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The similarity measure of generalized fuzzy numbers based on interval distance
Applied Mathematics Letters 25 (2012) 1528–1534
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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
The similarity measure of generalized fuzzy numbers based on interval distance M. Adabitabar Firozja a,∗ , G.H. Fath-Tabar b , Z. Eslampia a a
Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
b
Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Kashan, Kashan 87317-51167, Iran
article
info
Article history: Received 7 September 2011 Accepted 1 January 2012 Keywords: Fuzzy number Interval arithmetic Distance measure
abstract In this paper, we proposed a new interval distance of two fuzzy numbers that satisfy on metric properties. Also, this metric distance satisfies on some of the other properties. Then, we used this metric for similarity measure. Finality, we tested with some examples. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy systems are used to study a variety of problems: fuzzy linear systems [1,2], fuzzy differential equations [3–5], fuzzy linear programming [6,7], particle physics [8,9] and other topics. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Chang and Zadeh [10] and others. The distance measure is a very essential tool in various fields of study that measures the distance or difference between two points. Many researchers have worked also on distance measures of fuzzy numbers, [11–16] where more are concluded to be crisp numbers. In [11] Tran et al. introduce the distance on the interval numbers and then extend to fuzzy numbers. Diamond [12] defined a measure for fuzzy numbers in the Euclidean space. Yang et al. [13] modified the Diamond proposed measure with a function that captures more information about vagueness. In [14] a fuzzy distance measure for Gaussian type fuzzy numbers is defined which includes a normalization factor. Cheng in [15], also proposed a distance index based on the centroid points of the fuzzy numbers. Allahviranllo and Firozja [16] compute the distance between two fuzzy numbers on the interval numbers based on convex hall of endpoints. It is evident that, if we use a crisp number for a distance of two fuzzy (interval) numbers we lose much important information. Some researchers proposed the fuzzy distance of fuzzy numbers. Chakraborty and Chakraborty in [17] could compute the fuzzy distance between two generalized fuzzy numbers. They employed the fuzzy distance for the fuzzy numbers while their α -level is difference. Guha and Chakraborty [18] proposed the fuzzy distance for the generalized fuzzy numbers and the confidence level of the fuzzy numbers is considered. In this study, we define a new distance between two fuzzy numbers by their α -cut, an interval metric distance on fuzzy numbers. Many different methods exist for the similarity measure between fuzzy numbers [18–22]. For example, Chen and Chen [19] proposed a similarity measure based the center of gravity (COG) of generalized fuzzy numbers that is complicated and not useful for general LR-type. Also, Deng Yang presented a new similarity measure of generalized fuzzy numbers with the radius of gyration(ROG) points [22]. Guha and Chakraborty [18] proposed the similarity measure between two generalized fuzzy numbers based on the distance measure.
∗
Corresponding author. E-mail address: [email protected] (M. Adabitabar Firozja).
0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.01.009
M. Adabitabar Firozja et al. / Applied Mathematics Letters 25 (2012) 1528–1534
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In this paper, Section 2 briefly describes the basic notation and definitions of an LR-type fuzzy number, the α -cut and the basics of interval arithmetic. In Section 3, we describe an interval distance for the distance between two fuzzy numbers and prove the properties of the metric for new distance and computing the distance measure for trapezoidal (triangular) fuzzy numbers. Section 4, proposes a similarity measure between two fuzzy numbers, finally a short conclusion is given in Section 5. 2. Background fuzzy Definition 1. A˜ = (a1 , a2 , a3 , a4 )LR is a generalized left right fuzzy number (GLRFN) of Dubois and Prade [23], if its membership function satisfies the following:
a2 − x , L a2 − a1 1, µA˜ (x) = x − a3 R , a4 − a3 0,
a1 ≤ x ≤ a2 , a2 ≤ x ≤ a3 ,
(1)
a3 ≤ x ≤ a4 , otherwise
where L and R are strictly decreasing functions defined on [0, 1] and satisfying the conditions: L(t ) = R(t ) = 1
if t ≤ 0
L(t ) = R(t ) = 0
if t ≥ 1.
(2)
For a2 = a3 , we have the classical definition of left right fuzzy numbers (LRFN). Trapezoidal fuzzy numbers (TrFN) are special cases of GLRFN with L(t ) = R(t ) = 1 − t. Triangular fuzzy numbers (TFN) are also special cases of GLRFN with L(t ) = R(t ) = 1 − t and a2 = a3 . r-level interval of fuzzy number A˜ as [A˜ ]r = {x ∈ R | µA˜ (x) ≥ r } where [A˜ ]r = 1 −1 [A(r ), A(r )] = [a2 − (a2 − a1 )L− A (r ), a3 + (a4 − a3 )RA (r )]. Let F (R) denote the fuzzy numbers on R and I (R) denote the interval numbers on R such as [a, b] where a ≤ b. Suppose two interval numbers [a, b], [c , d] ∈ I (R) and τ ∈ R then 1. [a, b] + [c , d] = [a + c , b + d] 2. τ [a, b] = [min{τ a, τ b}, max{τ a, τ b}]. We consider the comparison between two interval numbers [a, b] and [c , d] is the following:
[a, b] ≤ [c , d] ←→ a ≤ c , b ≤ d and [a, b] = [c , d] ←→ a = c , b = d. Interval number [a, b] is nonnegative such that [a, b] ≥ [0, 0], and typing [a, b] ≥ 0. We used a ≤ b ↔ min{a, b} = a. For the ranking of two fuzzy numbers A˜ and B˜ we define: A˜ ≤ B˜ ↔
1
A(α)dα, 0
1
A¯ (α)dα ≤
1
0
B(α)dα, 0
1
B¯ (α)dα .
0
3. Interval distance between fuzzy numbers Let us consider two fuzzy numbers A˜ and B˜ with [A˜ ]r = [A(r ), A(r )], [B˜ ]r = [B(r ), B(r )] respectively. The distance between these two fuzzy numbers is defined by d(A˜ , B˜ ) = [d, d] ∈ I (R) where d = min{ min |A(r ) − B(r )|, min |A(r ) − B(r )|} r ∈[0,1]
r ∈[0,1]
and d = max{ max |A(r ) − B(r )|, max |A(r ) − B(r )|}. r ∈[0,1]
r ∈[0,1]
3.1. Metric properties The proposed interval distance between two fuzzy numbers satisfies the following metric properties: (i) d(A˜ , B˜ ) ≥ 0 (ii) d(A˜ , A˜ ) = [0, 0]
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(iii) d(A˜ , B˜ ) = d(B˜ , A˜ ) (iv) d(A˜ , B˜ ) = [0, 0] ↔ A˜ = B˜ (v) d(A˜ , B˜ ) + d(B˜ , C˜ ) ≥ d(A˜ , C˜ ). Proof. Proof of properties (i)–(iv) is very clear, therefore we prove property (v). Suppose A˜ , B˜ , C˜ are three fuzzy numbers with r-cut represented as defined in the following
[A˜ ]r = [A(r ), A(r )],
[B˜ ]r = [B(r ), B(r )],
[C˜ ]r = [C (r ), C (r )].
If AB
BC
d(A˜ , B˜ ) = [dAB , d ],
d(B˜ , C˜ ) = [dBC , d ],
AC
d(A˜ , C˜ ) = [dAC , d ]
we should prove by the new metric many situations arising considering the following shapes. Suppose ri are where i = 1, 2, . . . , 6, so that AB
1. d(A˜ , B˜ ) = [dAB , d ] dAB =
1 : |A(r1 ) − B(r1 )| or
2 : |A(r ) − B(r )| 1 1
,
AB
d
=
1 : |A(r2 ) − B(r2 )| or
2 : |A(r ) − B(r )|. 2 2
BC
2. d(B˜ , C˜ ) = [dBC , d ] dBC =
1 : |B(r3 ) − C (r3 )| or
2 : |B(r ) − C (r )| 3 3
,
d
BC
=
1 : |B(r4 ) − C (r4 )| or
2 : |B(r ) − C (r )|. 4 4
AC
3. d(A˜ , C˜ ) = [dAC , d ] d
AC
=
1 : |A(r5 ) − C (r5 )| or
2 : |A(r ) − C (r )| 5 5
,
d
AC
=
1 : |A(r6 ) − C (r6 )| or
2 : |A(r ) − C (r )|. 6 6 AC
AB
For the proof of d(A˜ , C˜ ) ≤ d(A˜ , B˜ ) + d(B˜ , C˜ ) it is sufficient that [dAC , d ] ≤ [dAB + dBC , d Therefore we have to prove that
BC
+ d ].
dAC ≤ dAB + dBC AC
d
AB
(3)
BC
≤d +d .
(4) AB
BC
= |A(r2 ) − B(r2 )|, d = |B(r4 ) − C (r4 )| and d = |A(r6 ) − C (r6 )| then d = |A(r6 ) − C (r6 )| = |A(r6 ) − B(r6 ) + B(r6 ) − C (r6 )| ≤ |A(r6 ) − B(r6 )| + |B(r6 ) − C (r6 )| ≤ AB BC |A(r2 ) − B(r2 )| + |B(r4 ) − C (r4 )| = d + d . AB BC And if d = |A(r1 ) − B(r1 )|, d = |B(r3 ) − C (r3 )| and dAC = |A(r5 ) − C (r5 )| then, dAB + dBC = |A(r1 ) − B(r1 )| + |B(r3 ) − C (r3 )| ≥ |A(r1 ) − B(r1 ) + B(r3 ) − C (r3 )| ≥ |A(r1 ) − C (r3 )| ≥ |A(r5 ) − C (r5 )|. Therefore the proof is completed. We have 8 situations for (3), (4) where we consider one case of any one. If d AC
AC
3.2. Other properties of the new metric This interval distance satisfies the following properties where we show in the following proposition: Proposition 2. If A˜ , B˜ , C˜ ∈ F (R) and a, b, τ ∈ R 1. 2. 3. 4. 5.
d(τ A˜ , τ B˜ ) = |τ |d(A˜ , B˜ ); d(A˜ + τ , B˜ + τ ) = d(A˜ , B˜ ); d(A˜ + C˜ , B˜ + C˜ ) = d(A˜ , B˜ ); d([a, b], [c , d]) = [min{|a − c |, |b − d|}, max{|a − c |, |b − d|}]; d(a, b) = |a − b|.
Proposition 3. If A˜ = (a1 , a2 , a3 , a4 ) and B˜ = (b1 , b2 , b3 , b4 ) are two trapezoidal fuzzy numbers, then d(A˜ , B˜ ) = [d, d¯ ] so that If 0 <
b1 −a1 a2 −a1 −b2 +b1
< 1 or 0 <
a4 −b4 a4 −a3 −b4 +b3
< 1 then d = 0 otherwise d = min{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |, |a4 − b4 |}.
And d = max{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |, |a4 − b4 |}.
M. Adabitabar Firozja et al. / Applied Mathematics Letters 25 (2012) 1528–1534
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Proof. If [A˜ ]r = [A(r ), A(r )] = [(a2 − a1 )r + a1 , a4 − (a4 − a3 )r ] and [B˜ ]r = [B(r ), B(r )] = [(b2 − b1 )r + b1 , b4 − (b4 − b3 )r ] by,
|A(r ) − B(r )| = |(a1 − b1 ) + (a2 − a1 − b2 + b1 )r | and |A(r ) − B(r )| = |(a4 − b4 ) − (a4 − a3 − b4 + b3 )r | and regarding that
min |ax + b| =
0≤x≤1
0<−
min{|b|, |a + b|, 0}
b
< 1,
a otherwise
min{|b|, |a + b|},
and max |ax + b| = max{|b|, |a + b|}.
0≤x≤1
If 0 <
b1 −a1 a2 −a1 −b2 +b1
< 1 or 0 <
a4 −b4 a4 −a3 −b4 +b3
< 1 then d = 0 otherwise d = min{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |, |a4 − b4 |}
and d = max{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |, |a4 − b4 |}. Proposition 4. If A˜ = (a1 , a2 , a3 ) and B˜ = (b1 , b2 , b3 ) are two triangular fuzzy numbers then, d(A˜ , B˜ ) = [d, d¯ ] so that If 0 <
b1 −a1 a2 −a1 −b2 +b1
< 1 or 0 <
a3 −b3 a3 −a2 −b3 +b2
< 1 then d = 0 otherwise
d = min{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |} and d = max{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |}. Proposition 5. Suppose d(A˜ , B˜ ) = [d, d], if d = 0 then d = 0. 4. Similarity measure between two fuzzy numbers The similarity measure is important for presenting the degree of similarity between two objects or concepts. In a real life decision making situation, experts mostly express their opinions linguistically. Such linguistic expressions are the captured by fuzzy set theory and denoted as fuzzy numbers for analytic purpose. Some researchers introduce the similarity measures between two fuzzy numbers. Chen [20] and Lee [21], proposed a similarity measure for two normal trapezoidal fuzzy numbers A˜ 1 = (a1 , a2 ; β1 , γ1 ) and A˜ 2 = (a3 , a4 ; β2 , γ2 ) as follows s(A˜ 1 , A˜ 2 )chen = 1 −
|(a1 − β1 ) − (a3 − β2 )| + |a1 − a3 | + |a2 − a4 | + |(a2 + γ1 ) − (a4 − γ2 )| 4
where |a| denotes the absolute value of the real number a. 1
s(A˜ 1 , A˜ 2 )Lee = 1 −
(|(a1 − β1 ) − (a3 − β2 )|p + |a1 − a3 |p + |a2 − a4 |p + |(a2 + γ1 ) − (a4 − γ2 )|p ) p max(U ) − min(U )
where U is the universe of discourse. Chen and Chen [19] proposed a similarity measure between two generalized trapezoidal fuzzy numbers A˜ 1 (a1 , a2 ; β1 , γ1 ; w1 ) and A˜ 2 = (a3 , a4 ; β2 , γ2 ; w2 ) as follows s(A˜ 1 , A˜ 2 )chen–chen =
1−
B(sA
where K = 1 − |x∗A1 − x∗A2 |
|(a1 − β1 ) − (a3 − β2 )| + |a1 − a3 | + |a2 − a4 | + |(a2 + γ1 ) − (a4 − γ2 )| 4 1
,sA2 )
×
=
×K
min(y∗ ,y∗ ) A1 A2 . ∗ max(y ,y∗ ) A1
A2
Here, (x∗A , y∗A ) center-of-gravity and sA length of A˜ = (a1 , a2 ; β, γ ; w), respectively. Where sA = (a2 + γ ) − (a1 − β) and B(sA1 , sA2 ) =
1, 0,
sA1 + sA2 > 0 sA1 + sA2 = 0.
(5)
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And
a2 −a1 w × + 2 a2 +γ −a1 +β
y∗A =
6
w,
,
a2 + γ ̸= a1 − β (6)
a2 + γ = a1 − β 2 ∗ yA (a2 + a1 ) + (a2 + γ + a1 + β)(w − y∗A )
x∗A =
.
2w
Guha–Chakraborty [18] proposed a similarity measure between two generalized trapezoidal fuzzy numbers A˜ 1 (a1 , a2 ; β1 , γ1 ; w1 ) and A˜ 2 = (a3 , a4 ; β2 , γ2 ; w2 ) for 0 ≤ d(A˜ 1 ; A˜ 2 ) ≤ 1 as follows s(A˜ 1 , A˜ 2 )Guha–Chakraborty = 1 − d(A˜ 1 ; A˜ 2 ) =
dRr=w
1−
dRr=w + σ
,1 −
dLr=w dRr=w + σ
;
σ dRr=w + σ
,
θ
=
dRr=w + σ
where w = min(w1 , w2 ) and with
[L(r ), R(r )] = η([A˜ 1 ]r − [A˜ 2 ]r ) + (1 − η)([A˜ 2 ]r − [A˜ 1 ]r )
(7)
and
η=
A1 (w) + A1 (w)
1,
A1 (w) + A1 (w)
0,
A2 (w) + A2 (w)
≥
2
2 A2 (w) + A2 (w)
<
2
(8)
2
[L(r ), R(r )], L(r ) ≥ 0 [0, |L(r )| ∨ R(r )], L(r ) < 0 < R(r ). w w And θ = dLr=w − max{ 0 dLr dr , 0}, σ = | 0 dRr dr − dRr=w |. [dLr=w , dRr=w ] =
(9)
Here we present a new similarity measure based interval distance measure. The relation which has been used here is real interval S (A˜ , B˜ ) =
1 1 + d¯
,1 −
d
(10)
1 + d¯
˜ where [d, d¯ ] is the interval distance measure between A˜ and B. 4.1. Properties The new similarity measure has the following properties: 1. 2. 3. 4.
[0, 0] < S (A˜ , B˜ ) ≤ [1, 1]. S (A˜ , B˜ ) = S (B˜ , A˜ ). If A˜ = B˜ then S (A˜ , B˜ ) = [1, 1]. If A˜ ≤ B˜ ≤ C˜ , then S (A˜ , C˜ ) ≤ min{S (A˜ , B˜ ), S (B˜ , C˜ )}.
Proofs. 1. Because, d ≥ 0 and d ≤ d then
1 1 +d
> 0 and 1 −
d 1+d¯
≤ 1 hence [0, 0] < S (A˜ , B˜ ) ≤ [1, 1].
2. Because, d(A˜ , B˜ ) = d(B˜ , A˜ ), hence proof is evident. 3. Regarding (iv), the property of metric proof is evident. 4. A˜ ≤ B˜ ≤ C˜ , then 1
A(r )dr ≤ 0
1
B(r )dr ≤ 0
1
C (r )dr
1
and
0
A(r )dr ≤
0
min |A(r ) − B(r )| ≤ min |A(r ) − C (r )|} r ∈[0,1]
r ∈[0,1]
⇒
B(r )dr ≤ 0
hence,
min |A(r ) − B(r )| ≤ min |A(r ) − C (r )|} r ∈[0,1] r ∈[0,1]
1
dAB ≤ dAC
1
C (r )dr 0
M. Adabitabar Firozja et al. / Applied Mathematics Letters 25 (2012) 1528–1534
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Table 1 The sets of fuzzy numbers.
Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7
A˜
B˜
(0.1, 0.2, 0.3, 0.4) (0.1, 0.2, 0.3, 0.4) (0.1, 0.2, 0.3, 0.4) (0.3, 0.3, 0.3) (0.2, 0.2, 0.2) (0.1, 0.2, 0.3) (0.1, 0.2, 0.3)
(0.1, 0.25, 0.4) (0.4, 0.55, 0.7) (0.4, 0.5, 0.6, 0.7) (0.3, 0.3, 0.3) (0.3, 0.3, 0.3) (0.3, 0.3, 0.3) (0.2, 0.3, 0.4)
Table 2 Comparison of the proposed similarity measure with the other similarity measure. Set
Lee
Chen
Chen and Chen
Chakraborty and Guha
New
Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7
0.9167 0.5 0.5 # 0 0.5 0.6667
0.975 0.7 0.7 1 0.9 0.9 0.9
0.8357 0.42 0.49 1 0.9 0.54 0.81
(0.825, 0.95, 1) (0.525, 0.65, 0.75, 0.877) (0.5, 0.6, 0.8, 0.9) (1, 1, 1) (0.9, 0.9, 0.9) (0.85, 0.9, 0.95) (0.8, 0.9, 1)
[0.952, 1] [0.741, 0.815] [0.77, 0.77] [1, 1] [0.9, 0.9] [0.833, 1] [0.91, 0.91]
and
max |A(r ) − B(r )| ≤ max |A(r ) − C (r )|} r ∈[0,1] r ∈[0,1]
⇒
AB
d
≤d
AC
max |A(r ) − B(r )| ≤ max |A(r ) − C (r )|}. r ∈[0,1]
r ∈[0,1]
Hence, d(A˜ , B˜ ) ≤ d(A˜ , C˜ ), and we can write d(B˜ , C˜ ) ≤ d(A˜ , C˜ ), respectively. Therefore, with inverse relation metric and similarity
S (A˜ , C˜ ) ≤ S (A˜ , B˜ )
⇒
˜ ˜ S (A, C ) ≤ S (B˜ , C˜ ).
S (A˜ , C˜ ) ≤ min{S (A˜ , B˜ ), S (B˜ , C˜ )}
Note. It is evident that with fuzzy data, the similarity measure should be positive, whereas with other existing methods it may be 0 ∈ S (A˜ , B˜ ) or S (A˜ , B˜ ) = 0 but, by this approach S (A˜ , B˜ ) ̸= 0. 4.2. Comparing with other methods In this section, we used 7 sets of normal trapezoidal or triangular fuzzy numbers given in Table 1 and proposed in Guha–Chakraborty [18]. We compare the similarity measure based on interval distance with the four methods presented by Chen [20], Lee [21], Chen and Chen [19] and Guha–Chakraborty [18]. A comparison between the results of the proposed similarity measure and the results of the other methods is shown in Table 2. Regarding Table 2, in this approach 1 ∈ S (A˜ , B˜ ) if and only if ∃ r ; A(r ) = B(r ) or A(r ) = B(r ). 5. Conclusions Many papers have proposed the fuzzy distance measure. In this paper, we developed a distance measure between fuzzy numbers as an interval number. We used this metric to define a similarity measure on fuzzy numbers as an interval number. The proposed distance or similarity measure can lead us to other concepts, such as ranking fuzzy numbers, nearest approximation, defuzzification and fuzzy decision making problems. References [1] [2] [3] [4]
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Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
The similarity measure of generalized fuzzy numbers based on interval distance M. Adabitabar Firozja a,∗ , G.H. Fath-Tabar b , Z. Eslampia a a
Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
b
Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Kashan, Kashan 87317-51167, Iran
article
info
Article history: Received 7 September 2011 Accepted 1 January 2012 Keywords: Fuzzy number Interval arithmetic Distance measure
abstract In this paper, we proposed a new interval distance of two fuzzy numbers that satisfy on metric properties. Also, this metric distance satisfies on some of the other properties. Then, we used this metric for similarity measure. Finality, we tested with some examples. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy systems are used to study a variety of problems: fuzzy linear systems [1,2], fuzzy differential equations [3–5], fuzzy linear programming [6,7], particle physics [8,9] and other topics. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Chang and Zadeh [10] and others. The distance measure is a very essential tool in various fields of study that measures the distance or difference between two points. Many researchers have worked also on distance measures of fuzzy numbers, [11–16] where more are concluded to be crisp numbers. In [11] Tran et al. introduce the distance on the interval numbers and then extend to fuzzy numbers. Diamond [12] defined a measure for fuzzy numbers in the Euclidean space. Yang et al. [13] modified the Diamond proposed measure with a function that captures more information about vagueness. In [14] a fuzzy distance measure for Gaussian type fuzzy numbers is defined which includes a normalization factor. Cheng in [15], also proposed a distance index based on the centroid points of the fuzzy numbers. Allahviranllo and Firozja [16] compute the distance between two fuzzy numbers on the interval numbers based on convex hall of endpoints. It is evident that, if we use a crisp number for a distance of two fuzzy (interval) numbers we lose much important information. Some researchers proposed the fuzzy distance of fuzzy numbers. Chakraborty and Chakraborty in [17] could compute the fuzzy distance between two generalized fuzzy numbers. They employed the fuzzy distance for the fuzzy numbers while their α -level is difference. Guha and Chakraborty [18] proposed the fuzzy distance for the generalized fuzzy numbers and the confidence level of the fuzzy numbers is considered. In this study, we define a new distance between two fuzzy numbers by their α -cut, an interval metric distance on fuzzy numbers. Many different methods exist for the similarity measure between fuzzy numbers [18–22]. For example, Chen and Chen [19] proposed a similarity measure based the center of gravity (COG) of generalized fuzzy numbers that is complicated and not useful for general LR-type. Also, Deng Yang presented a new similarity measure of generalized fuzzy numbers with the radius of gyration(ROG) points [22]. Guha and Chakraborty [18] proposed the similarity measure between two generalized fuzzy numbers based on the distance measure.
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Corresponding author. E-mail address: [email protected] (M. Adabitabar Firozja).
0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.01.009
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In this paper, Section 2 briefly describes the basic notation and definitions of an LR-type fuzzy number, the α -cut and the basics of interval arithmetic. In Section 3, we describe an interval distance for the distance between two fuzzy numbers and prove the properties of the metric for new distance and computing the distance measure for trapezoidal (triangular) fuzzy numbers. Section 4, proposes a similarity measure between two fuzzy numbers, finally a short conclusion is given in Section 5. 2. Background fuzzy Definition 1. A˜ = (a1 , a2 , a3 , a4 )LR is a generalized left right fuzzy number (GLRFN) of Dubois and Prade [23], if its membership function satisfies the following:
a2 − x , L a2 − a1 1, µA˜ (x) = x − a3 R , a4 − a3 0,
a1 ≤ x ≤ a2 , a2 ≤ x ≤ a3 ,
(1)
a3 ≤ x ≤ a4 , otherwise
where L and R are strictly decreasing functions defined on [0, 1] and satisfying the conditions: L(t ) = R(t ) = 1
if t ≤ 0
L(t ) = R(t ) = 0
if t ≥ 1.
(2)
For a2 = a3 , we have the classical definition of left right fuzzy numbers (LRFN). Trapezoidal fuzzy numbers (TrFN) are special cases of GLRFN with L(t ) = R(t ) = 1 − t. Triangular fuzzy numbers (TFN) are also special cases of GLRFN with L(t ) = R(t ) = 1 − t and a2 = a3 . r-level interval of fuzzy number A˜ as [A˜ ]r = {x ∈ R | µA˜ (x) ≥ r } where [A˜ ]r = 1 −1 [A(r ), A(r )] = [a2 − (a2 − a1 )L− A (r ), a3 + (a4 − a3 )RA (r )]. Let F (R) denote the fuzzy numbers on R and I (R) denote the interval numbers on R such as [a, b] where a ≤ b. Suppose two interval numbers [a, b], [c , d] ∈ I (R) and τ ∈ R then 1. [a, b] + [c , d] = [a + c , b + d] 2. τ [a, b] = [min{τ a, τ b}, max{τ a, τ b}]. We consider the comparison between two interval numbers [a, b] and [c , d] is the following:
[a, b] ≤ [c , d] ←→ a ≤ c , b ≤ d and [a, b] = [c , d] ←→ a = c , b = d. Interval number [a, b] is nonnegative such that [a, b] ≥ [0, 0], and typing [a, b] ≥ 0. We used a ≤ b ↔ min{a, b} = a. For the ranking of two fuzzy numbers A˜ and B˜ we define: A˜ ≤ B˜ ↔
1
A(α)dα, 0
1
A¯ (α)dα ≤
1
0
B(α)dα, 0
1
B¯ (α)dα .
0
3. Interval distance between fuzzy numbers Let us consider two fuzzy numbers A˜ and B˜ with [A˜ ]r = [A(r ), A(r )], [B˜ ]r = [B(r ), B(r )] respectively. The distance between these two fuzzy numbers is defined by d(A˜ , B˜ ) = [d, d] ∈ I (R) where d = min{ min |A(r ) − B(r )|, min |A(r ) − B(r )|} r ∈[0,1]
r ∈[0,1]
and d = max{ max |A(r ) − B(r )|, max |A(r ) − B(r )|}. r ∈[0,1]
r ∈[0,1]
3.1. Metric properties The proposed interval distance between two fuzzy numbers satisfies the following metric properties: (i) d(A˜ , B˜ ) ≥ 0 (ii) d(A˜ , A˜ ) = [0, 0]
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(iii) d(A˜ , B˜ ) = d(B˜ , A˜ ) (iv) d(A˜ , B˜ ) = [0, 0] ↔ A˜ = B˜ (v) d(A˜ , B˜ ) + d(B˜ , C˜ ) ≥ d(A˜ , C˜ ). Proof. Proof of properties (i)–(iv) is very clear, therefore we prove property (v). Suppose A˜ , B˜ , C˜ are three fuzzy numbers with r-cut represented as defined in the following
[A˜ ]r = [A(r ), A(r )],
[B˜ ]r = [B(r ), B(r )],
[C˜ ]r = [C (r ), C (r )].
If AB
BC
d(A˜ , B˜ ) = [dAB , d ],
d(B˜ , C˜ ) = [dBC , d ],
AC
d(A˜ , C˜ ) = [dAC , d ]
we should prove by the new metric many situations arising considering the following shapes. Suppose ri are where i = 1, 2, . . . , 6, so that AB
1. d(A˜ , B˜ ) = [dAB , d ] dAB =
1 : |A(r1 ) − B(r1 )| or
2 : |A(r ) − B(r )| 1 1
,
AB
d
=
1 : |A(r2 ) − B(r2 )| or
2 : |A(r ) − B(r )|. 2 2
BC
2. d(B˜ , C˜ ) = [dBC , d ] dBC =
1 : |B(r3 ) − C (r3 )| or
2 : |B(r ) − C (r )| 3 3
,
d
BC
=
1 : |B(r4 ) − C (r4 )| or
2 : |B(r ) − C (r )|. 4 4
AC
3. d(A˜ , C˜ ) = [dAC , d ] d
AC
=
1 : |A(r5 ) − C (r5 )| or
2 : |A(r ) − C (r )| 5 5
,
d
AC
=
1 : |A(r6 ) − C (r6 )| or
2 : |A(r ) − C (r )|. 6 6 AC
AB
For the proof of d(A˜ , C˜ ) ≤ d(A˜ , B˜ ) + d(B˜ , C˜ ) it is sufficient that [dAC , d ] ≤ [dAB + dBC , d Therefore we have to prove that
BC
+ d ].
dAC ≤ dAB + dBC AC
d
AB
(3)
BC
≤d +d .
(4) AB
BC
= |A(r2 ) − B(r2 )|, d = |B(r4 ) − C (r4 )| and d = |A(r6 ) − C (r6 )| then d = |A(r6 ) − C (r6 )| = |A(r6 ) − B(r6 ) + B(r6 ) − C (r6 )| ≤ |A(r6 ) − B(r6 )| + |B(r6 ) − C (r6 )| ≤ AB BC |A(r2 ) − B(r2 )| + |B(r4 ) − C (r4 )| = d + d . AB BC And if d = |A(r1 ) − B(r1 )|, d = |B(r3 ) − C (r3 )| and dAC = |A(r5 ) − C (r5 )| then, dAB + dBC = |A(r1 ) − B(r1 )| + |B(r3 ) − C (r3 )| ≥ |A(r1 ) − B(r1 ) + B(r3 ) − C (r3 )| ≥ |A(r1 ) − C (r3 )| ≥ |A(r5 ) − C (r5 )|. Therefore the proof is completed. We have 8 situations for (3), (4) where we consider one case of any one. If d AC
AC
3.2. Other properties of the new metric This interval distance satisfies the following properties where we show in the following proposition: Proposition 2. If A˜ , B˜ , C˜ ∈ F (R) and a, b, τ ∈ R 1. 2. 3. 4. 5.
d(τ A˜ , τ B˜ ) = |τ |d(A˜ , B˜ ); d(A˜ + τ , B˜ + τ ) = d(A˜ , B˜ ); d(A˜ + C˜ , B˜ + C˜ ) = d(A˜ , B˜ ); d([a, b], [c , d]) = [min{|a − c |, |b − d|}, max{|a − c |, |b − d|}]; d(a, b) = |a − b|.
Proposition 3. If A˜ = (a1 , a2 , a3 , a4 ) and B˜ = (b1 , b2 , b3 , b4 ) are two trapezoidal fuzzy numbers, then d(A˜ , B˜ ) = [d, d¯ ] so that If 0 <
b1 −a1 a2 −a1 −b2 +b1
< 1 or 0 <
a4 −b4 a4 −a3 −b4 +b3
< 1 then d = 0 otherwise d = min{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |, |a4 − b4 |}.
And d = max{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |, |a4 − b4 |}.
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Proof. If [A˜ ]r = [A(r ), A(r )] = [(a2 − a1 )r + a1 , a4 − (a4 − a3 )r ] and [B˜ ]r = [B(r ), B(r )] = [(b2 − b1 )r + b1 , b4 − (b4 − b3 )r ] by,
|A(r ) − B(r )| = |(a1 − b1 ) + (a2 − a1 − b2 + b1 )r | and |A(r ) − B(r )| = |(a4 − b4 ) − (a4 − a3 − b4 + b3 )r | and regarding that
min |ax + b| =
0≤x≤1
0<−
min{|b|, |a + b|, 0}
b
< 1,
a otherwise
min{|b|, |a + b|},
and max |ax + b| = max{|b|, |a + b|}.
0≤x≤1
If 0 <
b1 −a1 a2 −a1 −b2 +b1
< 1 or 0 <
a4 −b4 a4 −a3 −b4 +b3
< 1 then d = 0 otherwise d = min{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |, |a4 − b4 |}
and d = max{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |, |a4 − b4 |}. Proposition 4. If A˜ = (a1 , a2 , a3 ) and B˜ = (b1 , b2 , b3 ) are two triangular fuzzy numbers then, d(A˜ , B˜ ) = [d, d¯ ] so that If 0 <
b1 −a1 a2 −a1 −b2 +b1
< 1 or 0 <
a3 −b3 a3 −a2 −b3 +b2
< 1 then d = 0 otherwise
d = min{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |} and d = max{|a1 − b1 |, |a2 − b2 |, |a3 − b3 |}. Proposition 5. Suppose d(A˜ , B˜ ) = [d, d], if d = 0 then d = 0. 4. Similarity measure between two fuzzy numbers The similarity measure is important for presenting the degree of similarity between two objects or concepts. In a real life decision making situation, experts mostly express their opinions linguistically. Such linguistic expressions are the captured by fuzzy set theory and denoted as fuzzy numbers for analytic purpose. Some researchers introduce the similarity measures between two fuzzy numbers. Chen [20] and Lee [21], proposed a similarity measure for two normal trapezoidal fuzzy numbers A˜ 1 = (a1 , a2 ; β1 , γ1 ) and A˜ 2 = (a3 , a4 ; β2 , γ2 ) as follows s(A˜ 1 , A˜ 2 )chen = 1 −
|(a1 − β1 ) − (a3 − β2 )| + |a1 − a3 | + |a2 − a4 | + |(a2 + γ1 ) − (a4 − γ2 )| 4
where |a| denotes the absolute value of the real number a. 1
s(A˜ 1 , A˜ 2 )Lee = 1 −
(|(a1 − β1 ) − (a3 − β2 )|p + |a1 − a3 |p + |a2 − a4 |p + |(a2 + γ1 ) − (a4 − γ2 )|p ) p max(U ) − min(U )
where U is the universe of discourse. Chen and Chen [19] proposed a similarity measure between two generalized trapezoidal fuzzy numbers A˜ 1 (a1 , a2 ; β1 , γ1 ; w1 ) and A˜ 2 = (a3 , a4 ; β2 , γ2 ; w2 ) as follows s(A˜ 1 , A˜ 2 )chen–chen =
1−
B(sA
where K = 1 − |x∗A1 − x∗A2 |
|(a1 − β1 ) − (a3 − β2 )| + |a1 − a3 | + |a2 − a4 | + |(a2 + γ1 ) − (a4 − γ2 )| 4 1
,sA2 )
×
=
×K
min(y∗ ,y∗ ) A1 A2 . ∗ max(y ,y∗ ) A1
A2
Here, (x∗A , y∗A ) center-of-gravity and sA length of A˜ = (a1 , a2 ; β, γ ; w), respectively. Where sA = (a2 + γ ) − (a1 − β) and B(sA1 , sA2 ) =
1, 0,
sA1 + sA2 > 0 sA1 + sA2 = 0.
(5)
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And
a2 −a1 w × + 2 a2 +γ −a1 +β
y∗A =
6
w,
,
a2 + γ ̸= a1 − β (6)
a2 + γ = a1 − β 2 ∗ yA (a2 + a1 ) + (a2 + γ + a1 + β)(w − y∗A )
x∗A =
.
2w
Guha–Chakraborty [18] proposed a similarity measure between two generalized trapezoidal fuzzy numbers A˜ 1 (a1 , a2 ; β1 , γ1 ; w1 ) and A˜ 2 = (a3 , a4 ; β2 , γ2 ; w2 ) for 0 ≤ d(A˜ 1 ; A˜ 2 ) ≤ 1 as follows s(A˜ 1 , A˜ 2 )Guha–Chakraborty = 1 − d(A˜ 1 ; A˜ 2 ) =
dRr=w
1−
dRr=w + σ
,1 −
dLr=w dRr=w + σ
;
σ dRr=w + σ
,
θ
=
dRr=w + σ
where w = min(w1 , w2 ) and with
[L(r ), R(r )] = η([A˜ 1 ]r − [A˜ 2 ]r ) + (1 − η)([A˜ 2 ]r − [A˜ 1 ]r )
(7)
and
η=
A1 (w) + A1 (w)
1,
A1 (w) + A1 (w)
0,
A2 (w) + A2 (w)
≥
2
2 A2 (w) + A2 (w)
<
2
(8)
2
[L(r ), R(r )], L(r ) ≥ 0 [0, |L(r )| ∨ R(r )], L(r ) < 0 < R(r ). w w And θ = dLr=w − max{ 0 dLr dr , 0}, σ = | 0 dRr dr − dRr=w |. [dLr=w , dRr=w ] =
(9)
Here we present a new similarity measure based interval distance measure. The relation which has been used here is real interval S (A˜ , B˜ ) =
1 1 + d¯
,1 −
d
(10)
1 + d¯
˜ where [d, d¯ ] is the interval distance measure between A˜ and B. 4.1. Properties The new similarity measure has the following properties: 1. 2. 3. 4.
[0, 0] < S (A˜ , B˜ ) ≤ [1, 1]. S (A˜ , B˜ ) = S (B˜ , A˜ ). If A˜ = B˜ then S (A˜ , B˜ ) = [1, 1]. If A˜ ≤ B˜ ≤ C˜ , then S (A˜ , C˜ ) ≤ min{S (A˜ , B˜ ), S (B˜ , C˜ )}.
Proofs. 1. Because, d ≥ 0 and d ≤ d then
1 1 +d
> 0 and 1 −
d 1+d¯
≤ 1 hence [0, 0] < S (A˜ , B˜ ) ≤ [1, 1].
2. Because, d(A˜ , B˜ ) = d(B˜ , A˜ ), hence proof is evident. 3. Regarding (iv), the property of metric proof is evident. 4. A˜ ≤ B˜ ≤ C˜ , then 1
A(r )dr ≤ 0
1
B(r )dr ≤ 0
1
C (r )dr
1
and
0
A(r )dr ≤
0
min |A(r ) − B(r )| ≤ min |A(r ) − C (r )|} r ∈[0,1]
r ∈[0,1]
⇒
B(r )dr ≤ 0
hence,
min |A(r ) − B(r )| ≤ min |A(r ) − C (r )|} r ∈[0,1] r ∈[0,1]
1
dAB ≤ dAC
1
C (r )dr 0
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Table 1 The sets of fuzzy numbers.
Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7
A˜
B˜
(0.1, 0.2, 0.3, 0.4) (0.1, 0.2, 0.3, 0.4) (0.1, 0.2, 0.3, 0.4) (0.3, 0.3, 0.3) (0.2, 0.2, 0.2) (0.1, 0.2, 0.3) (0.1, 0.2, 0.3)
(0.1, 0.25, 0.4) (0.4, 0.55, 0.7) (0.4, 0.5, 0.6, 0.7) (0.3, 0.3, 0.3) (0.3, 0.3, 0.3) (0.3, 0.3, 0.3) (0.2, 0.3, 0.4)
Table 2 Comparison of the proposed similarity measure with the other similarity measure. Set
Lee
Chen
Chen and Chen
Chakraborty and Guha
New
Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7
0.9167 0.5 0.5 # 0 0.5 0.6667
0.975 0.7 0.7 1 0.9 0.9 0.9
0.8357 0.42 0.49 1 0.9 0.54 0.81
(0.825, 0.95, 1) (0.525, 0.65, 0.75, 0.877) (0.5, 0.6, 0.8, 0.9) (1, 1, 1) (0.9, 0.9, 0.9) (0.85, 0.9, 0.95) (0.8, 0.9, 1)
[0.952, 1] [0.741, 0.815] [0.77, 0.77] [1, 1] [0.9, 0.9] [0.833, 1] [0.91, 0.91]
and
max |A(r ) − B(r )| ≤ max |A(r ) − C (r )|} r ∈[0,1] r ∈[0,1]
⇒
AB
d
≤d
AC
max |A(r ) − B(r )| ≤ max |A(r ) − C (r )|}. r ∈[0,1]
r ∈[0,1]
Hence, d(A˜ , B˜ ) ≤ d(A˜ , C˜ ), and we can write d(B˜ , C˜ ) ≤ d(A˜ , C˜ ), respectively. Therefore, with inverse relation metric and similarity
S (A˜ , C˜ ) ≤ S (A˜ , B˜ )
⇒
˜ ˜ S (A, C ) ≤ S (B˜ , C˜ ).
S (A˜ , C˜ ) ≤ min{S (A˜ , B˜ ), S (B˜ , C˜ )}
Note. It is evident that with fuzzy data, the similarity measure should be positive, whereas with other existing methods it may be 0 ∈ S (A˜ , B˜ ) or S (A˜ , B˜ ) = 0 but, by this approach S (A˜ , B˜ ) ̸= 0. 4.2. Comparing with other methods In this section, we used 7 sets of normal trapezoidal or triangular fuzzy numbers given in Table 1 and proposed in Guha–Chakraborty [18]. We compare the similarity measure based on interval distance with the four methods presented by Chen [20], Lee [21], Chen and Chen [19] and Guha–Chakraborty [18]. A comparison between the results of the proposed similarity measure and the results of the other methods is shown in Table 2. Regarding Table 2, in this approach 1 ∈ S (A˜ , B˜ ) if and only if ∃ r ; A(r ) = B(r ) or A(r ) = B(r ). 5. Conclusions Many papers have proposed the fuzzy distance measure. In this paper, we developed a distance measure between fuzzy numbers as an interval number. We used this metric to define a similarity measure on fuzzy numbers as an interval number. The proposed distance or similarity measure can lead us to other concepts, such as ranking fuzzy numbers, nearest approximation, defuzzification and fuzzy decision making problems. References [1] [2] [3] [4]
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