A critical analysis of the eigenvalue method used to derive priorities in AHP

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A critical analysis of the eigenvalue methodused to derive priorities in AHP

Carlos A. Bana e Costa a,b,*,1, Jean-Claude Vansnick c

a CEG-IST, Centre for Management Studies of IST, Technical University of Lisbon, Lisbon, Portugalb Department of Management-Operational Research Group, London School of Economics, UK

c Universite de Mons-Hainaut, F.W.S.E., Place du Parc, 20-7000 Mons, Belgium

Available online 2 January 2007

Abstract

A lot of research has been devoted to the critical analysis of the Analytic Hierarchy Process (AHP), from various per-spectives. However, as far as we know, no one has addressed a fundamental problem, discussed in this paper, concerningthe meaning of the priority vector derived from the principal eigenvalue method used in AHP. The role of AHPs consis-tency ratio is also analysed. 2006 Elsevier B.V. All rights reserved.

Keywords: Decision analysis; Analytic Hierarchy Process; Eigenvalue method; Condition of order preservation

1. Introduction and objective of the analysis

Since Saaty (1977, 1980) introduced the AnalyticHierarchy Process (AHP), many applications inreal-world decision-making have been reported (cf.Zahedi, 1986; Golden et al., 1989; Shim, 1989; Var-gas, 1990; Saaty, 2000; Forman and Gass, 2001;Golden and Wasil, 2003; Vaidya and Kumar,2006). In parallel, AHP has often been criticised in

the literature, from several perspectives (see, forexample, Watson and Freeling, 1982, 1983; Beltonand Gear, 1983, 1985; French, 1988; Holder, 1990;

Dyer, 1990a,b; Barzilai and Golany, 1994; Saloand Hamalainen, 1997). A debate about the maincriticisms of AHP can be found in Belton and Stew-art (2002) and Smith and von Winterfeldt (2004).Saaty has frequently contested these critics (see,for example, Saaty et al., 1983; Saaty and Vargas,1984; Saaty, 1990, 1997; Saaty and Hu, 1998) and,in essence, has not modified his original method(see Saaty, 2005). Independently of our agreement

with some of those criticisms, the analysis of whichis beyond the scope of this paper, we believe that theelicitation of pairwise comparison judgements andthe possibility of expressing them verbally are cor-nerstones of the popularity of AHP.

There is, however, a key problem that, as far aswe know, has never before been addressed in the lit-erature. It concerns the meaning of the priority vec-tor derived from the principal eigenvalue methodused in AHP. The AHP uses a principal eigenvalue

0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2006.09.022

* Corresponding author. Address: CEG-IST, Centre for Man-agement Studies of IST, Technical University of Lisbon, Lisbon,Portugal.

E-mail addresses: [email protected] (C.A. Bana e Costa),[email protected] (J.-C. Vansnick).1 This author was supported by POCTI and LSE.

Available online at www.sciencedirect.com

European Journal of Operational Research 187 (2008) 14221428

www.elsevier.com/locate/ejor
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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method (EM) to derive priority vectors (Saaty andHu, 1998, p. 121). Following Saaty, the priority vec-tor has two meanings: The first is a numericalranking of the alternatives that indicates an orderof preference among them. The other is that the

ordering should also reflect intensity or cardinalpreference as indicated by the ratios of the numeri-cal values (. . .) (Saaty, 2003, p. 86). This secondmeaning requires, in our view, that these ratios pre-serve, whenever possible, the order of the respectivepreference intensities, which is not always the casefor AHP priority vectors. Indeed, the ratios ofAHP priority values can violate this order albeitthe ratios of alternative priority values, derivedfrom the same pairwise comparisons, preserve it.From our decision-aid perspective, this is a basicdrawback of AHP. Consider the following

condition:Condition of Order Preservation (COP): For all

alternatives x1, x2, x3, x4 such that x1 dominates2 x2

and x3 dominates x4, if the evaluators judgementsindicate the extent to which x1 dominates x2 isgreater that the extent to which x3 dominates x4,then the vector of priorities w should be such that,not only w(x1) > w(x2) and w(x3) > w(x4) (preserva-tion of order of preference) but also that w(x1)/w(x2) > w(x3)/w(x4) (preservation of order of inten-sity of preference).

For instance, if x1 strongly dominates x2 and x3moderately dominates x4, it is from our view funda-mental that, whenever possible, the vector of prior-ities w be such that w(x1)/w(x2) > w(x3)/w(x4);indeed, these judgements indicate that the intensityof preference of x1 over x2 is higher than the inten-sity of preference of x3 over x4.

We will prove with simple examples that theAHP priority vector does not necessarily satisfythe COP, even though it is possible to respect thiscondition. In such cases, alternative priority valuesthat satisfy COP can easily be found by a mathe-matical program including COP constrains. Theparticular program that we used is not importantin the scope of this paper, since our intention isnot at all to suggest an alternative procedure toAHP.

Note that a numerical scale that satisfies the COPdoes not always exist. In our constructive perspec-tive, it is essential to detect these situations and dis-

cuss them with the evaluator before proposing anypriority scale. A complementary objective of thispaper is to analyse if the consistency ratio used inAHP can reveal such situations.

The rest of this paper is organised in the following

manner: in Section 2, we review the principal eigen-value method used in AHP to derive priority vectors;in Sections 3 and 4, we present some examples inwhich it would be possible to satisfy the COP, how-ever, the AHP priority vectors violate it; in Section 5,we show that the AHP consistency ratio is not suit-able for detecting the existence (or the non-existence)of a numerical scale satisfying the COP; a briefconclusion is presented in Section 6.

2. Overview of the principal eigenvalue method (EM)

Let X= {x1, x2,. . .

, xn} be a set of elements and }a property or criterion that they have in common(Saaty, 1996, p. 24) for example, X could be a setof cars and } their comfort. How can we help a per-son J quantify the relative priority (or importance)that the elements of X have for her, in terms of }?

The EM used in AHP to derive priorities for theelements ofXrequires that a number denoted wij be assigned to each pair of elements (xi, xj) repre-senting, in the opinion ofJ, the ratio of the priorityof the dominant element (xi) relative to the priority

of the dominated element (xj) (Saaty, 1997). J isinvited to compare the elements pairwise and canexpress her judgements in two different ways:

either numerically, by giving a real numberbetween 1 (inclusive) and 10 (exclusive) (Saaty,1989) for example, if xi is a Chevrolet and xja Lada and if J judges the Chevrolet to be sixtimes more comfortable than the Lada, thanwij = 6;

or verbally, by choosing one of the followingexpressions: equal importance, moderate domi-nance, strong dominance, very strong domi-nance, extreme dominance, or an intermediatejudgement between two consecutive expressions;each verbal pairwise comparison elicited is thenautomatically converted into a number wij asexhibited in Table 1 for example, if xi is a Peu-geot and xjan Opel and ifJjudges the Peugeot tobe moderately more comfortable than the Opel,then wij = 3.

During the elicitation process, a positive recipro-

cal matrix, in which each element x1, x2,. . .

, xn of X

2 In this paper, dominance is used in the sense of strict

preference.

C.A. Bana e Costa, J.-C. Vansnick / European Journal of Operational Research 187 (2008) 14221428 1423


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is assigned one line and one column, can be filled byplacing the corresponding number at the intersec-tion of the line of xi with the column of xj

wij if xi dominates xj;

1=wij if xj dominates xi;

1 if xi does not dominate xj

and xj does not dominate xi:

8>>>>>:

For example, assuming that for all i, j2{1,2, . . . , n} xi dominates xj if and only if i< j, theformat of the positive reciprocal matrix will be

W

1 w12 w1n

1=w12 1 w2n

1=w1n 1=w2n 1

0BBBBBB@

1CCCCCCA:

In order to assign a priority (or a weight) toeach element xi a numerical value that we willdenote w(xi) the principal eigenvalue kmax ofmatrix W and its normalised eigenvector are calcu-lated: the components of this vector are the w(xi).This procedure has a very interesting property: ifthe judgements of J are such that wij wjk = wik forall i< j< k (cardinal consistency condition), thederived w(xi) are such that wij = w(xi)/w(xj) for alli< j.

However, cardinal consistency is seldomobserved in practice. Therefore, AHP makes useof a consistency test that prevents priorities frombeing accepted if the inconsistency level is high. Inorder to measure the deviation of matrix W from

consistency, a consistency index CI is defined as

kmax n/(n 1) and a random index RI (of ordern) is calculated as the average of the CI of manythousands reciprocal matrices (of order n) randomlygenerated from the scale 1 to 9, with reciprocalsforced. The values of RI for matrices of size

1,2,. . .

,10 can be found in Saaty (2005, p. 374).The ratio of CI to RI for the same order matrix iscalled the consistency ratio CR. According to Saaty(1980, p. 21), a consistency ratio of 0.10 or less isconsidered acceptable. That is, an inconsistencyis stated to be a matter of concern if CR exceeds0.1, in which case the pairwise comparisons shouldbe re-examined.

If the elements are to be compared according toseveral }, the AHP proposes that a hierarchy bebuilt with the general goal on top, the elements atthe bottom and the } at intermediate levels. The

procedure described above is then repeatedlyapplied bottom-up: to calculate a vector of prioritiesfor the elements with respect to each } situated atthe bottom intermediate level; to calculate a vectorof weights for each cluster of} at the different lev-els. All this judgemental information is then syn-thesised from bottom to top by successive additiveaggregations, in order to derive a vector of overallpriorities for the elements.

3. Examples in which the COP is violated by the

priority vector derived from the EM

We present in this section two examples provingthat the COP may be violated by the priority vectorgiven by the EM for each one of them, althoughscales exist that do respect it. Example 1 involvesverbal judgements and Example 2 involves numeri-cal judgements.

Example 1 (Case of verbal judgements). Let X={x1, x2, x3, x4, x5} be a set of alternatives betweenwhich the following pairwise comparisons wereformulated by a person J:

{x1, x2}: x1 dominates x2, equal to moderatedominance.{x1, x3}: x1 dominates x3, moderate dominance.{x1, x4}: x1 dominates x4, strong dominance.{x1, x5}: x1 dominates x5, extreme dominance.{x2, x3}: x2 dominates x3, equal to moderatedominance.{x2, x4}: x2 dominates x4, moderate to strongdominance.

{x2, x5}: x2 dominates x5, extreme dominance.

Table 1Converting verbal judgements into numbers

Verbal expressionsa Corresponding numbers

Equal 1Equal to moderate 2Moderate 3

Moderate to strong 4Strong 5Strong to very strong 6Very strong 7Very strong to extreme 8Extreme 9

a In Saaty (1996, 2005) the verbal expressions equal to mod-erate, moderate to strong, strong to very strong and verystrong to extreme are replaced by weak, moderate plus,strong plus and very, very strong, respectively.

1424 C.A. Bana e Costa, J.-C. Vansnick / European Journal of Operational Research 187 (2008) 14221428


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{x3, x4}: x3 dominates x4, equal to moderatedominance.{x3, x5}: x3 dominates x5, very strong to extremedominance.{x4, x5}: x4 dominates x5, very strong dominance.

From Table 1, the corresponding positive reci-procal matrix is

1 2 3 5 9

1=2 1 2 4 9

1=3 1=2 1 2 8

1=5 1=4 1=2 1 7

1=9 1=9 1=8 1=7 1

0BBBBBB@

1CCCCCCA

for which the normalised eigenvector correspondingto its principal eigenvalue is

0:426

0:281

0:165

0:101

0:027

0BBBBBB@

1CCCCCCA:

Consequently, given the judgements ofJ, the pri-orities obtained through the EM are

wx1 0:426;wx2 0:281;

wx3 0:165;

wx4 0:101;

wx5 0:027:

Then, in particular, w(x1)/w(x4) % 4.218 andw(x4)/w(x5) % 3.741, that is, w(x1)/w(x4) > w(x4)/w(x5). Given that J judged that x4 very stronglydominates x5 and x1 strongly dominates x4, the pri-ority vector given by the EM violates the COP. Yet,

for example, the scale w*

wx1 0:385;

wx2 0:275;

wx3 0:195;

wx4 0:125;

w

x5 0:020;respects the COP, as shown in Table 2. Let us alsopoint out that the value of the consistency ratiofor the judgements in Example 1 is 0.05, signifi-cantly smaller than the 0.10 threshold; therefore,in AHPs perspective the judgements need not berevised.

Example 2 (Case of numerical judgements). LetX= {x1, x2, x3, x4} be a set of alternatives betweenwhich the following pairwise comparisons were for-

mulated by a person J:

{x1, x2}: x1 dominates x2 2.5 times.{x1, x3}: x1 dominates x3 4 times.{x1, x4}: x1 dominates x4 9.5 times.{x2, x3}: x2 dominates x3 3 times.{x2, x4}: x2 dominates x4 6.5 times.{x3, x4}: x3 dominates x4 5 times.

The corresponding positive reciprocal matrix is

1 2:5 4 9:51=2:5 1 3 6:5

1=4 1=3 1 5

1=9:5 1=6:5 1=5 1

0BBB@

1CCCA

for which the normalised eigenvector correspondingto its maximal eigenvalue is

0:533

0:287

0:139

0:041

0BBB

@

1CCC

A

:

Table 2Example 1 values of the ratios w*(xi)/w*(xj)

Possible verbal judgements (xi, xj) pair(s) and respective w*(xi)/w*(xj) ratios

Equal to moderate (x1, x2): 1.40 (x2, x3): 1.41 (x3, x4): 1.56Moderate (x1, x3): 1.97Moderate to strong (x2, x4): 2.20Strong (x1, x4): 3.08Strong to very strong ;Very strong (x4, x5): 6.25Very strong to extreme (x3, x5): 9.75

Extreme (x2, x5): 13.75 (x1, x5): 19.25



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Consequently, given the judgements ofJ, the pri-orities obtained through the EM are

wx1 0:533;

wx2 0:287;

wx3 0:139;

wx4 0:041:

For all i, j2 {1,2,3,4} such that i< j, Table 3 pre-sents the numerical value wij given by J when shejudged how many times xi dominates xj, togetherwith the respective value of the ratio w(xi)/w(xj).

It is not surprising that the values of w(xi)/w(xj)are not the same as the numerical judgements wij(because the latter are not cardinally consistent) butit is surprising to verify that their order is notpreserved by the ratios. Indeed, w34 > w13 but w(x3)/w(x4) < w(x1)/w(x3). This proves that, again, the

priority vector given by the EM violates the COP.Yet, for example, the scale w*:

wx1 0:48;

wx2 0:32;

wx3 0:16;

wx4 0:04;

respects the COP. Indeed,

wx1

wx4 12 >

wx2

wx4 8 >

wx3

wx4 4 >

wx1

wx3

3 > w

x2wx3

2 > w

x1wx2

1:5:

Moreover, the value of the consistency ratio forthe judgements in Example 2 is 0.05, significantlysmaller than the 0.10 threshold; therefore in AHPsperspective the judgements need not be revised.

4. Analysis of one of Saatys examples

Example 3. In this section, we analyse the violation

of the COP in one of the examples presented in

Saaty (1977, pp. 254256) and Saaty (1980, pp. 4041) to empirically validate the EM. We refer to theexample of pairwise comparisons of the GNP ofseveral countries, in which, for a given matrix ofverbal judgements, the priorities given by the AHP

are remarkably close to the normalised GNP values.The countries are (Saatys notation) US, USSR,China, France, UK, Japan and W. Germany andthe matrix of judgements presented is

US

USSR

China

France

UK

Japan

W: Germany

US USSR China France UK Japan W: Germany

1 4 9 6 6 5 5

1=4 1 7 5 5 3 4

1=9 1=7 1 1=5 1=5 1=7 1=5

1=6 1=5 5 1 1 1=3 1=3

1=6 1=5 5 1 1 1=3 1=3

1=5 1=3 7 3 3 1 2

1=5 1=4 5 3 3 1=2 1

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA

The corresponding priorities are

wUS 0:427;

wUSSR 0:230;

wChina 0:021;

wFrance 0:052;

wUK 0:052;

wJapan 0:123;

wW: Germany 0:094:

These are the priorities appearing in Saaty(1980), which are a little different from those inSaaty (1977): 0.429, 0.231, 0.021, 0.053, 0.053,0.119, and 0.095, respectively. Nevertheless, in bothof these priority vectors the same five violations ofthe COP can be observed. We will analyse two ofthese hereafter.

(1) According to the matrix of judgements, USdominates USSR (4 times) more than Japandominates France (3 times). But, w(US)/w(USSR) % 1.857 and w(Japan)/w(France) %2.365, that is, w(US)/w(USSR) < w(Japan)/w(France).

(2) According to the matrix of judgements, Japandominates China (7 times) more than USdominates UK (6 times). But, w(Japan)/w(China) % 5.857 and w(US)/w(UK) % 8.212,that is, w(Japan)/w(China) < w(US)/w(UK).

In spite of this, it is possible to avoid all of theviolations of the COP, as for example with thefollowing priority vector of priorities w* (see Table

4):

Table 3Example 2 values of wij and w(xi)/w(xj)

wij w(xi)/w(xj)

{x1, x4} 9.5 13{x2, x4} 6.5 7{x3, x4} 5 3.39

{x1, x3} 4 3.83{x2, x3} 3 2.06{x1, x2} 2.5 1.86



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wUS 0:414;

wUSSR 0:217;

wChina 0:019;

wFrance 0:069;

wUK 0:069;

wJapan 0:117;

wW: Germany 0:095:

Let us also point out that the value of the consis-tency ratio for the judgements of this example is0.08.

5. Discussion about the consistency ratio (CR)

Example 4. In this section, we present an examplein which it is impossible to find a numerical scalesatisfying the COP and analyse the value of the CR.Let X= {x1, x2, x3, x4, x5} be a set of alternativesbetween which the following pairwise comparisonjudgements were formulated by a person J:

{x1, x2}: x1 dominates x2, equal to moderatedominance.{x1, x3}: x1 dominates x3, strong dominance.

{x1, x4}: x1 dominates x4, very strong dominance.

{x1, x5}: x1 dominates x5, extreme dominance.{x2, x3}: x2 dominates x3, equal to moderatedominance.{x2, x4}: x2 dominates x4, moderate dominance.{x2, x5}: x2 dominates x5, very strong dominance.

{x3,

x4}:

x3 dominates

x4, moderate dominance.{x3, x5}: x3 dominates x5, strong dominance.

{x4, x5}: x4 dominates x5, equal to moderatedominance.

For this set of judgements, it is impossible tosatisfy the COP. Indeed, one should simultaneouslyhave:

(1) w(x1)/w(x3) > w(x2)/w(x4), because, accordingto Js judgements, x1 dominates x3 (strongdominance) more than x2 dominates x4 (mod-erate dominance), and

(2) w(x3)/w(x4) > w(x1)/w(x2), because, accordingto Js judgements, x3 dominates x4 (moderatedominance) more than x1 dominates x2 (equalto moderate dominance).

This is impossible because the product, memberto member, of these two inequalities gives w(x1)/w(x4) > w(x1)/w(x4).

In our view, this shows that we are in face of areal case of judgemental inconsistency because,

contrary to Examples 13, the set of judgements inthe present example is incompatible with a numer-ical representation that guarantees order preserva-tion. And yet, the value of the CR corresponding tothese judgements is very small (0.03), which means,in the AHPs perspective, that these judgementswould not necessitate to be revised. Moreover, 0.03is smaller than the values of the consistency ratiosfor Examples 13 (0.05, 0.05 and 0.08) in which, asshown in Sections 3 and 4, scales exist that satisfythe COP, unlike to the present example in which an

inconsistency problem undoubtedly exists. Thisshows that the CR used in AHP is not suitable fordetecting the existence (or the non-existence) of anumerical scale satisfying the COP.

6. Conclusion

In this article, we have addressed the foundationsof AHP, by analysing the eigenvalue method (EM)used to derive a priority vector. Our main conclu-sion is that, although the EM is very elegant from

a mathematical viewpoint, the priority vector

Table 4Verification of the COP

Possible verbaljudgements

(xi, xj) pair(s) and respective w*(xi)/w*(xj) ratios

Equal to moderate (Japan, W. Germany): 1.23Moderate (W. Germany, France): 1.38

(W. Germany, UK): 1.38(Japan, France): 1.70(Japan, UK): 1.70(USSR, Japan): 1.85

Moderate to strong (US, USSR): 1.91(USSR, W. Germany): 2.28

Strong (USSR, France): 3.14(USSR, UK): 3.14(US, Japan): 3.54(UK, China): 3.63(France, China): 3.63(US, W. Germany): 4.36

Strong to very strong (US, France): 6.00(US, UK): 6.00

Very strong (Japan, China): 6.16(USSR, China): 11.42

Very strong toextreme

B

Extreme (US, China): 21.79



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derived from it can violate a condition of order pres-ervation that, in our opinion, is fundamental indecision aiding an activity in which it is essentialto respect values and judgements. In light of that,and independently of all other criticisms presented

in the literature, we consider that the EM has a seri-ous fundamental weakness that makes the use ofAHP as a decision support tool very problematic.As Saaty (2005, p. 346) points out, the purposeof decision-making is to help people make decisionsaccording to their own understanding, and. . . methods offered to help make better decisionsshould be closer to being descriptive and consider-ably transparent.

Finally, it is worthwhile to note that the criticismof the EM, presented in this paper, is also valid forany other method that has been (or may be)

conceived to derive a vector of priorities from apairwise comparison matrix on the basis of a math-ematical technique that does not integrate what wecall the COP, or does not automatically guaranteeits satisfaction.

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A critical analysis of the eigenvalue method used to derive priorities in AHP

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