A note on “Ranking generalized exponential trapezoidal fuzzy numbers based on variance”

Abstract

It is well known that the ranking of generalized fuzzy numbers depend upon the height of fuzzy numbers. In this note, it is shown that the method, proposed by Rezvani [Applied Mathematics and Computation 262 (2015) 191-198] for ranking of generalized exponential trapezoidal fuzzy numbers, is independent from height of fuzzy numbers. Hence, it is not genuine to use this method for ranking of generalized exponential trapezoidal fuzzy numbers.

Keywords

Exponential trapezoidal fuzzy numbers ranking method variance

1 Introduction

Rezvani [1, Theorem 1, pp. 192] proved that the probability density function f_A (x), corresponding to exponential trapezoidal fuzzy number A = (a, b, c, d) _E, is given as: $f_{A} (x) = {\begin{matrix} \frac{w e^{[\frac{(x - a)}{(b - a)}]}}{a (1 - e) - b + c + (e - 1) d}; & a \leq x \leq b, \\ \frac{w e}{a (1 - e) - b + c + (e - 1) d}; & b \leq x \leq c, \\ \frac{w e^{[\frac{(d - x)}{(d - c)}]}}{a (1 - e) - b + c + (e - 1) d}; & c \leq x \leq d . \end{matrix}$ (1)

Also, he used this f_A (x)to prove the remaining results [1; Theorem 2, pp. 193; Theorem 3, pp. 193] of the published paper [1].

Rezvani [1, Theorem 1, pp. 192] has used the following procedure to prove that f_A (x)is the probability density function corresponding to exponential trapezoidal fuzzy number A = (a, b, c, d) _E.

Step 1: Let f_A (x) = Cμ_A (x), where C is a non-negative real number and

$μ_{A} (x) = {\begin{matrix} {we}^{- [\frac{(b - x)}{(b - a)}]}; & a \leq x \leq b, \\ w; & b \leq x \leq c, \\ {we}^{- [\frac{(x - c)}{(d - c)}]}; & c \leq x \leq d . \end{matrix}$ is the membership function of exponential trapezoidal fuzzy number A = (a, b, c, d) _E.

Step 2: Since, f_A (x)is assumed as the probability density function. Therefore, $\int_{- \infty}^{\infty} f_{A} (x) dx$ should be 1. Also, $\begin{matrix} \int_{- \infty}^{\infty} f_{A} (x) dx = 1 \Rightarrow \int_{- \infty}^{\infty} C μ_{A} (x) dx = 1 \\ \Rightarrow \int_{a}^{b} {Ce}^{- [\frac{(b - x)}{(b - a)}]} dx + \int_{b}^{c} Cdx + \int_{c}^{d} {Ce}^{- [\frac{(x - c)}{(d - c)}]} dx = 1 \\ \Rightarrow C = \frac{e}{a (1 - e) - b + c + (e - 1) d} \end{matrix}$ (2)

Step 3: Putting the value of C, obtained from Step 2, in f_A (x) = Cμ_A (x), the probability density function f_A (x), defined in (1), is obtained.

2 Mathematical error in existing results

It is obvious from Step 2 that Rezvani [1] has used the wrong expression

$μ_{A} (x) = {\begin{matrix} e^{- [\frac{(b - x)}{(b - a)}]}; & a \leq x \leq b, \\ 1; & b \leq x \leq c, \\ e^{- [\frac{(x - c)}{(d - c)}]}; & c \leq x \leq d . \end{matrix}$ instead of the exact expression $μ_{A} (x) = {\begin{matrix} {we}^{- [\frac{(b - x)}{(b - a)}]}; & a \leq x \leq b, \\ w; & b \leq x \leq c, \\ {we}^{- [\frac{(x - c)}{(d - c)}]}; & c \leq x \leq d . \end{matrix}$ for calculating the value of C and hence to obtain the probability density function f_A (x) defined in (1), to prove the existing results [1; Theorem 2, pp. 193; Theorem 3, pp. 193] and to solve the numerical examples [Example 1, pp. 194; Example 2, pp. 195; Example 3, pp. 195; Example 4, pp. 195; Example 5, pp. 196; Example 6, pp. 196; Example 7, pp. 197].

3 Exact form of existing results

It can be easily verified that on using the exact expression μ_A (x) = ${\begin{matrix} {we}^{- [\frac{(b - x)}{(b - a)}]}; & a \leq x \leq b, \\ w; & b \leq x \leq c, \\ {we}^{- [\frac{(x - c)}{(d - c)}]}; & c \leq x \leq d . \end{matrix}$ instead of using the wrong expression $μ_{A} (x) = {\begin{matrix} e^{- [\frac{(b - x)}{(b - a)}]} & ; a \leq x \leq b, \\ 1 & ; b \leq x \leq c, \\ e^{- [\frac{(x - c)}{(d - c)}]} & ; c \leq x \leq d . \end{matrix}$ for calculating the value of C, the obtained exact value of C is $\frac{e}{w [a (1 - e) - b + c + (e - 1) d]}$ . Also, putting this value of C in f_A (x) = Cμ_A (x), the obtained exact probability density function f_A (x) corresponding to exponential trapezoidal fuzzy number A = (a, b, c, d) _E is $f_{A} (x) = {\begin{matrix} \frac{e^{[\frac{(x - a)}{(b - a)}]}}{a (1 - e) - b + c + (e - 1) d} & ; a \leq x \leq b, \\ \frac{e}{a (1 - e) - b + c + (e - 1) d} & ; b \leq x \leq c, \\ \frac{e^{[\frac{(d - x)}{(d - c)}]}}{a (1 - e) - b + c + (e - 1) d} & ; c \leq x \leq d . \end{matrix}$

Hence, the error occurring in the existing results [1; Theorem 1, pp. 192, Theorem 2, pp. 193; Theorem 3,pp. 193] and in the solution of numerical examples [1; Example 1, pp. 194; Example 2, pp. 195; Example 3, pp. 195; Example 4, pp. 195; Example 5, pp. 196; Example 6, pp. 196; Example 7, pp. 197] can be resolved by using the exact probability density function f_A (x) i.e., $f_{A} (x) = {\begin{matrix} \frac{e^{[\frac{(x - a)}{(b - a)}]}}{a (1 - e) - b + c + (e - 1) d} & ; a \leq x \leq b, \\ \frac{e}{a (1 - e) - b + c + (e - 1) d} & ; b \leq x \leq c, \\ \frac{e^{[\frac{(d - x)}{(d - c)}]}}{a (1 - e) - b + c + (e - 1) d} & ; c \leq x \leq d . \end{matrix}$ instead of using the wrong probability density function f_A (x) i.e., $f_{A} (x) = {\begin{matrix} \frac{{we}^{[\frac{(x - a)}{(b - a)}]}}{a (1 - e) - b + c + (e - 1) d} & ; a \leq x \leq b, \\ \frac{we}{a (1 - e) - b + c + (e - 1) d} & ; b \leq x \leq c, \\ \frac{{we}^{[\frac{(d - x)}{(d - c)}]}}{a (1 - e) - b + c + (e - 1) d} & ; c \leq x \leq d . \end{matrix}$

Further, it can be concluded that the existing results [1; Theorem 1, pp. 192, Theorem 2, pp. 193; Theorem 3, pp. 193] and existing solution the numerical examples [1; Example 1, pp. 194; Example 2, pp. 195; Example 3, pp. 195; Example 4, pp. 195; Example 5, pp. 196; Example 6, pp.196; Example 7, pp.197] are not valid. However, if w, present in existing results [1; Theorem 2, pp. 193; Theorem 3, pp. 193], is replaced by 1 then the existing results [1; Theorem 1, pp. 192, Theorem 2, pp. 193; Theorem 3, pp. 193] will be valid. Also, the exact solution of numerical examples [1; Example 1, pp. 194; Example 2, pp. 195; Example 3, pp. 195; Example 4, pp. 195; Example 5, pp. 196; Example 6, pp. 196; Example 7, pp. 197] can be obtained by using the exact expressions $\begin{matrix} C & = & \frac{e}{w [a (1 - e) - b + c + (e - 1) d]}, \\ μ & = & \frac{1}{a (1 - e) - b + c + (e - 1) d} \\ [\frac{1.72 b^{2} - 1.72 c^{2} - a^{2} + d^{2}}{2}], \end{matrix}$ and $\begin{matrix} σ^{2} & = & \frac{1}{a (1 - e) - b + c + (e - 1) d} \\ [\frac{1.72 b^{3} - 1.72 c^{3} - a^{3} + d^{3}}{3}] \\ - \frac{1}{{[a (1 - e) - b + c + (e - 1) d]}^{2}} \\ [\frac{(1.72 b^{2} - 1.72 c^{2} - a^{2} + d^{2})^{2}}{4}] \end{matrix}$ instead of using the wrong expressions $C = \frac{e}{a (1 - e) - b + c + (e - 1) d}$ , $\begin{matrix} μ & = & \frac{w}{a (1 - e) - b + c + (e - 1) d} \\ [\frac{1.72 b^{2} - 1.72 c^{2} - a^{2} + d^{2}}{2}], \end{matrix}$ and $\begin{matrix} σ^{2} & = & \frac{w}{a (1 - e) - b + c + (e - 1) d} \\ [\frac{1.72 b^{3} - 1.72 c^{3} - a^{3} + d^{3}}{3}] \\ - \frac{w^{2}}{{[a (1 - e) - b + c + (e - 1) d]}^{2}} \\ [\frac{(1.72 b^{2} - 1.72 c^{2} - a^{2} + d^{2})^{2}}{4}] \end{matrix}$

respectively for calculating the constants C_A, C_B, C_C; means μ_A, μ_B, μ_C and variance $σ_{A}^{2}, σ_{B}^{2}, σ_{C}^{2}$ of exponential trapezoidal fuzzy numbers A, Band C respectively.

4 Flaws in existing methods

If A = (a₁, b₁, c₁, d₁ ; w₁) _E and B = (a₂, b₂, c₂, d₂ ; w₂) _E are two generalized exponential trapezoidal fuzzy numbers then according to Rezvani [1], A > B if $σ_{A}^{2} > σ_{B}^{2}$ , A = B if $σ_{A}^{2} = σ_{B}^{2}$ and A < B if $σ_{A}^{2} < σ_{B}^{2}$ . It is obvious from the exact expression

$\begin{matrix} σ^{2} & = & \frac{1}{a (1 - e) - b + c + (e - 1) d} \\ [\frac{1.72 b^{3} - 1.72 c^{3} - a^{3} + d^{3}}{3}] \\ - \frac{1}{{[a (1 - e) - b + c + (e - 1) d]}^{2}} \\ [\frac{(1.72 b^{2} - 1.72 c^{2} - a^{2} + d^{2})^{2}}{4}] \end{matrix}$ presented in Section 3, that σ² is independent from height of fuzzy numbers. Hence the ranking of generalized exponential trapezoidal fuzzy numbers will also be independent of height of fuzzy numbers which contradicts the property that ranking of generalized exponential trapezoidal fuzzy numbers should be dependent on heights of fuzzy numbers.

5 Conclusion

It is obvious from the Section 4 that ranking of generalized exponential trapezoidal fuzzy numbers, obtained by using the existing method [1], is independent from height of generalized exponential trapezoidal fuzzy numbers. While, the ranking of generalized exponential trapezoidal fuzzy numbers should be dependent on its height. Hence, it is not genuine to use the existing method [1] for comparing the generalized exponential trapezoidal fuzzy numbers.

Footnotes

Acknowledgments

Dr. Amit Kumar would like to acknowledge the adolescent inner blessings of Mehar (lovely daughter of his cousin sister Dr. Parmpreet Kaur). He believes that MATA VAISHNO DEVI has appeared on earth in the form of Mehar and without her blessings it was not possible to think the ideas presented in this paper.

References

Rezvani

, Ranking generalized exponential trapezoidal fuzzy numbers based on variance, Applied Mathematics and Computation, 262 (2015), 191–198.