Dear Majid Baseer ,

Since the fuzzy preference relation is a total ordering relation satisfying reciprocal and transitive laws on fuzzy numbers, the relative preference relation satisfies a total ordering relation on fuzzy numbers as well. Normally, utilizing preference relation is more reasonable than defuzzification on ranking fuzzy numbers, because defuzzification does not present preference degree between two fuzzy numbers and loses some messages.

Article Ranking triangle and trapezoidal fuzzy numbers based on the ...

Regards,

Shafagat

Majid Baseer In fuzzy logic and control systems, two forms of fuzzy numbers are used: triangular fuzzy numbers and trapezoidal fuzzy numbers. Both sorts of fuzzy numbers are used to represent fuzzy sets, with a fuzzy number defining the membership function of a fuzzy set.

Three points define triangular fuzzy numbers: a left endpoint, a right endpoint, and a peak point. A triangular fuzzy number's membership function is specified as a triangle shape. In situations where the membership function is known to be symmetric and the peak point is known, these fuzzy numbers are employed.

Four points define trapezoidal fuzzy numbers: a left endpoint, a right endpoint, a left support point, and a right support point. A trapezoidal fuzzy number's membership function has a trapezoid form. In situations where the membership function is known to be asymmetric and the support points are known, these fuzzy numbers are employed.

The usage of triangular or trapezoidal fuzzy numbers is determined by the application and the information available about the membership function. In general, a triangular fuzzy number should be utilized if the membership function is known to be symmetric and the peak point is known. If the asymmetry of the membership function is known and the support points are known, then a trapezoidal fuzzy number should be used.

The findings may differ if we use a triangular fuzzy number instead of a trapezoidal fuzzy number, or vice versa. It will be determined by the particular application and the nature of the membership function. The results may not vary considerably in certain circumstances, while they may alter significantly in others. As a result, it is critical to carefully analyze the membership function's features and select the suitable type of fuzzy number.

The decision of whether to use triangular or trapezoidal fuzzy numbers in a given problem depends on the specific characteristics of the problem and the information available.

Triangular fuzzy numbers are typically used when the degree of membership for a particular value is known to be symmetric around a central value. For example, if the degree of membership for a particular value is known to be equal for values both above and below the central value, a triangular fuzzy number may be appropriate.

On the other hand, trapezoidal fuzzy numbers are typically used when the degree of membership for a particular value is known to be asymmetric. For example, if the degree of membership for a particular value is known to be higher for values above the central value than for values below the central value, a trapezoidal fuzzy number may be appropriate.

In terms of the effect on the results, using a triangular fuzzy number instead of a trapezoidal fuzzy number or vice versa can have a significant impact on the results depending on the specific problem and data. In general, the choice of fuzzy number will affect the degree of membership values assigned to the different data points, which in turn can affect the final results of any operations or calculations performed using these fuzzy numbers.

It's worth noting that the choice of fuzzy number is only one of the many factors that can affect the results of a fuzzy logic or fuzzy reasoning system. Other factors such as the specific membership functions used, the choice of fuzzy inference method, and the specific problem being solved can also have a significant impact on the results. Therefore, it's always recommend to evaluate the different options, and compare the results of the triangular and trapezoidal fuzzy numbers, and choose the one that best fits the problem and data.