# Table 1 Type-2 fuzzy membership functions

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Lower and upper membership functions

$$f_{1}$$

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$$f_{2}$$

$$\Delta = - 10\%$$

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

$$DS = 0.5$$

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

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

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

$$\Delta = 10\%$$

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

$$DS = 1$$

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

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

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