Table 1 Type-2 fuzzy membership functions

\(\Delta = - 10\%\)

\(\overline{\alpha }_{{M_{1}^{1} }} = \overline{\alpha }_{{M_{2}^{1} }} = \frac{{f_{1max} - 0.9f_{1} }}{{f_{1max} - f_{1min} }}\)

\(DS = 0.5\)

\(\underline {\alpha }_{{M_{1}^{2} }} = \underline {\alpha }_{{M_{3}^{2} }} = \frac{{f_{2max} - f_{2} }}{{f_{2max} - f_{2min} }}\)

\(\underline {\alpha }_{{M_{3}^{1} }} = \underline {\alpha }_{{M_{4}^{1} }} = \frac{{0.9f_{1} - f_{1min} }}{{f_{1max} - f_{1min} }}\)

\(\underline {\alpha }_{{M_{2}^{2} }} = \underline {\alpha }_{{M_{4}^{2} }} = \frac{{f_{2} - f_{2min} }}{{f_{2max} - f_{2min} }}\)

\(\Delta = 10\%\)

\(\underline {\alpha }_{{M_{1}^{1} }} = \underline {\alpha }_{{M_{2}^{1} }} = \frac{{f_{1max} - 1.1f_{1} }}{{f_{1max} - f_{1min} }}\)

\(DS = 1\)

\(\overline{\alpha }_{{M_{1}^{2} }} = \overline{\alpha }_{{M_{3}^{2} }} = \frac{{f_{2max} - f_{2} }}{{f_{2max} - f_{2min} }}\)

\(\overline{\alpha }_{{M_{3}^{1} }} = \overline{\alpha }_{{M_{4}^{1} }} = \frac{{1.1f_{1} - f_{1min} }}{{f_{1max} - f_{1min} }}\)

\(\overline{\alpha }_{{M_{2}^{2} }} = \overline{\alpha }_{{M_{4}^{2} }} = \frac{{f_{2} - f_{2min} }}{{f_{2max} - f_{2min} }}\)