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Rev.R.Acad.Cienc.Exact.Fis.Nat. (Esp)Vol. 93, N ." 4, pp395-423, 1999Monogrfico: Problemascomplejos de decisin. II

BASIC THEORY OF THE ANALYTIC HIERARCHY PROCESS: HOW TO M AKEA DECISIONT H O M A S L . S A A T YUniversity of Pittsburgh

1 . I N T R O D U C T I O NAn assumption arising from the practice of science andengineering since the middle ages is that because natureis physical, we should be able to relate all measurementto physical dim ensions. But that is not true. Huma n thinking and feeling exist in the physical world but they arenot matter or gravity or electromagnetism in the strictsense science understands them today. They are intangible. The human experience involves a very large numberof intangibles. In general and with few exceptions, intangibles cannot be measured on a physical scale. H owever,they can be measured in relative terms through c ompari

son with other tangibles or intangibles with respect toattributes they have in common (taken one at a time) anda ratio scale can be derived from them that yields theirrelative measurement values. The attr ibutes are themselves compared as to their importance with respect to stillhigher attributes, relative measures derived, and so on upto an overall goal.Ratio scales are fundamental for capturing proportionality. All order at its most sophisticated level involvesproportiona lity of its parts in mak ing up the wh ole, and inturn the proportionality of their smaller parts to make upthe parts and so on. Without such proportionalities there

would be no definable relation among the parts and theresulting structure or function of the system under studywould appear to us as arbitrary.When one speaks of relative measurement, those ofus trained in the physical sciences and in mathematicsare likely to think of scales used to measure objects. Forexample, on a scale such as the yard or the meter, eachwith i ts units , we divide the corresponding measurem entsof lengths to get the relative lengths. But that is not whatI mean by relative measurement. First, I ask what wouldI do if I did not have a scale to measure length to definethe relative length of two objects? Henri Lebesgue [13]

wrote:

It would seem that the principle of economy would alwaysrequire that we evaluate ratios directly and not as ratios ofmeasurements. However, in practice, all lengths are measuredin meters, all angles in d egrees, etc.; that is we em ploy auxiliaryunits and, as it seems, with only the disadvantage of having twomeasurements to make instead of one. Sometimes, this is because of experimental difficulties or impossibilities that preventthe direct comparison of lengths or angles. But there is alsoanother reason.In geometrical problems, one needs to compare tw o lengths,for example, and only those two. It is quite different in practicewhen one encounters a hundred lengths and may expect to haveto compare these lengths two at a time in all possible manners.

Thus it is desirable and economical procedure to measure eachnew length. One single measurement for each length, made asprecisely as possible, gives the ratio of the length in question toeach other length. This explains the fact that in practice comparisons are never, or almost never, made directly but throughcomparisons with a standard scale.But when we have no standard scales to measure thingsabsolutely, we must make comparisons and derive relative measurements from them. The question is how, andwhat have we learned in this process?

We should note that we are not talking about a proposed theory that we can accept or reject. Comparisons leading to relative measurement is a talent of our brains. Ithas been neglected in science because we have not learned to formalize it in harmony with the usual way of creating standard scales and comparing or measuring thingson them one at a time.

The cognitive psychologis t Blumenthal [5] writes:Absolute judgmen t is the identification of the magnitudeof some simple stimulus,..., whereas comparative judgmentis the identification of some relation between two stimuliboth present to the observer. Absolute judgment involvesthe relation between a single stimulus and some information

held in short-term memory - information about some former

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396 Thomas L. Saaty Rev.RAcad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93comparison stimuli or about some previously experiencedmeasurement scale... To make the judgment, a person mustcompare an immediate impression with memory impressionof similar stimuli....Thus relative measurement through comparative judgment is intrinsic to our thinking and should not be carriedby us as an appendage whose real function is not understood well or at all and should be kept outside. It is notdifficult to see that relative measurement predates and isnecessary for creating and understanding absolutemeasurement. Some of the work reported on here is nowwell known. But we need it for the subsequent discussionthat lays the foundation for relative measurement.

We know from the neurological sciences that sensedata are mixed with temperature and other informationby the thalamus, before they are recorded in memory. Inthe end what we sense is what we are, and not fully whatis out there. Performance tests that I have conducted onnumerous occasions with a diversity of audiences indicate that an individual not experienced in ranking objectsaccording to size, when comparing one object that is verysmall, with another that is three times larger, would saythat they are about the same size. This is particularly truewhen there are other sizeable objects in the collection.Only by being exposed to many objects and asked tomake careful distinctions in size that the individual willbegin to show an improved ability to sort and rank theobjects according to size. What the person does is to adjust his sensation and impression with what he or she observes. It is not the real objects that on e com pares, b ut theimpressions one forms about them. One needs such realexperiences to institute early in one's mind the possibility of comparing things in pairs. This applies equally tomore abstract ideas and their relative importance to ahigher order property or goal. He would then be able tosay that one idea is more important than another in termsof the satisfaction of the goal and whether, according tohis or her understanding and experience, it is much moreimportant or slightly more important. The lesser of thetwo is always used as the unit in terms of which the moreimportant one is compared as to how much more important it is, and also how many times more, because thefeeling of importance is converted to magnitudes on numerous sense experiences and thus there is transfer fromthe concrete to the abstract so that the two can be combined to make tradeoffs when needed, which happens frequently in daily experience. It is not possible to comparethe lesser element with the greater one, because it mustfirst be used as a unit to determine the magnitude of thegreater one. Thus there is bias in human thinking in usingthe smaller of two elements as the unit. It is impossible apriori to ask how much less the smaller element is thanthe larger without first involving it as the unit of measurement. Thus, priorities of many objects can only be derived on the basis of dominance, and their reciprocal isautomatically calculated to determine in a meaningfulway, the relative priorities of being dominated.

We have learned from many applications that wrongdecis ions may be made in some cases where only onestructure is used for the purpose of generating prioritiesfor the alternatives. In general one needs two or more offour separate structures: one for benefits, one for costs,one for opportunities and a fourth for risks. Because onemust ask what dominates what in the paired comparisons, and by how much (homogeneous elements withclusters and pivots are used for widely spread alternatives) , in the end one multiplies the benefits of each alternative by the opportunities it creates and divides by thecosts times the risks.

N U M B E R S A R E A S G O O D A S T H E S C A L E ST O W H I C H T H E Y B E L O N G

Numerical scales are our simplest way to express relations between things. There are, in addition to nominalscales which are invariant under one-to-one transformations and used to designate objects by assigning each ofthem a different name or symbol, four kinds of numericalscales that we use to deal with the world. These scalesare, from weakest to strongest: ordinal, interval, ratio,and absolute. We need to say a few words about each. Itis worth mentioning at this stage that when there are multiple criteria, it must be possible to combine the rankingswith respect to the different criteria, and not every scaleadmits the use of the arithmetic operations (addition andmultiplication) needed to do the combining. Furthermore , there are situations of interdependence among the alternatives that narrow the choice of scale further.Ordinal Scales: Invariant under strictly monotone increasing transformations x ^ y if and only if/ (x ) ^ f(y).

When X is preferred to y, it is assigned any numbergreater than the number assigned tox ( the same numberonly if X an d 3 are the same). A ssigning numb ers thatindicate order of preference among alternatives is a mapping into an ordinal scale. The only property that onewants to see preserved is monotonicity or simply, greaterthan. The larger (smaller) the number the higher (lower)the rank. For example, if apples are preferred to orangesand oranges are preferred to bananas, we have an infinitenumber of ways to assign numbers indicating thisranking. Below are four ways we might do it:

Apples377

1,000,000.6

Oranges225

1,000.3

Bananas1132.1

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Thomas L. Saaty Rev.RAcad.Cienc.Exact.Fis.Nat (Esp), 1999; 93 397Suppose we have one ordinal ranking on taste andhave another on juiciness:

TasteJuicinessApples

243

Oranges172

Bananas21

Can we add the numbers for both taste and juiciness todetermine which fruit is the most preferred? No. Themagnitudes of the numbers are ambiguous, and we canget many different answers that do not lead to a uniqueoutcome. Similarly, although it may not be so obvious,we cannot aggregate students' judgments on how muchthey enjoy a particular lecture by assigning a numberfrom 1 to 4 and expect to get a meaningful outcome forjudging the competence of the lecturer. Conclusion: Wecannot use ordinals in ranking when many criteria aretaken together to obtain a single overall ranking.

The next three scales are known ascardinal scalesb ecause the assigned values have meaning beyond a simpleorder.Interval Scales: Invariant under positive linear transformations ax+b, a>Q.Interval scales have an arbitrary origin and an arbitraryunit. The advantage of keeping the multiplier apositive

is that if we then take the ratio of the difference of tworeadings on an interval scale to the difference of anothertwo readings, we obtain a ratio scale which is definednext. It is important to know that one plus one is not always equal to two if the numbers belong to an intervalscale. The temperature scale is an example of an intervalscale with a choice as to what zero signifies. In theCelsius scale 0 indicates the freezing point and 100 degrees indicates the boiling point of water, which on theFahrenheit scale have the respective values, 32 and 212.To establish the unit in each scale, one makes a mark atthe 0 Celsius, 32 Fahrenheit level and a nother mark at the100 Celsius, 212 Fahrenheit level, then divides the resulting range into 100 equal parts for the Celsius and 180equal parts for the Fahrenheit. When we apply ordinaryarithmetic to such commonplace scales of measurement,we find that some of the things we do are really illegitimate, because our arithmetic operations result in meaningless information. For example, if we measure temperature on an interval scale such as a Fah renheit scale and ifwe add 20 degrees of Fahrenheit temperature to 30 degrees, we get 50, which is not 50 degrees Fahrenheit temperature - a much warmer temperature. It is meaningfulto take the average of interval scale readings but not theirsum. Thus (ax, +b)-(ax2 + b) =a(x, +X2) +2b ,whichdoes not have the form a x+b.However, ifweaverage bydividing by 2, we do get an interval scale value. We canalso multiply interval scale readings by positive numberswhose sum is equal to 1 and add to get an interval scale

result, a weighted average. One also cannot multiplynumbers from an interval scale, because the result is notan interval scale. Thus (ax, +b)(ax2 +b )=a x^Xj+ab{x^+X2)+b^which again does not have the form ax+b.In decision making where relative measurement findsits best applications because of the need for judgments,interval scales can only be used to rank alternatives withrespect to criteria but not to rank the criteria themselves.One cannot measure criteria or goals on an interval scaleand then use them for weighting alternatives because onethen obtains a product oftwointerval scales, which as wehave seen is not meaningful. Note, however, that the useof interval scales demands the use of tangible objective scales to evaluate alternatives with respect to intangible criteria. When no such tangible numerical indicatorexists,we must somehow find an absolute scale to definethe range in which the alternatives for that criterion canspread, and then assess the given alternatives one by oneas to where they fall in that range. In the absence of anumerical rang e indicator, such a ranking ofthealternatives cannot be made. Conclusion: Interval scales cannotlegitimately be considered for all our purposes - perhapsonly for the alternatives as m ulticriteria utility peop le tryto do by using ratio scales for the criteria. If there is feedback, this approach would not be valid because we wouldhave to add and multiply numbers from interval scales.

We note that the ratioof differences between intervalscale readings is meaningful if we have these readings -but how does one create them in the first place? One cantake the ratio of the difference between apples and oranges to the difference between apples and bananas to decide how much more apples are preferred to oranges thanto bananas. The question is, Can we m ake these comparisons more directly and more simply? We can, with a finer scale. Next we turn to ratio and absolute scales. These two are intimately related, as we shall see below.

Ratio Scales:Invariant under positive similarity transformations ax, a > 0.Length, weight, time, and many other physical attribu

tes can be measured on a ratio scale. Not only can oneadd and multiply numbers from the same ratio scale butone can also multiply numbers from two different ratioscales and still obtain a new ratio scale - something thatphysics does all the time. Ratios are important for gauging a response in proportion to a stimulus or an action inresponse to an idea or abelief. One cannot arbitrarilyassign numbers to things and claim that they are from aratio scale. One needs to be 100 sure that the numbersused belong to a ratio scale. The question is - how?The ratio measurements of objects on a ratio scale areabsolute numbers. If one object has a ratio measurementof six pounds and another of two, their ratio is 6/2 = 3.The larger object is three times heavier than the smallerone, which is an absolute quantity, indicating the simple

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398 Thomas L. Saaty Rev.RAcad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93ratio 3 to 1, whether the measurement is in pounds, kilograms, or other units. All ratio scales can be reduced to acomparison of objects on an absolute scale in the ratio ofX to 1, where the smaller (less dominant) object is 1 andthe larger (more dominant) object isx times the smallerobject, in which case the smaller object is 1/x as large asthe larger object - a reciprocal relation in which the larger object is the unit.

For a ratio scale, we havea x^ +ax2 =a(x^ -hX2) =ax^which belongs to the same ratio scale, and ax^bxj =abx.x^ which belongs to a new ratio scale.However, axj + bx2does not define a ratio scale, andthus we cannot add measurements from different ratioscales.Absolute Scales: Invariant under the identity transformation/(x) =X(is an identity).

There is no transformation on absolute numbers thatleaves them invariant under some operation other thanusing their given values. In other words, an absolutenumber says what it says and it cannot be said in anyother way. If there are five people in a room, there is noway to transform the number5through arithmetic operations and obtain another number that would meaningfully describe the number of people in the room. If oneobject is 5 times heavier than another object, there is noway it can be made different and still convey the idea thatit is 5 times heavier. This is what is meant by invarianceunder the identity transformation. Examples of absolutescales are all collections of numbers that indicate magnitude (how many as used in counting) and frequency (howoften). By abuse of thought, people often think of numbers measured on what we refer to as an objective scale, such as the interval scale of temperature or the ratioscale of weight, as absolute numbers.

It is interesting to note that when we count both menand sheep, to know the number of each in the group wemust express it in relative terms, a ratio of the number ofeach kind to the total number of both kinds. We can thensay, of the total, such and such percentage is of one kindand such and such percentage is of the other. Thus w e useratios to express the relative number of each kind of absolute number in terms ofalarger absolute num ber that isthe sum of the two and represents a higher order of generality. Conclusion: At the heart of dealing with a varietyof things and grouping them together is the notion ofproportionality formalized through ratio scales. In a sense,then, ratio scales are philosophically even more fundamental than absolute numbers when many things have tobe combined for overall understanding, and that is whatour mind does. Kuffler and Nichols [12] p. 57, write:...the surprising co nclusion is that the brain receives little information about the absolute level of uniform illumination...Signals arrive only from the cells with receptive fields situated close to the border...we perceive thedifference or contrast at the boundary and it is by that

standard that brightness in the uniformly illuminatedcentral area is judged.What we need is a theory based on absolute measurement that does not require objective scales, whichwould make it possible not only to measure intangiblesbut also to combine multidimensional measurements intoa unidimensional scale. It is obvious that we can do thisonly if the scales are relative, not absolute, and are amenable to arithmetic operations. It will be seen that, fromthe very idea of a relative scale, our scales must be ratioscales. It is on such a relative priority scale that the human mind (and other forms of existence) determines itsdegree of equilibrium on all the properties it subconsciously perceives at once. At each instant it takes in avariety of data from different dimensions and combinesthem into an overall assessment oftheorder and meaningthat serve our survival needs at that particular moment.These observations are the basis for what follows.A judgment or comparison is the numerical representation ofarelationship between two elements that share acommon parent.

3. THE PARADIGM CASE; CONSISTENCYWe will first show that when the judgments use measurements from a scale to form the ratios, the resultingmatrix is consistent and deriving the scale is an elemen

tary but fundamental operation. Later we generalize tothe inconsistent case where the numerical values of thejudgments are not taken from precise measurements butare ratios estimated according to knowledge and perception.Let us assume that n activities are being considered bya group of interested people and that their tasks area) to provide judgm ents on the relative importanceof these activities, andh) to ensure that the judgm ents are quantified to anextent that permits a quantitative interpretation of

the judgments among all activities.Our goal is to describe a method for deriving, fromthese quantified judgments (i.e., from the relative valuesassociated with pairs of activities), a set of weights to beassociated with individual activities in order to put theinformation resulting from a and b into usable form.LetA,,^2,.. . ,A,,be the activities. The quantified judgments on pairs of activities (A.,Aj)are represented by ann-by-n matrixA = (a.^)AiJ= 1,2, ...,/t).

The problem is to assign to the n activities A,,A2,...,A, aset of numerical weights vi^,, M , ..., vi',, that reflect therecorded quantified judgments.

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Thomas L. Saaty Rev.RAcad.Cienc.Exact.Fis.Nat .(Esp), 1999; 93 399First we get a simple question out of the way. ThematrixA may have several, or only few, non-zero entries

aj.Zeros are used when the judgm ent is unavailable. Thequestion arises: how many entries are necessary to ensurethe existence of a set of we ights that is meaningful in thecontex t of the prob lem ? The an swer is: it is sufficient thatthere be a set of entries that interconnects all activities inthe sense that for every two indices /, j there should besome chain of (positive) entries connecting / with j :

Note that aj itself is such a chain of length 1. (Such amatrix A = (aj) correspo nds to a strongly conne ctedgraph.) This gives precise meaning to the formulation oftask b.One of the most important aspects of the AHP is that itallows us to measure the overall consistency of the judgmentsaj. An extreme example of inconsis tent judgmentsis if we judge one activity to be more important than another and the second more important than the first, aj >1an dj-> 1. More subtle is the case when the judgmentsof three alternatives are not transitive. We might judgeone stone two times as heavy as the first, a third stonetwice as heavy as the seco nd, but the first and last to be ofequal weight. In that caseaj ^ ciik^kr This example leadsus to the

Definit ionA =(aj) is consistent ifci^jj/ = a- , i, j, k = 1, ...,n (1)

We see that such a matrix can be constructed from aset of n elements which form a chain (or more generally,a spanning tree, a connected graph without cycles thatincludes all n elements for its vertices) across the rowsand columns.To interpret our first theorem let us consider the following case. An adult and a child are compared accordingto their height. If the adult is estimated to be two and ahalf times taller, that may be demonstrated by marking

off several heights of the child end to end. However, ifwe have an absolute scale of measurement with the childmeasuring w, units and the adult W2 units, then the comparison would assign the adult the relative value W2/W,and the child Wj/w2, the reciprocal value. These ratiosyield the paired comparison values (Wi/w2)/l and1/(^2/^,), respectively, in which the height of the childserves as the unit of comparison. Such a representation isvalid only if w, and w^ belong to a ratio scale so that theratio w,/w2 is indepe ndent of the unit used, be it in in chesor in centimeters, for example. In this way, we can interpret all ratios as absolute numbers or dominance units.

Let us now form the matrix W whose rows consist ofthe ratios of the measu reme nts Wj of each of n items w ithrespect to all others.

/ w , /W =

w, W,/W2 Wi/W\

W 2 / W , W 2 / W 2 Wjjw,^

It is easy to prove the following theorem:heorem 1. A positive n by n matrix ha s the ratioform A = (w/Wj), i J = l,...,n, if, and only if it is consistent.1Corollary. If (1) is true then A is reciprocal a =

We observe that if W is the matrix above and w is thevector w = (Wj, . . . wj^then Ww = nw. This suggests.Theorem 2 . The matrix of ratios A = (w/Wj) is consistent if and only ifn is its principal eigenvalue and Aw =nw. Further, w > 0 is unique to within a multiplicativeconstant.

Proof The if part of the proof is clear. Now for theotherhalf. If A is consistent then n and w are one of itseigenvalues and its corresponding eigenvector, respectively. Now A has rank one because every row is a constant multiple of the first row. Thus all its eigenvalues except one are equal to zero. The sum of the eigenvalues ofa matrix is equal to its trace, the sum of the diagonalelements, and in this case, the trace of A is equal to n.Therefore, /i is a simple eigenvalue of A. It is also thelargest, or principal, eigenvalue of A. Alternatively, A =Dee^D'^ where D is a diagonal matrix withd- =w,and e= (1,..., )^.T herefore, A an d ee^ are similar and have thesame eigenvalues [20]. The characteristic equation of ^^^is obviously " - n"~^= 0, and the result follows.

The solution w of Aw = nw, the principal right eigenvector of A, consists of positive entries and is obviouslyunique to within a positive multiplicative constant (a similarity transformation) thus defining a ratio scale. Toensure uniqueness , we normalize w by dividing by thesum of its entries. Given the comparison matrix A, wecan directly recover w as the normalized version of anycolum n of A; A = wv, v =(1 /w, , . . . , l/wj. It is interestingto note that for A = (w/Wj), all the conclusions of thewell-known theorem of Perron are valid without recourseto that theorem. Perron's theorem says that a matrix ofpositive entries has a simple positive real eigenvaluewhich dominates all other eigenvalues in modulus and acorresponding eigenvector whose entries are positivethat is unique to within multiplication by a constant.

Here, we concern ourselves only with right eigenvectors because of the nature of dominance. In paired comparisons, the smaller element of a pair serves as the unitof comparison. There is no way of starting with the larger

http://rev.racad.cienc.exact.fis.nat/http://rev.racad.cienc.exact.fis.nat/http://rev.racad.cienc.exact.fis.nat/

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400 Thom as L. Saaty Rev.R.Acad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93of a pair and decomposing it to determine what fractionof it the smaller is without first using the smaller one as astandard for the decomposition.

If A is consistent, then a jmay be repre sented as aratio from an existing ratio scale, such as the kilogramscale for weight. It may also be represented by using ajudgment estimate as to how many times more the dominant member of the pair has a property for which no scaleexists, such as smell or customer satisfaction. Of course,if the measurements from an actual scale are used in thepairwise comp arisons, the derived scale of relative ma gnitudes is not a new scale - it is the same one used to dothe measuring. We note that any finite set of n readingsw,, ..., w, from a ratio s cale defin es th e princip al eig envector of a consistent n by n matrix W = (w/Wj).With regard to the order induced in w by W, in general, we would expect for an arbitrary positive matrix A ={aj), that if for some / and j a- ^ a . fo r all k, then w- ^Wj should hold. But when A is inconsistent, i.e., it doesnot satisfy (1), what is an appropriate order condition tobe satisfied by thea^p and how general can such a condition be? We now develop conditions for order preservation that are essentially observations on the behavior of aconsistent matrix later generalized to the inconsistentcase. The ratio (w/w^/l may be interpreted as assigningth ei** activity the unit value of a scale and the j ^ ^ activitythe absolute value w/Wi. In the consistent case, order relations on w-, / = 1, ..., n, can be inferred from thea-j asfollows: we factor out Wj from the first row, Wj from the

second and so on, leaving us with a matrix of identicalrows and w- ^ Wjis both necessary and sufficient for A -^ -implies w- ^Wj

Theorem 3 . If A is consistent, then1 ^ (A'"e).l im - > -:p >CW:> 0

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Thomas L. Saaty Rev.R.Acad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93 401and (i)-(v) are true.

Proof Follows from A" ' =n"'~^A wherenis the principal eigenvalue ofA, and A = (wjwj).It appears that the problem of constructing ratio scalesfrom aj has a natural principal eigenvalue structure. Ourtask is to extend this formulation to the case where A isno longer consistent.

4. S M A L L P E R T U R B A T I O N S A N D R A T I OS C A L E A P P R O X I M A T I O NBecause we are interested in the construction of an appropriate matrix W of ratios that serves as a good ap

proximation to a given reciprocal matrix A, we begin byassuming thatAitself is a perturbation of W .We need thefollowing kind of background information.For an unrepeated eigenvalue of a positive matrix A itis known [11,19,22] that a small perturbation A(8) of Agives rise to a perturbation (8) that is an alytic in theneighborhood ofg= 0 and small becauseA(8) is reciprocal. The following known theorem s give us a part of whatwe need.Theorem 4 . (Existence): If is a simple eigenvalueof A, then for small & > 0, there is an eigenvalue X(e,)of

A(E) with power series expansion in E:X(&) = A + &X^'^ + 8^^^^^ + ...

and corresponding right and left eigenvectors w(&) andvffij such thatwfej = w +fiw^'^4-e w ^ -H...v(g) = V + sv^'^ + sV^^ + ...

Let & -j be a perturbation of a reciprocal matrix A suchthat B = (aij + &j) is also positive [7].Theorem 5 . If a positive reciprocal ma trix A has theeigenvalues A,, A2' ' ^n^here the multiplicity ofXj is m-

nwith Y, ^j = ^5 then given s > 0 there is a 3(8) > 0 suchthat if\aj + & ij - aj\ ^ for all i andj the matrix B hasexactly mj eigenvalues in the circle IA " ^jl OJ,j= 1,... , n. In that case,Atakes the formA = WoE= DED~^ where W = (w/wj), E = (8-j),D a diagonalmatrix with w as diagonal vector, and o refers to the Ha-damard or elementwise product of the two matrices . Theprincipal eigenvalue of A coincides with that of E. Theprincipal eigenvector ofA is the elementwise product ofthe principal eigenvectors w = (w,, ..., w^, an d e =(1 ,..., 1)^ of W and of E respectively [20].The distinction we make between an arbitrary positivematrix and a reciprocal matrix is that we can control astep by step mo dification of a reciproca l matrix so that inthe representationA = WoE = DED~\ th e8.j, i,j=l,...,nare small. The purpose is to ensure that perturbing theprincipal eigenvalue and eigenvector of W yields theprincipal eigenvalue and eigenvector of A.Why do we need such a perturbation? Because we assume that there is an underlying ratio scale that we attempt to approximate. By improving the consistency of thematrix, we obtain an approximation of the underlyingscale by the principal eigenvector of the resulting matrix.

Theorem 7 . w is the principal eigenvector of a positivematrix A if and only if Ee = X^^^^e.Note that e is the principal eigenvector of E and E isaperturbation of the matrix e^e. When Ee ^ X^,^^etheprincipal eigenvector ofA is another vector w' / w andA = W'oE' where E'e = X^^^e.

If A is a consistent matrix, then it has one positiveeigenvalue X^= n and all other eigenvalues are zero. Fora suitable s > 0 there is a 3(8) > 0 such that for |0.j| < ethe perturbed matrix B has one eigenvalue in the circlel/i, - n| < e and the remaining eigenvalues fall in a circle1^.- 0 | < 8j = 2, ..., n.

C oro l l a ry , w is the principal eigenvector of a positivereciprocal matrix A = WoE, if and only if, Ee =X^^^e and

Assume that A is an arbitrary positive matrix that is asmall perturbation E of W = (w/Wj).Then we have

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402 Thomas L. Saaty Rev.R.Acad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93Theorem 8 . (Order preservation): A positive matrix Asatisfies condition (vj, if and only if the derived scale wis the principal eigenvector of A, i.e., Aw = A^ax^-Proof We give two proofs of this theorem, the firstis based on the well known theorem of Perron and thesecond, which is more appropriate for our purpose isbased on perturbation.Let

s,= A'e'TFeand

= - I

3)

4)^ = 1

The convergence of the components of i to the samelimit as the components of s^ is the standard Cesaro sum-mability. Since,Sv =- A'ee^A^e w as /c 0 0 5)

where w is the normalized principal right eigenvector ofA, we have1 Z^ A^et, .= - E -f

m " , e'A^ek= 1w as m 0 0 (6)

For the second proof, first assume that A has onlysimple eigenvalues. Using Sylvester 's formula:

f(A) = l^f(Xy^ , A _ = l ,

we have on writing/(A) = A^, dividing through by A^^^,multiplying on the left by (A --X^^^I) to obtain the characteristic polynomial of A then multiplying on the rightby e we obtain:A'e

k-^co Ali m -: ^ = cw ,for some constant c > 0Sylvester 's formula for multiple eigenvalues of multiph-citym. shows that one must consider derivatives of/(A) oforder no more than m;. However, it is easy to verify byinterchanging derivative and limit, that when each termis divided by A^,^^^ its value tends to zero as /:> oo, andthe result again follows.

Therefore, it is necessary to obtain the principal eigenvector w to capture order propertie s from A , but not sufficient to ensure that W =(w/Wj) is a good approximation

to A. The method we use to derive the scale w from apositive inconsistent matrix must also satisfy the following conditions on what constitutes a good numerical approximation to the aJ by ratios. The first two are localconditions on each aj, the second two are global conditions on all a j through the principal eig enva lue andeigenvector as functions of theaj.Four Condit ions for Good Approximat ions1. Reciprocity

The reciprocal condition is a local relation betweenpairs of elements: aj = l/aj,neede d to ensure that, asperturbations of ratios,aja ndaj-can be approximated byratios from a ratio scale that are themselves reciprocal. Itis a necessary condition for consistency.2. Homogeneity - Uniformly Bounded Above and Below

Homogeneity is also a local condition on each a-^T oensure consistency in the paired comparisons, the elements must be of the same order of magnitude whichmeans that our perceptions in comparing them, should beof nearly the same order of magnitude. Thus we requirethat the a jbe uniformly bounded above by a positiveconstant K and, because of the reciprocal condition, theyare automatically uniformly bounded below away fromzero:

l/K ^ a.j ^ K, K> 0, /,j = 1, ..., nIt is a fact that people are unable to directly comparewidely disparate objects such as an apple and a watermelon according to weight. If they are not comparable, it ispossible to aggregate them in such hom ogeneous clustersto make the com parisons by introducing hypothetical elements of gradually increasing or decreasing sizes withwhich they can be compared.For example (Figure 1), to compare an unripe cherrytomato with a watermelon, we compare i t with a smallgreen tomato and a lime in one cluster, then compare thelime with a grapefruit and a honeydew melon in a secondcluster, and finally compare the honeydew melon with asugar baby watermelon and an oblong watermelon in athied cluster. The relative measurements in the clusterscan be combined because we included the largest element (the cantaloupe) in the small cluster as the smallest element of the adjacent larger cluster. Then the relative weights of the elements in the second cluster are alldivided by the relative weight of the common elementand multiplied by its relative weight in the smaller cluster. In this manner, relative measurement of the elementsin the two clusters can be related and the two clusterscombined after obtaining relative measurement by paired

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Thomas L. Saaty Rev.RAcad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93 403

Unripe Cherry Tomato

L ime08 _..08

.65 X 1 = .65

Honeydew.10 1.10

5 .69x1=5 .69

.07

.08

.10

Smal l Green Tomato

Grapefru i t: 2 ^ = 2 7 5.08 ^

. 6 5 x 2 . 7 5 = 1 , 7 9

Sugar Baby Watermelon.30 = 3.10

5.69 X 3=17.07

.28

.22

.30

Lime

Ob long Wa te rme lon.60 = 6.10

5 .69x6=34 .14

.65

.70

.60

This means that 34,14/ ,07 = 487,7 unr ipe cherry toma toes are equal to the ob lon g wa term elon.

Figura 1. Clustering to Compare the very Small with the very Large.

comparisons in each cluster. The process is continuedfrom cluster to adjacent cluster. Here we see that in theend more than 487 cherry tomatoes make up a watermelon. This kind of clustering has to be done with respect to each criterion. For example, instead of size wecould have used relative greenness for clustering andcomparisons.3. Near Consistency

The near consistency condition which is global, is formed in terms of the (structural parameters)X^.^ andnofA andW.It is a less familiar and m ore intricate conditionthat we need to discuss at some length. The requirementthat comparisons be carried out on homogeneous elements ensures that the coefficients in the comparisonmatrix are not too large and generally of the same orderof magnitude, i.e., from 1 to 9. Knowing this constrainsthe size of the perturbations ,.y, whose sum as we shallsee below, is measured in terms of the near consistencycondition1^,,^-'^.

The object then is to apply this condition to developalgorithms to explore changing the judgments and theirapproximation by successively decreasing the inconsistency ofthe judgments and then approximating them withratios from the derived scale. The simplest such algo

rithm is one which identifies thata^jfor whicha-jWJw. ismaximum and indicates decreasing it in the direction ofwjwj. Another algorithm due to Marker [9] utilizes thegradient of thea^j.In the end, we obtain either a consistent matrix or a closer approximation to a consistent onedepending on whether the information available allowsfor making the proposed revisions ina^j.Because consistency is necessary and sufficient for Ato have the form A = {w-lw^, we use w to explore possible changes ina-j to modify Acloser to that form. Weform a consistent matrix W ' - (yv'Jw'^, whose elements

are approximations to the corresponding elements of A.We have a,-^ = (w-/wp z-^., 8j > O.What we have to dealwith is the converse of: given a problem, find a goodapproximation to its solution. It is, given a problem withits exact solution, use the properties of this solution torevise the problem, i.e. the judgments which give rise toa-j. Repeat the process to a level of admissible consistency, (see below)4. Uniform Continuity

Uniform continuity implies that w , / = 1, ..., n as afunction of aj should be relatively insensitive to smallchanges in the aj in order that the ratios Wj/wj remaingood approximations to the a-y. For example, it holds in

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404 Thomas L. Saaty Rev.RAcad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93w. as the ith component of the principal eigenvector because it is an algebraic function ofX^^ (whose value isshown to lie near n because of 3) ,and of thea-jandlla-j,which are bounded.Let us now turn to more elaboration ofthenear consistency condition in3) .We first show the interesting result,that inconsistency or violation of(1)by variousa^jcan becaptured by a single number /lj^--w, which measures thedeviation of alla^. from w-lwj.

Assume that the reciprocal condition a^. = IIa.jandboundedness \IK ^ J- ^ K,whereTT> 0 is a constant,hold. Leta-J= (1 +-^ ^J^j y> - 1 be a perturbation ofW =(w-/Wj),where w is the principal eigenvector of A.Theorem 9. 1^^^ ^ n.Proof Usinga-: =1/a--,andAw =X^^^w, we havecp ji ij^ max '

1 ^ li . 7)Theorem 10. A is consistentif and onlyif ^^^=n.Proof IfAis a consistent, then because of (1), eachrow ofA is a constant multiple of a given row. This implies that the rank ofAis on e, and all but one ofitseigenvaluesA, / = 1, ...,n ,are zero. However, it follows fromnour earlier argument that, ^ /I-=Trace(A) =n.Therefore/ = imax= ^ ' Conversely,X^^ = n , implies Sfj =0, andaj =wjwj.From (2) we can determine the magnitude of thegreatest perturbation by setting one of the terms equaltoX^^^-nand solving forojin the resulting qu adratic. Anaverage perturbation value is obtained by replacing^max~^ in the previous result by(X^^^-n)/(n - 1).A measure of inconsistency is obtained by taking theratio ofX^^^-nto its average value over a large n umber of

reciprocal matrices ofthesame order n, whose en tries arerandomly chosen in the interval [l/K, K].If this ratio issmall (e.g., 10% or less -for example 5% for 3 by 3matrices) [8, 15, 16], we accept the estimate ofw.Otherwise, we attempt to improve consistency and derive anew w. After each iteration, we assume that the newmatrix is a perturbation of W and its eigenvalue andeigenvector are perturbations of n and w, respectively.In his experimental work in the 1950's, the psychologist George Miller [14], found that in general, people(such as chess experts looking ahead a few moves to decide on a good next move) could deal with informationinvolving simultaneously only a few facts: seven plus orminus two. With more, they become confused and cannothandle the information. Since the individual needs to

maintain consistency in his decision matrix, he cannotconsider more than a few options at a time. This is inharmony with the established fact that for a reciprocalmatrix (though not in general) the principal eigenvalue isstable for small perturbations when n is small.

W e have seen that only order preserving derived scalesw are of interest. There are many ways to obtain wfromA. Most of them are error minimizing procedures such asthe method of least squares:z 8)

which also produces nonunique answers. Only the principal eigenvector satisfies order preserving requirementswhen there is inconsistency. We summarize with:Theorem 11. If a positive n by n matrix A is: recip rocal, homogeneous, and nearconsistent, then the scale wderived from A w - A^ ^wis orderpreserving,unique towithin a similarity transformation and uniformly continuous in thea-J, i, j = I, ..., n.

Similar results can be obtained whenA is nonnegative.Also we have extended this discrete approximation ofAby W to the continuous case ofAreciprocal kernel and itseigenfunction [17,18].5. SOME STRUCTURAL PROPERTIESOF POSITIVE RECIPROCAL MATRICESWe make the following observations on the structure ofreciprocal matrices. The elementwise product of two nby nreciprocal matrices is a reciprocal matrix. It followsthat the set of reciprocal matrices is closed under the operation Hadamard product. The matrixe^eis the identity:e'^e = e^eoe^e = ee^and A^ is the inverse ofA, AoPJ -PJoA =e^e.Thus the set G of n by n reciprocal matricesis an abelian group. Because every subgroup of anabelian group is normal, in particular, the set of n by nconsistent matrices is a normal subgroup(EoWoE^= W)of the group of positive reciprocal matrices.

Two matricesAandBareR -equivalent(AR B)if, andonly if, there are a vector wand positive constantsaandbsuch that{lia) Aw= (lib) Bw.The set of all consistentmatrices can be partitioned into disjoint equivalenceclasses. Given a consistent matrix Wand a perturbationmatrixEsuch thatEe = ae, a > 0a.constant, we use theHadamard product to define A' = W oE such that (lia)A'w = (lln) Ww. A' and Ware /^-equivalent. There is a1-1 correspondence between the set of all consistent matrices and the set of all matrices A' defined by suchHadamard products. An /^-equivalence class Q(W)is theset of all A' such that A'R W. The set of equivalenceclasses Q(W) forms a partition of the set of reciprocal

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Thomas L. Saaty Rev.RAca d.Cienc.Exact.Fis.Nat. (Esp), 1999; 93 405matrices. It is known that all the elements in Q(W) areconnected by perturbationsE ,E', E",..., corresponding toa fixed value of a > 0 such that{EoE'oE ..)e = ae.Thusgiven an arbitrary reciprocal matrix A, there exists anequivalence class to whichA belongs.

DeTurck [6] has proved that: The structure group Gofthe set of positive reciprocal nx nmatrices has2n connected components. It consists of nonnegative matriceswhich have exactly one nonzero entry in each row andcolumn. These matrices can be written as D 5, whereDis a diagonal matrix with positive diagonal entries and Sis a permutation matrix, and the negatives of such matrices. The connected component GQof the identity consists of diagonal matrices with positive entries on the diagonal. If A is a positive reciprocal matrix w ith principalright eigenvectorw= (w,,W2,...,wj^ andDBGQis a diagonal matrix with positive diagonal entries d^, dj, ...

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406 Thomas L. Saaty Rev.RAcad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93have similar influences or effects. One must also arrangethem in some rational order to trace the outcome of theseinfluences. Briefly, we see decision making as a processthat involves the following steps:1. Structure a problem with a model that shows theproblem 's key elements and their relationships.

2. Elicit judgm ents that reflect knowledge, feelings,or em otions.3. Represent those judgments with meaningfulnumbers.4. Use these numbers to calculate the priorities of theelements of the hierarchy.5. Synthesize these results to determine an overalloutcome.6. Analyze sensitivity to changes in judgm ent.The analytic hierarchy process (AHP), the decisionmaking process described in this paper, meets these criteria. It is about breaking a problem down and then aggregating the solutions of all the subproblems into a conclusion. It facilitates decision making by organizingperceptions, feelings, judgments, and memories into aframework that exhibits the forces that influence a decision. In the simple and most common case, the forces are

arranged from the more general and less controllable tothe more specific and controllable. The AHP is based onthe innate human ability to make sound judgments aboutsmall problems. It has been applied in a variety of decisions and planning projects in nearly 20 countries.Here rationalityis defined to be: Focusing on the goal of solving the problem; Knowing enough about a problem to develop a complete structure of relations and influences; Having enough knowledge and experience and access to the knowledge and experience of others toassess the priority of influence and dominance (importance, preference, or likelihood to the goal as appropriate) among the relations in the structure; Allowing for differences in opinion with an abilityto develop a best com promise.

How to Structure a HierarchyPerhaps the most creative part of decision making thathas a significant effect on the outcome is modeling theproblem. In the AHP, a problem is structured as a hierarchy. This is then followed by a process of prioritiz

ation, which we describe in detail later. Prioritization involves eliciting judgments in response to questions aboutthe dominance of one element over another when compared with respect to a property. The basic principle tofollow in creating this structure is always to see if onecan answer the following question: Can I compare theelements on a lower level using some or all of the elemen ts on the next higher level as criteria or attributes ofthe lower level elements?

A useful way to proceed in structuring a decision is tocome down from the goal as far as one can by decomposing it into the most general and most easily controlledfactors. One can then go up from the alternatives beginning with the simplest subcriteria that they must satisfyand aggregating the subcriteria into generic higher levelcriteria until the levels of the two processes are linked insuch a way as to make comparison possible.Here are some suggestions for an elaborate design ofahierarchy: (1) Identify the overall goal. What are you trying to accomplish? What is the main question? (2) Identify the subgoals of the overall goal. If relevant, identifytime horizons that affect the decision. (3) Identify criteriathat must be satisfied to fulfill the subgo als of the overallgoal. (4) Identify subcriteria under each criterion. Notethat criteria or subcriteria may be specified in terms ofranges of values of parameters or in terms of verbal intensities such as high, medium, low. (5) Identify the actors involved. (6) Identify the actors' goals. (7) Identify

the actors' policies. (8) Identify options or outcomes. (9)For yes-no decisions, take the most preferred outcomeand compare the benefits and costs of making the decision with those of not making it. (10) Do a benefit/costanalysis using marginal values. Because we are dealingwith dominance hierarchies, ask which alternative yieldsthe greatest benefit; for costs, which alternative costs themost, and for risks, which alternative is more risky. Wenow illustrate the process with an example in which opportunity is combined with other benefits and risk iscombined with other costs. We have numerous exam plesin which they are treated separately particularly when anew system is being designed. In other words only whenthe complexity requires separate considerations that oneuses different structures for each.An Example-The Hospice Problem

Westmoreland County Hospital in Western Pennsylvania, like hospitals in many other counties around thenation, has been concerned with the costs of the facilitiesand manpower involved in taking care of terminally illpatients. Normally these patients do not need as muchmedical attention as do other patients. Those who bestutilize the limited resources in a hospital are patients whorequire the medical attention of its specialists and advanced technology equipment - whose utilization depends on the demand of patients admitted into the hospi-

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Thomas L. Saaty Rev.R.Acad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93 407tal. The terminally ill need medical attention only episodically. Most of the time such patients need psychological support. Such support is best given by the patient's family, whose members are able to supply thelove and care the patients most need. For the mentalhealth of the patient, home therapy is a benefit. From themedical standpoint, especially during a crisis, the hospital provides a greater benefit. M ost patients need the helpof medical professionals only during a crisis. Some willalso need equipment and surgery. The planning association of the hospital wanted to develop alternatives andto choose the best one considering various criteria fromthe standpoint of the patient, the hospital, the community, and society at large. In this problem, we need toconsider the costs and benefits of the decision. Cost includes economic costs and all sorts of intangibles, suchas inconvenience and pain. Such disbenefits are not directly related to benefits as their mathematical inverses,because patients infinitely prefer the benefits of goodhealth to these intangible disbenefits. To study the problem, one needs to deal with benefits and with costs separately.Approaching the Problem

I met with representatives of the planning associationfor several hours to decide on the best alternative. Tomake a decision by considering benefits and costs, onemust first answer the question: In this problem, do thebenefits justify the costs? If they do, then either the benefits are so much more important than the costs that thedecision is based simply on benefits, or the two are soclose in value that both the benefits and the costs shouldbe considered. Then we use two hierarchies for the purpose and make the choice by forming ratios of the priorities of the alternatives (benefits /costs c-) from them.One asks which is most beneficial in the benefits hierarchy (Figure 2) and which is most costly in the costshierarchy (Figure 3). If the benefits do not justify thecosts, the costs alone determine the best alternative - thatwhich is the least costly. In this example, w e decided thatboth benefits and costs had to be considered in separatehierarchies. In a risk problem, a third hierarchy is used todetermine the most desired alternative with respect to allthree: benefits, costs, and risks. In this problem, we assumed risk to be the same for all contingencies. Whereasfor most decisions one uses only a single hierarchy, weconstructed two hierarchies for the hospice problem, onefor benefits or gains (which model of hospice care yieldsthe greater benefit) and one for costs or pains (whichmodel costs more).

The planning association thought the concepts ofbenefits and costs were too general to enable it to make adecision. Thus, the planners and I further subdividedeach (benefits and costs) into detailed subcriteria to enable the group to develop alternatives and to evaluate thefiner distinctions the members perceived between the

three alternatives. The alternatives were to care for terminally ill patients at the hospital, at home, or partly atthe hospital and partly at home.For each of the two hierarchies, benefits and costs, thegoal clearly had to be choosing the best hospice. Weplaced this goal at the top of each hierarchy. Then thegroup discussed and identified overall criteria for eachhierarchy; these criteria need not be the same for thebenefits as for the costs.The two hierarchies are fairly clear and straightforward in their description. They descend from the moregeneral criteria in the second level to secondary subcriteria in the third level and then to tertiary subcriteria inthe fourth level on to the alternatives at the bottom orfifth level.At the general criteria level, each of the hierarchies,benefits or costs, involved three major interests. The decision should benefit the recipient, the institution, and society as a whole, and their relative importance is theprime determinant as to which outcome is more likely tobe preferred. We located these three elements on the second level ofthebenefits hierarchy. A s the decision w ouldbenefit each party differently and the importance of thebenefits to each recipient affects the outcome, the groupthought that it was important to specify the types of benefit for the recipient and the institution. Recipients wantphysical, psycho-social and economic benefits, while the

institution wants only psychosocial and economic benefits. We located these benefits in the third level of thehierarchy. Each of these in turn needed further decomposition into specific items in terms of which the decisionalternatives could be evaluated. For example, while therecipient measures economic benefits in terms of reduced costs and improved productivity, the institutionneeded the more specific measurements of reducedlength of stay, better utilization of resources, and increased financial support from the community. Therewas no reason to decompose the societal benefits into athird level subcriteria, hence societal benefits connectsdirectly to the fourth level. The group considered threemodels for the decision alternatives, and located them onthe bottom or fifth level of the hierarchy: In M odel 1, thehospital provided full care to the patients; In Model 2, thefamily cares for the patient at home , and the hospital provides only emergency treatment (no nurses go to thehouse); and in Model 3, the hospital and the home sharepatient care (with visiting nurses going to the home).

In the costs hierarchy there were also three major interests in the second level that would incur costs or pains:community, institution, and society. In this decision thecosts incurred by the patient were not included as a separate factor. Patient and family could be thought of aspart of the community. We thought decomposition wasnecessary only for institutional costs. We included fivesuch costs in the third level: capital costs, operating

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408 ThomasL.Saaty

GOAL

GENERALCRITERIA

SECONDARYSUBCRITERIA

TERTIARYSUBCRITERIA

ALTERNATIVES

1Physical0.161

-Direct careof

Recipient Benefits0.64

patients0.02^Pal l ia t ive care0.14

Psycho-socia l0.44

Rev.RAcad.Cienc.ExactFis.Nat. Esp), 999; 93

CHOOSING BEST HOSPICEBenef i ts Hierarchy

-Vo lun tee rsupport0.02

1Economic0.041

-Reduced cos ts0.01' - Improved-Ne twork ing p roduc t i v i t yin fami l ies 0.030.06

-Re l ie fof post -death d ist ress0.12

-Emot iona l suppo r tt of am i lyand patient0.21' -A l lev ia t ionof gu i l t0.03

11Instititional Beneffits0.261

Psycho-socia l0.231

-Pub l i c i t yandpubl ic re la t ions0.19-Vo lun tee rrecru i tment0.03' -Professional

rec ru i tmen tandsuppo r t0.06

1Economic0.031

Societal Benefits0.10

-Reduced leng thof stay0.006-Dea th asasocial issue0.02

-Be t ter u t i l iza t ion -Rehum anizat ionofof resources med ica l , pro fessional0.023 and heal th inst i tu t ions0.08' - Increased f inancia lsuppo r t f romthec o m m u n i t y0.001

(Each alternative nnodel below isconnectedto every tert ia ry subcr i ter ion)MODEL 1 MODEL20.43 0.12 MODEL30.45

Un i tof beds wi th teamgiv ing home care as in ahospi ta lor nurs ing home)

Figura 2. Hospice Benefits Hierarchy.

Mixe d bed, cont ractua l ho me care(Part lyin hospi ta lfor emergencycareandpart ly in home when bet terno nursesgo to the house)

Hospi ta landhome care sharecase management (wi th v is i t ingnursesto theh o m e ;ifex t reme lysick patient goesto the hospital)

costs, education costs, bad debt costs, and recruitmentcosts. Educational costs applytoeducating thecommunity and training the staff. Recruitment costs apply tostaff and volunteers. Since boththecosts hierarchyandthe benefits hierarchy concern the same decision, theyboth have the same alternatives in their bottom levels,even thoughthecosts hierarchy has fewer levels.Judgments andComp arisons

A judgmentorcomparisonis the numerical representationofarelationship between two elements that shareacommon parent.The set of all such judgments can berepresented in a square matrix in which the set of elements is compared with itself. Each judgment representsthedominanceof anelementin thecolumnon theleft over an element in the row on top. It reflects theanswersto twoquestions: whichof the twoelementsismore important with respect to a higher level criterion,and howstrongly, usingthe 1- 9scale showninTable1fortheelementon the left overtheelementat the top ofthe matrix. If theelement on the left is less important

than thaton the top ofthe matrix,weenter the reciprocalvalue in the corresponding position in the matrix.It isimportant tonote thatthe lesser element is always usedastheunitand thegreaterone isestimatedas amultipleof that unit. From all the paired comparisons we calculatethe prioritiesandexhibit themon therightofthe matrix.Fora set of n elements in a matrixoneneeds n(n-l)/2comparisons because thereare n I's on thediagonalforcomparing elements with themselvesand of the remaining judgments, half are reciprocals. Thus we have(n^-n)/2judgments. Whenarough estimateisneededorwhen pressedfor timeorwhenavery reliable expertisproviding thejudgments, one occasionally elicits onlythe minimumofn-l judgments.

As usual withtheAHP,inboththecostand thebenefits models,wecomparedthecriteriaandsubc riteriaaccording to their relative importance with respectto theparent elementin theadjacent upper level.Forexample,in the first matrix ofcom parisonsof the three benefitscriteria with respectto thegoalofchoosingthebest hospice alternative, recipient benefits are moderately moreimportant than institutional benefitsand areassignedthe

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Thomas L.Saaty Rev.RAcad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93 409

GOAL

GENERALCRITERIA

SECONDARYSUBCRITERIA

TERTIARYSUBCRITERIA

Commun i t y Cos ts0.14

Capital0.14

CHOOSING BEST HOSPICECosts Hierarchy

Inst i tu t iona l Costs0.71

Operat ing0.40 Educat ion0.07

C o m m u n i t y0.01

Bad debt0.15

Tra in ing sta f f0.06

Societal Costs0.15

Recru i tment0.06

Staff0.05 Volunteers0.01

ALTERNATIVES(Each a l ternat ive model be lowisconnectedto every tert ia ry subcr i ter ion)

MODEL10.43Un i tof beds wi th teamgiv ing home care as in ahospi ta lor nurs ing home)

Figura3 Hospice Costs Hierarchy.

MODEL20.12Mixe d bed, cont ractua l hom e care(Part lyinhospi ta lforemergencycareandpart ly in home when bet terno nursesgo to the house)

MODEL30.45Hospi ta landhome care sharecase management (wi th v is i t ingnursesto theh o m e ;if ext remelysick patient goesto the hospital)

absolute number 3 in the (1,2) or first-row second-column position. Three signifies three times more. The reciprocal value is automatically entered in the (2,1) position, where institutional benefits on the left are compared

with recipient benefits at the top. Similarly a 5, corresponding to strong dominance or importance, is assignedto recipient benefits over social benefits in the (1,3) position, and a 3, corresponding to moderate dominance, is

Tabla 1 The Fundamental ScaleIntensity ofImportance13579

2,4, 6, 8

Reciprocalsofabo-1 ve

Rationals1.1-1.9

DefinitionEqual Importance.Moderate importance.Strong importance.Very strongordemonstrated importance.Extreme importance.For compromise betweentheabovevalues.If activity / has one of theabove nonzero num bers assignedto itwhen compared with activity j , then j has thereciprocal value when compared with/.Ratios arising from thescale.For tied activities.

ExplanationTwo activities contribute equallyto theobjective. iExperience andjudgment slightly favor one activity overanother.Experience and judgment strongly favor one activity overanother.An activityis favored very strongly over another;itsdominance demonstrated inpractice.The evidence favoring one activity over another is of thehighest possible orderof affirmation.Sometimes one needs to interpolate a compromise judg-ment numerically because thereis nogood wordtodescribeit.A comparison mandated by choosing thesmaller elementastheunittoestimatethelargerone as amultipleof thatunit.If consistency wereto beforced byobtainingnnumericalvaluesto span thematrix.When elementsare closeandnearly indistinguishable;moderateis 1.3 andextremeis 1.9.

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410 Thom as L. Saaty Rev.R.AcadCienc.ExacLFis.Nat (Esp), 1999; 93assigned to institutional benefits over social benefits inthe (2,3) position with corresponding reciprocals in thetranspose positions of the matrix.

A scale of absolute numbers used to assign numericalvalues to judgments made by comparing two elementswith the smaller element used as the unit and the largerone assigned a value from this scale as a multiple of thatunit.Importancehere is generic and can be replaced bypreference orlikelihood,the latter indicating that one canuse it for risk analysis as the C hina exam ple given later inthe paper shows.If there is a real ratio scale of measurement and it isdesired to use it instead ofthefundamental scale, one canuse these values if one accepts the linearity inherent in

the values ofthescale and does not w ish or is not allowedto (because for example, the problem belongs to someone else whose priorities may not be known at the time)interpret these values as priorities by putting them inranges and applying the fundamental scale to comparethe relative importance of these ranges. Otherwise, thevalues should be interpreted according to the fundamental scale.There are numerous examples in the literature thatserve to give validation to this scale, the protein examplebelow is one.Figure 4 below shows five areas to which the readercan apply to the paired comparison process in a matrixand use the 1-9 scale to test the validity of the procedure .We can approximate the priorities in the matrix by assuming that it is consistent. We normalize each columnand then take the average of the corresponding entries inthe columns.

The actual relative values of these areasare A= 0.47,B=0.05,C = 0.24, D =0.14, and E= 0.09 with which theanswer may be compared. By comparing pairwise morethan two alternatives in a decision p roblem, on e is able toobtain better values for the derived scale because of redundancy in the comparisons, which helps improve theoverall accuracy of the judgments.

Figura 4. Five Figures to Compare in Pairs to Reproduce TheirRelative Weights.

Note that the 1-9 scale can be extended to l-oo by aprocess of clustering as illustrated with the comparisonof the unripe cherry tomato and the water melon. Wenow return to our hospice example.Judgments in a matrix may not be consistent. In eliciting judgmen ts, one makes redundant comparisons to improve the validity of the answer, given that respondentsmay be uncertain or may make poor judgments in comparing some of the elements. Redundancy gives rise tomultiple comparisons of an element with other elementsand hence to numerical inconsistencies. For example,where we compare recipient benefits with institutionalbenefits and with societal benefits, we have the respectivejudgments 3 and 5. Nowifx =3yandx=5 zthen3 y=5z or y = 5/3z-Ifthejudges were consistent, institutionalbenefits would be assigned the value 5/3 instead of the 3given in the matrix. Thus the judgments are inconsistent.In fact, we are not sure which judgments are more accurate and which are the cause of the inconsistency. Inconsistency is inherent in the judgment process. Inconsistency may be considered a tolerable error in measurementonly when it is of a lower order of magnitude (10 percent) than the actual measurement itself; otherwise theinconsistency would bias the result by a sizable error

comparable to or exceeding the actual measurementitself.

Protein in FoodA: SteakB: PotatoesC: ApplesD: SoybeanE: Whole Wheat BreadF: Tasty CakeG: Fish

RELATIVE AMOUNT OF PROTEIN IN SEVEN FOODSWhat food has more protein?

A B C D E F G1 9 9 6 4 5 11/9 1 1 1/2 1/4 1/3 1/41/9 1 1 1/3 1/3 1/5 1/91/6 2 3 1 1/2 1 1/61 / 4 4 3 2 1 3 1/3

1/5 3 5 1 1/3 1 1/51 4 9 6 3 5 1

EstimatedValues0.3450.0310.0300.0650.1240.0780.328

ActualValues0.3700.0400.0000.0700.1100.0900.320

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Thomas L. Saaty Rev.R.Acad.Cienc.Exact.Fis.Nat (Esp), 1999; 93 411When the judgments are inconsistent, the decisionmaker may not know where the greatest inconsistency is.The AH P can show one by one in sequential order which

judgments are the most inconsistent, and that suggeststhe value that best improves consistency. However, thisrecommendation may not necessarily lead to a more accurate set of priorities that correspond to some underlying preference of the decision makers. Greater consistency does not imply greater accuracy and one should goabout improving consistency (if one can given the available knowledge) by making slight changes compatiblewith one's understanding. If one cannot reach an acceptable level of consistency, one should gather more information or reexamine the framework oftheh ierarchy[16].Under each matrix I have indicated a consistency ratio

(CR) comparing the inconsistency of the set of judgments in that matrix with what it would be if the judgments and the corresponding reciprocals were taken atrandom from the scale. For a 3-by-3 matrix this ratioshould be about five percent, for a 4-by-4 about eight percent, and for larger matrices, about 10 percent [15, 16].Priorities are numerical ranks measured on a ratioscale. A ratio scale is a set of positive numbers whoseratios remain the same if all the numbers are multipliedby an arbitrary positive number. An example is the scaleused to measure weight. The ratio of these weights is thesame in pounds and in kilograms. Here one scaleisjust a

constant multiple oftheother. The object of evaluation isto elicit judgments concerning relative importance of theelements of the hierarchy to create scales of priority ofinfluence.Because the benefits priorities ofthealternatives at thebottom level belong to a ratio scale and their costs priorities also belong to a ratio scale, and since the product orquotient (but not the sum or the difference) of two ratioscales is also a ratio scale, to derive the answer w e dividethe benefits priority of each alternative by its costs priority. We then choose the alternative with the largest ofthese ratios. It is also possible to allocate a resource proportionately among the alternatives.I will explain how priorities are developed from judgments and how they are synthesized down the hierarchyby a process of weighting and adding to go from local

priorities derived from judgments with respect to a singlecriterion toglobalpriorities derived from multiplicationby the priority of the criterion andoverallpriorities derived by adding the global priorities ofthesame element.The local priorities are listed on the right of each matrix.If the judgm ents are perfectly consistent, and hence CR =0, we obtain the local priorities by adding the values ineach row and dividing by the sum of all the judgments, orsimply by normalizing the judgments in any column, bydividing each entry by the sum of the entries in that column. If the judgments are inconsistent but have a tolerable level of inconsistency, we obtain the priorities byraising the matrix to large powers, which is known totake into consideration all intransitivities between the elements, such as those I showed above betweenx,y,andz[16].Again, we obtain the priorities from this matrix byadding the judgment values in each row and dividing bythe sum of all the judgments. To summarize, the globalpriorities at the level immediately under the goal areequal to the local priorities because the priority of thegoal is equal to one. The global priorities at the next levelare obtained by weighting the local priorities of this levelby the global priority at the level immediately above andso on. The overall priorities of the alternatives are obtained by weighting the local priorities by the global priorities of all the parent criteria or subcriteria in terms ofwhich they are compared and then adding. (If an elementin a set is not comparable with the others on some property and should be left out, the local priorities can beaugmented by adding a zero in the appropriate position.)In Table 2 we compare the criteria under benefits.

The process is repeated in all the matrices by askingthe appropriate dominance or importance question. Forexample, for the matrix comparing the subcriteria of theparent criterion institutional benefits (Table 3), psychosocial benefits are regarded as very strongly more important than economic benefits, and 7 is entered in the(1,2) position and 1/7 in the (2,1) position.In comparing the three models for patient care, weasked members of the planning association which model

they preferred with respect to each of the covering orparent secondary criterion in level 3 or with respect to thetertiary criteria in level 4. For example, for the sub-criterion direct care (located on the left-most branch inthe benefits hierarchy), we obtained a matrix of pairedTable 2. The judgments in this matrix are the responses to the question: Which criterion is more important with respect tochoosing the best hospice alternative and how strongly?

Choosing BestHospiceRecipient BenefitsInstitutional BenefitsSocietal Benefits

RecipientBenefits11/3

1/5

InstitutionalBenefits31

1/3

SocialBenefits531

Priorities.64.26.11CR. = .33

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412 Thomas L. Saaty Rev.RAcad.Cienc.Exact.Fis.Nat. (Esp), 1999; 93Table 3. The judgments in this matrix are the responses to the question: Which subcriterion yields the greater benefit with respect toinstitutional benefits and how strongly?

Institutional benefitsPsycho-socialEconomicPsycho-social11/7

Economic71Priorities.875.125

C.R. = .000Table 4. Thejudgments in this matrix are the responses to the question: Which model yields the greater benefit with respect to directcare of patient and how strongly?

Direct of patientModel I: Unit/TeamModel II: Mixed/Home CareModel III: Case Management

Model I11/51/3

Model II513

Model III31/31

Priorities.64.10.26

C.R. = .33comparisons (Table 4) in which Model 1 is preferredover Models 2 and 3 by 5 and 3 respectively and Model 3is preferred by 3 over Model 2. The group first made allthe comparisons using semantic terms for the fundamental scale and then translated them to the correspondingnumbers .

For the costs hierarchy, I again illustrate with threematrices. First the group compared the three major costcriteria and provided judgments in response to the question: which criterion is a more important determinant ofthe cost of a hospice model? Table 5 shows the judgments obtained.

datory because we needed to give back priorities derivedfrom using actual measurements. In the followingexample, we have two criteria price and repair cost.W e also have three i tems, A, B, and C, whose values areas follows:Normalized sum

TotalPrice Repair Cost Total Norm alizedA 200 150 350 .269B 300 50 350 .269C 500 100 600 .462

The group then compared the subcriteria under institutional costs and obtained the importance matrix shown inTable 6.Finally we compared the three models to find outwhich incurs the highest cost for each criterion or sub-criterion. Table 7 shows the results of comparing themwith respect to the costs of recruiting staff.Our procedure for synthesis involves multiplying thepriorities of the alternatives by those of the criteria andadding as shown below. This additive procedure is man-

Weighting, adding and then normalizingPrice(1000/1300)200/1000300/1000500/1000

Repair Cost(300/1300)150/30050/300100/300

Weightingand adding.269.269.462Note that the priority of each criterion is the quotient ofthe sum of the values of the items under it to the sum ofthe values of the items under both criteria.

Table 5. The judgments in this matrix are the responses to the question: Which criterion is a greater determinant of cost with respectto the care method and how strongly?Choosing Besthospice costs)

Community CostsInstitutional CostsSocietal Costs

Community151

Institutional1/511/5

Societal151

Priorities.14.71.14

C.R. = .000

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Thomas L. Saaty Rev.RAca d.Cienc.Exact.Fis.Nat. (Esp), 1999; 93 413Tab le 6. The judgmen ts in this matrix are the responses to the question: Which criterion incurs greater institutional costs and howstrongly?

Institutional costsCapitalOperatingEducationBad DebtRecruitment

Capital17471

Operating1/711/91/41/5

Education1/49121

Bad Debt1/741/211/3

Recruitment15131

Priorities.05.57.10.21.07

C.R. = .08As shown in Table 8, we divided the benefits prioritiesby the costs priorities for each alternative to obtain thebest alternative, model 3, the one with the largest valuefor the ratio. Table 8 shows two ways or modes of synthesizing the local priorities of the alternatives using theglobal priorities of their parent criteria: The distributivemode and the ideal mode. In the dis tr ibutive mode, theweights of the alternatives sum to one. It is used whenthere is dependence among the alternatives and a unitpriority is distributed among them. The ideal mode isused to obtain the single best alternative regardless ofwhat other alternatives there are. In the ideal mode, thelocal priorities of the alternatives under each criterion aredivided by the largest value among them. For each criterion one alternative becomes an ideal with value one.Synthesis is obtained by multiplying these values by thepriorities of their corresponding criteria and then adding.An alternative that is best for every criterion receives a

composite priority of one. All other alternatives receive asmaller value. The composite values also belong to a ratio scale. Adding an irrelevant alternative does notchange the ranking of the highly ranked alternatives. Inaddition, if new alternatives are introduced that are assigned greater values than the best alternative withoutkeeping the values assigned before to the existing alternatives, there can be no reversal in the ranks of the oldalternatives.In our example, for both modes the local priorities areweighted by the global priorities of the parent criteria and

synthesized and the benefit-to-cost ratios formed. In thiscase, both modes lead to the same outcome for hospice,which is model 3. As we shall see below, we need bothmodes to deal with the effect of adding (or deleting) alternatives on an already ranked set.

Model 3 has the largest ratio scale values of benefits tocosts in both the distributive and ideal modes, and thehospital selected it for treating terminal patients. Thisneed not always be the case. In this case, there is dependence of the personnel resources allocated to the threemodels because some of these resources would be shiftedbased on the decision. Therefore the distributive mode isthe appropriate method of synthesis. If the alternativeswere sufficiently distinct with no dependence in theirdefinition, the ideal mode would be the way to synthesize.I also performed marginal analysis to determine w herethe hospital should allocate additional resources for thegreatest marginal return. To perform marginal analysis, Ifirst ordered the alternatives by increasing cost prioritiesand then formed the benefit-to-cost ratios correspondingto the smallest cost, followed by the ratios of the differences of successive benefits to costs. If this differencein benefits is negative, the new alternative is droppedfrom consideration and the process continued. The alternative with the largest marginal ratio is then chosen. Forthe costs and corresponding benefits from the synthesisrows in Table 8, I obtained:Costs: .20Benefits: .12

Marginal Ratios:.1 2 _ . 4 5 -

.21

.45

.12

.59.43

33.20 .21 - .20.43 - .45. 5 9 - . 2 1 = -0.05

The third alternative is not a contender for resourcesbecause its marginal return is negative. The second alter-Ta ble 7 . The entries in this matrix respond to the question: Which m odel incurs greater cost with respect to institutional costs forrrecruiting staff and how strongly?

Institutional costs forrecruiting staffModel I: Unit/TeamModel II: Mixed/Home CareModel III: Case Management

Model I11/4

1/4

Model 11411

Model III411

Priorities.66.17.17C.R. = .000

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414 Thomas L. Saaty Rev.R.Acad.Cienc.ExacLFis.Nat (Esp), 1999; 93Tabla 8. The benefit/cost ratios of the three models given in the bottom row of the table are obtained for both the distributive andideal modes. Here one multiplies each of the six columns of priorities of a model by the column of criteria weights on the left andadds to obtain the synthesis of overall priorities, once for the benefits (top half of table) and once for the costs (bottom half of table)and forms the ratios of corresponding synthesis numbers to arrive at the benefit/cost ratio (bottom row of table)

BenefitsDirect Care of PatientPalliative CareVolunteer SupportNetworking in FamiliesRelief of Post Death StressEmotional Support of Familyand PatientAlleviation of GuiltReduced Economic Costs forPatientImproved ProductivityPublicity and Public RelationsVolunteer RecruitmentProfessional Recruitmentand SupportReduced Length of StayBetter Utiliation of ResourcesIncreased Monetary SupportDeath as a Social IssueRehumanization of InstitutionsSynthesis

CostsCommunity CostsInstitutional Capital CostsInstitutional Operating CostsInstitutional Costs forEducating the CommunityInstitutional Costs for TrainingStaffInstitutional Bad DebtInstitutional Costs ofRecruiting StaffInstitutional Costs ofRecruiting VolunteersSocietal CostsSynthesisBenefict/Cost Ratio

Priorities.02.14.02.06.12.21.03.01.03.19.03.06.006.023.001.02.08

.14.03.40

.01

.06.15

.05

.01.15

Distributive ModeModel 1

0.640.640.090.460.300.300.300.120.120.630.640.650.260.090.730.200.24

0.428

0.330.760.730.650.560.600.660.600.33

0.5830.734

Model 20.100.100.170.220.080.080.080.650.270.080.100.230.100.220.080.200.14

0.121

0.330.090.080.240.320.200.170.200.33

0.1920.630

Model 30.260.260.740.320.620.620.620.230.610.290.260.120.640.690.190.600.62

0.451

0.330.150.190.110.120.200.170.200.33

0.2242.013

Model 11.0001.0000.1221.0000.4840.4840.4840.1850.1971.0001.0001.0000.4060.1301.0000.3330.3870.424

1.0001.0001.0001.0001.0001.0001.0001.0001.0000.5230.811

Ideal ModeModel 2

0.1560.1560.2300.4780.1290.1290.1291.0000.4430.1270.1560.3540.4060.1301.0000.3330.2260.123

1.0000.1180.1100.3690.5710.3330.2580.3331.0000.2290.537

Model 30.4060.4061.0000.6961.0001.0001.0000.3541.0000.4600.4060.1851.0001.0000.2601.0001.0000.453

1.0000.1970.2600.1690.2140.3330.2580.3331.0000.2491.819

native is best. In fact, in addition to adopting the thirdmodel, the hospital management chose the second modelof hospice care for further development.A Second Example Combining Benefits, Costsand Risks -The Wisdom of a Trade War withChina over Intellectual Property Rights

This example was developed jointly with the author 'scolleague Professor Jen S. Shang in mid February 1995

to understand the issues when the media were voicingstrong conflicting c oncern s prior to the action to be takenin Beijing later in February. Many copies of the analysiswere sent to congressmen and senators and to the chiefU.S. negotiator in Washington, and to several newspapers in the U.S. and in China. A telephone call wasreceived from Mr. M ickey K an tor 's office, the chief U Snegotiator after the meeting in Beijing congratulating uson the analysis not to sanction China. The person callingsaid, Aren' t you glad we did not sanction China? Webelieve that this short and concise analysis may have had

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Thomas L. Saaty Rev.RAcad.Cienc.ExactFis.Nat. (Esp), 1999; 93 415some effect on that decision. The full write up of about 7pages is not included here because of space limitation.The reader will have no difficulty following the analysiscarried out in the three hierarchies shown below. I havekept the tense of the the next few paragraphs as it waswhen the paper was written to better convey the sense ofurgency in which it was written.

There are many and strong conflicting opinions aboutwhat to do with Chinese piracy of U.S. technology andmanagement know-how. Should the U.S. sanction Chinaon February 26? The basic arguments in favor of imposing tariffs derive from the U.S. perceived need not toallow China to become a runaway nation with an inwardoriented closed economy. Some also argue convincinglythat a nation whose economy will equal that of the U.S.in three decades must be taught to play by the rules. Wehave made a brief study of the decision to impose tariffson Chinese products in the U.S. It is not the immediatesmall injury to U.S . corporations from such an action thatis of major con cern, but what m ight happen in the future.The effect of the tariffs will be decisively more intangible with long-term results that can aggravate trade inthe Pacific.Our findings based on benefits, costs, and risks and onall the factors w e could bring to bear on the outcome is adefinite and very decisiveN o,which means that it is notin the best overall interest of the U.S. to take strong action against China. Since usually we are not told much

about what China says, we also summarize some arguments gleaned from Chinese newspapers. We explain ouranalysis and offer the reader the opportunity to perform asimilar evaluation based on the factors given here plusothers we may have overlooked. In our opinion, the costsare too high to treat China in the same style as an outlawnation even though China can and should do better as amember of the world community.To arrive at a rational decision, we considered the factors that influence the outcome of the decision, and arranged them in three hierarchies: one for the benefits ofimplementing such a sanction, one for the costs and a

third for the risks and uncertainties that can occur (seeFigure 5). Each hierarchy has a goal followed by the criteria that affect the performance of the goal. The alternatives are listed at the last level of the hierarchy. They are:Yes - to sanction China or No - not to sanction China.In each hierarchy, we sy nthesize the values forYesandforiVoby multiplying each alternative's priority with theimportance of its parent criterion, and adding to obtainthe overall result for Yes and for No. A user-friendlycomputer software program, Expert Choice, was used todo all the calculations. To combine the results from thethree hierarchies, we divide the benefit results forYesbythe costs and by the risks for Ye s to obtain the final outcome. We do the same for No and select YesorNod epending on which has the larger value. W hileYes'sbene

fits are high, the corresponding costs and risks are alsohigh. Its ratio is less than that of the No decision. Nodominates Yesboth when no risk is considered and alsowhen projected risk is taken into account. Including riskby using possible scenarios of the future can be a powerful tool in assessing the decision on the effect of the future.

To ensure that the outcom e not be construed as a resultof whimsical judgmen ts, we performed a com prehensivesensitivity analysis. Sensitivity analysis assists the decision maker to discover how changes in the priorities affect the recommended decision. The YesandNoweightsare fixed because they are our best judgments based onthe facts. So we fixed theYesand A o judgm ents as shownin Figure 6 and varied the importance of each factor.There is a wide range of admissible priority value that apolicy maker may choose for each factor. Our sensitivityanalysis covers all the reasonable priorities a politicianmight choose. We changed each factor's importancefrom its value indicated in the hierarchy to the near extreme values 0.2 and 0.8. This gave us six variations ineach hierarchy because there are three factors in each.With three hierarchies, we generated 216 (6^^6*6) datapoints. In this simulation, we found that it is only whenlong term negative competition is thought to be unimportant that sanctions would be justified. From Figure 6 depicting the 216 possibilities, we see that No dominatesYesappreciably. Regardless of the weights one assigns tothe factors, over90%ofthecases lead toNo ,not to sanction China.Deng Rong, the daughter of Deng X iaoping, the m ostsenior elder statesman of China, said recently sanctionsare never the best way to resolve a dispute. One shouldtalk things over and consider the interests of the people.Our analysis supports this attitude.7. GENERALIZATION TO THE CONTINUOUSCASE

The expression encountered for deriving a ratio scalefor pairwise comparisons in the finite case (9)with a.j=l/aj or a-j-j=1 (the reciprocal property),

(10)a-j)>0 and

7 = 1generalizes to the continuous case through Fredholm'sintegral equation of the second kind:Kis, t)w{t) dt = A^,, w{s) (11)

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416 Thomas L. Saaty

1Protect r ights and mainta in h igh Incent iveto m ake and se l l products in China0.696

Yes .80No .20

Benefits

Rev.R.Acad.Cienc.Exact.Fis.Nat. Esp), 1999; 93

Rule o f Law Bring China to res ponsib lef ree-t rad ing0.200Yes .60No .40

Overall Result : Yes 0.729No 0.271

1$ Bil l ion Tarrif fs make Chineseproducts more expensive0.094Yes .70No .30

Costs

1Help t rade def ic i t wi t h C hina0.117

Yes .50No .50

Retaliat ion0.280Yes .90No .10