#### 期刊菜单

Bilateral Matching Decision Based on Picture Fuzzy Sets
DOI: 10.12677/MSE.2022.114092, PDF, HTML, XML, 下载: 35  浏览: 85  国家自然科学基金支持

Abstract: Aiming at the problem of bilateral matching decision under fuzzy preference information of pictures, a bilateral matching decision method is presented. Firstly, TOPSIS method is used to transform fuzzy information of pictures into satisfaction. On this basis, considering one-to-one matching constraints, a bilateral matching decision model is constructed. Furthermore, according to the combined satisfaction matching method, the multi-objective planning model is transformed into a single-objective planning model, and the best bilateral matching scheme is obtained by solving this model. Finally, an example of product development requirements matching verifies the effectiveness and reliability of the proposed method.

1. 引言

2. 文献综述

3. 预备知识

3.1. 图片模糊集

$d\left({t}_{1},{t}_{2}\right)=\frac{1}{3}\left(|{\mu }_{1}-{\mu }_{2}|+|{\nu }_{1}-{\nu }_{2}|+|{ο}_{1}-{ο}_{2}|\right)$ (1)

3.2. 双边匹配

1) $\delta \left({H}_{a}\right)\in J$

2) $\delta \left({J}_{b}\right)\in H\cup \left\{{J}_{b}\right\}$

3) $\delta \left({J}_{b}\right)={H}_{a}$ 当且仅当 $\delta \left({H}_{a}\right)={J}_{b}$

$\delta$ 被称作双边匹配。在双边匹配 $\delta$ 中， $\delta \left({J}_{b}\right)={H}_{a}$$\left({H}_{a},{J}_{b}\right)$ 表示 ${H}_{a}$${J}_{b}$$\delta$ 中成功匹配， $\delta \left({J}_{b}\right)={J}_{b}$$\delta \left({H}_{a}\right)={H}_{a}$ 表示 ${H}_{a}$${J}_{b}$$\delta$ 中未能成功匹配。

4. 双边匹配模型

4.1. 问题描述

4.2. 主体满意度计算

${t}_{ab}^{+}=〈1,0,0〉$ (2)

${t}_{ab}^{-}=〈0,0,1〉$ (3)

$d\left({t}_{ab}^{H\to J},{t}_{ab}^{+}\right)=\frac{1}{3}\left(|{\mu }_{ab}^{H\to J}-1|+|{\nu }_{ab}^{H\to J}-0|+|{ο}_{ab}^{H\to J}-0|\right)$ (4)

$d\left({t}_{ab}^{H\to J},{t}_{ab}^{-}\right)=\frac{1}{3}\left(|{\mu }_{ab}^{H\to J}-0|+|{\nu }_{ab}^{H\to J}-0|+|{ο}_{ab}^{H\to J}-1|\right)$ (5)

${\eta }_{ab}^{H\to J}=\frac{d\left({t}_{ab}^{H\to J},{t}_{ab}^{-}\right)}{d\left({t}_{ab}^{H\to J},{t}_{ab}^{+}\right)+d\left({t}_{ab}^{H\to J},{t}_{ab}^{-}\right)}$ (6)

$d\left({t}_{ab}^{J\to H},{t}_{ab}^{+}\right)=\frac{1}{3}\left(|{\mu }_{ab}^{J\to H}-1|+|{\nu }_{ab}^{J\to H}-0|+|{ο}_{ab}^{J\to H}-0|\right)$ (7)

$d\left({t}_{ab}^{J\to H},{t}_{ab}^{-}\right)=\frac{1}{3}\left(|{\mu }_{ab}^{J\to H}-0|+|{\nu }_{ab}^{J\to H}-0|+|{ο}_{ab}^{J\to H}-1|\right)$ (8)

${\eta }_{ab}^{H\to J}=\frac{d\left({t}_{ab}^{H\to J},{t}_{ab}^{-}\right)}{d\left({t}_{ab}^{H\to J},{t}_{ab}^{+}\right)+d\left({t}_{ab}^{H\to J},{t}_{ab}^{-}\right)}$ (9)

4.3. 模型构建与求解

$\text{(M-1)}\left\{\begin{array}{l}\text{Max}{Z}_{1}=\underset{a=1}{\overset{m}{\sum }}\underset{b=1}{\overset{n}{\sum }}{\eta }_{ab}^{H\to J}{x}_{ab}\\ \text{Max}{Z}_{2}=\underset{a=1}{\overset{m}{\sum }}\underset{b=1}{\overset{n}{\sum }}{\eta }_{ab}^{J\to H}{x}_{ab}\\ \text{s}\text{.t}\text{.}\underset{a=1}{\overset{m}{\sum }}{x}_{ab}\le 1,\text{}b=1,2,\cdots ,n\\ \text{}\underset{b=1}{\overset{n}{\sum }}{x}_{ab}\le 1,\text{}a=1,2,\cdots ,m\\ {x}_{ab}=\left\{0,1\right\},a=1,2,\cdots ,m;b=1,2,\cdots ,n\end{array}$

$\text{(M-2)}\left\{\begin{array}{l}\text{Max}{Z}_{1}=\underset{a=1}{\overset{m}{\sum }}\underset{b=1}{\overset{n}{\sum }}\left[\kappa \left(\frac{{\eta }_{ab}^{H\to J}+{\eta }_{ab}^{J\to H}}{2}\right)+\left(1-\kappa \right)\sqrt{{\eta }_{ab}^{H\to J}×{\eta }_{ab}^{J\to H}}\right]{x}_{ab}\\ \text{s}\text{.t}\text{.}\underset{a=1}{\overset{m}{\sum }}{x}_{ab}\le 1,\text{}b=1,2,\cdots ,n\\ \text{}\underset{b=1}{\overset{n}{\sum }}{x}_{ab}\le 1,\text{}a=1,2,\cdots ,m\\ {x}_{ab}=\left\{0,1\right\},a=1,2,\cdots ,m;b=1,2,\cdots ,n\end{array}$

4.4. 双边匹配方法求解途径

5. 算例

Table 1. Picture fuzzy number matrix Φ = [ t a b H → J ] m × n

Table 2. Picture fuzzy number matrix Γ = [ t a b J → H ] m × n

Table 3. Satisfaction matrix σ = [ η a b H → J ] m × n

Table 4. Satisfaction matrix ς = [ η a b J → H ] m × n

$\text{(M-1)}\left\{\begin{array}{l}\text{Max}{Z}_{1}=\underset{a=1}{\overset{m}{\sum }}\underset{b=1}{\overset{n}{\sum }}{\eta }_{ab}^{H\to J}{x}_{ab}\\ \text{Max}{Z}_{2}=\underset{a=1}{\overset{m}{\sum }}\underset{b=1}{\overset{n}{\sum }}{\eta }_{ab}^{J\to H}{x}_{ab}\\ \text{s}\text{.t}\text{.}\underset{a=1}{\overset{m}{\sum }}{x}_{ab}\le 1,\text{}b=1,2,\cdots ,n\\ \text{}\underset{b=1}{\overset{n}{\sum }}{x}_{ab}\le 1,\text{}a=1,2,\cdots ,m\\ {x}_{ab}=\left\{0,1\right\},a=1,2,\cdots ,m;b=1,2,\cdots ,n\end{array}$

$\text{(M-2)}\left\{\begin{array}{l}\text{Max}{Z}_{1}=\underset{a=1}{\overset{m}{\sum }}\underset{b=1}{\overset{n}{\sum }}\left[\kappa \left(\frac{{\eta }_{ab}^{H\to J}+{\eta }_{ab}^{J\to H}}{\text{2}}\right)\right]{x}_{ab}\\ \text{s}\text{.t}\text{.}\underset{a=1}{\overset{m}{\sum }}{x}_{ab}\le 1,\text{}b=1,2,\cdots ,n\\ \text{}\underset{b=1}{\overset{n}{\sum }}{x}_{ab}\le 1,\text{}a=1,2,\cdots ,m\\ {x}_{ab}=\left\{0,1\right\},a=1,2,\cdots ,m;b=1,2,\cdots ,n\end{array}$

Table 5. Best bilateral matching scheme x a b

6. 结论

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