Appendix

Fuzzy Operator Formulas

The following is a list of formulas used to operate on the membership grades of fuzzy set A for selected general operations on fuzzy sets. In the following formulas, xn represents the new membership grades after applying the operators, and yn represents the original membership grades for the elements of fuzzy set A

Concentrate

Dilate

Intensify Contrast

Normalize

Complement

Standard

Sugeno[], w (-1, )

Yager[w], w (0, )

Intersection Formulas

The following is a list of the formulas used for various intersections between two fuzzy sets, A and B. The following formulas indicate how the membership grades for corresponding elements in fuzzy sets A and B should be combined. In the formulas, a represents the membership grade for an element in fuzzy set A, and b represents the membership grade for an element in fuzzy set B.

Standard

Hamacher[v], v (0, )

Frank[s], s (0, ) s 1.

Yager[w], w (0, )

DuboisPrade[], [0, 1]

Dombi[], (0, )

Weber[], (-1, )

Yu[], (-1 , )

Union Formulas

The following is a list of the formulas used for various unions between two fuzzy sets, A and B. The following formulas indicate how the membership grades for corresponding elements in fuzzy sets A and B should be combined. In the formulas, a represents the membership grade for an element in fuzzy set A, and b represents the membership grade for the corresponding element in fuzzy set B.

Standard

Hamacher[v], v (0, )

Frank[s], s (0, ) s 1

Yager[w], w (0, )

DuboisPrade[], [0, 1]

Dombi[], (0, )

Weber[], (-1 , )

Yu[], (-1 , )

Averaging Formulas

Following is a list of the formulas for taking the various averages used in this package. The averaging operations are denoted by the letter h, and represent the membership grades for corresponding elements in the n fuzzy sets being averaged.

Arithmetic Mean

Geometric Mean

Harmonic Mean

Generalized Mean[], (- , )

Miscellaneous Formulas

Gaussian Fuzzy Sets

where m is the mean, s is the width, and x is the element.

Bell Fuzzy Sets

where c is the center, crossover points are at c ± w, a slope at the crossover points of s/2w, and x is the element.

Sigmoid Fuzzy Sets

where s controls the slope at crossover point c, and x is the element.