#### 期刊菜单

Two-Sided Matching Decision under the Condition of Trapezoidal Intuitionistic Fuzzy Preference Information
DOI: 10.12677/MSE.2022.114074, PDF , HTML, XML, 下载: 64  浏览: 118  国家自然科学基金支持

Abstract: This paper creates a two-sided matching decision method based on the trapezoidal intuitionistic fuzzy number information. The two-sided matching problem with trapezoidal intuitionistic fuzzy numbers is described on the basis of the presented related theories of trapezoidal intuitionistic fuzzy numbers and two-sided matchings. Then, the multi-objective two-sided matching model to maximizing the satisfaction degrees of trapezoidal intuitionistic fuzzy numbers can be established. The multi-objective two-sided matching model can be transformed into a single-objective two- sided matching model according to the operational laws of trapezoidal intuitionistic fuzzy number, the defuzzification method of trapezoidal fuzzy number and the linear weighted method. Moreover, the “best” two-sided matching scheme can be determined through solving the model. The feasibility and utility of the displayed two-sided matching decision is explained by the matching example between inclusive finance companies and small micro-enterprises.

1. 引言

2. 文献综述

3. 理论基础

3.1. 定义

${\varphi }_{\stackrel{˜}{a}}\left(x\right)=\left\{\begin{array}{l}\left(x-\underset{_}{a}\right){\omega }_{\stackrel{˜}{a}}/\left({a}^{\prime }-\underset{_}{a}\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\underset{_}{a}\le x<{a}^{\prime }\\ {\omega }_{\stackrel{˜}{a}},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{a}^{\prime }\le x\le {a}^{″}\\ \left(\stackrel{¯}{a}-x\right){\omega }_{\stackrel{˜}{a}}/\left(\stackrel{¯}{a}-{a}^{″}\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{a}^{″}\stackrel{¯}{a}\end{array}$ (1)

${\phi }_{\stackrel{˜}{a}}\left(x\right)=\left\{\begin{array}{l}\left[{a}^{\prime }-x+{u}_{\stackrel{˜}{a}}\left(x-\underset{_}{a}\right)\right]/\left({a}^{\prime }-\underset{_}{a}\right)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\underset{_}{a}\le x<{a}^{\prime }\\ {u}_{\stackrel{˜}{a}}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{a}^{\prime }\le x\le {a}^{″}\\ \left[x-{a}^{″}+{u}_{\stackrel{˜}{a}}\left(\stackrel{¯}{a}-x\right)\right]/\left(\stackrel{¯}{a}-{a}^{″}\right)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{a}^{″}\le x\le \stackrel{¯}{a}\\ 1\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }x<\underset{_}{a},x>\stackrel{¯}{a}\end{array}$ (2)

${\pi }_{\stackrel{˜}{a}}\left(x\right)=1-{\varphi }_{\stackrel{˜}{a}}\left(x\right)-{\phi }_{\stackrel{¯}{a}}\left(x\right)$ (3)

$\underset{_}{a}\ge 0$$\stackrel{¯}{a}>0$ 则称 $\stackrel{˜}{a}=〈\left(\underset{_}{a},{a}^{\prime },{a}^{″},\stackrel{¯}{a}\right);{\omega }_{\stackrel{˜}{a}},{u}_{\stackrel{˜}{a}}〉$ 为正梯形直觉模糊数，记为 $\stackrel{˜}{a}>0$。同样地，如果 $\stackrel{¯}{a}\le 0$$\underset{_}{a}<0$，则称 $\stackrel{˜}{a}=〈\left(\underset{_}{a},{a}^{\prime },{a}^{″},\stackrel{¯}{a}\right);{\omega }_{\stackrel{˜}{a}},{u}_{\stackrel{˜}{a}}〉$ 为负梯形直觉模糊数，记为 $\stackrel{˜}{a}<0$。明显地，在 ${\omega }_{a}=1$${u}_{a}=0$ 的情况， ${\varphi }_{\stackrel{˜}{a}}+{\phi }_{\stackrel{˜}{a}}=1$，则 $\stackrel{˜}{a}=〈\left(\underset{_}{a},{a}^{\prime },{a}^{″},\stackrel{¯}{a}\right);{\omega }_{\stackrel{˜}{a}},{u}_{\stackrel{˜}{a}}〉$ 退化成梯形模糊数 $\stackrel{˜}{a}=〈\left(\underset{_}{a},{a}^{\prime },{a}^{″},\stackrel{¯}{a}\right);1,0〉$

$\stackrel{˜}{a}+\stackrel{˜}{b}=〈\left(\underset{_}{a}+\underset{_}{b},{a}^{\prime }+{b}^{\prime },{a}^{″}+{b}^{″},\stackrel{¯}{a}+\stackrel{¯}{b}\right);{\omega }_{\stackrel{˜}{a}}\wedge {\omega }_{\stackrel{˜}{b}},{u}_{\stackrel{˜}{a}}\vee {u}_{\stackrel{˜}{b}}〉$ (4)

$\stackrel{˜}{a}-\stackrel{˜}{b}=〈\left(\underset{_}{a}-\stackrel{¯}{b},{a}^{\prime }-{b}^{″},{a}^{″}-{b}^{\prime },\stackrel{¯}{a}-\underset{_}{b}\right);{\omega }_{\stackrel{˜}{a}}\wedge {\omega }_{\stackrel{˜}{b}},{u}_{\stackrel{˜}{a}}\vee {u}_{\stackrel{˜}{b}}〉$ (5)

$\stackrel{˜}{a}×\stackrel{˜}{b}=\left\{\begin{array}{l}〈\left(\underset{_}{a}\underset{_}{b},{a}_{1}{b}_{1},{a}_{2}{b}_{2},\stackrel{¯}{a}\stackrel{¯}{b}\right);{\omega }_{\stackrel{˜}{a}}\wedge {\omega }_{\stackrel{˜}{b}},{u}_{\stackrel{˜}{a}}\vee {u}_{\stackrel{˜}{b}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\stackrel{˜}{a}>0,\stackrel{˜}{b}>0\\ 〈\left(\underset{_}{a}\stackrel{¯}{b},{a}_{1}{b}_{2},{a}_{2}{b}_{1},\stackrel{¯}{a}\underset{_}{b}\right);{\omega }_{\stackrel{˜}{a}}\wedge {\omega }_{\stackrel{˜}{b}},{u}_{\stackrel{˜}{a}}\vee {u}_{\stackrel{˜}{b}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\stackrel{˜}{a}<0,\stackrel{˜}{b}>0\\ 〈\left(\stackrel{¯}{a}\stackrel{¯}{b},{a}_{2}{b}_{2},{a}_{1}{b}_{1},\underset{_}{a}\underset{_}{b}\right);{\omega }_{\stackrel{˜}{a}}\wedge {\omega }_{\stackrel{˜}{b}},{u}_{\stackrel{˜}{a}}\vee {u}_{\stackrel{˜}{b}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\stackrel{˜}{a}<0,\stackrel{˜}{b}<0\end{array}$ (6)

$\stackrel{˜}{a}÷\stackrel{˜}{b}=\left\{\begin{array}{l}〈\left(\underset{_}{a}/\stackrel{¯}{b},{a}_{1}/{b}_{2},{a}_{2}/{b}_{1},\stackrel{¯}{a}/\underset{_}{b}\right);{\omega }_{\stackrel{˜}{a}}\wedge {\omega }_{\stackrel{˜}{b}},{u}_{\stackrel{˜}{a}}\vee {u}_{\stackrel{˜}{b}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\stackrel{˜}{a}>0,\stackrel{˜}{b}>0\\ 〈\left(\stackrel{¯}{a}/\stackrel{¯}{b},{a}_{2}/{b}_{2},{a}_{1}/{b}_{1},\underset{_}{a}/\underset{_}{b}\right);{\omega }_{\stackrel{˜}{a}}\wedge {\omega }_{\stackrel{˜}{b}},{u}_{\stackrel{˜}{a}}\vee {u}_{\stackrel{˜}{b}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\stackrel{˜}{a}<0,\stackrel{˜}{b}>0\\ 〈\left(\stackrel{¯}{a}/\underset{_}{b},{a}_{2}/{b}_{1},{a}_{1}/{b}_{2},\underset{_}{a}/\stackrel{¯}{b}\right);{\omega }_{\stackrel{˜}{a}}\wedge {\omega }_{\stackrel{˜}{b}},{u}_{\stackrel{˜}{a}}\vee {u}_{\stackrel{˜}{b}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\stackrel{˜}{a}<0,\stackrel{˜}{b}<0\end{array}$ (7)

$\lambda \stackrel{˜}{a}=\left\{\begin{array}{l}〈\left(\lambda \underset{_}{a},\lambda {a}_{1},\lambda {a}_{2},\lambda \stackrel{¯}{a}\right);{\omega }_{\stackrel{˜}{a}},{u}_{\stackrel{˜}{a}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\lambda >0\\ 〈\left(\lambda \stackrel{¯}{a},\lambda {a}_{2},\lambda {a}_{1},\lambda \underset{_}{a}\right);{\omega }_{\stackrel{˜}{a}},{u}_{\stackrel{˜}{a}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\lambda <0\end{array}$ (8)

${\stackrel{˜}{a}}^{-1}=〈\left(1/\stackrel{¯}{a},1/{a}_{2},1/{a}_{1},1/\underset{_}{a}\right);{\omega }_{\stackrel{˜}{a}},{u}_{\stackrel{˜}{a}}〉,\text{ }\text{ }\text{ }\text{ }\text{ }\stackrel{˜}{a}\ne 0$ (9)

3.2. 梯形模糊数去模糊化方法

$C\left(\stackrel{⌢}{M}\right)=\frac{\underset{_}{a}+4{a}^{\prime }+4{a}^{″}+\stackrel{¯}{a}}{10}$ (10)

3.3. 双边匹配理论

4. 梯形直觉模糊偏好下的双边匹配决策

4.1. 梯形直觉模糊双边匹配问题

4.2. 建立梯形直觉模糊双边匹配模型

$\left(\text{M-1}\right)\left\{\begin{array}{l}\mathrm{max}\text{ }{D}_{{\tau }_{i}}=\underset{j=1}{\overset{n}{\sum }}〈\left({\underset{_}{a}}_{ij},{{a}^{\prime }}_{ij},{{a}^{″}}_{ij},{\stackrel{¯}{a}}_{ij}\right);{\omega }_{{\stackrel{˜}{a}}_{ij}},{u}_{{\stackrel{˜}{a}}_{ij}}〉{\rho }_{ij},i\in M\\ \mathrm{max}\text{ }{D}_{{\epsilon }_{j}}=\underset{i=1}{\overset{m}{\sum }}〈\left({\underset{_}{b}}_{ij},{{b}^{\prime }}_{ij},{{b}^{″}}_{ij},{\stackrel{¯}{b}}_{ij}\right);{\omega }_{{\stackrel{˜}{b}}_{ij}},{u}_{{\stackrel{˜}{b}}_{ij}}〉{\rho }_{ij},j\in N\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{ }\underset{j=1}{\overset{n}{\sum }}{\rho }_{ij}=1,i\in M;\underset{i=1}{\overset{m}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

4.3. 求解梯形直觉模糊双边匹配模型

$\left(\text{M-2}\right)\left\{\begin{array}{l}\mathrm{max}\text{ }{D}_{\tau }=\underset{i=1}{\overset{m}{\sum }}\underset{j=1}{\overset{n}{\sum }}〈\left({\underset{_}{a}}_{ij},{{a}^{\prime }}_{ij},{{a}^{″}}_{ij},{\stackrel{¯}{a}}_{ij}\right);{\omega }_{{\stackrel{˜}{a}}_{ij}},{u}_{{\stackrel{˜}{a}}_{ij}}〉{\rho }_{ij}\\ \mathrm{max}\text{ }{D}_{\epsilon }=\underset{i=1}{\overset{m}{\sum }}\underset{j=1}{\overset{n}{\sum }}〈\left({\underset{_}{b}}_{ij},{{b}^{\prime }}_{ij},{{b}^{″}}_{ij},{\stackrel{¯}{b}}_{ij}\right);{\omega }_{{\stackrel{˜}{b}}_{ij}},{u}_{{\stackrel{˜}{b}}_{ij}}〉{\rho }_{ij}\\ \text{s}\text{.t}\text{.}\text{ }\underset{j=1}{\overset{n}{\sum }}{\rho }_{ij}=1,i\in M;\underset{i=1}{\overset{m}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

$\left(\text{M-3}\right)\left\{\begin{array}{l}\mathrm{max}D=\underset{i=1}{\overset{m}{\sum }}\underset{j=1}{\overset{n}{\sum }}{\stackrel{˜}{c}}_{ij}{\rho }_{ij}\\ \text{s}\text{.t}\text{.}\text{ }\underset{j=1}{\overset{n}{\sum }}{\rho }_{ij}=1,i\in M;\underset{i=1}{\overset{m}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

$\left(\text{M-4}\right)\left\{\begin{array}{l}\mathrm{max}D=\underset{i=1}{\overset{m}{\sum }}\underset{j=1}{\overset{n}{\sum }}{\stackrel{⌢}{d}}_{ij}{\rho }_{ij}\\ \text{s}\text{.t}\text{.}\text{ }\underset{j=1}{\overset{n}{\sum }}{\rho }_{ij}=1,i\in M;\underset{i=1}{\overset{m}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

$\left(\text{M-5}\right)\left\{\begin{array}{l}\mathrm{max}\text{ }D=\underset{i=1}{\overset{m}{\sum }}\underset{j=1}{\overset{n}{\sum }}{f}_{ij}{\rho }_{ij}\\ \text{s}\text{.t}\text{.}\text{ }\underset{j=1}{\overset{n}{\sum }}{\rho }_{ij}=1,i\in M;\underset{i=1}{\overset{m}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

4.4. 梯形直觉模糊双边匹配决策的具体操作

5. 实例分析

$\stackrel{˜}{A}={\left[〈\left({\underset{_}{a}}_{ij},{{a}^{\prime }}_{ij},{{a}^{″}}_{ij},{\stackrel{¯}{a}}_{ij}\right);{\omega }_{{\stackrel{˜}{a}}_{ij}},{u}_{{\stackrel{˜}{a}}_{ij}}〉\right]}_{4×6}$ ；而另一方普惠金融公司( $\epsilon$ 方)的综合评价针对的是其信誉级别、管理

$\stackrel{˜}{B}={\left[〈\left({\underset{_}{b}}_{ij}\text{,}{{b}^{\prime }}_{ij}\text{,}{{b}^{″}}_{ij},{\stackrel{¯}{b}}_{ij}\right);{\omega }_{{\stackrel{˜}{b}}_{ij}},{u}_{{\stackrel{˜}{b}}_{ij}}〉\right]}_{4×6}$ ；其中，满意度从低到高用1-10分来表示。最终的决策过程由该咨询投资

Table 1. Trapezoidal intuitionistic fuzzy number matrix A ˜ = [ 〈 ( a _ i j , a ′ i j , a ″ i j , a ¯ i j ) ; ω a ˜ i j , u a ˜ i j 〉 ] 4 × 6

Table 2. Trapezoidal intuitionistic fuzzy number matrix B ˜ = [ 〈 ( b _ i j , b ′ i j , b ″ i j , b ¯ i j ) ; ω b ˜ i j , u b ˜ i j 〉 ] 4 × 6

$\left(\text{M-1}\right)\left\{\begin{array}{l}\mathrm{max}\text{ }{D}_{{\tau }_{i}}=\underset{j=1}{\overset{6}{\sum }}〈\left({\underset{_}{a}}_{ij},{{a}^{\prime }}_{ij},{{a}^{″}}_{ij},{\stackrel{¯}{a}}_{ij}\right);{\omega }_{{\stackrel{˜}{a}}_{ij}},{u}_{{\stackrel{˜}{a}}_{ij}}〉{\rho }_{ij},i\in M\\ \mathrm{max}\text{ }{D}_{{\epsilon }_{j}}=\underset{i=1}{\overset{4}{\sum }}〈\left({\underset{_}{b}}_{ij},{{b}^{\prime }}_{ij},{{b}^{″}}_{ij},{\stackrel{¯}{b}}_{ij}\right);{\omega }_{{\stackrel{˜}{a}}_{ij}},{u}_{{\stackrel{˜}{a}}_{ij}}〉{\rho }_{ij},j\in N\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{ }\underset{j=1}{\overset{6}{\sum }}{\rho }_{ij}=1,i\in M;\underset{i=1}{\overset{4}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

$\left(\text{M-3}\right)\left\{\begin{array}{l}\mathrm{max}\text{ }D=\underset{i=1}{\overset{4}{\sum }}\underset{j=1}{\overset{6}{\sum }}{\stackrel{˜}{c}}_{ij}{\rho }_{ij}\\ \text{s}\text{.t}\text{.}\text{ }\underset{j=1}{\overset{6}{\sum }}{\rho }_{ij}=1,i\in M;\underset{i=1}{\overset{4}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

Table 3. Trapezoidal intuitionistic fuzzy number coefficient matrix C ˜ = [ c ˜ i j ] 4 × 6

$\left(\text{M-4}\right)\left\{\begin{array}{l}\mathrm{max}\text{ }D=\underset{i=1}{\overset{4}{\sum }}\underset{j=1}{\overset{6}{\sum }}\left({\underset{_}{c}}_{ij},{{c}^{\prime }}_{ij},{{c}^{″}}_{ij},{\stackrel{¯}{c}}_{ij}\right)\left({\omega }_{{\stackrel{↔}{c}}_{ij}}-{u}_{{\stackrel{↔}{c}}_{ij}}\right){\rho }_{ij}\\ \text{s}\text{.t}\text{.}\text{ }\underset{i=1}{\overset{4}{\sum }}{\rho }_{ij}=1,i\in M;\underset{j=1}{\overset{6}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

Table 4. Trapezoidal fuzzy number coefficient matrix D ⌢ = [ d ⌢ i j ] 4 × 6

$\left(\text{M-5}\right)\left\{\begin{array}{l}\mathrm{max}\text{ }D=\underset{i=1}{\overset{4}{\sum }}\underset{j=1}{\overset{6}{\sum }}{f}_{ij}{\rho }_{ij}\\ \text{s}\text{.t}\text{.}\text{ }\underset{j=1}{\overset{n}{\sum }}{\rho }_{ij}=1,i\in M;\underset{i=1}{\overset{4}{\sum }}{\rho }_{ij}\le 1,j\in N;{\rho }_{ij}\in \left\{0,1\right\},i\in M,j\in N\end{array}$

Table 5. Coefficient matrix F = [ f i j ] 4 × 6

Table 6. Optimal bilateral matching matrix Θ * = [ ρ i j * ] 4 × 6

6. 结论

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