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A model of fuzzy team decision
Fuzzy Sets and Systems 2 (1979) 201-212. © North-Holland Publishing Company
A M O D E L OF F U Z Z Y T E A M D E C I S I O N H. Nojiri Department of Systems Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, 152, Japan This paper formulates a fuzzy team decision problem. The concept of fuzzy sets is introduced to formulate the team decision processes which contain fuzzy states, fuzzy information functions, fuzzy information signals, fuzzy decision functions and fuzzy actions. After a description of the basic notion of team theory, a model of fuzzy team decision is proposed.
Key words: Decision theory, Fuzzy decisions, Team decisions
1. Introduction Since its introduction by Zadeh in 1965 [15], the concept of fuzziness has been extended to algorithms, learning theory, automata, formal languages, pattern recognition, probability theory, information retrieval and the decision making process [5, 10]. The application of fuzzy set approach to special problems in decision making was suggested in 1969 [4], 1970 [1], 1975 [6] and 1976 [14]. For a survey on the subject we refer to Negoita and Ralescu [10]. It is our purpose here to introduce the notion of fuzzy sets defined by Zadeh [15] to the theory of teams by Marschak and Radner [7]. The Paper is divided into three parts. The first will describe the basic concepts of team theory. In the second part we will propose a model of fuzzy team decision. In the third part we will define the concept of a fuzzy information function and a fuzzy decision function.
2. The team decision problem Let a team consist of n members. Let X = {x} be the set of mutually exclusive, uncontrollable states of the environment. The uncertainty about x is expressed by a probability distribution on X. Let us assume that the set X is finite and X does not contain any states that have zero probability. An information received, or an observation made, by a team as a whole, called a team information, has n components. Denote the team information by y; then y = ( Y l , . . . , Y,), where y~, i = 1 , . . . , n, is ~the part of the team information received by the ith member and it is related to the state of the environment x by a function ~i; that is, if the environment is in state x, then member i will have information yi = ~(x). Let Y~ = {Yi} be the set of possible alternative information ~ignals that can be received by the ith member, and let Y={y} be the set of all possible alternative 201
H. Noiiri
202
information signals that can be received by the team: then { y } { y t } x { y 2 } x . . . x{y,}, Y-----X~=IYi. ±m action taken by a team as a whole, called a team action, also has n components; then a = ( a l , . . . , o~), where a~, i = 1 , . . . , n, is the action taken by the ith member. Let A~ = {o~} be the set of feasible actions that can be taken by the ith member, and let A = {a} be the set of feasible team actions that can be taken by the team as a whole; then {a}={al}x{az}x'.'x{a,,}, A----X'~=~A~, [7,13]. The ith member of a team takes his action a~ on the basis of his information signal, y~; thus a~ = o~(y~). The symbol a~(y~) denotes the action prescribed b.¢ the decision function t~ when signal y~ is received; then a decision function a~ is a function from Y, to the set A~ of feasible actions. T' -. a-tuple of decision functions can be denoted by a = ( ~ , . . . , a , ) and called the decision function (or decision rule) of the team; a is a function from Y to A ; then a = a ( y ) . The n-tuple r ) = ( ~ . . . . , ~!,) may be called the information structure of the team. ~i is a function from X to Y; then y = ri(x). Again in condensed notation, y = 71(x), a = a ( y ) = a[~l(x)], E7]. The expected utility of the team resulting from the use of an information structure ~1= (TIx,..., ~1,) and a team decision function a = ( a x , . . . , a~) is a ; to,
= F
(x,
= -~xto(x, a t [ ~ l l ( x ) ] , . . . ,
a,[vl,,(x)])¢r(x),
(1)
where to(-, • ) mapping from X x A to the real line denotes the utility function of the team and ~r(. ) mapping from X to [0, 1] denotes the probability function. It is assumed th: t all members have the same utiqty function and the same probability function. O(vl, a ; to, '.~) is a function of ~, a, to and ~r. It is assumed that the utility and probability function used are the true ones, thus they are not decision variables, as are 71 and a, and hence, for simplicity, we shall write D(~I, a ) in place of n ( ~ , a ; to, ~r). The problem then becomes one of choosing, for a given information structure ~', the best decision rules, that is the a = a * for which 1~(~') = Max 1~(~1', a).
(2)
ot
After each information structure is evaluated in terms of the best decision function that can be used with it, the obvious conclusion is to use the information structure, ~*, that maximizes l~(vl). The optimal pair of vecl::ors (~*, a * ) is called the optimal organization form [8]. These concepts are the basic ones of team theory which we shall need for our formulation. For more detailed discussions on the theory of teams, the reader may refer to the work of Marschak and Radner [7].
3. Formulation of fuzzy team decision problems In the team decision problem, the information variable y~ of the ith m e m b e r of a team is related to the state of the environment x by a function ~i. For a 6×ed
A model
of fuzzy team decision
203
information function r/i, each signal Yi is identified with a subset of X, namely, the set of states x for which qrh(x) = YiBy introducing the notion of fuzzy sets defined by Zadeh [1, 15, 17]. let us assume that each team member i receives a fuzzy information signal Yi about the state of the environment x. Then, each fuzzy information signal Yi (an element of Yi) is identified with a fuzzy subset of X, the set of states of the environment. For each member i, i = 1 , . . . , n, le~ Ci be a fuzzy set of information signals in 'the information space Y~ of the ith member. Then a fuzzy subset Ci in Y~ is characterized by a membership function /ac," Y~--* [0, 1] which associates with each information signal y~ of Y~ a, number /z~(yi) in the interval [0, 1] which represents the grade of membership of Yi in C~. Fuzzy sets C ~ , . . . , C, are defined by Ci = {(yi, ~c, (yi))},
yieY,
i---1,...,n
(3)
where the symbol i denotes the ith member of the team, Ci denotes a fuzzy set of information signals and n denotes the number of members in a team. An information tunction ~1~ of the ith member is a function from X to Y~ = {y,}. Then, the inverse mapping rl~-~(Ci) induces ~: fuzzy set B~ in X. For each n: :nber i, i = 1 , . . . , n, a fuzzy set Bi is defined by B, = {(x, ttB,(x))},
x~X
(4)
ttB, (x) =ttc, (yi),
y, E Y,
(5)
for all x in X which are mapped by rh into Yi. Let us consider now a conv,~rse problem and for each member i, i = 1, . . . . n, let Bi be a fuzzy set in X. Then, the information function ~1~ induces a fuzzy set Ci of information signals in Y~ whose membership function is given by tt~(yi) =
Sup
{ttB,(x)}
= 0,
if n~-~(),i) is not empty
(6)
otherwise,
where rl~-l(y~) is the set of points in X which :~re mapped by -q~ into y~. Let a fuzzy set B be the union of fuzzy sets B ~ , . . . , E,. Then, a fuzzy subset B in X is defined by B = B l t.) B 2 LI- • • U B . B = {(x,
x
(7) X
ttB (x) = max(~B,(x),..., tLm (x))
(8)
or, in abbreviated form ttB (x) =/xB~lx) v/zB~(x) v - - " v ttB. (x)
(9)
where the disjunction symbol v means taldng the lnaximum of tcB,(x), i = 1 , . . . , n. B denotes a fuzzy set of states, x, of the environment which are observed by the team as a whole. In order to define a fuzzy set of informaticn signals to the team, let C be the Cartesian product of n fuzzy sets C ~ , . . . , C,. Then, by applying the definition in
204
H. Noiiri
Ref. [19], C - C~ x - - • x C,, is defined by
C =--{(y, ~c(Y))},
ye Y
g,c(y) = ~ c ( y , , . . . ,
y~),
(10) y, • IT,,
i = 1,...,
n
= min[p,c,(y,),...,/~c.(Y.)]
(11)
or, in abbreviated form / ~ ( y ) - / ~ , c . ( y , ) A ~ ( y 2 ) A " " "A/J,c~ (Y.)
(12)
where the conjunction symbol ^ means taking the minimum of /~c,(Y~), i:1 . . . . , n and { y } = {Yl} x . - - x {y,}, Y = X[~=l Y~. We suppose that fuzzy sets C 1 , . . , , C. are noninteraeting. T h e notion of noninteracting of fuzzy sets corresponds to the notion of the independence of random variables [18]. The Eqs (10) and (11) mean that C = C l x . - - x C , is a fuzzy set of ordered pairs y = ( y ~ , . . . , y~), y~ ~ Y~, i = 1 , . . . , n, with the grade of membership of Y = ( Y ~ , . . . , Y , 0 in Y = Y ~ x - . - x Y , , given by ~ c ( y ) = ~c,(yl)A- • • ^ gc~(y,). In th~s sense, C = C~ x . • • x C, is an n-ary fuzzy relation in the product space Y = Y~ x . • • x Y , . C denotes a f u z z y set of ~infoivnation signals, y = ( y ~ , . . . , y,), to the team. In the team decision problem, the ith member of a team takes his action a~ on the basis of his information, y~; thus a~=m(y~). A decision function m is a function from Y~ to the set A~ of feasible actions. The n-tuple of decision functions can be denoted by a = ( a ~ , . . . , t~,) and called the decision rule of the team; a is a function from Y to A ; then a = a(y). Thus, let G be a fuzzy set of information signals in the information space Y~ of the ith member. Then, the decision function a, induces a fuzzy s e t / ~ of actions in the action space A, of the ith member. For each member i, i = 1 , . . . , n, a fuzzy set D~ of actions is defined by D, = {(a,, go~(a~))},
(13)
a,~A,
~o,(a~)=
Sup {/~q(Yi)} if aT~(ai) is not empty, y,~c'(~,~ = 0, otherwise,
(14)
where the supremum is taken over the set of points aT~(a~) in Y~ which are mapped by a, into a~. Let D be the Cartesian product of n fuzzy sets D ~ , . . . , D , , then D = D1 x D2 x . • • x D , is defined by D={(a,l~o(a))},
(15)
a~A
/ ~ D ( a ) = l L D ( a l , . . . , an), ~A~, = m i n [ ~ o , ( a ~ ) , . . . , t~o. (an)]
i= 1,...,
n
(16)
or, in abbreviated form /zo(a) = g,D,(a~) ^ ' ' " A btD. (an)
(17)
where the conjunction symbol ^ means taking the minimum of ~o,(a~); i = 1 , . . . , n, {a} = {a~} x . . . x {a~}, A ~ X,"_-, A~.
A model of fuzzy team decision
205
D denotes a fuzzy set of actions, a = ( a ~ , . . . , a , ) , in the product space A = A~ x . • • x A , which are taken by the team as a whole. In order to decide a set of actions to the team, a pair of thresholds may be introduced as follows [15]: 3" > ~ where 0 < 3", 6 < 1 which leads to a three-level logic as (1) a belongs to D, or 'true' if t~o (a) ~> 3' (2) a does not belong to D, or "false' if txo(a)~<6 (3) a has an indeterminate status relative to D, or ' u n d e t e r m i n e d ' if 6 < t ~ o ( a ) < 3', where D is a fuzzy set of actions to the team. Let us assume that the team takes a s e t / 9 , of actions defined by
(18) Then, the problem becomes one of choosing, for a given set Dr, the best action, that is the a - a* for which O(a*) =
Max O(a)
(19)
a~D,
O(a)= ~xoJ(x, a)'a'(x)
(20)
where O(a) denotes the expected utility, ¢o(., • ) mapping from X >"A to the real line denotes the utility function of the team and ~r(- ) mapping from X to [0, 1] is the probability function. The expected utility O ( a ) is said to represent the .<
.<
preference ordering K if for all actions ai and aj for i ~ j in Dr "ai if, ai if and only if [~(ai)<~O(ai). If the state of the environment is fuzzy and B is a fuzzy set in X, then Eq. (20) is replaced by
g~(a) -,Y,~oJ(x, a)t~n(x)Tr(x).
(21)
In Eq. (21), the utility o~(x, a) is scaled down in proportion to txB(x) in B and Eq. (21) expresses the expected t~tility over B [20]. If the team takes a fuzzy se'~ D of actions and B is a fuzzy set of states observed by the team, then the maximal expected utility of the team, given its information structure rl', is the expected utility maximized with respect to a" /~(,i') = Max O(~l', a )
(22)
O(~1', ,:~)= Zxco(x, a[rl'(x)])t~B×o(x, a[rl~(x)])Tr(x)
(23)
= Zx,,,(x,
where O(rl', a ) denotes the expected utility over the Cartesian product of B and D.
4. F~zy information hmct~ons and fuzzy derision functions In the theory of teams [7], information can be regarded as the outcome of information-gathering. For each member i, i = 1 . . . . , n, a given method of
206
H. Noiiri
information-gathering applied to the true state of environment x results in a particular signal Yi- To each x in X will con'espond a signal y~. Thus, to a given information-gathering method is associated a function ~ from X to Yv Let us assume that each m e m b e r i uses a given fuzzy method of informationgathering, namely, a fuzzy mapping ~1~ from X to Y~. Then, by applying the definition in [5], for each m e m b e r i, i -~ 1, .... , n, a fuzzy mapping ~li is defined as: Definition. A fuzzy mapping "q~ from X to Y~ is a fuzzy set on X x Y~ with membership function pa,(x, Yi)A fuzzy informagon function .q,(x) is a fuzzy set on the information space ~ of member i with membership function Pn.oo(Yi) = Pn,(x, yi),
i = 1 , . . . , n.
(24)
Its inverse r171(y~) is a fuzzy set on X with Pn,-'tyO(x) = Pn,(x, Yi),
i = 1 , . . . , n.
(25)
For each i, a fuzzy information function rh is said to be liner than a fuzzy
information function O~ if p~,~)(y~) ~ po,~(y~),
'¢x,
y~ e,X; Y~,
i = 1 , . . . , n.
(26)
Theorem 1. If rl~ is finer than O~, nT. ~ is finer than OT~. lProo|: T h e proof is trivial and omitted [5]. For each m e m b e r i, i = 1 , . . . , n, let B~ be a fuzzy set on X. Then, T h e fuzzy mapping ~ii induces a fuzzy set C~ of information signals in Y~ which is defined as C, = {(y,, Pc, (Y,))},
Y, ~ Y,
(27)
pc.(y~)-Sup(~a.(x)^p..(x. Y,)),
i= 1,...,n.
(28)
x~X
Let C be the Cartesian product of n fuzzy sets C ~ , . . . , C,. Then, C = Ci x C2 x- • • x C, is defined by C = {(y, Pc(Y))},
Y~ Y
Pc(Y) = P c ( Y , , - - - , Y,),
(29) Yi e Y~,
i= 1,..., n
(30)
= min[pc,(yx),. • •, Pc. (Y,)] where p~(y~), i = 1 , . . . , n, are given by the Eq. (28). C denotes a fuzzy set of information signals, y = ( y l , . . . , y,), to the team. T h e o r e m 2. For each i, i = 1 , . . . , n, let Bk, B~ be two fuzzy sets on X. Then
if
Bk D_Bi,
"rl~(Bk) ~ ~ (Bt)
n,(Bk 0 B~) = n~(Bk) O n~(B~) n~(Bk n B~) c_ n~(Bk) n n~(B~).
if rh is liner than 0~, then
Vh( Bk ) c_ 0~( Bk ).
(31) (32) (33) (34)
A model of fuzzy team decision
207
Proof of (32). Let H = Bk U B~. ~,~,u4)(Y,) = Sup(/.%~ (x) A/~,~, (X, y,)) x~X
= Sup((p,a~ ( x ) v p,a, (x)) ^/x,~, (x, y,)) Jc~X
= Sup[(/~a~ (x) A tt,~, (X, y,)) v (tta, (x) ^ ~,~, (x, y,))] x~X
= Sup(/x m (x) A p,,~,(x, y,)) v Sup(/xa, (x) A/~., (x, yi)). x~X
x~X
Proof of (33). Let K = Bk ¢'1B~, then
/.I,n,(K){Yi)-- Sup(KK (X) A/d,n, (X, Yi)) xEX
= Sup(/xsk (x)^ lgm(x) ^ lxni(x, y,)) xEX
~
x~X
= tL,,,~ao(y~) ^ w,~,ts~(y~). The proofs of (31) and (34) are trivial and omitted [5]. For each member i, i = 1 , . . . , n, a fuzzy decision function ai is defined as"
Definition. A fuzzy mapping a, from Y~ to Ai is a fuzzy set on Y~ x A, with membership function / ~ (y~, a~). A fuzzy decision function a~(y~) is a fuzzy set on A~ with membership function tt,,,t~.~(a,) = tL,,,(y,, a~),
(35)
i = 1 , . . . , n.
Its inverse a~-t(a~) is a fuzzy set on Y~ with
/z~, ,to~)(y~)= tt,~, (y,, a~),
i = 1,...,
n.
(36)
A fuzzy decision function a~ is said to be finer than a fuzzy decision function [3~
if /x=,~,~(a~)~ < t~a,
Vy~, a~ ~ Y~, A,,
i = 1 , . . . , n.
(37)
T h e o r e m 3. If a~ is finer than [3~, Ol? 1 is finer than [3=,! Proof. The proof is trivial and omitted [5]. Let C~ be a fuzzy set of information signals in the information space Yi. Then, a fuzzy set D~ of actions in the action .~pace Ai of member i is defined as
I.co,( a, ) = /.h~,co(a, ) -- Sup(ire, (y~) ^ ~ , (y~, a~)),
(38) i=l,...,n.
H. Noiiri
208 Theorem
4. For each i, i = 1 , . . . ,
n, let Ck, C~ be two f u z z y sets on Y~, then
a~(Ck U Cz) = a~(C~,) U a~(C~)
(39) (40)
a~(Ck N Ct) G a,(Ck) n a~(C,).
(41)
If ai is finer than/3i, then
(42)
if
Ck ~--C~,
ai(Ck) ~_a~(Cz)
t~(C~ ) c_ [3i(Ck).
If Ck ~- C~ and ai is finer than ~i, then
ai (Ca) c_ [3i(Ck).
(43)
ProoL The proof is analogous t o the proof of Theorem 2 and omitted [5]. Let D be the Cartesian product of n fuzzy sets D ~ , . . . , / 9 , , , 0 1 X 0 2 X . • • X O n is defined by D = {(a,/zo(a))},
aeA
Izo(a)= lxo(a~' " " ' a") =min[po,(al),...,/Xo.(ah)],
then D = (44) (45)
a~ ~Ai,
i = 1,..., n
where /zo,(a~), i = 1 , . . . , n, are given by the equation (38). D denotes a fuzzy set of actions, a = ( a ~ , . . . , a , ) , in the product space A = A1 x- • • x A , which are taken by the team as a whole.
5. An example We consider the similar examplc given in [7]. Let us suppose that an urn is known to contain two balls, of which each may be either red or oraage. The symbol red and orange denote the fuzzified states of red and orange. A single ball is drawn from the urn; its color (but not the identity of the ball) is observed. Thus the color is the fuzzy information signal. There a r e 2 2 = 4 possible original compositions of the urn, and two possible eutcomes of the sampling experiment; hence there might be said to be 2 x 4 = 8 alternative states. Of these, two would be considered impossible, since one could not draw a red ball if the t~rn contained only orange balls, and vice versa. The six remaining states are listed in Table 1. We must regard states 62 and x3 as distinct, even though the composition of the urn is the same in both, because the signals (the colors of the ball drawn) are different. The same is true of x4 and X5•
It will be supposed that a team consists of 2 members. Each member observes the states of the urn and takes the information signals of red or orange.. There are six states of the environment, X = { x l , x2, x3, x4, xa, x6}. Let {yl}={y~l, Y~2}, {Y2~-" {Y21, Y22}, Yll = T~I(Xl)= T~I(X3)= nl(XS), YI2 = ~1(X2) = O/~I(X4)= T~l(fg6), 921 = T~2(X2) = ~2(X4) "- T~2(X6), Y22 := 'i~2(Xl)-- ~2(X3) "= 7J2(XS); YlI - OV~lhge, y 1 2 = r e d , Y21 = l e d , Y22 = orange.
A model of fuzzy team decision
209
Table 1 Urn
State
Ball 1
Ball 2
Sample ball drawn
O
O
O
R R O O R
O O R R R
R O R O' R
xI x2 x3 x.t xs x6
Then, there are 4 team information signals V t = ( y 1 1 , Y21), = ( Y l l , Y22), y3=(yt2, Y20, Y4=(Y~2, Y22). Let member 1 take a fuzzy set C~ of information signals; C~ = { ( y ~ , 0.6), (Y12, 0.9)} and member 2 take a fuzzy set C2 of information signals; C2={(yzl, 0.8), (Y22, 0.7)}. Then B~ = {(x~, 0.6), (x2, 0.9), (x3, 0.6), (x4, 0.9), (xs, 0.6), (x6, 0.9)} and t32= {(x,, 0.7), (x2, 0.8), (x3, 0.7), (x4, 0.8), (xs, 0.7), (x6, 0.8)}. B = Ba f3 B2 = {(Xl, 0,6), (x2, 0.8), (x3, 0.6), (x4, 0.8), (Xs, 0.6), (X6, 0.8)}.
A fuzzy set of information signals to the team is defined by C = {(y, # c ( Y ) ) } , Y e Y IZc(Y) = t~c(Yl, Y2), Yl e Y 1 , y 2 e Y 2 #c(Yl) =/£c(Yll, Y21) = gc,(Yts) ^ gc,(Y21) = 0.6/~ 0.8 = 0.6 /~c(Y2) = ~c(Yll, Yz2) = gLc,(Yl l) A #Lc2(Y22) = 0.6 ^ 0.7 = 0.6 gc(Y :')= .' c(Y12, Y21)= ~c,(Y12)/~ Vtc~(Yzl) = 0.9/x 0.8 = 0.8 gc(Y 4) = go(Y12, Y22)= gc,(Yl2) A gc2(Y22) = 0.9 A 0.7 = 0.7 C = { ( y ' , 0.6), (y2, 0.6), (y3, 0.8), (y4, 0.7)} ) ti21 = o t 2 ( y z t ) , ti22 = a 2 ( Y 2 2 ) ; thus, there are 2 team Let a~, = a~(yt~) = a ,~" :~y~2, actions, al=(atl, a2~) and a 2 = ( 0 1 1 , 0 2 2 ) . Member 1 takes a fuzzy set D~ of actions; D~ = { ( t i l l , go,(a~t))}, member 2 takes a fuzzy set D2 of actions; D2 =
{(a"l, gOa(ti21)), (022, ~-I'O2(ti22))}
go,(a,~) =
Max
yt~at-t(att)
gc,(y,)=max[gc,(y,,),
ttc,(y,2)]
= max[0.6, 0.9] = 0.9 go:(o::)
=
~0~(a22)
=
Max
tzc2(Y2)=
gc~(Y2t)
=
0.8
Max
gc2(Y2) =
gc_,(Y22)
=
0.7
y2e,m2-t(az,)
y2e"'2- I(a22)
D, ={(a,1, 0.9)},
D 2 = ( ( a ~ , , 0.8), (a22, 0.7)}
H. Noiiri
210
A fuzzy set of actions to the team is defined by
a ~A tzo(a)=p.D(at, a2), a t ~ A t , a2~A2 = ~r,,(aO ^/xr,~(a2) ij,o(a 1) = / . / , O ( a l 1, a2t) = tzD,(al ) ^ ~o2(a21) = 0.9 ^ 0.8 = 0.8 ~ o ( a 2) = tzo(all, a22) =/Zo,(atl) A/zo2(a22) = 0.9 ^ 0.7 = 0.7
D = {(a, l.~o(a))},
then, D = {(a t, 0.8),
(a 2, 0.7)}.
It will be supposed that the team's utility payoff matrix is given in Table 2 and let ,r(xi) = 0 . 1 , ¢r(x2) = 0.3, ,r(x3) = 0.2, Ir(x,) = 0.1, 1r(xs) = 0.1, ,r(x6) = 0.2. Then, if 3, = 0 . 7 , Do7 ={a I lxo(a)>--O.7}={a t, a2}, 6
O ( a ' ) = Y. o(x,, a l ) c r ( x i ) = 0 . 2 i=l 0
O ( a 2) = )" ~o(x~, a2)w(x~) = 0.8 i=l
For a given set Do.7, the best action is a = a 2. If 3, =0.8, Do.s={alm,ta)>~O.8}={a'}.For a given set Do.s, the best action is
area 1.
If 3,=0.7; Do.7={al, a 2} and B = B~f'IB2, then the expected utility over B is given by 6
O ( a l ) = ~ o(x, at)o.a(x~)~(x~)=O.12 i=l 6
O ( a 2 ) = E o(xi, a2)l~e(xi)1r(xi)=0.60 i=l
For a given set Do7 and B, the best action is a = a 2. If the team takes a fuzzy set D of actions and B is a fuzzy set of states observed by the team, then the expected utility over the Cartesian product of B and D is given by 6 a')=
= 0.12 i=l 6 ~"~(~r~, 0/2) --" E ~tJ(Xi' a2[~'~(Xi)])~I'6B×D 2(X'i' Ol2['l~(Xi )])q'l'(Xi) " - 0 . 6 0 i=l 6 ff~('l~, £//3)__. E ~0(Xi' Ol3[n(Xi)])tgB×Da(Xi ' Ol3[n(Xi)])~l'r(Xi) -" 0.12 i=l 6 ~"~('i~, a 4) ='- E £0(Xi' a4[~](X'i)])~LI'B×D4(Xi ' 0~4[~'~(Xi)])'/T(Xi)-" 0.60 i=l
where D t = { ( a ~ , 0 . 8 ) , (a2,0.7)}, D 2 = { ( a ~ , 0 . 7 ) , (a2,0.8)}, D 3 = { ( a ~ , 0 . 8 ) , (a 2, 0.0)}, D4 = {(a ~, 0.0), (a 2, 0.8)} and B_ = {(x~, 0.6), (x2, 0.8), (x3, 0.6), (x4, 0.8), (x5, 0.6), (x6, 0.8)}.
A model of fuzzy team decision
211
Table 2. Payoff matiix States Actions
xi
x2
x3
x4
x5
x6
aI
1 0
0 1
0 1
0 1
1 0
0 1
a 2
For a given information structure ~1 = ( ~ , ~12), the best decision rules are a 2 and a4.
6. Conclusions This paper presents a model for the fuzzy team decision. By introducing the concept of fuzziness, we have formulated the team decision processes which contain fuzzy states, fuzzy information functions, fuzzy information signals, fuzzy decision functions and fuzzy actions. We then have presented an example showing the team decision processes which contain fuzzy information signals and fuzzy actions. References [1] R.E. Bellman and L A . Zadeh, Decision-making in a fuzzy environment, Management Sci. 17 (1970) B 141-B 164. [2] R.E. Bellman and M. Giertz, On the analytic formalism of the theory of fuzzy sets, Infermation Science 5 (1973) 149-156. [3] V. Balachandran and g.D. Deshmukh, A stochastic model of persuasive communication, Management Science 22 (1976) 829-840. [4] S.S.L. Chang, Fuzzy dynamic programming and the decision making process, in: Proc. 3rd Princeton Conf. on Information Sciences and Systems (1969) 200-20,~. [5] S.S.L. Chang and L.A. Zadeh, On fuzzy mapping and control, IEEE Trans. Syst. Man Cybern. SMC-2 (1972) 30-34. [6] L.W. Fung and K.S. Fu, An axiomatic approach to rational decision making in a fuzzy environment, in: L A . Zadeh et al. (eds.). Fuzzy $3ts and Their Applications to CognitLre and Decision Processes (Academic Press, New York, 1975) 227-256. [7] J. Marschak and R. Radner, Economic Theory of Teams (Yale University Press, New Haven, 1972). [8] K.R. MacCrimmon, Descriptive aspect~ ¢~f team theo~; Observation, communication and decision heuristics in information systems, Management Science 20 (1974) 1323-1334. [9] C.B. McGuire, Some team models of a sales organization, Management Sci. 7 (1961 ) 101-130. [10] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Birkhiiuser, Basel, 1975). [11] R. Radner, Team decision problems, Ann. Math. Statist. 33 (1962) 857-881. [12] P.L. Yu, A class of solutions for group decision problems, Management Sci. 19 (1973) 936-946. [13] C.C. Ying, A model of adaptive team decision, Operations Research 17 (1969) 800-811. [14] H.-J. Zimmermann, Description and optimization of fuzzy systems, Int. J. General Systems 2 (1976) 209-215. [15] L . A Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353. [16] L.A. Zadeh, Fuzzy sets and systems, in: Proc. Symp. System Theory (Polytechnic Press, Brooklyn, New York, 1965) 29-37.
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[17] L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23 (1968) 421-427. [ 18] L.A. Zadeh, Toward a theory of fuzzy systems, in: R.E. Kalman and N. DeClaris (eds.), Aspects of Network and System Theory (Holt, Rinehart and Winston, New York., 1971) 469-490. [19] L.A. Zaa~h, Outline of a new approach to the analysis cf complex systems and decision processes, IEEE Trans. Syst. Man Cybern. SMC-3 (1973) 28--44. [20] L.A. Zadeh, The concept of a linguistic variable and itg application to approximate reasoning, Information Sciences 8 (1975) 199-249 (Part 9; 8 (1975) 301-357 (Part !I); 9 (1975) 43-80 (Part liD.
A M O D E L OF F U Z Z Y T E A M D E C I S I O N H. Nojiri Department of Systems Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, 152, Japan This paper formulates a fuzzy team decision problem. The concept of fuzzy sets is introduced to formulate the team decision processes which contain fuzzy states, fuzzy information functions, fuzzy information signals, fuzzy decision functions and fuzzy actions. After a description of the basic notion of team theory, a model of fuzzy team decision is proposed.
Key words: Decision theory, Fuzzy decisions, Team decisions
1. Introduction Since its introduction by Zadeh in 1965 [15], the concept of fuzziness has been extended to algorithms, learning theory, automata, formal languages, pattern recognition, probability theory, information retrieval and the decision making process [5, 10]. The application of fuzzy set approach to special problems in decision making was suggested in 1969 [4], 1970 [1], 1975 [6] and 1976 [14]. For a survey on the subject we refer to Negoita and Ralescu [10]. It is our purpose here to introduce the notion of fuzzy sets defined by Zadeh [15] to the theory of teams by Marschak and Radner [7]. The Paper is divided into three parts. The first will describe the basic concepts of team theory. In the second part we will propose a model of fuzzy team decision. In the third part we will define the concept of a fuzzy information function and a fuzzy decision function.
2. The team decision problem Let a team consist of n members. Let X = {x} be the set of mutually exclusive, uncontrollable states of the environment. The uncertainty about x is expressed by a probability distribution on X. Let us assume that the set X is finite and X does not contain any states that have zero probability. An information received, or an observation made, by a team as a whole, called a team information, has n components. Denote the team information by y; then y = ( Y l , . . . , Y,), where y~, i = 1 , . . . , n, is ~the part of the team information received by the ith member and it is related to the state of the environment x by a function ~i; that is, if the environment is in state x, then member i will have information yi = ~(x). Let Y~ = {Yi} be the set of possible alternative information ~ignals that can be received by the ith member, and let Y={y} be the set of all possible alternative 201
H. Noiiri
202
information signals that can be received by the team: then { y } { y t } x { y 2 } x . . . x{y,}, Y-----X~=IYi. ±m action taken by a team as a whole, called a team action, also has n components; then a = ( a l , . . . , o~), where a~, i = 1 , . . . , n, is the action taken by the ith member. Let A~ = {o~} be the set of feasible actions that can be taken by the ith member, and let A = {a} be the set of feasible team actions that can be taken by the team as a whole; then {a}={al}x{az}x'.'x{a,,}, A----X'~=~A~, [7,13]. The ith member of a team takes his action a~ on the basis of his information signal, y~; thus a~ = o~(y~). The symbol a~(y~) denotes the action prescribed b.¢ the decision function t~ when signal y~ is received; then a decision function a~ is a function from Y, to the set A~ of feasible actions. T' -. a-tuple of decision functions can be denoted by a = ( ~ , . . . , a , ) and called the decision function (or decision rule) of the team; a is a function from Y to A ; then a = a ( y ) . The n-tuple r ) = ( ~ . . . . , ~!,) may be called the information structure of the team. ~i is a function from X to Y; then y = ri(x). Again in condensed notation, y = 71(x), a = a ( y ) = a[~l(x)], E7]. The expected utility of the team resulting from the use of an information structure ~1= (TIx,..., ~1,) and a team decision function a = ( a x , . . . , a~) is a ; to,
= F
(x,
= -~xto(x, a t [ ~ l l ( x ) ] , . . . ,
a,[vl,,(x)])¢r(x),
(1)
where to(-, • ) mapping from X x A to the real line denotes the utility function of the team and ~r(. ) mapping from X to [0, 1] denotes the probability function. It is assumed th: t all members have the same utiqty function and the same probability function. O(vl, a ; to, '.~) is a function of ~, a, to and ~r. It is assumed that the utility and probability function used are the true ones, thus they are not decision variables, as are 71 and a, and hence, for simplicity, we shall write D(~I, a ) in place of n ( ~ , a ; to, ~r). The problem then becomes one of choosing, for a given information structure ~', the best decision rules, that is the a = a * for which 1~(~') = Max 1~(~1', a).
(2)
ot
After each information structure is evaluated in terms of the best decision function that can be used with it, the obvious conclusion is to use the information structure, ~*, that maximizes l~(vl). The optimal pair of vecl::ors (~*, a * ) is called the optimal organization form [8]. These concepts are the basic ones of team theory which we shall need for our formulation. For more detailed discussions on the theory of teams, the reader may refer to the work of Marschak and Radner [7].
3. Formulation of fuzzy team decision problems In the team decision problem, the information variable y~ of the ith m e m b e r of a team is related to the state of the environment x by a function ~i. For a 6×ed
A model
of fuzzy team decision
203
information function r/i, each signal Yi is identified with a subset of X, namely, the set of states x for which qrh(x) = YiBy introducing the notion of fuzzy sets defined by Zadeh [1, 15, 17]. let us assume that each team member i receives a fuzzy information signal Yi about the state of the environment x. Then, each fuzzy information signal Yi (an element of Yi) is identified with a fuzzy subset of X, the set of states of the environment. For each member i, i = 1 , . . . , n, le~ Ci be a fuzzy set of information signals in 'the information space Y~ of the ith member. Then a fuzzy subset Ci in Y~ is characterized by a membership function /ac," Y~--* [0, 1] which associates with each information signal y~ of Y~ a, number /z~(yi) in the interval [0, 1] which represents the grade of membership of Yi in C~. Fuzzy sets C ~ , . . . , C, are defined by Ci = {(yi, ~c, (yi))},
yieY,
i---1,...,n
(3)
where the symbol i denotes the ith member of the team, Ci denotes a fuzzy set of information signals and n denotes the number of members in a team. An information tunction ~1~ of the ith member is a function from X to Y~ = {y,}. Then, the inverse mapping rl~-~(Ci) induces ~: fuzzy set B~ in X. For each n: :nber i, i = 1 , . . . , n, a fuzzy set Bi is defined by B, = {(x, ttB,(x))},
x~X
(4)
ttB, (x) =ttc, (yi),
y, E Y,
(5)
for all x in X which are mapped by rh into Yi. Let us consider now a conv,~rse problem and for each member i, i = 1, . . . . n, let Bi be a fuzzy set in X. Then, the information function ~1~ induces a fuzzy set Ci of information signals in Y~ whose membership function is given by tt~(yi) =
Sup
{ttB,(x)}
= 0,
if n~-~(),i) is not empty
(6)
otherwise,
where rl~-l(y~) is the set of points in X which :~re mapped by -q~ into y~. Let a fuzzy set B be the union of fuzzy sets B ~ , . . . , E,. Then, a fuzzy subset B in X is defined by B = B l t.) B 2 LI- • • U B . B = {(x,
x
(7) X
ttB (x) = max(~B,(x),..., tLm (x))
(8)
or, in abbreviated form ttB (x) =/xB~lx) v/zB~(x) v - - " v ttB. (x)
(9)
where the disjunction symbol v means taldng the lnaximum of tcB,(x), i = 1 , . . . , n. B denotes a fuzzy set of states, x, of the environment which are observed by the team as a whole. In order to define a fuzzy set of informaticn signals to the team, let C be the Cartesian product of n fuzzy sets C ~ , . . . , C,. Then, by applying the definition in
204
H. Noiiri
Ref. [19], C - C~ x - - • x C,, is defined by
C =--{(y, ~c(Y))},
ye Y
g,c(y) = ~ c ( y , , . . . ,
y~),
(10) y, • IT,,
i = 1,...,
n
= min[p,c,(y,),...,/~c.(Y.)]
(11)
or, in abbreviated form / ~ ( y ) - / ~ , c . ( y , ) A ~ ( y 2 ) A " " "A/J,c~ (Y.)
(12)
where the conjunction symbol ^ means taking the minimum of /~c,(Y~), i:1 . . . . , n and { y } = {Yl} x . - - x {y,}, Y = X[~=l Y~. We suppose that fuzzy sets C 1 , . . , , C. are noninteraeting. T h e notion of noninteracting of fuzzy sets corresponds to the notion of the independence of random variables [18]. The Eqs (10) and (11) mean that C = C l x . - - x C , is a fuzzy set of ordered pairs y = ( y ~ , . . . , y~), y~ ~ Y~, i = 1 , . . . , n, with the grade of membership of Y = ( Y ~ , . . . , Y , 0 in Y = Y ~ x - . - x Y , , given by ~ c ( y ) = ~c,(yl)A- • • ^ gc~(y,). In th~s sense, C = C~ x . • • x C, is an n-ary fuzzy relation in the product space Y = Y~ x . • • x Y , . C denotes a f u z z y set of ~infoivnation signals, y = ( y ~ , . . . , y,), to the team. In the team decision problem, the ith member of a team takes his action a~ on the basis of his information, y~; thus a~=m(y~). A decision function m is a function from Y~ to the set A~ of feasible actions. The n-tuple of decision functions can be denoted by a = ( a ~ , . . . , t~,) and called the decision rule of the team; a is a function from Y to A ; then a = a(y). Thus, let G be a fuzzy set of information signals in the information space Y~ of the ith member. Then, the decision function a, induces a fuzzy s e t / ~ of actions in the action space A, of the ith member. For each member i, i = 1 , . . . , n, a fuzzy set D~ of actions is defined by D, = {(a,, go~(a~))},
(13)
a,~A,
~o,(a~)=
Sup {/~q(Yi)} if aT~(ai) is not empty, y,~c'(~,~ = 0, otherwise,
(14)
where the supremum is taken over the set of points aT~(a~) in Y~ which are mapped by a, into a~. Let D be the Cartesian product of n fuzzy sets D ~ , . . . , D , , then D = D1 x D2 x . • • x D , is defined by D={(a,l~o(a))},
(15)
a~A
/ ~ D ( a ) = l L D ( a l , . . . , an), ~A~, = m i n [ ~ o , ( a ~ ) , . . . , t~o. (an)]
i= 1,...,
n
(16)
or, in abbreviated form /zo(a) = g,D,(a~) ^ ' ' " A btD. (an)
(17)
where the conjunction symbol ^ means taking the minimum of ~o,(a~); i = 1 , . . . , n, {a} = {a~} x . . . x {a~}, A ~ X,"_-, A~.
A model of fuzzy team decision
205
D denotes a fuzzy set of actions, a = ( a ~ , . . . , a , ) , in the product space A = A~ x . • • x A , which are taken by the team as a whole. In order to decide a set of actions to the team, a pair of thresholds may be introduced as follows [15]: 3" > ~ where 0 < 3", 6 < 1 which leads to a three-level logic as (1) a belongs to D, or 'true' if t~o (a) ~> 3' (2) a does not belong to D, or "false' if txo(a)~<6 (3) a has an indeterminate status relative to D, or ' u n d e t e r m i n e d ' if 6 < t ~ o ( a ) < 3', where D is a fuzzy set of actions to the team. Let us assume that the team takes a s e t / 9 , of actions defined by
(18) Then, the problem becomes one of choosing, for a given set Dr, the best action, that is the a - a* for which O(a*) =
Max O(a)
(19)
a~D,
O(a)= ~xoJ(x, a)'a'(x)
(20)
where O(a) denotes the expected utility, ¢o(., • ) mapping from X >"A to the real line denotes the utility function of the team and ~r(- ) mapping from X to [0, 1] is the probability function. The expected utility O ( a ) is said to represent the .<
.<
preference ordering K if for all actions ai and aj for i ~ j in Dr "ai if, ai if and only if [~(ai)<~O(ai). If the state of the environment is fuzzy and B is a fuzzy set in X, then Eq. (20) is replaced by
g~(a) -,Y,~oJ(x, a)t~n(x)Tr(x).
(21)
In Eq. (21), the utility o~(x, a) is scaled down in proportion to txB(x) in B and Eq. (21) expresses the expected t~tility over B [20]. If the team takes a fuzzy se'~ D of actions and B is a fuzzy set of states observed by the team, then the maximal expected utility of the team, given its information structure rl', is the expected utility maximized with respect to a" /~(,i') = Max O(~l', a )
(22)
O(~1', ,:~)= Zxco(x, a[rl'(x)])t~B×o(x, a[rl~(x)])Tr(x)
(23)
= Zx,,,(x,
where O(rl', a ) denotes the expected utility over the Cartesian product of B and D.
4. F~zy information hmct~ons and fuzzy derision functions In the theory of teams [7], information can be regarded as the outcome of information-gathering. For each member i, i = 1 . . . . , n, a given method of
206
H. Noiiri
information-gathering applied to the true state of environment x results in a particular signal Yi- To each x in X will con'espond a signal y~. Thus, to a given information-gathering method is associated a function ~ from X to Yv Let us assume that each m e m b e r i uses a given fuzzy method of informationgathering, namely, a fuzzy mapping ~1~ from X to Y~. Then, by applying the definition in [5], for each m e m b e r i, i -~ 1, .... , n, a fuzzy mapping ~li is defined as: Definition. A fuzzy mapping "q~ from X to Y~ is a fuzzy set on X x Y~ with membership function pa,(x, Yi)A fuzzy informagon function .q,(x) is a fuzzy set on the information space ~ of member i with membership function Pn.oo(Yi) = Pn,(x, yi),
i = 1 , . . . , n.
(24)
Its inverse r171(y~) is a fuzzy set on X with Pn,-'tyO(x) = Pn,(x, Yi),
i = 1 , . . . , n.
(25)
For each i, a fuzzy information function rh is said to be liner than a fuzzy
information function O~ if p~,~)(y~) ~ po,~(y~),
'¢x,
y~ e,X; Y~,
i = 1 , . . . , n.
(26)
Theorem 1. If rl~ is finer than O~, nT. ~ is finer than OT~. lProo|: T h e proof is trivial and omitted [5]. For each m e m b e r i, i = 1 , . . . , n, let B~ be a fuzzy set on X. Then, T h e fuzzy mapping ~ii induces a fuzzy set C~ of information signals in Y~ which is defined as C, = {(y,, Pc, (Y,))},
Y, ~ Y,
(27)
pc.(y~)-Sup(~a.(x)^p..(x. Y,)),
i= 1,...,n.
(28)
x~X
Let C be the Cartesian product of n fuzzy sets C ~ , . . . , C,. Then, C = Ci x C2 x- • • x C, is defined by C = {(y, Pc(Y))},
Y~ Y
Pc(Y) = P c ( Y , , - - - , Y,),
(29) Yi e Y~,
i= 1,..., n
(30)
= min[pc,(yx),. • •, Pc. (Y,)] where p~(y~), i = 1 , . . . , n, are given by the Eq. (28). C denotes a fuzzy set of information signals, y = ( y l , . . . , y,), to the team. T h e o r e m 2. For each i, i = 1 , . . . , n, let Bk, B~ be two fuzzy sets on X. Then
if
Bk D_Bi,
"rl~(Bk) ~ ~ (Bt)
n,(Bk 0 B~) = n~(Bk) O n~(B~) n~(Bk n B~) c_ n~(Bk) n n~(B~).
if rh is liner than 0~, then
Vh( Bk ) c_ 0~( Bk ).
(31) (32) (33) (34)
A model of fuzzy team decision
207
Proof of (32). Let H = Bk U B~. ~,~,u4)(Y,) = Sup(/.%~ (x) A/~,~, (X, y,)) x~X
= Sup((p,a~ ( x ) v p,a, (x)) ^/x,~, (x, y,)) Jc~X
= Sup[(/~a~ (x) A tt,~, (X, y,)) v (tta, (x) ^ ~,~, (x, y,))] x~X
= Sup(/x m (x) A p,,~,(x, y,)) v Sup(/xa, (x) A/~., (x, yi)). x~X
x~X
Proof of (33). Let K = Bk ¢'1B~, then
/.I,n,(K){Yi)-- Sup(KK (X) A/d,n, (X, Yi)) xEX
= Sup(/xsk (x)^ lgm(x) ^ lxni(x, y,)) xEX
~
x~X
= tL,,,~ao(y~) ^ w,~,ts~(y~). The proofs of (31) and (34) are trivial and omitted [5]. For each member i, i = 1 , . . . , n, a fuzzy decision function ai is defined as"
Definition. A fuzzy mapping a, from Y~ to Ai is a fuzzy set on Y~ x A, with membership function / ~ (y~, a~). A fuzzy decision function a~(y~) is a fuzzy set on A~ with membership function tt,,,t~.~(a,) = tL,,,(y,, a~),
(35)
i = 1 , . . . , n.
Its inverse a~-t(a~) is a fuzzy set on Y~ with
/z~, ,to~)(y~)= tt,~, (y,, a~),
i = 1,...,
n.
(36)
A fuzzy decision function a~ is said to be finer than a fuzzy decision function [3~
if /x=,~,~(a~)~ < t~a,
Vy~, a~ ~ Y~, A,,
i = 1 , . . . , n.
(37)
T h e o r e m 3. If a~ is finer than [3~, Ol? 1 is finer than [3=,! Proof. The proof is trivial and omitted [5]. Let C~ be a fuzzy set of information signals in the information space Yi. Then, a fuzzy set D~ of actions in the action .~pace Ai of member i is defined as
I.co,( a, ) = /.h~,co(a, ) -- Sup(ire, (y~) ^ ~ , (y~, a~)),
(38) i=l,...,n.
H. Noiiri
208 Theorem
4. For each i, i = 1 , . . . ,
n, let Ck, C~ be two f u z z y sets on Y~, then
a~(Ck U Cz) = a~(C~,) U a~(C~)
(39) (40)
a~(Ck N Ct) G a,(Ck) n a~(C,).
(41)
If ai is finer than/3i, then
(42)
if
Ck ~--C~,
ai(Ck) ~_a~(Cz)
t~(C~ ) c_ [3i(Ck).
If Ck ~- C~ and ai is finer than ~i, then
ai (Ca) c_ [3i(Ck).
(43)
ProoL The proof is analogous t o the proof of Theorem 2 and omitted [5]. Let D be the Cartesian product of n fuzzy sets D ~ , . . . , / 9 , , , 0 1 X 0 2 X . • • X O n is defined by D = {(a,/zo(a))},
aeA
Izo(a)= lxo(a~' " " ' a") =min[po,(al),...,/Xo.(ah)],
then D = (44) (45)
a~ ~Ai,
i = 1,..., n
where /zo,(a~), i = 1 , . . . , n, are given by the equation (38). D denotes a fuzzy set of actions, a = ( a ~ , . . . , a , ) , in the product space A = A1 x- • • x A , which are taken by the team as a whole.
5. An example We consider the similar examplc given in [7]. Let us suppose that an urn is known to contain two balls, of which each may be either red or oraage. The symbol red and orange denote the fuzzified states of red and orange. A single ball is drawn from the urn; its color (but not the identity of the ball) is observed. Thus the color is the fuzzy information signal. There a r e 2 2 = 4 possible original compositions of the urn, and two possible eutcomes of the sampling experiment; hence there might be said to be 2 x 4 = 8 alternative states. Of these, two would be considered impossible, since one could not draw a red ball if the t~rn contained only orange balls, and vice versa. The six remaining states are listed in Table 1. We must regard states 62 and x3 as distinct, even though the composition of the urn is the same in both, because the signals (the colors of the ball drawn) are different. The same is true of x4 and X5•
It will be supposed that a team consists of 2 members. Each member observes the states of the urn and takes the information signals of red or orange.. There are six states of the environment, X = { x l , x2, x3, x4, xa, x6}. Let {yl}={y~l, Y~2}, {Y2~-" {Y21, Y22}, Yll = T~I(Xl)= T~I(X3)= nl(XS), YI2 = ~1(X2) = O/~I(X4)= T~l(fg6), 921 = T~2(X2) = ~2(X4) "- T~2(X6), Y22 := 'i~2(Xl)-- ~2(X3) "= 7J2(XS); YlI - OV~lhge, y 1 2 = r e d , Y21 = l e d , Y22 = orange.
A model of fuzzy team decision
209
Table 1 Urn
State
Ball 1
Ball 2
Sample ball drawn
O
O
O
R R O O R
O O R R R
R O R O' R
xI x2 x3 x.t xs x6
Then, there are 4 team information signals V t = ( y 1 1 , Y21), = ( Y l l , Y22), y3=(yt2, Y20, Y4=(Y~2, Y22). Let member 1 take a fuzzy set C~ of information signals; C~ = { ( y ~ , 0.6), (Y12, 0.9)} and member 2 take a fuzzy set C2 of information signals; C2={(yzl, 0.8), (Y22, 0.7)}. Then B~ = {(x~, 0.6), (x2, 0.9), (x3, 0.6), (x4, 0.9), (xs, 0.6), (x6, 0.9)} and t32= {(x,, 0.7), (x2, 0.8), (x3, 0.7), (x4, 0.8), (xs, 0.7), (x6, 0.8)}. B = Ba f3 B2 = {(Xl, 0,6), (x2, 0.8), (x3, 0.6), (x4, 0.8), (Xs, 0.6), (X6, 0.8)}.
A fuzzy set of information signals to the team is defined by C = {(y, # c ( Y ) ) } , Y e Y IZc(Y) = t~c(Yl, Y2), Yl e Y 1 , y 2 e Y 2 #c(Yl) =/£c(Yll, Y21) = gc,(Yts) ^ gc,(Y21) = 0.6/~ 0.8 = 0.6 /~c(Y2) = ~c(Yll, Yz2) = gLc,(Yl l) A #Lc2(Y22) = 0.6 ^ 0.7 = 0.6 gc(Y :')= .' c(Y12, Y21)= ~c,(Y12)/~ Vtc~(Yzl) = 0.9/x 0.8 = 0.8 gc(Y 4) = go(Y12, Y22)= gc,(Yl2) A gc2(Y22) = 0.9 A 0.7 = 0.7 C = { ( y ' , 0.6), (y2, 0.6), (y3, 0.8), (y4, 0.7)} ) ti21 = o t 2 ( y z t ) , ti22 = a 2 ( Y 2 2 ) ; thus, there are 2 team Let a~, = a~(yt~) = a ,~" :~y~2, actions, al=(atl, a2~) and a 2 = ( 0 1 1 , 0 2 2 ) . Member 1 takes a fuzzy set D~ of actions; D~ = { ( t i l l , go,(a~t))}, member 2 takes a fuzzy set D2 of actions; D2 =
{(a"l, gOa(ti21)), (022, ~-I'O2(ti22))}
go,(a,~) =
Max
yt~at-t(att)
gc,(y,)=max[gc,(y,,),
ttc,(y,2)]
= max[0.6, 0.9] = 0.9 go:(o::)
=
~0~(a22)
=
Max
tzc2(Y2)=
gc~(Y2t)
=
0.8
Max
gc2(Y2) =
gc_,(Y22)
=
0.7
y2e,m2-t(az,)
y2e"'2- I(a22)
D, ={(a,1, 0.9)},
D 2 = ( ( a ~ , , 0.8), (a22, 0.7)}
H. Noiiri
210
A fuzzy set of actions to the team is defined by
a ~A tzo(a)=p.D(at, a2), a t ~ A t , a2~A2 = ~r,,(aO ^/xr,~(a2) ij,o(a 1) = / . / , O ( a l 1, a2t) = tzD,(al ) ^ ~o2(a21) = 0.9 ^ 0.8 = 0.8 ~ o ( a 2) = tzo(all, a22) =/Zo,(atl) A/zo2(a22) = 0.9 ^ 0.7 = 0.7
D = {(a, l.~o(a))},
then, D = {(a t, 0.8),
(a 2, 0.7)}.
It will be supposed that the team's utility payoff matrix is given in Table 2 and let ,r(xi) = 0 . 1 , ¢r(x2) = 0.3, ,r(x3) = 0.2, Ir(x,) = 0.1, 1r(xs) = 0.1, ,r(x6) = 0.2. Then, if 3, = 0 . 7 , Do7 ={a I lxo(a)>--O.7}={a t, a2}, 6
O ( a ' ) = Y. o(x,, a l ) c r ( x i ) = 0 . 2 i=l 0
O ( a 2) = )" ~o(x~, a2)w(x~) = 0.8 i=l
For a given set Do.7, the best action is a = a 2. If 3, =0.8, Do.s={alm,ta)>~O.8}={a'}.For a given set Do.s, the best action is
area 1.
If 3,=0.7; Do.7={al, a 2} and B = B~f'IB2, then the expected utility over B is given by 6
O ( a l ) = ~ o(x, at)o.a(x~)~(x~)=O.12 i=l 6
O ( a 2 ) = E o(xi, a2)l~e(xi)1r(xi)=0.60 i=l
For a given set Do7 and B, the best action is a = a 2. If the team takes a fuzzy set D of actions and B is a fuzzy set of states observed by the team, then the expected utility over the Cartesian product of B and D is given by 6 a')=
= 0.12 i=l 6 ~"~(~r~, 0/2) --" E ~tJ(Xi' a2[~'~(Xi)])~I'6B×D 2(X'i' Ol2['l~(Xi )])q'l'(Xi) " - 0 . 6 0 i=l 6 ff~('l~, £//3)__. E ~0(Xi' Ol3[n(Xi)])tgB×Da(Xi ' Ol3[n(Xi)])~l'r(Xi) -" 0.12 i=l 6 ~"~('i~, a 4) ='- E £0(Xi' a4[~](X'i)])~LI'B×D4(Xi ' 0~4[~'~(Xi)])'/T(Xi)-" 0.60 i=l
where D t = { ( a ~ , 0 . 8 ) , (a2,0.7)}, D 2 = { ( a ~ , 0 . 7 ) , (a2,0.8)}, D 3 = { ( a ~ , 0 . 8 ) , (a 2, 0.0)}, D4 = {(a ~, 0.0), (a 2, 0.8)} and B_ = {(x~, 0.6), (x2, 0.8), (x3, 0.6), (x4, 0.8), (x5, 0.6), (x6, 0.8)}.
A model of fuzzy team decision
211
Table 2. Payoff matiix States Actions
xi
x2
x3
x4
x5
x6
aI
1 0
0 1
0 1
0 1
1 0
0 1
a 2
For a given information structure ~1 = ( ~ , ~12), the best decision rules are a 2 and a4.
6. Conclusions This paper presents a model for the fuzzy team decision. By introducing the concept of fuzziness, we have formulated the team decision processes which contain fuzzy states, fuzzy information functions, fuzzy information signals, fuzzy decision functions and fuzzy actions. We then have presented an example showing the team decision processes which contain fuzzy information signals and fuzzy actions. References [1] R.E. Bellman and L A . Zadeh, Decision-making in a fuzzy environment, Management Sci. 17 (1970) B 141-B 164. [2] R.E. Bellman and M. Giertz, On the analytic formalism of the theory of fuzzy sets, Infermation Science 5 (1973) 149-156. [3] V. Balachandran and g.D. Deshmukh, A stochastic model of persuasive communication, Management Science 22 (1976) 829-840. [4] S.S.L. Chang, Fuzzy dynamic programming and the decision making process, in: Proc. 3rd Princeton Conf. on Information Sciences and Systems (1969) 200-20,~. [5] S.S.L. Chang and L.A. Zadeh, On fuzzy mapping and control, IEEE Trans. Syst. Man Cybern. SMC-2 (1972) 30-34. [6] L.W. Fung and K.S. Fu, An axiomatic approach to rational decision making in a fuzzy environment, in: L A . Zadeh et al. (eds.). Fuzzy $3ts and Their Applications to CognitLre and Decision Processes (Academic Press, New York, 1975) 227-256. [7] J. Marschak and R. Radner, Economic Theory of Teams (Yale University Press, New Haven, 1972). [8] K.R. MacCrimmon, Descriptive aspect~ ¢~f team theo~; Observation, communication and decision heuristics in information systems, Management Science 20 (1974) 1323-1334. [9] C.B. McGuire, Some team models of a sales organization, Management Sci. 7 (1961 ) 101-130. [10] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Birkhiiuser, Basel, 1975). [11] R. Radner, Team decision problems, Ann. Math. Statist. 33 (1962) 857-881. [12] P.L. Yu, A class of solutions for group decision problems, Management Sci. 19 (1973) 936-946. [13] C.C. Ying, A model of adaptive team decision, Operations Research 17 (1969) 800-811. [14] H.-J. Zimmermann, Description and optimization of fuzzy systems, Int. J. General Systems 2 (1976) 209-215. [15] L . A Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353. [16] L.A. Zadeh, Fuzzy sets and systems, in: Proc. Symp. System Theory (Polytechnic Press, Brooklyn, New York, 1965) 29-37.
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