EQUIVALENCE CLASS IN THE SET OF FUZZY NUMBERS AND ITS APPLICATION IN DECISION-MAKING PROBLEMS

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EQUIVALENCE CLASS IN THE SET OF FUZZY NUMBERS AND ITS APPLICATION IN DECISION-MAKING PROBLEMS

GEETANJALI PANDA, MOTILAL PANIGRAHI, AND SUDARSAN NANDA Received 4 November 2005; Revised 20 June 2006; Accepted 22 June 2006

An equivalence relation is defined in the set of fuzzy numbers. In a particular equivalence class, arithmetic operations of fuzzy numbers are introduced. A fuzzy matrix with respect to a particular class and its associated crisp matrices are also introduced. The concept of equivalence class is applied in fuzzy decision-making problems and justified through a numerical example.

1. Introduction

Since the discovery of fuzzy sets, the arithmetic operations of fuzzy numbers (Zadeh [7,8]) which may be viewed as a generalization of interval arithmetic (Moore [5]) have emerged as an important area of research within the theory of fuzzy sets (Mizumoto and Tanaka [4], Dubois and Prade [2]). The arithmetic operations of fuzzy numbers have been performed either by extension principle [7,8] or by using alpha cuts as discussed by Dubois and Prade [2]. If two triangular (linear) fuzzy numbers are added or subtracted by applying extension principle, then the result is again a triangular fuzzy number. How- ever, when we take other operations like multiplication or division, then the result is not a linear triangular fuzzy number. Thus these operations are not closed in the sense that operations of two same types (triangular/trapezoidal, etc.) of fuzzy numbers may not necessarily result in fuzzy number of that type (triangular/trapezoidal, etc.). Hence it is cumbersome to find the membership function of the arithmetical operations of large number of fuzzy numbers (may be of same type) by these principles. In the present paper, an attempt is made to overcome such type of diﬃculties. InSection 3, a relation between two fuzzy numbers is defined and it has been proved that this relation divides the whole set of fuzzy numbers into equivalence classes. InSection 4, arithmetic operations in a particular class are defined. These operations are diﬀerent from the arithmetic operations of fuzzy numbers, developed by extension principle or alpha cut. It has been proved and also verified through examples that these arithmetic operations are closed in their respective

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 74165, Pages1–19

DOI10.1155/IJMMS/2006/74165

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classes (e.g., product of two triangular fuzzy numbers in a particular class results in a triangular fuzzy number of the same class).

[μ]-fuzzy matrix and arithmetic operations of [μ]-fuzzy matrices are introduced in Section 5. Some properties such as definiteness, symmetry of [μ]-fuzzy matrices (these are useful to formulate quadratic programming) are discussed.

To make these concepts of equivalence class, arithmetic operations in a particular class and [μ]-fuzzy matrix useful in practical sense, an attempt is made to apply it in fuzzy decision-making problems. Fuzzy decision making was discovered by Bellman and Zadeh [1] and a lot of work has been done by many researchers like Zimmermann [9], Werners [6], and so forth to solve fuzzy decision-making problems using different types of defuzzification methods. In this paper, a different type of solution method for fuzzy decision-making problem has been developed without defuzzifying it. In any defuzzification method, solution of a fuzzy decision-making problem does not involve the aspiration level of the fuzzy numbers present in the problem. But in our solution method, each aspiration level corresponds to a solution. We have considered a fuzzy decision-making problem whose coefficients are fuzzy numbers belonging to a particular class. A general fuzzy decision-making problem in matrix form is formulated and its solution method is given inSection 6. The methodology is verified for fuzzy quadratic programming problem.

Throughout this paper, we have considered the set of all fuzzy numbers with compact support and also LR type.

2. Some preliminary results on fuzzy numbers

Definition 2.1 (fuzzy number). A fuzzy number is a fuzzy subset of the real lineRwhich is convex and normal.

LetFbe the set of all fuzzy numbers which are upper semicontinuous and have compact support. The support of a fuzzy numberawith membership functionμ_a:R→[0, 1]

is denoted by supp(a), where supp( a) =[la,ra] and

μa(t)=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

μ_aL(t), l_a≤t≤a_∗, 1, a_∗≤t≤a^∗, μ_aR(t), a^∗≤t≤r_a, 0, otherwise.

(2.1)

μ_aL : [l_a,a_∗]→[0, 1] is a continuous and strictly increasing function andμ_aR : [a^∗,r_a]→ [0, 1] is a continuous and strictly decreasing function.μa(t)=1 fort∈[a_∗,a^∗] and is called the core ofa, denoted by core( a). Symbolically, the fuzzy number awith compact support is represented bya= l_a;a_∗;a^∗;r_a.

Definition 2.2 (triangular fuzzy number). A fuzzy numbera= l_a;a_∗;a^∗;r_ais said to be a triangular fuzzy number ifa_∗=a^∗. Moreover, if the membership functionμ_ais such thatμaLandμaR are linear, thenais a linear triangular fuzzy number.

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Definition 2.3 (LR fuzzy number). A fuzzy numberais of LR type if there exist reference functionsLandR, scalarsα >0,β >0 with

μ_a(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ L

a_∗−t α

, t≤a_∗, 1, a_∗≤t≤a^∗, R

t−a^∗ β

, t≥a^∗.

(2.2)

[a_∗,a^∗] is the core ofa, αandβare the left and right spreads, respectively.L,R:R⁺→ [0, 1] are decreasing shape functions.L(0)=R(0)=1,L(1)=R(1)=0 (orL(t)>0,R(t)>

0, for all t,L(−∞)=R(+∞)=0). An LR-type fuzzy number a is represented by a= a_∗;a^∗;α,βLR. LetF^∗denote the set of all LR fuzzy numbers.

Definition 2.4 (ranking of fuzzy numbers [3]). For 0≤λ≤1, the λ-integral value of a fuzzy numbera, denoted by Iλ(a), is

Iλ(a) =λR(a) + (1 −λ)L(a), (2.3) where

R(a) = ¹

0μ⁻_aR¹(α)dα, L(a) = ¹

0μ⁻_aL¹(α)dα.

(2.4)

For two fuzzy numbersaandb,ab(less than or equal to in fuzzy sense) if and only if Iλ(a) ≤Iλ(b) for 0≤λ≤1.

The next section is devoted to define equivalence class in the set of fuzzy numbers.

3. Equivalence class in fuzzy numbers

Before introducing an equivalence relation, we define the domain of a fuzzy number inF andF^∗as follows.

(I) Supposea= la;a_∗;a^∗;raandb= lb;b_∗;b^∗;rbare two fuzzy numbers inFwith membership functionsμ_aandμ_b, respectively. The domains ofaandbareD_aandD_b, respectively, which are real lines and can be represented by

Da=

− ∞,la

∪ la,ra

∪ ra,∞

, D_b=

− ∞,l_b∪

l_b,r_b∪ r_b,∞

. (3.1)

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Let f :D_a→D_bbe defined as

f(t)=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

t+lb−la, t≤la, b_∗t−la

+lb

a_∗−t

a_∗−l_a , la≤t≤a_∗, b^∗t−a_∗+b_∗a^∗−t

a^∗−a_∗ , a_∗≤t≤a^∗, rb

t−a^∗+b^∗ra−t

r_a−a^∗ , a^∗≤t≤ra, t+r_b−r_a, t≥r_a.

(3.2)

(II) Supposeaandbare inF^∗,a= a_∗;a^∗;α,βLR, andb= b_∗;b^∗;α,βLR. Then D_a=

− ∞,a_∗−α∪

a_∗−α,a^∗+β∪

a^∗+β,∞ , D_b=

− ∞,b_∗−α∪

b_∗−α,b^∗+β∪

b^∗+β,∞

. (3.3)

In this case, f :Da→D_bcan be defined as in (3.2), just by replacinglabya_∗−α,lbby b_∗−α,rabya^∗+β, andrbbyb^∗+β.

It is not hard to see that f is bijective in both cases.

Definition 3.1. Fora,b∈F (orF^∗), define a relation^∼=betweenaandb(bis said to be related toaby the relation^∼₌) as

a^∼₌b iﬀμ_a(t)=μ_bf(t). (3.4) Example 3.2. Leta1; 2; 4,b2; 5; 7 ∈Fbe two linear triangular fuzzy numbers. Then

μa(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t−1, 1≤t≤2, 2− t

2, 2≤t≤4, 0, otherwise,

μ_b(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ t−2

3 , 2≤t≤5, 7−t

2 , 5≤t≤7, 0, otherwise,

f(t)=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

t+ 1, t≤1, 3t−1, 1≤t≤2, t+ 3, t≥2.

(3.5)

For allt,μa(t)=μ_b(f(t)). Hencea^∼=b.

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Example 3.3. a= 2; 1, 2L^∗R^∗,b= 5; 1, 2L^∗R^∗ are two fuzzy numbers inF^∗.L^∗(t)= (1 +t²)⁻¹,R^∗(t)=e⁻^t. Then

μ_a(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1

1 + (2−t)², t≤2, e⁻^(t⁻^2)/2, t≥2,

μ_b(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1

1 + (5−t)², t_≤5, e⁻^(t⁻^5)/2, t≥5,

(3.6)

f(t)=t+ 3 andμ_a(t)=μ_b(t+ 3). Hencea^∼₌b.

Theorem 3.4. ^∼=is an equivalence relation inF.

Proof. Leta, b, c∈F,a= l_a;a_∗;a^∗;r_a,b= l_b;b_∗;b^∗;r_b, andc= l_c;c_∗;c^∗;r_c. (i) Obviously,^∼₌is reflexive.

(ii) Leta^∼₌b, that is, μ_a(t)=μ_b(f(t)), where f is defined as in (3.2). Since f is bijective, f⁻¹:D_b→D_aexists and is defined as

f⁻¹(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t+la−lb, t≤lb, a_∗t−lb

+la b_∗−t

b_∗−lb , lb≤t≤b_∗, a^∗t−b_∗+a_∗b^∗−t

b^∗−b_∗ , b_∗≤t≤b^∗, ra

t−b^∗+a^∗rb−t

rb−b^∗ , b^∗≤t≤rb, t+r_a−r_b, t≥r_b.

(3.7)

It is easy to see that f⁻¹(f(t))=f f⁻¹(t)=tfor eacht. Also

μa(f⁻¹(t))=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ μaL

f⁻¹(t), la≤f⁻¹(t)≤a_∗,

1, a_∗≤a^∗,

μ_aR f⁻¹(t), a^∗_≤f⁻¹(t)_≤r_a,

0, otherwise

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=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ μ_bL

b_∗f⁻¹(t)−l_a+l_ba_∗−f⁻¹(t) a_∗−l_a

, la≤f⁻¹(t)≤a_∗,

1, a_∗≤f⁻¹(t)≤a^∗,

μ_bR rb

f⁻¹(t)−a^∗+b^∗ra−f⁻¹(t) ra−a^∗

, a^∗≤f⁻¹(t)≤ra,

0, otherwise

=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

μ_bL (t), lb≤t≤b_∗, 1, b_∗≤t≤b^∗, μ_bR (t), b^∗≤t≤rb, 0, otherwise

=μ_b(t).

(3.8) Henceb^∼₌a, that is, ^∼₌is symmetric.

(iii) Leta^∼₌bandb^∼₌c. f :D_a→D_bis as defined in (3.2) andg:D_b→D_cis

g(t)=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

t+l_c−l_b, t≤l_b, c_∗t−l_b+l_cb_∗−t

b_∗−l_b , l_b≤t≤b_∗, c^∗t−b_∗+c_∗b^∗−t

b^∗−b_∗ , b_∗≤t≤b^∗, r_ct−b^∗+c^∗r_b−t

r_b−b^∗ , b^∗≤t≤r_b, t+rc−rb, t≥rb,

(3.9)

such thatμ_a(t)=μ_b(f(t)),μ_b(t)=μ_c(g(t)). Soμ_a(t)=μ_c(g(f(t)))=μ_c((gof)(t)), where gof :Da→Dcis

gof(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

f(t) +lc−lb, f(t)≤lb, c_∗f(t)−l_b+l_cb_∗−f(t)

b_∗−l_b , lb≤f(t)≤b_∗, c^∗f(t)−b_∗+c_∗b^∗−f(t)

b^∗−b_∗ , b_∗≤ f(t)≤b^∗, rc

f(t)−b^∗+c^∗rb−f(t)

rb−b^∗ , b^∗≤f(t)≤rb, f(t) +rc−rb, f(t)≥rb,

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=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t+lc−la, t≤la, c_∗t−l_a+l_ca_∗−t

a_∗−l_a , l_a≤t≤a_∗, c^∗t−a_∗+c_∗a^∗−t

a^∗−a_∗ , a_∗≤t≤a^∗, r_ct−a^∗+c^∗r_a−t

r_a−a^∗ , a^∗≤t≤r_a, t+r_c−r_a, t≥r_a.

(3.10)

Hencea^∼₌c, that is, ^∼₌is transitive.

So^∼₌is an equivalence relation inF.

Theorem 3.5. ^∼₌is an equivalence relation inF^∗.

Proof. Proof of this result is similar to the proof ofTheorem 3.4.

The set of all the fuzzy numbers inF(orF^∗), which are equivalent to each other by the relation^∼₌, is known as an equivalence class and we denote it by [μ]. (In fact, the equivalence class [μ] contains all the fuzzy numbers with similar membership function μ.) Thus [μ] may be a triangular class, trapezoidal class, Gaussian class or a particular type of LR class, and so forth.

4. Arithmetic operations of fuzzy numbers in an equivalence class Supposea,b∈[μ],a= la;a_∗;a^∗;ra, andb= lb;b_∗;b^∗;rb ∈[μ],

supp(a) =

l_a,r_a, core(a) = a_∗,a^∗, domain ofa=Da=

− ∞,la

∪ la,ra

∪ ra,∞

, supp(b)=

l_b,r_b, core(b) = b_∗,b^∗, domain ofb=D_b=

− ∞,l_b∪

l_b,r_b∪ r_b,∞

.

(4.1)

We now define the arithmetic operation∗as follows:

a∗b=c, (4.2)

where the fuzzy numbercis defined as below:

supp(c) =supp(a) ∗supp(b)= la,ra

∗ lb,rb

= lc,rc

, core(c) =core(a) ∗core(b)=

a_∗,a^∗∗

b_∗,b^∗=

c_∗,c^∗. (4.3) Here∗denotes the arithmetic operation of intervals corresponding to fuzzy arithmetic operation “∗” (i.e., if∗represents fuzzy multiplication, then∗represents multiplication

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of intervals). The domain ofcdenoted byD_corD_a_∗_bis Dc=

− ∞,lc

∪ lc,rc

∪ rc,∞

. (4.4)

Theorem 4.1. Ifa,b∈[μ], thena∗b∈[μ].

Proof. Leta, b∈[μ] anda∗b=cis defined as above.

We defineg:D_a→D_a_∗_bby

g(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t+lc−la, t≤la, c_∗t−la

+lc

a_∗−t

a_∗−l_a , la≤t≤a_∗, c^∗t−a_∗+c_∗a^∗−t

a^∗−a_∗ , a_∗≤t≤a^∗, rc

t−a^∗+c^∗ra−t

r_a−a^∗ , a^∗≤t≤ra, t+r_c−r_a, t≥r_a.

(4.5)

The membership value of the fuzzy numbera∗bis defined as

μ_a_∗_bg(t)=μ_a(t). (4.6)

Sincegis a bijective mapping, so

μc(t)=μ_a_∗_b(t)=μa

g⁻¹(t). (4.7)

Hencea^∼=a∗b, that is,a∗b∈[μ]. This proves that∗is closed in [μ].

Let us consider an example of fuzzy multiplication. We denote the multiplication of two fuzzy numbers by⊗.

Example 4.2 (multiplication). Consider the following two fuzzy numbers:a= 1; 3; 6; 7, b= 2; 3; 5; 8 ∈[μ]. [μ] is a linear trapezoidal class. Then the membership functions ofa andbare

μa(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ t−1

2 , 1≤t≤3, 1, 3≤t≤6, 7−t, 6≤t≤7, 0, otherwise,

μ_b(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t−2, 2≤t≤3, 1, 3≤t≤5,

8−t

3 , 5≤t≤8, 0, otherwise,

(4.8)

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supp(a⊗b)=[2, 56], core(a⊗b) =[9, 30]. Now

g(t)=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

t+ 1, t≤1, 7t−3

2 , 1≤t≤3, 7t−12, 3≤t≤6, 26t−126, 6≤t≤7, t+ 49, t≥7.

(4.9)

We will evaluateμ_a_∗_bas below.

(i)t≤2⇒g⁻¹(t)≤1. Soμ_a_⊗_b(t)=μa(g⁻¹(t))=0.

(ii) 2≤t≤9⇒1≤g⁻¹(t)≤3. Soμ_a_⊗_b(t)=μ_a(g⁻¹(t))=(g⁻¹(t)−1)/2=((2t+ 3)/

7−1)/2=(t−2)/7.

(iii) 9≤t≤30⇒3≤g⁻¹(t)≤6. Soμ_a_⊗_b(t)=μa(g⁻¹(t))=1.

(iv) 30≤t≤56⇒6≤g⁻¹(t)≤7. Soμ_a_⊗_b(t)=μ_a(g⁻¹(t))=7−g⁻¹(t)=7−(t+ 126)/

26=(56−t)/26.

(v)t≥56⇒g⁻¹(t)≥7. Soμ_a_⊗_b(t)=μa(g⁻¹(t))=0.

On the other hand, we can defineh:D_b→D_a_⊗_bby

h(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t, t≤2,

7t−12, 2≤t≤3, 21t−45

2 , 3≤t≤5, 26t−40

3 , 5≤t≤8, t+ 48, t≥8.

(4.10)

(i)t≤2⇒h⁻¹(t)≤2. Soμ_a_⊗_b(t)=μ_b(h⁻¹(t))=0.

(ii) 2≤t≤9⇒2≤h⁻¹(t)≤3. Soμ_a_⊗_b(t)=μ_b(h⁻¹(t))=h⁻¹(t)−2=(t+ 12)/7− 2=(t−2)/(7).

(iii) 9≤t≤30⇒3≤h⁻¹(t)≤5. Soμ_a_⊗_b(t)=μ_b(h⁻¹(t))=1.

(iv) 30≤t≤56⇒5≤h⁻¹(t)≤8. So μ_a_⊗_b(t)=μ_b(h⁻¹(t))=(8−h⁻¹(t))/3=(8− (3t+ 40)/26)/3=(56−t)/26.

(v)t≥56⇒h⁻¹(t)≥8. Soμ_a_⊗_b(t)=μ_b(h⁻¹(t))=0.

From both cases, it is clear that

μ_a_⊗_b(t)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ t−2

7 , 2≤t≤9, 1, 9≤t≤30,

56−t

26 , 30≤t≤56, 0, otherwise.

(4.11)