Table 8 Sequences and equations to calculate fuzzy positive ideal solution and fuzzy negative ideal solution

Step 1: Distance from fuzzy positive ideal solution and fuzzy negative ideal solution

\(d_{i}^{*} =\)\(\sum\nolimits_{j = 1}^{n} {d({\tilde{\text{v}}}_{ij} ,{\tilde{\text{v}}}^{*}_{i} )}\)

\(\begin{aligned} d_{i}^{ - } & = \sum\nolimits_{j = 1}^{n} {d({\tilde{\text{v}}}_{ij} ,{\tilde{\text{v}}}^{*}_{i} )} \\ & \quad i = 1,2, \ldots ,m \\ \end{aligned}\) (Zhang et al. 2021) (6)

Step 2: Calculating closeness coefficient of alternatives and ranking them

\(\begin{aligned} {\text{C}}C_{i} & = \frac{{d_{i}^{ - } }}{{d_{i}^{*} + d_{i}^{ - } }} \\ & \quad i = 1,2, \ldots ,m \\ \end{aligned}\) (Zhang et al. 2021) (7)

Step 3: Comaprison

Alternatives closer to fuzzy positive ideal solution and further from fuzzy negative ideal solutions are the best alternative and their closeness coefficient is nearer to 1