A Fuzzy Distance between Two Fuzzy Numbers Saeid Abbasbandy and Saeide Hajighasemi Department of Mathematics, Science and Research Branch Islamic Azad University, Tehran, 14515/775, Iran Tel.: +98(912)1305326 (S. Abbasbandy)
[email protected]
Abstract. In this paper by using Hausdorff distance as a maximum distance between two fuzzy numbers, a new fuzzy distance is introduced between two fuzzy numbers. Several examples are used to show preference of the proposed fuzzy distance to others.
1
Introduction
The methods of measuring of distance between fuzzy numbers have became important due to the significant applications in diverse fields like remote sensing, data mining, pattern recognition and multivariate data analysis and so on. Several distance measures for precise numbers are well established in the literature. Several researchers focused on computing the distance between fuzzy numbers [1,2,3,6,8,9]. Usually the distance methods basically compute crisp distance values for fuzzy numbers. Naturally a logical question occurs to us: if the numbers themselves are not known exactly, how can the distance between them be an exact value? In view of this, Voxman [9] first introduced a fuzzy distance for fuzzy numbers. Therefore a distance measure for fuzzy numbers is that the distance between two uncertain numbers should also be an uncertain number, logically. Section 2 describes the basic notation and definitions of fuzzy numbers, support and α-cut of fuzzy numbers. Also the fuzzy distance of Voxman is described in Section 2.1. A new distance measure between fuzzy numbers is defined in Section 3 and a fuzzy distance measure in Section 4. Ambiguity and fuzziness of fuzzy distance measure are investigated in Section 4.1. Finally, conclusions are drawn in Section 5.
2
Preliminaries
A fuzzy set on a set X is a function μ : X → [0, 1]. The support of μ, supp μ is the closure of the set {x ∈ X | μ(x) > 0}. Definition 1. [9] A fuzzy number is a fuzzy set μ : IR → [0, 1] on IR satisfying (i) μ is upper semi-continuous;
Corresponding author.
E. H¨ ullermeier, R. Kruse, and F. Hoffmann (Eds.): IPMU 2010, Part II, CCIS 81, pp. 376–382, 2010. c Springer-Verlag Berlin Heidelberg 2010
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(ii) supp μ is a closed and bounded interval; (iii) if supp μ = [a, b], then there exist c, d, a ≤ c ≤ d ≤ b, such that μ is increasing on the interval [a, c], equal to 1 on the interval [c, d] and decreasing on the interval [d, b]. We let F denote the family of all fuzzy numbers. If μ ∈ F , then for each α, 0 < α ≤ 1, the α-cut of μ, is defined by μα = {x ∈ X | μ(x) ≥ α}. The α-cut representation of μ is the pair of functions, (L(α), R(α)), defined by ⎧ ⎨ inf{x | x ∈ μα } L(α) =
⎩
if α > 0,
inf{x | x ∈ supp μ}
if α = 0,
and ⎧ ⎨ sup{x | x ∈ μα } R(α) =
⎩
if α > 0,
sup{x | x ∈ supp μ}
if α = 0.
If μ is a fuzzy number then the compliment of μ, μc , is the fuzzy set defined by μc (x) = 1 − μ(x). If K is the set of compact subsets of IR2 , and A and B are two subsets of IR2 then the Hausdorff metric H : K × K → [0, ∞) is defined by [9] H(A, B) = max{sup dE (b, A), sup dE (a, B)}, a∈A
b∈B
where dE is the usual Euclidean metric for IR2 . Definition 2. The metric d∞ on F × F is defined by d∞ (μ, ν) = sup {H(μα , να )}. α∈[0,1]
Definition 3. The μ is a triangular fuzzy number, and We write μ = (x0 , σ, β), with defuzzifier x0 , and left fuzziness σ > 0 and right fuzziness β > 0 is a fuzzy set where the membership function is as ⎧1 (x − x0 + σ), ⎪ ⎪ ⎪σ ⎪ ⎨ μ(x) = β1 (x0 − x + β), ⎪ ⎪ ⎪ ⎪ ⎩ 0,
x0 − σ < x ≤ x0 , x0 ≤ x < x0 + β, otherwise.
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2.1
S. Abbasbandy and S. Hajighasemi
Fuzzy Distance Given by Voxman
Here, we briefly describe the fuzzy distance measure by Voxman [9]. The fuzzy distance function on F, Δ : F × F → F , is define by Δ(μ, ν)(z) =
sup min{ μ(x), ν(y)}.
|x−y|=z
For each pair of fuzzy numbers μ and ν, let Δμν denote the fuzzy number Δ(μ, ν). R L R If the α-cut representations of μ and ν are (AL 1 (α), A1 (α)) and (A2 (α), A2 (α)), respectively, then the α-cut representation of Δμν , (L(α), R(α)), is given by
L(α) = and
3
⎧ R ⎨ max {AL 2 (α)−A1 (α), 0}
if
1 L 2 (A1 (1)
1 L R + AR 1 (1)) ≤ 2 (A2 (1) + A2 (1)),
⎩
if
1 L 2 (A2 (1)
1 L R + AR 2 (1)) ≤ 2 (A1 (1) + A1 (1)),
R max {AL 1 (α)−A2 (α), 0}
L R L R(α) = max {AR 1 (α) − A2 (α), A2 (α) − A1 (α)}.
A New Distance between Two Fuzzy Numbers
Let μ and ν be two arbitrary fuzzy numbers with α-cut representations R L R (AL 1 (α), A1 (α)) and (A2 (α), A2 (α)), respectively. The distance between μ and ν is defined as 1 2 R L L [(1 − α)(AR (1) d(μ, ν) = 1 (α) − A2 (α)) + α(A1 (α) − A2 (α))]dα 0 +
1 1 2
R L L [α(AR 1 (α) − A2 (α)) + (1 − α)(A1 (α) − A2 (α))]dα .
In other words, right dominance has preference to the left dominance. Theorem 1. For fuzzy numbers μ, ν and ω, we have (i) d(μ, ν) ≥ 0 and d(μ, μ) = 0; (ii) d(μ, ν) = d(ν, μ); (iii) d(μ, ν) ≤ d(μ, ω) + d(ω, ν). Proof. We consider only (iii). Suppose μ and ν have α-cut representations as R before, and ω has α-cut representation (AL 3 (α), A3 (α)). By (1), we have 1 2 R R R [(1 − α) AR d(μ, ν) = 1 (α) − A3 (α) + A3 (α) − A2 (α) 0 L L L +α AL 1 (α) − A3 (α) + A3 (α) − A2 (α) ]dα
A Fuzzy Distance between Two Fuzzy Numbers
+
1 1 2
379
R R R [α AR 1 (α) − A3 (α) + A3 (α) − A2 (α) L L L +(1 − α) AL 1 (α) − A3 (α) + A3 (α)A2 (α) ]dα
1 2 R L L ≤ [(1 − α)(AR 1 (α) − A3 (α)) + α(A1 (α) − A3 (α))]dα 0 1 R R L L [α(A1 (α) − A3 (α) + (1 − α)(A1 (α) − A3 (α))]dα + 1 2 1 2 R L L [(1 − α)(AR + 3 (α) − A2 (α)) + α(A3 (α) − A2 (α))]dα 0 1 R L L [α(AR + 3 (α) − A2 (α) + (1 − α)(A3 (α) − A2 (α))]dα = d(μ, ω) + d(ω, ν). 1 2
Since we introduce this distance by dominance, similarity Hausdorff distance we can be proved these properties (i) (ii) (iii) (iv)
d(u + w, v + w) = d(u, v) for every u, v, w ∈ F , d(u + v, ˜ 0) ≤ d(u, ˜ 0) + d(v, ˜ 0) for every u, v ∈ F , d(λu, λv) = |λ|d(u, v) for every u, v ∈ F and λ ∈ IR, d(u + v, w + z) ≤ d(u, w) + d(v, z) for u, v, w, and z ∈ F .
Theorem 2. For two fuzzy numbers μ and ν, We have d(μ, ν) ≤
d∞ (μ, ν).
Proof. By definition d(., .) we have, 1 12 2 R L d(μ, ν) = (1 − α)(AR (α) − A (α))dα + α(AL 1 2 1 (α) − A2 (α))dα 0 0 1 1 R R L L + α(A1 (α) − A2 (α))dα + (1 − α)(A1 (α) − A2 (α))]dα . 1 1 2 2 R R By d∞ (μ, ν) = M , we have A1 (α) − A2 (α) ≤ M and L assumption L A1 (α) − A2 (α) ≤ M and mean value theorem for integrals, We obtain 12 12 d(μ, ν) ≤ M (1 − α)dα + M αdα 0
+M
1 1 2
αdα + M
1
(1 − α)dα + M
=M 0
0 1 1 2
(1 − α)dα
1
(α)dα = M 0
Therefore d(μ, ν) ≤ d∞ (μ, ν).
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S. Abbasbandy and S. Hajighasemi Table 1. Comparison of d and d∞ μ
ν
d(μ, ν) d∞ (μ, ν)
(4,3,1) (0,1,2)
27 8
4
(3,2,2) (4,3,1)
0.5
1
(2,1,1) (4,1,1)
2
2
(4,1,1) (6,2,2)
2.25
3
(2,1,4) (3,2,2)
0.125
1
(2,1,1) (6,1,1)
4
4
(3,2,2) (3,1,1)
0.25
1
See Table 1 for comparison between Hausdorff distance and d distance for some triangular fuzzy numbers. We can see that d(μ, ν) ≤ d∞ (μ, ν) in all examples.
4
New Fuzzy Distance between Two Fuzzy Numbers
R Let two fuzzy numbers μ and ν, with α-cut representation (AL 1 (α), A1 (α)) and L R (A2 (α), A2 (α)), respectively, are given. By d(., .) and d∞ (., .), we can introduce the fuzzy distance by a symmetric triangular fuzzy number as follows: d(μ, ν) + d∞ (μ, ν) d∞ (μ, ν) − d(μ, ν) d∞ (μ, ν) − d(μ, ν)
, , d(μ, ν) = , (2) 2 2 2
with α-cut representation (λα (μ, ν), ρα (μ, ν)). The proposed fuzzy distance (2) satisfies fuzzy distance properties followed in Kaleva and Seikkala [7]. Theorem 3. For fuzzy numbers μ, ν and ω, we have
ν) =
d(μ, 0 if only if μ = ν;
ν) = d(ν,
μ); d(μ, λα (μ, ν) ≤ λα (μ, ω) + λα (ω, ν) and ρα (μ, ν) ≤ ρα (μ, ω) + ρα (ω, ν).
1, x = 0,
Proof. (i) By definition of fuzzy zero, 0(x) = , from assumption 0, x = 0,
ν) =
d(μ, 0, we obtain d(μ, ν) + d∞ (μ, ν) = 0. Since d(μ, ν) and d∞ (μ, ν) are positive numbers, we have d(μ, ν) = d∞ (μ, ν) = 0 and hence μ = ν. Also, converse is obvious. (ii) By properties of d(., .) and d∞ (., .), it is obvious. (iii) By definition of λα (μ, ν), we have (i) (ii) (iii)
d (μ, ν) − d(μ, ν) α α ∞ = 1− d(μ, ν) + d∞ (μ, ν) λα (μ, ν) = d(μ, ν) + α 2 2 2
A Fuzzy Distance between Two Fuzzy Numbers
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α α d(μ, ω)+d(ω, ν) + d∞ (μ, ω)+d∞ (ω, ν) = λα (μ, ω)+λα (ω, ν), ≤ 1− 2 2 because (1 − α2 ) > 0. For ρ(μ, ν), we have the similar proof. 4.1
Ambiguity and Fuzziness of a Fuzzy Number
Delgado et al. [4,5] have extensively studied two attributes of fuzzy numbers, ambiguity and fuzziness. Ambiguity may be seen as a ’global spread’ of the membership function, whereas the fuzziness involve a comparison between the fuzzy set and its complement. These concepts are defined as follow :
1
S(α)[R(α) − L(α)]dα,
A(μ) = 0
1
S(α)[q − p]dα −
F (μ) = +
1 1 2
S(α)[Lc (α) − p]dα +
1 2
0
1 2
S(α)[q − Rc (α)]dα +
1
1 2 1 2
S(α)[L(α) − p]dα +
0
1
S(α)[R(α) − L(α)]dα
S(α)[Rc (α) − Lc (α)]dα
0
+
1 2
S(α)[q − R(α)]dα ,
0
where supp μ = [p, q] and (L(α), R(α)) be the α-cut representations of μ. Also μc be the complement of μ with α-cut representations (Lc (α), Rc (α)). The function S : [0, 1] → [0, 1] is an increasing function and S(0) = 0 and S(1) = 1, [9]. We 1 say that S is a regular reducing function if 0 S(α)dα = 12 . A routine calculation shows for S(α) = α, we have
1 2
F (μ) =
[R(α) − L(α)]dα +
0
1 1 2
[L(α) − R(α)]dα.
Table 2. Comparison of ambiguity and fuzziness μ
ν
ν)) F (d(μ,
ν)) A(Δ(μ, ν)) F (Δ(μ, ν)) A(d(μ,
(3,2,2) (4,3,1)
1 12
1 8
(2,1,1) (4,1,1)
0
0
1 8 7 48
3 16 7 32
(2,1,1) (6,1,1)
0
0
(3,2,2) (3,1,1)
1 8
3 16
(4,1,1) (6,2,2) (2,1,4) (3,2,2)
68 75 2 3 53 54 203 216 2 3 1 2
17 20
1 4 3
1 1 3 4
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S. Abbasbandy and S. Hajighasemi
ν) are less than of Table 2 shows that the ambiguity and the fuzziness of d(μ, the ambiguity and fuzziness of Δ(μ, ν), which is defined by Voxman [9], for some examples. We can see that, when the support of μ and ν are disjoint, then
ν)) = F (d(μ,
ν)) = 0. d(μ, ν) = d∞ (μ, ν) and in this case, A(d(μ,
5
Conclusions
Here, a new distance measure has been introduced for computing crisp distances for fuzzy numbers. It is reasonable, the distance between two uncertain numbers should also be an uncertain number. Voxman first introduced the concept of fuzzy distance for fuzzy numbers. In this paper, we introduce another fuzzy distance measure between two fuzzy numbers. However, the method proposed in this paper compute a fuzzy distance value with less ambiguity and fuzziness as compared to that of Voxman’s method, which has been shown by some examples.
Acknowledgements The authors would like to thank the anonymous referees for their constructive suggestions and comments.
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