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Scaling the membership function
320
European Journal of Operational Research 25 (1986) 320-329 North-Holland
Scaring the membership function T h o m a s L. S A A T Y
322 Mervis Hall, Unioersity of Pittsburgh, Pittsburgh, PA 15260, U.S.A.
1. Introduction
To scale the membership function, a method is needed that can assess the importance of elements with respect to a property or criterion that is common among them. It is conceivable that membership should be determined in terms of several criteria tangible and intangible. The approach needed is a multicriterion process whose measurements admit arithmetic operations that preserve the scaling property. Ratio scales is what one obtains through ratio comparisons of elements with respect to criteria. Below we give a set of axioms underlying the Analytic Hierarchy Process for constructing and composing ratio scales, an effective way for constructing the membership function. The following definition is due to Zadeh: Let X = ( x } be a set of points. A fuzzy set A in X is a set of ordered pairs: A={[x,~A(x)]
),
x~X,
where ~A(x) is the grade of membership of x ~ A. If the values of ~t(x) belong to a membership space M, then A is a mapping from X to M and ~A: X---' M is called the membership function of A. A is an ordinary set when M contains only two values 0 and 1. Note that X may be a set of concrete objects or alternatively of abstract criteria. To determine the degree of membership, an effective method of measurement is needed that works uniformly for both concrete and abstract sets X. There are two kinds of measurement, fundamental measurement and derived measurement. Fundamental measurement is direct and involves
Received October 1982; revised October 1984, April 1985.
some kind of scale. Since there is an infinite number of ways for defining X, it is rather hopeless to seek to determine the membership function through direct scaling. Of course, there are many cases where a scale may not be known but there is hope to develop one. For example it is conceivable that a scale can be improvised to measure how alert people can be. In the case of abstract elements such as criteria expressing intensity of perceived properties according to a common attribute which they share, such a group of individuals who are beginning their political career with respect to political acumen, the only possibility we have today is to use judgement from judges who are experienced with respect to that property. This is the case of a derived scale. The problem is how to record and use judgments in a meaningful fashion from which a scale could be derived. I believe that one can convince himself of the following two fundamental observations about the membership function. (1) Except when there is a scale of measurement that can be applied to determine the degree of belonging, membership must be determined by making relative comparisons among the elements. (2) A useful scale for comparisons exists and has been tested and validated in a large number of applications. It is the instrument used to translate judgment and observation to measurement. Axiomatization of this process proceeds as shown below. It is important to point out that a normalized solution is obtained that is unique and belongs to a ratio scale. What is most significant is that the solution by its very construction gives, along with the degree of membership, the correct rank order of the elements. The approach briefly described here is being used widely in other areas and needs careful
0377-2217/86/$3.50 © 1986, ElsevierScience Publishers B.V. (North-Holland)
T.L. Saaty / Scalingthe membershipfunction examination by fuzzy set experts regarding its viability [3,4,5].
2. Axioms for deriving a scale from fundamental measurement and for hierarchic composition Let ~ be a finite set of n elements called alternatives. Let ~ be a set of properties or attributes with respect to which elements in ~¢ are compared. Philosophers distinguish between properties and attributes. A property is a feature that an object or substantial individual possess even if we are ignorant of this fact. On the other hand an attribute is a feature we assign to some object: it is a concept. Here we assume that properties and attributes are interchangeable and generally refer to them as criteria. A criterion is a primitive. When two objects or elements in ~¢ are compared according to a criterion in So, we say that we are performing binary comparisons. Let > c be a binary relation on ~¢ representing ' m o r e preferred than' with respect to a criterion in cg. Let - c be the binary relation 'indifferent to' with respect to a criterion C in ~. Hence, given two elements, A~, A j ~ ¢ , either A , > c Aj or Aj> c Ai or A , - c Aj for all C ~ cg. A given family of binary relations > c with respect to a criterion C in ~ is a primitive. Let ~ be the set of mappings from .~e×~¢ to R ÷ (the set of positive reals). Let f: ~'--, ~ . Let P e l f ( C ) for C ~ cg. Pc assigns a positive real number to every pair (A~, Aj) ~ . ~ ¢ × ~ . Let Pc(A,, A j ) = a , j ~ R +, A~, A ~ ¢ . For each C ~ ~', the triple (.ac×..ae, R +, Pc) is a fundamental or primitive scale. A fundamental scale is a mapping of objects to a numerical system. Definition. For all A~, Aj ~ ¢
ifandonlyifPc(A~, Aj)=I.
Axiom 1 (Reciprocal). For all A,, A j ~ C~
¢ and
Pc(A,, Aj) = 1/ec( Aj, A,).
A=(a+j)=(Pc(A+, Aj)) be
pc(A,,
) >i ec(Aj,
does not hold for all A k e ~ . In other words, it m a y happen that Pc(A,, Aj)> 1, and for some A, e..a¢ we have
Pc(A,,
Pc(Aj,
A more restrictivecondition is the following Definition. The mapping Pc is said to be consistent if and only if
ec(A,,
for all i, j , and k.
If A ~ > c Aj we say that A, dominates Aj with respect to C ~ c~. Thus Pc represents the intensity with which an alternative dominates another.
Let
paired comparisons of the alternatives with respect to a criterion C ~ c¢. By Axiom 1, A is a positive reciprocal matrix. The object is to obtain a scale of relative dominance (or rank order) of the alternatives from the paired comparisons given in A. There is a natural way to derive the relative dominance of a set of alternatives from a pairwise comparison matrix A. Let Rgcn~ be the set of (n × n) positive reciprocal matrices A = ( a i j ) - (Pc(Ai, Aj)) for all C ~ c¢. Let ~b: Rgtn)-~ [0,1] n for A ~ Rgtn), ~b(A) is an n-dimensional vector whose components belong to [0,1]. The triple (RMt,~, [0,1] ~, ~k) is a derived scale. A derived scale is a mapping between two numerical relational systems. It is important to point out that the rank order implied by the derived scale ~b may not coincide with the order represented by the pairwise comparisons. Let ~k(A) be the ith component of ~k(A). It denotes the relative dominance of the ith alternative. By definition, for A~, Ai ~ , A, > c Aj implies Pc(A,, A j ) > 1. However, if Pc(A,, Aj)> 1, the derived scale could imply that ~kj(A)> ~ki(A). This occurs if row dominance does not hold, i.e., for A,, Aj ~.~¢ and C ~
ec(A,, aj)ec(Aj,
and C ~
A~> c Aj ifandonlyif Pc(A ~, Aj)> I, A~-cAj
321
the set of
(1)
Similarly the matrix A is consistent if and only if aijajk=aik for all i, j and k. If Pc is consistent, then Axiom 1 automatically follows.
2.1. Hierarchic axioms Definition. A partially Ordered set is any set S with a binary relation ~< which satisfies the following conditions: (a) Reflexive:
For all x ~ S, x ~< x,
T.L. Saaty / Scaling the membershipfunction
322
(b) Transitive:
F o r all x, y, z ~ S, if x ~
any relation x<~y (read, y intype we define x ~
Partially ordered sets with a finite n u m b e r of elements can be conveniently represented by a directed graph. Each element of the set is represented by a vertex so that an arc is directed from y to x if x < y. D e f i n i t i o n . A subset E of a p a r t i a l l y o r d e r e d set S
is said to be bounded from above (below) if there is an element s ~ S such that x ~ s ( > / s ) for every x ~ E. The element s is called an u p p e r (lower) b o u n d of E. W e say that E has a s u p r e m u m (infimum) if it has u p p e r (lower) b o u n d s and if the set of u p p e r (lower) b o u n d s U ( L ) has an element u~(l~) such that u~ >~ u for all u ~ U (/1 >/l for all
T h e notions of f u n d a m e n t a l a n d derived scales can be e x t e n d e d to x ~ L k, x - C L k ÷ ~ replacing C a n d ~¢ respectively. T h e derived scale resulting from c o m p a r i n g the elements in x - with respect to x is called a local derioed scale or local priorities. G i v e n Lk, Lk+l, C_,.~, let us d e n o t e the local d e r i v e d scale for y ~ x - a n d x ~ L k by q , k + l ( y / x ) , k = 2, 3 . . . . . h - 1. W i t h o u t loss of generality we m a y assume that
Y~. t p k + a ( y / x ) = 1. y~x-
C o n s i d e r the m a t r i x ~ b k ( L k / L k _ l ) whose colu m n s are local derived scales of elements in Lx. with respect to elements in L k_ 1. D e f i n i t i o n ( H i e r a r c h i c composition). T h e global derived scale (rank order) of any element in 0~' is o b t a i n e d from its c o m p o n e n t s in the c o r r e s p o n d ing vector of the following:
~p,(b) = 1,
q,z(L2)=~2(b-/b),
~pk(Lk)=tpk(Lk/Lk_,)q,k_,(Lk_,),
k= 3..... h
I~L). D e f i n i t i o n . Let ~
be a finite p a r t i a l l y o r d e r e d set with largest element b..Yea is a hierarchy if it satisfies the conditions: (1) There is a p a r t i t i o n of .9~ into sets called levels ( L k, k = 1 , 2 . . . . . h } , w h e r e L ~ = { b } . (2) x ~ L k implies x - _ L k + 1, where x-={ylxcoversy), k=1,2 ..... h-1. (3) x ~ L k implies x+_C L k _ l , where x ÷={ylycoversx}, k = 2 , 3 . . . . . h. D e f i n i t i o n . Given a positive real n u m b e r p >~ 1 a
n o n e m p t y set x-___ Lt.+~ is said to be p - h o m o g e neous with respect to x ~ L k if for every p a i r of elements y~, Y2 ~ x - , 1
- <~Pc(Ya, Y2) <~P; P in particular, the reciprocal a x i o m implies that
Pc(y,, x)=
I.
A x i o m 2. G i v e n a hierarchy.Yea, x ~ .Yeaand x ~ L k,
X-C_ Lk+ 1 is p - h o m o g e n e o u s for k = 1 . . . . . h - 1.
D e f i n i t i o n . A set d is said to be exterior dependent on a set c~ if a f u n d a m e n t a l scale can be
d e f i n e d on ~
with respect to every C ~ c~.
3 ( D e p e n d e n c e ) . Let .,~ be a hierarchy with levels L 1, L 2 . . . . . L h. F o r each L k, k = l , 2 . . . . . h - 1, Lk+ 1 is exterior d e p e n d e n t on L k. Axiom
D e f i n i t i o n . Let .~¢ be exterior d e p e n d e n t on ~'. T h e elements in .a¢ are said to be interior dependent with respect to C ~ c£ if for some A ~..a¢, . ~ is exterior d e p e n d e n t on A.
Let 3 a be a family of n o n e m p t y sets ~'1, ~'2 . . . . . ~',, where ~'i consists of the elements {eij, j = 1 . . . . . m i ) , i = 1, 2 . . . . . n. ~ is a system if (i) It is a directed g r a p h whose vertices are ~'~ and whose arcs are defined through (if) G i v e n two c o m p o n e n t s ~ a n d ~ ~ 5" there is an arc from ~i to ~ if ~ is exterior d e p e n d e n t on ~i. Let ~ a . c ( ~ ¢ ) be the derived scale of the elem e n t s of ~¢ with respect to A ~.~e for a criterion Definition.
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T.L. Saaty / Scaling the membership function
C e ~'. Let ffc(J~¢) be the derived scale of the elements of ~¢ with respect to a criterion C e ~'. Let D g _~~¢ be the set of elements of ~e exterior dependent on A e ~¢. Let
=
~.7_laii = n and trace(A) = EkXk = n, then hma X
~l~n. If Xmax = n,
n~max = ~.
E
B~Da
If the elements of . ~ are interior dependent with respect to C ~ ' , then e ~ c ( ~ ) 4 : q ~ c ( ~ ) where
= (,t,c(A), A Expectations are beliefs about the rank of alternatives derived from prior knowledge. Assume that a decision maker has a ranking of a finite set of laternatives ~ with respect to prior knowledge of criteria ~' arrived at intuitively. He may have expectations about the rank order.
Axiom 4 (Rational expectations). c~c ~ _ L~,
aijwjWi-1
i,j=l
.ja¢= L~.
= n+
Z
(aijw, w,-' +
1~ i ~ j ~ n =n+
E (Yig+a/Yij) • 1 ~i<~j<~n
Since y~j+y~]~ >/2, and n~kmax=n2, equality is uniquely obtained on putting y~i= 1, i.e., a~j = w~/wj. The condition a~jajk = a~k holds for all i, j and k, and the result follows.
Theorem 3. Let A = ( a , / ) E Rct,). There exists a function ~ = (~bl, q'2 . . . . . ~k,): R c t , ) ~ [0,1]" such that (i) a,j ~,(A)/%( A), (ii) The relative dominance of the ith alternative, ~i(A), is the ith component of the principal right eigenvector of A, (iii) Given two alternatives A i, Aj e.a¢, A~ >_ c Aj if and only if ~bi(A ) >! ~bj(A). =
3. Results from the axioms N o t e that if Pc is consistent, then Axiom 1 follows, i.e., consistency implies the reciprocal property. The first few theorems are based on this more restrictive property of consistency. Let R c t , ) ~ R g t , ~ be the set of all ( n X n ) consistent matrices.
Theorem 1. Let A
~
RMtn). A ~ Rct,, ~ if and only
tf r a n k ( A ) = i. Proof. If A ~ R c t , ) , then a~jajk = a~k for all i, j and k. Hence, given a row of A, (a~l, a~2 . . . . . a,, }, all other rows can be obtained from it by means of the relation ajk = aik/a~j and r a n k ( A ) = 1. Let us now assume that r a n k ( A ) = 1. Given a row ajh ( j 4= i, h = 1, 2 . . . . . n), ajh = Maih (h = 1, 2 . . . . . n) where M is a positive constant. Also, for any reciprocal matrix, a , = 1 ( i = 1, 2 . . . . . n). Thus, for i = h we have ay, = M a , = M and aj~ = a~a~h for all i, j and k, and A is consistent.
Theorem 2. Let A ~ RMt,). A ~ R c t , ) if and only if its principal eigenvalue hma x is equal to n. Proof. By T h e o r e m 1 we have r a n k ( A ) = 1. Also, all eigenvalues of A but one vanish. Since trace(A)
Proof. A ~ R c t , ) implies that aij = a~ka~ 1 for all k, and each i and j. Also by Theorem 1, we have r a n k ( A ) = 1 and we can write a~j = x i / x j, where x i, xj >/0 (i, j = 1, 2 . . . . . n). Multiplying A by the vector x T = (xl, x 2 . . . . . x , ) we have A x = nx. Dividing both sides of this expression by ET_ ~x~ and writing
w = x~ ~ x i i=1
we have A w = nw, and ~,7_lwi = 1. By Theorem 2 we have n as the largest positive real eigenvalue of A and w as its corresponding right eigenvector. Since aij = x i / x j = wi/wj for all i and j, we have d/i(A ) = w i = 1, 2 . . . . . n and (i) and (ii) follow. By Axiom 1, for A ~ R c ( n ), A~> c Aj if and only if a~j >/1 for all i and j, and hence we have ~p~(A)>~pj(A) for all i and j. It would appear that it is unnecessary to invoke the P e r r o n - F r o b e n i u s Theory to ensure the existence and uniqueness of a largest positive real eigenvalue and its eigenvector. We have already proved the existence of an essentially unique solu-
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T.L. Saaty / Scaling the membership function
(b) Transitive:
F o r all x, y, z ~ S, if x ~
(c) A n t i s y m m e t r i c : F o r all x, y ~ S, ifx~yand y~
Definition. F o r cludes x ) of this x ~
any type y is t
relation x ~
Partially ordered sets with a finite n u m b e r of elements can be conveniently represented b y a directed graph. Each element of the set is represented by a vertex so that an arc is directed from y to x if x < y.
Definition. A subset E of a partially o r d e r e d set S is said to be bounded from above (below) if there is an element s ~ S such that x ~< s ( >I s) for every x ~ E. The element s is called an u p p e r (lower) b o u n d of E. W e say that E has a s u p r e m u m (infimum) if it has u p p e r (lower) b o u n d s and if the set of upper (lower) b o u n d s U ( L ) has an element ul(ll) such that u I >/u for all u ~ U (l 1 >I l for all
T h e n o t i o n s of f u n d a m e n t a l a n d derived scales can be e x t e n d e d to x ~ L k, x - c L , + 1 replacing C a n d 0a¢ respectively. T h e derived scale resulting from c o m p a r i n g the elements in x - with respect to x is called a local derioed scale or local priorities. G i v e n L , , L k + l , G . g ' , let us d e n o t e the local derived scale for y ~ x - a n d x ~ L , by +k+l(y/x), k = 2, 3 . . . . . h - 1. W i t h o u t loss of generality we m a y assume that
S_, + , + , ( y / x ) = 1. y ~ .V-
C o n s i d e r the m a t r i x + , ( L k / L , _ I ) whose colu m n s are local derived scales of elements in L k with respect to elements in L k_ ~.
Definition ( H i e r a r c h i c composition). The global derived scale (rank order) of any element in ~ ' is o b t a i n e d from its c o m p o n e n t s in the c o r r e s p o n d ing vector of the following:
+,(b)=l, +2(Lz)=+2(b-/b), ~,(Lk)=~a(Lk/Lk_,)~a_,(La_~),
k = 3 ..... h
I~L). Definition. Let ,gff be a finite partially o r d e r e d set
Definition. A set ~ is said to be exterior dependent on a set c¢ if a f u n d a m e n t a l scale can be
with largest element b. ,,~ is a hierarchy if it
defined on ~
satisfies the conditions: (1) There is a p a r t i t i o n of ,gf,a into sets called levels { Lk, k = 1, 2 . . . . . h }, where L 1 = { b }. (2) x ~ L , implies x - _ c L k + 1, where x-={ylxcoversy}, k = l , 2. . . . . h - 1 . (3) x ~ L k implies x+G Lk_I, where x +={ylycoversx}, k = 2 , 3 . . . . . h.
Definition. Given a positive real n u m b e r P >/1 a n o n e m p t y set x - _ Lk+ ~ is said to be p - h o m o g e neous with respect to x ~ L k if for every p a i r of elements y~, Y2 ~ x - , 1
p
y2) p;
in particular, the reciprocal axiom implies that
Pc(Y,, Y , ) = 1. A x i o m 2. G i v e n a h i e r a r c h y , , ~ , x ~ ~ a n d x ~ L k, X - _ L , + 1 is p - h o m o g e n e o u s for k = 1 . . . . . h - 1.
with respect to every C ~ c¢.
A x i o m 3 ( D e p e n d e n c e ) . Let .,~ be a hierarchy with levels L~, L 2 . . . . . L h. F o r each L k, k = l , 2 . . . . . h - 1, L k*~ is exterior d e p e n d e n t on L k.
Definition. Let ~¢ be exterior d e p e n d e n t on ~ . T h e elements in ~ are said to be interior dependent with respect to C ~ cg if for s o m e A ~ ~¢, ~¢ is exterior d e p e n d e n t on A. Definition. Let 5 a be a family of n o n e m p t y sets ~G, ~'2 . . . . . ~g,, where ~gi consists of the elements ( e q , j = l . . . . . rn~}, i = 1 , 2 . . . . . n. 5" is a system if (i) It is a directed g r a p h whose vertices are c¢i a n d whose arcs are defined through (if) G i v e n two c o m p o n e n t s c¢ a n d ~ ~ 5" there is an arc from c¢, to ~ if ~ is exterior d e p e n d e n t on c¢i. Let + a . c ( d ) be the derived scale of the elem e n t s of oa¢ with respect to A ~.~¢ for a criterion
T.L. Saaty / Scaling the membership function
C ~ ~'. Let q , c ( ~ ¢) be the derived scale of the elements of ~¢ with respect to a criterion C ~ ~'. Let D A _ ~ be the set of elements of ~ exterior dependent on A ~ . Let
i
aijwjwi- I
i,j=l
B~D~
If the elements of ~¢ are interior dependent with respect to C ~ C ~ , then q~c(..~)4:+c(~¢) where
,t,c(~)= (%(A), A ~ : } . Expectations are beliefs about the rank of alternatives derived from prior knowledge. Assume that a decision m a k e r has a ranking of a finite set of laternatives ..~ with respect to prior knowledge of criteria ~' arrived at intuitively. He may have expectations about the rank order. Axiom 4 (Rational expectations). Cgc ~ - Ln,
--- E,"_la~ = n and t r a c e ( A ) = Ekkk = n, then ~max -----~k1 = n . If Xma~ = n, n~kmax =
E
323
~ ' = L~.
3. Results from the axioms Note that if Pc is consistent, then Axiom 1 follows, i.e., consistency implies the reciprocal property. The first few theorems are based on this m o r e restrictive property of consistency. Let R c t , ) C R g t , ~ be the set of all ( n x n ) consistent matrices. Theorem 1. Let A ~ RM~,, ~. A ~ Rc~,) if and only /f r a n k ( A ) = i. Proof. If A ~ Rc~,, ), then a,jaj~ = a~k for all i, j and k. Hence, given a row of A, ( ai~, a n . . . . . ai,, }, all other rows can be obtained from it by means of the relation ajk = a~k/a~j and rank(A) = 1. Let us now assume that r a n k ( A ) = 1. Given a row ajh ( j 4: i, h = 1, 2 . . . . . n), ajh = Maih (h = 1, 2 . . . . . n ) where M is a positive constant. Also, for any reciprocal matrix, a i i - - 1 ( i = l , 2 . . . . . n). Thus, for i = h we have a j i = Ma;~ = M and a j h = aj~a~h for all i, j and k, and A is consistent. Theorem 2. Let A ~ R~t¢,, ~. A ~ R c t , ~ if and only if its principal eigenvalue ~ max is equal to n. Proof. By T h e o r e m 1 we have r a n k ( A ) = 1. Also, all eigenvalues of A but one vanish. Since trace(A)
=n+
Y'.
(aijwjw, -1 + ajiw, wf -1 )
l <~i~j<~n
-.+
E
(y,j+l/y,j)
I <~i<~j<~n
Since y ~ j + y , ~ >t2, and n~,m~x=n 1, equality is uniquely obtained on putting y~j = 1, i.e., a o = w~/wj. The condition a,jajk = ai~. holds for all i, j and k, and the result follows. Theorem 3. Let A = ( a ~ j ) ~ Rc~,). There exists a function + = (~l, ~b2. . . . . q , ) : R c t , ) ~ [0,1] n such that (i) a,j = q , , ( A ) / ~ j ( A ) , (ii) The relative dominance of the ith alternative, q,i(A), is the ith component of the principal right eigenvector of A, (iii) Given two alternatives A,, Aj ~ s ~ , A i >_ c Aj if and only if +i(A) >1q~j(A). Proof. A ~ R c ( , ) implies that a~j = a~ka ~) for all k, and each i and j. Also by T h e o r e m 1, we have r a n k ( A ) = 1 and we can write a,j = x J x j , where x~, xj >/0 (i, j = 1, 2 . . . . . n). Multiplying A by the vector x T = ( x l, xz . . . . . x , ) we have A x = nx. Dividing both sides of this expression by ~-7=ix, and writing
w = x~ ~ x, i=1
we have A w = nw, and Y'.7=lw/--- 1. By T h e o r e m 2 we have n as the largest positive real eigenvalue of A and w as its corresponding right eigenvector. Since a~j = x J x . i = wi/wj for all i and j, we have q,~(A) = w. i = 1, 2 . . . . . n and (i) and (ii) follow. By Axiom 1, for A ~ R c ( n ) , A~>_. c Aj if and only if a~j >_-1 for all i and j, and hence we have +~(A)>:q,j(A) for all i and j. It would appear that it is unnecessary to invoke the P e r r o n - F r o b e n i u s Theory to ensure the existence and uniqueness of a largest positive real eigenvalue and its eigenvector, We have already proved the existence of an essentially unique solu-
T.L. Saaty / Sca6ng the membershipfunction
324
tion in the consistent case. A similar result follows using the perturbation argument given below.
principal eigenvector of A' satisfies Ihm,x - n I ~< c. Let A A = ~'B. Wilkinson has shown that for a sufficiently small ~', X m,~ can be given by a convergent power series [6]
Theorem 4. Let A G R c ~ ) , and let )t 1 = n and )t z = 0 be the eigenvalues of A with multiplicity 1 and ( n - 1), respectively. Given c > 0, there is a 8 = 8 ( e ) > 0 such that if
)t,,~x = n + k ( r + kz.r 2 + . . . .
l a o + r;i - a~j I = ] %./I ~< 8
h max
fori, j=l,2
..... n
Proof. Let E0 = ½n, and let ~ ~< n / 2 . The circles C1: I / ~ - n l = c and C2: I / ~ - 0 l = c are disjoint. Let f(/t, A) be the characteristic polynomial of A. Let t ) = m i n l f ( / ~ , A)[ for ~ on @. N o t e that min [ f(/.t, A ) l i s defined because f is a continuous function of /~, and ~ >/0 since the roots of f(/~, A) = 0 are the centers of the circles, f(/~, B) is a continous function of the 1 + n 2 variables # and a;i+~';j, i, j = l , 2 . . . . . n, and for some 8 > 0 , f ( # , B) 4:0 for/~ on any Cj, j = 1, 2, if I ~',jl ~< 8, i, j = l , 2 . . . . . n. From the theory of functions of a complex variable, the number of roots ~ of f ( # , B ) = 0 which lie inside ~ , j = 1, 2, is given by 1 f'(#,B) 2~'i f(/x, B) d~,
Let w be the right-eigenvector corresponding to the simple eigenvalue n of A. Since n is a simple eigenvalue, ( A - n l ) has at least one non-vanishing minor of order (n - 1). Suppose, without loss of generality, that this lies in the first (n - 1) rows of ( A - h i ) . Then from the theory of linear equations, the components of w may be taken to be ( A . I , A . 2 . . . . . A . . ) where A.; denotes the cofactor of the (n, i) element of ( A - n l ) , and is a polynominal in n of degree not greater than (n -
1).
The components of w' are polynomials in Xma~ and ~', and since the power series expansion of max is convergent for all sufficiently small ~-, each c o m p o n e n t of w' is represented by a convergent power series in -r. We have W ' ~--- W -'1"-TZ 1 -4- ' r 2 2 2 Jr" . . .
and j=l,2,
which is also a continuous function of the n z variables a , / a n d r,j with I r;jl ~< 8. For B = A , we have n l ( A ) = l and n 2 ( A ) = n - 1. Since % ( B ) , j = 1, 2, is continous, it cannot j u m p from n s ( A ) to % ( B ) and the two must be equal and have the value n l ( B ) = 1 and n 2 ( B ) = n - 1 for all B with [a;j + "fij- a;j[~< 8, i, j = 1, 2~...~
~ n as r ~ oo, and
IXm,x-- h i = O(T) ~
the matrix B = (a,j + 5 j ) has exactly 1 and (n - 1) eigenvalues in the circles [1~ - n [ < ~ and [1~ - 0 [ < ~, respectively.
%(B)-
Now,
n.
Theorem 5. Let A ~ R c¢,) and let w be its principal right eigenvector. Let A A = (8;j) be a matrix o f perturbations o f the entries o f A such that A' = A + A A ~ R M~,,), and let w' be its principal right eigenvector. Given c > 0, there exists a 8 > 0 such that [ 8ij [ <~8 for all i and j , then [ w" - w; [ <-Nc f o r all i = 1 , 2 . . . . . n. Proof. By Theorem 4, given ~ > 0, there exists a 8 > 0 such that if l S;jl~ 8 for all i and j, the
Iw'- wl= o(~-)~c. By Theorems 4 and 5, it follows that a small perturbation A' of A transforms the eigenvalue problem ( A - n I ) w = 0 to
( A ' - X.,axI)w' =0. Theorem 6 (Ratio estimation). Let A ~ R M t , ) , and let w be its principal right eigenvector. Let c;j = aijwjw i- 1, and let 1 - .r < cij < 1 + ~, "r > O, f o r all i a n d j . Given c > 0 and • < ~, there exists a 8 > 0 such that f o r all ( x I, x 2 . . . . . x , ) , x i > O , i = l , 2. . . . . n i f 1- 8 <
ai)
x,/xj
<1+8
f o r all i a n d j ,
(2)
<1+(
f o r all i a n d j .
(3)
then
w,/wj 1-(<
xi~.
T . L Saaty / S c a r i n g the mernbership function
Proof. Substituting a u ~ ~ for w,/wj in (3) we have
w;/wj _ _
_
1 =] 1
I X'/XJ
aij
1]
1% xi/x, <~
By definition have
I
x;/xj
1 + ~-
1/c,
(;i=
Note that ET.j.lauwjwt -1 - n 2 = n(n - 1)# is also a measure of the departure from consistency. It is also possible to show that ( A - n l ) w = 0 is transformed into ( A ' - hmaxI)w' = 0 by means of graph theoretic concepts.
< (1 + ~')8 + ~'. Given c > 0 and 0 < ~ ' < ( , there exists a 8 = (~ - ~')/(1 + ~')> 0 such that (2) implies (3). Theorem 7. Let A = (a;)) ~ RM(,, ). Let X ...... be its
principal eigenvalue and let w be its corresponding right eigenvector with Y~'=lwt = 1, then %max >1 n. Proof. Let a u = ~w~-1 c u, i, j = 1, 2 . . . . . n. Since
A w = X m~w, -bsT. j = i auwj = h . . . . we have ~
a u wj - n =
i,j=l
i,j
~ R M(n). Let h r n a x be the principal eigenvector of A, and let w be its corresponding right eigenvector with F~7=lw, = 1. Then
~ m u.,~ - - n
is a measure of the average departure from consistency. Proof. For A ~ R c(,) c R M(,,), by Theorem 2 we
have )~m,~=n, and hence we have # = 0 . For A ~ RM(,, ), let a u = ~ u w j w J for all i and j. We have n
n)kmax
E j=l
W.j
aij-~
= ~
i,j=l
j~l
~-ij q-
,
l~
= -1+
Theorem 10. I f h ~ 8cl,, I, the intensities of all paths from i to j are equal to a U.
k. Corollary I. I f A ~ Rc(,), the entry in the (i, j ) position can be represented as the intensity of paths of any length starting with i and terminating with j. Proof. Follows from the proof of Theorem 10.
Proof. From Theorem 10, the intensity of a path
of any length from i to j is equal to a u. An arbitrary entry of A k is given by
)k m a x - - n
n-1
and k. Hence, we have a , = auajkakt = 1 for all i = 1, 2 . . . . . n. By induction, if a ; g . . . at._,; = 1 for all i I ... i,,_1, then a,, . . . a t . - d a i . i = a t i a i , i = 1 and the result follows.
Corollary 2. I f A E R c ( , ) , the entry in the (i, j ) position is the average intensity of paths of length k from i to j, and Ak = n k - lA ( k > 1).
(,j'
~'ij = n -]-
~
Proof. A ~ R e ( , ) implies a u % k = ate. for all i, j
Proof. Follows from a U = atkakl for all i, j and
-
n--I
~kmax =
Definition. A cycle is a path of pairwise comparisons which terminates at its starting point.
the intensities of all cycles are equal to a , , i = 1, 2 . . . . . n.
Theorem 8. Let A
-
Definition. The intensity of judgments associated with a path from i to j called the path intensity is equal to the products of the intensities associated with the arcs of that path.
Theorem 9. I f A ~ Re(,),
~%-n.
By definition, the matrix ( ( ; y ) ~ RM(,, ). We have % = 1 for all i, and ct./> 0 for all i and j. Hence, we have F~t:j=1~u - n = F~t,j( u > 0 and the result follows.
i.¢~_
As c u --, 1, i.e., consistency is approached, /~ ~ 0. Also, ~ is convex in c u, since ( ~ u + 1 / c ; / ) is convex, and has its minimum at (u = 1, i, j = 1, 2 . . . . . n. Thus, # is small or large depending on ~u being near to or far from unity, respectively, i.e., near to or far from consistency, and the result follows.
for all i and j, and we
w;/wj I a;______~j _ 1 x;/x-----~j - 1 = % , x J % + I % - 11
..... - n =
325
1 n(n-
(,j+l) 1) l<~i~j~,
cu
tj
"
=
i i
i l ~ l i,=1
"""
i
ik_l=l
aihflgt2
•••
a,..
328
T.L. Saaty / Scalingthe membershipfunction
This theorem highlights the fact that the eigenvector gives the relative dominance (rank order) of each alternative over the other alternatives along paths of arbitrary length.
4. Concluding remarks
4.1. The axiom of expectations Here is a situation which presents itself in choice problems. An apple is preferred to an orange. More apples are then introduced. Should an apple continue to be preferred or does the presence of many apples now make an orange more preferred? Both situations may be considered desirable under appropriate cirucmstances. If the apples are replaced by gold bars and oranges by iron bars, clearly in economic terms gold would continue to be preferred; 'the more the better'. On the other hand if the objects being compared are fertilizer and food and at the moment fertilizer is needed, adding more fertilizer can change the preference 'when there is a lot of it around, you never want it very much'. The question is how can a theoretical approach be developed so that it can lead to both outcomes according to which one is desired? It is obvious that consciousness about a problem is in the mind of the user. A mathematical approach cannot by itself make such a distinction automatically. However, a powerful modeling approach is one which can incorporate in the model criteria or properties which make it possible to use the same mathematics but produce the desired kind of result for the particular problem. Such an approach affords us the greatest flexibility in modeling. As the basic component, we use a direct and justifiable method of computation within a framework which we can alter according to our expectations. Such a model would be useful because it can handle different exigencies which cannot all be included in a single method of calculation particularly when it is expected of the model to product two opposite kinds of answers. There are two other ideas to mention in this respect. The first is that a higher priority criterion in a hierarchy may have serveral alternatives to be judged under it while a lower priority criterion may have a few. It is desired to increase the priority of the elements in the larger set because if there are many of them they may each receive a
smaller composite priority than each of the few elements under the low priority criterion. Sometimes it is the other rarer elements that need to receive a higher priority. In the first case one can multiply the priority of each criterion by the relative number of elements under it and normalize the weights of the criteria. In the second case a complementary set of weights can be used. A more general approach would be the one described above to introduce an additional criterion in the hierarchy called 'importance of number of descendants'. The prioritization is also carried out under this criterion, which in turn has also been compared with the other criteria. A third illustration of the need to augment the structure according to expectation is that of three secretaries who apply for a job: one excellent at English but poor at typing, the second is balanced but not excellent at either and the third is excellent at typing and poor at English. The way to surface the balanced secretary is to add to the criteria, English and typing, a third one, balance. In this manner the balanced secretary could receive the overriding priority. To cover such situations which at first may appear paradoxical and whose variety can be infinite whether analyzed through an analytic hierarchy or by other means, we must usually modify the structure, hence, the need for Axiom 4.
4. 2. Dependence of criterion membership on element membership There are problems in which the criteria are themselves dependent on the alternatives and their membership function is determined in terms of the alternatives. Since the two form an interactive set, alternatives on criteria and criteria on alternatives, the feedback system of the Analytic Hierarchy Process is used to determine the final priorities [2]. 4.3. The membership function of a large number of
elements It should be noted that when the number of elements being compared is large, the task of deriving a membership function can be simplified by decomposing the criteria into subcriteria representing a spectrum of intensities (e.g., high, medium, low) for which a membership function is constructed with respect to each criterion and then
T.L. Saaty / Scaling the membership function
weighted by the membership value of that criterion. The membership of the elements or alternatives is derived by assigning to each the membership value from the appropriate intensity under each criterion and adding over all the criteria.
329
judgment and experience. The derived scales can be combined to preserve the ratio scale property of the individual scales. This approach can be validated for problems involving a known scale of measurement.
4.4. Conclusions
References We have given a theoretical basis for constructing a membership function by means of pairwise comparisons. It is the first theory which allows for intransitivity and more generally for inconsistency in the information used to make the comparisons, yet when the inconsistency is high it suggests improving the quality of this information in particular positions in the matrix. An advantage of this approach is that it enables one to measure membership with respect to criteria even in cases when scales have not been agreed upon. Furthermore, it makes it possible to use existing measurements along with newly derived ones based on
[1] Hardy, G.H., Divergent Series, Oxford University Press, London/New York, 1949. [2] Saaty, T.L., "Exploring the interface between hierarchies, multiple objectives and fuzzy sets", Fuzzy Sets and Systems 1 (1978) 57-68. [3] Saaty, T.L., The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [4] Saaty, T.L., and Vargas, L.G., "Inconsistency and rank preservation", Journal of Mathematical Psychology 28 (2) (1984) 205-214. [5] Saaty, T.L., "Measuring the fuzziness of sets", Journal of Cybernetics 4 (4) (1974) 53-61. [6] Wilkinson, J.H., The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965.
European Journal of Operational Research 25 (1986) 320-329 North-Holland
Scaring the membership function T h o m a s L. S A A T Y
322 Mervis Hall, Unioersity of Pittsburgh, Pittsburgh, PA 15260, U.S.A.
1. Introduction
To scale the membership function, a method is needed that can assess the importance of elements with respect to a property or criterion that is common among them. It is conceivable that membership should be determined in terms of several criteria tangible and intangible. The approach needed is a multicriterion process whose measurements admit arithmetic operations that preserve the scaling property. Ratio scales is what one obtains through ratio comparisons of elements with respect to criteria. Below we give a set of axioms underlying the Analytic Hierarchy Process for constructing and composing ratio scales, an effective way for constructing the membership function. The following definition is due to Zadeh: Let X = ( x } be a set of points. A fuzzy set A in X is a set of ordered pairs: A={[x,~A(x)]
),
x~X,
where ~A(x) is the grade of membership of x ~ A. If the values of ~t(x) belong to a membership space M, then A is a mapping from X to M and ~A: X---' M is called the membership function of A. A is an ordinary set when M contains only two values 0 and 1. Note that X may be a set of concrete objects or alternatively of abstract criteria. To determine the degree of membership, an effective method of measurement is needed that works uniformly for both concrete and abstract sets X. There are two kinds of measurement, fundamental measurement and derived measurement. Fundamental measurement is direct and involves
Received October 1982; revised October 1984, April 1985.
some kind of scale. Since there is an infinite number of ways for defining X, it is rather hopeless to seek to determine the membership function through direct scaling. Of course, there are many cases where a scale may not be known but there is hope to develop one. For example it is conceivable that a scale can be improvised to measure how alert people can be. In the case of abstract elements such as criteria expressing intensity of perceived properties according to a common attribute which they share, such a group of individuals who are beginning their political career with respect to political acumen, the only possibility we have today is to use judgement from judges who are experienced with respect to that property. This is the case of a derived scale. The problem is how to record and use judgments in a meaningful fashion from which a scale could be derived. I believe that one can convince himself of the following two fundamental observations about the membership function. (1) Except when there is a scale of measurement that can be applied to determine the degree of belonging, membership must be determined by making relative comparisons among the elements. (2) A useful scale for comparisons exists and has been tested and validated in a large number of applications. It is the instrument used to translate judgment and observation to measurement. Axiomatization of this process proceeds as shown below. It is important to point out that a normalized solution is obtained that is unique and belongs to a ratio scale. What is most significant is that the solution by its very construction gives, along with the degree of membership, the correct rank order of the elements. The approach briefly described here is being used widely in other areas and needs careful
0377-2217/86/$3.50 © 1986, ElsevierScience Publishers B.V. (North-Holland)
T.L. Saaty / Scalingthe membershipfunction examination by fuzzy set experts regarding its viability [3,4,5].
2. Axioms for deriving a scale from fundamental measurement and for hierarchic composition Let ~ be a finite set of n elements called alternatives. Let ~ be a set of properties or attributes with respect to which elements in ~¢ are compared. Philosophers distinguish between properties and attributes. A property is a feature that an object or substantial individual possess even if we are ignorant of this fact. On the other hand an attribute is a feature we assign to some object: it is a concept. Here we assume that properties and attributes are interchangeable and generally refer to them as criteria. A criterion is a primitive. When two objects or elements in ~¢ are compared according to a criterion in So, we say that we are performing binary comparisons. Let > c be a binary relation on ~¢ representing ' m o r e preferred than' with respect to a criterion in cg. Let - c be the binary relation 'indifferent to' with respect to a criterion C in ~. Hence, given two elements, A~, A j ~ ¢ , either A , > c Aj or Aj> c Ai or A , - c Aj for all C ~ cg. A given family of binary relations > c with respect to a criterion C in ~ is a primitive. Let ~ be the set of mappings from .~e×~¢ to R ÷ (the set of positive reals). Let f: ~'--, ~ . Let P e l f ( C ) for C ~ cg. Pc assigns a positive real number to every pair (A~, Aj) ~ . ~ ¢ × ~ . Let Pc(A,, A j ) = a , j ~ R +, A~, A ~ ¢ . For each C ~ ~', the triple (.ac×..ae, R +, Pc) is a fundamental or primitive scale. A fundamental scale is a mapping of objects to a numerical system. Definition. For all A~, Aj ~ ¢
ifandonlyifPc(A~, Aj)=I.
Axiom 1 (Reciprocal). For all A,, A j ~ C~
¢ and
Pc(A,, Aj) = 1/ec( Aj, A,).
A=(a+j)=(Pc(A+, Aj)) be
pc(A,,
) >i ec(Aj,
does not hold for all A k e ~ . In other words, it m a y happen that Pc(A,, Aj)> 1, and for some A, e..a¢ we have
Pc(A,,
Pc(Aj,
A more restrictivecondition is the following Definition. The mapping Pc is said to be consistent if and only if
ec(A,,
for all i, j , and k.
If A ~ > c Aj we say that A, dominates Aj with respect to C ~ c~. Thus Pc represents the intensity with which an alternative dominates another.
Let
paired comparisons of the alternatives with respect to a criterion C ~ c¢. By Axiom 1, A is a positive reciprocal matrix. The object is to obtain a scale of relative dominance (or rank order) of the alternatives from the paired comparisons given in A. There is a natural way to derive the relative dominance of a set of alternatives from a pairwise comparison matrix A. Let Rgcn~ be the set of (n × n) positive reciprocal matrices A = ( a i j ) - (Pc(Ai, Aj)) for all C ~ c¢. Let ~b: Rgtn)-~ [0,1] n for A ~ Rgtn), ~b(A) is an n-dimensional vector whose components belong to [0,1]. The triple (RMt,~, [0,1] ~, ~k) is a derived scale. A derived scale is a mapping between two numerical relational systems. It is important to point out that the rank order implied by the derived scale ~b may not coincide with the order represented by the pairwise comparisons. Let ~k(A) be the ith component of ~k(A). It denotes the relative dominance of the ith alternative. By definition, for A~, Ai ~ , A, > c Aj implies Pc(A,, A j ) > 1. However, if Pc(A,, Aj)> 1, the derived scale could imply that ~kj(A)> ~ki(A). This occurs if row dominance does not hold, i.e., for A,, Aj ~.~¢ and C ~
ec(A,, aj)ec(Aj,
and C ~
A~> c Aj ifandonlyif Pc(A ~, Aj)> I, A~-cAj
321
the set of
(1)
Similarly the matrix A is consistent if and only if aijajk=aik for all i, j and k. If Pc is consistent, then Axiom 1 automatically follows.
2.1. Hierarchic axioms Definition. A partially Ordered set is any set S with a binary relation ~< which satisfies the following conditions: (a) Reflexive:
For all x ~ S, x ~< x,
T.L. Saaty / Scaling the membershipfunction
322
(b) Transitive:
F o r all x, y, z ~ S, if x ~
any relation x<~y (read, y intype we define x ~
Partially ordered sets with a finite n u m b e r of elements can be conveniently represented by a directed graph. Each element of the set is represented by a vertex so that an arc is directed from y to x if x < y. D e f i n i t i o n . A subset E of a p a r t i a l l y o r d e r e d set S
is said to be bounded from above (below) if there is an element s ~ S such that x ~ s ( > / s ) for every x ~ E. The element s is called an u p p e r (lower) b o u n d of E. W e say that E has a s u p r e m u m (infimum) if it has u p p e r (lower) b o u n d s and if the set of u p p e r (lower) b o u n d s U ( L ) has an element u~(l~) such that u~ >~ u for all u ~ U (/1 >/l for all
T h e notions of f u n d a m e n t a l a n d derived scales can be e x t e n d e d to x ~ L k, x - C L k ÷ ~ replacing C a n d ~¢ respectively. T h e derived scale resulting from c o m p a r i n g the elements in x - with respect to x is called a local derioed scale or local priorities. G i v e n Lk, Lk+l, C_,.~, let us d e n o t e the local d e r i v e d scale for y ~ x - a n d x ~ L k by q , k + l ( y / x ) , k = 2, 3 . . . . . h - 1. W i t h o u t loss of generality we m a y assume that
Y~. t p k + a ( y / x ) = 1. y~x-
C o n s i d e r the m a t r i x ~ b k ( L k / L k _ l ) whose colu m n s are local derived scales of elements in Lx. with respect to elements in L k_ 1. D e f i n i t i o n ( H i e r a r c h i c composition). T h e global derived scale (rank order) of any element in 0~' is o b t a i n e d from its c o m p o n e n t s in the c o r r e s p o n d ing vector of the following:
~p,(b) = 1,
q,z(L2)=~2(b-/b),
~pk(Lk)=tpk(Lk/Lk_,)q,k_,(Lk_,),
k= 3..... h
I~L). D e f i n i t i o n . Let ~
be a finite p a r t i a l l y o r d e r e d set with largest element b..Yea is a hierarchy if it satisfies the conditions: (1) There is a p a r t i t i o n of .9~ into sets called levels ( L k, k = 1 , 2 . . . . . h } , w h e r e L ~ = { b } . (2) x ~ L k implies x - _ L k + 1, where x-={ylxcoversy), k=1,2 ..... h-1. (3) x ~ L k implies x+_C L k _ l , where x ÷={ylycoversx}, k = 2 , 3 . . . . . h. D e f i n i t i o n . Given a positive real n u m b e r p >~ 1 a
n o n e m p t y set x-___ Lt.+~ is said to be p - h o m o g e neous with respect to x ~ L k if for every p a i r of elements y~, Y2 ~ x - , 1
- <~Pc(Ya, Y2) <~P; P in particular, the reciprocal a x i o m implies that
Pc(y,, x)=
I.
A x i o m 2. G i v e n a hierarchy.Yea, x ~ .Yeaand x ~ L k,
X-C_ Lk+ 1 is p - h o m o g e n e o u s for k = 1 . . . . . h - 1.
D e f i n i t i o n . A set d is said to be exterior dependent on a set c~ if a f u n d a m e n t a l scale can be
d e f i n e d on ~
with respect to every C ~ c~.
3 ( D e p e n d e n c e ) . Let .,~ be a hierarchy with levels L 1, L 2 . . . . . L h. F o r each L k, k = l , 2 . . . . . h - 1, Lk+ 1 is exterior d e p e n d e n t on L k. Axiom
D e f i n i t i o n . Let .~¢ be exterior d e p e n d e n t on ~'. T h e elements in .a¢ are said to be interior dependent with respect to C ~ c£ if for some A ~..a¢, . ~ is exterior d e p e n d e n t on A.
Let 3 a be a family of n o n e m p t y sets ~'1, ~'2 . . . . . ~',, where ~'i consists of the elements {eij, j = 1 . . . . . m i ) , i = 1, 2 . . . . . n. ~ is a system if (i) It is a directed g r a p h whose vertices are ~'~ and whose arcs are defined through (if) G i v e n two c o m p o n e n t s ~ a n d ~ ~ 5" there is an arc from ~i to ~ if ~ is exterior d e p e n d e n t on ~i. Let ~ a . c ( ~ ¢ ) be the derived scale of the elem e n t s of ~¢ with respect to A ~.~e for a criterion Definition.
323
T.L. Saaty / Scaling the membership function
C e ~'. Let ffc(J~¢) be the derived scale of the elements of ~¢ with respect to a criterion C e ~'. Let D g _~~¢ be the set of elements of ~e exterior dependent on A e ~¢. Let
=
~.7_laii = n and trace(A) = EkXk = n, then hma X
~l~n. If Xmax = n,
n~max = ~.
E
B~Da
If the elements of . ~ are interior dependent with respect to C ~ ' , then e ~ c ( ~ ) 4 : q ~ c ( ~ ) where
= (,t,c(A), A Expectations are beliefs about the rank of alternatives derived from prior knowledge. Assume that a decision maker has a ranking of a finite set of laternatives ~ with respect to prior knowledge of criteria ~' arrived at intuitively. He may have expectations about the rank order.
Axiom 4 (Rational expectations). c~c ~ _ L~,
aijwjWi-1
i,j=l
.ja¢= L~.
= n+
Z
(aijw, w,-' +
1~ i ~ j ~ n =n+
E (Yig+a/Yij) • 1 ~i<~j<~n
Since y~j+y~]~ >/2, and n~kmax=n2, equality is uniquely obtained on putting y~i= 1, i.e., a~j = w~/wj. The condition a~jajk = a~k holds for all i, j and k, and the result follows.
Theorem 3. Let A = ( a , / ) E Rct,). There exists a function ~ = (~bl, q'2 . . . . . ~k,): R c t , ) ~ [0,1]" such that (i) a,j ~,(A)/%( A), (ii) The relative dominance of the ith alternative, ~i(A), is the ith component of the principal right eigenvector of A, (iii) Given two alternatives A i, Aj e.a¢, A~ >_ c Aj if and only if ~bi(A ) >! ~bj(A). =
3. Results from the axioms N o t e that if Pc is consistent, then Axiom 1 follows, i.e., consistency implies the reciprocal property. The first few theorems are based on this more restrictive property of consistency. Let R c t , ) ~ R g t , ~ be the set of all ( n X n ) consistent matrices.
Theorem 1. Let A
~
RMtn). A ~ Rct,, ~ if and only
tf r a n k ( A ) = i. Proof. If A ~ R c t , ) , then a~jajk = a~k for all i, j and k. Hence, given a row of A, (a~l, a~2 . . . . . a,, }, all other rows can be obtained from it by means of the relation ajk = aik/a~j and r a n k ( A ) = 1. Let us now assume that r a n k ( A ) = 1. Given a row ajh ( j 4= i, h = 1, 2 . . . . . n), ajh = Maih (h = 1, 2 . . . . . n) where M is a positive constant. Also, for any reciprocal matrix, a , = 1 ( i = 1, 2 . . . . . n). Thus, for i = h we have ay, = M a , = M and aj~ = a~a~h for all i, j and k, and A is consistent.
Theorem 2. Let A ~ RMt,). A ~ R c t , ) if and only if its principal eigenvalue hma x is equal to n. Proof. By T h e o r e m 1 we have r a n k ( A ) = 1. Also, all eigenvalues of A but one vanish. Since trace(A)
Proof. A ~ R c t , ) implies that aij = a~ka~ 1 for all k, and each i and j. Also by Theorem 1, we have r a n k ( A ) = 1 and we can write a~j = x i / x j, where x i, xj >/0 (i, j = 1, 2 . . . . . n). Multiplying A by the vector x T = (xl, x 2 . . . . . x , ) we have A x = nx. Dividing both sides of this expression by ET_ ~x~ and writing
w = x~ ~ x i i=1
we have A w = nw, and ~,7_lwi = 1. By Theorem 2 we have n as the largest positive real eigenvalue of A and w as its corresponding right eigenvector. Since aij = x i / x j = wi/wj for all i and j, we have d/i(A ) = w i = 1, 2 . . . . . n and (i) and (ii) follow. By Axiom 1, for A ~ R c ( n ), A~> c Aj if and only if a~j >/1 for all i and j, and hence we have ~p~(A)>~pj(A) for all i and j. It would appear that it is unnecessary to invoke the P e r r o n - F r o b e n i u s Theory to ensure the existence and uniqueness of a largest positive real eigenvalue and its eigenvector. We have already proved the existence of an essentially unique solu-
322
T.L. Saaty / Scaling the membership function
(b) Transitive:
F o r all x, y, z ~ S, if x ~
(c) A n t i s y m m e t r i c : F o r all x, y ~ S, ifx~yand y~
Definition. F o r cludes x ) of this x ~
any type y is t
relation x ~
Partially ordered sets with a finite n u m b e r of elements can be conveniently represented b y a directed graph. Each element of the set is represented by a vertex so that an arc is directed from y to x if x < y.
Definition. A subset E of a partially o r d e r e d set S is said to be bounded from above (below) if there is an element s ~ S such that x ~< s ( >I s) for every x ~ E. The element s is called an u p p e r (lower) b o u n d of E. W e say that E has a s u p r e m u m (infimum) if it has u p p e r (lower) b o u n d s and if the set of upper (lower) b o u n d s U ( L ) has an element ul(ll) such that u I >/u for all u ~ U (l 1 >I l for all
T h e n o t i o n s of f u n d a m e n t a l a n d derived scales can be e x t e n d e d to x ~ L k, x - c L , + 1 replacing C a n d 0a¢ respectively. T h e derived scale resulting from c o m p a r i n g the elements in x - with respect to x is called a local derioed scale or local priorities. G i v e n L , , L k + l , G . g ' , let us d e n o t e the local derived scale for y ~ x - a n d x ~ L , by +k+l(y/x), k = 2, 3 . . . . . h - 1. W i t h o u t loss of generality we m a y assume that
S_, + , + , ( y / x ) = 1. y ~ .V-
C o n s i d e r the m a t r i x + , ( L k / L , _ I ) whose colu m n s are local derived scales of elements in L k with respect to elements in L k_ ~.
Definition ( H i e r a r c h i c composition). The global derived scale (rank order) of any element in ~ ' is o b t a i n e d from its c o m p o n e n t s in the c o r r e s p o n d ing vector of the following:
+,(b)=l, +2(Lz)=+2(b-/b), ~,(Lk)=~a(Lk/Lk_,)~a_,(La_~),
k = 3 ..... h
I~L). Definition. Let ,gff be a finite partially o r d e r e d set
Definition. A set ~ is said to be exterior dependent on a set c¢ if a f u n d a m e n t a l scale can be
with largest element b. ,,~ is a hierarchy if it
defined on ~
satisfies the conditions: (1) There is a p a r t i t i o n of ,gf,a into sets called levels { Lk, k = 1, 2 . . . . . h }, where L 1 = { b }. (2) x ~ L , implies x - _ c L k + 1, where x-={ylxcoversy}, k = l , 2. . . . . h - 1 . (3) x ~ L k implies x+G Lk_I, where x +={ylycoversx}, k = 2 , 3 . . . . . h.
Definition. Given a positive real n u m b e r P >/1 a n o n e m p t y set x - _ Lk+ ~ is said to be p - h o m o g e neous with respect to x ~ L k if for every p a i r of elements y~, Y2 ~ x - , 1
p
y2) p;
in particular, the reciprocal axiom implies that
Pc(Y,, Y , ) = 1. A x i o m 2. G i v e n a h i e r a r c h y , , ~ , x ~ ~ a n d x ~ L k, X - _ L , + 1 is p - h o m o g e n e o u s for k = 1 . . . . . h - 1.
with respect to every C ~ c¢.
A x i o m 3 ( D e p e n d e n c e ) . Let .,~ be a hierarchy with levels L~, L 2 . . . . . L h. F o r each L k, k = l , 2 . . . . . h - 1, L k*~ is exterior d e p e n d e n t on L k.
Definition. Let ~¢ be exterior d e p e n d e n t on ~ . T h e elements in ~ are said to be interior dependent with respect to C ~ cg if for s o m e A ~ ~¢, ~¢ is exterior d e p e n d e n t on A. Definition. Let 5 a be a family of n o n e m p t y sets ~G, ~'2 . . . . . ~g,, where ~gi consists of the elements ( e q , j = l . . . . . rn~}, i = 1 , 2 . . . . . n. 5" is a system if (i) It is a directed g r a p h whose vertices are c¢i a n d whose arcs are defined through (if) G i v e n two c o m p o n e n t s c¢ a n d ~ ~ 5" there is an arc from c¢, to ~ if ~ is exterior d e p e n d e n t on c¢i. Let + a . c ( d ) be the derived scale of the elem e n t s of oa¢ with respect to A ~.~¢ for a criterion
T.L. Saaty / Scaling the membership function
C ~ ~'. Let q , c ( ~ ¢) be the derived scale of the elements of ~¢ with respect to a criterion C ~ ~'. Let D A _ ~ be the set of elements of ~ exterior dependent on A ~ . Let
i
aijwjwi- I
i,j=l
B~D~
If the elements of ~¢ are interior dependent with respect to C ~ C ~ , then q~c(..~)4:+c(~¢) where
,t,c(~)= (%(A), A ~ : } . Expectations are beliefs about the rank of alternatives derived from prior knowledge. Assume that a decision m a k e r has a ranking of a finite set of laternatives ..~ with respect to prior knowledge of criteria ~' arrived at intuitively. He may have expectations about the rank order. Axiom 4 (Rational expectations). Cgc ~ - Ln,
--- E,"_la~ = n and t r a c e ( A ) = Ekkk = n, then ~max -----~k1 = n . If Xma~ = n, n~kmax =
E
323
~ ' = L~.
3. Results from the axioms Note that if Pc is consistent, then Axiom 1 follows, i.e., consistency implies the reciprocal property. The first few theorems are based on this m o r e restrictive property of consistency. Let R c t , ) C R g t , ~ be the set of all ( n x n ) consistent matrices. Theorem 1. Let A ~ RM~,, ~. A ~ Rc~,) if and only /f r a n k ( A ) = i. Proof. If A ~ Rc~,, ), then a,jaj~ = a~k for all i, j and k. Hence, given a row of A, ( ai~, a n . . . . . ai,, }, all other rows can be obtained from it by means of the relation ajk = a~k/a~j and rank(A) = 1. Let us now assume that r a n k ( A ) = 1. Given a row ajh ( j 4: i, h = 1, 2 . . . . . n), ajh = Maih (h = 1, 2 . . . . . n ) where M is a positive constant. Also, for any reciprocal matrix, a i i - - 1 ( i = l , 2 . . . . . n). Thus, for i = h we have a j i = Ma;~ = M and a j h = aj~a~h for all i, j and k, and A is consistent. Theorem 2. Let A ~ R~t¢,, ~. A ~ R c t , ~ if and only if its principal eigenvalue ~ max is equal to n. Proof. By T h e o r e m 1 we have r a n k ( A ) = 1. Also, all eigenvalues of A but one vanish. Since trace(A)
=n+
Y'.
(aijwjw, -1 + ajiw, wf -1 )
l <~i~j<~n
-.+
E
(y,j+l/y,j)
I <~i<~j<~n
Since y ~ j + y , ~ >t2, and n~,m~x=n 1, equality is uniquely obtained on putting y~j = 1, i.e., a o = w~/wj. The condition a,jajk = ai~. holds for all i, j and k, and the result follows. Theorem 3. Let A = ( a ~ j ) ~ Rc~,). There exists a function + = (~l, ~b2. . . . . q , ) : R c t , ) ~ [0,1] n such that (i) a,j = q , , ( A ) / ~ j ( A ) , (ii) The relative dominance of the ith alternative, q,i(A), is the ith component of the principal right eigenvector of A, (iii) Given two alternatives A,, Aj ~ s ~ , A i >_ c Aj if and only if +i(A) >1q~j(A). Proof. A ~ R c ( , ) implies that a~j = a~ka ~) for all k, and each i and j. Also by T h e o r e m 1, we have r a n k ( A ) = 1 and we can write a,j = x J x j , where x~, xj >/0 (i, j = 1, 2 . . . . . n). Multiplying A by the vector x T = ( x l, xz . . . . . x , ) we have A x = nx. Dividing both sides of this expression by ~-7=ix, and writing
w = x~ ~ x, i=1
we have A w = nw, and Y'.7=lw/--- 1. By T h e o r e m 2 we have n as the largest positive real eigenvalue of A and w as its corresponding right eigenvector. Since a~j = x J x . i = wi/wj for all i and j, we have q,~(A) = w. i = 1, 2 . . . . . n and (i) and (ii) follow. By Axiom 1, for A ~ R c ( n ) , A~>_. c Aj if and only if a~j >_-1 for all i and j, and hence we have +~(A)>:q,j(A) for all i and j. It would appear that it is unnecessary to invoke the P e r r o n - F r o b e n i u s Theory to ensure the existence and uniqueness of a largest positive real eigenvalue and its eigenvector, We have already proved the existence of an essentially unique solu-
T.L. Saaty / Sca6ng the membershipfunction
324
tion in the consistent case. A similar result follows using the perturbation argument given below.
principal eigenvector of A' satisfies Ihm,x - n I ~< c. Let A A = ~'B. Wilkinson has shown that for a sufficiently small ~', X m,~ can be given by a convergent power series [6]
Theorem 4. Let A G R c ~ ) , and let )t 1 = n and )t z = 0 be the eigenvalues of A with multiplicity 1 and ( n - 1), respectively. Given c > 0, there is a 8 = 8 ( e ) > 0 such that if
)t,,~x = n + k ( r + kz.r 2 + . . . .
l a o + r;i - a~j I = ] %./I ~< 8
h max
fori, j=l,2
..... n
Proof. Let E0 = ½n, and let ~ ~< n / 2 . The circles C1: I / ~ - n l = c and C2: I / ~ - 0 l = c are disjoint. Let f(/t, A) be the characteristic polynomial of A. Let t ) = m i n l f ( / ~ , A)[ for ~ on @. N o t e that min [ f(/.t, A ) l i s defined because f is a continuous function of /~, and ~ >/0 since the roots of f(/~, A) = 0 are the centers of the circles, f(/~, B) is a continous function of the 1 + n 2 variables # and a;i+~';j, i, j = l , 2 . . . . . n, and for some 8 > 0 , f ( # , B) 4:0 for/~ on any Cj, j = 1, 2, if I ~',jl ~< 8, i, j = l , 2 . . . . . n. From the theory of functions of a complex variable, the number of roots ~ of f ( # , B ) = 0 which lie inside ~ , j = 1, 2, is given by 1 f'(#,B) 2~'i f(/x, B) d~,
Let w be the right-eigenvector corresponding to the simple eigenvalue n of A. Since n is a simple eigenvalue, ( A - n l ) has at least one non-vanishing minor of order (n - 1). Suppose, without loss of generality, that this lies in the first (n - 1) rows of ( A - h i ) . Then from the theory of linear equations, the components of w may be taken to be ( A . I , A . 2 . . . . . A . . ) where A.; denotes the cofactor of the (n, i) element of ( A - n l ) , and is a polynominal in n of degree not greater than (n -
1).
The components of w' are polynomials in Xma~ and ~', and since the power series expansion of max is convergent for all sufficiently small ~-, each c o m p o n e n t of w' is represented by a convergent power series in -r. We have W ' ~--- W -'1"-TZ 1 -4- ' r 2 2 2 Jr" . . .
and j=l,2,
which is also a continuous function of the n z variables a , / a n d r,j with I r;jl ~< 8. For B = A , we have n l ( A ) = l and n 2 ( A ) = n - 1. Since % ( B ) , j = 1, 2, is continous, it cannot j u m p from n s ( A ) to % ( B ) and the two must be equal and have the value n l ( B ) = 1 and n 2 ( B ) = n - 1 for all B with [a;j + "fij- a;j[~< 8, i, j = 1, 2~...~
~ n as r ~ oo, and
IXm,x-- h i = O(T) ~
the matrix B = (a,j + 5 j ) has exactly 1 and (n - 1) eigenvalues in the circles [1~ - n [ < ~ and [1~ - 0 [ < ~, respectively.
%(B)-
Now,
n.
Theorem 5. Let A ~ R c¢,) and let w be its principal right eigenvector. Let A A = (8;j) be a matrix o f perturbations o f the entries o f A such that A' = A + A A ~ R M~,,), and let w' be its principal right eigenvector. Given c > 0, there exists a 8 > 0 such that [ 8ij [ <~8 for all i and j , then [ w" - w; [ <-Nc f o r all i = 1 , 2 . . . . . n. Proof. By Theorem 4, given ~ > 0, there exists a 8 > 0 such that if l S;jl~ 8 for all i and j, the
Iw'- wl= o(~-)~c. By Theorems 4 and 5, it follows that a small perturbation A' of A transforms the eigenvalue problem ( A - n I ) w = 0 to
( A ' - X.,axI)w' =0. Theorem 6 (Ratio estimation). Let A ~ R M t , ) , and let w be its principal right eigenvector. Let c;j = aijwjw i- 1, and let 1 - .r < cij < 1 + ~, "r > O, f o r all i a n d j . Given c > 0 and • < ~, there exists a 8 > 0 such that f o r all ( x I, x 2 . . . . . x , ) , x i > O , i = l , 2. . . . . n i f 1- 8 <
ai)
x,/xj
<1+8
f o r all i a n d j ,
(2)
<1+(
f o r all i a n d j .
(3)
then
w,/wj 1-(<
xi~.
T . L Saaty / S c a r i n g the mernbership function
Proof. Substituting a u ~ ~ for w,/wj in (3) we have
w;/wj _ _
_
1 =] 1
I X'/XJ
aij
1]
1% xi/x, <~
By definition have
I
x;/xj
1 + ~-
1/c,
(;i=
Note that ET.j.lauwjwt -1 - n 2 = n(n - 1)# is also a measure of the departure from consistency. It is also possible to show that ( A - n l ) w = 0 is transformed into ( A ' - hmaxI)w' = 0 by means of graph theoretic concepts.
< (1 + ~')8 + ~'. Given c > 0 and 0 < ~ ' < ( , there exists a 8 = (~ - ~')/(1 + ~')> 0 such that (2) implies (3). Theorem 7. Let A = (a;)) ~ RM(,, ). Let X ...... be its
principal eigenvalue and let w be its corresponding right eigenvector with Y~'=lwt = 1, then %max >1 n. Proof. Let a u = ~w~-1 c u, i, j = 1, 2 . . . . . n. Since
A w = X m~w, -bsT. j = i auwj = h . . . . we have ~
a u wj - n =
i,j=l
i,j
~ R M(n). Let h r n a x be the principal eigenvector of A, and let w be its corresponding right eigenvector with F~7=lw, = 1. Then
~ m u.,~ - - n
is a measure of the average departure from consistency. Proof. For A ~ R c(,) c R M(,,), by Theorem 2 we
have )~m,~=n, and hence we have # = 0 . For A ~ RM(,, ), let a u = ~ u w j w J for all i and j. We have n
n)kmax
E j=l
W.j
aij-~
= ~
i,j=l
j~l
~-ij q-
,
l~
= -1+
Theorem 10. I f h ~ 8cl,, I, the intensities of all paths from i to j are equal to a U.
k. Corollary I. I f A ~ Rc(,), the entry in the (i, j ) position can be represented as the intensity of paths of any length starting with i and terminating with j. Proof. Follows from the proof of Theorem 10.
Proof. From Theorem 10, the intensity of a path
of any length from i to j is equal to a u. An arbitrary entry of A k is given by
)k m a x - - n
n-1
and k. Hence, we have a , = auajkakt = 1 for all i = 1, 2 . . . . . n. By induction, if a ; g . . . at._,; = 1 for all i I ... i,,_1, then a,, . . . a t . - d a i . i = a t i a i , i = 1 and the result follows.
Corollary 2. I f A E R c ( , ) , the entry in the (i, j ) position is the average intensity of paths of length k from i to j, and Ak = n k - lA ( k > 1).
(,j'
~'ij = n -]-
~
Proof. A ~ R e ( , ) implies a u % k = ate. for all i, j
Proof. Follows from a U = atkakl for all i, j and
-
n--I
~kmax =
Definition. A cycle is a path of pairwise comparisons which terminates at its starting point.
the intensities of all cycles are equal to a , , i = 1, 2 . . . . . n.
Theorem 8. Let A
-
Definition. The intensity of judgments associated with a path from i to j called the path intensity is equal to the products of the intensities associated with the arcs of that path.
Theorem 9. I f A ~ Re(,),
~%-n.
By definition, the matrix ( ( ; y ) ~ RM(,, ). We have % = 1 for all i, and ct./> 0 for all i and j. Hence, we have F~t:j=1~u - n = F~t,j( u > 0 and the result follows.
i.¢~_
As c u --, 1, i.e., consistency is approached, /~ ~ 0. Also, ~ is convex in c u, since ( ~ u + 1 / c ; / ) is convex, and has its minimum at (u = 1, i, j = 1, 2 . . . . . n. Thus, # is small or large depending on ~u being near to or far from unity, respectively, i.e., near to or far from consistency, and the result follows.
for all i and j, and we
w;/wj I a;______~j _ 1 x;/x-----~j - 1 = % , x J % + I % - 11
..... - n =
325
1 n(n-
(,j+l) 1) l<~i~j~,
cu
tj
"
=
i i
i l ~ l i,=1
"""
i
ik_l=l
aihflgt2
•••
a,..
328
T.L. Saaty / Scalingthe membershipfunction
This theorem highlights the fact that the eigenvector gives the relative dominance (rank order) of each alternative over the other alternatives along paths of arbitrary length.
4. Concluding remarks
4.1. The axiom of expectations Here is a situation which presents itself in choice problems. An apple is preferred to an orange. More apples are then introduced. Should an apple continue to be preferred or does the presence of many apples now make an orange more preferred? Both situations may be considered desirable under appropriate cirucmstances. If the apples are replaced by gold bars and oranges by iron bars, clearly in economic terms gold would continue to be preferred; 'the more the better'. On the other hand if the objects being compared are fertilizer and food and at the moment fertilizer is needed, adding more fertilizer can change the preference 'when there is a lot of it around, you never want it very much'. The question is how can a theoretical approach be developed so that it can lead to both outcomes according to which one is desired? It is obvious that consciousness about a problem is in the mind of the user. A mathematical approach cannot by itself make such a distinction automatically. However, a powerful modeling approach is one which can incorporate in the model criteria or properties which make it possible to use the same mathematics but produce the desired kind of result for the particular problem. Such an approach affords us the greatest flexibility in modeling. As the basic component, we use a direct and justifiable method of computation within a framework which we can alter according to our expectations. Such a model would be useful because it can handle different exigencies which cannot all be included in a single method of calculation particularly when it is expected of the model to product two opposite kinds of answers. There are two other ideas to mention in this respect. The first is that a higher priority criterion in a hierarchy may have serveral alternatives to be judged under it while a lower priority criterion may have a few. It is desired to increase the priority of the elements in the larger set because if there are many of them they may each receive a
smaller composite priority than each of the few elements under the low priority criterion. Sometimes it is the other rarer elements that need to receive a higher priority. In the first case one can multiply the priority of each criterion by the relative number of elements under it and normalize the weights of the criteria. In the second case a complementary set of weights can be used. A more general approach would be the one described above to introduce an additional criterion in the hierarchy called 'importance of number of descendants'. The prioritization is also carried out under this criterion, which in turn has also been compared with the other criteria. A third illustration of the need to augment the structure according to expectation is that of three secretaries who apply for a job: one excellent at English but poor at typing, the second is balanced but not excellent at either and the third is excellent at typing and poor at English. The way to surface the balanced secretary is to add to the criteria, English and typing, a third one, balance. In this manner the balanced secretary could receive the overriding priority. To cover such situations which at first may appear paradoxical and whose variety can be infinite whether analyzed through an analytic hierarchy or by other means, we must usually modify the structure, hence, the need for Axiom 4.
4. 2. Dependence of criterion membership on element membership There are problems in which the criteria are themselves dependent on the alternatives and their membership function is determined in terms of the alternatives. Since the two form an interactive set, alternatives on criteria and criteria on alternatives, the feedback system of the Analytic Hierarchy Process is used to determine the final priorities [2]. 4.3. The membership function of a large number of
elements It should be noted that when the number of elements being compared is large, the task of deriving a membership function can be simplified by decomposing the criteria into subcriteria representing a spectrum of intensities (e.g., high, medium, low) for which a membership function is constructed with respect to each criterion and then
T.L. Saaty / Scaling the membership function
weighted by the membership value of that criterion. The membership of the elements or alternatives is derived by assigning to each the membership value from the appropriate intensity under each criterion and adding over all the criteria.
329
judgment and experience. The derived scales can be combined to preserve the ratio scale property of the individual scales. This approach can be validated for problems involving a known scale of measurement.
4.4. Conclusions
References We have given a theoretical basis for constructing a membership function by means of pairwise comparisons. It is the first theory which allows for intransitivity and more generally for inconsistency in the information used to make the comparisons, yet when the inconsistency is high it suggests improving the quality of this information in particular positions in the matrix. An advantage of this approach is that it enables one to measure membership with respect to criteria even in cases when scales have not been agreed upon. Furthermore, it makes it possible to use existing measurements along with newly derived ones based on
[1] Hardy, G.H., Divergent Series, Oxford University Press, London/New York, 1949. [2] Saaty, T.L., "Exploring the interface between hierarchies, multiple objectives and fuzzy sets", Fuzzy Sets and Systems 1 (1978) 57-68. [3] Saaty, T.L., The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [4] Saaty, T.L., and Vargas, L.G., "Inconsistency and rank preservation", Journal of Mathematical Psychology 28 (2) (1984) 205-214. [5] Saaty, T.L., "Measuring the fuzziness of sets", Journal of Cybernetics 4 (4) (1974) 53-61. [6] Wilkinson, J.H., The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965.