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The Analytic Hierarchy Process in an uncertain environment: A simulation approach
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 91 (1996) 27-37
Theory
and Methodology
The Analytic Hierarchy Process in an uncertain environment: A simulation approach David
Hauser,
Pandu
Tadikamalla
*
Joseph M. Katz Graduate School of Business, 258 Merris Hall, Unit,ersity of Pittsburgh, Pittsburgh, PA 15260, USA
Received March 1994; revised August 1994
Abstract Traditionally, decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the Analytic Hierarchy Process (AHP). Since many circumstances exist which make such a convergence difficult, confidence in the outcome of an ensuing A H P synthesis may be reduced. Likewise, when a group of decision makers cannot arrive at a consensus regarding a judgment, some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments. The A H P utilizes point estimates in order to derive the relative weights of criteria, sub-criteria, and alternatives which govern a decision problem. However, when point estimates are difficult to determine, distributions describing feasible judgments may be more appropriate. Using simulation, we will demonstrate that levels of confidence can be developed, expected weights can be calculated and expected ranks can be determined. It will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis. Keywords: Analytic Hierarchy Process; Ratio scale; Interval scales; Simulation
I. Introduction With the Analytic Hierarchy Process (AHP), the overall p r i o r i t i z a t i o n o f a set o f a l t e r n a t i v e s to a p r o b l e m can b e s y n t h e s i z e d f r o m all s t a t e d pairwise r e l a t i o n s h i p s [3-5]. Y e t in o r d e r for o n e to synthesize the p r i o r i t i e s with t h e A H P , t h e pairwise relationships should be point estimate j u d g m e n t s , n o t d i s t r i b u t i o n s o r r a n g e s o f feasible j u d g m e n t s . T h e r e a r e i n s t a n c e s in p r o b l e m solving w h e r e t h e r e l a t i o n s h i p b e t w e e n two o r m o r e e l e m e n t s o f a p r o b l e m is d e f i n i t e a n d precise, * Corresponding author. Fax: (412) 648-1693.
relative to s o m e a t t r i b u t e . F o r e x a m p l e , the interest r a t e at o n e b a n k m a y b e 1.2 t i m e s t h e i n t e r e s t r a t e o f a n o t h e r bank. Such p a i r w i s e r e l a t i o n s h i p s can b e distinctly d e s c r i b e d . T h e r e a r e o t h e r instances where the relationship between various e n t i t i e s c a n n o t b e explicitly d e s c r i b e d with r e g a r d to a p a r t i c u l a r c r i t e r i o n , yet t h e r e l a t i o n s h i p m a y b e subjectively u n d e r s t o o d . F o r e x a m p l e , an individual m a y p r e f e r o n e v a c a t i o n spot m o d e r a t e l y m o r e t h a n a n o t h e r relative to t h e c r i t e r i o n f u n . T h i s p e r s o n m a y not a s s o c i a t e " m o d e r a t e l y m o r e t h a n " with " t h r e e times m o r e t h a n " b u t the subjective u n d e r s t a n d i n g o f t h e r e l a t i o n s h i p b e t w e e n t h e two v a c a t i o n a l t e r n a t i v e s is fairly precise.
0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-2217(95)00002-X
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D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
However, there are instances where the relationship between a pair of entities can only be described with some sort of probability distribution and cannot be described as a point estimate. Such circumstances include both truly uncertain environments, and the composite description of opinions of a group of decision makers which for whatever reason cannot agree. In the former circumstance, it may be known that one entity may cost somewhere between three and five times that of another entity. This particular pairwise relationship within this particular environment can be termed uncertain. In these sorts of environments, the information may be unclear. In the later circumstance, a group of decision makers may not be able to converge their opinions to a single judgment, rather each decision maker chooses to retain his or her own judgment. Hence the consensus may best be described by a probability distribution and can therefore also be termed uncertain. No matter what the cause of the uncertainty of the judgments may happen to be, careful and complete analysis of the problem hierarchy and the probability distributions describing the judgments of this structure can yield potentially more confidence in the results than traditional sensitivity analysis. Further, expected priorities can be derived and a comparison of and between the point estimate judgment approach and the judgment distribution approach may yield some useful conclusions regarding the original data. Likewise, analysis of inconsistency may reveal useful information regarding the overall importance of some uncertain judgments. Arbel and Vargas [1] studied the effect of interval judgments on the A H P and found that within these ranges, there exists relatively optimal point estimators which best categorize the intervals. Moreno-Jimenez and Vargas [2] studied the problem of determining the most probable ranking of alternatives that one should infer when decision makers use interval judgments rather than point estimates in the AHP. However this later study was analytical and was limited to the single reciprocal matrix. Most problems expand far beyond the single matrix into highly complex hierarchies and networks. This paper examines the scope and influences of interval judgments
beyond the reciprocal matrix and offers an approach for determining the expected rank and expected weight of each alternative.
2. The decision matrix
In order to derive a set of priorities of alternatives with respect to a governing criterion, the relationship between each pair of alternatives must be placed into a reciprocal matrix. We define a square matrix A = (aij), V i , j ~ [1,n] to be a reciprocal matrix with n alternatives where aij = 1/aji and aij indicates that the ith alternative is aij times more dominant than the jth alternative with respect to the criterion governing the matrix. Once the reciprocal matrix is determined, the priorities are calculated. From this matrix, the inconsistency ratio (IR) can also be calculated. Acceptable values of the IR must be less than 0.10 [5,6]. If a particular judgment of matrix A is described by an interval or by a distribution rather than by a point estimate, there are several ways of choosing a single representative value for the judgment from the feasible range: (1) use the arithmetic mean of the range as the point estimate, (2) use any value for the judgment within the range that yields the lowest IR value, (3) use the most-likely value, or (4) take the geometric mean of a large set of values chosen randomly from the distribution. However, each of these approaches are problematic since the representative value of the distribution may not yield an acceptable nor expected ranking. The first suggested approach may be problematic since the arithmetic mean of the range may yield an IR value larger than 0.10. Besides, there may be one ore more extreme judgments resulting in a highly skewed distribution for the judgment. In this case, the mean of the distribution may not be an appropriate value to use. The second suggested approach may also be problematic since all discrete values within the distribution may yield an IR greater than 0.10. Even if some discrete values within the distribution resuited in an IR for the matrix less than 0.10, the objective of the problem may not require in-
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
finitesimally small amounts of inconsistency. The most-likely value, in some cases, may not have a very high probability c o m p a r e d to the other values. The fourth suggested approach suffers from shortcomings similar to the first suggestion. However, the advantage to the fourth approach is that it is similar to group decision making in which the geometric mean of the judgments of all the decision makers is used as the representative point estimate. However, this representative value may also have an IR value greater than 0.10. Therefore another approach will be required.
29
! if
I : [
iil
!!
f
!
i i
3. T h e d i s t r i b u t i o n a l o n g t h e s c a l e
The scale used in the A H P for pairwise comparisons is the 1 to 9 ratio scale [3-5]. We note that the inverse relationships are simply the multiplicative inverses of the values from within the 1 to 9 scale, hence the actual scale used in the A H P is 1 / 9 to 1 to 9. Suppose all of the judgments of a reciprocal matrix are uniformly distributed within 100p% of the stated judgment. The feasible range of judgments for a point estim a t o r value x is x +_px. For all values between x-px and x + p x which are greater than 1, the A H P ratio scale and the x +_p x scale are equivalent. We call the x +_px distribution a virtual scale since this scale is used to describe the judgment interval but in general is not actually used by the A H P (see Fig. 1). The portion o f the virtual scale, x ++_px, that is less than 1 must be m a p p e d back onto the actual scale in order for it to retain the properties of a ratio scale and be useful to the AHP. Consider an example which leads one to this conclusion. Suppose we are interested in randomly generating a judgment from the uniform distribution x + px, where x = 1, and p = 0.5. Any value less than 1 on the ratio scale actually corresponds to the 1 / 9 to 1 portion of the ratio scale that is actually used in the AHP. For this particular example, the largest possible value that we may use is 1 + 0.5 × 1 = 1.5, which is slightly above "equal" and below "equal to m o d e r a t e . " The lowest possible value that we may use, since this range of values is symmetrical must also be slightly
I I
• 11 i
~r I
, ii
~,~, :
J',
'1,
Fig. 1. The virtual scale versus the actual scale.
above "equal" and below "equal to m o d e r a t e " but for the reciprocal relationship. Therefore we know that the 1 - 0.5 × 1 = 0.5 must signify the reciprocal of 1.5. Further, any value below 1 that is generated must be m a p p e d onto the actual scale. There are two possible errors which one may encounter if one fails to account for such a mapping: (1) a judgment of zero may be generated and the reciprocal judgment cannot be used, and (2) the width of the interval is different when considering an interval and a ratio scale and thus one cannot use interval scale generators with ratio scale numbers. Suppose v is a specific value chosen randomly from the uniformly distributed interval x _+px. If v < 1, then v is a m e m b e r of the set of points attainable from the virtual scale and v is not a m e m b e r of the set of attainable points from the actual scale. The actual scale equivalent of v can be m a p p e d via the one-to-one function /(v)
-
1
2-v
.
(l)
If L' > 1, then v is a m e m b e r of both the set of attainable points from the virtual scale and the set of attainable points from the actual scale. As such, v does not need to be altered when u >_ 1,
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
30
therefore the actual ratio scale equivalent of v can be mapped via the one-to-one function f ( v ) = v.
from the standpoint of a prospective student according to their desirability. Six independent characteristics were selected for the comparisons - learning, friends, school life, vocational training, college preparation, and music classes. See Fig. 2 for the hierarchy. The pairwise judgment matrices for this hierarchy were as shown in Tables 1 and 2 and the overall synthesis of priorities is shown in Table 3. According to these judgments, school A (one) should be chosen since school A received the greatest overall synthesized priority (see Table 3). However, suppose the weights of the criteria or the weights of the alternatives were in question. These questionable weights can be perturbed in order to measure the amount of change in the overall synthesized priorities of the alternatives. However, the magnitude or the extent to which one should perturb these weights is arbitrary. Hence such an analysis, however revealing it may be, lacks a basis in realism. The magnitude of the weight perturbations may be realistically impossible alterations and therefore the conclusions drawn from the sensitivity analysis may not be applicable. Suppose each of the point estimates describing the pairwise comparisons were uncertain. Let each point estimate be the midpoint of a uniform distribution the width of which is 50% of the value of the point estimate. Therefore any value within x + 0.25x is equally as valid as any other value in the range for describing the pairwise relationship. The width of the interval may coin-
(2)
Thus, the mapping of any point v along the virtual scale to the actual scale can be accomplished by the one-to-one function
f(v)
=
-
v
ifv
(3)
,
ifv>l where x - p x < v < x + p x
and f ( x - p x ) < f ( v )
<_/(x + px). Within each interval, a value is randomly generated based on the probability distribution. This selected value is mapped to the actual scale used in the A H P by (3). Note that when p = 0, the value which will be randomly generated is simply the point estimate x. In short, as p increases, the variance of the distribution increases and the number of inconsistent combinations of judgments also increases. Also as p increases, the rankings of the alternatives is likely to change more than in the case when p is small. Hence for each value of p, a level of confidence must be developed in order to describe the most probabilistic rankings.
4. A n a l y s i s
- the simulation
approach
Consider the following example from Saaty [5]. Three high schools, A, B, and C, were analyzed
II Ooal I
I Learg II Friends II SchooLiIq VocatioO II Preparation CoXgo II Music Training Classes
(One)
(Two)
School A (One)
(Three)
(Four)
(Five)
(Six)
I
I
(Two)
(Three)
Fig. 2. School selection example.
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37 Table 1 Comparison of characteristics with respect to overall satisfaction with school Relative to: The goal
One Two Three Four Five Six
One
Two
Three
Four
Five
Six
1.0
5.0 1.0
7.0 3.0 1.0
5.0 0.2 0.25 1.0
3.0 0.16667 0.2 0.2 1.0
1.0 0.16667 0.2 0.16667 1.0 1.0
31
Table 3 Synthesis utilizing the point estimates as the judgements Synthesis with respect to goal Overall inconsistency index = 0.09 One Two Three
0.398 0.357 0.244
Table 4 Composite of 500 runs based on a uniform probability distribution (frequency of each rank) Composite results
cide with the maximum difference between judgments of group members making a c o m m o n decision. It may even coincide with the maximum amount an individual decision has been known to deviate. For each of the matrices of the school example, discrete values for the judgments were randomly generated from the uniform distribution of the provided point estimator +25%. After all matrices were determined, the overall synthesized priorities were found and the rank was recorded. This process was repeated 500 times. See Table 4 for the details. Of the 500 runs, 96.4% resulted in the first alternative with the first rank. H e n c e we can conclude that the ranking of the first alternative via the point estimates as the judgments are 96.4% accurate. Even within this level of uncertainty (_+25%), we can have 100% confidence in the ranking of the third alternative (see Table 4).
Rank:
One
Two
Three
One
Two
Three
1 2 3
482 18 0
18 482 0
0 0 500
96.4% 3.6% 0.0%
3.6% 96.4% 0J)%
[/.(le~ 0.0% 100.0%
In the AHP, priorities with an IR greater than 0.10 are considered to have judgments which are too random-like [5,7]. Hence, of these 500 runs, only those with an IR less than 0.10 should be considered (see Table 5). Of the 500 repetitions, only 418 runs had an IR < 0.10. By examining these more restrictive results, the same conclusions regarding the confidence in the rankings can be drawn. Despite all the uncertainty in the judgments, the third alternative was ranked in the third position 100% of the time. Hence we may conclude that the ranking is fixed irrespective of the level of consistency demanded by the problem solver. Likewise, any group disagreement con-
Table 2 Comparison of schools with respect to the six characteristics Relative to: Criterion one
One Two Three
Relative to: Criterion three
One
Two
Three
1.0
0.33333 1.0
0.5 3.0 1.0
Relative to: Criterion two
One Two Three
One Two Three
Relative to: Criterion five
One
Two
Three
1.0
5.0 1.0
1.0 0.2 1.0
Relative to: Criterion four
One
Two
Three
1.0
1.0 1.0
1.0 1.0 1.0
One Two Three
One Two Three
One
Two
Three
1.0
0.5 1.0
1.0 2.0 1.0
Relative to: Criterion six
One
Two
Three
1.0
9.0 1.0
7.0 0.2 1.0
One Two Three
One
Two
Three
1.0
6J) 1.0
4.(I 0.3333 1.0
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
32
Table 5 Composite of the 418 runs with IR < 0.10 based on a uniform probability distribution (frequency of each rank) Consistency-limit results Rank
One
Two
Three
One
Two
Three
1 2 3
405 13 0
13 405 0
0 0 418
96.9% 3.1% 0.0%
3.1% 96.9% 0.0%
0.0% 0.0% 100.0%
cerning a judgment or set of judgments is clearly pointless with regard to the overall synthesis for this particular problem since very little will change in the outcome. Since a great deal of faith can be placed in the ranking of the alternatives for this problem within the given level of uncertainty of the environment, an analysis of the inconsistency is not too revealing (see Table 6). Across all 500 runs, the inconsistency index is clearly normally distributed around 0.09. We can conclude that most of the runs are below the 0.10 upper limit, however, that conclusion could have been drawn directly from the number of runs which met this criterion. The only use that the inconsistency information yields is a primitive measure of the feasible minimum level of inconsistency. Since we know the distribution of the inconsistency index, we can easily determine the likelihood that a particular level of inconsistency could be achieved. From this, we can conclude whether or not that level is attainable and worth the cost of reaching that level (see Table 6). Consider a more complicated and buoyant example. Suppose a particular problem consisted of five criteria and four alternatives. See Fig. 3 for the hierarchy. The reciprocal matrices of the point estimate judgments of this problem are shown in Tables 7 and 8 and the overall synthesis utilizing these matrices is shown in Table 9. In this problem, alternative number one is globally more preferred to the other four alternatives, although each alternative is closely weighted. However, suppose that the judgments from which the synthesis was derived were accurate only to within 25% above and 25% below the given point estimates. Given this particular circumstance, simulation was run. For each judg-
Table 6 Frequency distribution of the different levels of the IR across all 500 simulated runs based on a uniform probability distribution Composite results Inconsistency distribution 0.00) 0.01) 0.02) 0.03) 0.04) 0.05) 0.06) 0.07) 0.08) 0.09) 0.10) 0.11) 0.12) 0.13) 0.14) 0.15) 0.16) 0.17) 0.18) 0.19)
5 33 93 154 133 59 14 9
0.20) 0.21) 0.22) 0.23) 0.24) 0.25) 0.26) 0.27) 0.28) 0.29) 0.30) 0.31) 0.32) 0.33) 0.34) 0.35) 0.36) 0.37) 0.38) 0.39)
0.40) 0.41) 0.42) 0.43) 0.44) 0.45) 0.46) 0.47) 0.48) 0.49) 0.50) 0.51) 0.52) 0.53) 0.54) 0.55) 0.56) 0.57) 0.58) 0.59)
0.60) 0.61) 0.62) 0.63) 0.64) 0.65) 0.66) 0.67) 0.68) 0.69) 0.70) 0.71) 0.72) 0.73) 0.74) 0.75) 0.76) 0.77) 0.78) 0.79)
0.80) 0.81) 0.82) 0.83) 0.84) 0.85) 0.86) 0.87) 0.88) 0.89) 0.90) 0.91) 0.92) 0.93) 0.94) 0.95) 0.96) 0.97) 0.98) 0.99)
ment x given the reciprocal matrices, representative judgment points were randomly generated from within the uniformly distributed range of x + 0.25x (see Table 10). Of the 500 simulations, all 500 runs resulted in an IR < 0.10 hence each run is both feasible and quite realistic. Consider the results of Table 10. Although each alternative has seen each possible rank, it is clear that alternative one is inclined to be positioned in the first rank. Likewise, alternative two is inclined to be positioned in the fourth rank. However, how much confidence can we have in Table 7 The pairwise comparisons of the criteria relative to the goal Relative to: The goal One Two Three Four Five
One
Two
Three
Four
Five
1.0
1.0 1.0
1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0 1.0
D. Hauser, P~ Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
33
Table 8 Comparison of alternatives with respect to the five criteria Relative to: Criterion one
One Two Three Four
Relative to: Criterion four
One
Two
Three
Four
1.0
1.2 1.0
1.4 1.2 1.0
1.6 2.0 3.0 1.0
Relative to: Criterion two
One Two Three Four
One Two Three Four
One
Two
Three
Four
1.0
2.0 1.0
3.0 2.0 1.0
4.0 3.(1 2.0 1.0
Relative to: Criterion five
One
Two
Three
Four
1.0
0.83333 1.0
0.71429 0.83333 1.0
0.625 0.71429 0.83333 1.0
One Two Three Four
One
Tow
Thre
Four
1.0
0.5 1.0
0.33333 0.5 1.0
(I.25 0.33333 (1.5 1.0
Relative to: Criterion three
One Two Three Four
One
Two
Three
Four
1.0
1.0 1.0
1.0 1.0 1.0
1.0 1.0 1.0 1.0
these two rankings and what is the most probabilistic ranking of alternatives three and four? In order to address these two related questions, we must first develop the notion of expected rank. We can see from Table 10 that the first alternative will be of rank one 68.6% of the time. Can we infer that we have 68.6% confidence in that rank? When considering each alternative and each rank individually, this perception of the measure of confidence is accurate. However, we may be interested not in the rank of one alternative, rather we are interested in the ranks of all alternatives simultaneously.
I 1 21 I II one II Two U II F= II ivo II I I I I I I I I J Fig. 3. Example with five criteria and four altenratives.
If we have 68.6% confidence that the first alternative will be in the first rank, we cannot simply say that we have 1.6% confidence that the second alternative will be in the first rank, since there can only exist a single alternative for each Table 9 Synthesis utilizing the point estimates as the judgements Synthesis with respect to goal Overall inconsistency index = 0.01 One Two Three Four
0,262 0,238 0,247 0,253
Table 10 Composite of 500 runs based on a uniform probability distribution (frequency of each rank) Composite results
Rank: One Two Three Four 1 2 3 4
343 8 40 111 32 127 42 112 236 4 348 97
109 230 110 51
One
Two
Three Four
68.6% 1.6% 8.0% 22.2% 6.4% 25.4% 8.4% 22.4% 47.2% 0.8% 69.6% 19.4%
21.8% 46.0% 22.0% 10.2%
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
34
rank. However, a composite of the four different levels of confidence for each of the four alternatives will reveal the overall ranking of each alternative. We know from Table 10 that alternative one is of rank one 68.6% of the time, of rank two 22.2% of the time, of rank three 8.4% of the time and of rank four 0.8% of the time. Hence, these percentages could act as weights of the ordinal ranking positions. We define expected score to be the composite value representing the probabilities of each rank for each alternative. Consider the following equation for expected score. ESi = ~'~ ( P i , k ) ( k ) ,
Vi ~ [1,n],
(4)
k=l
where ESi is the expected score of the ith alternative, and Pi,k is the proportion of the trials that the ith alternative had rank k. There are two problems with this approach. First, the score is mathematically the expected rank. Rarely will the ranks computed in this manner be integral. Since a rank order is, by definition, the set of indices of an ordered list, the definition of (4) cannot be complete. Second, lower rankings are directly correlated with the larger weights, however in equation (4), the lower rankings are inversely correlated with the larger weights. Therefore rather than sum together the product of the fraction of time each rank occurred and the rank itself, we will sum together the product of the fraction of the time each rank occurred and n + 1 minus the rank itself. In this manner, the resultant will be inversely correlated with the weights and the larger weights we know will yield the lower ranks. Therefore we improve upon (4) and consider the function n
ES~ = ~] Pi,k(n + 1 -- k ) ,
Vi ~ [l,n],
(5)
k~l
where ES i is the expected score of the ith alternative, and Pi,k is the proportion of the trials that the ith alternative had rank k. Next, we define the expected weight to be the normalized expected scores. When the alternatives are placed in descending order of the ex-
Table 11 Expected results based on a uniform probability distribution Alternatives
Expected rank
Expected weight
One Two Three Four
1 4 3 2
0.3586 0.1400 0.2220 0.2794
pected weights, the results reveal the expected rank of alternatives. Hence, we define EWi
ESi z.,@ESk,
Vi ~ [l,n].
(6)
k=l
The expected weights of (6) are determined from the frequency that each rank occurred for each alternative. Hence these weights are statistical weights indicating a composite frequency or a mean of feasible weights around which we expect the actual weights to be scattered. The actual meaning and interpretation of ES~ of (5) is misleading. We understand that we cannot ever actually achieve a fractional rank, hence after we compute the expected weights of (6), we simply define the expected rank, ER~, to be the index of the ith alternative once sorted in non-increasing order of EW/. Continuing with earlier example, we calculate the expected weights of the four alternatives and derive the expected rankings (see Table 11). Confirming our earlier conclusion, we see that alternative one has the highest expected weight (the lowest expected ranking) and alternative two has the lowest expected weight (the highest expected ranking). Consider the expected weight of alternative four which has an expected rank of two. We see that the expected weight of alternative four (0.2794) is greater than the expected weight of the third ranking alternative (0.2220). Hence we can be assured that alternative four will be of rank two. Likewise, we can be assured that alternative three has the third rank since its expected weight (0.2220) is greater than the expected weight of the fourth ranking alternative (0.1400). In general, the frequency distributions of the rankings for an alternative could range from
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
highly skewed ("exponential-like") to "normal" or could be "uniform like". For example, we see that the frequency distributions of the first and second alternatives seems to be highly skewed indicating quite clearly that the most probable ranks for these alternatives are in the first and fourth positions, respectively. However, we see that the frequency distributions of the third and fourth alternatives seem to be "normally" distributed around the third and second ranks, respectively.
5. The probability distributions In two examples considered here, we have assumed uniform distributions for all judgments in the range of x + p x . This, probably, is an extreme case implying that the decision makers are not only unable to converge to a single estimate of their judgment but their judgments are uniformly distributed across the range. A more reasonable scenario is that some proportion of the people may converge to a "modal value" within the range considered. This modal value could be in the center of the range creating a symmetric distribution or it could be anywhere in the range creating skewed distributions. Beta distributions and triangular distributions are often used to represent different shapes of data with finite known range of values. In the A H P type applications where we can only assume to know the range and the mode, triangular distributions are reasonable and easier to use. We use symmetric triangular distributions for the second example considered earlier. Note that for a given range, ( a , b ) , the uniform distribution has a higher variance 2 O'uniform =
~(b -
a) 2
than the symmetrical triangular distribution 2
O'syrnmetrical triangular
= ~ ( b - a) 2.
Thus the use of triangular distributions, as opposed to the uniform distribution, should provide more "consistent" rankings.
35
Table 12 Composite of 500 runs based on a symmetrical triangular probability distribution (frequency of each rank) Composite results Rank One Two Three Four
One
1 2 3 4
77.0% 0.6% 3.4% 19.0% 18.6% 3.2% 25.2% 53.0% 4.2% 19.2% 53.8% 22.8% 0.2% 77.0% 17.6% 5.2%
385 3 93 16 21 96 1 385
17 126 269 88
95 265 114 26
Two
Three Four
Consider once again the second example from above. See Fig. 3 for the hierarchy and Tables 7 and 8 for the reciprocal matrices. The composite results of Tables 10 and 11 describe the outcomes and computations regarding the simulation of 500 runs utilizing a uniform probability distribution x ++_px. In contrast, consider the influence of a symmetric triangular probability distribution. Table 12 contains the composite results of the 500 simulated runs which utilized a triangular distribution centered about x with lower and upper extremes equal to x - p x and x + p x , respectively. Just as in the uniform scenario, all 500 runs of the triangular scenario resulted in an IR _< 0.10, hence all results are feasible. The conclusion drawn earlier regarding the rank is more than supported by the results of the triangular probability distribution simulations. From the uniform probability distribution results, we concluded that the first alternative was quite skewed; leaning toward the first ranking. From the triangular probability distribution, we see that the first alternative is extremely skewed; leaning toward the first ranking. Likewise, each of the other three rank-distributions are more skewed when the symmetrical triangular probability distribution was utilized than when the uniform probability distribution was utilized. Since the center point of the symmetrical triangular distribution is the point estimate, one would expect that the triangular probability distribution should result in an expected rank more closely aligned with the rank of the deterministic system utilizing solely the specified point estimates for each judgment. Consider the expected ranks of the triangular probability simulated results (see Table 13). The
D. Hauser, P. Tadikamalla /European Journal of Operational Research 91 (1996) 27-37
36
Table 13 Expected results based on a symmetrical triangular probability distribution Alternatives
Expected rank
Expected weight
One Two Three Four
1 4 3 2
0.3724 0.1274 0.2144 0.2858
expected rank of the triangular system matches the expected rank of the uniform system. Hence we can conclude that for this problem, an environment as uncertain as a uniform probability distribution depletes only a small measure of confidence in the resultant. The resulting rank is held constant from the point estimate approach to the symmetrical triangular probability distribution approach all the way to a uniform probability distribution approach. We must note that the expected weights of the triangular system are more distinct than the expected weights of the uniform system. In the uniform system, the first ranking alternatives yielded an expected weight of 0.3586 and the second ranking alternative yielded an expected weight of 0.2794. In the triangular system, the first ranking alternative yielded an expected weight of 0.3724 and the second ranking alternative yielded an expected weight of 0.2858. The differential is larger in the triangular system indicating that the symmetrical probability distribution results in more definitive figures and demonstrates that one may have a greater measure of confidence in the resulting rank than the uniform probability distribution. If an environment was completely certain, the rank is simple to determine and hence complete confidence in the result is assured. However, as uncertainty increases, the probability distribution decreases the confidence. If the uncertainty of the judgments can be described by a triangular probability distribution, we know the confidence in the result will be less than the measure of confidence we would have in a certain environment. If the uncertainty of the judgments is high and can be described by a uniform probability distribution, we know the confidence in a result
will be less than the measure of confidence we would have in a triangular probabilistic environment.
6. Conclusions
Traditional sensitivity analysis requires that the decision maker perturb in a deterministic fashion the weight of one or more criteria a n d / o r alternatives in order to note changes in the ranks of the alternatives [5]. However, by how much and in what fashion should one perturb the various weights is an important issue. The basic premise behind the A H P is that one notion can only be understood when it is compared relative to another notion and has little meaning otherwise [3-6]. Therefore one cannot perturb a weight of an element in a hierarchy and simply draw a conclusion from the changes in weights a n d / o r ranks of the alternatives unless the magnitude of the proposed perturbation has some meaning and is not arbitrary. Further, it may very well be that such a change is not truly possible in the real world. For instance, increasing the weight of a criterion of a hierarchy may have a substantial impact on the ranks of the alternatives; however the decision maker may not be able to control the weight of that criterion in real life. The precise interpretations of the weight perturbations that the decision maker institutes may be difficult to make. Then by how much should one perturb these weights and in what direction? The exact relationship between realistic variations in the judgments may not be known relative to perturbations in the weights themselves, hence simulation offers a more rich approach. Suppose one employed the traditional sensitivity analysis approach and found a different ranking for certain alterations in the weights of some of the elements of the hierarchy. What rank order for the alternatives can one expect? At best, the only conclusion that can be drawn from traditional sensitivity is some measure of confidence in the rank. For a partially or even for a completely uncertain environment, simulation is a better approach for providing a measure of confidence in a
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
rank, and for providing expected weights and ranks. References [1] Arbel, A., and Vargas, L.G., "The analytic hierarchy process with interval judgements", in: Proceedings of the MCDM Conference held in Washington DC, August 1990. [2] Moreno-Jimenez, J.M., and Vargas, L.G., "A probabilistic study of preference structures in the Analytic Hierarchy Process with interval judgements", Mathematical and Computer Modelling 17 (1993) 73-81.
37
[3] Saaty, T.L., "Scaling method for priorities in hierarchical structures", Journal of Mathematical Psychology 15 (1977) 234-281. [4] Saaty, T.L., "Axiomatic foundation of the Analytic Hierarchy Process", Management Science 32 (1986) 841-855. [5] Saaty, T.L., Multicriteria Decision Making: The Analytic Hierarchy Process, RWS Publications: Pittsburgh, PA, 1990. [6] Saaty, T.L., and Vargas, L.G., "Uncertainty and rank order in the Analytic Hierarchy Process", European Journal of Operational Research 32 (1987) 107-117. [7] Vargas, L.G., "Reciprocal matrices with random coefficients", Mathematical and Computer Modelling 3 (1982) 69-81.
European Journal of Operational Research 91 (1996) 27-37
Theory
and Methodology
The Analytic Hierarchy Process in an uncertain environment: A simulation approach David
Hauser,
Pandu
Tadikamalla
*
Joseph M. Katz Graduate School of Business, 258 Merris Hall, Unit,ersity of Pittsburgh, Pittsburgh, PA 15260, USA
Received March 1994; revised August 1994
Abstract Traditionally, decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the Analytic Hierarchy Process (AHP). Since many circumstances exist which make such a convergence difficult, confidence in the outcome of an ensuing A H P synthesis may be reduced. Likewise, when a group of decision makers cannot arrive at a consensus regarding a judgment, some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments. The A H P utilizes point estimates in order to derive the relative weights of criteria, sub-criteria, and alternatives which govern a decision problem. However, when point estimates are difficult to determine, distributions describing feasible judgments may be more appropriate. Using simulation, we will demonstrate that levels of confidence can be developed, expected weights can be calculated and expected ranks can be determined. It will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis. Keywords: Analytic Hierarchy Process; Ratio scale; Interval scales; Simulation
I. Introduction With the Analytic Hierarchy Process (AHP), the overall p r i o r i t i z a t i o n o f a set o f a l t e r n a t i v e s to a p r o b l e m can b e s y n t h e s i z e d f r o m all s t a t e d pairwise r e l a t i o n s h i p s [3-5]. Y e t in o r d e r for o n e to synthesize the p r i o r i t i e s with t h e A H P , t h e pairwise relationships should be point estimate j u d g m e n t s , n o t d i s t r i b u t i o n s o r r a n g e s o f feasible j u d g m e n t s . T h e r e a r e i n s t a n c e s in p r o b l e m solving w h e r e t h e r e l a t i o n s h i p b e t w e e n two o r m o r e e l e m e n t s o f a p r o b l e m is d e f i n i t e a n d precise, * Corresponding author. Fax: (412) 648-1693.
relative to s o m e a t t r i b u t e . F o r e x a m p l e , the interest r a t e at o n e b a n k m a y b e 1.2 t i m e s t h e i n t e r e s t r a t e o f a n o t h e r bank. Such p a i r w i s e r e l a t i o n s h i p s can b e distinctly d e s c r i b e d . T h e r e a r e o t h e r instances where the relationship between various e n t i t i e s c a n n o t b e explicitly d e s c r i b e d with r e g a r d to a p a r t i c u l a r c r i t e r i o n , yet t h e r e l a t i o n s h i p m a y b e subjectively u n d e r s t o o d . F o r e x a m p l e , an individual m a y p r e f e r o n e v a c a t i o n spot m o d e r a t e l y m o r e t h a n a n o t h e r relative to t h e c r i t e r i o n f u n . T h i s p e r s o n m a y not a s s o c i a t e " m o d e r a t e l y m o r e t h a n " with " t h r e e times m o r e t h a n " b u t the subjective u n d e r s t a n d i n g o f t h e r e l a t i o n s h i p b e t w e e n t h e two v a c a t i o n a l t e r n a t i v e s is fairly precise.
0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-2217(95)00002-X
28
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
However, there are instances where the relationship between a pair of entities can only be described with some sort of probability distribution and cannot be described as a point estimate. Such circumstances include both truly uncertain environments, and the composite description of opinions of a group of decision makers which for whatever reason cannot agree. In the former circumstance, it may be known that one entity may cost somewhere between three and five times that of another entity. This particular pairwise relationship within this particular environment can be termed uncertain. In these sorts of environments, the information may be unclear. In the later circumstance, a group of decision makers may not be able to converge their opinions to a single judgment, rather each decision maker chooses to retain his or her own judgment. Hence the consensus may best be described by a probability distribution and can therefore also be termed uncertain. No matter what the cause of the uncertainty of the judgments may happen to be, careful and complete analysis of the problem hierarchy and the probability distributions describing the judgments of this structure can yield potentially more confidence in the results than traditional sensitivity analysis. Further, expected priorities can be derived and a comparison of and between the point estimate judgment approach and the judgment distribution approach may yield some useful conclusions regarding the original data. Likewise, analysis of inconsistency may reveal useful information regarding the overall importance of some uncertain judgments. Arbel and Vargas [1] studied the effect of interval judgments on the A H P and found that within these ranges, there exists relatively optimal point estimators which best categorize the intervals. Moreno-Jimenez and Vargas [2] studied the problem of determining the most probable ranking of alternatives that one should infer when decision makers use interval judgments rather than point estimates in the AHP. However this later study was analytical and was limited to the single reciprocal matrix. Most problems expand far beyond the single matrix into highly complex hierarchies and networks. This paper examines the scope and influences of interval judgments
beyond the reciprocal matrix and offers an approach for determining the expected rank and expected weight of each alternative.
2. The decision matrix
In order to derive a set of priorities of alternatives with respect to a governing criterion, the relationship between each pair of alternatives must be placed into a reciprocal matrix. We define a square matrix A = (aij), V i , j ~ [1,n] to be a reciprocal matrix with n alternatives where aij = 1/aji and aij indicates that the ith alternative is aij times more dominant than the jth alternative with respect to the criterion governing the matrix. Once the reciprocal matrix is determined, the priorities are calculated. From this matrix, the inconsistency ratio (IR) can also be calculated. Acceptable values of the IR must be less than 0.10 [5,6]. If a particular judgment of matrix A is described by an interval or by a distribution rather than by a point estimate, there are several ways of choosing a single representative value for the judgment from the feasible range: (1) use the arithmetic mean of the range as the point estimate, (2) use any value for the judgment within the range that yields the lowest IR value, (3) use the most-likely value, or (4) take the geometric mean of a large set of values chosen randomly from the distribution. However, each of these approaches are problematic since the representative value of the distribution may not yield an acceptable nor expected ranking. The first suggested approach may be problematic since the arithmetic mean of the range may yield an IR value larger than 0.10. Besides, there may be one ore more extreme judgments resulting in a highly skewed distribution for the judgment. In this case, the mean of the distribution may not be an appropriate value to use. The second suggested approach may also be problematic since all discrete values within the distribution may yield an IR greater than 0.10. Even if some discrete values within the distribution resuited in an IR for the matrix less than 0.10, the objective of the problem may not require in-
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
finitesimally small amounts of inconsistency. The most-likely value, in some cases, may not have a very high probability c o m p a r e d to the other values. The fourth suggested approach suffers from shortcomings similar to the first suggestion. However, the advantage to the fourth approach is that it is similar to group decision making in which the geometric mean of the judgments of all the decision makers is used as the representative point estimate. However, this representative value may also have an IR value greater than 0.10. Therefore another approach will be required.
29
! if
I : [
iil
!!
f
!
i i
3. T h e d i s t r i b u t i o n a l o n g t h e s c a l e
The scale used in the A H P for pairwise comparisons is the 1 to 9 ratio scale [3-5]. We note that the inverse relationships are simply the multiplicative inverses of the values from within the 1 to 9 scale, hence the actual scale used in the A H P is 1 / 9 to 1 to 9. Suppose all of the judgments of a reciprocal matrix are uniformly distributed within 100p% of the stated judgment. The feasible range of judgments for a point estim a t o r value x is x +_px. For all values between x-px and x + p x which are greater than 1, the A H P ratio scale and the x +_p x scale are equivalent. We call the x +_px distribution a virtual scale since this scale is used to describe the judgment interval but in general is not actually used by the A H P (see Fig. 1). The portion o f the virtual scale, x ++_px, that is less than 1 must be m a p p e d back onto the actual scale in order for it to retain the properties of a ratio scale and be useful to the AHP. Consider an example which leads one to this conclusion. Suppose we are interested in randomly generating a judgment from the uniform distribution x + px, where x = 1, and p = 0.5. Any value less than 1 on the ratio scale actually corresponds to the 1 / 9 to 1 portion of the ratio scale that is actually used in the AHP. For this particular example, the largest possible value that we may use is 1 + 0.5 × 1 = 1.5, which is slightly above "equal" and below "equal to m o d e r a t e . " The lowest possible value that we may use, since this range of values is symmetrical must also be slightly
I I
• 11 i
~r I
, ii
~,~, :
J',
'1,
Fig. 1. The virtual scale versus the actual scale.
above "equal" and below "equal to m o d e r a t e " but for the reciprocal relationship. Therefore we know that the 1 - 0.5 × 1 = 0.5 must signify the reciprocal of 1.5. Further, any value below 1 that is generated must be m a p p e d onto the actual scale. There are two possible errors which one may encounter if one fails to account for such a mapping: (1) a judgment of zero may be generated and the reciprocal judgment cannot be used, and (2) the width of the interval is different when considering an interval and a ratio scale and thus one cannot use interval scale generators with ratio scale numbers. Suppose v is a specific value chosen randomly from the uniformly distributed interval x _+px. If v < 1, then v is a m e m b e r of the set of points attainable from the virtual scale and v is not a m e m b e r of the set of attainable points from the actual scale. The actual scale equivalent of v can be m a p p e d via the one-to-one function /(v)
-
1
2-v
.
(l)
If L' > 1, then v is a m e m b e r of both the set of attainable points from the virtual scale and the set of attainable points from the actual scale. As such, v does not need to be altered when u >_ 1,
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
30
therefore the actual ratio scale equivalent of v can be mapped via the one-to-one function f ( v ) = v.
from the standpoint of a prospective student according to their desirability. Six independent characteristics were selected for the comparisons - learning, friends, school life, vocational training, college preparation, and music classes. See Fig. 2 for the hierarchy. The pairwise judgment matrices for this hierarchy were as shown in Tables 1 and 2 and the overall synthesis of priorities is shown in Table 3. According to these judgments, school A (one) should be chosen since school A received the greatest overall synthesized priority (see Table 3). However, suppose the weights of the criteria or the weights of the alternatives were in question. These questionable weights can be perturbed in order to measure the amount of change in the overall synthesized priorities of the alternatives. However, the magnitude or the extent to which one should perturb these weights is arbitrary. Hence such an analysis, however revealing it may be, lacks a basis in realism. The magnitude of the weight perturbations may be realistically impossible alterations and therefore the conclusions drawn from the sensitivity analysis may not be applicable. Suppose each of the point estimates describing the pairwise comparisons were uncertain. Let each point estimate be the midpoint of a uniform distribution the width of which is 50% of the value of the point estimate. Therefore any value within x + 0.25x is equally as valid as any other value in the range for describing the pairwise relationship. The width of the interval may coin-
(2)
Thus, the mapping of any point v along the virtual scale to the actual scale can be accomplished by the one-to-one function
f(v)
=
-
v
ifv
(3)
,
ifv>l where x - p x < v < x + p x
and f ( x - p x ) < f ( v )
<_/(x + px). Within each interval, a value is randomly generated based on the probability distribution. This selected value is mapped to the actual scale used in the A H P by (3). Note that when p = 0, the value which will be randomly generated is simply the point estimate x. In short, as p increases, the variance of the distribution increases and the number of inconsistent combinations of judgments also increases. Also as p increases, the rankings of the alternatives is likely to change more than in the case when p is small. Hence for each value of p, a level of confidence must be developed in order to describe the most probabilistic rankings.
4. A n a l y s i s
- the simulation
approach
Consider the following example from Saaty [5]. Three high schools, A, B, and C, were analyzed
II Ooal I
I Learg II Friends II SchooLiIq VocatioO II Preparation CoXgo II Music Training Classes
(One)
(Two)
School A (One)
(Three)
(Four)
(Five)
(Six)
I
I
(Two)
(Three)
Fig. 2. School selection example.
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37 Table 1 Comparison of characteristics with respect to overall satisfaction with school Relative to: The goal
One Two Three Four Five Six
One
Two
Three
Four
Five
Six
1.0
5.0 1.0
7.0 3.0 1.0
5.0 0.2 0.25 1.0
3.0 0.16667 0.2 0.2 1.0
1.0 0.16667 0.2 0.16667 1.0 1.0
31
Table 3 Synthesis utilizing the point estimates as the judgements Synthesis with respect to goal Overall inconsistency index = 0.09 One Two Three
0.398 0.357 0.244
Table 4 Composite of 500 runs based on a uniform probability distribution (frequency of each rank) Composite results
cide with the maximum difference between judgments of group members making a c o m m o n decision. It may even coincide with the maximum amount an individual decision has been known to deviate. For each of the matrices of the school example, discrete values for the judgments were randomly generated from the uniform distribution of the provided point estimator +25%. After all matrices were determined, the overall synthesized priorities were found and the rank was recorded. This process was repeated 500 times. See Table 4 for the details. Of the 500 runs, 96.4% resulted in the first alternative with the first rank. H e n c e we can conclude that the ranking of the first alternative via the point estimates as the judgments are 96.4% accurate. Even within this level of uncertainty (_+25%), we can have 100% confidence in the ranking of the third alternative (see Table 4).
Rank:
One
Two
Three
One
Two
Three
1 2 3
482 18 0
18 482 0
0 0 500
96.4% 3.6% 0.0%
3.6% 96.4% 0J)%
[/.(le~ 0.0% 100.0%
In the AHP, priorities with an IR greater than 0.10 are considered to have judgments which are too random-like [5,7]. Hence, of these 500 runs, only those with an IR less than 0.10 should be considered (see Table 5). Of the 500 repetitions, only 418 runs had an IR < 0.10. By examining these more restrictive results, the same conclusions regarding the confidence in the rankings can be drawn. Despite all the uncertainty in the judgments, the third alternative was ranked in the third position 100% of the time. Hence we may conclude that the ranking is fixed irrespective of the level of consistency demanded by the problem solver. Likewise, any group disagreement con-
Table 2 Comparison of schools with respect to the six characteristics Relative to: Criterion one
One Two Three
Relative to: Criterion three
One
Two
Three
1.0
0.33333 1.0
0.5 3.0 1.0
Relative to: Criterion two
One Two Three
One Two Three
Relative to: Criterion five
One
Two
Three
1.0
5.0 1.0
1.0 0.2 1.0
Relative to: Criterion four
One
Two
Three
1.0
1.0 1.0
1.0 1.0 1.0
One Two Three
One Two Three
One
Two
Three
1.0
0.5 1.0
1.0 2.0 1.0
Relative to: Criterion six
One
Two
Three
1.0
9.0 1.0
7.0 0.2 1.0
One Two Three
One
Two
Three
1.0
6J) 1.0
4.(I 0.3333 1.0
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
32
Table 5 Composite of the 418 runs with IR < 0.10 based on a uniform probability distribution (frequency of each rank) Consistency-limit results Rank
One
Two
Three
One
Two
Three
1 2 3
405 13 0
13 405 0
0 0 418
96.9% 3.1% 0.0%
3.1% 96.9% 0.0%
0.0% 0.0% 100.0%
cerning a judgment or set of judgments is clearly pointless with regard to the overall synthesis for this particular problem since very little will change in the outcome. Since a great deal of faith can be placed in the ranking of the alternatives for this problem within the given level of uncertainty of the environment, an analysis of the inconsistency is not too revealing (see Table 6). Across all 500 runs, the inconsistency index is clearly normally distributed around 0.09. We can conclude that most of the runs are below the 0.10 upper limit, however, that conclusion could have been drawn directly from the number of runs which met this criterion. The only use that the inconsistency information yields is a primitive measure of the feasible minimum level of inconsistency. Since we know the distribution of the inconsistency index, we can easily determine the likelihood that a particular level of inconsistency could be achieved. From this, we can conclude whether or not that level is attainable and worth the cost of reaching that level (see Table 6). Consider a more complicated and buoyant example. Suppose a particular problem consisted of five criteria and four alternatives. See Fig. 3 for the hierarchy. The reciprocal matrices of the point estimate judgments of this problem are shown in Tables 7 and 8 and the overall synthesis utilizing these matrices is shown in Table 9. In this problem, alternative number one is globally more preferred to the other four alternatives, although each alternative is closely weighted. However, suppose that the judgments from which the synthesis was derived were accurate only to within 25% above and 25% below the given point estimates. Given this particular circumstance, simulation was run. For each judg-
Table 6 Frequency distribution of the different levels of the IR across all 500 simulated runs based on a uniform probability distribution Composite results Inconsistency distribution 0.00) 0.01) 0.02) 0.03) 0.04) 0.05) 0.06) 0.07) 0.08) 0.09) 0.10) 0.11) 0.12) 0.13) 0.14) 0.15) 0.16) 0.17) 0.18) 0.19)
5 33 93 154 133 59 14 9
0.20) 0.21) 0.22) 0.23) 0.24) 0.25) 0.26) 0.27) 0.28) 0.29) 0.30) 0.31) 0.32) 0.33) 0.34) 0.35) 0.36) 0.37) 0.38) 0.39)
0.40) 0.41) 0.42) 0.43) 0.44) 0.45) 0.46) 0.47) 0.48) 0.49) 0.50) 0.51) 0.52) 0.53) 0.54) 0.55) 0.56) 0.57) 0.58) 0.59)
0.60) 0.61) 0.62) 0.63) 0.64) 0.65) 0.66) 0.67) 0.68) 0.69) 0.70) 0.71) 0.72) 0.73) 0.74) 0.75) 0.76) 0.77) 0.78) 0.79)
0.80) 0.81) 0.82) 0.83) 0.84) 0.85) 0.86) 0.87) 0.88) 0.89) 0.90) 0.91) 0.92) 0.93) 0.94) 0.95) 0.96) 0.97) 0.98) 0.99)
ment x given the reciprocal matrices, representative judgment points were randomly generated from within the uniformly distributed range of x + 0.25x (see Table 10). Of the 500 simulations, all 500 runs resulted in an IR < 0.10 hence each run is both feasible and quite realistic. Consider the results of Table 10. Although each alternative has seen each possible rank, it is clear that alternative one is inclined to be positioned in the first rank. Likewise, alternative two is inclined to be positioned in the fourth rank. However, how much confidence can we have in Table 7 The pairwise comparisons of the criteria relative to the goal Relative to: The goal One Two Three Four Five
One
Two
Three
Four
Five
1.0
1.0 1.0
1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0 1.0
D. Hauser, P~ Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
33
Table 8 Comparison of alternatives with respect to the five criteria Relative to: Criterion one
One Two Three Four
Relative to: Criterion four
One
Two
Three
Four
1.0
1.2 1.0
1.4 1.2 1.0
1.6 2.0 3.0 1.0
Relative to: Criterion two
One Two Three Four
One Two Three Four
One
Two
Three
Four
1.0
2.0 1.0
3.0 2.0 1.0
4.0 3.(1 2.0 1.0
Relative to: Criterion five
One
Two
Three
Four
1.0
0.83333 1.0
0.71429 0.83333 1.0
0.625 0.71429 0.83333 1.0
One Two Three Four
One
Tow
Thre
Four
1.0
0.5 1.0
0.33333 0.5 1.0
(I.25 0.33333 (1.5 1.0
Relative to: Criterion three
One Two Three Four
One
Two
Three
Four
1.0
1.0 1.0
1.0 1.0 1.0
1.0 1.0 1.0 1.0
these two rankings and what is the most probabilistic ranking of alternatives three and four? In order to address these two related questions, we must first develop the notion of expected rank. We can see from Table 10 that the first alternative will be of rank one 68.6% of the time. Can we infer that we have 68.6% confidence in that rank? When considering each alternative and each rank individually, this perception of the measure of confidence is accurate. However, we may be interested not in the rank of one alternative, rather we are interested in the ranks of all alternatives simultaneously.
I 1 21 I II one II Two U II F= II ivo II I I I I I I I I J Fig. 3. Example with five criteria and four altenratives.
If we have 68.6% confidence that the first alternative will be in the first rank, we cannot simply say that we have 1.6% confidence that the second alternative will be in the first rank, since there can only exist a single alternative for each Table 9 Synthesis utilizing the point estimates as the judgements Synthesis with respect to goal Overall inconsistency index = 0.01 One Two Three Four
0,262 0,238 0,247 0,253
Table 10 Composite of 500 runs based on a uniform probability distribution (frequency of each rank) Composite results
Rank: One Two Three Four 1 2 3 4
343 8 40 111 32 127 42 112 236 4 348 97
109 230 110 51
One
Two
Three Four
68.6% 1.6% 8.0% 22.2% 6.4% 25.4% 8.4% 22.4% 47.2% 0.8% 69.6% 19.4%
21.8% 46.0% 22.0% 10.2%
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
34
rank. However, a composite of the four different levels of confidence for each of the four alternatives will reveal the overall ranking of each alternative. We know from Table 10 that alternative one is of rank one 68.6% of the time, of rank two 22.2% of the time, of rank three 8.4% of the time and of rank four 0.8% of the time. Hence, these percentages could act as weights of the ordinal ranking positions. We define expected score to be the composite value representing the probabilities of each rank for each alternative. Consider the following equation for expected score. ESi = ~'~ ( P i , k ) ( k ) ,
Vi ~ [1,n],
(4)
k=l
where ESi is the expected score of the ith alternative, and Pi,k is the proportion of the trials that the ith alternative had rank k. There are two problems with this approach. First, the score is mathematically the expected rank. Rarely will the ranks computed in this manner be integral. Since a rank order is, by definition, the set of indices of an ordered list, the definition of (4) cannot be complete. Second, lower rankings are directly correlated with the larger weights, however in equation (4), the lower rankings are inversely correlated with the larger weights. Therefore rather than sum together the product of the fraction of time each rank occurred and the rank itself, we will sum together the product of the fraction of the time each rank occurred and n + 1 minus the rank itself. In this manner, the resultant will be inversely correlated with the weights and the larger weights we know will yield the lower ranks. Therefore we improve upon (4) and consider the function n
ES~ = ~] Pi,k(n + 1 -- k ) ,
Vi ~ [l,n],
(5)
k~l
where ES i is the expected score of the ith alternative, and Pi,k is the proportion of the trials that the ith alternative had rank k. Next, we define the expected weight to be the normalized expected scores. When the alternatives are placed in descending order of the ex-
Table 11 Expected results based on a uniform probability distribution Alternatives
Expected rank
Expected weight
One Two Three Four
1 4 3 2
0.3586 0.1400 0.2220 0.2794
pected weights, the results reveal the expected rank of alternatives. Hence, we define EWi
ESi z.,@ESk,
Vi ~ [l,n].
(6)
k=l
The expected weights of (6) are determined from the frequency that each rank occurred for each alternative. Hence these weights are statistical weights indicating a composite frequency or a mean of feasible weights around which we expect the actual weights to be scattered. The actual meaning and interpretation of ES~ of (5) is misleading. We understand that we cannot ever actually achieve a fractional rank, hence after we compute the expected weights of (6), we simply define the expected rank, ER~, to be the index of the ith alternative once sorted in non-increasing order of EW/. Continuing with earlier example, we calculate the expected weights of the four alternatives and derive the expected rankings (see Table 11). Confirming our earlier conclusion, we see that alternative one has the highest expected weight (the lowest expected ranking) and alternative two has the lowest expected weight (the highest expected ranking). Consider the expected weight of alternative four which has an expected rank of two. We see that the expected weight of alternative four (0.2794) is greater than the expected weight of the third ranking alternative (0.2220). Hence we can be assured that alternative four will be of rank two. Likewise, we can be assured that alternative three has the third rank since its expected weight (0.2220) is greater than the expected weight of the fourth ranking alternative (0.1400). In general, the frequency distributions of the rankings for an alternative could range from
D. Hauser, P. Tadikamalla / European Journal of Operational Research 91 (1996) 27-37
highly skewed ("exponential-like") to "normal" or could be "uniform like". For example, we see that the frequency distributions of the first and second alternatives seems to be highly skewed indicating quite clearly that the most probable ranks for these alternatives are in the first and fourth positions, respectively. However, we see that the frequency distributions of the third and fourth alternatives seem to be "normally" distributed around the third and second ranks, respectively.
5. The probability distributions In two examples considered here, we have assumed uniform distributions for all judgments in the range of x + p x . This, probably, is an extreme case implying that the decision makers are not only unable to converge to a single estimate of their judgment but their judgments are uniformly distributed across the range. A more reasonable scenario is that some proportion of the people may converge to a "modal value" within the range considered. This modal value could be in the center of the range creating a symmetric distribution or it could be anywhere in the range creating skewed distributions. Beta distributions and triangular distributions are often used to represent different shapes of data with finite known range of values. In the A H P type applications where we can only assume to know the range and the mode, triangular distributions are reasonable and easier to use. We use symmetric triangular distributions for the second example considered earlier. Note that for a given range, ( a , b ) , the uniform distribution has a higher variance 2 O'uniform =
~(b -
a) 2
than the symmetrical triangular distribution 2
O'syrnmetrical triangular
= ~ ( b - a) 2.
Thus the use of triangular distributions, as opposed to the uniform distribution, should provide more "consistent" rankings.
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Table 12 Composite of 500 runs based on a symmetrical triangular probability distribution (frequency of each rank) Composite results Rank One Two Three Four
One
1 2 3 4
77.0% 0.6% 3.4% 19.0% 18.6% 3.2% 25.2% 53.0% 4.2% 19.2% 53.8% 22.8% 0.2% 77.0% 17.6% 5.2%
385 3 93 16 21 96 1 385
17 126 269 88
95 265 114 26
Two
Three Four
Consider once again the second example from above. See Fig. 3 for the hierarchy and Tables 7 and 8 for the reciprocal matrices. The composite results of Tables 10 and 11 describe the outcomes and computations regarding the simulation of 500 runs utilizing a uniform probability distribution x ++_px. In contrast, consider the influence of a symmetric triangular probability distribution. Table 12 contains the composite results of the 500 simulated runs which utilized a triangular distribution centered about x with lower and upper extremes equal to x - p x and x + p x , respectively. Just as in the uniform scenario, all 500 runs of the triangular scenario resulted in an IR _< 0.10, hence all results are feasible. The conclusion drawn earlier regarding the rank is more than supported by the results of the triangular probability distribution simulations. From the uniform probability distribution results, we concluded that the first alternative was quite skewed; leaning toward the first ranking. From the triangular probability distribution, we see that the first alternative is extremely skewed; leaning toward the first ranking. Likewise, each of the other three rank-distributions are more skewed when the symmetrical triangular probability distribution was utilized than when the uniform probability distribution was utilized. Since the center point of the symmetrical triangular distribution is the point estimate, one would expect that the triangular probability distribution should result in an expected rank more closely aligned with the rank of the deterministic system utilizing solely the specified point estimates for each judgment. Consider the expected ranks of the triangular probability simulated results (see Table 13). The
D. Hauser, P. Tadikamalla /European Journal of Operational Research 91 (1996) 27-37
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Table 13 Expected results based on a symmetrical triangular probability distribution Alternatives
Expected rank
Expected weight
One Two Three Four
1 4 3 2
0.3724 0.1274 0.2144 0.2858
expected rank of the triangular system matches the expected rank of the uniform system. Hence we can conclude that for this problem, an environment as uncertain as a uniform probability distribution depletes only a small measure of confidence in the resultant. The resulting rank is held constant from the point estimate approach to the symmetrical triangular probability distribution approach all the way to a uniform probability distribution approach. We must note that the expected weights of the triangular system are more distinct than the expected weights of the uniform system. In the uniform system, the first ranking alternatives yielded an expected weight of 0.3586 and the second ranking alternative yielded an expected weight of 0.2794. In the triangular system, the first ranking alternative yielded an expected weight of 0.3724 and the second ranking alternative yielded an expected weight of 0.2858. The differential is larger in the triangular system indicating that the symmetrical probability distribution results in more definitive figures and demonstrates that one may have a greater measure of confidence in the resulting rank than the uniform probability distribution. If an environment was completely certain, the rank is simple to determine and hence complete confidence in the result is assured. However, as uncertainty increases, the probability distribution decreases the confidence. If the uncertainty of the judgments can be described by a triangular probability distribution, we know the confidence in the result will be less than the measure of confidence we would have in a certain environment. If the uncertainty of the judgments is high and can be described by a uniform probability distribution, we know the confidence in a result
will be less than the measure of confidence we would have in a triangular probabilistic environment.
6. Conclusions
Traditional sensitivity analysis requires that the decision maker perturb in a deterministic fashion the weight of one or more criteria a n d / o r alternatives in order to note changes in the ranks of the alternatives [5]. However, by how much and in what fashion should one perturb the various weights is an important issue. The basic premise behind the A H P is that one notion can only be understood when it is compared relative to another notion and has little meaning otherwise [3-6]. Therefore one cannot perturb a weight of an element in a hierarchy and simply draw a conclusion from the changes in weights a n d / o r ranks of the alternatives unless the magnitude of the proposed perturbation has some meaning and is not arbitrary. Further, it may very well be that such a change is not truly possible in the real world. For instance, increasing the weight of a criterion of a hierarchy may have a substantial impact on the ranks of the alternatives; however the decision maker may not be able to control the weight of that criterion in real life. The precise interpretations of the weight perturbations that the decision maker institutes may be difficult to make. Then by how much should one perturb these weights and in what direction? The exact relationship between realistic variations in the judgments may not be known relative to perturbations in the weights themselves, hence simulation offers a more rich approach. Suppose one employed the traditional sensitivity analysis approach and found a different ranking for certain alterations in the weights of some of the elements of the hierarchy. What rank order for the alternatives can one expect? At best, the only conclusion that can be drawn from traditional sensitivity is some measure of confidence in the rank. For a partially or even for a completely uncertain environment, simulation is a better approach for providing a measure of confidence in a
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rank, and for providing expected weights and ranks. References [1] Arbel, A., and Vargas, L.G., "The analytic hierarchy process with interval judgements", in: Proceedings of the MCDM Conference held in Washington DC, August 1990. [2] Moreno-Jimenez, J.M., and Vargas, L.G., "A probabilistic study of preference structures in the Analytic Hierarchy Process with interval judgements", Mathematical and Computer Modelling 17 (1993) 73-81.
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[3] Saaty, T.L., "Scaling method for priorities in hierarchical structures", Journal of Mathematical Psychology 15 (1977) 234-281. [4] Saaty, T.L., "Axiomatic foundation of the Analytic Hierarchy Process", Management Science 32 (1986) 841-855. [5] Saaty, T.L., Multicriteria Decision Making: The Analytic Hierarchy Process, RWS Publications: Pittsburgh, PA, 1990. [6] Saaty, T.L., and Vargas, L.G., "Uncertainty and rank order in the Analytic Hierarchy Process", European Journal of Operational Research 32 (1987) 107-117. [7] Vargas, L.G., "Reciprocal matrices with random coefficients", Mathematical and Computer Modelling 3 (1982) 69-81.