Spatial fuzzy consumer's behavior : a multidimensional analysis

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Spatial fuzzy consumer’s behavior : a multidimensional analysis

Claude Ponsard

To cite this version:

Claude Ponsard. Spatial fuzzy consumer’s behavior : a multidimensional analysis. [Research Report]

Institut de mathématiques économiques (IME). 1983, 24 p., figures, bibliographie. �hal-01533737�

EQUIPE DE RECHERCHE ASSOCIEE AU C.N.R.S.

DOCUMENT DE TRAVAIL

INSTITUT DE MATHEMATIQUES ECONOMIQUES

UNIVERSITE DE DIJON

FACULTE DE SCIENCE ECON OMIQUE ET DE GESTION 4, BOULEVARD GABRIEL - 21000 DIJON

SPATIAL FUZZY CONSUMER'S BEHAVIOR A multidimensional Analysis

N° 63

Claude PONSARD June 1983

Regional Science Association Twenty-Third European Congress

Poitiers (France) 30 August - 2 September 1983

S P A T I A L F U Z Z Y C O N S U M E R ' S B E H A V I O R A multidimensional Analysis

C. PONSARD

Institute of Economic Mathematics

4, boulevard Gabriel - 21000 Dijon (France)

Abstract.

This paper is devoted to a multidimensional analysis of the consumer's behavior when the decisionmaker is acting in a fuzzy space and pointing out an imprecise attitude.

At first, the process of decisionmaking is described with the help of three relationships between the set of goods which are supplied in seve­

ral points, the set of their characteristics and the set of consumer's a priori possible behaviors. All these relations are fuzzy. The model applies the theory of fuzzy relations equations.

Then, the stages of the decision process are analyzed. Often fuzzy com­

portemental relations are like "black boxes". The mathematical solution of the model indicates in which conditions their valuations are possi­

ble.

The main interest of this method is to do without additive operations on subjective items and to use operators which are coherent with the fuzzy nature of the variables.

1. Introduction.

In Operational Research, especially in marketing and advertising stu­

dies, searchers have often to solve the following problem : what will be the behavior of a group of individuals in presence of a lot of consump­

tion goods which are supplied in some scattered supply points and have

numerous and various characteristics ?

The process of decisionmaking to be analyzed lays on subjective and qua­

litative relationships : some relations express the perception of real or fancied characteristics which are associated with goods and spatial structures of markets ; other relations formulate the consumers' atti­

tude with respect to these characteristics ; at last, other relations express the revealed behaviors as to the goods such as they have been appreciated and valued.

The aim of this paper is to present a model which fully takes into account the subjective and qualitative nature of the relationships bet­

ween man, space and goods. This model applies the theory of fuzzy rela­

tions equations. Initially, this theory was formulated by Sanchez (1974 and 1976). Then it was set out and developed by several authors (see especially Kaufmann 1977 ; Prevot 1977 and 1981 ; Viganza 1982 ; Pedrycz 1983). Recently Zu-Wei (1980) introduced the necessary and suf­

ficient conditions for the existence of solutions of fuzzy equations, without developing the proofs. They will be supplied in the Appendix next to this paper.

The following model is different, but complementary, of neoclassical fuzzy spatial models. In these ones, relationships between man, space and goods are expressed with fuzzy preferences relations which allow, under certain conditions, to define fuzzy utility functions (Ponsard 1979 and 1981 a). In this framework, consumer's partial equilibrium is the result of a fuzzy economic calculation : to maximize the fuzzy uti­

lity function with an elastic ressources constraint (Ponsard 1981 b).

The same solution is available for the producer's equilibrium analysis, by means of suitable adaptations (Ponsard 1982 a). At last, in the par­

ticular case where the objective is precise and the constraint alone is fuzzy, the economic calculation is technically simpler (Ponsard 1982 b).

3

Now the aim of the present approach is to apply a multidimensional ana­

lysis to the exploration of the consumer's decisionmaking mechanism in a fuzzy context. At first, the complete decisionmaking process is des­

cribed and formalized. Then, its different steps are analyzed in de­

tails. The conditions, which must be verified in order that the rela­

tions specifying each phasis of the process be compatible with each other, are proved. They express the economic conditions for the fuzzy behavior coherence.

Remark. In order to avoid any ambiguity in the notation of mathematical terms, ordinary (non fuzzy) concepts are underlined, whereas fuzzy con­

cepts are not.

2. The decisionmaking process.

The economic decisionmaker is an individual consumer or an homogeneous group of consumers. Its residential location is given. It enables to set out the delivered price system for the consumer. But consumption can take place either at the consumer's residence or at any other point. In this last case, the consumer's travelling expenses are incorporated in the delivered price.

I, t], be a set of t spots indexed s where the goods are supplied.

In order to express that a good can be supplied at several spots and a spot can supply several goods, we define a mapping, denoted by f, from P x S into rN+ such that :

C1,r] be a set of r goods indexed p, and

f : P x S (p,s)

■t>rN+

f (p,s) = i, i G I_, fN+

with i = (p - 1 )_S + 1, pS> 11 p E P

where _S = Card S_

We put f (r,t) = m. Thus _I = [1,m].

Thus, we define the set of located goods : ii x L = — with A = { xi ) > i ^ I_.

Example. The above mentioned formulation is complicated but it expresses a very simple relationship. It enables to indicate the nature of a given good and its supply spot with the help of one index only so that the symbols below are made easier.

Let four goods and three supply spots. Therefore = 3. We have (See Table 1).

Next, will be written in the form of a vector.

The located goods have numerous intrinsic characteristics. Also, a given good has many characteristics according to its supply spot.

indexed k. For example, k = 1 means the aptitude of a located good to satisfy a given need ; k = 2 designates the value which is associated with the state of contentment furnished by a consumption ; k = 3 indica­

tes the delivered price level such as the consumer appreciates it ; k = 4 means the quality imputed to a located good ; k = 5 designates the accessibility of goods depending on their supply spots location ; and so on.

possible behaviors that the consumer can have regarding the located goods and their characteristics. This set is not reduced to the purcha­

sing behavior alone ; it includes all the elements which characterize the spatial consumption behavior. For example, j = 1 designates the in­

tention of purchasing some quantities of goods at some supply spots ; j = 2 means the attachment to a trademark ; j = 3 indicates the propen­

sity to imitate other consumers' behavior ; j = 4 designates the prefe­

rence for a given supply spot ; and so on.

Let

At last, , with j € _J, = [1,n], the set of n a priori

4bis

Table 1

g1 (1,1) (1,2) (1,3)

X1 X2 X3

(2,1) (2,2) (2,3)

X4 X5 X6

g3 (3,1) (3,2) (3,3)

.../

X7 X8 X9

g4 (4,1) (4,2) (4,3)

X10 X11 X12

G x L X

5

The relationships between the sets X, X anc* — are iike "black boxes", in the sense that they are very imprecise. First, the consumer has only exceptionally a complete information relating to his environment. Gene­

rally, he knows imperfectly the space where he lives. Then, man is not a robot and the consumer has not a perfect aptitude to discriminate bet­

ween goods and supply spots. He does not necessarily associate an exact valuation to a given characteristic. He does not always know if a good supplied at a given spot has a wanted characteristic. Accordingly, the decisionmaking is subject to a weakened rationality.

Therefore, to consider the relationships between X.» X anc^ — as fuzzY bi­

nary relations and to analyze them with the help of fuzzy relations equations theory is pertinent.

The decisionmaking process includes three stages : treatment of informa­

tion by the consumer, valuation of characteristics and revealed beha­

vior.

Treatment of information.

At first, a good x^ has more or less a given characteristic depending on the consumer's appreciation and/or on the supply spot.

This treatment of data is dependent on the perception and the apprehen­

sion of the available information relating to spatial environment and real or fancied characteristics of goods. It also depends on learning by doing phenomenon.

It means that x^ is set in correspondence with y^ at a given degree, de­

noted by y (x^, y^). We put that y (x^, y^) €[0,1]. Thus, a fuzzy bina­

ry relation, denoted by A, between the elements of and X is defined.

Relation A is a fuzzy subset of X x Y such that :

To simplify, we put : y^ (x^, y^) = a ^ .

Thus means the valuation assigned to the located good x^ for the characteristic that it possesses more or less in the consumer's opi­

nion.

Valuation of characteristics

A characteristic y^ is more or less interesting for the consumer. It has a relative value depending on the importance attributed to a state of satisfaction. In other words, the consumer's attitude with respect to goods and supply spots characteristics leads to the valuation of these characteristics.

Thus y^ is set in correspondence with Zj at a given degree, denoted by y (y , z.), with y (y. , z.) G [0,1]. Thus a fuzzy binary relation, deno-

k J k J

ted by B, between the elements of and ~L_ is defined. Relation B is a fuzzy subset of V x Z such that :

B = l ^ y^{k ’ Z j ^ ’} e e : ^{P B * y k ’ z j '} 6 ^•

We put : PB (yk , z.) = bkj .

Thus means the valuation assigned to the characteristic for the state of satisfaction z . (a priori possible behavior) associated to it

J in the consumer's appreciation.

Revealed behavior.

At last, a located good x^ is more or less matter of a possible beha­

vior.

Thus x^ is set in correspondence with z^ at the degree y (x^, z^), with y (x ., z.) G [0,1]. A fuzzy binary relation, denoted by C, between the

^ J

elements of X. and Z_ is defined. Relation C is a fuzzy subset of X x Z

We put : VV (x., z.) = c. ..

K C ¹ ’ j ij

Thus Cjj is the valuation of the located good x^ with respect to the consumer's possible behavior z . . If we take again the above examples,

J

it reveals the intention of buying a very great, great, mean, small or null quantity of good x^, the degree of attachment to a trademark, the intensity of other consumers' imitation, the strengh of the preference for a given supply spot ; and so on.

The model of decisionmaking.

The complete model of decisionmaking is summarized on Figure 1.

Relation A expresses the step of information treatment, relation B the step of characteristics valuation and relation C represents the result (revealed behavior).

In usual consumption behavior models, especially in Expectancy-Value Theory (for a survey, see Martin 1976), X is a vector of non located goods and /\ a Boolean matrix : a good has or has not a given characte­

ristic. Moreover j3 and are matrices whose elements are measures, not valuations (fuzzy measures). At last, the authors put the relation : Ç = A x B which raises up numerous controversies concerning the additi­

vity property and the interpretation of the product.

All these models lay on a particular specification. The following ana­

lysis requires different operators.

The composite fuzzy relation of A and B is the "max-min" composite rela­

tion, denoted by B o A, which is expressed, with the previous symbols :

•‘b .a**- 2) = ^ [aik A bkj]

where v and a designate the maximum and the minimum of the membership functions respectively. In this formulation, the "min" operator expres­

ses the logical connective "and" (fuzzy intersection) for any couple of elements which are taken into A and B. The "max" operator means the re-

7bis

A c X x Y

0

0

Figure 1

suit which is obtained for all the couples (fuzzy union).

Usual operators are particular cases of them. Indeed, in the ordinary (non fuzzy) case, we have :

aik 6 [°’1 H d b kj 6 (°'1 }

and the composite relation of A and B is written :

PB c A ( x, z) = t aik . bk .

where (.) indicates the Boolean multiplication and t the Boolean sum of k

the products.

Illustration (I).

Let three finite sets :

X. = £ x-|> x² ’ x³ ’ x4 ) a se^ ^our located goods

— = { ^1’ ^ 2’ ^3 J 3 S6^ three characteristics

1_ - ^{z 2} | a se^ ^wo a Priori possible behaviors

and let A c X_ x Y_ and B c H Z be two fuzzy binary relations such that (see Table 2).

Then, B o A = C gives, by application of the max-min composite relation formula : (see Table 3).

3. Analysis of the decisionmaking process stages.

Now two interesting problems can be studied : in which conditions does equation B o A = C admit a solution B and a solution A ? In other words, is it possible to make explicit the valuation of characteristics if A and C are known ? In the same manner, is it possible to make explicit the step of information treatment if B and C are known ? The solution of the first problem will be explained thoroughly. The so­

lution of the second question will be given without proof because it follows the same arguments. Moreover, in practice, relation B is the

8bis

Table 2

Vi v2 y3

X1 0.1 0.7 0.3

X2 0.8 0.3 0.6

x3 0.6 0.5 0.4

x4 1 0.6 0.3

B = y

Table 3

C =

Z1 z2

X1 0.2 0.4

X2 0.7 0.5

X3 0.6 0.5

x4 0.7 0.5

, C c X. x 1

most difficult for empirical observation.

Solution B of equation B o A = C. General case.

The necessary and sufficient conditions for the existence of solutions B of equation B o A = C are set out by the following theorem.

Theorem. Equation B o A = C has a solution B = A' (S) C if and only if : (1) V a.. > V c . • , V k 6 K , M j 6 J , V i £ l

k 1 j J

i.e./] T = ( k 6 H I aik > Y cij \ * 0 » y 1 6 I J

(2) V U (b a c )\ » V Cii , V i £ 1 , y p €1. , V j € J.

k€ T v PJ / j J

where A' designates the transpose of A and a an operator such that c = a a b with c = 1 if a 4 b

c = b if a > b.

The proof is tedious. It is stated in the Appendix (see Definition (1) and Theorem (3)).

The theorem demonstrates the existence of solutions, but not its unique­

ness. In the Appendix, we shall prove that the above solution B is the greatest element in the set of possible solutions, i.e. the solution which overranks all the others. The non-uniqueness of the solution may appear as a deceptive result, but we must recall that the problem is put in a fuzzy context (for a complete mathematical statement, see Appendix, Theorems (1) to (5)).

Illustration (II).

We consider again illustration (I). Now A and C are given. We would com­

pute B.

First we verify if conditions (1) and (2) of the above theorem are sa­

tisfied.

9

(1) k aik > j c ij •

10

al2 II O C12 = 0.4 a21 = 0.8 >

C21 = 0.7 a31 = 0.6)/

C31 = 0.6 a41 II

C41 = 0.7 Condition (1) is satisfied.

(2) . V fA (a, k 6 T (p kp « c .) \ \ pj ) '/ j ij V c . . with T = {k € K I sik >/ y Cjj ]

For i = 1, we have : T* = ^{£ 2} j

For i = 2, 3, 4, we have : T 1 = {l ^ . We have :

T

Solution B = A' (a) C. (see Table 4)

Matrix B of illustration (I) is found again.

Particular case.

The solution of the preceding problem can be improved in the particular case where matrix A contains one and only one element for a row which is equal to 1, all the others being null. This formulation fits the hypo­

thesis in which the consumer, during the phasis of information treat­

ment, bounds his appreciation to a single characteristic of the located goods, for example the delivered price, and does not take into conside­

ration the others. Or, it expresses that the consumer's data treatment is such that the most appreciated characteristic alone is chosen, all

10bis

Table 4

X1 X2 X3 X4

0.1 0.8 0.6 1

0.7 0.3 0.5 0.6

0.3 0.6 0.4 0.3 a

Z1 Z2

X1 0.2 0.4

X2 0.7 0.5

X3 0.6 0.5

x4 0.7 0.5

Z1 *2

the others being neglected, in the valuation of each located good.

In such a context, the set of solutions B of equation B ° A = C has the structure of a complete lattice. In other words, there exist a greatest and a smallest element which bound the set of solutions. Indeed, we prove that the greatest element, denoted by B, is equal to A' @ C, as in the general case ; the smallest element, denoted by B, is equal to A A' C where the toplined terms designate the pseudocomplements and A 1 the transpose of matrix A (see Appendix, Definitions (2) and (3) and Theorems (6) and (7)).

Illustration (III).

We consider again illustration (I). Now the matrix A is such that the maximal elements of each row are equal to 1, all the others being null.

Matrix B is kept.

We have : (see Table 5)

Like in illustration (II), we verify that the necessary and sufficient conditions of a solution B of equation B o A = C are fulfilled.

We have :

(1) Maximal solution : B = A' (a) C. (see Table 6) (2) Minimal solution : B = A' (a) C. (see Table 7)

(3) In the two cases, we verify that matrix C is found again. It suffi­

ces to compute (A o Ü) and (A o B) respectively.

Solution A of equation B o A = C .

Now, if the phasis of data treatment is unknown, the equation must be solved into A, relations B and C being given.

Theorem : Equation B o A = C has a solution A = (B (a)C')' if and only if (1> k bkj > Ï cij . w i e i , w j £ j , w k e K.

i . e . O = t , T =[k<= K I bk j > y C i , j / 0 W j S J 11

U b i s Table 5

y1 *2 y3 Z1 z2

0 1 0 Z1 Z2

X1 0.2 0.4

0.7 0.5 y1

1 0 0

X2 0.7 0.5

0 y2 0.2 0.4

1 0 0 X3 0.7 0.5

0.2 0.5 y3

1 0 0

X4 0.7 0.5

Table 6

Z1 Z2

y1 0.7 0.5

B = y2 0.2 0.4

y3 1 1

Table

Z1 z2

y1 0.7 0.5

^ B = y2V 0.2 0.4

y3 0 0

(2) k'sT (p (bKp ° W j > ï cij - « J « J - » p e J , w i e i-

where C' designates the transpose of C and (B (a) C 1)' the transpose of (B © C ' ) .

From this result, the analysis is analogous to the study which was sta­

ted for the valuation of the characteristics in the general case.

Illustration (IV).

Let the following equation :

(see Table 8)

After conditions (1) and (2) of the theorem have been verified, we have (see Table 9)

4. Conclusion.

The consumer does not always know what he wants exactly. The fuzziness which distinguishes his behavior may foil empirical policies, for exam­

ple marketing programming, advertising campaign or shops location choi­

ces. Also applied economics needs a theoretical model likely to forma­

lize the fuzziness of behaviors and to analyse their effects. The par­

ticular case where behavior is not fuzzy is not opposite : it is simply an extremal case where membership functions are Boolean.

The conditions for the existence of solutions of composite fuzzy rela­

tions equations have a manifest economic significance. They express the conditions which must be fulfilled in order that behaviors reveal at least a weakened coherence. When they are not verified, the observed behaviors are incoherent and the model permits to know where the causes of this inconsistency are situated.

At last, the above model should be still more thoroughly examined. In the general case, the set of solutions has the structure of a sup-semi- lattice. The minimal solutions bounding inferiorly the set of solutions will require to be studied.

12

12bis

y1 y2 y3

1 0.3 1

1 0.6 1

A

Table 8

Z1 z2

0.7 0.2

0.3 0.8

0.1 0.3

B C

A C K N O W L E D G E M E N T S 13

The author would like to thank Mr. Prevot, associated with the Institute of Economic Mathematics, for fruitfull discussions he had with him and Mrs Penez for her assistance in translating this paper into English.

14

Mathematical Appendix

Definition (1) : A Brouwer's lattice is a lattice, denoted by J-, in which for any couple of given elements a and b the set of all elements x, x G J_, such that a a x b contains a greatest element, denoted by a a b, called the relative pseudocomplement of a in b.

If L_ = [0,1] and if a, b G L^, then : a a b = 1 i f a ^ b

a a b : b if a > b

The operator © is neither commutative nor associative.

The relative pseudocomplement of a in b is unique.

Definition (2) : Let E be a set and L_ = [0,1] the set in which the ele­

ments of the subsets of have their values.

Let A and B be two fuzzy subsets of £. By definition, B is the pseudo­

complement of A if :

V x G E : pB (x) = 1 - (x).

We note : B = A. Clearly, A can be the pseudocomplement of A itself. We write then :

y x G E : (x) = 1 - yA (x).

Remark. These two definitions must be carefully distinguished. In defi­

nition (1), J_ is endowed with the properties of a Brouwerian lattice, while in definition (2) L_ is viewed as a membership set.

Theorem (1). Let A c x and C c X_ x 1_ . We have : (A' (a) C) o A c C

where A' is the transpose of the relation A.

Proof.

We put W = (A1 (a) C) o A

We have : yw (x,z) = V ^ A (x,y) A V a c (y,z))

by definition of the operation o (max-min composite fuzzy relation).

Let still :

yw (x,z) =

y

01 _yc(x’z))j

= y ( V x,y) A ( y A (x,y) a y c ( x ’z) A

\ v t e x

t i x yA (t,y) a Uc (t,z)fj

By definition of the operation a, we have : yw (x,z) * y^(yA (x,y) A (V X,y)a Uc (x’z)) yw (x,z) y (x,z).

[Q.E.D.]

Corollary. If B = A 1 (a) C is the solution of the equation B o A = C, then (A' @ C) o A = C.

- trivial - 15

Theorem (2). Let A c x Y. ar|d B e x we have : B e A' C.

The proof is similar to that of theorem (1).

Theorem (3). The equation B o A = C has a solution B = A' (a) C if and only if :

(1) k aik > y cij . v k e k , \/ j . i.e.^— T = ^ k G K. | ai(< >/ ^ c „ j i 0 , M i G

(2⁾ _{k G T}

y

T \ P kp PJ / J ij . - ’ J -

Proof.

Necessary conditions. In virtue of the corollary of theorem (1), if B = A 1 (a) C is the solution of the equation B o A = C, the equality (A1 (a) C) o A = C is verified. We must prove that the conditions (1) and

16

(2) must be fulfilled in order that this equality be true.

We have :

B „ A = (A' © C) . A = J v ^a,k * (a (akp o. cp .))J

= ilf Ca-kA <vi °L k \ ik k1 v > 1 j < v ... <\ “ki ij km c >)mn /

= L k V. ik / fa-, A (a. . a c. .) a (ki ij \p ^a (a kp a c .)))|pj //J

= I y Ia -I A (a., a c. .) L k y ik Ik IJ _{A ( A} \p (a kp a c .)m pjvyj

[ ï {

because a, . = a..ki ik

. I a .. a c . . a

k I ik ij [a (a. a c .))]

Ip kp PJ //

because a A (a a b ) = a A b

= [ ( k a ik) A C ij A ( k f p A ( a k p 0,cpj)|]

If condition (1) is not verified, i.e. if 3 i G I /

/a.. < V c. . , k ik j ij \t k G K , \j j G J, there is no solution, - J Indeed, let . c . . = c . „ v ... v c . . v... v c . = M

J ij i1 ij m

We have : if a.. a c. . a ( Y / A (a a c .))I [k ik; ij ]k \p kp P J //

y a \ k ikj a M a / y ( ^k \p a ( a kp a c .))) pj '/

-- ( k aik)A (k(pA ( \ p a c Pj ^ k S k < c ij

Consequently : B 0 A = (A1 @ C)c A / C. Condition (1) is necessary.

If condition (2) is not fulfilled, i.e. if

3 i ej. /k ^ (akp a cp J ))< y Cij , « P 6 I , tfj 6 J,» i S I there is no solution.

We have :

17

a (a, a c .)]

P kp PJ 1

B o A = (A' a C) o A = ^ |a.k a^,

; [{k e ï (aik A (j <akp a cPj I ] v (uVei-T ( aik Afi(\p a cpj^)

where 1 - T = G _K | a ^ < V Cij^ > ^ ^ I

But’ k e x f i k A ( j (akp “ "pj’)) =

= k € T (aik A (ak1 ° V A---A <aki “ ^ A---A <akm ° c, j )

= k € 7 (aik A (aik “ cij> A (? (V

cpj^)

because a.. = a. .lk ki

= k ï r ( ai k A c i j A (î ' V ^ p j ' ) )

= (k 6 * aik) A cij A (k 6 % (È (akp “ ^ j 1))

By condition ( 1 ) : / a )• V c . . , t f k G T > y j G J , W i G I

k i k j i j ’ I ’ ° — ’ —

Hence : . a.. >/ V c . . = c.„ v...v c . . v...v c .

k t y ik J ij i1 ij in

We have : }! a., >. c. . W i G I . t f j G J

k G T i k ^ i j — ’ J —

"e have ; (k 6 T aik)A cij A (k I 7 (A (\ p 01 "pj»)) =

= °ij A (k €T (p (akp “ cp j >)) » j £ J , # i 6 I , # p 6 I

= M A il _{^k G T1}

C

= I C m ( A (ai Ot C . ) ] < M

k \P kp pj /

< c. ..

Finally, k gVT (aik * ( a ^ a cpJ))) IJ In the same manner, we have :

k€ I-T (aik A (è <akp a cpj>)) =

( k S j - T aik)A cij A fél-T (p (akp “ V ) )

We have : a . . < ^ c . . , V k £ I - T , If i 6 I , V i £ J

l k j i j - - - ’ J -

Hence : k aik < J c . . , tf j £ J , n £ _ I

k Ê I - T aik < Ci1 V -'-V cij V *--V cin We have : k ^ < c ^ , tf i G I, V j € J

™ US ’(k e J.-T ‘ik)* cij (k G I_-T (p (akp a cpj))j ^ k G 1/T aik <

W i G I , V p 6 I , tf j G J .

Consequently : B 0 A = ( A'(a) C) A £ C. Condition (2) is necessary.

Sufficient conditions. We suppose that conditions (1) and (2) are ful­

filled. We have :

= [ k ( aik ''(PM \ P % j > ) j

(k€T (aik A (p (akp “ cpj’)/} V [k 6I-T(aikA (p(akp “ cpj'»)))

y k G J< , M i GI_ , \j j 6 J , y p G_I_

But> k e T ^ ' l i ^ k p ^ p j 1)) =

k 6 T (aik A < a k1 ''•••A ( a ki 01 "ij' (akm a ^{cmn^j}

= k £ T (aik A (aik “ cij)A (pA (akp ° cpj>)) B o A = (A- a C) o A =

because a.. = a. .lk ki

: i/cm ia ii, A c - • A /A (a, k G T ^ ik ij ^p kp a c .)')) pj'IJ

because a A ( a a b ) = a A b

19

k€T aikj A Cij A ^kG T ( p ^3kp a Cp j ^ )

By condition (1) : V* a.. >/ V c . . , V k 6 T , t f i E I , W j G J

1 k lk j i j - — J —

We have : ^

ai k » ^{V c .} .

J iJ M i G _l , W j

V

kGT1 aik > Ci1 V ".. v c . . IJ

v . . .v c.

in By condition (2) ^V

• k G T (A\P ^(a,kp a c .)]

PJ /1 » ^y c .. J IJ

We have

i 6 ^ , tf j 6 J , tf p 6 I A (a. a c .)]

PJ I ^{> C. V . . .V} c . . V . . . V c.

' l1 IJ in

, c i A (ai a c .)) >/ c.. , V i G I, Wp 6 I, y j 6 J k t T l P kp PJ / ij

Consequently : ^ a . ^ c.. - ^ (*(akp a cpJ))j

M i £ ]_ , M p G X»

= c . . IJ

Therefore we have : B o A = V . /a (a

c. . V ij But : kG I-T

[ k G \ - T ( aik A (p (akP a Cp j})) fa.. a (a (a a c .)]) =

\ lk ^p kp P J //

^k GL-T 3ikj Cij (k£ ^ P a °PJ < c . . ij

Thus : c. . V

ij

M i G l J j E j , W p G _I

(k e^x-T^ik A

(p

M i G I , tf p G I , tf j G J

= c . . IJ

Consequently : B o A = (A1 (a) C)c>A = = Conditions (1) and (2) are sufficient

[Q.E.D.]

Theorem (4). The equation Bo A = C has a solution A = (B (oT) C ')' if and only if :

( D y b. . > V c. . , tf i 6 I , j G J , U S K .

k k j ' i i j — J — —

i.e^ „,.==fr T =

(2> k 6 T (i (bkp “ W ) > i cij ■ J 6 2. » P e J . W i 6 I •

The proof is similar to that of theorem (3).

Remark. The forms of theorems (3) and (4) are slightly different from Zu-Wei's formulations (1980) in which a third condition is put. In fact, the proofs can be derived from more concise statements.

Theorem (5). Let S(B) be the set of fuzzy relations such that B o A = C.

Then (A' © C) is the greatest element in S(B).

Proof.

9 *

We put B = A 1 {a) C and we suppose that there exists B G S(B) such

* v that B 3 B .

As the max-min composite relation keeps the inclusion relation, we have

* ?

B o A 3 B o A.

But this relation is inconsistent with theorem(l) which states that :

* V

B o A c B oA

Therefore, B is the greatest element in S(B).V

[Q.E.D.]

Definition (3) : A binary relation /\ c X x Y is called an "elementary rowed relation" if and only if :

tfx G X_ : (1) 3 ! y = y , y £ V , such that y^ (x, y ) = 1

where 3 ! is the quantificator of unique existence which means "it exists one and only one".

(2) V y / y , y G V : y^ (x, y) = 0.

20

k G K lb

kj >/ ij

Therefore this relation is not fuzzy. The expression "elementary rowed relation" simply means "a single element for each row equal to 1, all the others being equal to zero". Each row is an elementary vector.

Theorem (6). Let A c X x V an elementary rowed relation and B e y x Z a fuzzy relation. We have :

B o A = B o A Proof

We put : C = B o/\ c X. x Z^ We have : M (x, z) €

By definition : y^— \ (x, z) = y (x, z) = 1 - y (x, z).

D O H C C

In virtue of De Morgan theorems, we have :

( x , z ) U X Z : lic ( x , z ) = A (x> y ) V ( y , z )^

A being an elementary rowed relation, for x = X| given, there exists a unique element y(x,|) such that :

yA ( v y = 1 and yA ( v y (v ) = 0 y y ^ y (x ) , yft (x, y) = 0 and yft (x, y) = 1 Therefore, we have :

yc (x, z) = K ( x r y C x , , ) ) ^ ^ ) , z)j A ^ ( Xj y) ^V y-(y, z )^

21

y t y ( x ^

= K§ (y (X1)’ Z )

X x Z : y (x, z) = V

- - c y ^{K (x,} y) A (y, z)j

On the other hand , M (x, z) £ X. x Z. : W B . A <X’ z) = y (Ufl (x’ y) ' WB (y’ Z)j

= y (*,))* ug (y ( x ^ , zj^ (*■ y) n Mg <-y ’ z ')

y i _{y (x1)}

= 1 A Ug (y ( x ^ , z) V ( ° A yg (y,z)

y / y ( x ^

= Mg (y (x,), z J.

[Q.E.D.]

Theorem (7) : Let S(B) be the set of the solutions B of the equation B o A_ = C where h_ is an elementary rowed relation. Then : (1) S(B) has the structure of a complete Brouwerian lattice, the ordering relation of this lattice being the fuzzy inclusion relation.

(2) B = _A' (a) C is the greatest element of the lattice.V A --- —=.

(3) B = _A' (a) C is the smallest element of the lattice.

Proof.

(1) Let B^ G S(B) and B^ G S(B). It is easy to prove that the properties of a lattice are verified. Indeed, it suffices to point out that B^ and B2 fulfill the following properties : commutativity, associativity, idempotence, and absorption. These properties are dual for the fuzzy union and the fuzzy intersection.

(2) B = A_' (a) C by virtue of theorem (5).

A --- —

(3) B = (a) C. Since h_ is an elementary rowed relation, in virtue of theorem (6) we have :

Applying theorem (2) to A and B, we have : B c A' © C

or, by the definition of the pseudocomplement : B ^ A 1 © C .

Thus, _A' C being included into B is the minimal solution and the least element of the lattice.

23

B is

[Q.E.D.]

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