![](https://csdnimg.cn/release/download_crawler_static/15482522/bg4.jpg)
where f
½1
is the pseudo-inverse of f defined as follows
f
½1
ðyÞ¼
f
1
ðyÞ; if y 2½0; f ð0Þ
0; if y 2ðf ð0Þ; 1:
(
The function f is called the additive generator of >.
It is not difficult to see that the Product t-norm >ðx; yÞ¼xy is a (continuous) Archimedean t-norm, with additive gener-
ator f
P
ðxÞ¼logðxÞ, but the minimum t-norm is not Archimedean. More common continuous Archimedean t-norms such as
Yager t-norm, Dombi t-norm have been discussed in [16].
Definition 6 [13]. Let > be a t-norm. A family of fuzzy variables fn
i
; i 2 Ig is called T-independent if for any subset
fi
1
; i
2
; ...; i
n
gI and n P 2,
Posfn
i
k
2 B
k
; k ¼ 1; 2; ...; ng¼>
n
k¼1
Posfn
i
k
2 B
k
gð3Þ
for any subsets B
1
; B
2
; ...; B
n
of R.
Particularly, fuzzy variables n
1
; ...; n
n
are T-independent if
Posfn
k
2 B
k
; k ¼ 1; 2; ...; ng¼>
n
k¼1
Posfn
k
2 B
k
gð4Þ
for any subsets B
1
; B
2
; ...; B
n
of R. Furthermore, regarding the T-independence between two families of fuzzy variables, we
have the following definitions.
Definition 7. Let > be a t-norm. We say two families of fuzzy variables fn
i
; i 2 Ig and ff
j
; j 2 Jg are mutually T-independent
iff for any fi
1
; ...; i
n
gI and fj
1
; ...; j
m
gJ with n; m P 1, fuzzy vectors ðn
i
1
; ...; n
i
n
Þ and ðf
j
1
; ...; f
j
m
Þ are T-independent.
For T-independent fuzzy variables n
k
with possibility distributions
l
k
for k ¼ 1; 2; ...; m, and a function g : R
m
! R,we
can determine the possibility distribution of gðn
1
; n
2
; ...; n
m
Þ as
l
gð n
1
;n
2
;...;n
m
Þ
ðxÞ¼ sup
x
1
;x
2
;...x
m
2R
>
m
k¼1
l
k
ðx
k
Þjx ¼ gðx
1
; x
2
; ...; x
m
Þ
;
which is usually called (generalized) Zadeh’s extension principle in the literature.
Example 1. Assume that n
1
; ...; n
n
are T-independent fuzzy variables, where > is a continuous Archimedean t-norm with
additive generator f. Determine the possibility distribution of the arithmetic mean
n
1
þþn
n
n
.
By Zadeh’s extension principle, the possibility distributions of ð n
1
þþn
n
Þ=n is
l
1
n
ðn
1
þþn
n
Þ
ðzÞ¼ sup
x
1
þþx
n
¼nz
>ð
l
n
1
ðx
1
Þ; ...;
l
n
n
ðx
n
ÞÞ: ð5Þ
Furthermore, since > is a continuous Archimedean t-norm with additive generator f, we know f and f
½1
are both continuous
and nonincreasing. It follows from Eqs. (2) and (5) that
l
1
n
ðn
1
þþn
n
Þ
ðrÞ
¼ sup
x
1
þþx
n
¼nr
f
½1
X
n
k¼1
f ð
l
n
k
ðx
k
ÞÞ
!
¼ f
½1
inf
x
1
þþx
n
¼nr
X
n
k¼1
f ð
l
n
k
ðx
k
ÞÞ
!
: ð6Þ
Furthermore, we plot in Fig. 1 the possibility distributions of arithmetic means
n
1
þþn
n
n
of T-independent and identically
distributed fuzzy variables under a continuous Archimedean t-norm (product t-norm) for different n’s to illustrate the pos-
sibility distribution evolution. From Fig. 1 we can see that the spread of the arithmetic mean distribution becomes narrower
and narrower as n increases. This property of T-independent fuzzy variables under continuous Archimedean t-norms plays a
pivotal role in support of our major results and will be rigorously proved later in Section 3.
Note that the family of t-norm contains a large number of operators, and in fact it is not true that any t-norm (not even
any Archimedean t-norm) can be utilized as a T-independence operator. For instance, consider two independent events
A; B; PosfAg¼PosfBg¼0:5 with Yager t-norm (k ¼ 1):>ðx; yÞ¼maxfx þ y 1; 0g as the independence-operator. We then
have PosfA; Bg¼>ðPosfAg; PosfBgÞ ¼ 0. Apparently, A; B are inconsistent events, which contradicts the independence
between A and B. Hence, for the sake of rationality, we give the following criterion of independence-consistency for t-norms
being selected safely as independence-operators.
Definition 8. A t-norm > is said to be independence-consistent, if >ða; bÞ > 0, for any a; b 2ð0; 1.
Using this criterion, we can verify the independence-consistency of the t-norms. For instance, the product t-norm and the
Dombi t-norm ðk 2ð0; 1ÞÞ:
S. Wang, Y.-K. Liu / Applied Mathematical Modelling 39 (2015) 1536–1553
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