A note on ‘‘Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP”

doi:10.1016/j.ins.2016.01.054

Information Sciences

Volumes 346–347, 10 June 2016, Pages 1-5

https://doi.org/10.1016/j.ins.2016.01.054 Get rights and content

Abstract

Wang and Chen (2008) stated and proved some results (Proposition 5.1, Proposition 5.2, Information Sciences, 178 (2008), 3759) and used these results to propose a method to construct fuzzy linguistic preference relation matrices. In this paper, it is pointed out that Wang and Chen have used some mathematical incorrect assumptions for proving these results. Hence, the statement and proof of these results as well as the method, proposed by Wang and Chen, are not valid. Further, the exact results are stated and proved.

Introduction

Wang and Chen [1, pp. 3759] stated and proved the following results.

Proposition 5.1

[1, pp. 3759]. Given that a set of alternatives, $X = {x_{1}, \dots, x_{n}}$ associated with a fuzzy reciprocal multiplicative preference matrix $\tilde{A} = ({\tilde{a}}_{i j})$ with ${\tilde{a}}_{i j} \in [1 / 9, 9]$ , and the corresponding fuzzy reciprocal linguistic preference relation, $\tilde{P} = ({\tilde{p}}_{i j})$ with ${\tilde{p}}_{i j} \in [0, 1]$ , verifies the additive reciprocal, then, the following statements are equivalent.

(1)
$p_{i j}^{L} + p_{j i}^{R} = 1 \forall i, j \in {1, \dots, n} .$
(2)
$p_{i j}^{M} + p_{j i}^{M} = 1 \forall i, j \in {1, \dots, n} .$
(3)
$p_{i j}^{R} + p_{j i}^{L} = 1 \forall i, j \in {1, \dots, n} .$

Proof

[1, pp. 3759] Since, $\tilde{A} = ({\tilde{a}}_{i j})$ is a reciprocal matrix,

So, ${\tilde{a}}_{j i} = {\tilde{a}}_{i j}^{- 1} = \frac{1}{{\tilde{a}}_{i j}} \Rightarrow {\tilde{a}}_{i j} \otimes {\tilde{a}}_{j i} \approx 1$

Taking logarithms on both sides yields $\begin{matrix} lo g_{9} {\tilde{a}}_{i j} \oplus lo g_{9} {\tilde{a}}_{j i} \approx 0 \forall i, j \in {1, \dots, n} \end{matrix}$

Adding 2 and dividing 2 on both sides, $\begin{matrix} \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{i j}) \oplus \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{j i}) \approx \tilde{1} \forall i, j \in {1, \dots, n} \end{matrix}$

Assuming $\begin{matrix} {\tilde{p}}_{i j} = \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{i j}) and {\tilde{p}}_{j i} = \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{j i}) \end{matrix}$ $\begin{matrix} \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{i j}) \oplus \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{j i}) \approx \tilde{1} \forall i, j \in {1, \dots, n} \end{matrix}$ $\begin{matrix} \Rightarrow {\tilde{p}}_{i j} \oplus {\tilde{p}}_{j i} \approx \tilde{1} \forall i, j \in {1, \dots, n} \end{matrix}$ $\begin{matrix} \Rightarrow (p_{i j}^{L}, p_{i j}^{M}, p_{i j}^{R}) \oplus {\tilde{p}}_{j i} \approx \tilde{1} = (1, 1, 1) \end{matrix}$ $\begin{matrix} \Rightarrow {\tilde{p}}_{j i} \approx 1 Θ (p_{i j}^{L}, p_{i j}^{M}, p_{i j}^{R}) = (1 - p_{i j}^{R}, 1 - p_{i j}^{M}, 1 - p_{i j}^{L}) \end{matrix}$ $\begin{matrix} (p_{j i}^{L}, p_{j i}^{M}, p_{j i}^{R}) \approx (1 - p_{i j}^{R}, 1 - p_{i j}^{M}, 1 - p_{i j}^{L} \end{matrix}$ $\begin{matrix} p_{i j}^{L} + p_{j i}^{R} = 1, p_{i j}^{M} + p_{j i}^{M} = 1, p_{i j}^{R} + p_{j i}^{L} = 1 \forall i, j, k \end{matrix}$

Proposition 5.2

[1, pp. 3759] For a reciprocal fuzzy linguistic preference relation $\tilde{P} = ({\tilde{p}}_{i j}) = (p_{i j}^{L}, p_{i j}^{M}, p_{i j}^{R})$ to be consistent, verifies the additive consistency, then, the following statements must be equivalent:

(a)
$p_{i j}^{L} + p_{j k}^{L} + p_{k i}^{R} = \frac{3}{2} \forall i < j < k .$
(b)
$p_{i j}^{M} + p_{j k}^{M} + p_{k i}^{M} = \frac{3}{2} \forall i < j < k .$
(c)
$p_{i j}^{R} + p_{j k}^{R} + p_{k i}^{L} = \frac{3}{2} \forall i < j < k .$

Proof

[1, pp. 3759] Since, $\tilde{A} = ({\tilde{a}}_{i j})$ is a consistent matrix. So, $\begin{matrix} {\tilde{a}}_{i j} \otimes {\tilde{a}}_{j k} ≅ {\tilde{a}}_{i k} \forall i, j, k . \end{matrix}$

Taking logarithms on both sides yields $\begin{matrix} \log_{9} {\tilde{a}}_{i j} \oplus \log_{9} {\tilde{a}}_{j k} = \log_{9} {\tilde{a}}_{i k} \forall i, j, k, \end{matrix}$ $\begin{matrix} lo g_{9} {\tilde{a}}_{i j} \oplus lo g_{9} {\tilde{a}}_{j k} Θ lo g_{9} {\tilde{a}}_{i k} = \tilde{0}, \end{matrix}$ $\begin{matrix} lo g_{9} {\tilde{a}}_{i j} \oplus lo g_{9} {\tilde{a}}_{j k} \oplus lo g_{9} {\tilde{a}}_{k i} = \tilde{0} . \end{matrix}$

Adding 3 and dividing by 2 on both sides $\begin{matrix} \frac{1}{2} (1 \oplus \log_{9} {\tilde{a}}_{i j}) \oplus \frac{1}{2} (1 \oplus \log_{9} {\tilde{a}}_{j k}) \oplus \frac{1}{2} (1 \oplus \log_{9} {\tilde{a}}_{k i}) = \frac{\tilde{3}}{\tilde{2}} \forall i, j, k . \end{matrix}$

Assuming $\begin{matrix} {\tilde{p}}_{i j} = \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{i j}), {\tilde{p}}_{j k} = \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{j k}), {\tilde{p}}_{k i} = \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{k i}) \end{matrix}$ $\begin{matrix} \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{i j}) \oplus \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{j k}) \oplus \frac{1}{2} (1 \oplus lo g_{9} {\tilde{a}}_{k i}) = \frac{\tilde{3}}{\tilde{2}} \forall i, j, k . \\ \Rightarrow {\tilde{p}}_{i j} \oplus {\tilde{p}}_{j k} \oplus {\tilde{p}}_{k i} = \frac{\tilde{3}}{\tilde{2}} = (\frac{3}{2}, \frac{3}{2}, \frac{3}{2}) \end{matrix}$ $\begin{matrix} \Rightarrow (p_{i j}^{L}, p_{i j}^{M}, p_{i j}^{R}) \oplus (p_{j k}^{L}, p_{j k}^{M}, p_{j k}^{R}) \oplus {\tilde{p}}_{k i} = (\frac{3}{2}, \frac{3}{2}, \frac{3}{2}) \end{matrix}$ $\begin{matrix} \Rightarrow {\tilde{p}}_{k i} = (\frac{3}{2}, \frac{3}{2}, \frac{3}{2}) Θ (p_{i j}^{L} + p_{j k}^{L}, p_{i j}^{M} + p_{j k}^{M}, p_{i j}^{R} + p_{j k}^{R}), \end{matrix}$ $\begin{matrix} \Rightarrow {\tilde{p}}_{k i} = (p_{k i}^{L}, p_{k i}^{M}, p_{k i}^{R}) = (\frac{3}{2} - p_{i j}^{R} - p_{j k}^{R}, \frac{3}{2} - p_{i j}^{M} - p_{j k}^{M}, \frac{3}{2} - p_{i j}^{L} - p_{j k}^{L}), \end{matrix}$ $\begin{matrix} ∴ p_{i j}^{L} + p_{j k}^{L} + p_{k i}^{R} = \frac{3}{2}, p_{i j}^{M} + p_{j k}^{M} + p_{k i}^{M} = \frac{3}{2}, p_{i j}^{R} + p_{j k}^{R} + p_{k i}^{L} = \frac{3}{2} . \end{matrix}$

Thus the expressions (a)–(c) are obtained.

Section snippets

Flaws in existing results

In this section, the flaws in the existing results [[1], Proposition 5.1, Proposition 5.2, pp. 3759] are pointed out.

1.
Wang and Chen [1] have used the following mathematical incorrect assumptions for proving the result stated in Proposition 5.1 [1, pp. 3759].
- (i)
  It is obvious from Eq. 4 [1, pp. 3757] that if ${\tilde{a}}_{i j} = (a_{i j}^{L}, a_{i j}^{M}, a_{i j}^{R}) and {\tilde{a}}_{j i} = \frac{1}{{\tilde{a}}_{i j}} = (\frac{1}{a_{i j}^{R}}, \frac{1}{a_{i j}^{M}}, \frac{1}{a_{i j}^{L}})$ are two positive triangular fuzzy numbers then $\begin{matrix} {\tilde{a}}_{i j} \otimes {\tilde{a}}_{j i} = (a_{i j}^{L}, a_{i j}^{M}, a_{i j}^{R}) \otimes (\frac{1}{a_{i j}^{R}}, \frac{1}{a_{i j}^{M}}, \frac{1}{a_{i j}^{L}}) \approx (\frac{a_{i j}^{L}}{a_{i j}^{R}}, 1, \frac{a_{i j}^{R}}{a_{i j}^{L}}) \neq (1, 1, 1) \end{matrix}$
However, in the

Exact results with valid proof

In this section, the exact results are stated and proved

Proposition 5.3

Given that a set of alternatives, $X = {x_{1}, \dots, x_{n}}$ associated with a fuzzy reciprocal multiplicative preference matrix $\tilde{A} = ({\tilde{a}}_{i j})$ with ${\tilde{a}}_{i j} \in [1 / 9, 9]$ , and the corresponding fuzzy reciprocal linguistic preference relation, $\tilde{P} = ({\tilde{p}}_{i j})$ with ${\tilde{p}}_{i j} \in [0, 1]$ , verifies the additive reciprocal, then, the following statements are equivalent.

(a)
$p_{j i}^{L} + p_{i j}^{L} = 1, \forall \forall i, j \in {1, \dots, n}$
(b)
$p_{j i}^{M} + p_{j i}^{M} = 1, \forall i, j \in {1, \dots, n}$
(c)
$p_{j i}^{R} + p_{i j}^{R} = 1, \forall i, j \in {1, \dots, n} .$

Proof

Since, $\tilde{A} = ({\tilde{a}}_{i j})$ is a reciprocal matrix, $\begin{matrix} So {\tilde{a}}_{j i} = \frac{1}{a} \end{matrix}$

Conclusions

In this paper, it is shown that the existing results [[1], Proposition 5.1, Proposition 5.2, pp. 3759] are not valid. Also, the exact results are stated and proved.

Acknowledgment

I, Dr. Amit Kumar, would like to acknowledge the adolescent blessings of Mehar (lovely daughter of my cousin sister Dr. Parmpreet Kaur). I believe that Mata Vaishno Devi has appeared on the earth in the form of Mehar and without Mehar's blessings it was not possible to think the ideas presented in this manuscript. The first author would like to acknowledge the financial support given to her by Department of Science and Technology under Department of Electronics and Information Technology

Reference (1)

T.C. Wang et al.
Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP
Inf. Sci.
(2008)

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A note on ‘‘Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP”

Abstract

Introduction

Section snippets

Flaws in existing results

Exact results with valid proof

Conclusions

Acknowledgment

Inf. Sci.