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Fuzzy adaptive learning model of decision-making process
Fuzzy Sets and Systems 18 (1986) 273-282 North-Holland
FUZZY ADAPTIVE LEARNING D E C I S I O N - M A K I N G PROCESS
273
MODEL OF
A.F. BLISHUN Institute of Automated Management Systems of The Ministry of Fishing Industry, Moscow, USSR Received February 1985 Revised March 1985 A mathematical model is described which generalizes fuzzy information to adopt to a new situation. This model is able to modify a priori information obtained from experts. Preferences of experts are realized by means of a set of strictly unformed criteria which can be described in terms of a composite linguistic variable. Using a decision maker's selection of a particular alternative from a set of possible ones the model improves its performance. It replaces a priori linguistic rules by refined linguistic rules. These rules correspond to the choice of the decision-maker.
Keywords: Linguistic variable, Fuzzy scale, Multi-attribute decision, Adaptive learning.
"An exact description of any real physical situation is virtually impossible. This is a fact we have had to accept and adjust to." (R. Bellman, M. Giertz, 1973) 1. Introduction The management of complex systems often needs nonformal reasoning on the basis of intuitional subjective knowledge. During the automation of the management such reasoning cannot be described in an adequate quantitative form. In particular, there are many problems of modelling of a decision-making process with the following characteristics: presence of decision-maker, importance of subjective factors, dynamics of subjective factors. The solution of the problems mentioned above requires suitable mathematical models which are able to generalize fuzzy information to adopt to the new situation. These models have to be capable of improving their performance during operation. In this paper an adaptive model of this type is suggested. While making a decision the selection of a particular alternative from a set of possible ones is realized. When analysing the situation, possible alternatives can be mapped in attribute space or semantic space. In semantic space an alternative displays the attribute with certain degree. This degree is measured by means of a scale of measurements. It is well known, however, that in most cases a human's decision is vaguely or imprecisely defined so that the grade of attribute is measured by the expert very approximately. 0165-0114/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
274
A.F. Blishun
Therefore, a method of scaling must satisfy the following conditions [6]: (a) it should allow for a moderate uncertainty in judgments about grades of attribute without changing the judgment value on the scale; (b) strong changes of judgment should be reflected in appropriate variations up and down the scale; (c) the results of the model should not change drastically by making small changes in the judgment values. The fuzzy scale satisfies the above conditions. Definition 1. Let U be a nonempty set of empirical objects, R i (i = 1 , . . . , n) a set of relations over U, L _ R (the set of real numbers), T a set of fuzzy subsets of L, St (i -- 1 , . . . , n) a set of relations over T. Let Ri and Si be ki-place relations. Then a mapping M : U--* T is a fuzzy scale if R,(ul .....
uk,) = S , ( m ( u l ) , . . . , m(Uk,))
for all i e l = {1. . . . . n} and ( u l . . . . . Uk,) • U k'. It is clear that from relations between judgment values on fuzzy scale we are able to draw a conclusion about empirical relations between empirical objects as follows: empirical objects ul . . . . . Uk, are in relation R if and only if corresponding values m ( u l ) , . . . , m(Uk,) on the fuzzy scale are in relation S. In order to fit the above problems into a certain mathematical model it seems reasonable to assume the following conditions concerning the fuzzy scale. Hypothesis I. The fuzzy scale must ensure the evaluation of any grade of attribute.
Hypothesis II.
In the case of a finite number of elements of the fuzzy scale neighbouring values given an equal change of grade of attribute, i.e. the 'physical' distances of each two adjacent elements of the fuzzy scale in semantic space are equal [1, 5, 7]. Depending on the situation and the personal perceptions of experts the grade of the same attribute can be evaluated differently. Therefore while identifying real values of features with semantic images it is essentially a form of mapping between feature space (U) and semantic space (P). The mapping has to preserve the relative place of objects in semantic space. A natural way to formalize the concept of fuzzy mapping is via the concept of fuzzy property [1]. Defmition 2. A property p defined on an object u is a function p : U--* [0, 1] such that p ( u ) > 0 iff u has the property p. Definitions 1 and 2 follow by the fact that grades of attribute correspond to an ordered set P = {p(u) : u e U}. Consequently, each element of the fuzzy scale corresponds to a fuzzy subset of P. According to Definition 1 the attribute can be
Fuzzy adaptive learning model
275
)a,
/
/
/
,J
/ / Fig. 1.
interpreted as a linguistic variable whose values are the labels of fuzzy subsets of P. Figure 1 shows the correspondence of judgements about grades of attribute in semantic space and feature space. When fuzzy scales are used reception of supplementary information or change of personal opinion entails their adjustment by means of either a displacement of fuzzy scale elements or a change of mapping p. This paper deals with the first case only.
2. Formulation of the problem Let a rational decision choice be realized on the basis of a collective of strictly unformed criteria, which cannot be described in an adequate analytical form. In this case preferences are described by the following linguistic statement type: "if values of features u l , . . . , u n which characterize an alternative u i are avaluated by terms tl~, . . . , t~i, then u i satisfies the j-th criterion with value t~+j,i".
Every alternative can be described in terms of a composite linguistic variable L = ( L I. . . . .
L ......
Ln+k, Ln+k+l),
where Li is a unary linguistic variable with term set T~, with universal set U~, and with basic variable ui, tit e Tt c F(U~). Values from term set ti~ e T~, n + 1 ~< i ~< k + n + 1, point out the grade of criterion satisfaction. T~ consists of linguistic values of linguistic variable.
A.F. Blishun
276
A set of values of L is separated in two subsets: MI=
{tli, ....
tni, tn+j,i), n + l <~j <<-n + k}[-T 1,
M 2 = {(t,+Li, . . . .
tn+k+l,i)}i=~ 2.
M1 characterizes local criteria and M 2 characterizes the global criterion. M = M~ U M2 consists of a priori information which is obtained from an expert. Each t, t e M, can be represented as a fuzzy point in a subspace of D, where D is represented as the Cartesian product of feature space and criteria space [4]. In reality, any value of feature u~ has correspondent degree of property Pi. Hence u i defines the point (Pl(Ul) . . . . . pn(un)) in the semantic space/'1 x . • • x P . and if t i E M then t; defines the fuzzy point t~(p(.)). For an alternative u ~the grade of global criterion satisfaction w ( u i) is computed as
w(ui)=t,UM2(tlUMtui°tl)°tr.
(1)
Assume that a set of alternatives R ' = {ui= ( u ~ , . . . , u/)} is given to the decision-maker. He chooses a set of alternatives R" from the given set R': R" ~_R'. Using the preference u~> uJ, u j e R ' \ R " , u~e R", it is necessary to replace M by some At which confirms the above preference in the following way: if(u;) > ff(u j)
3. A l g o r i t h m
for all u i e R '', u J e R ' \ R
".
of solution
A block diagram of fuzzy adaptive learning algorithm is shown in Figure 2. Let us describe this algorithm in detail. I. Compression o f M Defmition 3. Let M ~_ {t i = (t~, . ..,tn+k+l),t)ETiUO i i t 1, t 2 ~ M,
),
1 t I = (t~. . . . . t~, /n+h) , ]1 ~ k, t2= (t 2, • "'' tn+j2)' 2 J2 ~< k.
Let Z be the truth-value function• We say that t 2 implies t 1 (denoted t 1 ~_ t 2) if the formula i=1
( Z ( t ) = t2i) v ---~Z(t I -'/: O) & Z(tZi 4 0))
is valid. In this step all pairs (t ~, P) are analysed. If t i implies tJ then t i is deleted.
277
Fuzzy adaptive learning model
Begin
i
J
Modification of values of criteria
Ys
End
VIII I
Modification of M
N~I
!
Compression ( of M
[ Modification of E 1, E 2
Yes Fuzzy Bongard Model
Generation and comparison of alternatives H
Construction of tables for learning
III
Determining of preferences of S z,S2
Form,' :tion
of masks
IV
Formation of ! estimating sequences
I Generation
Identification of logical features
V
] of alternatives S 1, S 2
No,
Fig. 2.
278
A.F. Blishun
II. Construction o f tables for learning Let us consider two classes of pairs of alternatives: K , = {(u i, u/):ui e R ", u/ e R ' \ R " } , K2 = {(u/, u i) : u i ~ R", u j e R ' \ R " } . Tables for learning K r, K2r are formulated when K1, K 2 have the following properties: (:) r ~ # 0 , r E # 0 ; (2) Vi 3j: w(u j) < w(ui), u i ~ R", uJ e R ' \ R " . Each pair of alternatives (u ~, u/) ~ K~ U K2 defines a row of the table for learning
¢; = ( o ~ , . . . ,
v~),
where -1 v~=
if value of feature u, is not defined,
1 + pt(u~) - p(u~) 2
otherwise.
Then we have
KT= {v'J = (v~ . . . . .
vg},
I ~ = {vJ ~ = (v~ . . . . .
v~)}.
III. Formation of masks Defmition 4. An n-tuple of numbers n I = (n{ . . . . , n~) is called the mask of n-tuple vO = (o~, . . . . v °) if nt/ is defined as follows. i n/I=
-
ifv~/>0"5+e' if 0 < v ~ < 0 . 5 -
e,
0< e<0.5.
otherwise, Delifition 5. Let u j = (u~ . . . . . u~) and u i= ( u i : , . . . , u~) be two n-tuples of values of associated features Ul. . . . , un. The l-th feature is called (a) an essential feature, if p(u~) #p(u}); (b) an e-essential feature, if ~o(u~)-p(u~) I > e. The set of all masks N = {n I} over K r U K r defines possible index sets of 2e-essential features. IV. Identification o f useful logical features Delifition 6. A logical feature of the table K = {v o = (v ij, . . . . v~)} is defined as follows:
¢pi,,~(vq) = /~ F(Dn,,(v~), D~(vi/)), v#¢K
Fuzzy adaptivelearningmodel
279
where
o,/(v'9
ns e N ,
= (v~)"~,® . . . ® (v~)"'.,
fl-v~/, (vi/)'~,=~vi/, 1.1,
N={nl};
~)=v
or ^,
if n{< O, if n{>O, if n{= 0;
and F is a logical function of two variables. Each table K T (i = 1, 2) is separated into working and verifying tables Ki~, Kf~: T K T = K,T U K,T~, K,~ f) K,oT = O;
T r,~, K,oT -# O.
Let C~, C~ (l = 1, 2) be the calculated values of logical features o v e r KiT, Kiro, e~ the maximal admissible distinction of working and verifying tables, and e2 the minimal admissible distinction of tables for learning K~r, K r. Then the value of a logical feature over K1r is given as follows: ~, = C ~ ^ C'o.
Definition 7. A logical feature is called an useful logical feature, if (1) A/~/< El, l = 1, 2; (2) A ~ I 2 > e2; w h e r e a ~ , = C'~ v C'~ - C'~ ^ C'~, At,12 = ~ , v ~2 - Ul ^ ~2. Definition 8. Useful logical features 01 . . . . . either/z~ < P2 or/z~- < ~ti-, where r
,,;-=
v
0r form a basis of logical features, if
r
"- =
!=1,2.
If the number of useful logical features in some basis is minimal, then the basis is a minimal basis. In this step all useful logical features are identified.
V. Generation of alternatives If a preference relation obtained by means of (1) does not coincide with a preference relation obtained by means of useful logical features then modification of M is necessary. The above preferences are defined on a set of alternatives which are generated in the following way. Each set t ~ M can be represented as a strict point in the semantic space. Elements of T~, Ir, I =g,, are evenly distributed over the interval [1,g/]. In this case, 1 corresponds to the best value from T~, g,. corresponds to the worst value from T~. An experience of this representation is confirmed by the second hypothesis and unimodal membership function /tt(p). The above simplification allows using in the algorithm only the operations on fuzzy sets of type 1. The algorithm which can deal with fuzzy sets of type 2 has been considered in [2]. We obtain the following condition of generation of pairs of alternatives (s 1, s2),
A.F. Blishun
280
s ~= (sl . . . . .
sa.), s 2 = (s~ . . . . .
s;.), s i t [1, g,]:
m a x Is/1 - s/Zl = 1. i
VI. Determining preference of generated alternatives
$1, S 2
F o r s 1, s 2, the set of c o m p a r i s o n values is c o m p u t e d as
s~
(sl ~,
1~
where S/12 =
1 + (s~i - s l ) / ( g ,
-
1)
2 T h e d e g r e e of m e m b e r s h i p of s ~2 in the l-th class is given by the equation
/U/(s12) =
k (1 - I * ; ( s
12) - / u / I ) ,
i=1
where
#~= o/~Kr c~i(v), If/Ul(S lz)
>/U2($12), t h e n
l = 1, 2. s ~ > s 2. If/ua(s 12)
VII. Formation and modification of estimating sequences of s 1, s z Definition 9. L e t s = ( s l , . . . , s,) be an alternative. A sequence of n u m b e r s E = (el . . . . . e,+k÷~) is called an estimating sequence of s if: (1) e~ =si for i = 1 . . . . . n; (2) for i = n + l ..... n+k, ei =
max t i; t:s~=tj, j = l ..... n t~(tl,...,tn+k+l)EM
(3) e,+k+ 1 = t~+k+l, such that (a) t = ( $ 1 , . . " , Sn, S nt + l , . . . , S nt + k , t n + k + l ) (b) I s ; - e , l < 0 . 5 , i = n + l .... ,n+k.
E M;
L e t E 1, E 2 be estimating sequences of s 1, S 2 and E 1 > E 2 if sl+k+l < s2~+k+~, E ~ < E 2 if Sn+k+ 11 > Sn+k+1.2 L e t critieria be c o r r e c t e d only in E 1, E 2. L e t e 3 be a correction step value e of criterion guesses for t e M. T h e set I = { i x . . . . , in} contains n u m b e r s of features which characterize the i-th local criterion, that is I = {j ] 3 t = ( q , . . . ,
tn+e)EM:tj~0
and tn+i--#O, j<-n}.
T h e n the modification of local criteria is given by 1 e , + , -- g.+, ^ ( l v (e,+i~ F,(gn+i _ 1))), 2 _ ( l v 2 e,+i--gn+i A (en+ i - e(g,+ i _ 1))),
281
F u z z y adaptive learning m o d e l
where ei
ife~+n< en+i 2 andVl•Is~>~s2; • 1 > 2 I - 1 ~ _2 - 3 lfen+i en+iandVtE-at"~at; otherwise.
e=
The modification of the global criterion is given by e n+k+l I = g n + k + l A (1 v ( e n1+ k + 1 + e ( g n + k + 1 -- 1))), eZn+k+l = g n + k + l A
(1 v
(e2+/+1 -
e(g.+,+, -
1))),
where E~
(i 3 -- 3
if E 1 > E 2 and s I ( S 2, if E 1 < E 2 and s I >s 2, otherwise.
VIII. Modification of M Let Ne~, N~ be sets of all numbers of lists from M, which are used in the construction of E 1, E2;/Vex2 = Nle \N~, N~ 1 = N~ \N~. We substitute M by 37/: /~{pi
~-~
(D~ . . . . .
iO/+k+l)},
where I p~ if i ~ N~2 U N~ 1, if p~ = 0, /~ [ e} if i e Ne12, Le2 i f i e N ~ 1.
4. Conclusion This paper describes a model which is able to modify a priori linguistic rules obtained from an expert. The corrected rules confirm the choice of the decision-maker. Apparently it gives a possibility to describe a numerical example in detail and to consider the model in all theoretical aspects, which will be done in a next paper.
References [1] A.D. Allen, Measuring the empirical properties of sets, IEEE Trans. Systems Man Cybernet. 4 (1974) 66-73. [2] A.F. Blishun, Formation of preference relation in fuzzy descriptions, Techn. Cybernet. 2 (1981) 204-210 (in Russian). [3] A.F. Blishun, Modelling of decision making process in fuzzy environment based on similarity of concept sets, Thesis, Computing Center of the Academy of Science, Moscow (1981) (in Russian).
282
A.F. Blishun
[4] J. Efstathiou and V. Rajkovic, Multiattfibute decision making using a fuzzy heuristic approach, IEEE Trans. Systems Man Cybernet. 9 (1979) 326-333. [5] C.E. Osgood, G.J. Suci and P.H. Tannenbaum, The measurement of meaning (Urbana, 1957) 290-304. [6] T.L. Saaty, Measuring the fuzziness of sets, J. Cybernet. 4 (4) (1974) 53-61. [7] U. Thole, H.-J. Zimmermann and P. Zysno, On the suitability of minimum and product operators for the intersection of fuzzy sets, Fuzzy Sets and Systems 2 (1979) 167-180.
FUZZY ADAPTIVE LEARNING D E C I S I O N - M A K I N G PROCESS
273
MODEL OF
A.F. BLISHUN Institute of Automated Management Systems of The Ministry of Fishing Industry, Moscow, USSR Received February 1985 Revised March 1985 A mathematical model is described which generalizes fuzzy information to adopt to a new situation. This model is able to modify a priori information obtained from experts. Preferences of experts are realized by means of a set of strictly unformed criteria which can be described in terms of a composite linguistic variable. Using a decision maker's selection of a particular alternative from a set of possible ones the model improves its performance. It replaces a priori linguistic rules by refined linguistic rules. These rules correspond to the choice of the decision-maker.
Keywords: Linguistic variable, Fuzzy scale, Multi-attribute decision, Adaptive learning.
"An exact description of any real physical situation is virtually impossible. This is a fact we have had to accept and adjust to." (R. Bellman, M. Giertz, 1973) 1. Introduction The management of complex systems often needs nonformal reasoning on the basis of intuitional subjective knowledge. During the automation of the management such reasoning cannot be described in an adequate quantitative form. In particular, there are many problems of modelling of a decision-making process with the following characteristics: presence of decision-maker, importance of subjective factors, dynamics of subjective factors. The solution of the problems mentioned above requires suitable mathematical models which are able to generalize fuzzy information to adopt to the new situation. These models have to be capable of improving their performance during operation. In this paper an adaptive model of this type is suggested. While making a decision the selection of a particular alternative from a set of possible ones is realized. When analysing the situation, possible alternatives can be mapped in attribute space or semantic space. In semantic space an alternative displays the attribute with certain degree. This degree is measured by means of a scale of measurements. It is well known, however, that in most cases a human's decision is vaguely or imprecisely defined so that the grade of attribute is measured by the expert very approximately. 0165-0114/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
274
A.F. Blishun
Therefore, a method of scaling must satisfy the following conditions [6]: (a) it should allow for a moderate uncertainty in judgments about grades of attribute without changing the judgment value on the scale; (b) strong changes of judgment should be reflected in appropriate variations up and down the scale; (c) the results of the model should not change drastically by making small changes in the judgment values. The fuzzy scale satisfies the above conditions. Definition 1. Let U be a nonempty set of empirical objects, R i (i = 1 , . . . , n) a set of relations over U, L _ R (the set of real numbers), T a set of fuzzy subsets of L, St (i -- 1 , . . . , n) a set of relations over T. Let Ri and Si be ki-place relations. Then a mapping M : U--* T is a fuzzy scale if R,(ul .....
uk,) = S , ( m ( u l ) , . . . , m(Uk,))
for all i e l = {1. . . . . n} and ( u l . . . . . Uk,) • U k'. It is clear that from relations between judgment values on fuzzy scale we are able to draw a conclusion about empirical relations between empirical objects as follows: empirical objects ul . . . . . Uk, are in relation R if and only if corresponding values m ( u l ) , . . . , m(Uk,) on the fuzzy scale are in relation S. In order to fit the above problems into a certain mathematical model it seems reasonable to assume the following conditions concerning the fuzzy scale. Hypothesis I. The fuzzy scale must ensure the evaluation of any grade of attribute.
Hypothesis II.
In the case of a finite number of elements of the fuzzy scale neighbouring values given an equal change of grade of attribute, i.e. the 'physical' distances of each two adjacent elements of the fuzzy scale in semantic space are equal [1, 5, 7]. Depending on the situation and the personal perceptions of experts the grade of the same attribute can be evaluated differently. Therefore while identifying real values of features with semantic images it is essentially a form of mapping between feature space (U) and semantic space (P). The mapping has to preserve the relative place of objects in semantic space. A natural way to formalize the concept of fuzzy mapping is via the concept of fuzzy property [1]. Defmition 2. A property p defined on an object u is a function p : U--* [0, 1] such that p ( u ) > 0 iff u has the property p. Definitions 1 and 2 follow by the fact that grades of attribute correspond to an ordered set P = {p(u) : u e U}. Consequently, each element of the fuzzy scale corresponds to a fuzzy subset of P. According to Definition 1 the attribute can be
Fuzzy adaptive learning model
275
)a,
/
/
/
,J
/ / Fig. 1.
interpreted as a linguistic variable whose values are the labels of fuzzy subsets of P. Figure 1 shows the correspondence of judgements about grades of attribute in semantic space and feature space. When fuzzy scales are used reception of supplementary information or change of personal opinion entails their adjustment by means of either a displacement of fuzzy scale elements or a change of mapping p. This paper deals with the first case only.
2. Formulation of the problem Let a rational decision choice be realized on the basis of a collective of strictly unformed criteria, which cannot be described in an adequate analytical form. In this case preferences are described by the following linguistic statement type: "if values of features u l , . . . , u n which characterize an alternative u i are avaluated by terms tl~, . . . , t~i, then u i satisfies the j-th criterion with value t~+j,i".
Every alternative can be described in terms of a composite linguistic variable L = ( L I. . . . .
L ......
Ln+k, Ln+k+l),
where Li is a unary linguistic variable with term set T~, with universal set U~, and with basic variable ui, tit e Tt c F(U~). Values from term set ti~ e T~, n + 1 ~< i ~< k + n + 1, point out the grade of criterion satisfaction. T~ consists of linguistic values of linguistic variable.
A.F. Blishun
276
A set of values of L is separated in two subsets: MI=
{tli, ....
tni, tn+j,i), n + l <~j <<-n + k}[-T 1,
M 2 = {(t,+Li, . . . .
tn+k+l,i)}i=~ 2.
M1 characterizes local criteria and M 2 characterizes the global criterion. M = M~ U M2 consists of a priori information which is obtained from an expert. Each t, t e M, can be represented as a fuzzy point in a subspace of D, where D is represented as the Cartesian product of feature space and criteria space [4]. In reality, any value of feature u~ has correspondent degree of property Pi. Hence u i defines the point (Pl(Ul) . . . . . pn(un)) in the semantic space/'1 x . • • x P . and if t i E M then t; defines the fuzzy point t~(p(.)). For an alternative u ~the grade of global criterion satisfaction w ( u i) is computed as
w(ui)=t,UM2(tlUMtui°tl)°tr.
(1)
Assume that a set of alternatives R ' = {ui= ( u ~ , . . . , u/)} is given to the decision-maker. He chooses a set of alternatives R" from the given set R': R" ~_R'. Using the preference u~> uJ, u j e R ' \ R " , u~e R", it is necessary to replace M by some At which confirms the above preference in the following way: if(u;) > ff(u j)
3. A l g o r i t h m
for all u i e R '', u J e R ' \ R
".
of solution
A block diagram of fuzzy adaptive learning algorithm is shown in Figure 2. Let us describe this algorithm in detail. I. Compression o f M Defmition 3. Let M ~_ {t i = (t~, . ..,tn+k+l),t)ETiUO i i t 1, t 2 ~ M,
),
1 t I = (t~. . . . . t~, /n+h) , ]1 ~ k, t2= (t 2, • "'' tn+j2)' 2 J2 ~< k.
Let Z be the truth-value function• We say that t 2 implies t 1 (denoted t 1 ~_ t 2) if the formula i=1
( Z ( t ) = t2i) v ---~Z(t I -'/: O) & Z(tZi 4 0))
is valid. In this step all pairs (t ~, P) are analysed. If t i implies tJ then t i is deleted.
277
Fuzzy adaptive learning model
Begin
i
J
Modification of values of criteria
Ys
End
VIII I
Modification of M
N~I
!
Compression ( of M
[ Modification of E 1, E 2
Yes Fuzzy Bongard Model
Generation and comparison of alternatives H
Construction of tables for learning
III
Determining of preferences of S z,S2
Form,' :tion
of masks
IV
Formation of ! estimating sequences
I Generation
Identification of logical features
V
] of alternatives S 1, S 2
No,
Fig. 2.
278
A.F. Blishun
II. Construction o f tables for learning Let us consider two classes of pairs of alternatives: K , = {(u i, u/):ui e R ", u/ e R ' \ R " } , K2 = {(u/, u i) : u i ~ R", u j e R ' \ R " } . Tables for learning K r, K2r are formulated when K1, K 2 have the following properties: (:) r ~ # 0 , r E # 0 ; (2) Vi 3j: w(u j) < w(ui), u i ~ R", uJ e R ' \ R " . Each pair of alternatives (u ~, u/) ~ K~ U K2 defines a row of the table for learning
¢; = ( o ~ , . . . ,
v~),
where -1 v~=
if value of feature u, is not defined,
1 + pt(u~) - p(u~) 2
otherwise.
Then we have
KT= {v'J = (v~ . . . . .
vg},
I ~ = {vJ ~ = (v~ . . . . .
v~)}.
III. Formation of masks Defmition 4. An n-tuple of numbers n I = (n{ . . . . , n~) is called the mask of n-tuple vO = (o~, . . . . v °) if nt/ is defined as follows. i n/I=
-
ifv~/>0"5+e' if 0 < v ~ < 0 . 5 -
e,
0< e<0.5.
otherwise, Delifition 5. Let u j = (u~ . . . . . u~) and u i= ( u i : , . . . , u~) be two n-tuples of values of associated features Ul. . . . , un. The l-th feature is called (a) an essential feature, if p(u~) #p(u}); (b) an e-essential feature, if ~o(u~)-p(u~) I > e. The set of all masks N = {n I} over K r U K r defines possible index sets of 2e-essential features. IV. Identification o f useful logical features Delifition 6. A logical feature of the table K = {v o = (v ij, . . . . v~)} is defined as follows:
¢pi,,~(vq) = /~ F(Dn,,(v~), D~(vi/)), v#¢K
Fuzzy adaptivelearningmodel
279
where
o,/(v'9
ns e N ,
= (v~)"~,® . . . ® (v~)"'.,
fl-v~/, (vi/)'~,=~vi/, 1.1,
N={nl};
~)=v
or ^,
if n{< O, if n{>O, if n{= 0;
and F is a logical function of two variables. Each table K T (i = 1, 2) is separated into working and verifying tables Ki~, Kf~: T K T = K,T U K,T~, K,~ f) K,oT = O;
T r,~, K,oT -# O.
Let C~, C~ (l = 1, 2) be the calculated values of logical features o v e r KiT, Kiro, e~ the maximal admissible distinction of working and verifying tables, and e2 the minimal admissible distinction of tables for learning K~r, K r. Then the value of a logical feature over K1r is given as follows: ~, = C ~ ^ C'o.
Definition 7. A logical feature is called an useful logical feature, if (1) A/~/< El, l = 1, 2; (2) A ~ I 2 > e2; w h e r e a ~ , = C'~ v C'~ - C'~ ^ C'~, At,12 = ~ , v ~2 - Ul ^ ~2. Definition 8. Useful logical features 01 . . . . . either/z~ < P2 or/z~- < ~ti-, where r
,,;-=
v
0r form a basis of logical features, if
r
"- =
!=1,2.
If the number of useful logical features in some basis is minimal, then the basis is a minimal basis. In this step all useful logical features are identified.
V. Generation of alternatives If a preference relation obtained by means of (1) does not coincide with a preference relation obtained by means of useful logical features then modification of M is necessary. The above preferences are defined on a set of alternatives which are generated in the following way. Each set t ~ M can be represented as a strict point in the semantic space. Elements of T~, Ir, I =g,, are evenly distributed over the interval [1,g/]. In this case, 1 corresponds to the best value from T~, g,. corresponds to the worst value from T~. An experience of this representation is confirmed by the second hypothesis and unimodal membership function /tt(p). The above simplification allows using in the algorithm only the operations on fuzzy sets of type 1. The algorithm which can deal with fuzzy sets of type 2 has been considered in [2]. We obtain the following condition of generation of pairs of alternatives (s 1, s2),
A.F. Blishun
280
s ~= (sl . . . . .
sa.), s 2 = (s~ . . . . .
s;.), s i t [1, g,]:
m a x Is/1 - s/Zl = 1. i
VI. Determining preference of generated alternatives
$1, S 2
F o r s 1, s 2, the set of c o m p a r i s o n values is c o m p u t e d as
s~
(sl ~,
1~
where S/12 =
1 + (s~i - s l ) / ( g ,
-
1)
2 T h e d e g r e e of m e m b e r s h i p of s ~2 in the l-th class is given by the equation
/U/(s12) =
k (1 - I * ; ( s
12) - / u / I ) ,
i=1
where
#~= o/~Kr c~i(v), If/Ul(S lz)
>/U2($12), t h e n
l = 1, 2. s ~ > s 2. If/ua(s 12)
VII. Formation and modification of estimating sequences of s 1, s z Definition 9. L e t s = ( s l , . . . , s,) be an alternative. A sequence of n u m b e r s E = (el . . . . . e,+k÷~) is called an estimating sequence of s if: (1) e~ =si for i = 1 . . . . . n; (2) for i = n + l ..... n+k, ei =
max t i; t:s~=tj, j = l ..... n t~(tl,...,tn+k+l)EM
(3) e,+k+ 1 = t~+k+l, such that (a) t = ( $ 1 , . . " , Sn, S nt + l , . . . , S nt + k , t n + k + l ) (b) I s ; - e , l < 0 . 5 , i = n + l .... ,n+k.
E M;
L e t E 1, E 2 be estimating sequences of s 1, S 2 and E 1 > E 2 if sl+k+l < s2~+k+~, E ~ < E 2 if Sn+k+ 11 > Sn+k+1.2 L e t critieria be c o r r e c t e d only in E 1, E 2. L e t e 3 be a correction step value e of criterion guesses for t e M. T h e set I = { i x . . . . , in} contains n u m b e r s of features which characterize the i-th local criterion, that is I = {j ] 3 t = ( q , . . . ,
tn+e)EM:tj~0
and tn+i--#O, j<-n}.
T h e n the modification of local criteria is given by 1 e , + , -- g.+, ^ ( l v (e,+i~ F,(gn+i _ 1))), 2 _ ( l v 2 e,+i--gn+i A (en+ i - e(g,+ i _ 1))),
281
F u z z y adaptive learning m o d e l
where ei
ife~+n< en+i 2 andVl•Is~>~s2; • 1 > 2 I - 1 ~ _2 - 3 lfen+i en+iandVtE-at"~at; otherwise.
e=
The modification of the global criterion is given by e n+k+l I = g n + k + l A (1 v ( e n1+ k + 1 + e ( g n + k + 1 -- 1))), eZn+k+l = g n + k + l A
(1 v
(e2+/+1 -
e(g.+,+, -
1))),
where E~
(i 3 -- 3
if E 1 > E 2 and s I ( S 2, if E 1 < E 2 and s I >s 2, otherwise.
VIII. Modification of M Let Ne~, N~ be sets of all numbers of lists from M, which are used in the construction of E 1, E2;/Vex2 = Nle \N~, N~ 1 = N~ \N~. We substitute M by 37/: /~{pi
~-~
(D~ . . . . .
iO/+k+l)},
where I p~ if i ~ N~2 U N~ 1, if p~ = 0, /~ [ e} if i e Ne12, Le2 i f i e N ~ 1.
4. Conclusion This paper describes a model which is able to modify a priori linguistic rules obtained from an expert. The corrected rules confirm the choice of the decision-maker. Apparently it gives a possibility to describe a numerical example in detail and to consider the model in all theoretical aspects, which will be done in a next paper.
References [1] A.D. Allen, Measuring the empirical properties of sets, IEEE Trans. Systems Man Cybernet. 4 (1974) 66-73. [2] A.F. Blishun, Formation of preference relation in fuzzy descriptions, Techn. Cybernet. 2 (1981) 204-210 (in Russian). [3] A.F. Blishun, Modelling of decision making process in fuzzy environment based on similarity of concept sets, Thesis, Computing Center of the Academy of Science, Moscow (1981) (in Russian).
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[4] J. Efstathiou and V. Rajkovic, Multiattfibute decision making using a fuzzy heuristic approach, IEEE Trans. Systems Man Cybernet. 9 (1979) 326-333. [5] C.E. Osgood, G.J. Suci and P.H. Tannenbaum, The measurement of meaning (Urbana, 1957) 290-304. [6] T.L. Saaty, Measuring the fuzziness of sets, J. Cybernet. 4 (4) (1974) 53-61. [7] U. Thole, H.-J. Zimmermann and P. Zysno, On the suitability of minimum and product operators for the intersection of fuzzy sets, Fuzzy Sets and Systems 2 (1979) 167-180.