Note on "Wash criterion in analytic hierarchy process"

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Short Communication

Note on ‘‘Wash criterion in analytic hierarchy process’’

Jennifer Shu-Jen Lin

a

, Shuo-Yan Chou

b,*

, Wayne T. Chouhuang

b

, C.P. Hsu

c

a

Institute of Technological and Vocational Education, National Taipei University of Technology, Taipei, Taiwan

b

Department of Industrial Management, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei, Taiwan 106, Taiwan

c

Institute of Management Technology, National Chiao Tung University, Taiwan Received 31 March 2006; accepted 22 December 2006

Available online 30 January 2007

Abstract

The paper of Finan and Hurley – published in Computers and Operations Research (2002) – was re-examined, where they discussed a seemingly contradictory phenomenon resulting from the ignoring of wash criteria in the analytic hierarchy process (AHP). With that, they raised a serious challenge to the AHP methodology. However, by reviewing their argu-ments and example data, analyses regarding to their propositions and numerical example are presented in this paper to counter their challenge.

Keywords: Decision analysis; Analytic hierarchy process (AHP); Rank reversal; Wash criterion

1. Introduction

The analytic hierarchy process (AHP) methodol-ogy utilizes pair-wise comparisons to derive the weights for multiple criteria and subsequently the rank-order of the alternatives for decision making.

Belton and Gear (1983) identiﬁed a rank reversal phenomenon in AHP and generated a substantive amount of discussion in the discipline. Researchers including the creator of AHP,Saaty (1977), all con-sented to the rank reversal phenomenon, though still debating its impact. The rank reversal is a phenome-non associated with the resulting of a diﬀerent alter-native rank order when new alteralter-natives are added to

an existing hierarchy. Saaty (1987) suggested that once a new alternative is added, one should introduce new functional criteria or use structural information to modify the weights of the existing functional crite-ria to avoid reversal. In contrast, there are also numerous researches aiming at resolving the phe-nomenon of rank reversal by modifying AHP theory or principles. Saaty (1995) demonstrated that rank reversal will not be a ﬂaw and can be resolved in four ways – absolute measurement, relative measurement, the criteria depending on the alternatives but not on their numbers, and the criteria depending on both the criteria and the alternatives – depending on the char-acteristics of the criteria. Deﬁnitely, rank reversal is not a mathematical or theoretical predicament but a practical phenomenon in the decision making pro-cess for various problems.

* _{Corresponding author. Tel.: +886 2 2737 6327.}

E-mail address:[email protected](S.-Y. Chou).

European Journal of Operational Research 185 (2008) 444–447

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This paper focuses onFinan and Hurley’s (2002)

paper where they utilized the notion of wash (non-discriminating) criteria to establish two propositions with respect to AHP. They proved that for a two-level AHP model when the comparison matrix is perfectly consistent, ignoring a wash criterion would not change the rank order for the alternatives. They stated that they could not prove or disprove for the case of an imperfectly consistent decision maker (DM). They then constructed a three-level example to illustrate that rank reversal does occur when a wash criterion is ignored. By demonstrating the occurrence of rank reversal with a counter-example, they tried to lay out a contradiction in the AHP methodology and thus a ﬂaw in the methodology. They claimed that their discovery posted a serious challenge to the AHP methodology.

In a later research,Liberatore and Nydick (2004)

claimed that after removing wash criteria, the rela-tive weights should be reevaluated and hence no rank reversal problems. Furthermore, Saaty and Vargas (2006)asserted that wash criteria could not be blindly deleted.

In this paper, the open question for the case of an imperfect consistent DM posted byFinan and Hur-ley (2002)is solved and thus a more general case is shown. In addition, the ‘‘counter-example’’ by Finan and Hurley is in fact based on the parameter values which violate the basic assumption of the AHP methodology.

2. Review of Finan and Hurley

In the paper by Finan and Hurley, they ﬁrst assumed that the DM begins with n + 1 criteria indexed by the set J ¼ f0; 1; . . . ; ng and m choice alternatives indexed by I¼ f1; 2; . . . ; mg. The DM’s problem is to decide a rank-order of the m alternatives. The wash criterion is indexed by 0 and the reduced set of criteria by J ¼ f1; . . . ; ng.

Finan and Hurley (2002)assumed the DM to be perfectly consistent and denoted the set of weights by cj for the full criteria set J and by cj for the reduced criteria set, such that

cj¼ ð1 c0Þcj for j¼ 1; 2; . . . ; n: ð1Þ To see this, they assumed that the elements of the pairwise comparison matrix for the full criteria set has elements aijand noted that

ci cj ¼ aij¼ ci cj for all i; j P 1: ð2Þ

It can be deducted from the above equation that

Finan and Hurley (2002)must have ﬁrst determined c0; c1; . . . ; cn, with cj>0 andPn_j¼0cj¼ 1 in order to derive aij. When the DM is perfectly consistent, Eq.

(1)does hold, and can be expressed as cj¼

cj 1 c0

for j¼ 1; . . . ; n: ð3Þ

They denoted the AHP hierarchy with t levels of criteria as H(t). With the AHP model being H(1), for Saaty or the additive method (SAHP), they derived Proposition 1 to show that the rank order is unaﬀected by eliminating the wash criterion. For the multiplicative procedure (MAHP), they derived Proposition 2 and established the same conclusion. However, an H(2) example was then used to demon-strate that ignoring a wash criterion does result in rank reversal. They claimed and we quote: ‘‘In view of the fact that every hierarchy with multiple levels of criteria can, in principle, be modeled as a hierar-chy with a single level of criteria, it must be that our methods for collapsing a hierarchy with multiple levels of criteria are incorrect. In sum, we view our results as a serious challenge to the AHP methodol-ogy’’. In the following sections, contrary to their claim, the validity of their Proposition 1 is ﬁrst dis-cussed. Then, the ‘‘counter-example’’ by Finan and Hurley is examined and dismissed.

3. Limitation of the propositions in Finan and Hurley The open question raised by Finan and Hurley (2002)on pp. 1028, line 27–28, stated: ‘‘We cannot prove the same result in the case of an imperfectly consistent DM’’. In the following, the analysis for an imperfectly consistent DM is established. It is clear that if the equality of

1 b1 . . . bn 1=b1 . . . aij 1=bn 2 6 6 6 6 4 3 7 7 7 7 5 c0 c1 . . . cn 2 6 6 6 6 4 3 7 7 7 7 5¼ knþ1 c0 c1 . . . cn 2 6 6 6 6 4 3 7 7 7 7 5 ð4Þ

holds, then so does

aij 2 6 4 3 7 5 c1 1c0 . . . cn 1c0 2 6 4 3 7 5 ¼ kn c1 1c0 . . . cn 1c0 2 6 4 3 7 5; ð5Þ

where knþ1and knare the maximum eigenvalues for the (n + 1)· (n + 1) and (n · n) pairwise compari-son matrices, respectively.

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From Eq.(4), we know that c0þ Xn j¼1 bjcj¼ knþ1c0; and ð6Þ c0 bi þX n j¼1 aijcj¼ knþ1ci for i¼ 1; 2; . . . ; n: ð7Þ

For the time being, Eq.(5)is assumed to hold in order to derive the necessary condition, which yields that Xn j¼1 aij cj 1 c0 ¼ kn ci 1 c0 for i¼ 1; 2; . . . ; n: ð8Þ By Eqs. (7) and (8), it can be established that ðknþ1 knÞbjcj¼ c0 for j¼ 1; 2; . . . ; n and Eq. (6) can be used to derive the following relation k2_nþ1 ð1 þ knÞknþ1þ kn n ¼ 0: ð9Þ From Eq. (9), if baijc is perfectly consistent, by

Saaty (1977, 1980), then kn= n such that knþ1¼ nþ 1, again by Saaty (1977, 1980), and the ðn þ 1Þ ðn þ 1Þ pairwise comparison matrix is per-fectly consistent, too.

Based on the above discussion, we may construct a counter-example where a ðn þ 1Þ ðn þ 1Þ pair-wise comparison matrix, Mðnþ1Þðnþ1Þ, is not perfectly consistent, andbaijc, where baijc is Mðnþ1Þðnþ1Þafter deleting the ﬁrst row and the ﬁrst column, is per-fectly consistent, such that Finan and Hurley’s assertion with Eq.(3) is invalid. Consequently, the proof of Proposition 1 inFinan and Hurley (2002)

is not applicable for the more general case where the DM is not perfectly consistent. In other words, the wash criterion can only be ignored in a very spe-cial case as employed in their Proposition 1.

The counter-example for the case of an imperfect consistent DM assumes baijc ¼ I3 and M44 ¼ 1 2 3 4 1=2 1 1 1 1=3 1 1 1 1=4 1 1 1 2 6 6 6 4 3 7 7 7 5: ð10Þ As the eigenvalues for M₄₄ are 4.0458, 0:0229 0:4299i and 0, the normalized principal eigenvector is therefore ½0:4912; 0:1865; 0:1662; 0:1561T. By the method of Finan and Hurley (2002), i.e., Eq.(3), it follows that

0:1865 1 0:4912; 0:1662 1 0:4912; 0:1561 1 0:4912 T ¼ ½0:3665; 0:3267; 0:3068T: ð11Þ

By the method ofFinan and Hurley (2002), the nor-malized principal eigenvector of baijc with aij¼ 1 for 1 6 i, j 6 3, will be ½0:3665; 0:3267; 0:3068T. However, the normalized principal eigenvector for thisbaijc should be ½1=3; 1=3; 1=3

T

, and hence, con-tradicting what Propositions 1 was trying to estab-lish for the case of an imperfectly consistent DM.

In addition, Finan and Hurley assumed (as given in the above Eq. (2)) that

aij¼ ci cj

for all i; j P 1; ð12Þ

which is questionably odd. From Saaty (1977, 1980), the entries in the pairwise comparison matrix should be in the set 1; 2; . . . ; 9;1

2; 1 3; . . . ; 1 9 . How-ever, the computational result ofci

cj cannot be guar-anteed to lie in the set 1; 2; . . . ; 9;1

2; 1 3; . . . ; 1 9 . Consequently, their Propositions 1 and 2 are based on variables with questionable values.

4. Questionable numerical example by Finan and Hurley

Consider the same numerical example of Finan and Hurley (2002) with the following data:

Goal G

Main criteria J J0

Main criteria weights 0.55 0.45

Subcriteria J0 J1 J2 J0₁ J0₂

Subcriteria weights 0.6 0.2 0.2 0.5 0.5

Option A1 0.5 0.8 0.4 0.2 0.6

Option A2 0.5 0.2 0.6 0.8 0.4

In Finan and Hurley (2002), they assumed that the comparison matrix is perfectly consistent. Based on that, three questionable sets of data exist in their table: (a) main criteria weights 0.55 and 0.45, (b) the weights for alternatives A1 and A2 for subcriteria J2 being 0.4 and 0.6, and (c) the weights for alterna-tives A1 and A2 for subcriteria J0₂being 0.6 and 0.4. The perfectly consistent comparison matrix with principal eigenvector ½0:55; 0:45T will be

1 11=9

9=11 1

. However, the entries a12¼ 11=9 and a21¼ 9=11 are not in the set f1; 2; . . . ; 9;1

2; 1 3; . . . ;

1

9g. This means that 0.55 and 0.45 are cho-sen arbitrarily rather than derived from the required set of data. Similarly, ½0:4; 0:6T and ½0:6; 0:4T are not principal eigenvectors for perfectly consistent

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comparison matrices with entries in f1; 2; . . . ; 9;1 2; 1 3; . . . ; 1 9g.

To better examine the issue of rank reversal, the three sets of relative weights of numeral example in

Finan and Hurley (2002)are modiﬁed to reﬂect the correct AHP. For the values 11/9 and 9/11 that models a DM’s view, the closest values from the allowable entry set 1; 2; . . . ; 9;1

2; 1 3; . . . ; 1 9 will be 1. The eigenvector ½0:55; 0:45T therefore becomes ½0:5; 0:5T. For the ratios 0.4/0.6 and 0.6/0.4, the closest ones to be changed to are: 1 and 1, or 1/2 and 2. Since the ratio 0.6/0.4 is of an equal distance to 1 and 2, the ratio 0.4/0.6 is analyzed where 0:4 0:6 1 2¼ 1 6<1 0:4 0:6¼ 1 3: ð13Þ

Hence, the weight vector½0:4; 0:6T is changed to ½1=3; 2=3T accordingly, and similarly, ½0:6; 0:4T to ½2=3; 1=3T.

Based on our above analysis for modiﬁcation, by the Saaty method, it can be found that the ﬁnal weights of A1and A2, J0should be changed from

0:477 0:523 to 0:48 0:52

if with wash criteria and should be changed from 0:51

0:49 to 0:5 0:5 if with-out wash criteria. With the modiﬁcation of data to ﬁt the requirement of AHP, the counter-example constructed by Finan and Hurley (2002) does not result in rank reversal for alternatives A1and A2. 5. Conclusion

Rank reversal phenomena which were used by

Finan and Hurley (2002) to construct a series of

arguments to challenge the AHP methodology are shown to be ﬂawed both in the process and in data. Though there may still be issues that need to be addressed about the AHP, however, not with a wrongful case.

Acknowledgements

The authors would like to express their sincere gratitude to the two anonymous referees whose con-structive suggestions and helpful comments improve the quality of the paper signiﬁcantly.

References

Belton, V., Gear, A.E., 1983. On a shortcoming of Saaty’s method of analytic hierarchy. Omega 11, 227–230.

Finan, J.S., Hurley, W.J., 2002. The analytic hierarchy process: Can wash criteria be ignored. Computers and Operations Research 29, 1025–1030.

Liberatore, M.J., Nydick, R.L., 2004. Wash criteria and the analytical hierarchy process. Computers and Operations Research 31, 889–892.

Saaty, T.L., 1977. A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology 15, 234–281. Saaty, T.L., 1980. The Analytic Hierarchy Process.

McGraw-Hill, New York.

Saaty, T.L., 1995. Decision Making for Leaders: The Analytical Hierarchy Process for Decisions in a Complex World, third ed. RWS Publications, Pittsburgh, PA.

Saaty, T.L., 1987. Rank generation preservation and reversal in the analytic hierarchy decision process. Decision Sciences 18, 157–177.

Saaty, T.L., Vargas, L.G., 2006. The analytic hierarchy process: Wash criteria should not be ignored. International Journal of Management and Decision Making 7, 180–188.