Table 9 Obtained results by proposed approach using different values of \(\alpha\) and \(\beta\).

\(\alpha =\beta =0.5\)

0.4930

0.3844

0.3874

0.4056

\(\aleph _{1}>\aleph _{4}>\aleph _{3}>\aleph _{2}\)

\(\alpha =0.5,\beta =0.0\)

0.4710

0.3391

0.3469

0.3663

\(\aleph _{1}>\aleph _{4}>\aleph _{3}>\aleph _{2}\)

\(\alpha =1.0,\beta =0.0\)

0.5322

0.4004

0.4028

0.4251

\(\aleph _{1}>\aleph _{4}>\aleph _{3}>\aleph _{2}\)

\(\alpha =0.0,\beta =0.5\)

0.4390

0.3373

0.3508

0.3676

\(\aleph _{1}>\aleph _{4}>\aleph _{3}>\aleph _{2}\)

\(\alpha =0.0,\beta =1.0\)

0.4972

0.3976

0.4046

0.4221

\(\aleph _{1}>\aleph _{4}>\aleph _{3}>\aleph _{2}\)

\(\alpha =\beta =1.0\)

0.5445

0.4374

0.4359

0.4550

\(\aleph _{1}>\aleph _{4}>\aleph _{2}>\aleph _{3}\)

\(\alpha =2.0,\beta =1.0\)

0.5814

0.4675

0.4661

0.4878

\(\aleph _{1}>\aleph _{4}>\aleph _{2}>\aleph _{3}\)

\(\alpha =5.0,\beta =1.0\)

0.6553

0.5303

0.5271

0.5593

\(\aleph _{1}>\aleph _{4}>\aleph _{2}>\aleph _{3}\)

\(\alpha =1.0,\beta =2.0\)

0.5744

0.4721

0.4644

0.4865

\(\aleph _{1}>\aleph _{4}>\aleph _{2}>\aleph _{3}\)

\(\alpha =1.0,\beta =5.0\)

0.6391

0.5389

0.5201

0.5539

\(\aleph _{1}>\aleph _{4}>\aleph _{2}>\aleph _{3}\)

\(\alpha =\beta =2.0\)

0.5961

0.4890

0.4828

0.5057

\(\aleph _{1}>\aleph _{4}>\aleph _{2}>\aleph _{3}\)

\(\alpha =\beta =5.0\)

0.6753

0.5623

0.5490

0.5867

\(\aleph _{1}>\aleph _{4}>\aleph _{2}>\aleph _{3}\)

\(\alpha =\beta =10\)

0.7385

0.6162

0.5974

0.6519

\(\aleph _{1}>\aleph _{4}>\aleph _{2}>\aleph _{3}\)