#### 期刊菜单

Evaluation Method of Impoverished Students in Western Ethnic Areas Based on Intuitionistic Trapezoidal Fuzzy TOPSIS Algorithm

Abstract: In view of the fact that the definition of impoverished students in universities of the western ethnic areas is vague and difficult to be precise, the intuitionistic trapezoidal fuzzy number and TOPSIS algorithm are used to comprehensively evaluate the accuracy and objectivity of identification of impoverished students. Firstly, the evaluation index system and the table of hybrid fuzzy decision value are constructed, and the intuitionistic trapezoidal fuzzy number scoring function is defined to obtain the hybrid fuzzy decision matrix. Secondly, the analytic hierarchy process is used to determine the weights of each index. And according to the TOPSIS algorithm, the students are comprehensively compared and sorted. Finally, the validity and objectivity of the evaluation method are illustrated by numerical examples.

1. 引言

2013年，习近平总书记在湖南省湘西土家族苗族自治州考察时首次提出“精准扶贫”，教育扶贫作为精准扶贫的重要内容，引起了党和国家的高度重视。贫困大学生在我国高校普遍存在，西部民族地区地处偏僻，高校贫困生现象尤为突出，人数呈现逐年上升的趋势。

2. 直觉梯形模糊数

2.1. 直觉梯形模糊数的定义

${b}_{1}\le {a}_{1}\le {b}_{2}\le {a}_{2}\le {b}_{3}\le {a}_{3}\le {b}_{4}\le {a}_{4}$

2.2. 直觉梯形模糊数的得分函数定义

1) ${E}_{\stackrel{˜}{t}}\left(\stackrel{˜}{\alpha }\right)=\frac{{a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}}{4}$$\stackrel{˜}{\alpha }$ 的隶属度期望值；

2) ${E}_{\stackrel{˜}{f}}\left(\stackrel{˜}{\alpha }\right)=\frac{{b}_{1}+{b}_{2}+{b}_{3}+{b}_{4}}{4}$$\stackrel{˜}{\alpha }$ 的非隶属度期望值。

$S\left(\stackrel{˜}{\alpha }\right)=\left({E}_{\stackrel{˜}{t}}\left(\stackrel{˜}{\alpha }\right)-{E}_{\stackrel{˜}{f}}\left(\stackrel{˜}{\alpha }\right)\right)\left({E}_{\stackrel{˜}{t}}\left(\stackrel{˜}{\alpha }\right)+{E}_{\stackrel{˜}{f}}\left(\stackrel{˜}{\alpha }\right)\right)$ (1)

$\stackrel{˜}{\alpha }=〈\left(\text{1},\text{1},\text{1},\text{1}\right),\left(\text{0},\text{0},\text{0},\text{0}\right)〉$ ，则 $S\left(\stackrel{˜}{\alpha }\right)=\text{1}$

${a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}={b}_{1}+{b}_{2}+{b}_{3}+{b}_{4}$ ，则 $S\left(\stackrel{˜}{\alpha }\right)=0$

$\stackrel{˜}{\alpha }=〈\left(\text{0},\text{0},\text{0},\text{0}\right),\left(\text{1},\text{1},\text{1},\text{1}\right)〉$ ，则 $S\left(\stackrel{˜}{\alpha }\right)=-\text{1}$

1) 若 $S\left({\stackrel{˜}{\alpha }}_{1}\right) ，则 ${\stackrel{˜}{\alpha }}_{1}<{\stackrel{˜}{\alpha }}_{2}$

2) 若 $S\left({\stackrel{˜}{\alpha }}_{1}\right)=S\left({\stackrel{˜}{\alpha }}_{2}\right)$ ，则 ${\stackrel{˜}{\alpha }}_{1}={\stackrel{˜}{\alpha }}_{2}$

3) 若 $S\left({\stackrel{˜}{\alpha }}_{1}\right)>S\left({\stackrel{˜}{\alpha }}_{2}\right)$ ，则 ${\stackrel{˜}{\alpha }}_{1}>{\stackrel{˜}{\alpha }}_{2}$

3. 基于直觉梯形模糊TOPSIS的贫困生评定模型建立

3.1. 西部民族地区高校贫困生评定指标体系的建立

Table 1. Evaluation index system for impoverished students in universities of western ethnic areas

3.2. 评估模型建立

Step 1 构建混合模糊决策矩阵

$P=\left\{{p}_{1},{p}_{2},\cdots ,{p}_{m}\right\}$ 为m个贫困生组成的集合， $Q=\left\{{q}_{1},{q}_{2},\cdots ,{q}_{n}\right\}$ 为n个决策属性。由专家给出评

Step 2 构造得分函数矩阵

Step 3 基于层次分析法确定各指标权重

1) 建立判断矩阵

Table 2. 1 - 9 scale and its meaning

2) 一致性检验

$CI=\frac{{\lambda }_{\mathrm{max}}\left(A\right)-n}{n-1}$ (2)

i) 若CI = 0，则成对比较矩阵A具有完全的一致性；

ii) 若CI接近0，则成对比较矩阵A具有满意的一致性；

iii) CI越大，成对比较矩阵A的不一致性越高。

$CR=\frac{CI}{RI}$ (3)

Table 3. Average random consistency index

$CR<0.1$ ，则认为成对比较矩阵A的不一致程度在可以接受的范围之内，通过了一致性检验。

3) 得到各指标的权向量w

Step 4 基于TOPSIS法确定各学生排序

1) 指标值规范化

${z}_{ij}=\frac{{y}_{ij}}{\sqrt{\underset{i=1}{\overset{m}{\sum }}{y}_{ij}^{2}}}$ (4)

${z}_{ij}=\frac{1/{y}_{ij}}{\sqrt{\underset{i=1}{\overset{m}{\sum }}{\left(1/{y}_{ij}\right)}^{2}}}$ (5)

2) 构建加权规范阵

${c}_{ij}={w}_{j}\cdot {z}_{ij}$ (6)

3) 确定正负理想解

4) 计算各位学生到正负理想的距离

${d}_{j}^{+}=\sqrt{\underset{j=1}{\overset{m}{\sum }}{\left({c}_{ij}-{c}_{j}^{+}\right)}^{2}},i=1,2,\cdots ,m$ (9)

${d}_{j}^{-}=\sqrt{\underset{j=1}{\overset{m}{\sum }}{\left({c}_{ij}-{c}_{j}^{-}\right)}^{2}},i=1,2,\cdots ,m$ (10)

5) 计算综合评价指数 ${f}_{i}^{*}$ 并排序

${f}_{i}^{*}=\frac{{\text{d}}_{i}^{-}}{{\text{d}}_{i}^{-}+{\text{d}}_{i}^{+}},i=1,2,\cdots ,m$ (11)

4. 数值算例分析

Table 4. Hybrid fuzzy decision value table of five students under various indicators

$\left[S\left({\stackrel{˜}{a}}_{ij}\right)\right]=\left[\begin{array}{r}\hfill 0.059375\\ \hfill 0.033594\\ \hfill 0.055000\\ \hfill 0.060469\\ \hfill 0.093750\end{array}\text{}\begin{array}{r}\hfill 0.043125\\ \hfill 0.028906\\ \hfill 0.011875\\ \hfill 0.038281\\ \hfill 0.055000\end{array}\text{}\begin{array}{r}\hfill 0.022500\\ \hfill 0.027500\\ \hfill 0.070000\\ \hfill 0.046875\\ \hfill 0.039375\end{array}\text{}\begin{array}{r}\hfill 0.065000\\ \hfill 0.117773\\ \hfill 0.065000\\ \hfill 0.065625\\ \hfill 0.065625\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{r}\hfill 0.039375\\ \hfill 0.031875\\ \hfill 0.078125\\ \hfill 0.070000\\ \hfill 0.060000\end{array}\right]$

$S=\left[\begin{array}{c}\begin{array}{c}1.20\\ 1.50\\ 1.10\\ 1.20\\ 1.40\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}\text{\hspace{0.17em}}\begin{array}{r}\hfill 0.059375\\ \hfill 0.033594\\ \hfill 0.055000\\ \hfill 0.060469\\ \hfill 0.093750\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{cccc}1& 0& 1& 1\\ 0& 1& 1& 1\\ 1& 0& 1& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}500\\ 580\\ 550\\ 540\\ 560\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{r}\hfill 0.043125\\ \hfill 0.028906\\ \hfill 0.011875\\ \hfill 0.038281\\ \hfill 0.055000\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{r}\hfill 0.022500\\ \hfill 0.027500\\ \hfill 0.070000\\ \hfill 0.046875\\ \hfill 0.039375\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{r}\hfill 0.065000\\ \hfill 0.117773\\ \hfill 0.065000\\ \hfill 0.065625\\ \hfill 0.065625\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{r}\hfill 0.039375\\ \hfill 0.031875\\ \hfill 0.078125\\ \hfill 0.070000\\ \hfill 0.060000\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}1\\ 1\\ 0\\ 0\\ 1\end{array}\right]$

${A}_{1}=\left[\begin{array}{cccccc}1& 1& 1& 1/3& 1& 6/5\\ 1& 1& 1/2& 1/2& 2& 4/3\\ 1& 2& 1& 1& 2& 5/4\\ 3& 2& 1& 1& 3& 2\\ 1& 1/2& 1/2& 1/3& 1& 1\\ 5/6& 3/4& 4/5& 1/2& 1& 1\end{array}\right]$${A}_{\text{2}}=\left[\begin{array}{cccccc}1& 5/4& 5/4& 5/3& 5/3& 1/2\\ 4/5& 1& 4/3& \text{2}& 5/4& 3/4\\ 4/5& 3/4& 1& 4/3& \text{1}& 5/8\\ 3/5& 1/2& 3/4& 1& 5/6& 1/3\\ 3/5& 4/5& 1& 6/5& 1& 1/2\\ \text{2}& 4/3& 8/5& \text{3}& \text{2}& 1\end{array}\right]$

${w}_{1}={\left[0.135803,\text{ }0.146002,0.208071,0.283990,0.103017,0.123116\right]}^{\text{T}}$

${w}_{2}={\left[0.181076,0.177610,0.140611,\text{ }0.099271,0.128252,0.273180\right]}^{\text{T}}$

Table 5. Weight value table of secondary indicators

$Z=\left[\begin{array}{ccccccc}\begin{array}{r}\hfill 0.468048\\ \hfill 0.374438\\ \hfill 0.510598\\ \hfill 0.468048\\ \hfill 0.401184\end{array}& \begin{array}{r}\hfill 0.387651\\ \hfill 0.685151\\ \hfill 0.418487\\ \hfill 0.380640\\ \hfill 0.245512\end{array}& \begin{array}{r}\hfill 0.486554\\ \hfill 0.419443\\ \hfill 0.442322\\ \hfill 0.450513\\ \hfill 0.434423\end{array}& \begin{array}{r}\hfill 0.233775\\ \hfill 0.348767\\ \hfill 0.848973\\ \hfill 0.263355\\ \hfill 0.183301\end{array}& \begin{array}{r}\hfill 0.226018\\ \hfill 0.276244\\ \hfill 0.703166\\ \hfill 0.470870\\ \hfill 0.395531\end{array}& \begin{array}{r}\hfill 0.369570\\ \hfill 0.669623\\ \hfill 0.369570\\ \hfill 0.373123\\ \hfill 0.373123\end{array}& \begin{array}{r}\hfill 0.300494\\ \hfill 0.243257\\ \hfill 0.596219\\ \hfill 0.534212\\ \hfill 0.457896\end{array}\end{array}\right]$

${C}^{1}=\left[\begin{array}{cccccc}\begin{array}{r}\hfill 0.031781\\ \hfill 0.025425\\ \hfill 0.034670\\ \hfill 0.031781\\ \hfill 0.027241\end{array}& \begin{array}{r}\hfill 0.028299\\ \hfill 0.050017\\ \hfill 0.030550\\ \hfill 0.027787\\ \hfill 0.017923\end{array}& \begin{array}{r}\hfill 0.104036\\ \hfill 0.000000\\ \hfill 0.104036\\ \hfill 0.000000\\ \hfill 0.000000\end{array}& \begin{array}{r}\hfill 0.000000\\ \hfill 0.141995\\ \hfill 0.000000\\ \hfill 0.000000\\ \hfill 0.141995\end{array}& \begin{array}{r}\hfill 0.051509\\ \hfill 0.051509\\ \hfill 0.051509\\ \hfill 0.051509\\ \hfill 0.000000\end{array}& \begin{array}{r}\hfill 0.061558\\ \hfill 0.061558\\ \hfill 0.000000\\ \hfill 0.000000\\ \hfill 0.000000\end{array}\end{array}\right]$

${C}^{2}=\left[\begin{array}{cccccc}\begin{array}{r}\hfill 0.044052\\ \hfill 0.037975\\ \hfill 0.040047\\ \hfill 0.040788\\ \hfill 0.039332\end{array}& \begin{array}{r}\hfill 0.020760\\ \hfill 0.030972\\ \hfill 0.075393\\ \hfill 0.023387\\ \hfill 0.016278\end{array}& \begin{array}{r}\hfill 0.015890\\ \hfill 0.019422\\ \hfill 0.049437\\ \hfill 0.033105\\ \hfill 0.027808\end{array}& \begin{array}{r}\hfill 0.018344\\ \hfill 0.033237\\ \hfill 0.018344\\ \hfill 0.018520\\ \hfill 0.018520\end{array}& \begin{array}{r}\hfill 0.019269\\ \hfill 0.015599\\ \hfill 0.038233\\ \hfill 0.034257\\ \hfill 0.029363\end{array}& \begin{array}{r}\hfill 0.136590\\ \hfill 0.136590\\ \hfill 0.000000\\ \hfill 0.000000\\ \hfill 0.136590\end{array}\end{array}\right]$

${c}_{j1}^{+}=\left[0.034670,0.050017,\text{ }0.104036,0.141995,0.051509,0.061558\right]$

${c}_{j2}^{+}=\left[0.044052,0.075393,0.049437,\text{ }0.033237,\text{ }0.038233,0.136590\right]$

${c}_{j1}^{-}=\left[0.025425,0.017923,\text{ }0.000000,\text{ }0.000000,\text{ }0.000000,\text{ }0.000000\right]$

${c}_{j2}^{-}=\left[0.037975,\text{ }0.016278,\text{ }0.015890,\text{ }0.018344,0.015599,0.000000\right]$

Table 6. Distance value and comprehensive evaluation index table

5. 数值算例分析

NOTES

*通讯作者。

 [1] 宋美喆. 基于模糊综合评价方法的高校贫困生认定研究[J]. 黑龙江高教研究, 2016(7): 16-20. [2] 薛单. 高校助学金等级评定模型研究及系统开发[D]: [硕士学位论文]. 北京: 北京交通大学, 2010. [3] 胡景. TOPSIS算法在高校贫困生认定中的应用[J]. 新乡学院学报, 2014, 31(2): 5-7. [4] 冯春明, 杨玉杰. 层次分析法在高校贫困生评定工作中的应用[J]. 聊城大学学报(自然科学版), 2014, 27(4): 105-110. [5] 彭德军, 杨婧宇, 沈有建. 基于变权AHP法的贫困生评定[J]. 海南师范大学学报(自然科学版), 2016, 29(3): 256-262. [6] 柴政, 屈莉莉, 彭贵宾. 高校贫困生精准资助的神经网络模型[J]. 数学的实践与认识, 2018, 48(16): 85-91. [7] 谭浩, 田爱奎, 郑睿. 基于高校学生消费数据的贫困生评价分析[J]. 电脑知识与技术, 2017, 13(21): 220-221+235. [8] 冯春苑. 一种改进的Apriori算法在认定大学贫困生中的研究[D]: [硕士学位论文]. 广州: 暨南大学, 2016. [9] Nehi, H.M. and Maleki, H.R. (2005) Intuitionistic Fuzzy Numbers and It’s Applications in Fuzzy Optimization Problem. Proceedings of the 9th WSEAS International Conference on Systems, Athens, Greece, 11-13 July 2005, 1-5. [10] 戴厚平, 罗宜武. 基于直觉模糊多属性决策的公路工程评标方法[J]. 数学的实践与认识, 2013, 43(16): 46-52. [11] 廖炎平, 刘莉, 邢超. TOPSIS中不同规范化方法的研究[J]. 北京理工大学学报, 2012, 32(8): 871-875+880.