Both the Analytic Hierarchy Process (AHP) and the Comprehensive Evaluation Method (TOPSIS) introduced earlier rely on some subjective factors (for example, the Analytic Hierarchy Process needs to find out the importance between two indicators to construct a judgment matrix, and the comprehensive analysis method needs to find literature to directly add weights to each indicator), none of them can completely rely on the data itself, so how do we give evaluation indicators when we cannot find evidence from literature or other places? This is what we use today. Talk about the entropy weight method.

modeling steps

data standardization

Due to the different types of data indicators, the magnitude of the data may be different. The data indicators may have negative values and may have both positive indicators (the bigger the better) and negative indicators (the smaller the better), which requires the data to be standardized first. Data standardization uses different formulas to separate positive and negative indicators.

Positive indicator standardization

Standardized data = (original data - minimum value in original data) / (maximum value of original data) - minimum value of original data)

Negative indicator normalization

Standardized data = (maximum value in original data - original data) / (maximum value of original data - minimum value of original data)

Obviously, after normalization, all the data fall between 0 and 1, and the smaller the negative index, the better, and the larger the better as the positive index.

Note: The reason for standardization is that there may be one or more of the three situations we mentioned above, such as different data magnitudes, negative data, and negative indicators. If there are none of these three situations, Then standardization is not required.

Find the entropy weight and variation coefficient of the index

Denotes $a_{ij}$ the standardized index, i represents the i-th scheme or the i-th person, and j represents the j-th index, then the probability of a certain index data appearing is,

$p_{ij}=\frac{a_{ij}}{\sum_{i=1}^{n}a_{ij}}$ ,

Note : The denominator is the sum of all data of the i-th scheme or i-th indicator.

The formula for calculating the entropy value is

The entropy formula of the jth index is

$e_{j}=-\frac{1}{lnn}\sum_{i=1}^{n}p_{ij}lnp_{ij}$ ,

The coefficient of variation is

$g_{j}=1-e_{j}$ .

Obviously, the larger the entropy value and the smaller the coefficient of variation, the more orderly the index is, and the smaller the information content of the index. The smaller the entropy value and the larger the coefficient of variation, the more disordered the index and the greater the information content of the index.

Calculate the weight according to the coefficient of variation

Calculate the weight of the jth indicator

$w_{j}=\frac{g_{j}}{\sum_{i=1}^{m}g_{j}}$ ,

Note: The denominator here is the sum of the coefficients of variation of the indicators.

Obviously, the greater the coefficient of variation of the index, the greater the weight of the index

Calculate the overall score according to the weight

The comprehensive evaluation value of the i-th object is

$s_{i}=\sum_{j=1}^{m}w_{j}p_{ij}$ ,

The larger the evaluation value, the better. The evaluation value obtained by the entropy weight method is completely derived from the original data, without any other subjective factors.

Entropy weight method solution steps