General multiple-objective decision functions and linguistically quantified statements

Int. J. Man-Machine Studies (1984) 21,389-400

General multiple-objective decision functions and linguistically quantified statements RONALD R. YAGER

Machine Intelligence Institute, Iona College, New Rochelle, New York 10801, U.S.A. (Received 5 March 1983) The concept of linguistically quantified propositions is used to develop a whole family of forms for the representation of multiple-objective decision functions.

Introduction Linguistically quantified statements are propositions of the form "Most people are tall" or "Several fat people are old". These types of statements provide a means of aggregating information. As Yager (1983)a, b, c) noted, there exist at least two ways of aggregating information via quantified statements. The first method, which corresponds to a consensory or summarization type aggregation, corresponds to the approach suggested by Zadeh (1978, 1983) and has been called the algebraic approach to aggregation by Yager (1983a). The second method, which was developed by Yager (1983a), corresponds to a competitive type aggregation and is accomplished via the substitution method. The competition type aggregation is very useful in situations in which the truth of the aggregated statements is highly dependent upon all the constituents being satisfied. This type of aggregation is very prevalent in multi-objective decision-making in the form of statements such as "Most objectives are satisfied by x". The consensory or summarization aggregation is very closely related to probability theory. In this short article a general formulation is provided for competitive type aggregations such as those found in multiple-objective decision problems. In these situations the quantifier is implicitly considered a monotonically non-decreasing fuzzy quantifier,t 50% of the objectives are satisfied is implicitly saying "At least 50% of the objectives are satisfied".

Problem formulation As a prototypical example of the use of competitive aggregation we can consider the multi-objective decision problem. Let X be the collection of potential solution alternatives to our problem. Let E = {E,, E2, 9 En} be a collection of objectives which are t A fuzzy quantifier Q is a fuzzy subset of the real numbers representingsuch concepts as "about seven", "most", "more than 50%". It is called monotonicallynon-decreasingif x > y ~Q(x)---Q(y). 389 0020-7373/84/110389+ 12503.00/0

(~ 1984Academic Press Inc. (London) Limited

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of concern in the decision. As has been well established, one could represent each objective as a fuzzy subset of X such that for each x c X, Ei(x) indicates the degree to which x satisfies the criteria specified by Ei (Bellman & Zadeh, 1970; Zimmerman, 1975). One could then formulate an overall decision objective function D as a fuzzy subset of X. This can be stated as the requirement that Q objectives are satisfied by the solution. Thus, for each x c X this overall function becomes the linguistic statement D(x) = Q objectives are satisfied by x where Q is a linguistic quantifier. Formally, this can be written as D(x) = Q E's are Ax, where Ax is the fuzzy subset "satisfied by x". We can then represent A~ as a fuzzy subset of E where, Ax(E,) = E,(x) indicates the degree to which Ei is satisfied by x. An enhancement to this situation would involve the case where associated with each E~ e E we have a measure of how important that objective is to the decision maker. Let b~ ~ [0, l] be the importance of objective Ei. Then the overall decision criteria D can be represented by the proposition D(x) = Q important objectives are satisfied by x. Formally this can be written as D ( x ) - - Q BEs are A~, where B is a fuzzy subset such that B(E~) = b,. In either case, for each x, D(x) becomes the truth or validity of the associated quantified proposition. Having obtained D(x) for each x c X, by a procedure to be discussed shortly, the optimal solution x* can then be selected as D(x*) = max D(x). xEX

In the following the procedures will be described for obtaining the validity of a generic quantified statement of the form Q Es are A or Q BEs are A, which are the constituent elements of the above approach. Let E = {El, E 2 , . . . , E,} be a collection of objects, such as objectives in a decision. Let Q be a monotonoically non-decreasing quantifier, such as most, at least 5, at least 50%, etc. As suggested by Zadeh (1983) there exist at least two kinds of these quantifiers. The first kind are called absolute quantifiers; examples are, at least three, all, at least one, etc., these quantifiers can be represented as fuzzy subsets of the non-negative numbers. The second kind are called proportional or relative quantifiers; examples are, most, at least half, almost all. These quantifiers can be represented as fuzzy subsets of the unit interval. Here QI and Q2will be used to indicate kind one and two quantifiers, respectively. Q will be used in situations where it is not necessary to distinguish between the two different types of quantifiers. Thus, in general a quantifier is a mapping Q: R-~ [0, 1].

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It is called monotonically non-decreasing if rt > r2=~Q(rl) -> Q(r2). There exist two basic formulations of quantified statements: Q Es are A

I

Q (BEs) are A.

II

and

Distinguishing between the kind 1 and 2 quantifiers four formulations are obtained: QI Es are A,

I'

Q2 Es are A,

I"

QI(BE)s are A

II'

Q2(BE)s areA.

II"

and

In the above, A and B are fuzzy subsets of E. As noted earlier, a prototypicai example of the Type I class of statements is "Most objectives are satisfied by x". In this case E = "set of objectives", A = "'satisfied by x", where A(E~)= degree of satisfaction of Ei by x, E(xi) and Q = "most". An example of the Type II class of statements would be "Almost all important objectives are satisfied by x". In this, B = " i m p o r t a n t objectives", described as a fuzzy subset of E, where B(Ei) is the importance of Ei. Parentically, it is noted that for each x, B could be different. The problem of interest is given a quantified proposition P, involving a Q, E, A and, perhaps, B, determine the truth of the proposition, which will be denoted as V(P). In the multi-objective decision problem one would have a whole collection of propositions {P~}, one for each x c X and in order to determine D(x) = V(Px), V(Px) is needed, for all x e X. In the following only the generic case P will be considered. Generally, one can consider a quantified statement P as true if there exists some subset C of E, called a considered or selected set of elements, such that the following apply. (!) The number of elements in C satisfies the condition of Q (and perhaps B), that is, there are Q elements in C. (2) Each element in C satisfies the property A. That is, for each Ei ~ C, A(Ei) is true. It is this second condition which imposes the competitive aspect of the aggregation for we desire all elements in C to satisfy A. Thus for any quantified statement P one can consider any subset of E, any C c E and evaluate the degree to which it satisfies the statement P, defined by the above two considerations. This satisfaction is denoted as Vp(C). The overall validity or truth of

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the quantified statement P, V(P), then becomes the best for any C, hence V(P) = Max {Vp(C)}. C~2 E

The means for implementing this idea will be considered for the four different categories of quantified statements separately. Before proceeding, some ideas from multivalued logic must first be introduced.

Multivalued logic operators A t-norm, T, is a binary operation which generalizes the " a n d " operation in multivalued logic (Dubois, 1981; Yager, 1982) (a " a n d " b =T(a, b)), T:[0, 1] •

1]-~ [0, 1]

such that (1)

T(a,

(2)

T(a, b) =T(b, a),

(3)

T(a, b)>-T(c,d) if a>-c and b>-d and

(4)

T(a, b, c ) = T(a, T(b, c))= T(T(a, b), c).

1) =

a,

Examples of T norms are (1)

T(a, b ) = Min (a, b ) = a ^ b,

(2)

T(a, b) = a.b and

(3)

T(a,b)=Tp(a,b)=l-Min(l,((1-a)P+(l-b)P)'/P),

p>-l.

A t co-norm S is a binary operation which generalizes the "or" operation in muhivalued logic (Dubois, 1981 ; Yager, 1982) [a "or" b = S(a, b)], S:[0, 1] •

1]~[0, I].

S satisfies properties 2-4 of the above, plus 1. T(a, 0) = a. Examples of t co-norms are (1)

S(a, b) = Max (a, b) = a v b,

(2)

S(a,b)=a+b-a.band

(3)

S(a,b)=Sp(a,b)=Min(l,(aP+bP)'/P),

p>-l.

In general, for any t norm T there exists a dual t co-norm S such that S(a, b ) = 1-T((I-a), (I-b)). The operation I will be used to indicate a generalized implication or ply operation in multivalued logic, [(a ~ b) = I(a, b)] (Yager, 1980; Bandler & Kohout, 1980):

I:[o, I] •

1]-, [o, I].

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Such that (i) I(a, I ) = 1, (2)

I ( O , a ) = 1,

(3)

I(1, a) = a,

(4)

I(a, b)->I(a, c) if b->c and

(5)

I(a, b)->I(c, b) if a-< c.

Some examples of I operations are (1)

l ( a , b ) = b a and

(2)

I(a, b ) = S ( ( 1 - a), b), where S is any t co-norm.

General competitive solution Each of the four categories of quantified statements will now be considered in turn. (I) P = " Q i Es are A " Recalling that A is a fuzzy subset of E and Q is an absolute fuzzy quantifier, a subset of R. For any selected subset C of X, the degree of validity of "Q, Es are A", Vp(C), is equivalent to the degree of validity of "Qt Es are in C" and "for all E~s e E if E~ is contained in C than A is satisfied by E~". Thus Vp = V(Q, [C) and V(AIC) = T~(V(QI [C), V(AIC)).

V(Qt [C) is the truth of the statement that Q Es are considered in the set C, w(mlc) is the truth of the fact that for all E~ c E if C(E~) than A(Ei) and Tl is any T-norm. Let ]C] be the cardinality of the set C, the number of elements in C, then V(QI [C) = Ql (C) where Q,(IcI) is the membership grade of the number of elements in C, IcI, in the fuzzy subset Q~. This follows directly from the idea of compatibility (Bellman & Zadeh, 1977). The truth of the statement for all E~s ~ E if C(E~) then A(Ei), V(AI C), is obtained as

V(AIC) = (Cl-, a,) and (r

a2) a n d . . . ( c , - * a , ) , n

V(AIC)-- T (I(c,, a,), i~l

where ci =C(E,), a~ =A(E~), I is any implication operator and T is any t-norm. Since c~ = 1 if E ~ C and c~ =0 if E ~ C and from the facts that I (1, a~) = a~ and I (0, a,)= I and T (1, a ) = a,

V(AIC)- T [A(E,)] E~eC

Therefore Vp(C)=T~(Q~(ICI), T (A(Ei))). EI~C

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Hence

V(p)=Max[Vp(C)]=Max[TI(QI(ICI),T (A(Ei))) ]. C~E

CcE

E~eC

In the following it will be assumed the elements A(Ei) are arranged in descending order, i < j ~ A ( E , ) -> A(Ej). THEOREM. Assume the elements {ACE1), ACE2),..., ACE,) are arranged in decreasing

order, A(Ei)>-A(Ej) if i
Max

i = 1,2,...,n

T~

l(i),j~ (A

.

Proof. Let C ~ be the collection of all subsets of E with i elements. Q

For any set C with i elements TEj~c[A(Ej)] is the largest if C = {El, E2,. 9 9 E~} because of the assumed ordering. Hence V(P)= Max

i = 1,2,...,n

1

1

~j

Corollary. If T~ = T = Min, then V(P)= Max [Q,(i) AA(E,)], i = 1,2,...,n

where Ei is the ith largest element in A.

Corollary. If T~ = T = product, V(P)= Max [Q,(i) "jOl (A(E~))], i = 1,2...,n where A(E~) -> A(E:) - . . .

--- A(E,).

In the framework of multi-object decisions where for each x, A(Ei) = Ei(x) we get, for T1 = T = Min, D(x) = Max [Ql(i) ^ E~(x)], i = 1,2,...,n

and E~(x) is the ith largest element in the set {El(x), E2(x) . . . . . En(x)}. For T1 = T = product, D(x)=

Max [Ql(i). l~I (Ej(x))].

i = 1,2,...,n

j=l

THEOREM. For the quantifier "at least one" then, for any T and T~,

Vp= Max A( E,) = A( EI). / = 1,2,...,n

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Proof. For the quantifier "at least one",

Max T, V(P) = i=l. 2......

Ql(r)-- 0

if r < l ,

Ql(r)=l

if r > ! and

(A(Ei)

= Max (A(EI) i=l.2...... j=t

I

=A(E,).

Thus, for multiple-objective decision problems the quantifier "at least one solution" always gives us the form D(x)=

Max [El(X)].

i = 1,2,...,n

THEOgEM. For the quantifier, Q~ = "All", for any T~ and T, n

V(p) = T [A(E,)]. j=l

Under the special cases: T=min, V(p)=

Min

A ( E i ) = A ( E . ) and

i = 1,2,...,n

T = product, V ( p ) = fi A(Ei). i=1

Proof For Q~ = "All", Ql(r)=l

for r = n and

Qt(r)=0

forr
Since

[(

Max T (A(Ej)) V(P)=i=l.2 ...... Tl Q.(i), j=~

,

V(p) = T (A(E,)). i=l

In the framework o f multiple-objective decision problems the use of "for all" leads to the following forms: D(x)=

Min

[Ei(x)],

when T = m i n and

i~l,2,...,n

D(x) -~ fi El(X),

when T = product.

i=1

Note that Mini=l,2 ...... [Ei(x)] is the form originally suggested by Bellman & Zadeh (1970).

( H ) P = " Q2 Es are A" For any selected subset C o f X, the validity o f Q2 Es are A is equivalent to the validity of "Q: Es are in C" and " F o r all Eise E if Ei is contained in C than A is satisfied by

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Ei". Thus,

ve(c) = V(Q21C) and V(AIC) = T,(V(Q21C), V(AIC)).

V(AIC) is calculated as before, but it must be shown how to obtain V(Q21C)with Q2 a proportional quantifier. Let ]CI be the cardinality of the set C, let n be the cardinatlity of the set E, then V(Q21 c)

=

Q2(Icl/n).

Using the results of the preceding part, for any t-norm T, where T~ = T, V(p) = Max [T(Q2(il n), A(E~), A ( E 2 ) , . . . , A(E,))]. i= 1,2,...,n

P = "Q, B E s a r e A " For any selected subset C of X, the validity of P, Vp(C), is equivalent to the validity of "Q~ BEs are in C " a n d " for all E~s e E, if Ei is contained in C and E~ is B then A is satisfied by E~". Thus, in this case, (III)

vp(c)= V(Q,IB, C) and

V(AIB, C)

= T(V(Q~ IB, C), V(AI B, C)), where V(QIIB, C) is the truth of the statement that Q~BEs are considered in the set C, and V(A1B, C) is the truth of the fact that for "all Ei e E if C(E~) and B(E~) then A(E~)". T~ is any t-norm. To calculate V(Q~IB, C), the truth that Q~ BEs are in the set C, one proceeds as follows: Q, BEs in the set C ~ Q ~ Es are B and are in C. Q~ Es are B and C. Let H=BnC, Since C is crisp any t-norm T2 can be used for N and the same result is obtained: H(E,) = T2(b,, c,) = b~ ^ ci = bi. c, = hi, where the notation B(Ei) = b, and C(E~) = ci and H(E~) = b, is used. Since H is a fuzzy subset, there exists a number of methods for calculating the cardinality of H and in turn the degree of truth that there are Q~ Es in H. The simplest way of calculating the cardinality of H is to form the sigma count of H (De Luca & Termini, 1972), thus

Inl = ~ hi = ~ c,^ b,. i=1

i=1

In this situation V(Q1 I B, C) = Q~(IHI), the membership grade of IH[ in Q~, which is a crisp number.

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397

The truth of the statement for all Eis~ E if C(E,) and B(Ei) then A(Ei) is obtained as follows: n

V(A[B,C) = Ta(l(h,, a,)), i=1

where hi = T2(ci, bi), and a~ = A(Ei) and respectively. Thus, in this situation,

Letting T~ =

T 2 =- T 3 =

T3

and I are any t-norm and implication,

Min = ^,

Vp(C)=Q,(L,~,

c,^bi) ^l(c,^ bl, a,)A l(c2^b2, a2)^...,

Vp(C) = Q, ( ~~, t ci^b,)^ i =Man, .2.....n

[I(c,^b,,a,)].

If I is selected, such that I(x, y) = yX, V~

^ ,c'^. bi) ,

i= ,.2.

[a~,^b,'].

Selecting I(x, y) = ( 1 - x) v y,

VP(C)=Q'( L ci^bi) Selecting

T t =

T2 =

T3 =

,=,.2.....,min[((l-(ciAbi))^ai)].

product and I = y ~,

In all the above, a~ = A(Ei), b~---B(Ei) and c~ = C(E~). Note that since V(P) = Max [Vp(C)], Cc_E

this can be formulated as an integer programming problem. For example, with T~ = T2 = T3 Min and I - 3 , V(P) = Max

j

i

I

ci ^ bi ^ i = Min [a~ '^b' 1,2,...,n

,

such that cic{0, 1} i = 1 , 2 , . . . , n.

(IV)

P = "Q2 BEs are A" For any selected subset C of X, the validity of P, Vp(C) is equivalent to the validity o f "Q2 BEs are in C " a n d " for all Eis ~ E if Ei is contained in C and E~ is B then A is satisfied by E,".

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Thus, again V~(C) = V(Q2 IB, C) and V(A IB, C), where V(Q21B, C) is the truth of the statement that Q~ BEs are considered in the set C and V (A/B, C) is that truth of the statement for all Ei e E if C(E~) and B(E~) then A(E,). To calculate V(Q21B, C) one may proceed as follows: Q2 BEs in the set C=>Q2 BEs rl n are in the set BNC. Let H = B f - ) C , let r~ =Y-i=l hi and let r2=Y.~=l b~, then V(Q21 B, C) = Q2(rl [r2). The calculation of V(AIB, C) is similar to the previous case, giving

,T3(I(T2(c,, b,), a,))

Vp(C) = T . Q2 ,=1

,E b, t Again the general formulation is V(P) = MaXccE [Vp(C)]. Note that with T 1 = T 2 = T 3 = ^ and I(x, y) = yX,

V(P)=Maex

l^

Q2 '=-I~ -

Min i = 1,2,...,n

[a~'^~']t With T~ = T: = T3 = product and I(x, y) = yX ..

vp = Max

9

2

CCE

9 11 ,,~,.c, . i= I

L \,z

/

For the quantifier Q2= All, Q2(r) = 0 if r < 1 and Q2(r) = 1 if r_> 1, Vp = i

(l(b,, a,)].

/=1

Under this situation, with I(x, y) = yX, Vp=

Min [a~,]

ifT=Min,

i = 1,2,...,n

Vp = I~I a~,

if T = product.

i=1

These are the forms that were originally suggested by Yager (1978). With Ti = Tz = T3 = Min and Q2 = "all" and I(x, y) --- Max ( 1 - x, y), Vp=

Min i=1,

[(l-bi)va,],

2,...,n

which is the form suggested by Yager (1981).

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More generally, with Q2 = " a l l " the following is a general set of multiple objective formulations:

T

D(x) =

[I(b,, A,(x))],

i= 1,2,....n

where T is any t-norm operator, I is any implication operation, b~ is the importance of objective i and A~(x) is the satisfaction of objective i by alternative x. The following result established the relationship between quantified statements with importances and those without. THEOREM. The general solution for Q Es are A is a special case of Q BEs are A with all objectives assumed to have importance l, B( E,) = bi = I. Proof. For Q BEs are A, V,(C) = T .

=~ T2(c,, b,) ,

Q, i

I

b,), a,)) . "=

With bi = 1, Vp(C) = TI

ci , Ta ( I C . G, au)

i i

I

,

i=l

which is the general solution for Q Es are A. For Q = Q2 the same result can easily be shown.

Conclusion Using the concept of linguistic statements, a means of formulating a general class of multiple-objective decision functions has been provided. A question that deserves further study is the effect of the choice of the t-norm operator and the implication operator in this formulation.

References BANDLER, W. & KOHOUT, L. J. (1980). Fuzzy power sets and fuzzy implication operations. Fuzzy Sets and Systems, 4, 13-30. BELLMAN, R. E. & ZADEH, L. A. (1970). Decision making in a fuzzy environment. Management Science, 17, 144-164. BELLMAN, R. E. & ZADEH, L. A. (1977). Local and fuzzy logics. In DUNN, J. M. & EPSTEIN, G., Eds, Modern Uses of Multiple-Valued Logic, pp. 103-165. Dordrecht: D. Reidel. DE LUCA, A. & TERMINI, S. (1972). A definition of non-probabilistic entropy in the setting of fuzzy sets theory. Information and Control, 20, 301-312. DUBOIS, D. (1981). Triangular norms for fuzzy sets. Proceedings of2nd International Seminar on Fuzzy Sets, Linz, pp. 39-68. YAGER, R. R. (1978). Fuzzy decision making including unequal objectives. Fuzzy Sets and Systems, 1, 87-95. YAGER, R. R. (1980). An approach to inference in approximate reasoning. International Journal of Man- Machine Studies, 13, 323-328. YAGER, R. R. (1981). A new methodology for ordinal multiple aspect decisions based on fuzzy sets. Decision Sciences, 12, 589-600.

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YAGER R. R. (1982). Generalized probabilities of fuzzy events from fuzzy belief structures. Information Science, 28, 45-62. YAGER R. R. (1983a). Quantified propositions in a linguistic logic. International Journal of Man-Machine Studies, 9, 195-227. YAGER R. R. (1983b). Aggregating evidence using quantified statements. Technical Report # MII-301, Machine Intelligence Institute, Iona College. YAGER R. R. (1983c). Quantifiers in the formulation of multiple-objective decision functions. Information Science, 31. ZADEH L. A. (1978). PRUF--a meaning representation language for natural languages. Inter. national Journal of Man-Machine Studies, 10, 395-460. ZADEH L. A. (1983). A computational approach to fuzzy quantifiers in natural languages. Computers and Mathematics with Applications, 9, 149-184. ZIMMERMAN, H. J. (1975). Description and optimization of fuzzy systems. Journal of General Systems, 2, 209-216.