Erratum to: A new measure of divergence with its application to multi-criteria decision making under fuzzy environment

1 Erratum to: Neural Comput & Applic DOI 10.1007/s00521-016-2311-y

The original article has been published online with some errors. The errors are corrected with this erratum.

• 1. In Eq. (12) and in all other places, where the factor $\frac{1}{{\sqrt {e - 1} }}$ occurs, it is in error. The correct factor is $\frac{1}{\sqrt e - 1}$.

• 2. Just before Eq. (18), there is in error. It has ‘−’ sign. The correction is for the sign to be ‘+’.

• 3. Equation (20) should be:

  $$\begin{aligned} & = \frac{1}{{n\left( {\sqrt e - 1} \right)}}\left[ {\sum\limits_{{x_{j} \in X_{1} }} {\left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left( {\frac{{\mu_{A} (x_{j} ) + \mu_{B} (x_{j} )}}{2}} \right)\exp \left( {\frac{{2 - \mu_{A} (x_{j} ) - \mu_{B} (x_{j} )}}{2}} \right)} \hfill \\ {\quad + \left( {\frac{{2 - \mu_{A} (x_{j} ) - \mu_{B} (x_{j} )}}{2}} \right)\exp \left( {\frac{{\mu_{A} (x_{j} ) + \mu_{B} (x_{j} )}}{2}} \right)} \hfill \\ \end{array} } \right\}} \hfill \\ {\quad - \left\{ {\frac{{\begin{array}{*{20}l} {\left( {\mu_{A} (x_{j} )\exp \left( {1 - \mu_{A} (x_{j} )} \right) + \left( {1 - \mu_{A} (x_{j} )} \right)\exp \left( {\mu_{A} (x_{j} )} \right)} \right)} \hfill \\ {\quad { + }\left( {\mu_{B} (x_{j} )\exp \left( {1 - \mu_{B} (x_{j} )} \right) + \left( {1 - \mu_{B} (x_{j} )} \right)\exp \left( {\mu_{B} (x_{j} )} \right)} \right)} \hfill \\ \end{array} }}{2}} \right\}} \hfill \\ \end{array} } \right\}} } \right. \\ & \quad + \left. {\sum\limits_{{x_{j} \in X_{2} }} {\left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left( {\frac{{\mu_{A} (x_{j} ) + \mu_{A} (x_{j} )}}{2}} \right)\exp \left( {\frac{{2 - \mu_{A} (x_{j} ) - \mu_{A} (x_{j} )}}{2}} \right)} \hfill \\ {\quad + \left( {\frac{{2 - \mu_{A} (x_{j} ) - \mu_{A} (x_{j} )}}{2}} \right)\exp \left( {\frac{{\mu_{A} (x_{j} ) + \mu_{A} (x_{j} )}}{2}} \right)} \hfill \\ \end{array} } \right\}} \hfill \\ { - \left\{ {\frac{{\begin{array}{*{20}l} {\left( {\mu_{A} (x_{j} )\exp \left( {1 - \mu_{A} (x_{j} )} \right) + \left( {1 - \mu_{A} (x_{j} )} \right)\exp \left( {\mu_{A} (x_{j} )} \right)} \right)} \hfill \\ {\quad { + }\left( {\mu_{A} (x_{j} )\exp \left( {1 - \mu_{A} (x_{j} )} \right) + \left( {1 - \mu_{A} (x_{j} )} \right)\exp \left( {\mu_{A} (x_{j} )} \right)} \right)} \hfill \\ \end{array} }}{2}} \right\}} \hfill \\ \end{array} } \right\}} } \right]. \\ \end{aligned}$$

• 4. The expression following Eq. (21) is in error, the correct form is as follows:

  $$\begin{aligned} & = \frac{1}{{n\left( {\sqrt e - 1} \right)}}\sum\limits_{j = 1}^{n} {\left[ {\left[ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left( {\frac{{\mu_{A} (x_{j} ) + 1 - \mu_{B} (x_{j} )}}{2}} \right)\exp \left( {\frac{{1 - \mu_{A} (x_{j} ) + \mu_{B} (x_{j} )}}{2}} \right)} \hfill \\ {\quad + \left( {\frac{{1 - \mu_{A} (x_{j} ) + \mu_{B} (x_{j} )}}{2}} \right)\exp \left( {\frac{{\mu_{A} (x_{j} ) + 1 - \mu_{B} (x_{j} )}}{2}} \right)} \hfill \\ \end{array} } \right\}} \hfill \\ {\quad - \left\{ {\frac{{\begin{array}{*{20}l} {\left( {\mu_{A} (x_{j} )\exp \left( {1 - \mu_{A} (x_{j} )} \right) + \left( {1 - \mu_{A} (x_{j} )} \right)\exp \left( {\mu_{A} (x_{j} )} \right)} \right)} \hfill \\ {\quad { + }\left( {\left( {\mu_{B} (x_{j} )} \right)\exp \left( {1 - \mu_{B} (x_{j} )} \right) + \left( {1 - \mu_{B} (x_{j} )} \right)\exp \left( {\mu_{B} (x_{j} )} \right)} \right)} \hfill \\ \end{array} }}{2}} \right\}} \hfill \\ \end{array} } \right]} \right.} \\ & \quad - \left. {\left[ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left( {\frac{{1 - \mu_{A} (x_{j} ) + \mu_{B} (x_{j} )}}{2}} \right)\exp \left( {\frac{{1 + \mu_{A} (x_{j} ) - \mu_{B} (x_{j} )}}{2}} \right)} \hfill \\ {\quad + \left( {\frac{{1 + \mu_{A} (x_{j} ) - \mu_{B} (x_{j} )}}{2}} \right)\exp \left( {\frac{{1 - \mu_{A} (x_{j} ) + \mu_{B} (x_{j} )}}{2}} \right)} \hfill \\ \end{array} } \right\}} \hfill \\ {\quad - \left\{ {\frac{{\begin{array}{*{20}l} {\left( {\left( {1 - \mu_{A} (x_{j} )} \right)\exp \left( {\mu_{A} (x_{j} )} \right) + \mu_{A} (x_{j} )\exp \left( {1 - \mu_{A} (x_{j} )} \right)} \right)} \hfill \\ {\quad { + }\left( {\left( {1 - \mu_{B} (x_{j} )} \right)\exp \left( {\mu_{B} (x_{j} )} \right) + \left( {\mu_{B} (x_{j} )} \right)\exp \left( {1 - \mu_{B} (x_{j} )} \right)} \right)} \hfill \\ \end{array} }}{2}} \right\}} \hfill \\ \end{array} } \right]} \right]. \\ \end{aligned}$$

• 5. The expression following Eq. (22) is in error, the correct form is as follows:

  $$\begin{aligned} & = \frac{1}{{n\left( {\sqrt e - 1} \right)}}\left[ {\sum\limits_{{x_{j} \in X_{1} }} {\left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left( {\frac{{\mu_{B} (x_{j} ) + \mu_{C} (x_{j} )}}{2}} \right)\exp \left( {\frac{{2 - \mu_{B} (x_{j} ) - \mu_{C} (x_{j} )}}{2}} \right)} \hfill \\ {\quad + \left( {\frac{{2 - \mu_{B} (x_{j} ) - \mu_{C} (x_{j} )}}{2}} \right)\exp \left( {\frac{{\mu_{B} (x_{j} ) + \mu_{C} (x_{j} )}}{2}} \right)} \hfill \\ \end{array} } \right\}} \hfill \\ {\quad - \left\{ {\frac{{\begin{array}{*{20}l} {\left( {\mu_{B} (x_{j} )\exp \left( {1 - \mu_{B} (x_{j} )} \right) + \left( {1 - \mu_{B} (x_{j} )} \right)\exp \left( {\mu_{B} (x_{j} )} \right)} \right)} \hfill \\ {\quad { + }\left( {\mu_{C} (x_{j} )\exp \left( {1 - \mu_{C} (x_{j} )} \right) + \left( {1 - \mu_{C} (x_{j} )} \right)\exp \left( {\mu_{C} (x_{j} )} \right)} \right)} \hfill \\ \end{array} }}{2}} \right\}} \hfill \\ \end{array} } \right\}} } \right. \\ & \quad + \left. {\sum\limits_{{x_{j} \in X_{2} }} {\left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left( {\frac{{\mu_{A} (x_{j} ) + \mu_{C} (x_{j} )}}{2}} \right)\exp \left( {\frac{{2 - \mu_{A} (x_{j} ) - \mu_{C} (x_{j} )}}{2}} \right)} \hfill \\ {\quad + \left( {\frac{{2 - \mu_{A} (x_{j} ) - \mu_{C} (x_{j} )}}{2}} \right)\exp \left( {\frac{{\mu_{A} (x_{j} ) + \mu_{C} (x_{j} )}}{2}} \right)} \hfill \\ \end{array} } \right\}} \hfill \\ {\quad - \left\{ {\frac{{\begin{array}{*{20}l} {\left( {\mu_{A} (x_{j} )\exp \left( {1 - \mu_{A} (x_{j} )} \right) + \left( {1 - \mu_{A} (x_{j} )} \right)\exp \left( {\mu_{A} (x_{j} )} \right)} \right)} \hfill \\ {\quad { + }\left( {\mu_{C} (x_{j} )\exp \left( {1 - \mu_{C} (x_{j} )} \right) + \left( {1 - \mu_{C} (x_{j} )} \right)\exp \left( {\mu_{C} (x_{j} )} \right)} \right)} \hfill \\ \end{array} }}{2}} \right\}} \hfill \\ \end{array} } \right\}} } \right]. \\ \end{aligned}$$

• 6. Equation (24) is in error, the correct form is as follows:

  $$\begin{aligned} & = \frac{1}{{n\left( {\sqrt e - 1} \right)}}\left[ {\sum\limits_{{x_{j} \in X_{1} }} {\left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left( {\frac{{\mu_{B} (x_{j} ) + \mu_{C} (x_{j} )}}{2}} \right)\exp \left( {\frac{{2 - \mu_{B} (x_{j} ) - \mu_{C} (x_{j} )}}{2}} \right)} \hfill \\ {\quad + \left( {\frac{{2 - \mu_{B} (x_{j} ) - \mu_{C} (x_{j} )}}{2}} \right)\exp \left( {\frac{{\mu_{B} (x_{j} ) + \mu_{C} (x_{j} )}}{2}} \right)} \hfill \\ \end{array} } \right\}} \hfill \\ {\quad - \left\{ {\frac{{\begin{array}{*{20}l} {\left( {\mu_{B} (x_{j} )\exp \left( {1 - \mu_{B} (x_{j} )} \right) + \left( {1 - \mu_{B} (x_{j} )} \right)\exp \left( {\mu_{B} (x_{j} )} \right)} \right)} \hfill \\ {\quad { + }\left( {\mu_{C} (x_{j} )\exp \left( {1 - \mu_{C} (x_{j} )} \right) + \left( {1 - \mu_{C} (x_{j} )} \right)\exp \left( {\mu_{C} (x_{j} )} \right)} \right)} \hfill \\ \end{array} }}{2}} \right\}} \hfill \\ \end{array} } \right\}} } \right. \\ & \quad + \left. {\sum\limits_{{x_{j} \in X_{2} }} {\left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left( {\frac{{\mu_{A} (x_{j} ) + \mu_{C} (x_{j} )}}{2}} \right)\exp \left( {\frac{{2 - \mu_{A} (x_{j} ) - \mu_{C} (x_{j} )}}{2}} \right)} \hfill \\ {\quad + \left( {\frac{{2 - \mu_{A} (x_{j} ) - \mu_{C} (x_{j} )}}{2}} \right)\exp \left( {\frac{{\mu_{A} (x_{j} ) + \mu_{C} (x_{j} )}}{2}} \right)} \hfill \\ \end{array} } \right\}} \hfill \\ {\quad - \left\{ {\frac{{\begin{array}{*{20}l} {\left( {\mu_{A} (x_{j} )\exp \left( {1 - \mu_{A} (x_{j} )} \right) + \left( {1 - \mu_{A} (x_{j} )} \right)\exp \left( {\mu_{A} (x_{j} )} \right)} \right)} \hfill \\ {\quad { + }\left( {\mu_{C} (x_{j} )\exp \left( {1 - \mu_{C} (x_{j} )} \right) + \left( {1 - \mu_{C} (x_{j} )} \right)\exp \left( {\mu_{C} (x_{j} )} \right)} \right)} \hfill \\ \end{array} }}{2}} \right\}} \hfill \\ \end{array} } \right\}} } \right]. \\ \end{aligned}$$

• 7. In Eqs. (28) and (29), ‘$\frac{1}{n}$’ should be replaced by ‘$\frac{1}{{n\left( {\sqrt e - 1} \right)}}$’.

• 8. The correct forms of Tables 1, 2 and 3 are as follows:

  Table 1 Values of ${\text{FJED}}(M^{ + } |M_{i} )$

  Table 2 Values of ${\text{FJED}}(M^{ - } |M_{i} )$

  Table 3 Values of ${\text{FJED}}\left( {M_{i} } \right)$

• 9. In Step 1 of numerical example, first expression for $M^{ + }$ is correct, and the error is the next line. It gives in fact $M^{ - }$. Thus, Step 1 in corrected form is to be:

  • Step 1 Obtaining the $M^{ + }$ and $M^{ - }$ given by

    $$\begin{aligned} & M^{ + } = \left\{ {\left\langle {C_{1} ,\,0.8} \right\rangle ,\left\langle {C_{2} ,\,0.7} \right\rangle ,\left\langle {C_{3} ,\,0.7} \right\rangle ,\left\langle {C_{4} ,\,0.6} \right\rangle } \right\}, \\ & M^{ - } = \left\{ {\left\langle {C_{1} ,\,0.5} \right\rangle ,\left\langle {C_{2} ,\,0.4} \right\rangle ,\left\langle {C_{3} ,\,0.3} \right\rangle ,\left\langle {C_{4} ,\,0.2} \right\rangle } \right\}. \\ \end{aligned}$$