A NOTE ON BLACKWELL'S THEOREM FOR FUZZY VARIABLES

Abstract

Recently, some results of Blackwell's Theorem in which interarrival times are characterized as fuzzy variables under t-norm-based fuzzy operations are discussed by Hong. However, these results are invalid. In this note, we give counter examples of these results.

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.