Ranking of interval type 2 fuzzy numbers using correlation coefficient and Mellin transform

Abstract

Ranking of fuzzy numbers exhibits a significant role to solve the multiple attributes group decision making (MAGDM) problems. One of the commonly used indices for MAGDM problems is the correlation coefficient. Type-2 fuzzy numbers (T2FNs) are considered to be more effective than type-1 fuzzy numbers (T1FNs) to handle uncertainties in MAGDM problems. Moreover, T2FNs allow additional freedom to reflect uncertainty. In this paper, we propose a new approach to rank the interval type-2 fuzzy numbers (IT2FNs) using the correlation coefficient, where the correlation coefficient is obtained using the Mellin transform. Initially, we define the correlation coefficient for interval type-2 trapezoidal fuzzy numbers (IT2TrFNs) using Mellin transform and prove some of its major features. Then a new group decision-making method has been recommended using the proposed ranking concept. Eventually, three examples are given to demonstrate the usefulness of the suggested method. Finally, a comparative study is conducted to depict the feasibility of the developed ranking technique.

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Appendix

Appendix

If \(B = - A\) then \(M_{{x_{B} }}^{L} (2) = - M_{{x_{A} }}^{L} (2)\) and \(M_{{x_{B} }}^{U} (2) = - M_{{x_{A} }}^{U} (2)\).

Proof

Consider \(A = \left\langle {A^{L} ,\,A^{U} } \right\rangle = \left\{ {\left( {a_{\,1}^{L} ,b_{\,1}^{L} ,c_{\,1}^{L} ,d_{\,1}^{L} ;h_{\,A}^{L} } \right),\,\,\left( {a_{\,1}^{U} ,b_{\,1}^{U} ,c_{\,1}^{U} ,d_{\,1}^{U} ;h_{\,A}^{U} } \right)} \right\}\) then \(- A = \left\{ {\left( { - d_{\,1}^{L} , - c_{\,1}^{L} , - b_{\,1}^{L} , - a_{\,1}^{L} ;h_{\,A}^{L} } \right),\,\,\,\left( { - d_{\,1}^{U} , - c_{\,1}^{U} , - b_{\,1}^{U} , - a_{\,1}^{U} ;h_{\,A}^{U} } \right)} \right\}\). Since Mellin transform \(M_{x} (s)\) is defined for all x > 0, therefore, we transform all the values of \(- A\) to make all positive. Now we shift the origin by \(\alpha\) therefore the new values \(\alpha - A = \left\{ {\left( {\alpha - d_{\,1}^{L} ,\alpha - c_{\,1}^{L} ,\alpha - b_{\,1}^{L} ,\alpha - a_{\,1}^{L} ;h_{\,A}^{L} } \right),\,\,\left( {\alpha - d_{\,1}^{U} ,\alpha - c_{\,1}^{U} ,\alpha - b_{\,1}^{U} ,\alpha - a_{\,1}^{U} ;h_{\,A}^{U} } \right)} \right\} = \left\{ {\left( {q_{\,1}^{L} ,q_{\,2}^{L} ,q_{\,3}^{L} ,q_{\,4}^{L} ;h_{\,A}^{L} } \right),\,\,\left( {l_{\,1}^{U} ,l_{\,2}^{U} ,l_{\,3}^{U} ,l_{\,4}^{U} ;h_{\,A}^{U} } \right)} \right\}\). Now the Mellin transform of \(\alpha - A\) will be \(M_{{x_{\alpha - A} }}^{L} (2) = \frac{1}{3}\left( {\sum\limits_{i = 1}^{4} {q_{\,i}^{L} } + \frac{{q_{\,1}^{L} q_{\,2}^{L} - q_{\,3}^{L} q_{\,4}^{L} }}{{q_{\,4}^{L} + q_{\,3}^{L} - q_{\,2}^{L} - q_{\,1}^{L} }}} \right)\) and \(M_{{x_{\alpha - A} }}^{U} (2) = \frac{1}{3}\left( {\sum\limits_{i = 1}^{4} {l_{\,i}^{U} } + \frac{{l_{\,1}^{U} l_{\,2}^{U} + l_{\,3}^{U} l_{\,4}^{U} }}{{l_{\,4}^{U} + l_{\,3}^{U} - l_{\,2}^{U} - l_{\,1}^{U} }}} \right)\).

Now putting the values of \(q_{\,i}^{L}\) we get

$$M_{{x_{\alpha - A} }}^{L} (2) = \frac{1}{3}\left( {\sum\limits_{i = 1}^{4} {q_{\,i}^{L} } + \frac{{q_{\,1}^{L} q_{\,2}^{L} - q_{\,3}^{L} q_{\,4}^{L} }}{{q_{\,4}^{L} + q_{\,3}^{L} - q_{\,2}^{L} - q_{\,1}^{L} }}} \right) = \frac{1}{3}\left( {4\alpha - (a_{\,1}^{L} + b_{\,1}^{L} + c_{\,1}^{L} + d_{\,1}^{L} ) + \frac{{ - \alpha (d_{\,1}^{L} + c^{L} - b_{\,1}^{L} - a_{\,1}^{L} ) + a_{\,1}^{L} b_{\,1}^{L} - c_{\,1}^{L} d_{\,1}^{L} }}{{d_{\,1}^{L} + c_{\,1}^{L} - b_{\,1}^{L} - a_{\,1}^{L} }}} \right) = \frac{1}{3}\left( {3\alpha - (a_{\,1}^{L} + b_{\,1}^{L} + c_{\,1}^{L} + d_{\,1}^{L} ) + \frac{{a_{\,1}^{L} b_{\,1}^{L} - c_{\,1}^{L} d_{\,1}^{L} }}{{d_{\,1}^{L} + c_{\,1}^{L} - b_{\,1}^{L} - a_{\,1}^{L} }}} \right) = \alpha - \frac{1}{3}\left( {(a_{\,1}^{L} + b_{\,1}^{L} + c_{\,1}^{L} + d_{\,1}^{L} ) + \frac{{a_{\,1}^{L} b_{\,1}^{L} - c_{\,1}^{L} d_{\,1}^{L} }}{{d_{\,1}^{L} + c_{\,1}^{L} - b_{\,1}^{L} - a_{\,1}^{L} }}} \right) = \alpha - M_{{x_{A} }}^{L} (2)$$

Therefore \(M_{{x_{\alpha - A} }}^{L} (2) = \alpha - M_{{x_{A} }}^{L} (2)\). Now to calculate the value of \(M_{{x_{\, - A} }}^{L} (2)\) we put \(\alpha = 0\)and we get \(M_{{x_{ - A} }}^{L} (2) = - M_{{x_{A} }}^{L} (2)\).

Similarly, we can show that \(M_{{x_{ - A} }}^{U} (2) = - M_{{x_{A} }}^{U} (2)\).

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De, A., Das, S. & Kar, S. Ranking of interval type 2 fuzzy numbers using correlation coefficient and Mellin transform. OPSEARCH (2021). https://doi.org/10.1007/s12597-020-00504-2

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Keywords

  • MAGDM
  • Generalized type-2 fuzzy number
  • Mellin transform
  • Correlation coefficient