Homayoun
`
`Reference 36
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 1
`
`

`

`JOURNAL
`
`01: MATHEMATICAL
`
`PSYCHOLOGY
`
`15,234-281
`
`(1977)
`
`A Scaling Method
`
`for Priorities
`
`in Hierarchical
`
`Structures
`
`THOMAS
`
`L.
`
`SAATY
`
`University
`
`of Pmmsylvania,
`
`Wharton
`
`School, Philadelphia,
`
`Pennsylvania
`
`19174
`
`the principal
`using
`ratios
`of scaling
`a method
`investigate
`is to
`this paper
`of
`The purpose
`is
`the matrix
`data
`of
`Consistency
`matrix.
`comparison
`eigenvector
`of a positive
`pairwise
`the nonprincipal
`eigen-
`the average
`of
`defined
`and measured
`by an expression
`involving
`and
`sufficient
`condition
`for
`consistency.
`values. We
`show
`that
`hmax = n
`is a necessary
`variance
`in
`judgmental
`errors.
`A scale of
`We also show
`that
`twice
`this measure
`is
`the
`with
`a discussion
`of how
`it compares
`with
`numbers
`from
`1 to 9 is
`introduced
`together
`other
`scales. To
`illustrate
`the
`theory,
`it
`is then
`applied
`to some
`examples
`for which
`the
`answer
`is known,
`offering
`the opportunity
`for
`validating
`the approach.
`The
`discussion
`is
`by
`then
`extended
`to multiple
`criterion
`decision
`making
`formally
`introducing
`the notion
`of a hierarchy,
`investigating
`some
`properties
`of hierarchies,
`and applying
`the eigenvalue
`approach
`to scaling
`complex
`problems
`structured
`hierarchically
`to obtain
`a unidimensional
`composite
`vector
`for
`scaling
`the elements
`falling
`in any
`single
`level
`of
`the
`hierarchy.
`A brief
`discussion
`is also
`included
`regarding
`how
`the hierarchy
`serves
`as a useful
`tool
`decomposing
`a large-scale
`problem,
`in order
`to make measurement
`possible
`despite
`now-classical
`observation
`that
`the mind
`is limited
`to 7 + 2 factors
`for simultaneous
`parison.
`
`for
`the
`com-
`
`1. INTRODUCTION
`
`for a set of activities
`to derive weights
`is how
`theory
`of decision
`problem
`A fundamental
`to several criteria.
`judged
`according
`is usually
`Importance
`according
`to
`importance.
`the activities.
`The
`criteria may,
`for
`Each criterion may be shared by some or by all
`example,
`be objectives which
`the activities have been devised
`to fulfill. This
`is a process
`of multiple
`criterion
`decision making which we study here
`through
`a theory of measure-
`ment
`in a hierarchical
`structure.
`to allocate
`for example,
`The object
`is to use the weights which we call priorities,
`activities
`by
`important
`a resource
`among
`the activities
`or simply
`implement
`the most
`the
`relative
`is to find
`rank
`if precise weights
`cannot
`be obtained.
`The problem
`then
`strength
`or priorities
`of each activity with
`respect
`to each objective
`and
`then compose
`the result obtained
`for each objective
`to obtain a single overall priority
`for all the activities.
`Frequently
`the objectives
`themselves must be prioritized
`or ranked
`in
`terms of yet
`another
`set of (higher-level)
`objectives.
`The priorities
`thus obtained
`are
`then used as
`weighting
`factors
`for
`the priorities
`just derived
`for
`the activities.
`In many applications
`we have noted
`that
`the process has
`to be continued
`by comparing
`the higher-level
`objectives
`in terms of still higher
`ones and so on up
`to a single overall objective.
`(The
`top
`level need not have a single element
`in which
`case one would
`have to assume
`rather
`than derive weights
`for
`the elements
`in
`that
`level.) The arrangement
`of the activities;
`234
`
`Inc.
`reserved.
`
`ISSN
`
`0022-2496
`
`Copyright
`All
`rights
`
`by Academic
`Q 1977
`of reproduction
`in any
`
`Press,
`form
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 2
`
`

`

`HIERARCHICAL
`
`STRUCTURES
`
`235
`
`first set of objectives, second set, and so on to the single element objective defines a
`hierarchical structure.
`the weights of the
`for scaling
`The paper
`is concerned with developing a method
`to an element (e.g. criterion or
`elements
`in each level of the hierarchy with
`respect
`objective) of the next higher level. We construct a matrix of pairwise comparisons of the
`activities whose entries indicate the strength with which one element dominates another
`as far as the criterion with
`respect to which
`they are compared is concerned.
`l,..., n, where n is the number of activities,
`If, for example, the weights are wi , i =
`then an entry a($ is an estimate of wi/wj
`. This scaling
`formulation
`is translated
`into
`a largest eigenvalue problem. The Perron-Frobenius
`theory (Gantmacher, 1960) ensures
`the existence of a largest real positive eigenvalue for matrices with positive entries whose
`associated eigenvector
`is the vector of weights. This vector
`is normalized by having its
`entries sum to unity.
`It is unique.
`Thus
`the activities
`in the lowest
`criterion
`in the next level derived
`to that criterion.
`for that
`The weight vectors at any one level are combined as the columns of a matrix
`level. The weight matrix of a level is multiplied on the right by the weight matrix
`(or
`vector) of the next higher level. If the highest level of the hierarchy consists of a single
`objective, then these multiplications will result in a single vector of weights which will
`indicate the relative priority of the entities of the lowest level for accomplishing
`the highest
`objective of the hierarchy.
`If one decision is required,
`the option with
`the highest weight
`is selected; otherwise,
`the resources are distributed
`to the options in proportion
`to their
`weights
`in the final vector. Other optimization problems with
`constraints have been
`considered elsewhere.
`into
`Special emphasis
`is placed in this work on the integration of human judgments
`decisions and on the measurement of the consistency of judgments. From a theoretical
`standpoint consistency
`is a necessary condition
`for representing a real-life problem with
`a scale; however,
`it is not sufficient. The actual validation of a derived scale in practice rests
`with statistical measures, with
`intuition, and with pragmatic justification of the results.
`
`respect
`level have a vector of weights with
`from a matrix of pairwise comparisons with
`
`to each
`respect
`
`2. RATIO
`
`SCALES
`
`FROM
`
`RECIPROCAL
`
`PAIRWISE
`
`COMPARISON
`
`MATRICES
`
`to their relative
`in pairs according
`to compare a set of n objects
`Suppose we wish
`the objects by A, ,..., A, and their
`weights
`(assumed
`to belong to a ratio scale). Denote
`weights by w1 ,..., w,
`. The pairwise
`comparisons may be represented by a matrix
`as follows:
`
`A,
`
`A,
`
`.a.
`
`A,
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 3
`
`

`

`236
`
`THOMAS
`
`L.
`
`SAATY
`
`This matrix has positive entries everywhere and satisfies the reciprocal property aji =
`l/aij . It is called a reciprocal matrix. We note that if we multiply
`this matrix by the
`transpose of the vector wT = (wr
`,..., w,) we obtain the vector rzw.
`Our problem
`takes the form
`
`Am = nw.
`
`that w was given. But if we only had A and wanted
`the assumption
`We started out with
`to recover w we would have to solve the system
`(A - n1)w = 0 in the unknown w.
`This has a nonzero solution
`if and only if n is an eigenvalue of A, i.e., it is a root of the
`characteristic equation of A. But A has unit rank since every row
`is a constant multiple
`of the first row. Thus all the eigenvalues hi , i =
`l,..., ?t, of A are zero except one. Also,
`it is known
`that
`
`gr ha = tr(A) = sum of the diagonal elements = n.
`
`Therefore only one of the hi , which we call Amax , equals n; and
`Ai = 0, hi # nmax .
`is any column of A. These solutions differ by a multi-
`The solution w of this problem
`plicative constant. However,
`it is desirable
`to have this solution normalized so that its
`components
`sum
`to unity. The result
`is a unique solution no matter which column
`is used. We have recovered
`the scale from the matrix of ratios.
`The matrix A satisfies the “cardinal”
`consistency property a,ai, = aiL and is called
`consistent. For example if we are given any row of A, we can determine
`the rest of the
`entries from this relation. This also holds for any set of tl entries whose graph is a spanning
`cycle of the graph of the matrix.
`the scale is not known
`in which
`Now
`suppose
`that we are dealing with a situation
`but we have estimates of the ratios in the matrix.
`In this case the cardinal consistency
`relation
`(elementwise
`dominance) above need not hold, nor need an ordinal
`relation
`of the form A, > Ai , Aj > A, imply A, > A, hold (where
`the Ai are rows of A).
`As a realistic
`representation
`of the situation
`in preference comparisons, we wish
`to account for
`inconsistency
`in judgments because, despite their best efforts, people’s
`feelings and preferences
`remain inconsistent and intransitive.
`imply small
`We know
`that
`in any matrix, small perturbations
`in the coefficients
`perturbations
`in the eigenvalues. Thus the problem Aw = nw becomes A’w’ = hmaxw’.
`We also know
`from
`the theorem of Perron-Frobenius
`that a matrix of positive entries
`has a real positive eigenvalue (of multiplicity
`1) whose modulus exceeds those of all
`other eigenvalues. The corresponding
`eigenvector
`solution has nonnegative entries
`and when normalized
`it is unique. Some of the remaining eigenvalues may be complex.
`Suppose then that we have a reciprocal matrix. What can we say about an overall
`estimate of inconsistency
`for both small and large perturbations of its entries ? In other
`words how close is h,,,
`to n and w’ to w ? If they are not close, we may either revise
`the estimates
`in the matrix or take several matrices
`from which
`the solution vector zu’
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 4
`
`

`

`HIERARCHICAL
`
`STRUCTURES
`
`237
`
`may be improved. Note that improving consistency does not mean getting an answer
`closer to the “real” life solution. It only means that the ratio estimates in the matrix,
`as a sample collection, are closer to being logically related than to being randomly chosen.
`From here on we use A = (uU) for the estimated matrix and w for the eigenvector.
`There should be no confusion in dropping the primes.
`It turns out that a reciprocal matrix A with positive entries is consistent if and only if
`h max = a (Theorem 1 below). With inconsistency Am, > n always. One can also show
`that ordinal consistency is preserved, i.e., if A, > Ai (or aik > aik , k = l,..., n) then
`wa 3 w3 (Theorem 2 below). We now establish (Amax - n)/(n - 1) as a measure of the
`consistency or reliability of information by an individual to be of the form wi/wj . We
`assume that because of possible error the estimate has the form wi/wi Eij where Eij > 0.
`First we note that to study the sensitivity of the eigenvector to perturbations in aij we
`cannot make a precise statement about a perturbation dw =
`in the vector
`(dw,
`,..., dw,)
`w = (WI )...) WJ because everywhere we deal with w, it appears in the form of ratios
`w,/Wj or with perturbations (mostly multiplicative) of this ratio. Thus, we cannot hope
`to obtain a simple measure of the absolute error in w.
`From general considerations one can show that the larger the order of the matrix the
`less significant are small perturbations or a few large perturbations on the eigenvector.
`If the order of the matrix is small, the effect of a large array perturbation on the eigen-
`vector can be relatively large. We may assume that when the consistency index shows
`that perturbations from consistency are large and hence the result is unreliable, the
`information available cannot be used to derive a reliable answer. If it is possible to
`improve the consistency to a point where its reliability indicated by the index is accep-
`table, i.e., the value of the index is small (as compared with its value from a randomly
`generated reciprocal matrix of the same order), we can carry out the following type of
`perturbation analysis.
`The choice of perturbation most appropriate for describing the effect of inconsistency
`on the eigenvector depends on what is thought to be the psychological process which
`goes on in the individual. Mathematically, general perturbations in the ratios may be
`reduced to the multiplicative form mentioned above. Other perturbations of interest
`can be reduced to the general form aij = (Wi/Wj) l ii . For example,
`
`(Wilwi) + %j = (wd/wj)(l + (wj/Wi) %j)*
`
`Starting with the relation
`
`from the ith component of Aw = hmaxw, we consider the two real-valued parameters
`Amax and p, the average of hi , i > 2 (even though they can occur as complex conjugate
`numbers),
`
`~=-(l/(n-l))~Xi=(hmax--n)i(n-l)~O,
`
`hmx=~1
`
`i=2
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 5
`
`

`

`238
`
`THOMAS
`
`L. SAATY
`
`is always an, near its
`It is desired to have p near zero, thus also to have hmax, which
`lower bound n, and thereby obtain consistency. Now we show
`that (hmax - n)/(n - 1)
`is related to the statistical
`root mean square error. To see this, we have from
`
`that
`
`and therefore
`
`iJ=
`
`x nlax - n
`n-1
`
`=
`
`+
`
`a,i -EC .
`wj
`
`Let aij = (wi/wj) Q, cij > 0. Clearly, we have consistency at cii
`imposing the reciprocal relation aj, = I/C+ , we have:
`
`z.Z 1. Now by
`
`p=-1-t
`
`l
`c , n(n - 1) l$i<j<?l (% + $13
`
`
`
`which -+O as cij -+ 1. Also, p is convex in the Eij since Eij + (l/cu) is convex (and has
`its minimum at cU = I), and the sum of convex functions is convex. Thus, p is small
`or large depending on cij being near or far from unity, respectively; i.e., near or far from
`consistency.
`If we write fij = 1 + Sij , we have
`
`/I2 = (l/fZ(fZ - 1)) 1 SFj - (S.ff/l + 6,).
`1q<j<n
`
`Let us assume that ( Sij / < 1 (and hence that S&/(1 + Sij) is small compared with Se).
`This is a reasonable assumption for an unbiased judge who is limited by the “natural”
`greatest lower bound --I on Sii (since aii must be greater than zero) and would tend to
`estimate symmetrically about zero in the interval (-1, 1). Now, p -+ 0 as Sij --f 0.
`Multiplication by 2 gives the variance of the 6, . Thus, 2~ is this variance.
`Suppose now we wish to develop a test of a hypothesis of consistency. Perfect con-
`sistency is stated in the null hypothesis as
`
`H 0 : p = 0.
`
`We test it versus its logical one-sided alternative
`
`HI : y > 0.
`
`The appropriate test statistic is
`
`m = (X,,
`
`- n)/(n -
`
`l),
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 6
`
`

`

`HIERARCHICAL
`
`STRUCTURES
`
`239
`
`is the maximum observed eigenvalue of the matrix whose elements, aij,
`where X,,
`contain random error. Developing a statistical measure for consistency requires finding
`the distribution of the statistic;m. While its specific form is beyond the scope of this
`paper, we observe that m follows a nonnegative probability distribution whose variance
`is twice its mean P and appears to be quite similar to the x2 distribution if we assume that
`all 6, are IV(0, 02) on (-1, 1). Analytically one may have to experiment with other
`distributions such as the j? distribution.
`For our purposes, without knowing the distribution, we use the conventional ratio
`(5 - &/(2%)r)‘12 with p,, = 0, i.e., we use (%/2)1/2 in a qualitative test to confirm the
`null hypothesis when the test statistic is, say, ,<I. Thus when % > 2 it is possible that
`inconsistency is indicated.
`There are several advantages of the eigenvalue method in developing a ratio scale as
`compared with direct estimates of the scale or with least-square methods. For example,
`when compared with the former, it captures more information through redundancy of
`information obtained from pairwise comparisons and the use of reciprocals. When
`compared with either method, it addresses the question of the consistency by a single
`numerical index and points to the reliability of the data and to revisions in the matrix.
`There is no easy way to study the sensitivity of the eigenvector w to errors in A.
`Apart from experiments and the many illustrations, particularly when the order of the
`matrix is large, one may use the following formula, complicated because of the many
`calculations it entails (Wilkinson, 1965):
`
`wr corresponds to h,,
`
`.
`
`(WjT(AA) WJ(A, - Aj) W/Wj) wj ,
`
`Llw, = i
`i=2
`Note that this equation requires the computation of the eigenvalues & , i = 1, 2,..., n,
`with h, = X,,,
`, the right and left eigenvectors of A, wi , and vi , i = 1,2 ,..., n. We have
`shown that wi is generally insensitive to small perturbations in A for our approach, since
`near consistency h, is well separated from hi and vjTwj is never arbitrarily small.
`As already mentioned, it is easy to prove that the solution of the problem Aw = nw
`when A is consistent is given by the normalized row sums or any normalized column
`of A. In addition, the solution to Aw = X,,w when hm, is close to n may be approxi-
`mated by normalizing each column of A and taking the average over the resulting rows.
`This yields a vector a; in this case one can readily obtain an estimate for h,sx by com-
`puting Am, dividing each of the components of the resulting vector by the corresponding
`component of Ed, and averaging the results.
`There are several useful results relating to the eigenvalue procedure. We mention a few
`of them here giving references where necessary.
`
`(asj) be an n x n matrix
`Let A =
`I.
`THEQREM
`then A is consistent
`if and only if h,,,
`= n.
`
`of positive coeficients with aii = a;‘;
`
`Proof.
`
`From
`
`X =
`
`f aijwjwil,
`j=l
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 7
`
`

`

`240
`
`we have
`
`THOMAS
`
`L. SAATY
`
`nh - n =
`
`aijwjw,l.
`
`1
`id=1
`ifj
`
`It is obvious that aii = wi/wj yields X = n and also A,, = n since the sum of the
`eigenvalues is equal to n, the trace of A.
`To prove the converse, note that in the foregoing expression we have only two terms
`involving aii . They are a,iwiw;l and wiwyl/aij . Their sum takes the form JJ + (l/y).
`To see that n is the minimum value of h ma;r attained uniquely at a, = wi/wi we note
`that for all these terms we have y + (1 /y) 2 2. Equality is uniquely obtained on putting
`y = 1, i.e., aSj = Wi/Wj . Thus, when Amax = n we have
`
`n2-722
`
`i 2=n2-n,
`i&l
`ifi
`
`from which it follows that aij = wilwj must hold.
`
`COROLLARY.
`
`For a positive matrix with retiprocal entries we have
`
`A msx 2 n.
`
`If A is inconsistent then we would expect that in some cases aij 3 akl need not imply
`However, since wi , i = I,..., n, is determined by the value of an entire
`(Wi/Wj) > (wk/wt).
`row, we would expect, for example, that if we have ordinal preferences among the
`activities, the following should hold:
`
`2 (Preservation of Ordinal Consistency). If (ol ,..., 0,) is an ordinal scale
`THEOREM
`on the activities C, ,..., C, , where oi > ok implies aij > akj , j = l,..., n, then oi > ok
`implies wi > wk .
`
`Proof.
`
`Indeed, we have from Aw = hmaxw, that
`
`h maxwi =
`
`il
`
`a,jwj > gl akjq
`
`= &UXXW~ 9
`
`with
`
`wi 2 wk.
`
`Because of its substantial importance, we briefly give the essential facts for the problem
`of existence and uniqueness of a solution to Aw = hmaxw. If A is positive, the following
`theorem of Perron assures the existence of a solution.
`
`3. A positive matrix A has a real positive, simple “dominant” characteristic
`THEOREM
`number AmaX to which corresponds a characteristic vector w = (wl , w2 , . . . , w,,) of
`the
`matrix A with positive coordinates wi > 0 (i = 1, 2,..., n).
`
`When A is simply nonnegative, the theorem of Frobenius assures a similar result if A
`is irreducible.
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 8
`
`

`

`HIERARCHICAL STRUCTURES
`
`241
`
`DEFINITION
`
`1. A matrix is irreducible if it cannot be decomposed into the form
`
`0
`A ) 3
`where A, and A, are square matrices and 0 is the zero matrix.
`The following theorem gives the equivalence of the matrix property of irreducibility and
`the strong connectedness of the directed graph of the matrix.
`
`(“,1,
`
`THEXEM
`graph G(A)
`
`4. An n x n complex matrix A
`is strongly connected.
`
`is irreducible
`
`if and only
`
`ij
`
`its directed
`
`We now state the general existence and uniqueness theorem.
`
`THEOREM 5 (Perron-Frobenius). Let A > 0 be irreducible.
`
`Then
`
`(i) A has a positive ea&nvalue
`eigenvalue of A.
`(ii) The eigenvector of A corresponding
`and is essentially unique.
`(iii)
`
`The number h,,,
`
`is given by.
`
`h,,, which
`
`is not exceeded in modulus by any other
`
`to the eigenvalue hm, has positive components
`
`COROLLARY.
`of A satisjies
`
`Let A > 0 be irreducible,
`
`and
`
`let x 3 0 arbitrary.
`
`Then
`
`the Perron
`
`root
`
`A well-known theorem of Wielandt (1950) . m matrix theory yields a stronger result
`than the following, which may be taken as a corollary to it:
`is a nonnegative irrducible matrix, then the value of Amax increases with any
`If A
`element aii of A.
`This corollary does not say explicitly how Amax increases with aij . However, an
`interesting observation for our purpose is that while an increase in aii gives rise to an
`increase in Amax , this increase is partly offset by a decrease in aji = llaij which is one
`of the requirements in filling out the comparison matrix A.
`It is known that the normalized row of the limiting matrix of Ak corresponds to the
`normalized eigenvector of Aw = h maxw. There are several ways of proving this. The
`simpler proofs require special assumptions on the eigenvectors of A.
`
`2. We define the norm of the matrix A by // A I/ 3
`DEFINITION
`the sum of all entries of A), where
`
`(Ae)rd
`
`(i.e., it is
`
`e=
`
`1
`1
`.
`
`. i! i
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 9
`
`

`

`242
`
`THOMAS
`
`L. SAATY
`
`irreducible matrix A is primitive
`3. A nonnegative
`DEFINITION
`is an integer p > 1 such that Ap > 0.
`
`if and only if there
`
`THEOREM
`
`6. For a primitive matrix A
`
`where C is a constant and wmax is the normalized
`
`eigenvector corresponding
`
`to h,,,
`
`.
`
`The following theorem asserts that the ratios of normalized eigenvector components
`remain the same when any row and corresponding column are deleted from a consistent
`matrix of pairwise comparisons.
`
`A by deleting
`is obtainedfiom
`If A is a positive consistent matrix and A’
`7.
`THEOREM
`ith column
`then A is consistent and its corresponding
`ez@nvector
`is obtained
`the ith row and
`from that of A by putting wi = 0 and normalizing
`the components.
`
`j = I,..., n. Thus
`Proof. Given any row of A, e.g., the first, we have aii = alj/ali,
`the ith row of A depends on the ith column entry in its first row being given. Conversely,
`aj, = a,,/alj
`. Thus no entry in A’ depends on the ith row or ith column of A and hence
`is also consistent. Since their entries coincide except in the ith row and ith column
`A’
`of A and since the solution of an eigenvalue problem with a consistent matrix is obtained
`from any normalized column, the theorem follows.
`
`is a matrix of pairwise comparisons and
`In the general case, if A = (aij)
`Remark.
`I,..., n, i # k, j # k, aij = 0, i = k or j = k, and
`A’ = (a;J with aii = aij , i, f =
`if the normalized eigenvector solutions of Aw = hmaxw and A’w’ = hmaxw’ are w and
`w’, respectively, then wk’ = 0 but wa’/wo’ # w,/ws , for all 01 and /3. In other words
`leaving one activity out of a pairwise comparison matrix does not distribute its weight
`proportionately among the other activities. The reason can be seen from the limiting
`relations which show that each activity is involved with the others in a complicated way.
`
`Here we are only interested in numerical entries of the eigenvector. In
`EXAMPLE.
`measurement of the relative wealth of nations illustration given later, the USSR, which
`occupies the second entry, is in the first comparison but is taken out in the second,
`retaining the others. No proportionality equivalence is observed.
`
`U.S.
`
`0.427
`0.504
`
`USSR
`
`0.230
`0.0
`
`China
`
`France
`
`U.K.
`
`Japan W. Germanv
`
`0.021
`0.0258
`
`0.052
`0.0728
`
`0.052
`0.0728
`
`0.123
`0.184
`
`0.094
`0.140
`
`The following theorem shows that seeking an order type of relationship between aij
`and wi/wj involves all of A and its powers in a complicated fashion.
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 10
`
`

`

`HIERARCHICAL
`
`STRUCTURES
`
`243
`
`THEOREM
`whenever
`
`8. For a primitive matrix A we have aij 3 akl if and only ifwi/wj
`
`2 w,Jw,
`
`,
`
`holds. (A pth subscript on a vector indicates the use of its pth entry.)
`
`Proof.
`
`In a typical case
`
`from which we have
`aij = &i
`eo,
`
`It also follows
`
`that
`ak2 =
`
`aknWq
`
`,
`
`aij > ukz t hwi > $
`wj
`
`+
`
`Thus,
`
`the theorem
`
`is true whenever
`
`the following
`
`a,&!, - L c akqwq .
`WZ q+z
`inequality holds:
`
`(llwj)
`
`C
`P#i
`
`QipW,
`
`3
`
`(liwz)
`
`akqwa
`
`-
`
`C
`QZZ
`
`Using Theorem 6 we replace every w, by
`
`the proof.
`yielding
`comparisons but the aif are not
`Assume
`that our mind in fact works with pairwise
`estimates of WJWj but of some function of the latter, aij(wJwj).
`For example, Stevens
`(1959) observed
`that aij as perceived
`for prothetic phenomena
`takes the form (wJw,uI)“,
`where a lies somewhere between 0.3 (in the case of loudness estimation) and 4 (in the
`case of electric shock estimation). For metathetic phenomena, Stevens points out that
`the power
`law need not apply, i.e., a = 1.
`I,..., n,
`i =
`Thus
`it is of interest
`to study
`the general form of the solution gi(wi),
`of an eigenvalue problem satisfying
`the generalized consistency
`condition of the form
`
`If the matrix A = (aij(wJwj))
`9 (The Eigenvalue Power Law).
`THEOREM
`satisfies the generalized consistency condition, then the eigenvalue problem
`
`of order n
`
`f(%>
`
`f(%k)
`
`=
`
`f(Qik)-
`
`i
`
`aij(wi/wj>
`
`gf(wj)
`
`=
`
`ngi(wi),
`
`i =
`
`1,***~ %
`
`has the e&nvertor
`
`solution (~~a,..., w,~) = (gl(wl),..., gn(wn)).
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 11
`
`

`

`244
`
`THOMAS
`
`L.
`
`SAATY
`
`Proof. Substituting the relation
`
`%04wJ = Ei(W,)/gi(euj)
`
`satisfied by the solutiongi(wJ, i = I,..., n, of the eigenvalue problem into the consistency
`condition we have
`
`f M~iY&(~~)l f E&(~h&%Jl = f ki(aT&4
`
`* ~,~~~>/~iWl~
`
`or if we put
`
`we have
`
`x = &e%)/&(fu~)
`
`and
`
`Y = ~dw%dWk)~
`
`This functional equation has the general solution
`
`f(x)f(r)
`
`=f(xr)-
`
`f(x)
`
`= xa.
`
`Thus generalizing the consistency condition for A implies that a generalization of
`the corresponding eigenvalue problem (with A,, = n) is solvable if we replace u,~
`by a constant power a of its argument. But we know that when a = 1, aii = WilWj ;
`thus, in general, aii = (wi/wUi)a, which implies that
`
`and hence,
`
`&(~i)/&4
`
`= (WilW?,
`
`i,j=
`
`1 >***, n,
`
`g,(w,> = w = &i),
`
`i = l,..., 71.
`
`Thus the solution of a pairwise comparison eigenvalue problem satisfying consistency
`produces estimates of a power of the underlying scale rather than the scale itself. In
`applications where knowledge, rather than our senses, is used to obtain the data, one
`would expect the power to be equal to unity and, hence, we have an estimate of the
`underlying scale itself. This observation may be useful in social applications.
`
`3. THE
`
`SCALE
`
`We now discuss the scale we recommend for use which has been successfully tested
`and compared with other scales.
`The judgments elicited from prople are taken qualitatively and corresponding scale
`values are assigned to them. In general, we do not expect “cardinal” consistency to hold
`everywhere in the matrix because people’s feelings do not conform to an exact formula.
`Nor do we expect “ordinal” consistency, as people’s judgments may not be transitive.
`However, to improve consistency in the numerical judgments, whatever value aii is
`assigned in comparing the ith activity with the jth, the reciprocal value is assigned to
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 12
`
`

`

`HIERARCHICAL
`
`STRUCTURES
`
`245
`
`value represents
`record whichever
`first
`. Usually we
`l/a,
`aji . Thus we put aji =
`dominance greater than unity. Roughly speaking,
`if one activity
`is judged
`to be (Y times
`stronger
`than another, then we record the latter as only l/a times as strong as the former.
`It can be easily seen that when we have consistency,
`the matrix has unit rank and it is
`sufficient
`to know one row of the matrix
`to construct
`the remaining entries. For example,
`if we know
`the first
`row
`then a,j = a,JaIi
`(under
`the rational assumption of course,
`that aIi # 0 for all i).
`judgments need not be even ordinally consistent
`It is useful to repeat that reported
`and, hence, they need not be transitive;
`i.e., if the relative importance of C, is greater
`than that of C’s and the relative
`importance of C, is greater
`than that of C, , then the
`relation of importance of C, need not be greater than that of C’s, a common occurrence
`in human judgments. An interesting
`illustration
`is afforded by tournaments
`regarding
`inconsistency or lack of transitivity
`of preferences. A team C, may lose against another
`team C, which has lost to a third
`team Cs ; yet C, may have won against Cs . Thus,
`team behavior
`is inconsistent-a
`fact which has to be accepted in the formulation, and
`nothing can be done about it.
`We now
`turn to a question of what numerical scale to use in the pairwise comparison
`matrices. Whatever problem we deal with we must use numbers
`that are sensible. From
`these the eigenvalue process would provide a scale. As we said earlier, the best argument
`in favor of a scale is if it can be used to reproduce
`results already known
`in physics,
`economics, or in whatever area there is already a scale. The scale we propose is useful for
`small values of rz < 10.
`Our choice of scale hinges on the following observation. Roughly,
`satisfy
`the requirements:
`
`the scale should
`
`they
`in feelings when
`to represent people’s differences
`It should be possible
`1.
`make comparisons.
`It should represent as much as possible all distinct shades of feeling
`that people have.
`2.
`If we denote the scale values by x1 , x2 ,..., xP , then let
`
`xi+1 - xi = 1,
`
`i=l
`
`,*.., p - 1.
`
`that the subject must be aware of all gradations at the same time,
`Since we require
`the psychological experiments
`(Miller, 1956) which show
`that an
`and we agree with
`individual cannot simultaneously
`compare more than seven objects (plus or minus two)
`without being confused, we are led to choose ap = 7 + 2. Using a unit difference between
`successive scale values is all that we allow, and using the fact that x1 = 1 for the identity
`comparison,
`it follows
`that the scale values will range from 1 to 9.
`for
`As a preliminary
`step toward
`the construction of an intensity scale of importance
`activities, we have broken down
`the importance
`ranks as shown
`in the following
`scale
`(Table 1). In using this scale the reader should recall that we assume that the individual
`providing
`the judgment has knowledge about the relative values of the elements being
`compared whose
`ratio is 21, and that the numerical
`ratios he forms are nearest-integer
`approximations
`scaled in such a way
`that the highest ratio corresponds
`to 9. We have
`assumed that an element with weight zero is eliminated from comparison. This, of course,
`
`PATENT OWNER DIRECTSTREAM, LLC
`EX. 2148, p. 13
`
`

`

`246
`
`THOMAS
`
`L.
`
`SAATY
`
`TABLE
`
`1
`
`The Scale and
`
`Its Desceiption
`
`Intensity
`importance
`
`of
`
`Definition
`
`Explanation
`
`I”
`
`3
`
`5
`
`7
`
`9
`
`Equal
`
`importance
`
`Weak
`over
`
`importance
`another
`
`of one
`
`Essential
`
`or strong
`
`importance
`
`Demonstrated
`
`importance
`
`Absolute
`
`importance
`
`2,4,
`
`6 8
`
`Intermediate
`the
`two
`
`values
`adjacent
`
`between
`judgments
`
`Reciprocals
`above
`
`of
`nonzero
`
`If activity
`nonzero
`when
`then
`when
`
`the above
`i has one of
`numbers
`assigned
`to
`compared
`with
`activityj,
`j has
`the
`reciprocal
`value
`compared
`with
`i
`
`it
`
`Rationals
`
`Ratios
`
`arising
`
`from
`
`the scale
`
`activities
`Two
`the objective
`
`contribute
`
`equally
`
`to
`
`Experience
`one activity
`
`judgment
`and
`over
`another
`
`Experience
`one activity
`
`judgment
`and
`over
`another
`
`slightly
`
`favor
`
`strongly
`
`favor
`
`An activity
`dominance
`
`favored
`is strongly
`is demonstrated
`
`its
`and
`in practice.
`
`one activity
`favoring
`evidence
`The
`another
`is of
`the highest
`possible
`order
`of affirmation
`
`over
`
`When
`
`compromise
`
`is needed
`
`If consistency
`obtaining
`the matrix
`
`to be
`were
`n numerical
`
`forced
`values
`
`by
`to span
`
`o On
`dominance
`
`occassion
`between
`
`in 2 by 2 problems,
`two
`nearly
`equal
`
`we have
`activities.
`
`used
`
`1 + E, 0 < E Q 3
`
`to
`
`indicate
`
`very
`
`slight
`
`comparison. Reciprocals of all
`that zero may not be used for pairwise
`imply
`does
`scaled ratios that are >l are entered in the transpose positions (not taken as judgments).
`Note that the eigenvector solution of the problem
`remains the same if we multiply
`the
`unit entries on the main diagonal, for example, by a constant greater than 1.
`At first glance one would
`like to have a scale extend as far out as possible. On