The coordination index of finite collection of fuzzy sets

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 107 (1999) 177-185 www.elsevier.com/locate/fss

The coordination index of finite collection of fuzzy sets Teimuraz Tsabadze Ministry for Fuel & Energy of Georgia, 10 Lermontov Str., TbilisL 380007, Georgia Received August 1996; received in revised form October 1997

Abstract

This paper introduces a concept of the coordination index of finite collection of fuzzy sets. The uniqueness of the representation of this index in metric lattice of fuzzy sets with continuous isotone estimation has been proved. The correlation between isotone estimations of fuzzy set and its nearest usual set has been determined. © 1999 Elsevier Science B.V. All rights reserved.

Keywords: Finite collection of fuzzy sets; Metric lattice of fuzzy sets; lsotone estimation; Regulation; Representative; Coordination index

1. Introduction

The uncommon interest in fuzzy sets is explained not only by their applied significance but also by the beauty of pure mathematical constructions. The theory of fuzzy sets, as a strict mathematical discipline, is based on the concept of membership functions. However, these functions themselves are always hypotheses and it is impossible to measure their adequacy to corresponding concepts by means of the theory alone. These hypotheses reflect subjective presentation of experts about the peculiarities of investigated objects. Group expert evaluations of investigated objects are used in most different applications. In this case we deal with finite collection of fuzzy sets. Here the problem appears as to how to determine the degree of coordination, the closeness of these evaluations which are subjective by their nature. Let us have only one example. Assume that we have some project with its vector of goals. The problem is to determine coordination's degree of evaluations of this vector by claimants in project team. It is doubtless that the ability to determine this degree allows us to collect the most effective project team. That is why the attempt of strict determination of coordination index of finite collection of fuzzy sets to our mind has practical significance and is of certain theoretical interest. In the process of working over the paper, pure mathematical interest arose from the problem to determine the correlation between isotone estimations of fuzzy set and its nearest usual set. This problem has been solved. Moreover, it allowed us to determine the evaluation of distance between two fuzzy sets by means 0165-0114/99/$ - see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 3 2 3 - 0

178

T. Tsabadze/Fuzzy Sets and Systems 107 (1999) 177-185

of the distance between their nearest usual sets. Such evaluation may be useful for the determination of the degree of closeness between two fuzzy sets in the case of absence of full information about their membership functions. Thus, the aim of this paper is to present some theoretical results which also have applied significance and are obtained in metric lattice of fuzzy sets. The article consists of five sections. The second section concludes the necessary and sufficient information for understanding the paper. In the third main section of the paper, the concept of the coordination index of finite collection of fuzzy sets is given for the first time. The uniqueness of the representation of this index in metric lattice of fuzzy sets with continuous isotone estimation is shown. In the fourth section, the evaluation of distance between two fuzzy sets by means of the distance between their nearest usual sets is stated. In the fifth and final section of the paper examples of the introduced concepts for certain particular domains of definition of fuzzy sets are set.

2. Essential notions Henceforth, the abbreviation FS means fuzzy set. 7J(X) = {# I # :X ~ [a; b] c ~} - lattice of all FS in X. - m i n i m a l element of hu(X): # 0 ( x ) = a

VxEX.

U - maximal element of kg(X): # u ( x ) = b

VxEX.

Maximally fuzzy set - A0.5: #Aos(X) = 1/2(#0(X) + #u(x)) A=B

¢* #A(x)=#~(x)

AC_B ¢¢, #A(X)<...pB(x )

VX EX.

VxEX, A,BE ~(X). VxEX, A,BE~(X).

A C B <=> #A(X)<-,.#~(X) V x E X and E]XOI#A(XO)<#B(XO), A , B E ~ ( X ) . Union of FS A and B: #Au~(x)=max{#A(x),ps(x)} VxEX. Intersection of FS A and B: #Arts(x)= min{#A(X),#B(X)} Vx EX. The distributivity of union and intersection of FS holds in ku(X): AN(BUC)=(AMB)U(ANC),

AU(BMC)=(AUB)M(AUC).

(2.1)

Complement of FS A is FS ,4 : #d(x) = #O(x) + #u(x) - #A(X) Vx E X. Nearest set to FS A is the usual set A [1]:

#A'(X) =

#u(x) #0(X)

if #A(X)>#Ao.5(X) •xEX. otherwise

(2.2)

One can easily check that A=A,

A@B=dUL

AfiB=dnL

A,BE~(X).

(2.3)

We say that the function v : ~u(X) --* ~+ is isotone estimation on 7~(X) if [2]:

v(A u ~ ) + v(A n ~ ) = v(A) + v(~)

(2.4)

T. Tsabadze/Fuzzy Sets and Systems 107 (1999) 177-185

179

and

AC_B ~ v(A)<~v(B).

(2.5)

We say that the isotone estimation v is continuous if for each a E [v(~); v(U)] there exists A E ~u(X) such that v(A)= a. We say that the isotone estimation v is symmetrical if

v(A) + v(A) = v({3) + v(U).

(2.6)

The isotone estimation v determines the metric on ~ ( X ) :

p(A,B) = v(A U B ) - v(A NB).

(2.7)

~ ( X ) with isotone estimation v and metric (2.7) is called metric lattice of FS.

3. The coordination index of finite collection of fuzzy sets Henceforth the abbreviation FC means finite collection. Definition 3.1. FC {A5} is the regulation of FC {Aj} if for each x E X finite sets {/~A,(X)} and {#A;(X)} coincide to within order

a n d #A,I(X)~IAA~2(X)~'''~#A~,(X), j =

1,m, m = 2 , 3 . . . . .

Thus, a regulation presents FC of nested FS. It is easy to see that the equality

~-~p(B, Aj) = Z p(B,A~) j=l

(3.1)

j=l

holds in metric lattice for any B E 7~(X) and FC {Aj}, j = 1,m, m = 2 , 3 . . . . .

Definition 3.2. In metric lattice FS A* is the representative of FC {Ai}, j = 1, m, m = 2, 3 .... if

~-~p(A*,Aj)<~ ~-~p(B, Aj) ,/=l

VBE ~u(X).

(3.2)

,/=l

From (3.2) it follows that representatives of FC of FS and its regulation coincide.

Theorem 3.1. In the metric lattice of FS representative A* of FC {Aj), j = 1,m, m = 2, 3 .... is determined

in the following way: A Im/'2 CA*C_ ~ -Am/2+l

ifm is even,

A*

if m is odd.

I =A(m+l)/2

Proof. At first, we show the validity of correlation /

/

ApC_A* C_Ap+l,

p E {1,2 . . . . . m -

1}.

(3.3)

Let us assume that A* g A ~ and A'kC_A*, k = l , t , l<~t, l, t E { 1 , 2 . . . . . m}. Let l<(m 4- 1)/2. For any j E { 1 , 2 . . . . . l - 1} by (2.7), (2.1) and (2.4) we have

p(A*,A~ ) - p(A* U A~I,A~) = v(a* ) - v(A* UA~) + 2v(A* UAS) - 2v(A* ) >~v(A* ) - v(A* UA'I).

T. TsabadzelFuzzy Sets and Systems 107 (1999) 177-185

180

So

p(A*,A~ ) - p(A* U A~,A~ ) >~-p(A*,A* UA~t).

(3.4)

For any j C {l, l + 1. . . . . m}, similarly, we obtain that

o(A*,A~) - o(A* UA'I,A }) = o(A*,A* UA~).

(3.5)

From (3.4) and (3.5) it follows that l--1

/--I

j=l

j=l

j=l

j=l

p(A* UA~,A~)~>(1 - l)p(A*,A* UA~),

p(A* UA~,A~)= (m - l + 1)p(A*,A* UAtt). As a result o f the last two expressions we have

-~p(A*,A~) - ~.~ p(A* UA~t,A~) >>.(m - 2l + 2)p(A*,A* UA~) > 0, j=l

j=l

but this is impossible because o f (3.2). N o w let t/> l >/(m + 1)/2. In this case, in the same way we have

~-~ p( A*,A~ ) - ~-~ p( A* N At,Aj.' ) >~ -I ( j=l

- m )p( A*,A* U A : ) > O,

j=l

but this also contradicts (3.2). It remains to show that A~ C_A* C ' Let A* c A ' 1. From here it follows that _ A m.

p(A*,A~) - ~-~p(A~I,A~) >O, j=!

j=l

but it is contrary to (3.2). The proof of case A* C A 'm is similar. So (3.3) is proved. From here it follows that ,

.t

p(A ,A'j) = ~ j=!

Iv(A*) - v(Aj)l.

j=l

It is known that the last expression reaches a minimal value when v(A*) is the median o f nondecreasing numerical sequence {v(A~)}, j = 1, m, m = 2, 3, .... Thus, if m is even, then V(Am/2 ~ ) ~ v(A • ) <<.V(Az/2+ t 1), while if m is odd, v(A*)=v(Aim+l)/2). Taking into consideration that we deal with nested sets, we are able to conclude that v(A) <<.v(B) ~ A C_B, v(A) = v(B) ~ A = B and the proof is completed. [] Corollary. I f A* and B* are the representatives of FC {Aj } and {Bj ), respectively, then A* U B* and A* n B* are the representatives of FC {Aj UBj} and {Aj NBj} respectively, j = 1,m, m = 2 , 3 . . . . . Proof. Let A*,B* and C c ~u(X) be the representatives of FC {Aj}, {Bj} and {Aj UBj}, respectively, and m is odd. By theorem C =A~m+l)/2 UB~m+I)/2 =~ C = A * UB*.

T. Tsabadze/Fuzzy Sets and Systems 107 (1999) 177-185

181

/ UB/m/2+l . On the other hand, Arm,2CA* CAtm/2+ Now, let m be even. By theorem A '~ m , 2 U B m~/ 2 C _ C C _ A m/2+l and B~/2 C B* C_B~m/2+l. The last two expressions give A~m/ZU ~m/2 R! C_,.~*~lB* C_Atm/2+l ""~t~ lm R/ 2 +' l ' ..encer~"----~* . ~ u'B* is the representative of {Aj U Bj}. In the same way one can show that A* NB* is the representative of {Aj UBj}. [] Definition 3.3. The FC o f FS is symmetrical if in its regulation the first [(2m + 1),/4] ~ sets are equal to (~ and the last [(2m + 1)/4] sets are equal to U, m = 2, 3 . . . . . In the sequel we need the following result. L e m m a 3.1. In metric lattice of FS for any FC {A/}, j = 1, m, m = 2, 3,... the correlation

0 <~~

p(A*,Aj) <~[(2m + 1 )/4]p(0, U),

(3.6)

/-1

is satisfied. In addition, 0 is reached if and only if all FS of FC are equal to each other, while [(2m + 1 )/4]p(~, U) is reached if and only if FC of FS is symmetrical. Proof. It is easy to show that m

Z p ( A * , A j ) = O ¢~ A*=Aj,

j = l , m , m=2,3 .....

j=l

Let m be odd. Then by (2.7), (3.1) and Theorem 3.1 (rn-- 1 )/2

j= I

j=(m+3)/2

j= 1

It is not difficult to see that the last expression reaches a maximal value when for V j E {(m + 3)/2, (m + 5 ) / 2 , . . . , m } A~ = U and for V j E {1,2 . . . . . (m - 1)/2} Aj = ~ (here A* may be arbitrary). Thus we have that m

0<~ Zp(A*,Aj)<,(m - 1)/2p(~, U),

(3.7)

i=1

and the peak is reached when FC of FS is symmetrical. / fm/Z" From here we obtain that Now let m be even. By Theorem 3.1 Arm,2CA*_ Q_Am/2+ I. Let A* --A -

j= 1

j=m/2+ 1

j= 1

and the peak of this expression is reached when A(.1 = U, j = m/2 + 1, m and Aj = ~, j = 1, m/2. Note that in this case A* = ~. Hence we have that

0 <~~ p(A*,Aj) <~m/2p(O, U). /= 1

1 Here and fiu'ther o n s y m b o l [ ] denotes a w h o l e part o f a number.

(3.8)

T. Tsabadze/FuzzySets and Systems 107 (1999) 177-185

182

For cases Aim~2 CA* CAtm/2+l we obtain (3.8) again; in addition, the peak is reached when FC of FS is symmetrical (note that when A* =A'm/2+ 1 then A* = U, while for Arm~2CA* CA'/2+1, A* is arbitrary). Both (3.7) and (3.8) give (3.6). It remains to show that if ~ " = l p(A*,Aj) reaches the peak, then FC of FS is symmetrical. We show it for even m and A* =Am~2+ t 1. In this case,

f i p(A*,Aj ) = m/2( v( U ) - v(O) ), j=l

and it is obvious that FC of FS is symmetrical. For cases

/ A~m/2CA* CAm~z+ 1 and odd m the proof is similar.

[] Let us introduce the main definition of this paper. Definition 3.4. In metric lattice of FS, the functional s: y(x)

× ...

--+

m times

is the coordination index of FC {Aj}, j = 1,m, m = 2 , 3 .... if it satisfies the following postulates: P1. S{Aj} = 0 if and only if FC {Aj} is symmetrical; P2. S{Aj} reaches maximal value if and only if all FS of FC are equal to each other; m m P3. S{Aj} >~S{Bj} if ~j=lp(A*,Aj)<~ Y~j=Ip(B*,Bj); in addition S{Aj} = S{Bj} if and only if P4.

ET_I p(A*,Aj)= E7_1 p(B*,Bj); S{Aj UBj} + S{A i nBj} = S{Aj) + S{Bj}.

Theorem 3.2. m = 2 , 3 .... if

In metric lattice of FS functional S{Aj} is the coordination index of FC {Aj}, j = 1,m,

S{Aj}=k(p(O,U)-[(2m+l)/4]-'xfip(A*,Aj)),j=,

k>0.

(3.9)

Moreover, if isotone estimation v is continuous, this representation is unique. ProoL Let (3.9) be true. Validity of P1 and P2 follows from Lemma 3.1. Validity of P3 follows from (3.9) directly. By (2.7), (2.4), (3.1) and Theorem 3.1 and its corollary one can check that m

m

m

m

j=l

j--I

j--1

i--1

This fact with corollary of Theorem 3.1 and (3.9) proves validity of P4. Let isotone estimation v be continuous. According to postulates P1, P2, P3 and Lemma 3.1, a coordination index represents a monotone decreasing function f(}--~-7=lp(A*,Aj)) with domain of definition from 0 until [(2m + a)/4]p(O,U). Let FC {Aj} and { B j ) j = 1,m, m = 2 , 3 .... be such that A~ C A2 . . . . . Am-~ CAm; B1 CAt C_B2 . . . . . Bin-1 C_AmC_Bm. Then by (2.7) and (2.4)

72 Tsabadze/Fuzzy Sets and Systems 107 (1999) 177-185

183

= f(v(Am UBm) - v(AI UBt)) + f(v(Am NBm) - v(Al ABI )) = f(v(Bm) - v(AI )) + f(v(Am) - v(B1 )). On the other hand

f\_[~_l p(A*,A/)

+ f \/=' p(B*,B/)

)

= f ( v ( A m ) - v(Al))+ f ( v ( B m ) - v(Bl)).

From the last two equations by postulate P4 we obtain that f ( v ( A m ) - v(N1 )) ÷ f ( v ( B m ) - v(B1 )) = f ( ~ ( B m ) - v(AI )) ÷ f(V(Am ) - v(B1 )).

(3.10)

Let a, bE ~+ be such that v(O)<~a
kl >0.

j= I

By last expression, postulate P2 and Lemma 3.1 we determine .£~a× = kl[(Zm + 1)/4]p(0, U) and having taken k = kt [(2m + 1)/4] we obtain (3.9).

[]

Corollary. In metric lattice of FS with symmetrical isotone estimation

s{Aj} =S{A-i}

j=l,m,

m : 2 , 3 .....

Proof. By (2.6) and (2.7) one can easily show that p(A,B)=p(A,B) VA, BE ~P(X). This fact with (3.9) proves the corollary. []

4. Some estimation of distance between fuzzy sets ~

~

Let us set down the following problem. It is required to evaluate a metric p(A,B) given that only p(A, B) is known (here A,BE 7/(X); A,B are nearest sets to A and B, respectively). For solving this problem we need the following result having however an independent meaning too.

Lemma 4.1. For any A E 7/(X) symmetrical isotone estimations of sets O, A, A and U are connected by the following inequality:

v((O) + v(A) <~2v(a) <~v(,4) + v(U), where A is determined by (2.2). Proof. Let us present FS A as a union of two FS: A =A1 UA2, where

/ZA,(X) =

~A(X) if l~A(X)<<,l~O.5(x), #~(X) otherwise,

184

T. Tsabadze/Fuzzy Sets and Systems 107 (1999) 177-185

/tA~(x) =

/tA(X) if /tA(X)>/tO.5(X)

(

/t0(x )

xEX.

otherwise

One can be easily convinced that A2=A, A2C,,]2, AIUAC_.41, A U ~ = A z U A , A A ~ = A I , ,41U-42=U, A1 N.4= 0. By these correlations and (2.4)-(2.6) we have 2v(A) - v(A) - v(0) -- v(A) + v(J) - v(A) - v(O) = ~(A2 [--JA2) -1- v(A1) - F ( A I ) - v(A2) -+- v ( U ) - v ( 0 ) = v(A2 [--JA2) - v ( A 2 ) -4- 2(v(A1) - v(0)) ~>0

=~ v(0) + v(A) ~<2v(A) = 2v(A1) + 2v(A2) - 2v(0) ~<2v(A2) + v(Al) + v(Al tAA) - v(.~) - v(0)

<. v(2) + V(Al ) + V(2~ ) - v(O) = v(A) + v(U).

[]

The following proposition solves the problem raised. Proposition 4.1. In metric lattice of FS with symmetrical isotone estimation for VA,B E ~(X)

O<~2p(A,B)<~p(J,[~) + p(O, U).

(4.1)

Proof. By (2.7), (2.3) and Lemma 4.1 we have

0 <~2p(A,B) = 2(v(A UB) - v(A AB))<~v(AUB) + v(U) - v(AAB) - v(O) = p ( 2 , ~ ) + p(O, u ) .

[]

5. Illustrations

Here we set the examples of the introduced coordination index for particular domains of definition of FS and isotone estimations. Let ~ P ( X ) = { # I # : X ~ [0; 1]} and X be the finite set {Xl,X2..... Xu}, N = 1,2 ..... We determine the capacity of FS A is the following way: N

P(A) = Z

#A(xi) xi E X,

A

~ ~IJ(X).

i--1

One can easily check that P satisfies the conditions (2.4)-(2.6) and therefore, we deal with symmetrical isotone estimation. By (2.7) the estimation P determines the following metric: N

p(A,B) = Z

I#A(Xi) --/tB(xi))

xi EX, A,B E ~P(X).

i--1

Note that this metric coincides with the Hemming distance. The coordination index of FC {Aj}, j = 1, m, m = 2, 3 ..... As. E P ( X ) yields the next expression:

T. Tsabadze/Fuzzy Sets and Systems 107 (1999) 177-185

185

Now let X be the interval [a; b] c ~ and {#} are integrable functions on this interval. Let us determine the

capacity of FS A in the following way: P(A)=

#A(x)dx

xEX, A E ~ ( X ) .

One can be easily convinced that P is the symmetrical isotone estimation on 7J(X). The metric is

p(A,B)--

I#A(X)-- #e(X)l dx x EX, A,B E ~(X)

and the coordination index of FC {Aj}, j = 1,m, m = 2 , 3 ..... Aj E ~ ( X ) presents the next expression:

S{A/}=k

b-a-[(2m+l)/4]

-I

k>0. j= I

a

References [1] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, vol. 1, Academic Press, New York, 1975. [2] A. Averkin, J. Batirshin, A. Blishun, V. Silov, V. Tarasov, Fuzzy Sets in the Models of Control and Artificial Intelligence (Nauka Press, Moscow, 1986).