#### Doug I. Jones

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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Shannon and Weaver (1947) proposed the entropy concept, and this concept has been highlighted by Zeleny (1982) for deciding the weights of attributes. Entropy is the measure of uncertainty in the information using probability methods. It indicates that a broad distribution represents more uncertainty than a sharply peaked distribution does.

To determine the weights by the entropy method, the normalized decision matrix we call $R_{i j}$ is considered. The equation used is
$$e_{j}=-k \sum_{i=1}^{n} R_{i j} \ln \left(R_{i j}\right)$$
where $k=1 / \ln (n)$ is a constant that guarantees that $0<e_{j}<1$. The value of $n$ refers to the number of alternatives. The degree of divergence $\left(d_{j}\right)$ of the average information contained by each attribute can be calculated as
$$d_{j}=1-e_{j}$$
The more divergent the performance rating $R_{i j p}$ for all $i$ and $j$, then the higher the corresponding $d_{j}$, the more important the attribute $B_{j}$ is considered to be.
The weights are found by Equation $4.4$ :
$$w_{j}=\frac{\left(1-e_{j}\right)}{\sum\left(1-e_{j}\right)}$$
Let us illustrate an example to obtain entropy weights.

Here, each criterion (attribute) is given a weight, and the sum of all weights must be equal to 1 . If criteria are equally weighted, then we merely need to sum the alternative values. Each alternative is assessed with regard to every criterion (attribute). The overall or composite performance score of an alternative is given simply by Equation $4.5$ with $m$ criteria.
$$P_{i}=\frac{\sum_{j=1}^{m} w_{j} m_{i j}}{m}$$
It was previously thought that all the units in the criteria must be identical units of measure such as dollars, pounds, and seconds. A normalization process can make the values unitless. So, we recommend normalizing the data as shown in Equation 4.6:
$$P_{i}=\frac{\sum_{j=1}^{m} w_{j} m_{i j \text { Normalized }}}{m}$$
where $\left(m_{i j \text { Normalized }}\right.$ ) represents the normalized value of $m_{i j}$ and $P_{i}$ is the overall or composite score of the alternative $A_{i}$. The alternative with the highest value of $P_{i}$ is considered the best alternative.

## 商业数学代考

Shannon 和 Weaver (1947) 提出了熵的概念, Zeleny (1982) 强调了这个概念来确定属性的 权重。熵是使用概率方法测量信息中的不确定性。它表明广泛分布比尖峰分布代表更多的不 确定性。

$$e_{j}=-k \sum_{i=1}^{n} R_{i j} \ln \left(R_{i j}\right)$$

$$d_{j}=1-e_{j}$$

$$w_{j}=\frac{\left(1-e_{j}\right)}{\sum\left(1-e_{j}\right)}$$

$$P_{i}=\frac{\sum_{j=1}^{m} w_{j} m_{i j}}{m}$$

$$P_{i}=\frac{\sum_{j=1}^{m} w_{j} m_{i j \text { Normalized }}}{m}$$

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