Prioritizing mountain catchment areas

Journal of Environmental Management (1991) 32, 357-366

Prioritizing Mountain Catchment Areas D. M. Murray Somkon Industrial Mathematics, PO Box 7254, Stellenbosch 7600, Republic of South Africa and K. yon G a d o w

Faculty of Forestry, University of Stellenbosch, Stellenbosch 7600, Republic of South Africa Received 15 February 1990

A methodology is developed for the ranking of mountain catchment (watershed) areas based on the actual numerical attribute values of each area as well as on subjective evaluations of the attributes themselves. It is shown, in particular, how the pairwise comparisons of the catchment area attributes serve to generate weights to weigh the rankings of the attribute values. Mountain catchment areas which are candidates for proclamation as protected areas can therefore be judged against the priority vector of the already proclaimed areas.

Keywords: pairwise comparisons, priority vector, ranking multi-criterion decision, weights, measurement scale.

1. Introduction The problem addressed here is that of comparing various mountain catchment areas (also known as watershed areas) when allocating resources for the management of a class of protected areas. For this application, the emphasis is not so much on the allocation of limited resources, but rather on the development of a decision aid that can be used to determine objectively whether an unproclaimed area has such a high management priority that it needs to be proclaimed. For this decision aid, data for existing proclaimed areas were used, based on the following attributes: 1. 2. 3. 4. 5.

Runoff (m3/km 2 x 106). Dam capacity ( m 3 • 106). Population. Gross geographical product (GGP) (South African Rand 1000). Conservation status (fraction between 0.3 and 1.0). 357

0301~4797/91/040357+ 10 $03.00/0

9 1991 AcademicPress Limited

358

Prioritizing mountain catchment areas

therefore consist o f the numerical values of the five attributes of each area as well as of each forestry expert's evaluation of the relative importance of the attributes themselves. The desired result is the prioritization of the areas as accurately as is justified by the data. 2. D a t a

Table 1 gives the values of the five attributes for each of the 10 areas. The priority for management increases with an increase in runoff, dam capacity, population, and G G P , but increases with decreasing conservation status. Table 2 illustrates the problem inherent in this multi-criterion decision application, namely that the five attributes rank the 10 areas differently. Furthermore, the attributes are on different scales, and this means that the numerical values cannot be used directly to obtain a combined priority index (in whichever quantitative way it is expressed) for each area. The crudest way of combining the attribute values is simply to rank the averages (or equivalently, the sums) of the ranks of each area in Table 2. The purpose of this study is to obtain a more sensitive and meaningful prioritization by taking into account the relative importance of the attributes as determined by the collectwe expert judgement of forestry officials.

TABLE 1. Numerical attribute values of the Mountain Catchment Areas Area 1. 2. 3. 4. 5. 6. 7. 8. 9. t0.

Cedarberg Hawequas Lang. West Rooiberg Kammanassie Outeniqua Drakensbergl Drakensberg2 ST and Boom Bredasdorp

Runoff

Dam capacity

Population

GGP

Conservation Status

0.31 0.6 0-3 0-01 0"033 0.6 0"417 0-417 0"063 0-124

151-454 1254.84 43-269 0 93.97 32.87 358.9 2958-9 67"53 7-68

47933 595704 73656 6000 14012 61640 581185 5017288 30000 140050

125-12 411 210-3 8"5 52.46 49.9 598-33 12518.4 32.5 97-6

27/33 24/33 26/33 33/33 30/33 27/33 30/33 30/33 24/33 18/33

TABLE2. Rankings of the numerical attribute values of Table 1 Area 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Cedarberg Hawequas Lang. West Rooiberg Kammanassie Outeniqua Drakensbergl Drakensberg2 ST and Boom Bredasdorp

Runoff

Dam capacity

Population

GGP

Conservation status

5 1 6 10 9 2 3 4 8 7

4 2 7 10 5 8 3 1 6 9

7 2 5 10 9 6 3 1 8 4

5 3 4 10 7 8 2 1 9 6

5 2 4 10 7 6 8 9 3 1

D. M. Murray and K. von Gadow

359

TABLE 3. The order in which the pairwise comparisons of the attributes were done Pair i j 1 5 4 3 4 1 2 5 3 2

2 3 1 2 5 3 4 1 4 5

Attribute i

Attribute j

Runoff Conservation status Gross geographical product Population Gross geographical product Runoff Dam capacity Conservation status Population Dam capacity

Dam capacity Population Runoff Dam capacity Conservation status Population Gross geographical product Runoff Gross geographical product Conservation status

Each of the experts made pairwise comparisons of the attributes in the order shown in Table 3. Attribute i was compared to attribute j with respect to m a n a g e m e n t priority. This evaluation was done on a three-point ordinal scale, where - 1, 0, or 1, respectively, indicate that the evaluator considers attribute i to be less, equally, or more important than attribute j. Table 4 shows the evaluations produced by the 28 experts. The particular three-point scale was selected for this application because it was impossible to distinguish between the attributes more sensitively. Any attempt to do so would have resulted in artificially precise data. The three-point scale yields evaluations on an ordinal scale without an inherent scaling factor. There can therefore be no unique and theoretically founded method to convert the ordinal measurements to priority values, such as weights, for the attributes. Since five parameters (a priority rating for each attribute) must be determined from 10 evaluations (and considerable redundancy therefore exists), it is reasonable to expect the parameters to be determined on a more sensitive measurement scale than an ordinal scale. This study is partly concerned, in fact, with establishing on how sensitive a scale the parameters can be determined. 3. M a t h e m a t i c a l

formulation

Let the pairwise evaluations be denoted by a,j and the parameters to be estimated by w,, i = 1. . . . . n. This means that au is a somewhat inaccurate estimate (or measurement) of the quantity k ( w , - wj), with k an unknown scale factor. The inaccuracy is caused, in the first instance, by the subjective judgement which cannot be perfectly consistent, and, in the second, by the truncation of k(w;-wj) to either - 1, 0, or 1. The model equations in terms of the measurements and parameters which must therefore be satisfied, are

a ~ k ( w ; - % ) = O, i, j = 1 . . . . . n and i
(1)

This means that there are n ( n - 1 ) / 2 , and for n = 5 therefore 10 equations in the five unknowns w,, i = 1 , . . . ,n. A different approach to equation (1) would be instructive. Suppose one started with the w,, i = 1. . . . . n, as real physical quantities measured on the most sensitive scale, for example as the volumes of different trees, and then determined the evaluations (or

Prioritizing mountain catchment areas

360

TABLE 4. Evaluations by the 28 experts Pair

Evaluation by expert 1

12 5 3 4 1 3 2 4 5

1 0 -1 1

1

3

-1 -1

2 5 3 2

4 1 4 5

0 0 0 -1

2

3

4

5

6

1 0 1 1

1 1

1 1

1 0 -1 1

0 1 0 0

1

-1

1 l

-1 -1

-1

1

1 1 0 1

1 1 1 -1

0 -1 0 1

-1 -1 -1

-1

7

8

9

10

11

12

13

14

1

1 1

1 1

0 1 -1 1

1 1 1 1

1 1

1

1 0 -1 1

1 0 0 1

1

0 1

-1 -1 -1

1

0

1 0 0 -1

0 1 0 -1

-1 1 -1 -1

0 1 0 -1

-1 0

-1

-1

-1 1

-1 0 0 -1

1

-1

1

1 0 -1

25

-1 -1

1

-1 0 -1

-1

1

0

1 1 -1

1 1 0 -1

-1 -1

26

27

28

-1

1

-1 1

continued

Evaluation by expert 15

16

17

18

19

20

21

22

23

24

1

1

1

1

0

1

1

0 1 1 1

0 -1 0 -1

1

0

0

0

-1

-1

1

1

0

0

1 1 0

0 1 -1 -1

0 1 1

1 2

0

1

1

1

1

1

1

1

5 4 3 4

1

1

0 -1 0 -1

1

I

0 -1 0 -1

I

-1 0 1 -1

-I

0

-I

3 1 2 5

-1

-1

1

TABLE 4. Evaluations by the 28 experts Pair

1

-1 -1 -1

-1 -1 -1

-1 0 -1

-1 1 -1

1

3

1

1

1

I

I

2 5 3 2

4 1 4 5

1 1 0 -1

1 0 1 -1

0 0 0 -1

0 0 0 -1

0 0 0 -1

-I

-1 0 1 -1

-1 1 0 1

-1 0 0 -1

-1 1 1 -1

1 1 1 -1 -1 0 0

-1 -1 -1 1

0 1 -1 -1

-1 1 0 0

-I

1

1

-I

m e a s u r e m e n t s ) a,j by pairwise c o m p a r i s o n s on the scale - 1, 0, o r 1 d e p e n d i n g on w, being less than, equal to, or g r e a t e r t h a n wj. To express av in terms o f w, a n d wj, a scale f a c t o r is clearly r e q u i r e d to t r a n s f o r m f r o m the units o f w, a n d t h e r e f o r e f r o m w,-wj to the scale o f the e v a l u a t i o n s a,j. T h e r e m u s t also be a positive t r u n c a t i o n t o l e r a n c e at to d i s t i n g u i s h between " e q u a l l y i m p o r t a n t " a n d "less (or m o r e ) i m p o r t a n t . " It therefore follows t h a t

atj =

-1 0 1

, , ,

k(w i - % ) < -o~
(2)

W h e n these a,j are t h u s o b t a i n e d , the p r o b l e m once a g a i n b e c o m e s t h a t o f d e t e r m i n i n g the w,, i--- 1. . . . . n f r o m the system (1). A l t h o u g h the c o m p a r i s o n s in T a b l e 4 r e m i n d s one o f the a n a l y t i c h i e r a r c h y process ( A H P ) d e v e l o p e d by S a a t y (1980), the a p p r o a c h f o l l o w e d here is quite different f r o m A H P . A c c o r d i n g to A H P , each e v a l u a t o r w o u l d have to p r o v i d e p a i r w i s e c o m p a r i s o n s o f the alternative c a t c h m e n t a r e a s with respect to each o f the attributes. This is

361

D. M. Murray and K. von Gadow

essentially AHP's way of enabling the evaluator to transform the numerical attribute values, with different scales and ranges, to a common scale. Thereafter, A H P requires the pairwise comparisons of the attributes before synthesizing all the comparisons to produce a weight vector for each evaluator. These vectors, or the comparisons themselves, must still be aggregated to produce a single weight vector. In the approach developed here, the catchment areas need not be compared with respect to the attributes, because, for each area, their numerical attribute values are available. These values are brought to a common scale by using their ranks. The threepoint ordinal scale, used for the pairwise attribute comparisons in Table 4, was consistent with the degree to which the evaluators could distinguish between the relative importance of the attributes. The complete system (1), with the order of Table 2, is actually the matrix equation 1

-1 -1

W1 W2 W3 W4 W5

1

-1

1 -1

1 1

1

-1

a12 a53

17/41 a32 =

-1

-1

1 1

,

(3)

a13 a24

-1

1

a45

a51

-1

6[34

1

-1

a25

where, without loss of generality, the scale factor k can, by redefining the w,, be taken as unity. System (3) is compactly written as A w = a,

(4)

with the meaning of the symbols self-evident. This overdetermined system immediately suggests the least squares solution (Gill et al., 1981) from

ArA w = A r a.

(5)

The matrix ArA is singular, however, as observed from

-

ArA=

-

4

-1 -1 -1

-1

-1

-

4 -1 -1

-1 4 -1

-

(6)

for which the rows are linearly dependent. System (4) can therefore not be solved with the ordinary least squares method. This singularity is not a defect of the model equations (1) or of the least squares method. It reflects the fact that with the pairwise comparisons there is still one unspecified degree of freedom, for instance a normalization. This is the case with comparisons of the form w,/wj on the proportional scale, as well as of the form w , - wj on

362

Prioritizing mountain catchment areas

the interval scale, or as truncated here to an ordinal scale. This means that a further condition, for instance a normalization of the form

~:, w~= K,

(7)

must be adjoined to system (4). The arbitrary scale factor that was absorbed in the w,, i = 1. . . . ,n, of system (3) has therefore reappeared in a different guise in system (7). The least squares solution of system (4) with the linear constraint (7) is given by

(8) where L is a Lagrange multiplier and C a row vector o f ones (Fletcher, 1981). The effect of a particular value of K is seen in system (8). From that, it is evident that the solution with, say, K = N can be derived from the solution with K = 0 by adding N / n to each wi. The differences w i - w s, i , j = 1 . . . . . n, i < j , are therefore invariant with respect to the selection of K. It must be emphasized that K is altogether arbitrary. There is no reason why K = 1 must necessarily hold. The conclusion to be drawn for this application, is that an interval scale is the most sensitive one with which the priority parameters w;, i = 1 , . . . ,n can be determined. It is indeed more sensitive than the ordinal scale on which the av were determined. 4. The mountain catchment data

To normalize equation (7) we take E, w, = 0, to avoid the impression that positive w, must necessarily be weights, as well as to emphasize that the w, are given on an interval scale. The computed parameters values of Table 5 are obtained by using the data of Table 4 as 28 right-hand sides in system (3). There are therefore 28 different possibilities of the vector a in system (8). The single decimal numbers were multiplied by 10 to make the entries in Table 5 less cluttered. The columns in the body of Table 5, the least squares solutions of system (3), subject to the normalizing constraint system (7), are the priority vectors (wt,w 2. . . . . w5)r corresponding to the different (overdetermined) evaluations in Table 4. A single priority vector must still be determined. The most obvious way is to take the row means of Table 5. This yields the priority vector (0.264, - 0 . 4 2 1 , -0-064, -0-229, 0.45) r. Because o f the linearity of system (8) the same result would have been obtained had the row means of the evaluations in Table 4 been used as a single evaluation vector a in (8). The data of Table 4 can yield an aggregate evaluation vector in yet another way. Table 6 shows the number of appearances of - 1 , 0 and 1 for the au in Table 4. Significantly, the maximum values in Table 6 mostly are much greater than the next highest values. This is indicative of a high degree of consensus among the experts. The aggregate vector as given in the right-hand column o f Table 6 now consists of the values - 1 , 0, or 1, depending on which value of a,j appears most frequently. This evaluation vector, as well as the mean o f that of Table 4, is given in Table 7. The residues in Table 7 are the estimated amounts by which the inaccurate measurements, or evaluations, must be adjusted to get consistent measurements. The reference standard deviation 60 is the estimated standard deviation of the measurements. The priority vector computed with the aggregate evaluation vector yields essentially the same result as before, namely that the ranking of the attributes, in decreasing order

D. M. Murray and K. von Gadow

363

TABLE 5. Computed parameter values ( x 10) for the 28 evaluation vectors Attribute

Evaluation vector 1

1

2

2 3 4 5

-6 4 -4 4

2

3 4

4

6

4 6 2 -4

-4 2 -8 6

-2 0 4 -6

0 -6 -6 6

-8

4

5

9

11

12

13

14

7 4

4

6

8

2

0

2

2

-2 -2 -2 8

-8 0 0 4

-6 0 -2 4

-6 -4 -2 6

-4 0 -8 4

-6 -2 -2 8

-8 0 0 8

-2 -2 -6 8

-8 -2 2 6

-2

8

10

6

TABLE 5. Computed parameter values ( x 10) for the 28 evaluation vectors--continued Attribute

Evaluation vector 15

16

17

18

19

20

21

22

23

24

25

26

1

2

6

6

6

6

2

6

0

0

4

0

4

2 3 4 5

2 -6 -6 8

0 -4 -8 6

-4 -2 -4 4

-4 -4 -4 6

-6 -2 -4 6

-6 4 -4 4

-8 -2 0 4

-8 6 -2 4

-6 6 6 -6

-2 -8 -2 8

-6 0 6 0

-2 -2 -6 6

27 -2

-4 -2 0 8

28 0

-6 4 -4 6

o f i m p o r t a n c e , is (5,1,3,4,2) a n d t h a t the differences b e t w e e n the p r i o r i t y values (when r a n k e d ) are essentially the same. W h e n the a g g r e g a t e e v a l u a t i o n v e c t o r is p e r t u r b e d slightly b y c h a n g i n g the ones to zeros in the two p o s i t i o n s where the residues a r e - 0 - 6 , the p e r t u r b e d e v a l u a t i o n v e c t o r o f T a b l e 7 is o b t a i n e d . This yields the p r i o r i t y v e c t o r as indicated. This v e c t o r gives e q u a l l y high priorities to a t t r i b u t e s 5 a n d 1 a n d e q u a l l y low p r i o r i t i e s to 3, 4 a n d 5. Since the p e r t u r b e d e v a l u a t i o n v e c t o r differs o n l y slightly f r o m the a g g r e g a t e e v a l u a t i o n vector, it m u s t be c o n c l u d e d t h a t the d a t a c a n n o t j u s t i f y a m o r e precise c a l c u l a t i o n o f the p r i o r i t y v e c t o r t h a n to say t h a t the o r d e r in d e c r e a s i n g p r i o r i t y is (5,1,3,4,2) a n d t h a t the

TABLE 6. Number of appearances of -- 1, 0, and 1 of the a,~ in the evaluations of Table 1 Number of Maximum number at

a~

- 1

0

1

al2 a53 a41

1 4 18

4 10 5

23 14 5

a32

4

7

17

1

a45 a13

22 4 10 3 4 24

2 5 10 9 17 2

4 19 8 16 7 2

- 1 1 0 1 0 - 1

a24

as~ a34 a25

1 1 - 1

Prioritizing mountain catchment areas

364

TABLE 7. Comparison of evaluation vectors and calculated priority vectors Evaluation Vector according to Table 4

Perturbation

a,j

Residues

ao

Residues

aq

Residues

0.7857 0.3571 - 0.4643 0.4643 -0.6429 0.5357 -0.0714 0.4643 0.1071 -0.7857

-0-1 0.157 - 0.029 -0.1017 -0.036 -0.207 -0.121 - 0.279 0-057 -0.086

1 1 - 1 1 - 1 1 0 1 0 - 1

0 0 0.2 -0.6 -0-2 -0.4 -0-2 - 0-6 0.2 -0.4

1 1 - 1 0 - 1 1 0 0 0 - 1

0 0 0 0 0 0 0 0 0 0

~0 wI w2 w3 w4 w5

Table 6

0.197 0.264 - 0.421 - 0'064 - 0"229 0"45

0.40 0.4 - 0.6 - 0"2 - 0"4 0"8

0 0.6 - 0-4 - 0-4 - 0-4 0'6

differences r e m a i n m o r e o r less c o n s t a n t except for a larger difference between the s e c o n d a n d third p r i o r i t y t h a n b e t w e e n the others. 5. A t t r i b u t e v a l u e s

T o c o m b i n e the a t t r i b u t e values o f T a b l e 1 with the p r i o r i t i e s o f the a t t r i b u t e s in T a b l e 7, it is necessary to c o n v e r t the n u m e r i c a l a t t r i b u t e values to s o m e c o m m o n scale. Since the p r i o r i t y p a r a m e t e r s c a n n o t be d e t e r m i n e d a n y m o r e sensitively t h a n o n a n interval scale, the a t t r i b u t e values s h o u l d be h a n d l e d in a similar way. T h e a p p r o a c h used here converts the a t t r i b u t e values to an o r d i n a l scale a n d weights the r a n k s o f the a t t r i b u t e values a c c o r d i n g to the p r i o r i t y v e c t o r for each c a t c h m e n t area. U n d e r the c i r c u m s t a n c e s the w e i g h t e d n u m e r i c a l value o f each a r e a can only be i n t e r p r e t e d on a n o r d i n a l o r an i n t e r v a l scale. T a b l e 8 gives five c o l u m n s o f the r a n k s o f the a t t r i b u t e values, followed b y c o l u m n s o f r a n k i n g s o f differently w e i g h t e d a t t r i b u t e ranks. T h e p r i o r i t y vectors o f T a b l e 7 c a n n o t be used as weights w i t h o u t first being c o n v e r t e d to positive values. This c a n be achieved b y a d d i n g a n a r b i t r a r y c o n s t a n t at to all five elements o f the p r i o r i t y v e c t o r w. Here, once again, the a r b i t r a r i n e s s o f the n u m b e r K i n e q u a t i o n (7) is evident. This aspect merely r e - e m p h a s i z e s the interval scale's characteristic, n a m e l y t h a t the origin is a r b i t r a r y . Since there is n o u n i q u e a which s h o u l d be a d d e d to the p r i o r i t y elements, the influence o f v a r i o u s p o s s i b l e choices o f values for at o n the u l t i m a t e r a n k i n g s h o u l d be e x a m i n e d . F o r very large ct we merely o b t a i n the r a n k i n g o f the s u m o f the r a n k s as s h o w n in T a b l e 8. O t h e r values o f ct are being e x a m i n e d as well to d e t e r m i n e the sensitivity o f the r a n k i n g s w i t h respect to weighting perturbations.

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D. M. Murray and K. von Gadow

TABLE 8. Rankings of attribute values and rankings of combined rankings with w= (0-264, - 0.421, - 0.064, - 0'229, 0'45) r of Table 7 Ranking r k k

Area name

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Cedarberg Hawequas Lang. West Rooiberg Kammanassie Outeniqua Drakensbergl Drakensberg2 ST and Boom Bredasdorp

~Zkrk(Ct + W)

1

2

3

4

5

~krk

Ct= 0"5

ct=l

Ct=5

5 1 6 10 9 2 3 4 8 7

4 2 7 10 5 8 3 1 6 9

7 2 5 10 9 6 3 1 8 4

5 3 4 10 7 8 2 1 9 6

5 2 4 10 7 6 8 9 3 1

4 1 5 10 9 7 3 2 8 6

7 1 4 10 9 6 3 5 8 2

6 1 5 10 9 7 3 2 8 4

5 1 4 10 9 7 3 2 8 6

The ranking which corresponds to a = 0.5 is p r o b a b l y the best solution, since the two m o s t i m p o r t a n t attributes, with this value o f ~t, play the m o s t i m p o r t a n t role. The priority ranking, in decreasing order, Hawequas Bredasdorp Drakensbergl Lang. West Drakensberg2 Outeniqua Cedarberg ST and B o o m Kammanassie Rooiberg o f m o u n t a i n catchment areas is therefore the required solution for the priorities o f the different areas. Finally, to use these results as a decision aid, there m u s t be a rule according to which an unproclaimed area can be c o m p a r e d with the existing ones. Such a rule could be that the m a n a g e m e n t priority o f a new area m a y be high e n o u g h to have it proclaimed as a m o u n t a i n catchment area when its priority according to the m e t h o d s described here, is higher than those o f a certain n u m b e r o f existing areas. W h a t can in fact be clearly observed f r o m the results above, is that ST and Boom, K a m m a n a s s i e and R o o i b e r g consistently are the three areas with the lowest priority. The p r o c l a m a t i o n o f a new area with higher priority than these is therefore justified. 6. C o n c l u s i o n s

According to the m e t h o d described here, an unproclaimed area's attribute values can be c o m p a r e d with those o f the existing areas in order to achieve a priority position for the new area with respect to the existing ones.

366

Prioritizing mountain catchment areas

The ideal would be a priority index for a new area determined on the basis o f its attribute values, without the need for a c o m p a r i s o n o f the new and the existing areas. N o t e n o u g h information and d a t a is available to determine a value function with which to transform the attributes to a c o m m o n scale for this application, however. A priority index p r o p o r t i o n a l to the resources that m u s t be m a d e available for the m a n a g e m e n t o f the area would be useful. I n view o f the consensus with the data o f Table 4, a similar degree o f agreement in refining the priority index m i g h t be possible. References

Fletcher, R. (1981). Practical Methods of Optimization-- Volume 2: Constrained Optimization. New York: John Wiley and Sons. Gill, P. E., Murray, W. and Wright, M. H. (1981). Practical Optimization. London: Academic Press. Saaty, T. L. (1980). The Analytic Hierarchy Process. New York: McGraw-Hill.