparison with base case)
\
t~ z
25 _o ,~
\
v
¢5 '35
Z
_o
(~20
20 1-
15 ,~ 10 m .J
ffl
0
18
21
24
WET
27 30 33 36 SULFATEDEPOSITION LIMIT
39
42
45
Figure 4. Deviation model results 235
92
E C E j ( 1 - R ~ )lij -~- E N E k t i k + B G i < D i j=l k=l
(4)
where R 7 denotes optimal solution values. Noting the strict inequalities, the over-achievement deviations, denoted V~, are simply 235
92
V/= D i - E CEj(1 - R7 )tij -- E NEgtik - BOi. j=l
k=l
(5) In certain cases, V~>/0 even when R~ = 0, for all j, thus indicating that the deposition standard for constraint i is already met at the existing emission levels of all sources. The resulting multiobjective deviation model can be written: 235
Minimize
where U~= decision variable corresponding to violation deviation, and W,u, W,v = objective function weights. By certain basic principles of linear programruing (e.g. see Hadley, 1963, p. 168), U~ and V~ cannot be simultaneously positive in an optimal solution, hence for each constraint i, U~ will be positive signifying a violation of the nominal deposition standard D i, or V/will be positive signifying a resulting deposition rate less than Di, or both U~ and V~ will be zero, indicating a deposition rate exactly equal to D i. The second objective function acts to minimize the weighted sum of these deviations. In the 'weighting method' (Cohon, 1978) the weights serve a dual purpose in that they permit the generation of tradeoffs between objective one (cost minimization) and objective two, and as well, enable the assignation of constraint-specific deviation tradeoffs.
zl "+ Z2 = E CjRj j=l 138
Application of deviation model
+ ~ (W/uUi + W/vv/),
(6)
i=1 235
subject to
92
E CEj(1 - Rj)tij + ~_, NEktik j=l
k=l
+ B G i - D i - U~+ V / = 0
Rj f> 0 V/>i 0
Vi,
Vj,
(7) (8)
Vi,
(9)
As would be expected, cost differences between the deviation approach and the previous 'hard standard' approach increase dramatically as nominal deposition limit (in fact, now a target) is decreased. As is shown in Figure 4, this is especially true for the lower values of D i. Note that cost comparisons with the base case are not possible below the feasibility limit of the original model (i.e. 24.9 kg(wet SO4)/ha-yr ). Unlike the basic
371
J.H. Ellis / Multiobjective MP models for acid rain control
a p p r o a c h however, the d e v i a t i o n m o d e l can never b e c o m e infeasible. O n c e F . C j R j reaches its maxim u m t h r o u g h the process of progressively decreasing D i, objective n u m b e r one of the d e v i a t i o n m o d e l stays c o n s t a n t a n d the sum of aggregate d e v i a t i o n s ( a n d n u m b e r of violations) increases. This explains w h y d e p o s i t i o n violations are shown for n o m i n a l limits b e l o w 24.9 in F i g u r e 4. T h e difference in total costs shown in F i g u r e 4 p e r t a i n s solely to the U.S. Of special n o t e is the result that U.S. a b a t e m e n t costs (not r e d u c t i o n s ) are equal to total ( i . e . U . S . plus C a n a d a ) costs, b e c a u s e the i n t r o d u c t i o n of d e p o s i t i o n d e v i a t i o n s e l i m i n a t e s C a n a d i a n SO 2 r e m o v a l costs. W i t h o u t deviations, the need to meet d e p o s i t i o n limits at selected U.S. r e c e p t o r s results in large C a n a d i a n a b a t e m e n t costs. W h e n selective violations are p e r m i t t e d , we see that the t r a d e o f f b e t w e e n aggregate cost a n d aggregate d e p o s i t i o n d e v i a t i o n s favors e l i m i n a t i n g C a n a d i a n source removals. This of course, results in higher d e p o s i t i o n rates for C a n a d i a n receptors that receive p r e d o m i n a n t l y C a n a d i a n emissions, b u t n o t to the p o i n t of exc e e d i n g the target limits (i.e. n o d e p o s i t i o n violations occur at C a n a d i a n receptors). These characteristics are s t r o n g l y d e p e n d e n t u p o n the weights chosen, Wi u a n d W, v which, as n o t e d earlier, were i d e n t i c a l for all receptors. If, for e x a m p l e , the weights for certain C a n a d i a n r e c e p t o r s were subs t a n t i a l l y increased, then C a n a d i a n SO 2 a b a t e m e n t w o u l d be forced, especially for those receptors at which d e p o s i t i o n originates p r e d o m i n a n t l y f r o m C a n a d i a n sources. T h e a b a t e m e n t cost red u c t i o n s d o however exact a p e n a l t y in increasing n u m b e r s o f n o m i n a l s t a n d a r d violations as D i decreases, as shown in F i g u r e 4. W h e t h e r the p o t e n t i a l a b a t e m e n t cost r e d u c t i o n s are d e s i r a b l e is clearly a f u n c t i o n of w h e t h e r violations at selected r e c e p t o r l o c a t i o n s (assigned t h r o u g h the weights W, u, Wi v) are p e r m i s s i b l e a n d furthermore, w h e t h e r the m a g n i t u d e s of the a n t i c i p a t e d violations are acceptable. I n o t h e r words, F i g u r e 4 tells o n l y a p a r t of the story. E q u a l l y as i m p o r t a n t are the a s s o c i a t e d e n v i r o n m e n t a l (i.e. d e p o s i t i o n a l ) disbenefits themselves, as d e s c r i b e d in T a b l e 2. W e see for example, the interesting result that at n o m i n a l Di = 24.0, 21 of the 138 limits are exc e e d e d (as o b t a i n e d f r o m F i g u r e 4) such that the m e a n exceedence is 2.5 kg(wet S O 4 ) / h a - y r ; 11 of those 21 receptors violate the s t a n d a r d b y 1.8 kg(wet S O 4 ) / h a - y r or less, a n d finally, the m a x i -
Table 2 Statistics of deposition deviations Nominal/ Statistics of deposition deviations (exceedences) deposition Maximum Median Mean Variance Coefficient limit a of variation 45.0 42.0 39.0 36.0 33.0 30.0 27.0 24.0 b 21.0 b 18.0 b
8.7(19) 11.7(28) 14.1(36) 10.8(30) 12.0(36) 6.9(23) 7.5(28) 7.8(33) 7.8(37) 9.0(50)
2.6(6) 5.6(13) 4.5(12) 3.9(11) 3.2(10) 2.3(8) 2.1(8) 1.8(8) 1.8(9) 1.2(7)
3.5(8) 13.8 6.5(15) 13.8 5.7(15) 19.4 4.9(14) 10.9 4.1(12) 10.9 2.9(10) 4.7 2.9(11) 4.9 2.5(10) 6.1 2.7(13) 7.0 2.6(14) 6.8
1.1 0.6 0.8 0.7 0.8 0.7 0.8 1.0 1.0 1.0
all deposition units = kg(wet(SO4)/ha-yr.) (variance units = (dep. units)2). b infeasible for the original (i.e. non-deviation) model. (.) statistics expressed as a percentage of the nominal limit.
m u m v i o l a t i o n is 7.8 kg(wet S O 4 ) / h a - y r . It occurs at the P u r d u e Ag. F a r m r e c e p t o r in I n d i a n a . T h e l o c a t i o n s of the m a x i m u m violations p r o vide useful i n f o r m a t i o n as well. In using ten different values for D i, the m a x i m u m violations occur at o n l y 4 different locations. T h u s we begin to see p r o n o u n c e d consistency in the results, as these f o u r r e c e p t o r l o c a t i o n s shown in T a b l e 3 regularly seem to p l a y i m p o r t a n t roles. This leads to some p o t e n t i a l l y i m p o r t a n t observations. G i v e n the focal c o n c e r n for these receptors, it w o u l d a p p e a r p r u d e n t to pose, at least, two kinds of questions: (1) are the transfer coefficients for these receptors a c c e p t a b l y representative? (both in a b s o l u t e Table 3 Locations of maximum violations Nominal deposition limit ~
Locations of maximum violations
45.0 42.0 39.0 36.0 33.0 30.0 27.0 24.0 21.0 18.0
Deep Creek Lake, MD (53.7) Deep Creek Lake, MD (53.7) Deep Creek Lake, MD (53.1) Penn State, PA (46.8) Penn State, PA (45.0) Penn State, PA (36.9) Stilwell Lake, NY (34.5) Purdue Ag. Farm, IN (31.8) Purdue Ag. Farm, IN; Stilwell Lake, NY (28.8) Purdue Ag. Farm, IN; Stilwell Lake, NY (27.0)
a deposition units = kg(wet SO4)/ha-yr. (-) maximum deposition rate (nominal limit + violation).
372
J.H. Ellis / Multiobjective M P models for acid rain control
terms and relative to other transfer coefficients in the model); and, (2) if so, can we afford to meet, at these receptors (and perhaps others like them) deposition standards as low as can be achieved for the majority of other receptors in the system? The desire to answer the first question constitutes a strong motivation for explicitly representing pollutant transport and transformation processes as stochastic phenomena in optimization models, as is the case in Fortin and McBean (1983), Ellis (1984), Fronza and Melli (1984), Ellis et al. (1985b, 1986), Guldmann (1986) and Fuessle et al. (1987). Implicit in the second question is one of the extremely sensitive tradeoffs associated with acid rain control, i.e., must we, to some extent, sacrifice environmental quality in certain regions so as to obtain affordable abatement strategies overall? Notwithstanding the ever-present externality issue, this clearly is a difficult argument to sell, for among many objections, it purports to sacrifice precisely those regions that may be least able to assimilate higher wet sulfate loadings. Furthermore, it can (and has) been argued that affordability is a highly ambiguous metric, given our modelling context of global cost minimization. There is merit in the argument that global (i.e. system-wide) costs are meaningless for it is solely the distribution of cost that is important. These issues, at their most fundamental level, involve difficult questions of equity which leads to the following multiobjective model that addresses one such aspect of equity.
Emission and deposition reduction deviations However defined, equity considerations will play a dominant role in designing an abatement strategy which has any realistic chance of political acceptance. Modelling methodologies which do not explicitly attempt, in some sense, to embody equity have severely impaired utility. Complicating the issue of course is the confounding, multiobjective nature of equity, in all its varied and diverse manifestations. We describe here, a simple and admittedly crude representation of one aspect of equity that establishes as metrics, the fractional emission reductions and fractional deposition reductions defined over every state and province in the system. A description of the multiobjective
model and its underlying motivation requires two definitions. Within each state and province, -1
J~Js
" " J~Js
/ -1
Fd=
E
CEjRjtij
~ iE1s j = l
E CEjtij i
,
j=l
(11) where F ~ - - f r a c t i o n a l emission reduction for s t a t e / province S, •Is --subset of controllable sources j in s t a t e / province S, Fd --fractional deposition reduction for s t a t e / province S, I s = subset of receptor locations i in state/province S. The important difference between (10) and (11) is that whereas F~ is determined by emission reductions wholly within each state or province, Fsd is determined by emission reductions at sources both within state/province S and all other sources as well. It might be argued then, that an abatement strategy which results in F~ considerably larger than Fd for state/province S is imposing an inequitable or, at least, impolitic situation: state/province S is, proportionately, removing much more SO2 emission than it is receiving in return, in terms of in-state deposition reductions. Of course, the arguments become much more complicated when one realizes that a relatively large value of F~ benefits downwind receptors in other states, and so on. Furthermore, on an annual average basis, easternmost sources generally do not contribute significantly to deposition at receptors in more western state/provinces hence the easternmost states/provinces inescapably have
Fff > F~. A convenient way to pull some of these observations together involves the use of the following model:
Minimize
zl
+ Z2 = E C j R j
J + ~-'.(W~Us+ W~Vs) S
(12)
J.H. Ellis / Multiobjective MP models for acid rain control
subject to
ECEj(1
-
Rj)tij +
ENE~t,k
+ BG, <~Dg Vi,
(13) -1
-lz z-j..j} i~Is j
z
iEls j
- Us + Vs = 0
vs,
(14)
Rj >10
v j,
(15)
Us , Vs>lO
VS,
(16)
Like the previous model, the second objective (z2) serves to minimize the sum of deviations, with those deviations now corresponding to positive (Us) or negative (Vs) differences between F~ and Fd. The global (but unattainable) minimum of z 2 equal to zero corresponds to the case wherein the fractional emission and deposition fractions are exactly equal for every state and province.
Application of the equity model The costs (in terms of ~CjRj) of attaining a balance of F~ and Fff are shown in Figure 5. As
373
expected, lower values of D i result in larger cost differences between the basic and 'equity' solutions. It is therefore more costly to achieve our version of equity for more stringent maximum allowable deposition rates. This continues only up to a point however. Once below Di = 27 kg(wet SO4)/ha-yr, cost differences drop dramatically. The reason for this behavior is that when approaching the limit of feasibility, the need to satisfy the 'hard' deposition constraints (13) completely dominates the solution. There no longer exists room to move in the sense of removal allocations which will minimize zl, satisfy (13), as well as minimize z 2 (equations (14) are always feasible). If constraints (13) were replaced with their deviation counterparts (7), and a third objective added in the form of the second summation in (6), then the aforementioned dramatic decreases would not occur. This would also permit the generation of useful tradeoffs between abatement cost, deposition deviations and deviations from equity as embodied in F~ and Fd. We note as well that the emission source and receptor subsets Js and I s need not be state or province-specific. Rather, S could represent a national delineation between the U.S. and Canada, or any categorization: politically-motivated, electric utility pool-motivated, or otherwise. Aside from cost, other aspects of the equity formulation are potentially informative. Specifi-
cost comparison -'equity' vs. base case-
20( 18(]
-10
16(] 7
140 12(]
U.S.
lO(]
Can.-----
.5
4
80 604020-
li3
g"
2~
2~, 27 30 33 36 WET SULFATE DEPOSITION LIMIT F i g u r e 5. E q u i t y m o d e l results
39
42
45
m
374
J.H. Ellis / Multiobjective MP models for acid rain control
Table 4 Equity deviation results State/province Arkansas a Georgia Illinois Indiana Kansas a Louisiana Maine a Maryland Michigan Minnesota Missouri Nebraska New Jersey New Mexico a New York North Carolina North Dakota a Ohio Oklahoma ~ Pennsylvania South Dakota a Tennessee Texas Vermont a Virginia West Virginia Manitoba Nova Scotia Ontario Quebec Saskatchewan a
Deposition limit (kg(wet SO4)/ha-yr) 45.0
42.0
39.0
36.0
33.0
30.0
27.0
24.9
V
V
V
V
V
V V
V
V
V V V
V V V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V V V
V V
V V
V V
V V
V V
V
V
V
V
V
V
V
V
V
V
V
V
V
V U
V U
V U
V U V U
V U V U
V U V U
V V V V V
V
V V
V V
V V
V
V
V
V
V
V
V
V
V
V
V V
U V
U V
U V
U V
U V
V
V
V
V
V
V
V
V
V
V
V
V
V
V V V U V V V V V
U
V
V V V U
a Mandatory 'V' entries for all deposition limits (no controllable sources) V -= fractional emission reduction < fractional deposition reduction; U -= fractional emission reduction > fractional deposition reduction.
cally, we c a n e x a m i n e at w h i c h states o r p r o v i n c e s is F~ < F~d, o r F~ > F~a, o r lastly, F• = F~d. T h i s i n f o r m a t i o n is p r o v i d e d in T a b l e 4. R e c a l l t h a t w h e n F~ < Fsa, Us = 0 a n d Vs > 0. S i m i l a r l y w h e n F~ > Fsa, t h e n Vs = 0 a n d Us :~ O. N o t c o u n t i n g states a n d p r o v i n c e s w i t h o u t c o n t r o l l a b l e s o u r c e s (i.e. Us m u s t e q u a l zero), t h e m a j o r i t y o f e n t r i e s s h o w t h a t F~ < Fsd t h u s i n d i c a t i n g t h a t f r a c t i o n a l deposition reductions exceed fractional emission r e d u c t i o n s . T h i s w a s u n e x p e c t e d for s o m e o f t h e states a n d p r o v i n c e s l i s t e d in T a b l e 4. G i v e n r e l a t i v e l y large e x i s t i n g e m i s s i o n levels a n d ' u p w i n d ' l o c a t i o n s ( t h e a n n u a l a v e r a g e a d v e c t i v e w i n d di-
r e c t i o n for e a s t e r n N o r t h A m e r i c a is w e s t to east), o n e m i g h t e x p e c t I l l i n o i s a n d I n d i a n a to h a v e Fff > Fsd f o r t h e l o w e r v a l u e s o f D i, b u t s u c h is n o t t h e case. W e see Fff = F d w h e n D, is 33 k g ( w e t S O 4 ) / h a - y r o r b e l o w . F o r the l a r g e r v a l u e s o f D~, a s s i g n m e n t o f Vs > 0 f o r I l l i n o i s a n d I n d i a n a i n d i cates that their prescribed emission reductions substantially benefit instate receptors. Additional s u p p o r t f o r this o b s e r v a t i o n is p r o v i d e d in T a b l e 5 w h i c h shows, for e a c h level o f D i, t h o s e r e c e p t o r s w h i c h w e r e b i n d i n g w i t h r e s p e c t to d e p o s i t i o n . W h e n b i n d i n g r e c e p t o r l o c a t i o n s are n o t in I n d i a n a , e m i s s i o n r e d u c t i o n s are d r i v e n , in part, b y t h e
J.H. Ellis / Multiobjective MP mode&for acid rain control
375
Table 5 Binding receptor locations for the equity model Receptor
Deposition limit (kg(wet SO 4)/ha-yr) 45.0
Purdue Ag. Farm, IN Rockport, IN Deep Creek Lake, M D Brookhaven, NY Stilwell, NY Wooster, OH Penn State, PA
X
X
42.0
39.0
36.0
33.0
30.0 X X X
X
X
X X X X X X
X X
X
X X X X
X
X X X
X X
27.0
24.9
X
X
X ~- binding receptor.
need to meet the deposition limit at other, out-ofstate receptors. This necessarily results in large in-state deposition reductions to the extent that F~ < Fsd for Illinois and Indiana. Pennsylvania, West Virginia and to a lesser extent, Ohio exhibit expected behavior in that F~ > Fsd. For Pennsylvania and Ohio however, this applies only until D i decreases to the limit of feasibility (24.9). At that limit we then see F~ < Fsd. The switch from Us to Vs is due to relative increases of in-state deposition reductions for Pennsylvania and Ohio that, for the most part, arise from out-of-state emission reductions. Before D~ ---24.9 is reached (e.g. at D i = 27.0) emission reductions for Pennsylvania are already at their respective upper bounds. Those for Ohio are very close to their bounds. Thus, further deposition reductions generally originate elsewhere. Indeed as described above, at the limit of feasibility with respect to deposition, the equity constraints become irrelevant and all source removal levels are set equal to their respective maximums, albeit at very high marginal costs.
Aggregate reduction targets The final multiobjective model uses the basic formulation and an extension based upon the principles of the constraint method of multiobjective programming (see generally, Cohon and Marks, 1973; Cohon, 1978). Appended to the basic model (equations (1)-(3)) is the constraint Y'~ CEjRj = G
(17)
jeJ
where G represents a user-specified aggregate
emission reduction and J a specified subset of sources, in this case, U.S. sources. In effect therefore, three objectives are considered. The purpose of this formulation is to provide a means of analyzing the ramifications and sensitivities of some of the typical acid rain control bills which have been proposed in the U.S. congress in recent years. Such bills prescribe an aggregate emission tonnage reduction, hence the motivation for constraint 17. While these approaches may be politically acceptable, only minimal consideration is given to the environmental improvements they purport to effect, the blanket assumption being that emission reductions will result in proportionate deposition reductions. For five levels of G: 4, 6, 8, 10 and 12 million ton SO 2 reductions, Figure 6 shows the tradeoffs between abatement cost and D i, Where cost remains more-or-less constant over a range of Di, indicates that the solution is being driven by constraint 17. In these plateau-like regions then, the aggregate reduction level G, and hence aggregate cost exceeds what is necessary to meet deposition limits /9,.. The presence of intersecting tradeoff curves reveals yet another informative characteristic of the solutions, and generally, the way in which the model works. For example, at about D~ = 27 kg(wet SO 4)/ha-yr, the 8 million ton and 10 million ton aggregate reduction tradeoff curves intersect, with the associated abatement cost approximately equal to 40 billion dollars. In both solutions, constraint (17) must be satisfied and several deposition constraints (equations (2)) are binding. With respect to total, aggregate cost per ton of sulfur removed, the 8 million ton reduction solution prescribes a relatively inefficient set of removal levels as compared to the 10 million ton
J.H. Ellis / Multiobjective MP models for acid rain control
376
\ 12 M \.
'Aggregate U.S. Emission R e d u c t i o n '
60
\
~-~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
o~ X, 8M \, \
=t
20
r
o
24
r 25'.5
l 27
MAXIMUM
r 28-5
l 30
31.5
3~3
34.5
36
37.5
ALLOWABLE DEPOSITION
Figure 6. Specified emission reduction model results
case, which achieves an additional 2 million tons of emission reduction for the same cost. From a deposition viewpoint however, the 10 million ton reduction solution appears inefficient, in that it satisfies a maximum deposition limitation that could be satisfied with an aggregate emission reduction that is 2 million tons less. This simple comparison serves only to demonstrate certain aspects of the model's behaviour. Obviously, many other questions can be brought to bear, based upon more realistic bases for comparison. For example, there is a certain incompatibility between models which act to prescribe aggregate emission reductions and those which are strictly deposition-constrained. In fact, the former are generally motivated by the desire to avoid the latter. The desire to avoid deposition-constrained approaches is, more precisely, the desire to avoid the use of transfer coefficients and their attendant uncertainties. Thus, a more conventional comparison might involve the 8 million ton reduction solution and its minimum feasib|e deposition limit (i.e. about 27 kg(wet SO4)/ha-yr, as shown in Figure 6) and the 10 million ton reduction and its corresponding minimum limit (25.8). Yet another modelling possibility combines the foregoing model with deviation versions of the deposition constraints (i.e. equations (7)). It is also possible to construct a deviation version of (17) so as to investigate the potential savings associated
with variations about G. It is simply Z CEjRj - Vc + Vc = O.
08)
jET_J
Another objective function component would be constructed in the form of Minimize
W~ U6+ wV vG
(19)
where W~ and Wv are prescribed weights, as before. This might be especially informative for the case where G -- 12 million tons, given that this aggregate reduction level also roughly corresponds to the reduction associated with the minimum feasible deposition limit (24.9) of the basic deposition-constrained model.
Closure
We hopefully have shown that the use of a relatively simple mathematical tool can provide insight into some important aspects of developing acid rain abatement strategies. Certain of these insights are of course, easily anticipated, such as the profound effects of receptor placement on strategy characteristics, similarly, the substantial cost reductions possible through the introduction of allowing deposition limit violations at selected receptors. Other results are much less intuitive,
J.H. Ellis / Multiobjective M P models for acid rain control
such as the d o m i n a n t role p l a y e d b y a subset of four receptors in the system. A s well, the a t t e m p t to b a l a n c e f r a c t i o n a l emission a n d d e p o s i t i o n red u c t i o n s y i e l d e d interesting a n d u n e x p e c t e d shifts in r e m o v a l efforts as a f u n c t i o n of m a x i m u m p r e s c r i b e d d e p o s i t i o n limit. L a s t l y we saw, f r o m a d e p o s i t i o n - c o n s t r a i n e d perspective, the p o t e n t i a l inefficiencies a s s o c i a t e d with strategies b a s e d u p o n aggregate emission r e d u c t i o n stipulations. W e close with the r e a f f i r m a t i o n t h a t the m o d e l s a n d m e t h o d o l o g y d e s c r i b e d herein c a n n o t m a k e policy. T h e y exist to serve as tools for the p o l i c y analyst. T h e i r a d v a n t a g e s lie p r i m a r i l y in the a b i l i t y to pull together, in a c o m p l i c a t e d , highlyi n t e r r e l a t e d setting, d i s p a r a t e a n d o f t e n diverse elements of a system. It is clear that a c i d rain c o n t r o l is precisely such a system. S y s t e m a n a l y t i c a p p r o a c h e s a n d especially m u l t i o b j e c t i v e app r o a c h e s can p r o v e to b e that n e e d e d b r i d g e between, say, h a r d science a n d the d e c i s i o n - m a k i n g process. A t a m i n i m u m , such a p p r o a c h e s s h o u l d possess the a t t r i b u t e s of multiobjectivity, stochasticity of p o l l u t a n t flux processes, the effects of coal m a r k e t / p r i c e e q u i l i b r i a - - a l l w r a p p e d in an interactive p a c k a g e suitable for use b y the decis i o n - m a k i n g c o m m u n i t y itself. R e s e a r c h c u r r e n t l y moves f o r w a r d t o w a r d that goal.
Acknowledgements Sincere t h a n k s to J e r r y C o h o n , Ben H o b b s , N a n c y Kete, C h u c k ReVelle a n d the reviewers for their n u m e r o u s helpful c o m m e n t s a n d criticisms. T h a n k s also to Ms. K e t e a n d F e l i x S e e b a c h e r for p e r f o r m i n g m a n y of the c o m p u t e r r u n s a n d to L a n e E n g l a n d for e x p e r t l y w o r d p r o c e s s i n g the m a n u s c r i p t . T h e s u p p o r t of the N a t i o n a l Science F o u n d a t i o n u n d e r g r a n t N o . ECE-8504582 is gratefully a c k n o w l e d g e d . This research was cond u c t e d using the C o r n e l l N a t i o n a l S u p e r c o m p u t e r Facility, a resource of the C e n t e r for T h e o r y a n d S i m u l a t i o n in Science a n d E n g i n e e r i n g at C o r n e l l University, which is f u n d e d in p a r t b y the N a tional Science F o u n d a t i o n , N e w Y o r k State a n d the I B M C o r p o r a t i o n .
377
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