Green supplier selection of electric vehicle charging based on Choquet integral and type-2 fuzzy uncertainty

Abstract

In this paper, a framework under interval type-2 fuzzy (IT2F) environment is proposed to select the optimal green supplier of electric vehicle charging facility (EVCF). In the primary stage, a decision committee consisting of senior executives and experts is established, and qualified suppliers are also selected. The second stage aims to solve the problem of inherent uncertainties and criteria interactions. So, firstly, IT2F numbers are adopted for the performance evaluation since the criteria value cannot be adequately represented by type-1 fuzzy numbers. Then, $\lambda$-fuzzy measure is adopted to measure the fuzzy densities of criteria by considering the interactions. After that, these fuzzy densities and aggregated IT2F matrix are as inputs to a proposed IT2F Choquet integral (IT2FCI) operator to evaluate the suppliers. Finally, to illustrate the validity of the proposed framework, a case study with a sensitivity analysis is presented. The weighting results indicate that the criterion of “production cost” owns the largest fuzzy density of 0.65, and the sorting results show that none of these alternatives are optimal in all criteria and the results are relatively stable for a change in fuzzy density.

Appendices

Appendix A: Basic operation of interval type-2 fuzzy numbers

$$ \begin{aligned} \tilde{\tilde{A}}_{1} \oplus \tilde{\tilde{A}}_{2} = (\tilde{A}_{1}^{U} ,\tilde{A}_{1}^{L} ) \oplus (\tilde{A}_{2}^{U} ,\tilde{A}_{2}^{L} ) = \hfill \\ \left( {\begin{array}{*{20}c} {a_{11}^{U} + a_{21}^{U} ,a_{12}^{U} + a_{22}^{U} ,a_{13}^{U} + a_{23}^{U} ,a_{14}^{U} + a_{24}^{U} ;\hbox{min} \left( {H_{1} (\tilde{A}_{1}^{U} ),H_{1} (\tilde{A}_{2}^{U} )} \right),\hbox{min} \left( {H_{2} (\tilde{A}_{1}^{U} ),H_{2} (\tilde{A}_{2}^{U} )} \right),} \\ {a_{11}^{L} + a_{21}^{L} ,a_{12}^{L} + a_{22}^{L} ,a_{13}^{L} + a_{23}^{L} ,a_{14}^{L} + a_{24}^{L} ;\hbox{min} \left( {H_{1} (\tilde{A}_{1}^{L} ),H_{1} (\tilde{A}_{2}^{L} )} \right),\hbox{min} \left( {H_{2} (\tilde{A}_{1}^{L} ),H_{2} (\tilde{A}_{2}^{L} )} \right)} \\ \end{array} } \right) \hfill \\ \end{aligned} $$

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$$ \begin{aligned} \tilde{\tilde{A}}_{1} \otimes \tilde{\tilde{A}}_{2} = (\tilde{A}_{1}^{U} ,\tilde{A}_{1}^{L} ) \otimes (\tilde{A}_{2}^{U} ,\tilde{A}_{2}^{L} ) = \hfill \\ \left( {\begin{array}{*{20}c} {a_{11}^{U} \times a_{21}^{U} ,a_{12}^{U} \times a_{22}^{U} ,a_{13}^{U} \times a_{23}^{U} ,a_{14}^{U} \times a_{24}^{U} ;\hbox{min} \left( {H_{1} (\tilde{A}_{1}^{U} ),H_{1} (\tilde{A}_{2}^{U} )} \right),\hbox{min} \left( {H_{2} (\tilde{A}_{1}^{U} ),H_{2} (\tilde{A}_{2}^{U} )} \right),} \\ {a_{11}^{L} \times a_{21}^{L} ,a_{12}^{L} \times a_{22}^{L} ,a_{13}^{L} \times a_{23}^{L} ,a_{14}^{L} \times a_{24}^{L} ;\hbox{min} \left( {H_{1} (\tilde{A}_{1}^{L} ),H_{1} (\tilde{A}_{2}^{L} )} \right),\hbox{min} \left( {H_{2} (\tilde{A}_{1}^{L} ),H_{2} (\tilde{A}_{2}^{L} )} \right)} \\ \end{array} } \right) \hfill \\ \end{aligned} $$

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$$ k \otimes \tilde{\tilde{A}}_{1} = \left( {\begin{array}{*{20}c} {\left( {ka_{11}^{U} ,ka_{12}^{U} ,ka_{13}^{U} ,ka_{14}^{U} ;H_{1} (\tilde{A}_{1}^{U} ),H_{2} (\tilde{A}_{1}^{U} )} \right),} \\ {\left( {ka_{11}^{L} ,ka_{12}^{L} ,ka_{13}^{L} ,ka_{14}^{L} ;H_{1} (\tilde{A}_{1}^{L} ),H_{2} (\tilde{A}_{1}^{L} )} \right)} \\ \end{array} } \right) $$

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Appendix B: Meaning of different $\lambda$ values

By parameter $\lambda$, the interactions among criteria can be represented as follows:

If $\lambda = 0$, there is no interaction among $A$ and $B$.

If $\lambda > 0$, then $g(A \cup B) > g(A) + g(B)$, which implies that the set $\{ A,B\}$ has multiplicative interaction.

If $\lambda < 0$, then $g(A \cup B) < g(A) + g(B)$, which implies that the set $\{ A,B\}$ has substitutive interaction.