Question D of the National Undergraduate Mathematical Model in 2004-Recruitment of Civil Servants. (Attached question link: 2004 National Mathematical Modeling Competition for College Students-Douding. com)

As shown in the figure below, the re-examination results (the expert group's evaluation of candidates' specialties) are given in four grades of ABCD.

Quantifying grades is a key step in modeling. So how do you quantify these four grades? This requires the membership method in fuzzy mathematics.

We set the corresponding comment set to {A (very good), B (good), C (general), D (poor)}, and the corresponding values are 5, 4, 3, 2. According to the actual situation, choose a larger membership function of Cauchy distribution as shown in the figure below.

By the known conditions are

Then bring it into (1) to get the membership function.

After calculation, the quantitative values of the expert group's evaluation {A (very good), B (good), C (average) and D (poor)} of the candidates are.

Therefore, the evaluation is quantitative. It is worth mentioning that this result is universal.

The tool used here is membership function, also known as membership function or fuzzy meta-function, which is a function used in fuzzy sets and a generalization of indicator function in general sets.

The indicative function value of the element can be 0 or 1.

The membership function of an element is a numerical value between 0 and 1, which indicates the true degree that an element belongs to a fuzzy set.

EXX: If the set S={ the weight exceeds 120kg}, Xiaoming 123kg belongs to the set S, and its index function is1; The weight of small gan 100kg (self-deception here) does not belong to set S, and the index function is 0. But for fuzzy sets, there may not be such a clear definition. Assuming that Xiao Pang is a fuzzy set, the membership function value of people with weight 120kg may be 0.9, and that of people with weight 100 kg may be 0.8.

Next, the definition of membership function in mathematics is given:

DEF: For set X, the membership function on set X is a function that maps set X to the unit real number interval [0, 1].

In the above-mentioned solving process, the membership function of large Cauchy distribution is used. In fact, in addition to the membership function of the large Cauchy distribution, there are small and intermediate types.

QAQ: What does this have to do with Cauchy or Cauchy distribution?

QAQ: Why do you choose the membership function of large Cauchy distribution?