Mathematical modeling evaluation category - Topsis model

There is a very common type of problem in mathematical modeling: selecting the optimal solution, which is called an evaluation problem. For example: Ctrip, Meituan and Fliggy, which of the three travel platforms is more suitable for novices to travel? Which one of Suzhou, Hangzhou or Nanjing is more suitable for Dragon Boat Festival travel? Which classmate in the class won the scholarship and so on. To make a choice, you first need to know what evaluation indicators there are. Taking the selection of a tourist destination as an example, you can search for relevant articles on CNKI or brainstorm within the group or use Internet search engine resources to get the criteria for everyone to consider when choosing a tourist destination: Scenery , humanities, crowding, etc. Score the plan in each evaluation indicator dimension, setting a total score of 5. "There is heaven above, and there are Suzhou and Hangzhou below." It can be said that the scenery of Suzhou and Hangzhou is very good, so they are given 5 points for scenery. In terms of culture, Nanjing is given 5 points for its rich historical heritage as the ancient capital of six dynasties. The score data of each plan in this type of evaluation question is given by itself based on the data, which is more suitable for the analytic hierarchy process. Whether to obtain a scholarship can be screened based on the scores in each subject. If the data exists objectively, you can use the topsis method mentioned below.

The TOPSIS method (Technique for Order Preference by Similarity to Ideal Solution) can be translated as the approximate ideal solution ranking method, and is often referred to as the superior and inferior solution distance method in China. The TOPSIS method is a commonly used method for comprehensive evaluation using original data. Its basic principle is to sort by detecting the distance between the evaluation object and the optimal solution and the worst solution. If the evaluation object is closest to the optimal solution and the worst solution at the same time, If it is far away from the worst solution, it is the best; otherwise it is not the best. Each index value of the optimal solution reaches the optimal value of each evaluation index. Each index value of the worst solution reaches the worst value of each evaluation index. Taking scholarships as an example, assume that whether you get a scholarship is only related to your scores in three subjects: Chinese, mathematics, and English. Your scores are 80, 90, and 100, and your best scores are 100, 100, and 100, and your worst scores are 50, 60, 50. Then the distance between you and the optimal solution is ?; the distance between you and the worst solution is .

step1: Positive indicator.

The specific indicators that will be encountered during evaluation can be divided into four categories. ① Very large indicators, also known as benefit indicators. The larger the value, the better, including performance, income, etc. ② Very small indicators, also known as It is a cost-based indicator, the smaller the value, the better, including expenses, casualties, etc. ③ Intermediate indicator, the value has an intermediate optimal point, such as the closer the pH value is to 7, the better, and the closer the blood pressure is to the ideal blood pressure (systolic blood pressure 120 mmHg, Diastolic blood pressure (diastolic blood pressure 80 mmHg), the better ④Interval-type index, the value is best within an interval. For example, the optimal population size of a city is between 10 million and 12 million (the numbers are only used as examples and have no practical significance).

Different types of indicators need to be forwarded according to different formulas, that is, all indicators are converted into extremely large sizes.

Very small conversion is the easiest, just use max-x directly. If the variable x is a positive number, you can also directly take the reciprocal. For example, the maximum cost is 3000, and the cost corresponding to the x variable is 1000. The converted value should be 3000-1000=2000, or directly take the reciprocal to 1/1000.

The intermediate conversion formula is. Taking the pH value as an example, the optimal solution is 7. A set of data has three variables 7, 8, and 9, then , , . so . Take i=2, the original data is 8, and the converted bit is 1-(8-7)/2=1/2.

Interval conversion is more complicated. If { } is a set of intermediate indicator sequences, and the best interval is [a, b], then the forwarding formula is as follows:

Taking human body temperature as an example, the original data are 35.2, 35.8, 36.6, 37.1, 37.8, 38.4. The optimal interval is 36 to 37, then a=36, b=37, M=max(36-35.2, 38.4-37)=1.4. Substitute into the above formula to get the converted data.

step2: Forward matrix standardization

Assuming there are n objects to be evaluated and m forward evaluation indicators, a forward matrix can be constructed. is the score of the first object after forwarding on the second evaluation index.

Denote the standardized matrix as Z, then each element in it is equal to the value of the element in the corresponding matrix X divided by the square sum of the element in the column, that is, .

step3: Calculate scores and normalize

The standardized matrix of n evaluation objects and m evaluation indicators is as follows:

Define the maximum value as the element in each column The set of maximum values

Define the minimum value as the set of minimum values ??of elements in each column

Then the distance between the i-th evaluation object and the maximum value is the calculated distance between j indicators and the maximum value. The following summation:

Similarly, the distance between the i-th evaluation object and the minimum value is the summation after calculating the distance between j indicators and the minimum value:

Then, the distance between the i-th evaluation object and the minimum value is The unnormalized score of i evaluation objects is, that is, the distance between z and the minimum value divided by the sum of the distance between z and the maximum value and the distance between z and the minimum value.

Because the distance is non-negative, it is obvious that the value is between 0 and 1. The larger the value, the larger the value, the closer to the optimal solution.

The score after normalization is , which should satisfy .

Normalization and standardization are essentially intended to eliminate the influence of dimensions. As a result, it is easier to compare sizes after normalization.

After obtaining the scores of all solutions, it is recommended to visually display the sorted scores. Excel can be used to draw a column chart.

As shown in the figure above, Option 5 has the highest score, so Option 5 should be selected.

The above process is the basic topsis model. This model defaults to the same weight of all indicators. The analytic hierarchy process or the entropy weight method can be used to determine the indicator weight and construct a weighted topsis model.

Source of information:

The above information comes from station b (up owner: mathematical modeling learning exchange)/video/BV1gJ411k7X4from=search&seid=6343799996011307859.

Thanks to the up owner for organizing it. The video is very detailed and suitable for beginners to get started~