the theory of ratio scale estimation saaty's ahp

MANAGEMENT SCIENCEVol 33, No 11. November 1987

Printed w USA

THE THEORY OF RATIO SCALE ESTIMATION:SAATY'S ANALYTIC HIERARCHY PROCESS*

PATRICK T. HARKER AND LUIS G. VARGASThe Wharton School, Umversily of Pennsylvania, Philadelphta, Pennsylvania 19104

Graduate School of Business, University of Pittsburgh, Ptttsburgh, Pennsylvania 15260

The Analytic Hierarchy Process developed by Saaty (1980) has proven to be an extremelyuseful method for decision making and planning. However, some researchers in these areashave raised concerns over the theoretical basis underlymg this process. This paper addressescurrently debated issues concerning the theoretical foundations of the Analytic HierarchyProcess. We also illustrate through proof and through examples the validity or fallaciousness ofthese criticisms.(DECISION THEORY; ANALYTIC HIERARCHY PROCESS; RATIO SCALES)

1. Introduction

Ever since Saaty's development of the Analytic Hierarchy Process (AHP) in the1970s, numerous books and papers have been written concerning its theory and itsapplications. The three major books dealing with the theory and application of thismethod are: The Analytic Hierarchy Process {Saaty 1980), Decision Making for Leaders(Saaty 1982), and The Logic of Priorities (Saaty and Vargas 1982). Some ofthe articlesappearing in the operations research/management science (OR/MS) literature includethe papers by Saaty (1977, 1983), Wind and Saaty (1980), Cook et al. (1984), Saaty andGholamnezhad (1982), Vargas (1983), Saaty and Vargas (1979), Mitchell and Soye(1983), Wasil et al. (1983), Golden et al. (1986), Hamalainen (1985), and Harker(1986). In addition to these publications, several government agencies, consulting firmsand corporations are currently using the AHP on a routine basis to analyze complexpolicy/planning issues; the paper by Zahedi (1986) provides an excellent review of thisliterature. Recent articles (Belton and Gear 1983, 1985; Dyer and Wendell 1985),however, have attacked the AHP on the grounds that it lacks a firm theoretical basisand, hence, it should not be considered a "correct" method to analyze decisions or, ingeneral, systems. Furthermore, the acceptance of this method has been slowed by whatwe believe to be (a) misunderstandings of its theoretical foundations, and (b) a reluc-tance to move away from traditional methods of analysis such as the Delphi technique,multi-attribute utility theory and Edward's SMART technique. In this paper we shallfocus on the former, namely, the misunderstandings ofthe theoretical basis ofthe AHP,and hopefully will address the latter through our discussion of the AHP's merits andshortcomings.

To begin, let us briefly define what the AHP is and its essential elements. The AHP isa comprehensive framework which is designed to cope with the intuitive, the rational,and the irrational when we make multiobjective, multicriterion and multiactor deci-sions with and without certainty for any number of alternatives. It is a method forderiving ratio scales used to integrate our procedure for representing the elements ofany problem. It organizes the basic rationality by breaking down a problem into itssmaller constituent parts and then calls for only simple pairwise comparison judgmentsto develop priorities in each hierarchy.

* Accepted by Ambar G. Rao, former Departmental Editor; received December 1985. This paper has beenwith the authors 1 \ months for I revision.

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1384 PATRICK T, HARKER AND LUIS G, VARGAS

There are three principles which one can recognize in problem solving: (I) Decom-position, (II) Comparative Judgments, and (III) Synthesis of Priorities.

The decomposition principle calls for structuring the hierarchy to capture the basicelements of the problem. The principle of comparative judgment calls for setting up amatrix to carry out pairwise comparisons ofthe relative importance ofthe elements in alevel with respect to the elements in the level immediately above it. This matrix is usedto generate a ratio scale. Finally, the principle of synthesis of priorities is used togenerate the global or composite priority of the elements at the lowest level of thehierarchy.

In addition, the AHP contains an intrinsic measure of inconsistency for each matrixand for the whole hierarchy. Knowledge of inconsistency enables one to determinethose judgments that need reassessment. In this sense, the AHP deviates radically fromthe more traditional decision analytic methods in that the latter have no formal way ofdealing with inconsistencies. Rather, these methods rely on the decision analyst's adhoc rules for the reconciliation of inconsistencies in judgments; Belton (1986) recentlymade this point in comparing the AHP and simple multi-attribute value functions.

There are essentially four areas in which the AHP has been criticized: lack of anaxiomatic foundation, ambiguity ofthe questions that the decision maker must answer,the scale used to measure the intensity of preference, and the Principle of HierarchicalComposition and rank reversal. Our purpose in this paper is to address theoreticalissues related to whether or not the AHP is a valid and acceptable method of ehcitingand analyzing subjective judgments, and not whether subjectivity should be incorpo-rated into OR/MS analyses of any type. We shall assume that the reader recognizes thenecessity of such subjective judgment in the modelling of complex socio-economicphenomena as well as within the traditional boundaries of decision analysis.

This pap»er summarizes the results of many published and unpublished papers; thus,many of the proofs are omitted but mention will be made to the appropriate referenceswhere these proofs can be found. The next section reviews the axiomatic foundation ofthe AHP. §3 addresses the issue of iht frame of reference in which subjective judgmentsare made in terms of these axioms. In both of these sections comparisons and contrastsare made with traditional utility theory and other measurement theories. §4 presentsarguments for and against the measurement scale used in the AHP, emphasizing the 1to 9 scale proposed by Saaty (1980), The cardinal measurement of preferences, asdefined in the axioms, is shown to be fully represented by the eigenvector methodproposed by Saaty. The arguments are based on a graph theoretic approach presentedin §5. In §6 we use a similar argument to show the validity of hierarchic compositionand the notion of systems with feedback (see Saaty 1980, Chapter 8). Finally, in §7 wereview other problems which concern the method and outline a plan of research toaddress these problems and extend the basic methodology.

2. Axiomatic Treatment of the AHP

The first major criticism of the AHP is that this method lacks an axiomatic founda-tion. However, Saaty (1986) has provided such a foundation. The purpose of thissection is to review Saaty's axioms and to interpret their meanings in light ofthe currentcriticisms of this method.

To begin with, Saaty (1986) defines a set of primitive notions in which the axioms arebased; they are:

(1) Attributes or Properties. A is the finite set of n elements called alternatives andCt is the set of properties or attributes v«th which the elements of A are compared.Philosophers often make a distinction between properties and attributes, but we shalltreat them as interchangeable and refer to them as criteria.

RATIO SCALE ESTIMATION THEORY 1385

(2) Binary Relation. When two objects are compared according to a property, wesay that one is performing binary comparisons. The binary relation >c represents"more preferred than" according to a property C. The binary relation ~ c represents"indifferent to" according to the property C.

(3) Fundamental Scale. Let P denote the set of mappings from A X A io R^,f.<l-*P, and Pc e / ( C ) for C G ©. Thus, every pair {A,, Aj) E A X A canhe assigned apositive real number Pc{A,, Aj) = a,j that represents the relative intensity with which anindividual perceives a property CG £ in an element^, G ^ in relation to other^^ G A:

A, >cAj if and only ifPdA,, Aj)> I.A, ~cAj if and only if Pc(A,,Aj) = 1.Using these primitive notions, Saaty (1986) has defined four axioms on which the

AHP is based. Rather than stating these axioms in their full mathematical form, weshall paraphrase them in order to state tbeir basic meaning.

Axiom 1 (The Reciprocal Condition). Given any two alternatives iA,,Aj)E.AX A,the intensity of preference of^, over^^y is inversely related to tbe intensity of preferenceof AJ o\eT A,:

PAA., AJ) = l/PAAj, A,), VA,, AjSA, CEd.

DEHNITION I. A hierarchy ^ i s a partially ordered set witb largest element b whichsatisfies the conditions:

(1) Tbere exists a partition of .^into levels [Lk, fc = 1, 2 h}, Li = {b}.(2) If X is an element ofthe Ath level (x £ L^), then the set of elements "below" x (jc~

= {y\x covers y}, k = 1,2, . . ., h = 1) is a subset ofthe {k -i- l)st level (JC" C Lk+^).(3) If X is an element ofthe Atb level, then the set of elements "above" x (x^ = {y\y

covers x}, A: = 2, 3, . . . , A) is a subset ofthe (k - l)st level {x^ £ i yt-i)-DEFINITION 2. Given a positive real number p > 1 a nonempty set x~ ^ L^+i is said

to be p-homogeneous with respect io X E Li^ if

1 /p < Pc(yi ,y2)^P for all >',, ;»2 e ^".

Axiom 2 (p-homogeneity). Given a hierarchy ^, x E ^ and x G Z.* C ^ , thenx~ c £ . , is p-homogeneous for all A: = \,2, . .. ,h- 1.

This axiom is somewhat equivalent to tbe archimedean property. Krantz et al. (1971,pp. 25-26) write:

. . . a rattier odd axiom is usually stated as part of each system. It is Archimedean because itcorresponds to the Archimedean property of reat numbers: for any positive numtier x, nomatter how small, and any number y, no matter how targe, there exists an integer n such thatfix's, y. This simply means that any two positive numtjers are comparable, i.e.. their ratio is notinfinite. Another way to say this, one which generalizes more readily to qualitative structures, isthat the set of integers for which «x < y is a finite s e t . . . Then the Archimedean axiom saysthat for any b, the set of integers n for which fc > na is finite.

It is evident that since the Archimedean property is true ofthe real numbers, it must also betrue within the empirical relational system; it is a necessary axiom. What is surprising is that it isa needed axiom. In the few cases where the independence of axioms has been studied, theArchimedean axiom has been found to be independent of others; and no one seems to havesuggested a more satisfactory substitute. It can be deleted if quite strong structural assumptionsare made . . . , but with our relatively weak structural assumptions, we do not know how toeliminate it in favor of more desirable necessary axioms.

The objection to it as a necessary axiom is that either it is trivially true in a finite structure. . . or it is unclear what constitutes empirical evidence against it since it may not be possible toexhibit an infinite standard sequence.

Tbus,(1) When comparing two elements according to a criterion a bounded scale is suffi-

cient, and

1386 PATRICK T. HARKER AND LUIS G. VARGAS

(2) Ratio comparisons follow directly from the archimedean property as no othernumerical comparison does.

DEHNITION 3. A set of .yl is said to be outer dependent on a set 6 if a fundamentalscale can be defined on A with respect to every C E £.

DEHNITION 4. Let A be outer dependent on £. The elements in A are said to beinner dependent (independent) with respect to C E £ if for some (all) E ^ , .^ is outerdependent (independent) on A.

Axiom 3 (Dependence). Let ^ b e a hierarchy with levels, Li, Li,. . ., L/,. For eachL,,k=l,2,...,h-\.

(1) Lk+] is outer dependent on L/,.(2) Lfc+i is inner dependent with respect to all x E Z. -(3) Lk is outer dependent of L^+i.DEHNITION 5. Expectations are beliefs about the rank of alternatives derived from

prior knowledge.Axiom 4 (Expectations). All criteria and alternatives are represented in the hierar-

chy; i.e., (E C ^— Lh, A = Lh. Thus, all expectations must be represented (or excluded)in terms of criteria and alternatives in the structure, and assigned priorities compatiblewith the expectations.

The first axiom derives from the intuitive idea that i(A is five times heavier than B,then it must be the case that B is one fifth as heavy as A. The second axiom which dealswith homogeneity simply states that individuals are only capable of expressing mean-ingful intensities of preference if the elements are comparable. Clearly, it is very diffi-cult to make meaningful comparisons of the weights of the sun and an atom; ourcognitive capabilities do not permit such comparisons between vastly differing objects.Thus, this axiom simply states that the elements of a particular level in a hierarchy.fmust be comparable. If one is faced with a situation in which comparisons must bemade between objects which are very different, the objects should be clustered intohomogeneous bundles and then the bundles can be compared at a higher level of .^.

The third axiom deals with the ability to make comparisons in a hierarchical struc-ture. Simply stated, this axiom says that such comparisons are possible when a set ofelements L^+i is to be compared in terms of an element in the next highest level (e.g.,comparing alternatives in terms of a particular criteria or subcriteria). Obviously, thereare situations where the alternatives depend on the alternatives and vice versa. In thisgeneral system, called a system with feedback in the AHP literature, there is not onlythe concept of outer dependence but also the concept inner dependence as defined bySaaty (1986). Thus the theory can be generalized to more than hierarchical structures,as we shall discuss in §6; the interested reader is also directed to the recent paper bySaaty and Takizawa (1986) where these concepts are explored in detail and an exampleof a system with interdependence is given.

The final point to be made before closing this section concerns Axiom 4. The mainpurpose of this axiom is to deal with issues related to the addition of new alternatives toA. or when alternatives are removed from the set. Consider for a moment the casewhere a copy of one of the alternatives is added to A. Axiom 4 essentially states thatnew criteria must be added to ©. For example, Saaty and Takizawa (1986) argue that ifan apple and an orange were being compared and one adds another apple to the set, anew criterion such as "the number of elements of a certain type (number of apples andnumber of oranges)" should be added to the hierarchy to preserve one's expectations,and thus one should alter the criteria set and the priorities assigned to them. If onerigidly adheres to the previous set of criteria and priorities, then an exact replica or copyof an alternative should never be added according to Axiom 4.

This axiom of expectations also implies that any alternative Aj which is not a truealternative, in the sense that some other alternative .^, E ^ is completely equivalent to


Aj over all criteria, should be removed from the choice problem since it adds nothing tothe choice set. For example, were we comparing alternative A (e.g., a blue Mercedes)and alternative B (e.g., a Chrysler) and a copy of A with respect to the given criteria (e.g.,a red Mercedes where color is not a criterion), this copy should be removed since thepreferences otA versus B will automatically give us the preferences of .B versus all copiesof A by the indiflFerence relationship ~ c ; i.e., A must be indiflFerent to all its copies.Thus, only the real choices should remain in the set A. However, in order to believethat two alternatives are copies, we must believe that there exist no criteria on whichone could differentiate between these two alternatives, or that £ is complete. If one suchcriterion did exist, such as "the number of elements," its addition to £ would cause thetwo alternatives not to be copies.

If ^c B, and no criteria are added to £ when copies to A are added to JL, one couldalways find the number of copies of A that must be added to JL to have B ^c-^ . which isnot a true reflection of our preferences. Hence, if copies are added to JL, then newcriteria must also be added to A.

Therefore, an exact replica or a copy with respect to S should only be added if it addsa truly new alternative to the set A by either altering the set of criteria £, or thepriorities assigned to the criteria and the alternatives.

3. The Frame of Reference

One of the most common criticisms of the AHP concerns the ambiguity of thequestion that the decision maker must answer. For example, Watson and Freeling(1983) write:

Our present criticism ofthe AHP is concerned only with this stage of analysis [of applying themethod to real-world problems] . . . What sort of question needs to be asked to elicit thenumbers in this matrix [of pairwise comparisons]? It would seem that they have to be of theform: "Which is more significant, purchase price or maintenance cost per year?" . . . If thisquestion is asked without further explanation, it is, we maintain, meaningless.

This problem of ambiguity is not a flaw of the AHP, but in fact arises out of afundamental question concerning the frame of reference in which one makes thenecessary subjective judgments. The meaning (of lack of meaning) of a question ulti-mately depends upon the cognitive environment in which one exists. One's belief as tothe meaning of terms such as "more important" or "strongly more important" is afunction of the cognitive frame of reference in which one currently resides. Thesedefinitions will vary from day to day and from individual to individual because thisreference frame varies over time and individuals. While it is true that a poorly wordedquestion yields poor results and that better wording of a question can significantlyincrease the effectiveness of the methods, no method or no perfect question will everremove ambiguity completely due to the reliance on the individual's frame of reference.Watson and Freeling (1983) and others have criticized the mode of questioning out-lined in Saaty's theory while not fully comprehending the above-mentioned issue orunderstanding that excessive ambiguity not explicable within the context of the frameof reference is not a failure ofthe method being used, but rather a failure ofthe analystor decision maker to fully comprehend the issue at hand and state questions whichmeaningfully address it.

Since according to theory, one's reference frame should not influence the way inwhich the question is answered, why should the frame of reference matter? For exam-ple, Stevens' law of measurement (1957) by ratio scales states that the value of this ratioshould be invariant to the actual value of the numerator and denominator, i.e., onlytheir ratio matter. However, even Stevens (1957, p. 153) recognizes that this assump-tion is only a first order approximation. In assessing classical utility functions, numer-

1388 PATRICK T, HARKER AND LUIS G, VARGAS

ous experiments by Tversky and others (Tversky and Kahneman 1981; Hershey et al.1982; Hershey and Schoemaker 1983; Schoemaker and Wait 1982; ICrantz et al. 1971)have shown conclusively that one's frame of reference matters. For example, whetherone is asked to adjust probabilities [probabUity equivalence] or the sure amount [cer-tainty equivalence] in eliciting a von Neumann-Morgenstem utility function, althoughthey are theoretically equivalent methods, they lead to different utility measurements(McCord and de Neufville 1984). Thus, the dependence of utility measurement onone's frame of reference is a well-established phenomenon.

Even more striking evidence to support the claim that the frame of reference matterscan be found in the geography literature. A favorite example which is often used toexplain the AHP is Saaty's measurement of distance from Philadelphia to a set of othercities in the world (Saaty 1980, pp. 41-42). If we were to choose any other city, forexample London, and ask the same relative distance questions from London, we shouldobtain the same relative distance measurement between Philadelphia and London aswe did when using Philadelphia as our anchor point. However, the research by Profes-sor GoUedge at the University of California, Santa Barbara, clearly shows that thisshould not be the case (see Smithsonian 1984). People's cognitive maps are fundamen-tally related to their anchor point or frame of reference, such as home and worklocations. Spatial relations are highly distorted as we move away from one's frame ofreference. Thus, a simple question such as "how far is points as compared to point B inreference to an anchor point" may be viewed as being an extremely ambiguous ques-tion if and/or B are far removed from one's reference point.

Finally, some colleagues have pointed out that there exist enough examples in theutility theory literature to show that the frame of reference introduces biases. However,these examples are not immediately applicable to the AHP because the theoreticalcomputations and the mode of questioning are fundamentally different from the lot-tery-type question of multi-attribute utility theory (MAUT). It is an open researchquestion as to what bias means in the context of the AHP. Tversky and Kahnemanhave been the leaders in the area of defining judgmental bias, but Kruglanski and hiscolleagues at Tel Aviv University, Israel, have begun to question the definition of whatexactly bias means in their theory of "Lay Epistemology" (see Kruglanski and Ajzen1983 and Snow 1984). Kruglanski's argument states that there are no secure criteria onwhich to judge bias, and that bias does not necessarily lead to errors in judgment. AsSnow (1984, p. 523) notes: "Human-machine interactions designed on the assumptionthat the human side is rational, or that the human side is generally subject to certainbiases, may also often be wrong. Thus, one can only recognize that reference points andthe mode of questioning will, in general, affect the results ofthe decision and that oneshould not attempt to correct for these effects in any way other than to make thequestions as clear as possible."

Therefore, ambiguity is a phenomenon of all preference eliciting methods and, to alarge extent, the reduction of ambiguity in the AHP is in the hands of the decisionanalyst. Clear definitions of the criteria, subcriteria and alternatives is essential in alldecision aids and should obviously be of major concern to users of AHP or any othermethodology. Thus, ambiguity is prevalent in all decision methodologies, not just theAHP, and simple corrections to account for bias as suggested by some researchersshould be avoided due to the type of problems raised by Kruglanski.

4. The Scale of Measure

One of the most controversial areas of the AHP concerns the scale used to measurethe intensity of preferences, Pc{A,,Aj), between two altematives.,4,, yE A. This scale isdefined to be a ratio scale, and it is assumed that the (cardinal) intensity of preference


between two alternatives can be expressed using the scale. The use of a ratio scale in theAHP is a major departure from traditional methods of decision analysis which typicallyemploy interval measures. Although there has been some criticism concerning the useof a ratio versus an interval scale, there is a large body of literature in psychology whichreadily accepts the use of a ratio scale in measuring the relative intensity of stimuli(Stevens 1957; Stevens and Galanter 1964; Krantz 1972). The major focus ofthecriticism lies in Saaty's 1 to 9 scale. This section addresses this issue. However, it shouldbe noted from the outset that if one accepts the notion of a ratio scale and the axioms of§2, then Saaty's eigenvector and hierarchical composition methods are the correctmethod of synthesizing the pairwise judgments. Thus, the belief which some critics(Dyer and Wendell 1985) hold that the AHP is ad hoc is in fact false. The scale ofmeasure may be subject to debate, but the mathematics behind the AHP follow directlyfrom the axioms in §2.

The first point that should be made concerning the scale of measure is that this issuearises in all preference inducing methods. For example, although the assumption oftransitivity of preferences in utility theory leads one to conclude that utility is invariantup to a linear transformation in the von Neumann-Morgenstem case, which impliesthat utihty measurements are not affected by the actual numerical scale used in themeasurement process, actual utility measurements have been shown to be highly sensi-tive to the scale employed. For example, Hershey et al. (1982), Hershey and Schoe-maker (1983), and others have shown that a linear transformation ofthe interval scalecan drastically alter the results which are generated. Thus, the numerical values used onthe scale do seem to affect an individual's stated preferences, and we cannot say thatawy method of preference revelation is entirely independent ofthe scale of measure.

The second point concerns the use of a ratio scale. Stevens (1957) and Stevens andGalanter (1964) have argued that ratio scales are an appropriate means with which toelicit response to stimuli.

Given that all methods are in some way scale dependent, what is the correct scale foruse in the AHP? Saaty (1980) has proposed that we use a ratio scale between 1 and 9,although as we have discussed, this scale is open to debate. The experiments reported bySaaty (1980) and the experience of many users of the AHP tend to support the view thatthe 1 to 9 scale captures fairly well the preferences of an individual. However, the scalecan be altered to suit an individual's needs. Take, for example, the issue of placing anupper limit on the ratio scale (such as 9). Why not allow for an unbounded scale?Although there are many convincing philosophical arguments to support the viewembodied in Axiom 2 that humans cannot fully comprehend the notion of infinity orinfinite preference for an altemative when compared with some other alternative, themathematical structure of the AHP can even deal with this situation. Consider thefollowing matrix of pairwise comparisons between two alternatives:

1 a

I/a 1

In the next section we show that the scale of relative measurements ofthe alternatives isgiven by the principal right eigenvector

_J. \_\l+a' \+al'

The limit of this eigenvector as a - • oo (unbounded or infinite preference) is (1,0); thatis, lexicographic-type preferences. Thus, the mathematics of the AHP is capable ofdealing with a ratio scale in the interval [1, oo). The recognition of this fact lays to restthe argument that the 1 to 9 scale causes inconsistencies in judgment to occur because


of having to remain in this interval; consistency can be easily enforced by expanding thescale, although the reasons why one may not wish to do this (Saaty 1986) are sufficientto choose Axiom 2.

Thus, any bounded ratio scale would be in accordance with the axioms. However,one must choose a particular scale in order to implement the AHP; this choice is madeon empirical grounds. Clearly, one scale may be appropriate for one application andmay not be appropriate for another. In this situation, a different scale could and shouldbe chosen for each application. For general applications where there is no a priorireason for choosing one scale over another, Saaty (1980) has chosen the scale [1,9] andsupports this choice with several experiments which lend evidence as to the appropri-ateness of the scale. Let us now consider the issue in some detail.

The 1 to 9 scale may be able to capture a great deal of information and has proven tobe extremely useful due to the fact that the AHP is somewhat scale independent. TheAHP, through the use of the eigenvector and normalization procedures, is a highlynonlinear operator on the scale of measure. Thus, a simple linear 1 to 9 scale can easilyrepresent a highly nonlinear cognitive scale when the AHP is used in conjunction vnththis linear scale. Furthermore, the AHP requires an individual to make n{n — l)/2pairwise comparisons when comparing n alternatives, versus the (n - 1) comparisonswhich would have to be made if cardinal transitivity (or consistency) is enforced. Thejudgments obtained by using the AHP contain redundant information so that even ifany one judgment is forced to be inconsistent due to the 1 to 9 scale, the final weightsare not substantially affected. Vargas' (1984) work on the stability of positive reciprocalmatrices lends further evidence to the conclusion that any inconsistencies caused by the1 to 9 scale do not significantly affect the final weights.

In order to illustrate that the 1 to 9 scale of measurement does capture perception,consider the distance measurement presented in Saaty (1980, pp. 41-42):

IDistancefiom

Philadelphia

CairoTokyoChicagoSan FranciscoLondonMontreal

Cairo

1M———

Tokyo

1———

Chicago

B(VS-A)A1

B(S-VS)S

San Francisco

MM—1

M

London

MM——1

Montreal

VSA

B(E-M)B(S-VS)B{S-VS)

1

whereE—Equal DistanceA/^—Moderate DistanceS—Strong DistanceVS—Very Strong DistanceA—Absolute Distance, andB{E~M), for example, represents Between Equal and Moderate.The 1 to 9 scale, when stated verbally, is given in Table 1 (Saaty 1980, p. 54).

TABLE 1

Pc(A,,Aj) Definition

13579

Equal DistanceModerate DistanceStrong DistanceVery Strong DistanceAbsolute Distance


TABLE 2

(1)(2)(3)(4)(5)

City

CairoTokyoChicago

Scale

1-91-51-15x^

San FranciscoLondonMontreal

p

Equal

11111

(I)

0,2550,3930.0360,1230,1650,028

0,979

Moderate

(2)

0,2450,3260,0510,1480,1800,051

0,957

3259

V3

TABLE 3

(3)

0,2500.4650,0240,0990,1460,016

0,850

Strong

538

25

(4)

0,2440,5540,0060,0720,1200,003

0,495

Very Strong

74

1149

(5)

0,2270,2820,0830,1540,1800,074

0,836

Absolute

95

1581

ActualNormalized

Distance

0,2780,3610,0320,1320,1770,019

Let us consider the use ofthe following set of alternative scales shown in Table 2.Using the above scales, the estimated distances are shown in Table 3. p is the correla-

tion coefficient between the estimated relative distances and the actual relative dis-tances. In this simple example, the 1 to 9 scale outperforms the two linear and the twononlinear transformations of this scale.

There is grovidng evidence as shown by various experiments and by the use of theAHP in practice that the 1 to 9 scale can accurately portray an individual's intensity ofpreference. However, it must be pointed out that any ratio scale can be used in thismethod; the choice of a scale such as the 1 to 9 scale is a result of experimental evidence.The experimentation with various scales of measure is an area in which research mustbe directed so that a consensus throughout the research community can be reachedconcerning the scale numerical values. Until this research is completed, however, the1 to 9 scale has proven to be an acceptable scale and is recommended for use inthe AHP.

5. Cardinal Measurement of Preferences: The Eigenvector Approach

Given the axioms of §2 and the scale of measure discussed in the previous section, thenext task in developing a cardinal measure of preferences involves the synthesis ofthepairwise comparisons into an overall weighting ofthe attributes. That is, how does onetake the information Pc(A,, Aj) for all A,,AjE A, and create a set of weights w, for allA, G Al This question is the subject of this section.

The pairwise comparisons Pc(A,, Aj), following the axioms of §2, form a positivereciprocal matrix which we denote by


where a,j ^ Pc{^i, Aj). Several alternative methods for synthesizing the informationcontained in A have been suggested. In addition to Saaty's method (1980), there is themethod of least squares (Cogger and Yu 1983; Jensen 1983; McKeekin 1979) whichfind the vector of weights w = {wi, W2, ..., Wn)^ by minimizing the Euclidian metric2"^=i (Oy — wJWj)^ and the method of logarithmic least squares (De Graan 1980;Fitchner 1983; Williams and Crawford 1980) which minimizes

2 flog ay-log ^ ) .

However, the eigenvector method is the only method which we believe is correct whenA is an inconsistent matrix; that is, when A does not satisfy the relation a,jaji, = a,/, for all/, J and k. Mirkin (1979) has proven that the eigenvector is the appropriate methodwhen the matrix is consistent. However, since inconsistency is a fact of life in measuringpreferences due to one's inability to (and, in fact, desire not to) dismiss judgmentswhich are not perfectly consistent (see the recent paper by Snyden 1985, for an interest-ing discussion of the "economic" causes of inconsistencies in judgments), the debatebetween which procedure is the correct method to use when A is inconsistent is of greatimportance. In order to show that the eigenvector method is the correct method to use,we begin with an illustrative example.

Consider the matrix

A = [ 1 1 3\l/a i 1

Any n X n positive matrix can be thought of as a strongly connected, directed graphwhere the labels on the arcs connecting the n nodes are the elements fly. Figure 1illustrates the graph for the matrix A. Furthermore, let a = 6 in which case A isconsistent. In terms of the graph depicted in Figure 1, consistency is recognized bynoting that all cycles from each node have the same intensity, where the intensity of apath is defined as the product of the intensities associated with the arcs of that path.Therefore, the weights of the alternatives are any column of the matrix A, and since aratio scale is invariant under similarity transformations, we can normalize these weightsto unity. Thus, we have {w,, Wi, w^ = (0.6, 0.3, 0.1).

There is, however, an alternative interpretation of this simple result. Consider the


length of all paths of length 1 from each node. The intensities associated with each pathare given by the matrix A. Now consider paths of length 2. From node one to node onethere are the paths (1-1-1) of intensity l,(l-2-l)ofintensity 2(1/2) = l , and( l -3 - l )o fintensity (1/6)6 = 1. Thus the intensity of node one as compared with node one takinginto account the "second-order" effects of the other alternatives is 1 -I- 1 + 1 = 3.Similarly, the intensity of preference of 1 over 2 taking into account second-ordereffects is given by the sum ofthe intensities of all paths of length 2, the paths (1-1-2)with intensity 1(2) = 2, (1-3-2) vidth intensity 6(1/3) = 2, and (1-2-2) with intensity1(2) = 2, giving an overall intensity of 2 + 2 + 2 = 6. By now, the readers familiar withgraph theory should have noticed that A corresponds to a type of node-arc incidencematrix which we call an intensity incidence matrix, and that the impact of node / onnode 7 on paths of length k is given by the Ath power of A, A'', which is analogous tofinding the adjacency of nodes in a graph using paths of length k. For our example wehave

/ 3= 1.5

\0.5

631

18\

3/

/ 9^ ' = 4 . 5

\ l .5

1893

5427

9

and so on. Each element a!*' ofA'^ represents the overall intensity of A, over Aj alongpaths of length k. As k increases, more and more of the total interactions betweenalternatives are taken into account. Thus, theyth column of ^* represent the overallintensity of preferences along paths of length k for each A, when viewed from node 7;and the normalized column represents the relative importance of each alternative alongpaths of length k. In the consistent case, each column of 4* when normalized yields thesame relative preferences as one would obtain by normalizing the columns of ' . For^^^we have

3 3/5 = 0.6 18 18/30 = 0.6

1.5—1.5/5 = 0.3, 9 - ^ 9/30 = 0.3,

0.5 0.5/5 = 0.1 3 3/30 = 0.1and so on.

What happens when A is an inconsistent matrix? To illustrate this case, let a = 4 inour example. Then, the paths of length 1 yield different relative intensities, which canbe seen from the normalized columns of A. We have

0.517 0.600 0.5000.286 0.300 0.3750.143 0.100 0.125

That is, the relative importance of each alternative depends on the node used asreference. Considering the paths of length 2, we include more ofthe interactions. Thiscan be seen from A^ and its normalized columns:

3 5.333 1475 3 8

0.667 1.167 3

/0.554 0.561 0.560(Normalized y ) 0.323 0.316 0.320

\0.123 0.123 0.120

This same line of argument for paths of lengths 3 and 4 yield the following normalizedcolumns:

1394 PATRICK T. HARKBR AND LUIS G, VARGAS

/O.f0.2

VO.l

/O.5584 0.5581 0.5588(Normalized ^^) (0.3198 0.3198 0.3193

.1218 0.1221 0.1219

/0.5584 0.5584 O.5584v(Normalized 4") 0.3196 0.3196 0.3196 .

\0.1220 0.1220 0.1220/

Therefore, the normalized path intensities as A; -•• oo are equivalent to the normalizedcolumns of/4*, and converge to the alternative weights w = (0.558, 0.320, 0.122).

If one accepts this intuitive notion that the normalized columns of 4* are the relativeimportance of each alternative when "viewed from a particular alternative" then Saaty(1986) has shown the following result.

THEOREM 1. Let A be a positive reciprocal matrix.(i) If A is consistent then the principal eigenvector of A is given by any of its columns.(ii) If A is inconsistent then the principal eigenvector is given by the limit of the

normalized intensities of paths of length k.

for all h = 1, 2, . . . , /J.

Therefore, this process of considering paths of lengths 1, 2, 3 , . . . , in order to captureall interactions among alternatives is in fact the eigenvector method.

The main reason why this controversy over the eigenvector method (EM), the leastsquares method (LSM), and the logarithmic least squares method (LLSM) has occurredis due to the issue of rank reversal (Belton and Gear 1983, 1985). Unlike traditionalmethods of decision analysis which do not deal directly with inconsistent judgments,rank reversal is a possibility in the current situation due to these inconsistencies. In asingle matrix A which we have been considering in this section, it seems intuitive fromthe graph theoretic argument that EM would retain rank ordering in an inconsistentmatrix while the other methods may or may not, Saaty and Vargas (1984a, b) haveaddressed this issue at length with the type of graph theoretic argument presentedabove. The dominance of alternatives along paths of length k contained in A'^ has beenshown by Saaty and Vargas (1984a) to be the preferred method of deriving the cardinalordering of a set of alternatives. The eigenvector of Theorem 2 is in fact the relativedominance of an alternative.

Before closing this section, the relationship between Axiom 2 (p-Homogeneity) andthe synthesis of pairwise comparisons should be stated. Given a scale (e.g., the 1 to 9scale) homogeneity states that all Pc(Ai, Aj) should be contained in the interval definedby the scale. If this were not the case, the mathematical argument presented in thissection would still be valid, but the results would be meaningless. If 9 is "absoluteimportance" on our scale, what meaning would 10 have and hence, what meaningwould the resulting weights have? Thus, homogeneity of alternatives is necessary for theweights w, to have meaning in the context of our scale of measure.

6. Preferences in General Network Structures and the Issue of Rank Reversal

While there has been some controversy concerning the eigenvector method, mostanalysts recognize the validity of this method through the type of argument presented inthe last section, and through the numerous successful applications of the method.However, some believe that the notion of hierarchical composition of levels of criteriaand alternatives (a) is too simplistic, and that (b) rank reversals can occur (Belton andGear 1983). In this section we address these issues.


Consider a simple hierarchy in which there are three criteria Ci, C2 and C3, and twoalternatives Ai and A2 which are compared with respect to the three criteria. Further-more, let us assume that not only the alternatives influence the criteria but that thecriteria also influence the alternatives. Thus, we have what Saaty (1980, Chapter 8) hasdefined as a system with feedback. This new system is no longer a simple hierarchy, buta complex network of interactions in which the choice of an alternative affects acriterion and the choice of a criterion affects an alternative. Therefore, we no longerhave a simplistic system, but a highly complex one, and the criticism voiced in (a) isimmediately ill-founded. Returning to our example, let the weights for each alternativewith respect to the criteria be as follows:

Criteria wi wy

123

0.50.40.8

0.50.60.2

Also, let the weights obtained by comparing the criteria according to a particularalternative (i.e., the feedback loop ofthe hierarchy) be given by

Alternative

0.30.3

0.30.5

0.40.2

Figure 2 depicts the network associated with this system. The reader somewhat familiarwith the AHP should immediately recognize this network as a generalization of astandard hierarchy.

What are the overall weights associated with each criterion and each alternative? Letus construct the incidence matrix for this network:

C, C2 A2

W =c,CJ

C3Al

A2

000

0.50.5

000

0.40.6

000

0.80.2

0.30.30.400

0.30.50.200

1 ' contains the "first-order" impacts of each alternative and criterion on all otheralternatives and criteria. It is analogous io A^ in the previous section. The third-orderimpacts which take into account the impacts of all elements along paths of length 3

FIGURE 2


000

0.5540.446

000

0.5540.446

000

0.5540.446

0.3000.3890.311

00

000

.300

.389

.31100

have the same type of graph-theoretic interpretation as A'' and can be found by com-puting W^. Continuing the same line of argument as in §5, we have

lim

Thus, the overall weights of the criteria are (0.300, 0.389, 0.311) and the compositeweights of the alternatives are (0.554, 0.446).

The theory behind the use of this "supermatrix" Wean be found in Saaty (1980).where he presents a slightly different exposition of this approach to systems withfeedback when he compares H^*+' with the transition probability matrix of a Markovprocess. There does exist a theory for expanding the AHP beyond simple tree struc-tures, and this theory does allow one to easily model complex systems, such as theassessment of economic input-output tables by Saaty and Vargas (1979).

The second problem which has been raised concerning the AHP involves the prob-lem of rank reversals. In a hierarchy, several authors have shown that rank reversalsoccur even if all judgments are consistent; the following example from Dyer andWendell (1985) illustrates this point.

Consider four alternatives, A^, A^, Ay and A^, which must be ranked according tofour criteria, Ci, C2, C3 and C4. Let us assume that performance measures are availableaccording to the four criteria as shown in Table 4.

I>yer and Wendell (1985, p. 20) attempt to make their example realistic by assumingthat the alternatives are mutually exclusive investment opportunities, each yieldingreturns for four years which are considered to be the criteria, and each requiring thesame initial capital outlay. Let Table 4 be the returns of the alternatives during the fouryears. Which alternative should be chosen to maximize the return from the investment?

If the criteria are assumed to be difference independent (see Dyer and Sarin 1979).the preference ranking is obtained by adding the scores according to the criteria. Notethat this difference independence is itself a subjective concept. In the previous examplewe have

Alternatives

A^AlA,A,

Scores

14201818

Rank

4122

Dyer and Wendell (1985) proceed to show that, assuming that the criteria are equallyimportant and that hierarchic composition is used to determine the composite score of

TABLE 4

Alternatives

A,A2AyA,

c,

1984

Q

9111

Criteria

C3

1948

c.

3155


the alternatives, if one first ranks the first three alternatives and then all four, rankreversal takes place. Let us consider the case of three alternatives. We have

A}

For the

.4}

A,

c,1/189/188/18

9/111/111/11

four alternatives, we have

c,1/229/228/224/22

Cz

9/121/121/121/12

Ci

I/I49/144/14

Q

1/229/224/228/22

Ct

3/91/95/9

Ct

3/141/145/145/14

CompositeScore

0,3200,3360,344

CompositeScore

0,2640,2430,2460,246

Rank

321

Rank

1422

and alternatives Ai and A2 have reversed ranking. The authors conclude that becausethis simple example cannot be handled by hierarchic composition, the AHP producesarbitrary rankings. In this example, rank reversals are only due to a misuse ofthe theoryrather than a faulty axiomatic base, as we shall now illustrate.

An important assumption underlying the use ofthe Principle of Hierarchic Compo-sition is that the weights of the criteria are independent from the alternatives consid-ered. If this assumption is violated, then the system with feedback approach must beused. In this case the Principle is violated and we must construct a supermatrix W asfollows:

c,C2

C3

c.Al

A2

A3

A,

C, C2 C3 Q

0

RELATIVE WEIGHTSOF ALTERNATIVES

ACCORDING TOCRITERIA

Al A2 AJ A4

RELATIVE WEIGHTSOF CRITERIA

ACCORDING TOALTERNATIVES

_ _

r\(J

Note that in the example, the criteria (time periods) should not have been assumedequally important if hierarchic composition is used, because the returns are not thesame under each time period. The question is, what should the weights of the criteriabe so that the relative weights of the alternatives coincide with the relative scores ob-tained under the assumption of difference independence. The answer is given byli \ In the example we have

Ci C2 C3 Ct Al A2 A] At

CtE

A,

A2

1/229/228/224/22

9/121/121/121/12

0

1/229/224/228/22

- -3/141/145/145/14

1/149/141/143/14

1 "

9/201/209/203/20

0

8/181/184/185/18

4/184/188/185/18

and


c,C2

c,

A,

A2

A,A,

c,

0.20000.2857

0.25710.2571

0

0.20000.28570.25710.2571

C3

0.20000.28570.2571

0.2571

C4

0.20000.28570.2571

0.2571

A,

0.31430.17140.3143

0.200

A2

0.31430.1714

0.31430.200

0

Ay

0.31430.1714

0.31430.200

A,'

0.31430.1714

0.31430.200

Note that the relative scores for the alternatives are given by

RelativeAlternatives

A,A2A,A,

Scores

14201818

ScoFes

0.20000.28570.25710.2571

which coincide with the weights for the alternatives given byThis result can be summarized in the following proposition. Let Jibe a finite set of«

alternatives and let £ be a finite set of m criteria. Let Oy be the value or score of the / thalternative according to they th criterion measured on an absolute scale. Let the criteriabe additive, i.e., the overall score of value ofthe /th alternative is given by the sum ofthescores of the alternative under all the criteria, 2jli ay. Let ^ be the matrix whosecolumns are the relative weights ofthe alternatives according to the criteria (i.e., 2".i a,,= 1,7= 1,2- • • ,m) and let .S be a matrix whose columns are the relative weights ofthecriteria according to the alternatives (i.e., 2;';i bj, = I, i = 1, 2, . . . , « ) . Let H^be amatrix defined as follows

where

' ail ai2

2, a,, 2, a,2

^2 ] a22

2, a,, 2,a,2

"nl aril

,2, a,, 2,a,2

2, a,mand

•y „ /^1 '^tm I nxm

jOtij Zj a2j Lj anj

Ot\2 0(22 ^n2

\ Z j OLxj l^jOl2j

' Columns may not add to unity because of rounding.


PROPOSITION. The relative weights ofthe criteria independent from alternatives aregiven by

limim BiABf = ( 1 ^ , J = 1, 2 , . . . , m]

and the relative weights ofthe alternatives with respect to all the criteria are given by

= (^^^,i= 1 ,2 , . . . ,« ) .

Therefore, one must be very careful and not misuse hierarchic composition and mustrecognize when general network structures are necessary.

Another example of rank reversal is due to Belton and Gear (1983). In their example,three alternatives A, B and C are compared against three criteria Ci, C2 and C3, thejudgment matrices being:

C. A B C w C2 A B C w

ABC

Cy

ABC

191

A

19/81/8

1/91

1/9

B

8/91

1/9

19I

C

891

1/119/111/11

w

8/119/11l/ll

ABC

GOAL

c,C2Cy

11/91/9

c,

• 111

911

c.

111

911

Cy

111

9/11l /Hl/Il

w

1/31/31/3

The overall weighu for A, B and C after applying Saaty's principle of hierarchic com-position (which we shall discuss in a moment) are:

WA = (1/3X1/11) + (1/3X9/11) + (l/3)(8/18) = 0.45,

^Ks = (1/3X9/11 + 1/11-1-9/18) =0.47,

ff'c = (1/3X1/11 + 1/11 + 1/18) =0.08.

Now, Belton and Gear add an alternative D and keep the same judgments for ^ , B andC:

c,ABCD

A

I919

yielding the

M

B

1/91

1/91

c

1919

rankings

^A = 0.37,

c,

ABCD

D

1/91

1/91

Wg

W

1/209/201/209/20

A

19/81/89/8

= 0.29,

B

8/91

1/91

w

C

8919

C,

ABCD

D

8/91

1/91

c = 0.06,

A

11/91/91/9

w

8/279/271/279/27

and

B

9111

WD

C

9111

= 0.29,

D

9111

w

9/121/121/121/12

reversing rank between A and B. However, the new alternative £> is a copy of 5 as canbe seen by the equality of their columns in the matrices given above. The special case ofAxiom 4 discussed in §2 has ruled out the addition of such an alternative because it doesnot add anything to the choice set. This same type of behavior is exhibited in discrete


choice modeling, as evidenced by the well-known "red bus-blue bus" example in thetransportation literature. In this example a logit model is estimated for the modal splitbetween auto and bus (the red bus) and a 50-50 split is observed. If a blue bus is nowadded, the same model used, and if it is assumed that the color ofthe bus does not affectchoice, then a 5-5-5 split is observed. However, intuition suggests that the split shouldbe 50-25-25. In order to prevent this rank reversal from occurring, nested logit modelswere created. That is, the copy ofthe bus is removed from the choice set at the auto-buslevel and moved to another level ofthe hierarchy. This example serves to illustrate thatthe deletion of copies and the addition of criteria to differentiate alternatives (the buscolor) is essential in both the AHP and in some utihty models. Thus, Behon and Gearscounterexample is vacuous when these facts are recognized.

On a deeper philosophical level, Saaty and Vargas (1984c) have argued that reversalof rank does make sense. Luce and Raiffa's (1957) famous example of a gentlemanchoosing salmon over steak at a new restaurant until he hears that they also serve snailsand frog legs and then chooses steak, points to the fact that rank reversal is a fact of life.Our definition of a copy in §2 carefully states that the set of criteria £ must be consid-ered to be a complete set, that is, no new criteria can be added. The explanationtypically given for this restaurant patron's seemingly erratic behavior is that uponhearing that the restaurant also has snails and frog legs, he feels that it must be a goodplace, and thus the steak is most hkely of high quality. That is, a new criterion ofthequality of the restaurant was added to the set of criteria S. Therefore, new alternativesand the possibility of rank reversal must be dealt with carefully because one can neverfully know the set £. However, if £ is believed to be complete, then copies must bedeleted because they add nothing to the altemative set.

Finally, the Principle of Hierarchic Composition is in fact a special case ofthe generalsystem with feedback. Saaty (1980, Chapter 8) has shown that hierarchic composition isobtained if the general principle of composition of systems with feedback is applied to ahierarchy. That is, hierarchic composition arises naturally out of the general systemwhen this system is defined appropriately. For example, consider the situation depictedin Figure 2 where the weights for the three criteria are given exogeneously; i.e.. theseweights are formed independently of the two alternatives. Let us assume that theseweights are (Ci, C2, C3) = (0.3, 0.4, 0.3). The supermatrix then becomes:

C. C2 C3 A, A2

and its hmit is given by:1 by

im

C2IV= C2

A,A2

y;f/2k+] _

000

0.5_0.5

" 000

0.550.45

000

0.40.6

000

0 0.30 0.40 0.3

0.8 00.2 0

000

0.55 0.550.45 0.45

0.30.40.300 _

0.3 (0.4 (0.3 (00

3.33.43.3000.45 0.45 0.45 0 0 _

The weights obtained in the supermatrix limit are precisely the weights obtained b;hierarchical composition:

= 0.3(0.5) + 0.4(0.4) -I- 0.3(0.8) = 0.55,

= 0.3(0.5) + 0.4(0.6) + 0.3(0.2) = 0.45.


Therefore, the Principle of Hierarchic Composition follows naturally from a restrictedform of a general system with feedback.

The critical issues which arise in using the supermatrix technique are: (a) what type ofquestions must one ask in order to compare criteria with respect to alternatives (theupper right-hand section ofthe supermatrix), and (b) are these questions meaningful?Obviously, decision makers may have some difficulty answering such questions inspecific applications, but this is not always the case. Consider the investment examplediscussed above; the question becomes: for a given investment, in which period did theinvestment perform better and by how much. Thus for investment A i the matrix ofcomparisons is:

PERIODS

c,C2CiCt

c,1

C2

1/91

Ci

191

G

1/331/31

Eigenvector

1/149/141/143/14

For example, alternative A1 did nine times better in period C2 than in Ci, and thus thecomparison of Ci versus C2 is | .

Note that the new investment is not a copy or even a near copy of any of the otherthree alternatives, and thus rank reversal cannot be explained as a violation of Axiom 4.This example is simply a case of violation of the linear hierarchical structure typicallyassociated with the AHP and points to the need for the supermatrix approach in certaininstances.

In the case of the example by Belton and Gear, the new alternative is a copy ofalternative B and thus does not contribute new information to the process. This type ofrank reversal is explained by the notion of structural or outer dependence (Saaty 1987).In the case ofthe investment example, the dependence ofthe criteria on the alternativesis not only due to the structural dependence of criteria on alternatives (they existtogether in the hierarchy and are thus dependent), but is also due to & functional or innerdependence (criteria must be judged with respect to alternatives). However, the secondexample exhibits pure structural dependence. Note that in both cases an appropriatelydefined supermatrix will derive the "intuitively" correct weights. If one simply ignoresthese dependencies, one is assuming the criteria weights are independent of the alter-natives and thus the Principle of Hierarchic Composition is valid. However, thesedependencies are valid in the modelling of real-life decision problems and one shouldconsider the use of supermatrices; much more work is necessary to clarify and applythis technique and its relationship to structural and functional dependencies. Ignoranceor dismissal of these notions of dependence in most classical decision analysis methodsreminds one of Davis's (1986, p. 75) quote: "Impossibilities are converted to possibili-ties by clarifying the structural background, by altering the context, by embedding thecontext in a wider context," but as he states later: "Why does one want to do it?"

7. Conclusion

The major criticisms which have been launched at the AHP are, in our opinion, not''alid. The AHP is based upon a firm theoretical foundation and, as examples in theliterature and in the day-to-day operations of various govemment agencies, corpora-lions and consulting firms illustrate, the AHP is a viable, usable decision-making tool.However, this does not mean that the AHP is the method, just as linear programmingcannot solve all OR/MS problems.


One should also not view the AHP as a subset of or perturbation to the traditionalmethods of dedsion analysis such as MAUT. The main controversy in the dedsionanalysis community has arisen by not recognizing that the AHP is based upon anentirely different set of axioms. For example, the fact that the AHP can exhibit depen-dence of irrelevant alternatives (rank reversal) has ruled out this method's use by manyin the OR/MS community. However, Axiom 4 excludes this possibility in the AHP ifthe method is carefuUy employed; i.e., independence of alternatives is never assumed orviewed in the AHP as necessary or even highly desirable. One may say that this hne ofargument is invalid since it contradicts the axioms of MAUT. However, MAUT con-tradicts a basic tenet ofthe AHP which is equally desirable; namely, the fact that peopleare inconsistent and that their inconsistency should be dealt with in a formal ratherthan in an ad hoc manner. Therefore, one must not judge the AHP according to theaxiom of MAUT, but rather should view the two methods as different and possiblycomplementary.

Finally, several interesting areas of research exist in the AHP: clarification of thesupermatrix technique and structural/functional dependencies, better exploitation ofthe inconsistency capabilities of the AHP in MAUT, SMART and other decisionanalysis methods, and the development of effident implementations of this method. Insum, the AHP is an area in which many interesting research problems can lead to manyuseful extensions. The AHP should not be viewed as a final product, but as a lively areafor intellectual progress.^

^ This work was partially funded by the U.S. Army and by the first author's NSF grant ECE-8552773; theirsupport is gratefully acknowledged. Also, the comments of an anonymous referee and the Associate Editor arcwarmly appreciated.

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the theory of ratio scale estimation saaty's ahp

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