An Improved ARAS Approach with T-Spherical Fuzzy Information and Its Application in Multi-attribute Group Decision-Making

Abstract

The additive ratio assessment system (ARAS) method is an effective technique for simplifying complex decision problems by determining the optimal alternative through the relative index (utility degree) to the ideal solution. However, there are still some shortcomings in the existing researches on the extension of this method when it is utilized in different decision environments, such as ignoring the correlation relationship between attributes, the lack of flexibility in the utilization of the decision process, and the relative index to the ideal solution may be scaled up or down with the ratio form. In order to overcome these disadvantages, this paper proposes the novel T-spherical fuzzy (TSF) cross entropy (TSFCE) measure and T-spherical Aczel-Alsina Heronian mean (TSFAAHM) aggregation operators and uses them to improve the ARAS method in the TSF environment. For the TSF multiple attribute group decision-making (MAGDM) problems, a group decision making model based on the improved ARAS is designed. In this model, the experts’ weights are obtained by the TSFCE-based similarity measure. The attribute combined weights are calculated by fusing the objective weights obtained by TSFCE-based entropy measure and the subjective weights got by the extended stepwise weight assessment ratio analysis (SWARA) integrated with TSFCE. In the improved ARAS method, the T-spherical Aczel-Alsina Weighted Heronian mean (TSFAAWHM) operator can capture the correlation relationship between the attributes. Compared with the relative index, the TSFCE can reflect the difference between the alternatives and the ideal solution to obtain a more stable solution ranking. Lastly, an illustrative example about the sustainable supplier selection of power battery echelon utilization (PBEU) for a 5G base station is used to demonstrate the proposed method. The effectiveness, practicability and superiority of proposed method are illustrated by parameters influence and methods comparison analysis.

Appendix

The following symmetric form of T-spherical fuzzy cross-entropy measures SCE_i(δ₁,δ₂) and seven TSF CE measures on TSFNs are all based on the Yang and Pang [17] definitions. Symmetric form of T-spherical fuzzy cross-entropy measures SCE_i(δ₁,δ₂) (i = 1,2,…,8) can be defined as

$$SCE_{i} (\delta_{1} ,\delta_{2} ) = CE_{i} (\delta_{1} ,\delta_{2} ) + CE_{i} (\delta_{2} ,\delta_{1} )$$

$$CE_{1} (\delta_{1} ,\delta_{2} ) = \tau_{1} \log_{2} \frac{{2\tau_{1} }}{{\tau_{1} + \tau_{2} }} + \eta_{1} \log_{2} \frac{{2\eta_{1} }}{{\eta_{1} + \eta_{2} }} + \vartheta_{1} \log_{2} \frac{{2\vartheta_{1} }}{{\vartheta_{1} + \vartheta_{2} }}$$

$$\begin{gathered} CE_{2} (\delta_{1} ,\delta_{2} ) = \tau_{1} \log_{2} \frac{{2\tau_{1} }}{{\tau_{1} + \tau_{2} }} + \left( {1 - \tau_{1} } \right)\log_{2} \frac{{1 - \tau_{1} }}{{1 - 0.5(\tau_{1} + \tau_{2} )}} \hfill \\ \, + \eta_{1} \log_{2} \frac{{2\eta_{1} }}{{\eta_{1} + \eta_{2} }} + \left( {1 - \eta_{1} } \right)\log_{2} \frac{{1 - \eta_{1} }}{{1 - 0.5(\eta_{1} + \eta_{2} )}} \hfill \\ \, + \vartheta_{1} \log_{2} \frac{{2\vartheta_{1} }}{{\vartheta_{1} + \vartheta_{2} }} + \left( {1 - \vartheta_{1} } \right)\log_{2} \frac{{1 - \vartheta_{1} }}{{1 - 0.5(\vartheta_{1} + \vartheta_{2} )}} \hfill \\ \end{gathered}$$

$$CE_{3} (\delta_{1} ,\delta_{2} ) = \left( {\left( {\tau_{1} \log_{2} \frac{{2\tau_{1} }}{{\tau_{1} + \tau_{2} }}} \right)^{\lambda } + \left( {\eta_{1} \log_{2} \frac{{2\eta_{1} }}{{\eta_{1} + \eta_{2} }}} \right)^{\lambda } + \left( {\vartheta_{1} \log_{2} \frac{{2\vartheta_{1} }}{{\vartheta_{1} + \vartheta_{2} }}} \right)^{\lambda } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}}$$

$$CE_{4} (\delta_{1} ,\delta_{2} ) = \left( \begin{gathered} \left( {\tau_{1} \log_{2} \frac{{2\tau_{1} }}{{\tau_{1} + \tau_{2} }}} \right)^{\lambda } + \left( {\left( {1 - \tau_{1} } \right)\log_{2} \frac{{1 - \tau_{1} }}{{1 - 0.5(\tau_{1} + \tau_{2} )}}} \right)^{\lambda } \hfill \\ + \left( {\eta_{1} \log_{2} \frac{{2\eta_{1} }}{{\eta_{1} + \eta_{2} }}} \right)^{\lambda } + \left( {\left( {1 - \eta_{1} } \right)\log_{2} \frac{{1 - \eta_{1} }}{{1 - 0.5(\eta_{1} + \eta_{2} )}}} \right)^{\lambda } \hfill \\ + \left( {\vartheta_{1} \log_{2} \frac{{2\vartheta_{1} }}{{\vartheta_{1} + \vartheta_{2} }}} \right)^{\lambda } + \left( {\left( {1 - \vartheta_{1} } \right)\log_{2} \frac{{1 - \vartheta_{1} }}{{1 - 0.5(\vartheta_{1} + \vartheta_{2} )}}} \right)^{\lambda } \hfill \\ \end{gathered} \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}}$$

$$\begin{gathered} CE_{5} (\delta_{1} ,\delta_{2} ) = \frac{{1 + \tau_{1} - \eta_{1} - \vartheta_{1} }}{2}\log_{2} \left( {\frac{{2(1 + \tau_{1} - \eta_{1} - \vartheta_{1} )}}{{2 + \tau_{1} - \eta_{1} - \vartheta_{1} + \tau_{2} - \eta_{2} - \vartheta_{2} }}} \right) \hfill \\ \, + \frac{{1 - \tau_{1} + \eta_{1} + \vartheta_{1} }}{2}\log_{2} \left( {\frac{{2(1 - \tau_{1} + \eta_{1} + \vartheta_{1} )}}{{2 - \tau_{1} + \eta_{1} + \vartheta_{1} - \tau_{2} + \eta_{2} + \vartheta_{2} }}} \right) \hfill \\ \end{gathered}$$

$$CE_{6} (\delta_{1} ,\delta_{2} ) = \frac{{\tau_{1}^{q} + \tau_{2}^{q} }}{2} - \left( {\frac{{\tau_{1} + \tau_{2} }}{2}} \right)^{q} + \frac{{\eta_{1}^{q} + \eta_{2}^{q} }}{2} - \left( {\frac{{\eta_{1} + \eta_{2} }}{2}} \right)^{q} + \frac{{\vartheta_{1}^{q} + \vartheta_{2}^{q} }}{2} - \left( {\frac{{\vartheta_{1} + \vartheta_{2} }}{2}} \right)^{q}$$

$$\begin{gathered} CE_{7} (\delta_{1} ,\delta_{2} ) = \left( {\frac{1}{{1 - 2^{1 - q} }}\left( {\frac{{\tau_{1}^{q} + \tau_{2}^{q} }}{2} - \left( {\frac{{\tau_{1} + \tau_{2} }}{2}} \right)^{q} } \right)^{\lambda } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} + \left( {\frac{1}{{1 - 2^{1 - q} }}\left( {\frac{{\eta_{1}^{q} + \eta_{2}^{q} }}{2} - \left( {\frac{{\eta_{1} + \eta_{2} }}{2}} \right)^{q} } \right)^{\lambda } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \hfill \\ \, + \left( {\frac{1}{{1 - 2^{1 - q} }}\left( {\frac{{\vartheta_{1}^{q} + \vartheta_{2}^{q} }}{2} - \left( {\frac{{\vartheta_{1} + \vartheta_{2} }}{2}} \right)^{q} } \right)^{\lambda } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \hfill \\ \end{gathered}$$