(p). I ([a, b], [c, d]) = (D. (p). I ([a, b], [c, d])). 1 p. (8). EUSFLAT-LFA 2011. July 2011. Aix-les-Bains, France. © 2011. The authors - Published by Atlantis Press. 770 ...
EUSFLAT-LFA 2011
July 2011
Aix-les-Bains, France
Ranking of fuzzy numbers based on alpha-distance S. Khezerloo1 T. Allahviranloo2 M. Khezerloo1 1
2
Young Researcher Club, Ardabil Branch, Islamic Azad University, Ardabil, Iran. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
2. Preliminaries
Abstract
A fuzzy set A˜ = (a1 , a2 , a3 , a4 ) is a generalized left right fuzzy numbers (GLRFN) of Dubois and Prade [8, 16], if its membership function satisfies the following: −x L( aa22−a ), a1 ≤ x ≤ a2 , 1 1, a2 ≤ x ≤ a3 , µA˜ (x) = (1) x−a3 R( ), a3 ≤ x ≤ a 4 , a4 −a3 0, otherwise
This paper presents a new approach to compare fuzzy numbers using α-distance. Initially, the metric distance on the interval numbers based on the convex hull of the endpoints is proposed and it is extended to fuzzy numbers. All the properties of the α-distance are proved in details. Finally, the ranking of fuzzy numbers by the α-distance is discussed. In addition, the proposed method is compared with some known ones, the validity of the new method is illustrated by applying its to several group of fuzzy numbers.
Keywords: Metric Distance; Ranking.
where L and R are strictly decreasing functions defined on [0, 1] and satisfying the conditions: L(t) = R(t) = 1 L(t) = R(t) = 0
Fuzzy Numbers;
if if
t≤0 t≥1
(2)
For a2 = a3 , we have the classical definition of left right fuzzy numbers (LRFN) of Dubois and Prade [8, 16]. 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 . A GLRFN A˜ is denoted as A˜ = (a1 , a2 , a3 , a4 )LR and an α−level interval of fuzzy number A˜ as:
1. Introduction The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [3]. Additional related material can be found in [1, 10, 11]. Many researchers have developed methods to compare and to rank fuzzy numbers, too [9, 13, 14, 16, 19]. Cheng [4] has proposed some methods, and recently many ranking methods have been presented by Ma, Kandel and Friedman [12] and Chu and Tsao [15] . In [18] a fuzzy distance measure for Gaussian type fuzzy numbers is defined. In this study a new metric distance on fuzzy numbers is introduced and then it is used for ranking fuzzy numbers by comparison with two crisp numbers max(M) and min(m). The rest of this paper is organized as follows: Section 2 contains preliminaries on fuzzy concepts. A metric distance measure for interval numbers with its properties is introduced in Section 3. Then, in Section 4, α−distance for fuzzy numbers is defined and its properties are discussed in detail. In the Section 5, we use a procedure for ranking fuzzy numbers based on the α-distance. For comparing the proposed ranking method with some other approaches, some numerical examples are provided in Section 6. Finally, the paper ends with conclusions in Section 7.
˜ α = [Al (α), Ar (α)] [A] −1 = [a2 − (a2 − a1 )L−1 A (α), a3 + (a4 − a3 )RA (α)] (3) ˜ are two GLRFN then: If A˜ and B ˜ B ˜ = C˜ if and only if [A] ˜ α +[B] ˜ α = [C] ˜ α (4) A+ and ˜ = C˜ if and only if [A] ˜ α · [B] ˜ α = [C] ˜ α (5) A˜ · B where for interval numbers, we have [a, b] + [c, d] = [a + c, b + d]
(6)
and [a, b]·[c, d] = [min{ac, ad, bc, bd}, max{ac, ad, bc, bd}] (7) 3. Distance for intervals Let f (x) = (b − a)x + a and g(x) = (d − c)x + c where x ∈ [0, 1]. The distance between two intervals [a, b] and [c, d], a ≤ b, c ≤ d, is denoted by (p) dI ([a, b], [c, d]) such that: (p)
(p)
1
dI ([a, b], [c, d]) = (DI ([a, b], [c, d])) p © 2011. The authors - Published by Atlantis Press
770
(8)
and
(one of the intervals collapses to a single point). (p)
DI ([a, b], [c, d]) =k f (x) − g(x) kpLp
(9)
a+b Proposition 6. is the nearest number 2 to [a, b] by using this metric. Proposition 7. If p = 2 then r 1 2 (p) (a + b2 + ab) dI ([a, b], 0) = 3
where k . k is the usual norm in the Lp space on [0, 1] (p > 1). This distance takes into account every point in both intervals with the convex combination of the lower and upper bound values of the both intervals. It is different from most of the existing distance measures for interval numbers (Bardossy et al.[1], Diamond [5], Diamond and Korner [6], and Diamond and Tanaka [7]). Tran and Duckstein [16] proposed a distance measure for intervals which also considers every point of the two intervals. But it was not a metric distance because d([a, b], [a, b]) 6= 0. Now we show that our proposed distance is a metric distance on interval numbers.
is equal to the distance measure of Tran and Duckstein [16]. The proofs of Propositions 1, 2, 3, 4, 5, 6 and 7 are clear. Proposition 8. (p)
(p)
dI ([a, b] + [c, d], [a, b] + [e, z]) = dI ([c, d], [e, z]) Proof: (p)
dI ([a, b] + [c, d], [a, b] + [e, z]) (p) = dI ([a + c, b + d], [a + e, b + z]) R1 = ( 0 ((b + d − a − c)x + a + c)
3.1. Metric properties This proposed distance measure satisfies the following metric properties:
(p)
2. [a, b] = [c, d] if and only if dI ([a, b], [c, d]) = 0 (p)
(p)
= dI ([c, d], [e, z])
(p)
3. dI ([a, b], [c, d]) = dI ([c, d], [a, b]) (p)
4. dI ([a, b], [c, d]) (p) dI ([a, b], [e, z])
+
(p)
dI ([c, d], [e, z])
Definition 1. If [a, b] and [c, d] are two intervals and M is a crisp number, then by (8) and (9), we have
≥
The proofs follow immediately from the properties of the Lp − norm.
(p)
1
(p)
dI ([a, b].[c, d], M ) = (DI ([a, b].[c, d], M )) p (11) where
(p)
3.2. Other properties of dI
(p)
This distance satisfies the following properties if a, b, c, d, e and z are real numbers:
DI ([a, b].[c, d], M ) =
Z
0
1
(p)
DI ([a, b].((d−c)x+c), M )dx (12)
If c = d = 1 then
Proposition 1. If λ ≥ 0, then (p)
(10)
1
− ((b + z − a − e)x + a + e)p dx) p R1 1 = ( 0 ((d − c)x + c) − ((z − e)x + e)p dx) p
(p)
1. dI ([a, b], [c, d]) ≥ 0
(p)
(p)
dI ([a, b].[c, d], M ) = dI ([a, b], M )
(p)
dI ([λa, λb], [λc, λd]) = |λ|dI ([a, b], [c, d])
4. Metric for fuzzy numbers
Proposition 2. (p)
˜ by using α−distance between two GLRFNs A˜ and B the metric defined in Section 3, can be defined as:
(p)
dI ([a + λ, b + λ], [c + λ, d + λ]) = dI ([a, b], [c, d]) Proposition 3.
(p)
(p) dI (a, b)
(p)
1
˜ B) ˜ = (D (A, ˜ B)) ˜ p dF (A, F
= |a − b|
(13)
such that
(both intervals collapse to a single point).
(p) ˜ ˜ (p) ˜ α ˜ α DF (A, B) = supα∈[0,1] DI ([A] , [B] )
Proposition 4. If p = 2 then
(14)
(p)
˜ B) ˜ is a metric on It can be proved that dF (A, GLRFNs.
(p)
d qI ([a, b], [c, d]) = 1 2 2 3 ((a − c) + (b − d) + (a − c)(b − d))
4.1. The properties of α−distance
Proposition 5. If p = 2 then (p) d qI ([a, b], c) = 1 2 3 ((a − c) +
The proposed distance ’α−distance’ has the following metric properties: (p) ˜ ˜ 1. dF (A, B) ≥ 0
(b − c)2 + (a − c)(b − c)) 771
2. 3. 4.
˜ if and only if d(p) (A, ˜ B) ˜ =0 A˜ = B F (p) ˜ ˜ (p) ˜ ˜ dF (A, B) = dF (B, A) (p) ˜ ˜ (p) ˜ ˜ (p) ˜ ˜ dF (A, B) + dF (B, C) ≥ dF (A, C)
Proposition 11. (p) ˜ E˜ + F˜ ) = d(p) (G, ˜ F˜ ) dF (E˜ + G, F
Proof: By considering Proposition (8)
Proofs: (p) ˜ α , [B] ˜ α ≥ 0 then D(p) (A, ˜ B) ˜ ≥ 1. Since DI [A] F (p) ˜ ˜ 0 hence d (A, B) ≥ 0.
1 (p) ˜ ˜ E ˜ + F˜ ) = D(p) (E˜ + G, ˜ E ˜ + F˜ ) p dF (E + G, F
˜ if and only if [A] ˜ α = [B] ˜α 2. We know that, A˜ = B (p) for all α ∈ [0, 1]. But, since dI ([., .], [., .]) is a ˜ α = [B] ˜ α if and only metric distance, then [A] (p) ˜ α ˜ α if DI ([A] , [B] ) = 0 for all α ∈ [0, 1]. Hence, ˜ if and only if D(p) (A, ˜ B) ˜ = 0 and then A˜ = B F (p) ˜ ˜ dF (A, B) = 0.
1 (p) ˜ α ˜ α , [E] ˜ α + [F˜ ]α ) p = supα∈[0,1] DI ([E] + [G]
F
p1 (p) ˜ α ˜ α = supα∈[0,1] DI ([G] , [F ] ) p1 (p) ˜ ˜ = DF (G, F) (p) ˜ ˜ = dF (G, F)
3. From equations (13) and (14), 1 (p) ˜ ˜ (p) ˜ α ˜ α p dF (A, B) = supα∈[0,1] DI ([A] , [B]
(16) Then the proof is completed. Proposition 12.
p1 (p) ˜ α ˜ α = supα∈[0,1] DI ([B] , [A] )
(p)
dF (a, b) = |a − b|
p1 (p) ˜ ˜ = DF (B, A)
(both fuzzy numbers collapse to a single point). Proposition 13.
(p) ˜ ˜ = dF (B, A).
(p)
4. From equations (13) and (14), p1 (p) ˜ ˜ (p) ˜ ˜ dF (A, C) = DF (A, C)
(both fuzzy numbers collapse to an interval).
p1 (p) = supα∈[0,1] DI ([A]α , [C]α )
(p)
= supα∈[0,1] DI ([A]α , [C]α )
p1
≤ supα∈[0,1]
α
([A] , [B] )
1
(17)
(p)
α∈[0,1]
(18) Then regarding (12) (p)
˜B ˜ , M ) = supα∈[0,1] DF (A.
(p) + dI ([B]α , [C]α ) α
(p)
(p)
(p)
(p) dI
(p)
˜ B, ˜ M ) = (D (A. ˜ B, ˜ M )) p dF (A. F
˜B ˜ , M ) = sup D ([A]α .[B]α , M )dα DF (A. I
≤ supα∈[0,1] dI ([A]α , [B]α )
˜ are two fuzzy numbers Definition 2. If A˜ and B and M is a crisp number then by (13), (14)
where
(p) = supα∈[0,1] dI ([A]α , [C]α )
(p)
dF ([a, b], [c, d]) = dI ([a, b], [c, d])
(15)
R1
(p)
DI ([A]α .((Bl (α) − Br (α))x + Br (α)), M )dx (19) (p) ˜ ˜ (p) ˜ ˜ If B = 1 then dF (A.B , M ) = dF (A, M ).
0
(p) + supα∈[0,1] dI ([B]α , [C]α )
5. Ranking fuzzy numbers by α-distance
(p) ˜ ˜ (p) ˜ ˜ = dF (A, B) + dF (B, C)
Ranking of fuzzy numbers has been studied by many researchers. Some researchers introduced distances and then compared fuzzy numbers with them ([2, 4, 16, 19]). Other researchers used the nearest symmetric (nonsymmetric) triangular fuzzy number or symmetric (nonsymmetric) trapezoidal fuzzy number for comparing fuzzy numbers, ([9, 13]). Each method has its own shortcoming. In this section, we propose an approach for rank˜ based on the αing two fuzzy numbers A˜ and B ˜ distance. For ranking two fuzzy numbers A˜ and B, ˜ ˜ if sup(supp(A)) < inf (supp(B)), it is evident that
Therefore, the proofs are completed. 4.2. Other properties of α−distance This distance satisfies the following properties: Proposition 9. If λ ≥ 0, then (p) ˜ λF˜ ) = |λ|d(p) (E, ˜ F˜ ) dF (λE, F
Proposition 10. (p) ˜ (p) ˜ ˜ dF (E + λ, F˜ + λ) = dF (E, F)
772
˜ and that the degree of A˜ < B ˜ is 1 and the deA˜ < B ˜ ˜ gree of A > B is 0; therefore, we consider two fuzzy ˜ such that supp(A)∩supp( ˜ ˜ 6= φ. numbers A˜ and B B) Tran and Duckstein [16] proposed a similar distance for ranking but it was not a metric distance because d([a, b], [a, b]) 6= 0. The proposed method in this pa˜ is based per for ranking two fuzzy numbers A˜ and B on the comparison of the degree of distance between ˜ and crisp max(M) and the fuzzy numbers A˜ and B crisp min(m) where: ˜ M ≥ max(suppA˜ ∪ suppB) & (20) ˜ ∪ suppB) ˜ m ≤ min(suppA
(a): (p) ˜ (p) ˜ M ) ≥ γd (B, M) γd (A, ˜ ⇔ A˜ - B or γ (p) (A, ˜ m) ≤ γ (p) (B, ˜ m) d d (23) such that the degree of ranking is defined as: ˜ B) ˜ (A
˜ B) ˜ (A
M and (p)
(p) (p) ˜ 1 ˜ γd (A, M ) = γd (B, M ) = 2 & (p) ˜ (p) ˜ m) = 12 γd (A, m) = γd (B, (28)
In this case, we have the maximum ambiguity for ranking, therefore the degree of ranking is defined as follows:
if only if
˜ B) ˜ (A=
γ(λ.M+(1−λ)m)
˜ B) ˜ (A>
= γ(λ.M+(1−λ)m) ˜ B) ˜ (A
γ(λM+(1−λ)m)
= 1,
′′
′
M ≥ max{M, M } and
′′
′
m ≤ min{m, m }
λ ∈ [0, 1]. Therefore
˜ and B ˜ - C˜ then for Proposition 19. If A˜ - B each λ ∈ [0, 1],
(p)
(p)
′′
(p)
′′
(p)
′′
(p)
′′
(p)
′′
(35)
≥ dF (˜ u, m )dF (˜ z, M ) D: For an arbitrary finite subset Λ of E and (˜ u, v˜) ∈ Λ2 , inf supp(˜ u) > supsupp(˜ v), we should have u˜ % v˜.
˜ M ≥ max(suppA˜ ∪ suppB), ˜ ∪ suppB), ˜ m ≤ min(suppA ˜ ∪ suppC), ˜ M ′ ≥ max(suppB
Proof: The proof is obvious by the definition of ranking.
˜ ∪ suppC), ˜ m′ ≤ min(suppB M ≥ max(M, M ),
(p)
′′
≥ dF (˜ v , m )dF (˜ z, M )
where
′
′′
≥ dF (˜ v , M )dF (˜ z, m )
˜ C) ˜ (A
v˜, then u˜ + z˜ > v˜ + z˜ when z˜ 6= 0.
(p)
Proof: If u ˜ > v˜ then
=
max(supp v˜) < min(supp u˜)
(α))x+zr (α)) , M) supα∈[0,1] DI ([u]α . ((zl (α)−zrM ′ (p)
(α))x+zr (α)) supα∈[0,1] DI ([v]α . ((zl (α)−zrM , M) ′
If
It is clear that if z˜ 6= 0 then k(α, x) =
max(supp (˜ v + z˜)) < min(supp (˜ u + z˜))
((zl (α) − zr (α))x + zr (α)) M′
then
hence,
0 < k(α, x) < 1
u ˜ + z˜ > v˜ + z˜ hence
G: Let u˜ , v˜, u˜z˜ and v˜z˜ be elements of E and
((zl (α)−zr (α))x+zr (α)) ,M) ′ M ((zl (α)−zr (α))x+zr (α)) (p) α supα∈[0,1] DI ([v] . ,M) ′ M (p)
supα∈[0,1] DI ([u]α .
z˜ ≥ 0(min(supp z˜) ≥ 0) u˜ - v˜, implies u ˜z˜ - v˜z˜.
(p)
supα∈[0,1] DI ([u]α .k(α,x),M) (p)
Proof: It is sufficient to show that (p)
′
(p)
supα∈[0,1] DI ([v]α .k(α,x),M)
′
(p)
(p)
(p)
M ) supα∈[0,1] k(α,x)DI ([v]α , k(α,x)
(38)
(p)
supα∈[0,1] k(α,x)DI ([u]α ,T ) (p)
supα∈[0,1] k(α,x)DI ([u]α ,T )
where ′
M ≥ max(supp u ˜ ∪ supp v˜) m ≤ min(supp u ˜ ∪ supp v˜)
with M ≥ max(supp u˜ ∪ supp v˜). Therefore
′
m ≤ min(supp z˜)
It is sufficient to show that (p) ′ ′ (p) dF (˜ uz˜, M M, s) ≥ dF (˜ v z˜, M M ) (p)
(p)
′
(p)
(p)
DF (˜ v z˜, M ′ M ) (39)
(p)
(p)
′
′
(43)
Remark 1. If u ˜ - v˜ then −˜ u % −˜ v.
6. Numerical example
′
′
(p)
DI ([u]α .((zl (α) − zr (α))x + zr (α)), M M )dx (40)
Considering (13), it is sufficient to show that (p)
′
(42)
DF (˜ uz˜, M M ) ≥ DF (˜ v z˜, M M )
DF (˜ uz˜, M M ) = supα∈[0,1]
(p)
≥1
or
′
We demonstrate that the proofs of ′ ′ (p) (p) dF (˜ uz˜, M M ) ≥ dF (˜ v z˜, M M ) and that ′ ′ (p) (p) of dF (˜ v z˜, m m) ≥ dF (˜ uz˜, m m) are similar. By (19) (p)
′
DF (˜ uz˜, M M )
dF (˜ v z˜, m m) ≥ dF (˜ uz˜, m m)
0
=1
Because, regarding the assumption, u˜ v˜ implies (p) supα∈[0,1] DI ([u]α , M ) ≥1 (41) (p) supα∈[0,1] DI ([v]α , M )
M ≥ max(supp z˜)
R1
≥
′
dF (˜ v z˜, M M )dF (˜ uz˜, m m)
=
M supα∈[0,1] k(α,x)DI ([u]α , k(α,x) )
′
dF (˜ uz˜, M M )dF (˜ v z˜, m m) ≥ (p)
=
The results of comparing our proposed method with some other ranking methods ([16]) are summed up in Figure 1 and Table 1. Figure 1: Figures of ranking.
′
DF (˜ uz˜, M M ) ≥ DF (˜ v z˜, M M )
Regarding to Figure 1 and Table 1 we have:
It is clear that (p)
′
DF (˜ uz˜, M M ) (p) DF (˜ v z˜, M ′ M )
(a): A1 = (.4, .9, 1), A2 = (.4, .7, 1), A3 = (.4, .5, 1) then A3 - A2 - A1
≥1
if (p)
′
(p)
′
supα∈[0,1] DI ([u]α .((zl (α) − zr (α))x + zr (α)), M M ) supα∈[0,1] DI ([v]α .((zl (α) − zr (α))x + zr (α)), M M )
(b): A1 = (.2, .5, .8), A2 = (.4, .5, .6) then ≥1
775
A1 ≈ A2
1
1 A2
A2
A1
A1
0.8
0.8
A3 A3
A1
0.6
0.6
0.4
0.4
0.2
0.2
0 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1.1
˜1 , A˜2 , A ˜3 ) Figure 1: (A
A1
A2
A2
0
0.2
0.4
0.6
0.8
1
˜1 , A ˜2 ) Figure 5: (A
1
7. Conclusions 0.8
In this paper, a metric distance on interval numbers was introduced using the α− cut concept. Subsequently, it was extended to fuzzy numbers. Comparison of fuzzy numbers was made based on the degree of distance between the fuzzy numbers and crisp max(M) and crisp min(m). If we add a fuzzy number to the set of fuzzy numbers, then M and m, as well as, may change, but there will be no changes in the ranking, which is the very useful property in the industrial problems. By some numerical examples, it was shown that the proposed method is more consistent with institution than the previous ranking methods.
A2
0.6
A2
0.4 A1 A1 0.2
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
˜1 , A˜2 ) Figure 2: (A 1
0.8
A3 A1 0.6 A2 A1 0.4
A2
References
A3
0.2
0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.9
1
[1] A. Bardossy and L. Duckstein, Fuzzy Rule-Based Modeling with Applications to Geophysical, Biological and Engineering Systems, CRC press, Boe, Raton, 1995. [2] G. Bortolan and R. Degan. A review of some methods for ranking fuzzy sets, Fuzzy Sets and Systems, 15: 1–19, 1985. [3] S.L. Chang and L.A. Zadeh, On fuzzy mapping and control, IEEE Trans. Syst. Man Cybernet. 2: 30–34, 1972. [4] C.H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems, 95: 307–317, 1998. [5] P. Diamond, Fuzzy least squares, Information Sciences 46: 141–157, 1988. [6] P. Diamond and R. Korner, Extended fuzzy linear models and least squares estimates, Comput. Math. Appl. 33: 15–32, 1997. [7] P. Diamond and H. Tanaka, Fuzzy regression analysis, in: R. Slowinski (Ed), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Boston, 1998. [8] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York , 1980. [9] P. Fortemps and M. Roubens, Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems, 82: 319–330, 1996.
˜1 , A˜2 , A ˜3 ) Figure 3: ((A 1
0.8
A1
0.6
A1
A2
A2
A3
0.4 A3
0.2
0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
˜1 , A˜2 , A ˜3 ) Figure 4: (A
(c): A1 = (.5, .7, .9), A2 (.3, .4, .7, .9) then
=
(.3, .7, .9), A3
=
A3 - A2 - A1 (d): A1 = (.3, .5, .8, .9), A2 = (.3, .5, .9), A3 = (.3, .5, .7) then A3 - A2 - A1 (e): A1 = (.3, .3, 1), A2 = (.1, .8, .8) then A1 - A2 776
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Table 1: Comparison of fuzzy numbers by some methods. M ethods Y ager
F1 F2 F3
Ba − Kw Bald − Gui
Kerre Jain
Dub − P ra
Lee − Li
F or − Rou T ran − Duc
1:P g r:a k=1 k=2 k = 1/2 PD P SD ND N SD U.m U.G P.m P.G F0 Dmax ; f = x Dmin ; f = x Dmax ; f = 1 Dmin ; f = 1
A1 0.760 0.900 0.800 1.000 0.420 0.550 0.280 1.000 0.900 0.840 0.950 1.000 0.740 0.630 0.260 0.760 ... 0.800 ... 0.800 0.187 0.838 0.231 0.808
a A2 0.700 0.760 0.700 0.740 0.330 0.400 0.230 0.860 0.760 0.650 0.860 0.740 0.230 0.380 0.000 0.700 ... 0.700 ... 0.700 0.308 0.704 0.316 0.707
A3 0.630 0.660 0.600 0.600 0.330 0.340 0.220 0.760 0.660 0.540 0.780 0.600 0.160 0.180 0.000 0.630 .... 0.600 ... 0.600 0.442 0.573 0.416 0.611
b A1 0.500 0.610 0.600 1.000 0.270 0.300 0.200 0.910 0.730 0.600 0.830 1.000 0.730 0.270 0.000 0.500 0.120 0.500 0.090 0.500 0.505 0.505 0.510 0.510
A2 0.500 0.540 0.500 1.000 0.270 0.240 0.230 0.910 0.670 0.480 0.800 1.000 0.240 0.760 0.000 0.500 0.040 0.500 0.030 0.500 0.501 0.501 0.501 0.501
dF
γd (., M );
(2)
0.409
0.500
0.591
0.500
0.500
dF
γd (., m);
(2)
0.591
0.500
0.409
0.500
0.500
Table 1.(continued) A1 0.700 0.750 0.700 1.000 0.370 0.420 0.270 1.000 0.820 0.710 0.890 1.000 0.500 0.670 0.000 0.700 ... 0.700 ... 0.700 0.304 0.702 0.307 0.703 0.357 0.643
777
c A2 0.630 0.750 0.650 1.000 0.270 0.350 0.190 0.910 0.820 0.710 0.890 1.000 0.500 0.350 0.000 0.630 ... 0.650 ... 0.650 0.342 0.671 0.365 0.658 0.464 0.536
A3 0.570 0.750 0.570 1.000 0.270 0.350 0.190 0.750 0.820 0.710 0.890 1.000 0.500 0.000 0.000 0.570 ... 0.580 ... 0.575 0.457 0.585 0.445 0.590 0.510 0.490
A1 0.620 0.810 0.620 1.000 0.450 0.350 0.310 1.000 0.900 0.820 0.940 1.000 0.800 0.500 0.000 0.620 ... 0.630 ... 0.625 0.395 0.650 0.398 0.639 0.522 0.478
d A2 0.560 0.640 0.540 1.000 0.370 0.400 0.280 0.850 0.690 0.560 0.800 1.000 0.200 0.500 0.000 0.560 ... 0.550 ... 0.550 0.473 0.539 0.462 0.560 0.536 0.464
A3 0.500 0.580 0.500 1.000 0.270 0.280 0.210 0.750 0.640 0.450 0.770 1.000 0.000 0.500 0.000 0.500 ... 0.500 ... 0.500 0.502 0.502 0.504 0.504 0.643 0.357
e A1 0.610 0.660 0.580 0.840 0.420 0.440 0.340 0.960 0.660 0.530 0.780 0.840 0.540 0.540 0.000 0.610 ... 0.530 ... 0.490 0.574 0.451 0.531 0.512 0.544 0.456
A2 0.530 0.690 0.560 1.000 0.330 0.370 0.240 0.890 0.690 0.510 0.810 1.000 0.460 0.460 0.160 0.530 ... 0.580 ... 0.610 0.355 0.673 0.417 0.628 0.453 0.547
