Table 4 Effects of parameter \({\varvec{q}}\) on decision results using Q-ROFHSEWA operator.

\(q=1\)

\(S({\mathcal{L}}_{1})\)= \(-\) 0.0357, \(S({\mathcal{L}}_{2})\) = 0.0116, \(S({\mathcal{L}}_{3})\) = 0.1660

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=2\)

\(S({\mathcal{L}}_{1})\)= 0.0196, \(S({\mathcal{L}}_{2})\) = 0.0544, \(S({\mathcal{L}}_{3})\) = 0.2293

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=3\)

\(S({\mathcal{L}}_{1})\)= 0.0574, \(S({\mathcal{L}}_{2})\) = 0.0815, \(S({\mathcal{L}}_{3})\) = 0.2181

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=4\)

\(S({\mathcal{L}}_{1})\)= 0.0677, \(S({\mathcal{L}}_{2})\) = 0.0860, \(S({\mathcal{L}}_{3})\) = 0.1828

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=5\)

\(S({\mathcal{L}}_{1})\)= 0.0648, \(S({\mathcal{L}}_{2})\) = 0.0790, \(S({\mathcal{L}}_{3})\) = 0.1466

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=6\)

\(S({\mathcal{L}}_{1})\)= 0.0575, \(S({\mathcal{L}}_{2})\) = 0.0683, \(S({\mathcal{L}}_{3})\) = 0.1159

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=7\)

\(S({\mathcal{L}}_{1})\)= 0.0496, \(S({\mathcal{L}}_{2})\) = 0.0575, \(S({\mathcal{L}}_{3})\) = 0.0917

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=8\)

\(S({\mathcal{L}}_{1})\)= 0.0425, \(S({\mathcal{L}}_{2})\) = 0.0479, \(S({\mathcal{L}}_{3})\) = 0.0730

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=9\)

\(S({\mathcal{L}}_{1})\)= 0.0365, \(S({\mathcal{L}}_{2})\) = 0.0399, \(S({\mathcal{L}}_{3})\) = 0.0585

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)

\(q=10\)

\(S({\mathcal{L}}_{1})\)= 0.0314, \(S({\mathcal{L}}_{2})\) = 0.0333, \(S({\mathcal{L}}_{3})\) = 0.0473

\({\mathrm{\aleph }}^{(3)}>{\mathrm{\aleph }}^{(2)}>{\mathrm{\aleph }}^{(1)}\)