12
PUBLIKACIJE ELEKTROTEHNICKOG FAKULTETA UNIVERZITETA U BEOGRADU PUBLICATIONSDE LA FACULTED'ELECTROTECHNIQUEDE L'UNIVERSITE A. BELGRADE SERIJA: MA'I'EMA TIKA I F IZ IK A - SERlE: MA THEMATIQUES ET PH YSIQUE N2 461 - K2 497 (1974) 483. WEIGHTED CONJUNCTIVE AND DISJUNCTIVE MEANS AND THEIR APPLICATION IN SYSTEM EVALUATION* Jozo J. Dujmovic ABSTRACT. In this paper a definition of weighted conjunctive and disjunctive means is proposed, and their basic properties investigated. These means are derived from weighted power means for use in formal models for estimation of the true value of a class of complex statements in continuous logic, and for application to solving evaluation and comparison problems of arbitrary complex systems. 1. Introduction For a given sequence of non-negative real numbers x = (Xl' ... , xn) and an extended real number r (rE Rx,), the power means are defined, according to [1], by expression (1) ( In ) . Ilr M~] (X) = -,;;~ x/ (O<:r]<+oo), ( n ) lln = OX; ;~1 (r = 0), (r = - (0), (r = + (0). In numerous applications relevant to the evaluation and comparison of arbitrary complex systems, the .case (2) x;EI, i= 1, . .., n, 1= [0, 1] is of interest. Here x; represents the true value of the corresponding statement in the continuous logic, so that the extreme true values 0 and 1 correspond to the false and true statement respectively. In the foIlowing text we shaIl consider only the case when restrictions (2) hold for the sequence x. According to [2, p. 76], (1) and (2), inequality (3) * Prese .ted June 1, 1974 by P. M. VASIC. 10. 147

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PUBLIKACIJE ELEKTROTEHNICKOG FAKULTETA UNIVERZITETA U BEOGRADUPUBLICATIONSDE LA FACULTED'ELECTROTECHNIQUEDE L'UNIVERSITE A. BELGRADE

SERIJA: MA'I'EMA TIKA I F I Z I K A - SERlE: MA THEMATIQUES ET PH Y S I Q UE

N2 461 - K2 497 (1974)

483. WEIGHTED CONJUNCTIVE AND DISJUNCTIVE MEANS ANDTHEIR APPLICATION IN SYSTEM EVALUATION*

Jozo J. Dujmovic

ABSTRACT. In this paper a definition of weighted conjunctive and disjunctivemeans is proposed, and their basic properties investigated. These means arederived from weighted power means for use in formal models for estimation ofthe true value of a class of complex statements in continuous logic, and forapplication to solving evaluation and comparison problems of arbitrary complexsystems.

1. Introduction

For a given sequence of non-negative real numbers x = (Xl' . . . , xn) andan extended real number r (rE Rx,), the power means are defined, accordingto [1], by expression

(1) (In

). Ilr

M~] (X) = -,;;~ x/ (O<:r]<+oo),

(n

)lln

= OX;;~1

(r = 0),

(r = -(0),

(r = + (0).

In numerous applications relevant to the evaluation and comparison ofarbitrary complex systems, the .case

(2) x;EI, i= 1, . .., n, 1= [0, 1]

is of interest. Here x; represents the true value of the corresponding statementin the continuous logic, so that the extreme true values 0 and 1 correspondto the false and true statement respectively. In the foIlowing text we shaIlconsider only the case when restrictions (2) hold for the sequence x.

According to [2, p. 76], (1) and (2), inequality

(3)

* Prese .ted June 1, 1974 by P. M. VASIC.

10. 147

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148 J. J. Dujmovic

is vali d for - 00-;;;,r -;;;,+ 00, so that the following is easy to note:

a) The position of M~] (x) within the interval [M~-OO](x), M~+oo](x)], isdetermined by the parameter r.

b) The boundary functions M~-oo](x) and M~+OO](x) represent the logicalfunctions conjunction and disjunction respectively, in the manner in whichthey are defined in continuous logic [3], so that it follows that there are somecommon points between power means and functions in continuous logic.

Starting from the above statement, an attempt is made in this paper toformulate, on the basis of weighted power means, a special class of means,called weighted conjunctive and disjunctive means, the application of whichenables the realization of formal models for estimation of the true value of aspecial class of complex statements in a continuous logic, which appear whenproblems of complex systems evaluation are involved. Though algorithms forevaluation of complex systems, based on weighted conjunctive and disjunctivemeans, are beyond the scope of the present paper, attention is called to certainpossible applications, so that a short illustrative example is given at the endof the paper.

2. Average value of a mean, characteristic function,conjunction and disjunction degrees

Averaging of non-negative real numbers from a given interval [a, b],O-;;;'a<b<+oo is very frequently applied. However, in a majority of cases itis sufficient to observe the averaging on the interval 1 = [0, 1]; the resultsobtained are easily extended to arbitrary intervals. It is suitable to describethe averaging on interval I by means of characteristic indicators which will bedefined in this text.

Let a generalized LOSONCZI'Smean be given [4]:

xIEI, <J\:I--?R+U{O}(i= 1, ..., n), F:I--?R (strictly monotone), F-1:R--?I,x = (xl' . . . , xn), <I>= (<1>1' . . . , <l>n)and let

1 1 1

An =.r dXI J dX2 . . . J max (Xl' ..., Xn)dXn'o 0 0

1 1 1

an = .r dXI J dX2 . . .Jmin (Xl' . . . , Xn)dXn,000

Definition 1. The average value of mean Mn (x; <1» on interval I is determined by

(4)1 1 1

Mn=J dXIJ dx2...J Mn(x; <I»dxn.000

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Weighted conjunctive and disjunctive means and their application. . . 149

Since min (Xl' . . . , Xn) ;;;:Mn (X; <1»;;;:max (Xl' . . . , xn), from (4) it foHows that

an ;;;:Mn;;;: An.

Definition 2. A characteristic function m: R:---+ I, determined by

(5) m(p)= lim Mn,n ++oo

is associated with generalized mean, where p = (Pi' ..., h) is the vector of adju-stable parameters appearing in a generalized mean Mn (x; <1».

EXAMPLES.If cfIj(x;) =XjP, F(Xj) =Xjq (p, qERoo), i~ 1,..., n, then m:R002 ~I. Ifcfl=(l,..., 1),F (Xj) = xi' (rE Roo), i = 1, . . . , n, then function m: R 00~ I has only one argument (r f-+m (r»).If the mean does not have adjustable parameters, then the characteristic function degeneratesinto a constant characteristic value (for example, it will be shown that the characteristic valueof the arithmetic mean is ma ~ 1/2, while that of the geometric mean is mg = l/e).

Definition 3. The position of Mn with respect to the lower bound of the[an, An] is determined by parameter c (c E I), called the conjunctiondefined by

intervaldegree,

(6) n>1.

Definition 4. The position of Mn with respect to the upper bound of the interval[an, An] is determined by parameter d(dEI), called the disjunction degree,defined by

(7) n> 1.

Since, according to [5]

(8) 1an=-,

n+l

nAn=-,

n+l

from (6), (7) and (8) it foHows

(9) n-(n + 1) Mnc= .

n-l

(10)

so that from (5), (9) and (10) the connection between conjunction degree,disjunction degrec and the charactcristic functicn is easily noted:

(11) lim c= I-m(p),n---++oo

lim d=m(p).n---++oo

In further text we shall consider only the powerwhen <1>=(1, ..., 1), F(x)=x/(rERoo), i= 1, ..., n,and Mn <erc reduced to M~] (x) and M~] re~pectively.

means, i.e. the caseso that Mn (x; <1»

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150 J. J. Dujrnovic

Calculation of aver2.ge value M~] on the basis of (4) leads to binomialintegrals and is quite involved, so that the practical calculation of M~] ispossible only by numerical integration on a computer. The following Can beanalytically calculated:

M[-OO]

-It -an,

(I 2)-[0] ( n )

nMn =-,

n+1

(I3)

1t/4

M~2]=

2 ~3[J (I + cos-2 Z)3/2dZ-7t/4J

= 0.554598,

o

(I 4) M&-I]= : (I -In 2),

I I-[-I]

f fxy [ xy

I ( I I)]M3 =3 dx - 1-- n 1+-+- dy=0.362881,

x+y x+y x yo 0

All above expressions were derived following brief calculation.

Theorem 1. The conjunction degree c and the disjunction degree d represent nor-malized and mutually complementary values which may be expressed, for the caseof power means, as the corresponding functions of parameter r, c=Cn(r), d=Dn (r)and they have the following properties:

(15) O~Cn(r)~I, O~Dn(r)~I, Cn(r) + Dn(r) = I,

Cn ( + 00) = Dn ( - 00) = 0, Cn ( - 00) = Dn ( + 00) = 1.

These properties follow directly from (6) and (7). Therefore, the value ofparameter r can be determined from the desired value of the conjunctiondegree or the disjunction degree:

(16)

where C;;I and D;;I represent functions inverse to Cn and Dn respectively.

Theorem 2. The conjunction and disjunction degrees of the geometric mean (r = 0)are determined by

Cn(O)=~_n+l (~ )n, Ci(O»Ci+l(O), i>l,n-I n-I n+ I

Dn(O)=n+I (~ )n_~, Di(O)<Di+l(O), i>1n-I n+ In-I

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Weighted conjunctive and disjunctive means and their application. . . 151-_.~

and have the following properties:

C~ (0) = 1 - lie, C2(0) = 2/3, 1 - lie ~ Cn(0) ~ 2/3

D~(O)=lle, D2(0)=1/3, 1/3~Dn(0)~lle.

The conjunction and disjunction degrees of the square mean (r = 2) are determined by

Cn(2) = 0.376775 (n = 2) Dn (2) = 0.623225 (n = 2)

= 0.390804 (n = 3) = 0.609196 (n = 3).

The conjunction and disjunction degrees of the harmonic mean (r = - 1) aredetermined by

Cn( -1) =In 16- 2 = 0.77259 (n = 2), Cn (- 1) = 0.774238 (n = 3)

Dn(-I)=3-ln 16=0.22741 (n=2), Dn(-I)=O.225762 (n=3).

This theorem is easily verified from (9), (10), (12), (13), and (14).

Theorem 3. The conjunction and disjunction degrees of the arithmetic mean areequal and constant:

(17)

Expression (17) ensues from (9), (10) and (12).The characteristic function of the power means is determined, according

to (11) by-[]

m (r) = D~ (r) = 1 - C~ (r) = M ~,

so that on the basis of the preceding theorems it follows:

m(- 00)=0, m(O)= lie, m (1) = 1/2, m(+oo)=1.

Since for positive and non-identically equal values of x, the power mean M~] (x)is a strictly increasing function of r, it follows that r f-+m (r) is also a strictlyincreasing function:

m(s)<m(t); - 00 ~s<t~ + 00.

3. Conjunctive and disjunctive means

The mapping defined by (16), C;;!, D;;!: I"::"R~, may be inserted intothe power means defining thus corresponding new means.

Definition 5. The conjunctive means (CM), whose parameter is the conjunctiondegree c, are determined by

(18)

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152 J. J. Dujmovic

The disjunctive means (DM), whose parameter is the disjunction degree d,are defined by

(19)

Theorem 4. CM and DM have the following properties:

a) If c= 1 and d=O, it follows that r= - 00, soto the conjunction:

that CM and DM reduce

[I] [0] .Ln (x) = N n (x) = mm (Xl' ..., xn).

b) If c = 0 and d = 1, it follows that r = + 00, so that CM and DM reduceto the disjunction:

L~O](x) = N~'] (x) = max (xl' ..., xn).

c) For 0 ~ c, d ~ 1, the following holds:

L~] (x) = N~'-c] (x), L~'-d] (x) =Nr:!](x),

L~'](x) ~ L~] (x) ~L~O](x) N~O](x) ~ N~d](x) ~N~'](x).

d) Since M~] (x) is (for arbitrary non-negative real numbers x) a monotonicallyincreasing function of parameter r, it follows that L~] (x) is a monotonically decreasing function of the conjunction degree c, and that N~d](x) is a monotonicallyincreasing function of the disjunction degree d.

All these properties are easily verified on the basis of (6), (7), (18), (19)and [1, p. 67].

4. Weighted conjunctive and disjunctive means

The mapping defined by functions c H>-C;' (c) and dH>-D;;t (d) may beused to derive weighted conjunctive and weighted disjunctive means fromweighted power means.

Definition 6. For a sequence of weights

i= 1, ..., n,n

L Wj=l,;=1

the weighted conjunctive means (WCM), containing parameter c, are determined by

(20)

= min (x" ..., xn)

=max(x" ..., xn)

(c=l)

(c = 0),

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Weighted conjunctive and disjunctive means and their application. . . 153----

and weighted disjunctive means (WDM), containing parameter d, are determined by

(21) N~d](X; W)=(~IWixf;;l(d)Y/D;;I(d) (d=l=Dn(O),O<d<l)

n W.=OXi

I

i~1

= min (XI' . . . , xn)

= maX (XI' . . . , xn)

(d = 0)

(d= 1).

If W=W=(n-l, n-l, ..., n-I), WCM and WDM reduce to CM and DMrespectively:

L~C](x; W) = L~] (x), N~d](x; W) = N~d](x),

and the relationships

L~](x; W)=N~I-c](X; W), N~d](x; W)=L~I-d](X; W)

hold in a similar manner as before.In the following text we shall consider those properties of WCM and WDM

which represent a basis for various applications.

5. Quasi-conjunction, quasi-disjunction andconjunctive-disjunctive undetermination

With reference to the problem of complex systems evaluation and comparison,it was found suitable to use WCM and WDM for estimation of the true valueof a class of complex statements consisting of n elementary statements of giventrue values xI' ..., xn' In that case, for O<c, d< 1, the true values of allelementary statements contribute to the true value of the complex statement.This represents the essential difference in comparison to log:c functions obtainedby a superposition of conjunction, disjunction and negation, wherein the truevalue of a complex statement is affected by the true values of only thosestatements which, in some subsets of elementary statements, have extreme truevalues. Besides, WCM and W DM enable continuous adjustment of the influenceof true values of elementary statements on the true value of a complex state-ment, by means of weights WI"", Wn, representing the relative degrees ofsignificance of true values of elementary statements.

If the conjunction degree c and disjunction degree d are treated asquantitative parameters of "the logical properties" of WCM and WDM, thenfrom (15) and (17) it follows that logical properties of the arithmetic meanare at a "half-way" between the logical properties of conjunction and ofdisjunction, so that in this way a class of functions similar to conjunctionand a class similar to disjunction can be separated. It is of practical interest,first of all, to set up a terminology for these classes.

Definition 7. For a class of functions L~] (x; W) and N~dJ(x; W), the followingterms are introduced:

- quasi-conjunction (QC), for 1/2<c<1 and 0<d<1/2

- conjunctive-disjunctive undetermination (A), for c = d = 1/2

- quasi-disjunction (QD), for O<c< 1/2 and 1/2<d< 1.

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154 J. J. Dujrnovi6

According to this definition, the logical properties of WCM and WDMare illustrated in Fig. 1.

c d1

1/2IIII

aA~~--J

i'--- ~ /tO

.I

QUASI-CONJUNCTION Q.UASI-DISJUNCTIOH

CONJUNCTIOH DISJUNCTIONCONJUNC'fIVE - DISJUNCTIVE

UNDETERMINATION

Fig. 1

Theorem 5. Quasi-conjunction is a class of functions with the following properties:

a) If C~C;l(O), i.e. d<:;;,D;I(O), then L~c](x; W), N~](x; W)=ftO only ifall elementary statements are partially true (i.e. x;>O, i = 1, ..., n).

b) If x;>O, i= 1, ..., n, then L~](x; W)=N~d](X; W)=min(xl' ..., xn)only if XI = x2 = . . . = xn' otherwise L~] (x; W), N~] (x; W) >min (xl' . . . , xn).

c) Greater true values of quasi-conjunctive complex statements can be achievedonly by a coincidence of predominantly true elementary statements. That is whypredominantly false elementary statements (for which x; < 1/2) dominantly influencethe true value of quasi-conjunctive complex statements, so that in some cases theappearance of only one predominantly false elementary statement can, to aconsiderable extent, decrease the true value of a complex statement.

This theorem is easy to verify according to definitions 6 and 7.

Theorem 6. Quasi-disjunction is a class of functions with the following properties:a) L~] (x; W) = N~] (x; W) = °

only if Xl = x2 = . . . = xn = °otherwise

L~](x; W), N~d](X; W»O.

b) L~c\x; W) = N~d](X; W) = max (xl' ..., xn) only if Xl = x2 = . . . ="xn,otherwise L~](x; W), N~d](X; W)<max(xl' ..., xn).

c) Greater true values of quasi-disjunctive complex statements are obtained,if any elementary statement is sufficiently true (or more of them), whose relativesignificance (i.e. the corresponding weight) is not too small. From this ensues thedominant influence of predominantly true elementary statements (for which x; > 1/2)on the true value of quasi-disjunctive complex statements; even a single predominantly'true elementary statement may, in certain cases, to a considerable extent, increasethe resulting true value of a complex statement.

This theorem follows from definitions 6 and 7.

Discussion. It should be noted in theorems 5 and 6 that properties a)represent conditions common for conjunction and QC, and disjunction and QD.Properties b) and c) are characteristic for QC and QD, and show that QC and QDcan rightly be treated as "weaker forms" of conjunction and disjunction. Fromtheorems 5, 6 and Fig. I it is easy to see that logical properties of WCMand WDM represent a mixture of "conjunctive properties" expressed through

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Name of the basic WCM (WDM)

Conjunction

Strong quasi-conjunction

Medium quasi-conjunction

Weak quasi-conjunction

Conjunctive - disjunctive un determination

Weak quasi-disjunction-----

Medium quasi-disjunction-'Strong quasi-disjunction. -Disjunction

----------- ~-

Weighted conjunctive and disjunctive means and their application. . .- ---

155

the conjunction degree c and "disjunctive properties" expressed through thedisjunction degree d. In the case of QC, conjunctive properties dominate,while in the case of QD disjunctive properties are predominant. In the caseof A, disjunctive and conjunctive properties are equally present, expressedthrough additive supplementing of the true values of elementary statements.In the general case, by selecting parameters c and d, conjunctive and disjunctiveproperties in WCM and WDM may be mixed in arbitrary quantities.

6. The system of nine basic weighted conjunctive and disjunctive means

For practical applications of WCM and WDM by teams of experts solvingthe problems of evaluation and comparison of complex systems, it is ofinterest to single out a finite set of various WCM and WDM with differentand clearly defined logical properties, within which the experts may selectfunctions which correspond best to particular applications. The purpose of thisprocedure is to facilitate the choice of adequate values of conjunction (disjun-ction) degrees. Since the discretization of conjunction and disjunction degreesmust be carried out, it is obvious that the best variety of logical propertiesof WCM and WDM is obtained, if such WCM and WDM are adopted towhich correspond equidistant values of the conjunction (disjunction) degree.Starting from the limited possibilities of an expert - classifier [6,7] it is suitableto adopt a system of nine basic WCM (WDM) shown in Table I.

Table 1

I

dI

SymbolI

1-~.ooo-I__o.o()O._~

I

0.875 I 0.125 C+

0.750 i 0.250 -~1_--_-,

0.625 : 0.375 C-i

050'01--0.50()"-::1-

-~:~~I ~:~~~-

I

I 0.125

)

' 0.875

0.000 1.000

~-~--

D-

DA

c

-~ ----- ---

---D+

D

This system of nine basic WCM (WDM) is connected to parameter r ofweighted power means in the manner illustrated in Fig. 2, for n = 2. Fromthis figure it can be seen that the harmonic mean is close to CA, the geometricmean close to C-, while the square mean is practically identical with D-.

In practical applications of the quoted nine basic WCM (WDM), some-times the need arises to amplify or to weaken the accepted basic values ofconjunction (disjunction) degrees. In order to meet this requirement, it issufficient to insert, between every two adjacent basic WCM (WDM) anauxiliary WCM (WDM) whose conjunction (disjunction) degree is the arithmetic

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156 J. J. Dujrnovic~-

mean of the conjunction (disjunction) degrees of the mentioned adjacentbasic WCM (WDM). In such a way, the following extended sequence ofadopted WCM (WDM) is obtained:

C, C++, C+, C+-, CA, C-+, C-, C--, A,

D--, D-, D-+, DA, D+-, D+, D++, D.

Here C++, C+-, C-+, C--, D--, D-+, D+- and D++ are the insertedauxiliary WCM (WDM).

In applications performed so far, these 17 weighted conjunctive (disjunctive)means completely satisfied all imposed requirements.

12 r

1110 9.521987654 }.929

32 }.Q1L-

1 1.0000.261

o -D.720-1

~n-

-2-3 -3.510'

--- C+-4

-5

-6-7

-8-9-10-11-12

d0.750 0.875 1. 00

FUNCTION d M 01 (d)

Fig. 2

7. Example of WCM and WDM application

WCM and WDM may be used in ?ll problems which involve or can bereduced to averaging marks or points. Fer example, if-xi is a mark representingthe true value of the statement "the candidate entirely satisfIes all requirementsfrom the i-th field" (i = I, . . . , n), then the total mark X of the same candidaterepresents the true value of the statement "the cand:date entirely satisfies all

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Weighted conjunctive and disjunctive means and their application. . . 157

imposed requirements". The total mark may be:}btained in different ways,depending upon the type of activity awaiting the candidate under consideration.From each given type of activity the specific criterion for evaluation andselection of a candidate may be derived. In the general case, the candidate canbe human, machine, or an arbitrary complex &ystem.

In the following example we shall present a short illustration of anevaluation procedure, and show how the criterion for evaluation of candidatesaffects the choice of the corresponding WCM (WDM) from the system of ninebasic conjunctive (disjunctive) means, in order to calculate the total mark Xof th~ candidate from the given elementary marks Xl' . . ., Xn'

a) A candidate, having high ability for any narrow field (marked

Xl' . . ., Xn), is required who would work exclusively in the field for which hehas maximum affinity and capability. The expression for these requirementsare obviously either D or D+.

b) The candidate will, most probably, work in one of the narrow fields.The corresponding WCM (WDM) is DA or D-.

c) A candidate of high total capability is required, i.e. the tendency istoward a candidate able enough in particular specialized fields, but not markedlyweak in some other fields, under condition that positive and negative exceptionsin the candidate's marks do not influence his total mark decisively. The same<:riterion can be adopted in cases when sufficient information regarding thetype of future activity of the candidate is lacking, so that the tendency is tominimize the possible error in evaluation. Since this criterion is neither conjun-ctively nor disjunctively polarized, the corresponding WCM (WDM) is A.

x

E

v

Fig. 3

d) The candidate will predominantly work in interdisciplinary activities i.e.he will be simultaneously working in several narrow fields. According to a weakor medium conjunctive polarization on this criterion, the corresponding WCM{WDM) are C- or CA.

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158 J. J. Dujmovic-.--- ---------------------

e) The candidate will have strictly multidisciplinary activity, not allowingany weak capabilities in any narrow fields. Obviously, this criterion is stronglyconjunctively polarized ("the chain is as strong as its weakest link"), so that theexpression of these requirements may be C+ or C.

If five candidates, whose total marks X, Y, Z, U and V are determinedaccording to the above criteria, were to form a team wherein efficient workis possible only if each candidate satisfies, to a greater extent, conditions of hisparticular criterion, then the total mark E, for a team as a whole, can becalculated applying CA according to the diagram shown in Fig. 3. The resultantmark E will represent the true value of the statement "the team entirely satisfiesall imposed requirements". Obviously, by applying the obtained total mark, itis possible to compare several competitive teams through very complex criteria.However, the detailed algorithms for evaluation of arbitrary complex systems,developed on the basis of WCM and WDM, are beyond the scope of thepresent paper.

8. Conclusion

The following facts, useful for estimating the role of WCM and WDM insystems evaluation, may be stressed in the conclusion.

1. The evaluation criteria described above, whose essential characteristic isimprecision in definition, which may be interpreted as fuzziness [8] (expressedthrough words such as "predominant", "to a greater extent" and so on), entirelycorrespond to real situations encountered by teams of experts evaluating complexsystems in the presence of incomplete information concerning the system andits forms of activity.

2. By superposition of elementary criteria it is possible to form morecomplex criteria of an arbitrary order of complexity. Corresponding WCM (WDM)enable the formal modelling of elementary criteria, while, by superpositionof WCM (WDM), formal models of criteria of arbitrary complexity are realized.

3. The application of WCM (WDM) enables approximate formal modellingof the real reasoning process of system evaluation in a fuzzy environment. Thatis why it is possible to use the entire quantity of information available to experts,and therefore to obtain maximum precision of the resulting formal model forsystem evaluation.

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