Table 19 Ranking results of the proposed method and other decision-making methods.

IVPQ-WA operator (\(S\left( {Q_{i} } \right)\))

0.0201

0.2705

0.2801

0.1045

0.1411

\(o_{3} \succ o_{2} \succ o_{5} \succ o_{4} \succ o_{1}\)

IVQ-ROFWMSM operator (\(S\left( {Q_{i} } \right)\))

0.0132

0.2613

0.2541

0.0912

0.1194

\(o_{2} \succ o_{3} \succ o_{5} \succ o_{4} \succ o_{1}\)

IVQ-ROFWDMSM operator (\(S\left( {Q_{i} } \right)\))

0.0135

0.2675

0.2655

0.0978

0.1276

\(o_{2} \succ o_{3} \succ o_{5} \succ o_{4} \succ o_{1}\)

IVQ-ROF VIKOR (\(\Omega_{i}\))

0.5137

0.6607

0.2550

0.5118

0.5000

\(o_{3} \succ o_{5} \succ o_{4} \succ o_{1} \succ o_{2}\)

IVQ-FF TOPSIS (\(CD\left( {o_{i} } \right)\))

0.5600

0.6471

0.6647

0.6335

0.6240

\(o_{3} \succ o_{2} \succ o_{4} \succ o_{5} \succ o_{1}\)

The proposed method (\({\mathbb{R}}\left( {o_{i} } \right)\))

0.0250

0.0302

0.0305

− 0.0305

0.0300

\(o_{3} \succ o_{2} \succ o_{5} \succ o_{1} \succ o_{4}\)