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A simulation-based evaluation of the approximate and the exact eigenvector methods employed in AHP
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 95 (1996) 656-662
Theory and Methodology
A simulation-based evaluation of the approximate and the exact eigenvector methods employed in AHP N. V i n o d K u m a r , L . S . G a n e s h
*
Industrial Engineering and Management Division, Department of Humanities and Social Sciences, Indian Institute of Technology, Madras 600 036, India Received August 1994; revised September 1995
Abstract
The Analytic Hierarchy Process (AHP) is a popular multi-criteria decision making method. It provides ratio-scale measurements of the priorities of elements in various levels of a hierarchy. These priorities are obtained through pairwise comparisons of elements in one level with reference to each element in the immediate higher level. Saaty has suggested two methods, viz., the approximate eigenvector method (AEV) and the exact eigenvector method (EEV) for calculating the priorities. Research on AHP has led to conflicting views on the use of these two methods. This paper presents a simulation analysis to evaluate them. The simulation analysis uses the concept of approximating a continuous pairwise comparison (CPC) matrix by its closest discretized pairwise comparison (DPC) matrix. This exercise supports Saaty's theoretical statements and empirically concludes that the EEV method is to be preferred over the AEV method for the calculation of priority vectors.
Keywords: Analytic Hierarchy Process; Exact eigenvector method; Approximate eigenvector method; Continuous pairwise comparison (CPC) matrix; Discretized pairwise comparison (DPC) matrix
1. Introduction
Multi-Criteria Decision Making (MCDM) is a prominent area of research in normative decision theory. The Analytic Hierarchy Process (AHP) is one of the popular methods employed in MCDM. It provides ratio-scale measurements of the priorities of
* Corresponding author.
elements in various levels of a hierarchy. These priorities are obtained through pairwise comparisons of elements in one level with reference to each element in the immediate higher level. The reviews by Vargas (1990), Shim (1989), and Zahedi (1986) list various application areas of AHP. It has also been criticised, especially by Schenkerman (1994), Murphy (1993), Dyer (1990a,b), Belton (1986), and Belton and Gear (1983). Many of these criticisms have been responded to by Saaty (1986, 1987, 1990, 1994) and Harker and Vargas (1987, 1990) and Vargas (1994).
0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 5 ) 0 0 3 0 2 - 9
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
One of the contentious issues in AHP is the calculation of a priority vector from a pairwise comparison matrix. The two commonly employed methods for the purpose are the Exact Eigenvector (EEV) Method and the Approximate Eigenvector (AEV) method. The EEV method uses Gantmacher's principles to obtain the priority vector and it is supported by mathematical proofs (Saaty, 1980, 1990). In this paper, the mathematical notations and proofs are not presented. Saaty (1980) also proposed the AEV method. This is more popularly known as the Geometric Mean Method or the Method of Least Squares. Research on AHP has featured many arguments on the appropriateness of these methods. Crawford and Williams (1985) carried out a series of simulation exercises and favored the AEV method. They used a continuous scale for evaluating the two methods. Saaty (1990) pointed out that the EEV and the AEV methods provide different rankings using the same pairwise comparison matrix. He used the nine-point discrete scale proposed by him earlier (Saaty, 1980) and suggested the use of the EEV method on the basis of theoretical arguments. Triantaphyllou and Mann (1990, 1992), and many users of AHP have preferred the AEV method for its computational simplicity. A further evaluation of these two methods becomes necessary since Crawford and Williams (1985) have carried out their evaluation using a continuous scale whereas Saaty (1990) has based his on the nine-point discrete scale commonly adopted by AHP users. This paper presents a simulation exercise to evaluate the two methods. It uses Triantaphyllou and Mann's (1990) concept of approximation of a consistent CPC matrix by its closest DPC matrix with reference to Saaty's nine-point scale. The approximation of consistent CPCs by discrete numbers in Saaty's nine-point scale is based on the common assumption that human beings are rational and can distinguish between alternatives in a consistent manner. The need for a simulation-based evaluation of the two methods also arises because the experts using AHP have access only to the nine-point discrete scale. Although, using a continuous scale in their minds they are capable of making fine distinctions between a pair of alternatives, the nine-point discrete scale does not allow such fine distinctions. Hence it
657
becomes necessary to evaluate how close the approximate discretized pairwise comparison values are to the hypothesized continuous-scale values. The 'closeness' is affected by the choice of the method used for calculating the priority vectors, i.e. the AEV and the EEV methods. The need for a simulationbased evaluation of these two methods should now be obvious. The organization of the paper is as follows. In Section 2, both the methods are explained briefly. In Section 3, the concepts of CPCs and DPCs are explained in detail. In Section 4, the simulation exercise and the results are discussed. In Section 5, conclusions of the paper are presented.
2. The exact and the approximate eigenvector methods Priority vectors of the alternatives in an AHP hierarchy are usually calculated from the pairwise comparison matrices using the exact and the approximate eigenvector methods. There are other methods such as simple additive normalisation and multiplicative normalisation, but, Saaty (1980) has preferred the exact and the approximate eigenvector methods over other methods. 2.1. The exact eigenvector m e t h o d
The eigenvector for a consistent reciprocal square matrix can be obtained by normalization of any column vector of the matrix. The conditions for a consistent reciprocal square matrix are as follows: (i) aii = 1/a~i for all i and j. (ii) aij = aik * aki for all k other than i and j. The AEV and the EEV methods yield the same eigenvectors in the case of a consistent reciprocal square matrix. Saaty (1980) has shown that in the case of a reciprocal inconsistent square matrix (a pairwise comparison matrix) the principal eigenvector should be considered as the priority vector of that matrix. From the maximum eigenvalue, the consistency ratio and the consistency index of the matrix are obtained. The computation of the maximum eigenvalue and the corresponding eigenvector is described below.
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N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
Table 1 Pairwise comparison matrix for four alternatives and the corresponding priority vector (principal eigenvector) 1 V 1.000 I 1.277 1.186 0.567
l 2 3 4
2 0.783 1.000 0.928 0.444
3 0.843 1.080 1.000 0.478
The equation A w = Aw
is iterated till a column vector w satisfying this equation is obtained. The normalized w column vector corresponds to the principal eigenvector of the matrix A. The iteration starts with an initial unit vector w, where, A is a reciprocal square matrix, w is the principal eigenvector of A, and A is the maximum eigenvalue of A. 2.2. The approximate eigenvector method
Let aij be an element of matrix A. For each row calculate Eq. (1). wi=
ai~
for all i = 1 . . . . . n.
(1)
Normalization of the column vector thus obtained from the above equation gives the AEV for the corresponding matrix. Crawford and Williams (1985) have proved the validity of this method.
3. Concept of the closest DPC matrix The concept of the closest DPC matrix has been suggested by Triantaphyllou and Mann (1990). They have applied it to deal with scales for pairwise comparisons, and applicability of pairwise compari-
4 1.763-] 2.252 / 2.091| 1.000J
priority vector F 0.2481 "] | 0.3171/ ~ 0.2941 | L_0.1407 ]
son matrices and the eigenvector method for the estimation of membership values in a fuzzy set (Triantaphyllou and Mann, 1990, 1992; Triantaphyllou et al., 1993). The rationality assumption has already been mentioned in Section 1. All other fundamental assumptions of the AHP, such as the inability of experts to distinguish among altematives when the number of alternatives is 7 + 2 (Miller, 1956), the requirement that alternatives be mutually exclusive, etc., are also considered here. In the simulation exercise, however, we have considered upto 15 alternatives in order to compare the two methods comprehensively. Table I showsthe cardinal values for the alternatives in a CPC matrix. Since the matrix is consistent, both the AEV and the EEV methods yield the same priority vector (principal eigenvector) shown in the table. But, an expert cannot provide judgements using a continuous scale as he has to choose only from among the options available in Saaty's discrete scale, 1 which has only the values ~, ~1 , . . . , ~ 1, 1, 2 . . . . . 9. As the experts are assumed to be rational, they ought to choose the closest available values from the ninepoint scale for approximating the corresponding pairwise comparisons in the continuous scale. The approximate matrix thus obtained is called the closest DPC matrix, and it is shown in Table 2 for the CPC matrix of Table 1.
Table 2 Closest DPC matrix of the matrix in Table 1, and the corresponding priority vectors using the EEV and the AEV methods I 2 3 4 EEV AEV
[ 0
2 3 4
l.OOO 1.000 0.500
0
1.000 1.000 0.500
0
l.OOO 1.000 0.500
2.000 2.000 I 1.000_.[
0.2857 L 0.2857 I 0.1429 _J
0.2857 / 0.2857] 0.1429_1
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
The above concept of approximation using the closest DPCs is the key input in the simulation exercise to follow.
bi
n
659
is the priority of element i in the priority vector obtained from the approximate DPC matrix of the simulated consistent CPC matrix. is the size of the reciprocal pairwise comparison matrix.
4. Simulation exercise and results
The simulation exercise is carried out by obtaining the closest DPC matrix from a consistent CPC matrix. The 'closeness' of the two priority vectors obtained by applying the AEV method to the CPC and the DPC matrices is compared with the 'closeness' of the two priority vectors obtained by applying the EEV method on the two matrices. The following indices are used to measure the 'closeness'. The root mean square deviation (RMS) This is measured in the following way:
i ~(ei--bi)2 RMS =
(2)
i= 1 n
where: RMS is the Root Mean Square deviation. is the priority of element i, using either the ei AEV or the EEV method, in the priority vector obtained from a simulated consistent CPC matrix.
The median absolute deviation about the median (MADM) This is not a commonly used statistical measure of variation, but earlier researchers (Saaty, 1980) have used it to compare various scales in the context of AHP. It is expressed in the following way: MADM=med(Ici-bil-med{Ici-bil}),
(3)
where MADM is the Median Absolute Deviation about the Median, and med{I c / - bil} is the median of the set of absolute deviations. The simulation exercise has been camed out for five runs of 1000 matrices each, for every matrix of size 2 to 15. The overall mean of the five RMS and MADM values corresponding to the five simulation runs has been calculated. The RMS values for the five simulation runs employing the EEV method and the average RMS deviation are shown in Table 3. The MADM values for the five simulation runs employing the EEV method and the average MADM value are shown in Table 4. Similarly, the RMS and
Table 3 The RMS values for the exact eigenvector method Size of the matrix
RMS values in simulation runs 1
2
3
4
5
2 3 4 5 6 7 8 9 10 1I 12 13 14 15
0.03050 0.02250 0.01750 0.01329 0.01077 0.00910 0.00774 0.00656 0.00574 0.00514 0.00455 0.00415 0.00375 0.00344
0.02890 0.02227 0.01693 0.01361 0.01133 0.00890 0.00774 0.00662 0.00566 0.00513 0.00461 0.00410 0.00375 0.00349
0.03055 0.02335 0.01723 0.01323 0.01067 0.00910 0.00766 0.00658 0.00586 0.00511 0.00454 0.00414 0.00375 0.00342
0.03111 0.02302 0.01713 0.01367 0.01095 0.00900 0.00757 0.00654 0.00577 0.00504 0.00463 0.00414 0.00379 0.00346
0.03107 0.02257 0.01741 0.01341 0.01093 0.00920 0.00777 0.00656 0.00573 0.00505 0.00458 0.00409 0.00375 0.00345
Average RMS value 0.03043 0.02274 0.01724 0.01344 0.01093 0.00906 0.00769 0.00658 0.00575 0.00509 0.00458 0.00412 0.00376 0.00345
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N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
Table 4 The MADM values for the exact eigenvector method Size of the matrix
MADM values in simulation runs 1
2
3
4
5
Average MADM value
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.00000 0.00456 0.00774 0.00476 0.00376 0.00325 0.00290 0.00235 0.00211 0.00196 0.00170 0.00161 0.00143 0.00130
0.00000 0.00483 0.00781 0.00487 0.00390 0.00311 0.00288 0.00240 0.00211 0.00194 0.00173 0.00155 0.00142 0.00134
0.00000 0.00481 0.00748 0.00494 0.00373 0.00327 0.00277 0.00244 0.00218 0.00188 0.00169 0.00156 0.00144 0.00129
0.00000 0.00472 0.00758 0.00481 0.00371 0.00329 0.00273 0.00240 0.00215 0.00184 0.00175 0.00157 0.00145 0.00133
0.00000 0.00471 0.00771 0.00492 0.00377 0.00340 0.00291 0.00248 0.00211 0.00191 0.00172 0.00158 0.00144 0.00131
0.00000 0.00473 0.00766 0.00486 0.00377 0.00326 0.00284 0.00241 0.00213 0.00191 0.00172 0.00157 0.00143 0.00131
MADM values for the AEV method are shown in Tables 5 and 6. The results of the simulation exercise clearly indicate the superiority of the exact eigenvector method over the approximate eigenvector method. For example consider a 9 X 9 matrix. The average RMS and MADM values for such a matrix when exact eigenvector method is employed are 0.00658 and 0.00241 respectively. The average RMS and MADM values when approximate eigenvector method is employed
for the same matrix are 0.01227 and 0.00342 respectively. Similar differences can be observed for other matrices also. This clearly indicates that the exact eigenvector method gives better results than the approximate eigenvector method. This has already been stated by Saaty (1990). However his work has not sufficiently discouraged users of AHP from employing the AEV method. The results of this simulation exercise run contrary to the claim of Crawford and Williams (1985) that the AEV method provides a
Table 5 The RMS values for the approximate eigenvector method Size of the matrix
RMS values in simulation runs 1
2
3
4
5
Average RMS value
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.11150 0.07455 0.04470 0.02612 0.01458 0.00910 0.00969 0.01222 0.01508 0.01719 0.01912 0.02064 0.02164 0.02282
0.10960 0.07289 0.04530 0.02588 0.01489 0.00890 0.00990 0.01219 0.01523 0.01692 0.01917 0.02066 0.02179 0.02252
0.11130 0.07300 0.04515 0.02610 0.01 447 0.00910 0.00971 0.01227 0.01498 0.01736 0.01906 0.02028 0.02110 0.02331
0.11557 0.07240 0.04458 0.02586 0.01466 0.00890 0.00977 0.01232 0.01516 0.01722 0.01927 0.02046 0.02187 0.02278
0.11369 0.07235 0.04559 0.02594 0.01451 0.00920 0.00991 0.01237 0.01475 0.01696 0.01889 0.02079 0.02195 0.02284
0.11232 0.07304 0.04506 0.02598 0.01462 0.00904 0.00979 0.01227 0.01504 0.01713 0.01910 0.02056 0.02167 0.02285
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
better estimate of the priority vector than the EEV method. The exercise has also revealed that as the size of the matrix increases, the RMS and MADM values decrease for both the EEV and the AEV methods. This is because as the size of the matrix increases, the spread of the values between 0 and 1 decreases. This in tum decreases the RMS and MADM values. The simulation exercise has been carried out using the C language. Rank reversals have not been observed in any of the simulation runs. Although earlier researchers (Crawford and Williams, 1985; Saaty, 1990) have claimed to have observed rank reversals we have only observed rank ties while employing the AEV and the EEV methods. An explanation of the meaning of rank reversal and rank tie is necessary here. Rankings of the alternatives are obtained from the CPC matrix using either the AEV or the EEV method. Similarly rankings for the corresponding DPC matrix are also obtained using either the AEV or the EEV method. If the rankings obtained from the CPC matrix and its corresponding DPC matrix are not in the same order, then rank reversal is said to have occurred. For example, employing the AEV method, let the original rankings of four decision alternatives of a CPC matrix be 2, 3, 1, and 4 respectively. Let the rankings of the same alternatives of the correspond-
661
ing DPC matrix (again, using the AEV method) be 3, 2, 1, and 4 respectively. The ranks of the decision alternatives 1 and 2 are interchanged, i.e., from the original ranking of 2 and 3 to 3 and 2. This is the rank reversal phenomenon in AHP and has been explained in greater detail in Belton and Gear (1983) and Dyer (1990). Employing the EEV method may also lead to rank reversal. Readers may please note that in the case of the CPC matrix, employing the AEV and the EEV methods leads to the same priority vector. The rank tie phenomenon can best be explained through an example. Using the AEV method, let the original rankings of four decision alternatives of a CPC matrix be 3, 2, 1, and 4 respectively. Let the rankings of the same alternatives of the corresponding DPC matrix (again, using the AEV method) be 2, 2, 1, and 4 respectively. Here, decision alternatives 1 and 2 have original rankings 3 and 2 respectively. After approximating the CPC matrix by the corresponding DPC matrix the rankings of the same alternatives are 2 and 2. This is called as rank tie. We have analysed for both rank reversals and rank ties in the simulation exercise and we have encountered only rank ties. Moreover, the AEV and the EEV methods yield the same results in the case of rank ties, and therefore the results are not presented here. We are not denying the possibility of
Table 6 The MADM values for the approximate eigenvector method Size of the matrix
MADM values in simulation runs 1
2
3
4
5
Average MADM value
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.00000 0.01675 0.01512 0.00752 0.00422 0.00324 0.00298 0.00335 0.00371 0.00391 0.00403 0.00411 0.00416 0.00435
0.00000 0.01653 0.01535 0.00742 0.00420 0.00315 0.00305 0.00345 0.00374 0.00387 0.00411 0.00418 0.00421 0.00424
0.00000 0.01638 0.01531 0.00741 0.00417 0.00326 0.00301 0.00350 0.00364 0.00388 0.00418 0.00416 0.00429 0.00423
0.00000 0.01635 0.01512 0.00754 0.00426 0.00326 0.00299 0.00337 0.00372 0.00390 0.00408 0.00423 0.00421 0.00428
0.00000 0.01580 0.01553 0.00759 0.00414 0.00341 0.00313 0.00341 0.00374 0.00392 0.00426 0.00436 0.00425 0.00430
0.00000 0.01636 0.01528 0.00749 0.00420 0.00326 0.00303 0.00342 0.00371 0.00389 0.00413 0.00421 0.00422 0.00428
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N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
rank reversals occurring while employing the two methods. The illustrative example provided by Saaty (1990) indicates that rank reversal is associated with the AEV method. The rank tie phenomenon is independent of the type of method employed for obtaining the priority vector and it has more to do with continuous and discrete scales. It may be possible to eliminate the rank tie phenomenon by using appropriate scales.
5. Conclusions In this paper a simulation exercise has been carfled out to evaluate the AEV and the EEV methods used in AHP for obtaining eigenvectors of pairwise comparison matrices. Both the methods have been suggested by Saaty (1980). This exercise shows the superiority of the EEV method over the the AEV method. This result is contrary to the findings of Crawford and Williams (1985) that the eigenvectors estimated by the AEV method are superior to those obtained by using the EEV method. The simulation analysis uses the concept of approximating a continuous pairwise comparison (CPC) matrix by its closest discretized pairwise comparison (DPC) matrix. This concept is more rational compared to Crawford and William's (1985) perturbations approach. If users of AHP do not choose the closest DPC matrix, then the situation will become more complex and rank reversals may arise. In such situations the EEV method is more preferable as Saaty (1990) has shown. This paper empirically concludes that the EEV method should be preferred over the AEV method.
References Belton, V. (1986), " A comparison of the Analytic Hierarchy Process and a simple multi-attribute value function", European Journal of Operational Research 26, 7-21. Belton, V., and Gear, T. (1983), "'On a shortcoming of Saaty's method of analytic hierarchies", Omega 11,228-230. Crawford, G., and Williams, C., (1985), "'A note on the analysis
of subjective judgement matrices", Journal of Mathematical Psychology 29, 387-405. Dyer, J.S. (1990a), "Remarks on the Analytic Hierarchy Process", Management Science 36, 249-258. Dyer, J.S. (1990b), " A clarification of 'Remarks on the Analytic Hierarchy Process' ", Management Science 36, 274-275. Harker, P.T., and Vargas, L.G. (1987), "The theory of ratio scale estimation: Saaty's Analytic Hierarchy Process", Management Science 33, 1383-1403. Harker, P.T., and Vargas, L.G. (1990), "Reply to 'Remarks on the Analytic Hierarchy Process by J.S. Dyer' ", Management Science 36, 269-273. Miller, G.A. (1956), "The magical number seven plus or minus two: Some limits on our capacity for processing information", Psychology Review 63, 81-97. Murphy, C.K. (1993), "Limits on the Analytic Hierarchy Process from its consistency index", European Journal of Operational Research 65, 138-139. Saaty, T.L. (1980), The Analytic Hierarchy Process, McGraw-Hill, New York. Saaty, T.L. (1986), "Axiomatic foundations of the Analytic Hierarchy Process", Management Science 20, 355-360. Saaty, T.L. (1987),"Rank generation, preservation and reversal in the Analytic Hierarchy decision process", Decision Sciences 18, 157-177. Saaty, T.L. (1990), "Eigenvector and logarithmic least squares", European Journal of Operational Research 48, 156-159. Saaty, T.L. (1994), "Homogeneity and clustering in AHP ensures the validity of the scale", European Journal of Operational Research 72, 598-601. Schenkerman, S. (1994),"Avoiding rank reversal in AHP decision-support models", European Journal of Operational Research 74, 407-419. Shim, J.P. (1989), "Bibliographical research on the Analytic Hierarchy Process", Socio-Economic Planning Sciences 23, 163-167. Triantaphyllou, E., and Mann, S.H. (1990), "An evaluation of the eigenvalue approach for determining the membership values in fuzzy sets", Fuzzy Sets and Systems 35, 295-301. Triantaphyllou, E., and Mann, S.H. (1992), " A computational evaluation of the original and revised Analytic Hierarchy Process", Working Paper. Triantaphyllou, E., et al. (1993), "On the evaluation and application of different scales for quantifying pairwise comparisons in fuzzy sets", Multi-Criteria Decision Analysis. Vargas, L.G. (1990), "An overview of the Analytic Hierarchy Process and its applications", European Journal of Operational Research 48, 2-8. Vargas, L.G. (1994), "'Reply to Schenkerman's avoiding rank reversal in AHP decision support models' ', European Journal of Operational Research 74, 420-425. Zahedi, F. (1986), "The Analytic Hierarchy Process - A survey of the method and its applications", Interfaces 16/4, 96-108.
European Journal of Operational Research 95 (1996) 656-662
Theory and Methodology
A simulation-based evaluation of the approximate and the exact eigenvector methods employed in AHP N. V i n o d K u m a r , L . S . G a n e s h
*
Industrial Engineering and Management Division, Department of Humanities and Social Sciences, Indian Institute of Technology, Madras 600 036, India Received August 1994; revised September 1995
Abstract
The Analytic Hierarchy Process (AHP) is a popular multi-criteria decision making method. It provides ratio-scale measurements of the priorities of elements in various levels of a hierarchy. These priorities are obtained through pairwise comparisons of elements in one level with reference to each element in the immediate higher level. Saaty has suggested two methods, viz., the approximate eigenvector method (AEV) and the exact eigenvector method (EEV) for calculating the priorities. Research on AHP has led to conflicting views on the use of these two methods. This paper presents a simulation analysis to evaluate them. The simulation analysis uses the concept of approximating a continuous pairwise comparison (CPC) matrix by its closest discretized pairwise comparison (DPC) matrix. This exercise supports Saaty's theoretical statements and empirically concludes that the EEV method is to be preferred over the AEV method for the calculation of priority vectors.
Keywords: Analytic Hierarchy Process; Exact eigenvector method; Approximate eigenvector method; Continuous pairwise comparison (CPC) matrix; Discretized pairwise comparison (DPC) matrix
1. Introduction
Multi-Criteria Decision Making (MCDM) is a prominent area of research in normative decision theory. The Analytic Hierarchy Process (AHP) is one of the popular methods employed in MCDM. It provides ratio-scale measurements of the priorities of
* Corresponding author.
elements in various levels of a hierarchy. These priorities are obtained through pairwise comparisons of elements in one level with reference to each element in the immediate higher level. The reviews by Vargas (1990), Shim (1989), and Zahedi (1986) list various application areas of AHP. It has also been criticised, especially by Schenkerman (1994), Murphy (1993), Dyer (1990a,b), Belton (1986), and Belton and Gear (1983). Many of these criticisms have been responded to by Saaty (1986, 1987, 1990, 1994) and Harker and Vargas (1987, 1990) and Vargas (1994).
0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 5 ) 0 0 3 0 2 - 9
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
One of the contentious issues in AHP is the calculation of a priority vector from a pairwise comparison matrix. The two commonly employed methods for the purpose are the Exact Eigenvector (EEV) Method and the Approximate Eigenvector (AEV) method. The EEV method uses Gantmacher's principles to obtain the priority vector and it is supported by mathematical proofs (Saaty, 1980, 1990). In this paper, the mathematical notations and proofs are not presented. Saaty (1980) also proposed the AEV method. This is more popularly known as the Geometric Mean Method or the Method of Least Squares. Research on AHP has featured many arguments on the appropriateness of these methods. Crawford and Williams (1985) carried out a series of simulation exercises and favored the AEV method. They used a continuous scale for evaluating the two methods. Saaty (1990) pointed out that the EEV and the AEV methods provide different rankings using the same pairwise comparison matrix. He used the nine-point discrete scale proposed by him earlier (Saaty, 1980) and suggested the use of the EEV method on the basis of theoretical arguments. Triantaphyllou and Mann (1990, 1992), and many users of AHP have preferred the AEV method for its computational simplicity. A further evaluation of these two methods becomes necessary since Crawford and Williams (1985) have carried out their evaluation using a continuous scale whereas Saaty (1990) has based his on the nine-point discrete scale commonly adopted by AHP users. This paper presents a simulation exercise to evaluate the two methods. It uses Triantaphyllou and Mann's (1990) concept of approximation of a consistent CPC matrix by its closest DPC matrix with reference to Saaty's nine-point scale. The approximation of consistent CPCs by discrete numbers in Saaty's nine-point scale is based on the common assumption that human beings are rational and can distinguish between alternatives in a consistent manner. The need for a simulation-based evaluation of the two methods also arises because the experts using AHP have access only to the nine-point discrete scale. Although, using a continuous scale in their minds they are capable of making fine distinctions between a pair of alternatives, the nine-point discrete scale does not allow such fine distinctions. Hence it
657
becomes necessary to evaluate how close the approximate discretized pairwise comparison values are to the hypothesized continuous-scale values. The 'closeness' is affected by the choice of the method used for calculating the priority vectors, i.e. the AEV and the EEV methods. The need for a simulationbased evaluation of these two methods should now be obvious. The organization of the paper is as follows. In Section 2, both the methods are explained briefly. In Section 3, the concepts of CPCs and DPCs are explained in detail. In Section 4, the simulation exercise and the results are discussed. In Section 5, conclusions of the paper are presented.
2. The exact and the approximate eigenvector methods Priority vectors of the alternatives in an AHP hierarchy are usually calculated from the pairwise comparison matrices using the exact and the approximate eigenvector methods. There are other methods such as simple additive normalisation and multiplicative normalisation, but, Saaty (1980) has preferred the exact and the approximate eigenvector methods over other methods. 2.1. The exact eigenvector m e t h o d
The eigenvector for a consistent reciprocal square matrix can be obtained by normalization of any column vector of the matrix. The conditions for a consistent reciprocal square matrix are as follows: (i) aii = 1/a~i for all i and j. (ii) aij = aik * aki for all k other than i and j. The AEV and the EEV methods yield the same eigenvectors in the case of a consistent reciprocal square matrix. Saaty (1980) has shown that in the case of a reciprocal inconsistent square matrix (a pairwise comparison matrix) the principal eigenvector should be considered as the priority vector of that matrix. From the maximum eigenvalue, the consistency ratio and the consistency index of the matrix are obtained. The computation of the maximum eigenvalue and the corresponding eigenvector is described below.
658
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
Table 1 Pairwise comparison matrix for four alternatives and the corresponding priority vector (principal eigenvector) 1 V 1.000 I 1.277 1.186 0.567
l 2 3 4
2 0.783 1.000 0.928 0.444
3 0.843 1.080 1.000 0.478
The equation A w = Aw
is iterated till a column vector w satisfying this equation is obtained. The normalized w column vector corresponds to the principal eigenvector of the matrix A. The iteration starts with an initial unit vector w, where, A is a reciprocal square matrix, w is the principal eigenvector of A, and A is the maximum eigenvalue of A. 2.2. The approximate eigenvector method
Let aij be an element of matrix A. For each row calculate Eq. (1). wi=
ai~
for all i = 1 . . . . . n.
(1)
Normalization of the column vector thus obtained from the above equation gives the AEV for the corresponding matrix. Crawford and Williams (1985) have proved the validity of this method.
3. Concept of the closest DPC matrix The concept of the closest DPC matrix has been suggested by Triantaphyllou and Mann (1990). They have applied it to deal with scales for pairwise comparisons, and applicability of pairwise compari-
4 1.763-] 2.252 / 2.091| 1.000J
priority vector F 0.2481 "] | 0.3171/ ~ 0.2941 | L_0.1407 ]
son matrices and the eigenvector method for the estimation of membership values in a fuzzy set (Triantaphyllou and Mann, 1990, 1992; Triantaphyllou et al., 1993). The rationality assumption has already been mentioned in Section 1. All other fundamental assumptions of the AHP, such as the inability of experts to distinguish among altematives when the number of alternatives is 7 + 2 (Miller, 1956), the requirement that alternatives be mutually exclusive, etc., are also considered here. In the simulation exercise, however, we have considered upto 15 alternatives in order to compare the two methods comprehensively. Table I showsthe cardinal values for the alternatives in a CPC matrix. Since the matrix is consistent, both the AEV and the EEV methods yield the same priority vector (principal eigenvector) shown in the table. But, an expert cannot provide judgements using a continuous scale as he has to choose only from among the options available in Saaty's discrete scale, 1 which has only the values ~, ~1 , . . . , ~ 1, 1, 2 . . . . . 9. As the experts are assumed to be rational, they ought to choose the closest available values from the ninepoint scale for approximating the corresponding pairwise comparisons in the continuous scale. The approximate matrix thus obtained is called the closest DPC matrix, and it is shown in Table 2 for the CPC matrix of Table 1.
Table 2 Closest DPC matrix of the matrix in Table 1, and the corresponding priority vectors using the EEV and the AEV methods I 2 3 4 EEV AEV
[ 0
2 3 4
l.OOO 1.000 0.500
0
1.000 1.000 0.500
0
l.OOO 1.000 0.500
2.000 2.000 I 1.000_.[
0.2857 L 0.2857 I 0.1429 _J
0.2857 / 0.2857] 0.1429_1
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
The above concept of approximation using the closest DPCs is the key input in the simulation exercise to follow.
bi
n
659
is the priority of element i in the priority vector obtained from the approximate DPC matrix of the simulated consistent CPC matrix. is the size of the reciprocal pairwise comparison matrix.
4. Simulation exercise and results
The simulation exercise is carried out by obtaining the closest DPC matrix from a consistent CPC matrix. The 'closeness' of the two priority vectors obtained by applying the AEV method to the CPC and the DPC matrices is compared with the 'closeness' of the two priority vectors obtained by applying the EEV method on the two matrices. The following indices are used to measure the 'closeness'. The root mean square deviation (RMS) This is measured in the following way:
i ~(ei--bi)2 RMS =
(2)
i= 1 n
where: RMS is the Root Mean Square deviation. is the priority of element i, using either the ei AEV or the EEV method, in the priority vector obtained from a simulated consistent CPC matrix.
The median absolute deviation about the median (MADM) This is not a commonly used statistical measure of variation, but earlier researchers (Saaty, 1980) have used it to compare various scales in the context of AHP. It is expressed in the following way: MADM=med(Ici-bil-med{Ici-bil}),
(3)
where MADM is the Median Absolute Deviation about the Median, and med{I c / - bil} is the median of the set of absolute deviations. The simulation exercise has been camed out for five runs of 1000 matrices each, for every matrix of size 2 to 15. The overall mean of the five RMS and MADM values corresponding to the five simulation runs has been calculated. The RMS values for the five simulation runs employing the EEV method and the average RMS deviation are shown in Table 3. The MADM values for the five simulation runs employing the EEV method and the average MADM value are shown in Table 4. Similarly, the RMS and
Table 3 The RMS values for the exact eigenvector method Size of the matrix
RMS values in simulation runs 1
2
3
4
5
2 3 4 5 6 7 8 9 10 1I 12 13 14 15
0.03050 0.02250 0.01750 0.01329 0.01077 0.00910 0.00774 0.00656 0.00574 0.00514 0.00455 0.00415 0.00375 0.00344
0.02890 0.02227 0.01693 0.01361 0.01133 0.00890 0.00774 0.00662 0.00566 0.00513 0.00461 0.00410 0.00375 0.00349
0.03055 0.02335 0.01723 0.01323 0.01067 0.00910 0.00766 0.00658 0.00586 0.00511 0.00454 0.00414 0.00375 0.00342
0.03111 0.02302 0.01713 0.01367 0.01095 0.00900 0.00757 0.00654 0.00577 0.00504 0.00463 0.00414 0.00379 0.00346
0.03107 0.02257 0.01741 0.01341 0.01093 0.00920 0.00777 0.00656 0.00573 0.00505 0.00458 0.00409 0.00375 0.00345
Average RMS value 0.03043 0.02274 0.01724 0.01344 0.01093 0.00906 0.00769 0.00658 0.00575 0.00509 0.00458 0.00412 0.00376 0.00345
660
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
Table 4 The MADM values for the exact eigenvector method Size of the matrix
MADM values in simulation runs 1
2
3
4
5
Average MADM value
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.00000 0.00456 0.00774 0.00476 0.00376 0.00325 0.00290 0.00235 0.00211 0.00196 0.00170 0.00161 0.00143 0.00130
0.00000 0.00483 0.00781 0.00487 0.00390 0.00311 0.00288 0.00240 0.00211 0.00194 0.00173 0.00155 0.00142 0.00134
0.00000 0.00481 0.00748 0.00494 0.00373 0.00327 0.00277 0.00244 0.00218 0.00188 0.00169 0.00156 0.00144 0.00129
0.00000 0.00472 0.00758 0.00481 0.00371 0.00329 0.00273 0.00240 0.00215 0.00184 0.00175 0.00157 0.00145 0.00133
0.00000 0.00471 0.00771 0.00492 0.00377 0.00340 0.00291 0.00248 0.00211 0.00191 0.00172 0.00158 0.00144 0.00131
0.00000 0.00473 0.00766 0.00486 0.00377 0.00326 0.00284 0.00241 0.00213 0.00191 0.00172 0.00157 0.00143 0.00131
MADM values for the AEV method are shown in Tables 5 and 6. The results of the simulation exercise clearly indicate the superiority of the exact eigenvector method over the approximate eigenvector method. For example consider a 9 X 9 matrix. The average RMS and MADM values for such a matrix when exact eigenvector method is employed are 0.00658 and 0.00241 respectively. The average RMS and MADM values when approximate eigenvector method is employed
for the same matrix are 0.01227 and 0.00342 respectively. Similar differences can be observed for other matrices also. This clearly indicates that the exact eigenvector method gives better results than the approximate eigenvector method. This has already been stated by Saaty (1990). However his work has not sufficiently discouraged users of AHP from employing the AEV method. The results of this simulation exercise run contrary to the claim of Crawford and Williams (1985) that the AEV method provides a
Table 5 The RMS values for the approximate eigenvector method Size of the matrix
RMS values in simulation runs 1
2
3
4
5
Average RMS value
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.11150 0.07455 0.04470 0.02612 0.01458 0.00910 0.00969 0.01222 0.01508 0.01719 0.01912 0.02064 0.02164 0.02282
0.10960 0.07289 0.04530 0.02588 0.01489 0.00890 0.00990 0.01219 0.01523 0.01692 0.01917 0.02066 0.02179 0.02252
0.11130 0.07300 0.04515 0.02610 0.01 447 0.00910 0.00971 0.01227 0.01498 0.01736 0.01906 0.02028 0.02110 0.02331
0.11557 0.07240 0.04458 0.02586 0.01466 0.00890 0.00977 0.01232 0.01516 0.01722 0.01927 0.02046 0.02187 0.02278
0.11369 0.07235 0.04559 0.02594 0.01451 0.00920 0.00991 0.01237 0.01475 0.01696 0.01889 0.02079 0.02195 0.02284
0.11232 0.07304 0.04506 0.02598 0.01462 0.00904 0.00979 0.01227 0.01504 0.01713 0.01910 0.02056 0.02167 0.02285
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
better estimate of the priority vector than the EEV method. The exercise has also revealed that as the size of the matrix increases, the RMS and MADM values decrease for both the EEV and the AEV methods. This is because as the size of the matrix increases, the spread of the values between 0 and 1 decreases. This in tum decreases the RMS and MADM values. The simulation exercise has been carried out using the C language. Rank reversals have not been observed in any of the simulation runs. Although earlier researchers (Crawford and Williams, 1985; Saaty, 1990) have claimed to have observed rank reversals we have only observed rank ties while employing the AEV and the EEV methods. An explanation of the meaning of rank reversal and rank tie is necessary here. Rankings of the alternatives are obtained from the CPC matrix using either the AEV or the EEV method. Similarly rankings for the corresponding DPC matrix are also obtained using either the AEV or the EEV method. If the rankings obtained from the CPC matrix and its corresponding DPC matrix are not in the same order, then rank reversal is said to have occurred. For example, employing the AEV method, let the original rankings of four decision alternatives of a CPC matrix be 2, 3, 1, and 4 respectively. Let the rankings of the same alternatives of the correspond-
661
ing DPC matrix (again, using the AEV method) be 3, 2, 1, and 4 respectively. The ranks of the decision alternatives 1 and 2 are interchanged, i.e., from the original ranking of 2 and 3 to 3 and 2. This is the rank reversal phenomenon in AHP and has been explained in greater detail in Belton and Gear (1983) and Dyer (1990). Employing the EEV method may also lead to rank reversal. Readers may please note that in the case of the CPC matrix, employing the AEV and the EEV methods leads to the same priority vector. The rank tie phenomenon can best be explained through an example. Using the AEV method, let the original rankings of four decision alternatives of a CPC matrix be 3, 2, 1, and 4 respectively. Let the rankings of the same alternatives of the corresponding DPC matrix (again, using the AEV method) be 2, 2, 1, and 4 respectively. Here, decision alternatives 1 and 2 have original rankings 3 and 2 respectively. After approximating the CPC matrix by the corresponding DPC matrix the rankings of the same alternatives are 2 and 2. This is called as rank tie. We have analysed for both rank reversals and rank ties in the simulation exercise and we have encountered only rank ties. Moreover, the AEV and the EEV methods yield the same results in the case of rank ties, and therefore the results are not presented here. We are not denying the possibility of
Table 6 The MADM values for the approximate eigenvector method Size of the matrix
MADM values in simulation runs 1
2
3
4
5
Average MADM value
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.00000 0.01675 0.01512 0.00752 0.00422 0.00324 0.00298 0.00335 0.00371 0.00391 0.00403 0.00411 0.00416 0.00435
0.00000 0.01653 0.01535 0.00742 0.00420 0.00315 0.00305 0.00345 0.00374 0.00387 0.00411 0.00418 0.00421 0.00424
0.00000 0.01638 0.01531 0.00741 0.00417 0.00326 0.00301 0.00350 0.00364 0.00388 0.00418 0.00416 0.00429 0.00423
0.00000 0.01635 0.01512 0.00754 0.00426 0.00326 0.00299 0.00337 0.00372 0.00390 0.00408 0.00423 0.00421 0.00428
0.00000 0.01580 0.01553 0.00759 0.00414 0.00341 0.00313 0.00341 0.00374 0.00392 0.00426 0.00436 0.00425 0.00430
0.00000 0.01636 0.01528 0.00749 0.00420 0.00326 0.00303 0.00342 0.00371 0.00389 0.00413 0.00421 0.00422 0.00428
662
N. Vinod Kumar, L.S. Ganesh / European Journal of Operational Research 95 (1996) 656-662
rank reversals occurring while employing the two methods. The illustrative example provided by Saaty (1990) indicates that rank reversal is associated with the AEV method. The rank tie phenomenon is independent of the type of method employed for obtaining the priority vector and it has more to do with continuous and discrete scales. It may be possible to eliminate the rank tie phenomenon by using appropriate scales.
5. Conclusions In this paper a simulation exercise has been carfled out to evaluate the AEV and the EEV methods used in AHP for obtaining eigenvectors of pairwise comparison matrices. Both the methods have been suggested by Saaty (1980). This exercise shows the superiority of the EEV method over the the AEV method. This result is contrary to the findings of Crawford and Williams (1985) that the eigenvectors estimated by the AEV method are superior to those obtained by using the EEV method. The simulation analysis uses the concept of approximating a continuous pairwise comparison (CPC) matrix by its closest discretized pairwise comparison (DPC) matrix. This concept is more rational compared to Crawford and William's (1985) perturbations approach. If users of AHP do not choose the closest DPC matrix, then the situation will become more complex and rank reversals may arise. In such situations the EEV method is more preferable as Saaty (1990) has shown. This paper empirically concludes that the EEV method should be preferred over the AEV method.
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