1. Introduction
Recently, Hong [1] discussed Blackwell's Theorem in which inter-arrival times are characterized as fuzzy variables under t-norm-based fuzzy operations. He obtained Blackwell's Theorem for T-related fuzzy variables with respect to necessity measure holds true where T is an Archimedean t-norm and derived fuzzy Blackwell's Theorem based on the expected value of fuzzy variables. However, these results are invalid. Hong [2] provided a corrected version of a result. In this note, we give some counter examples of these results.
2. Counterexamples
Hong [1] obtained the following results of Blackwell's Theorem for fuzzy variables.
Theorem 2.1 (Hong [1]). Let T be a continuous Archimedean t-norm. Let ξi = (a, α, β)LR, i = 1, 2, ⋯ , be a sequence of T-related L-R fuzzy variables with 0 ≤ α < a and N(t) be a fuzzy renewal variable. If c/a is a natural number then for any ϵ > 0,
Theorem 2.2 (Hong [1]). Let T be a continuous Archimedean t-norm. Let ξi = (a, α, β)LR, i = 1, 2, ⋯ , denote a sequence of T-related L-R fuzzy variables with 0 ≤ α < a and N(t) be a fuzzy renewal variable. If c/a is a natural number then
However, these results are invalid. The following example shows that Theorem 2.1 is wrong.
Example 2.3. Let ξ = ξi = (1, 1/2, 1/2)LR, i = 1, 2, ⋯ with L(x) = R(x) = 1-x. Let T be an Archimedean t-norm. Let c = 1. We then have for, 0 < ϵ < 1,
Let tn = n + (1 - δ) for small δ > 0 and k = n. Let
Then we have
and
Then
From (1) and (2), we have
and since δ > 0 is arbitrary and limδ→0μξ(1 - δ) = 1,
Therefore, Theorem 2.1 is wrong.
The following example shows that Theorem 2.2 is wrong.
Example 2.4. Let ξ = ξi = (1, 1/2, 0)LR, i = 1, 2, ⋯ with L(x) = R(x) = 1-x. Let T be an continuous Archimedean t-norm. Let c = 1. Let tn = n + (1 - δ) for small δ > 0 and k = n. Let
Then, from (1) in Example 1,
By taking ξi = 1, i = 1, 2, ⋯ , Pos(N(t + 1) - N(t) = 1) = 1, t > 0 is easy to check. We also note that
Then we have, for n > 1,
and hence
Therefore, Theorem 2.2 is wrong.
참고문헌
- D.H. Hong, Blackwell's Theorem for T-related fuzzy variables, Information Sciences 180 (2010), 1769-1777. https://doi.org/10.1016/j.ins.2010.01.006
- D.H. Hong, Erratum to "Blackwell's Theorem for T-related fuzzy variables", Information Sciences 250 (2013), 227-228. https://doi.org/10.1016/j.ins.2013.05.036