﻿ 基于TOPSIS的语言真值直觉模糊多属性决策
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 智能系统学报  2017, Vol. 12 Issue (4): 504-510  DOI: 10.11992/tis.201608008 0

### 引用本文

XU Yingying, ZOU Li, HUANG Zhixin, et al. Linguistic truth-valued intuitionistic fuzzy multi-attribute decision making based on TOPSIS[J]. CAAI Transactions on Intelligent Systems, 2017, 12(4), 504-510. DOI: 10.11992/tis.201608008.

### 文章历史

1. 辽宁师范大学 计算机与信息技术学院, 辽宁 大连 116081;
2. 辽宁师范大学 数学学院, 辽宁 大连 116081

Linguistic truth-valued intuitionistic fuzzy multi-attribute decision making based on TOPSIS
XU Yingying1, ZOU Li1, HUANG Zhixin2, PAN Chang1
1. School of Computer and Information Technology, Liaoning Normal University, Dalian 116081, China;
2. School of Mathematics, Liaoning Normal University, Dalian 116081, China
Abstract: For multi-attribute decision making problems with fuzzy linguistic-valued information, in this paper, we propose a linguistic truth-valued intuitionistic fuzzy multi-attribute decision making approach based on the technique for order performance by similarity to ideal solution (TOPSIS), in combination with the traditional TOPSIS approach. On the basis of linguistic truth-valued intuitionistic fuzzy algebra, in our approach, we used linguistic truth-valued intuitionistic fuzzy pairs to express fuzzy linguistic-valued information that is both comparable and incomparable. We define the normalized distance algorithm for linguistic truth-valued intuitionistic fuzzy pairs and discuss its related properties. We propose linguistic truth-valued intuitionistic fuzzy positive and negative ideal points by calculating the distances between the attribute values of every scheme with positive and negative ideal points to obtain their relative degree of closeness. From the ranking result of the relative degree of closeness, we can determine the best scheme. We give an example to illustrate the reasonability and effectiveness of our proposed decision-making approach.
Key words: TOPSIS    linguistic truth-valued intuitionistic fuzzy pairs    normalized distance    ideal point    multi-attribute decision making

1 预备知识

1)((hi, t), (hj, f)) ≥ ((hk, t), (hl, f))，当且仅当ikjl

2)((hi, t), (hj, f))与((hk, t), (hl, f))不可比，当且仅当ikjl，或ikjl

1)((hi, t), (hj, f))∪((hk, t), (hl, f))=((hmax(i, k), t), (hmax(j, l), f));

2)((hi, t), (hj, f))∩((hk, t), (hl, f))=((hmin(i, k), t), (hmin(j, l), f))。

 图 1 十元语言真值直觉模糊格的哈斯图 Fig.1 Hasse diagram of 10-element linguistic truth-valued intuitionistic fuzzy lattice
2 语言真值直觉模糊对之间的归一化距离及正、负理想点

 $\begin{array}{*{20}{c}} {{d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_i},t} \right),\left( {{h_j},f} \right)} \right),\left( {\left( {{h_k},t} \right),\left( {{h_l},f} \right)} \right)} \right) = }\\ {\frac{{\left| {i - k} \right| + \left| {j - l} \right|}}{{2n - 2}}} \end{array}$

 $\begin{array}{*{20}{c}} {{d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_1},t} \right),\left( {{h_1},f} \right)} \right),\left( {\left( {{h_3},t} \right),\left( {{h_4},f} \right)} \right)} \right) = }\\ {\frac{{\left| {1 - 3} \right| + \left| {1 - 4} \right|}}{8} = \frac{5}{8}} \end{array}$

1)0≤dLIF(((hi, t), (hj, f)), ((hk, t), (hl, f)))≤1;

2)dLIF(((hi, t), (hj, f)), ((hk, t), (hl, f)))=0当且仅当((hi, t), (hj, f))=((hk, t), (hl, f));

3)dLIF(((hi, t), (hj, f)), ((hk, t), (hl, f)))=dLIF(((hk, t), (hl, f)), ((hi, t), (hj, f)));

4) 若((hi, t), (hj, f))≥((hk, t), (hl, f))≥((hp, t), (hq, f))，则有dLIF(((hi, t), (hj, f)), ((hp, t), (hq, f)))≥dLIF(((hi, t), (hj, f)), ((hk, t), (hl, f)))且dLIF(((hi, t), (hj, f)), ((hp, t), (hq, f)))≥dLIF(((hk, t), (hl, f)), ((hp, t), (hq, f)))。

 $\begin{array}{*{20}{c}} {{d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_i},t} \right),\left( {{h_j},f} \right)} \right),\left( {\left( {{h_i},t} \right),\left( {{h_j},f} \right)} \right)} \right) = }\\ {\frac{{\left| {i - i} \right| + \left| {j - j} \right|}}{{2n - 2}} = 0} \end{array}$

 $\begin{array}{*{20}{c}} {{d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_n},t} \right),\left( {{h_n},f} \right)} \right),\left( {\left( {{h_1},t} \right),\left( {{h_1},f} \right)} \right)} \right) = }\\ {\frac{{\left| {n - 1} \right| + \left| {n - 1} \right|}}{{2n - 2}} = \frac{{2n - 2}}{{2n - 2}} = 1} \end{array}$

2) 对任意((hi, t), (hj, f)), ((hk, t), (hl, f))∈LI2n，由定义3可得，两个语言真值直觉模糊对之间的归一化距离为

 $\begin{array}{*{20}{c}} {{d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_i},t} \right),\left( {{h_j},f} \right)} \right),\left( {\left( {{h_k},t} \right),\left( {{h_l},f} \right)} \right)} \right) = }\\ {\frac{{\left| {i - k} \right| + \left| {j - l} \right|}}{{2n - 2}}} \end{array}$

dLIF(((hi, t), (hj, f)), ((hk, t), (hl, f)))=0，则i=kj=l，即((hi, t), (hj, f))=((hk, t), (hl, f))。

dLIF(((hi, t), (hj, f)), ((hk, t), (hl, f)))=dLIF(((hi, t), (hj, f)), ((hi, t), (hj, f)))=$\frac{{\left| {i - i} \right| + \left| {j - j} \right|}}{{2n - 2}} = 0$

3) 根据定义3可知：

 $\begin{array}{l} {d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_i},t} \right),\left( {{h_j},f} \right)} \right),\left( {\left( {{h_k},t} \right),\left( {{h_l},f} \right)} \right)} \right) = \\ \frac{{\left| {i - k} \right| + \left| {j - l} \right|}}{{2n - 2}}; \end{array}$
 $\begin{array}{l} {d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_k},t} \right),\left( {{h_l},f} \right)} \right),\left( {\left( {{h_i},t} \right),\left( {{h_j},f} \right)} \right)} \right) = \\ \frac{{\left| {k - i} \right| + \left| {l - j} \right|}}{{2n - 2}}; \end{array}$

4) 对任意(hi, t), (hj, f))，((hk, t), (hl, f))，((hp, t), (hq, f))∈LI2n，由定义3可得：

 $\begin{array}{*{20}{c}} {{d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_i},t} \right),\left( {{h_j},f} \right)} \right),\left( {\left( {{h_p},t} \right),\left( {{h_q},f} \right)} \right)} \right) = }\\ {\frac{{\left| {i - p} \right| + \left| {j - q} \right|}}{{2n - 2}} = \frac{{\left| {i + j} \right| - \left| {p + q} \right|}}{{2n - 2}}} \end{array}$
 $\begin{array}{*{20}{c}} {{d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_i},t} \right),\left( {{h_j},f} \right)} \right),\left( {\left( {{h_k},t} \right),\left( {{h_l},f} \right)} \right)} \right) = }\\ {\frac{{\left| {i - k} \right| + \left| {j - l} \right|}}{{2n - 2}} = \frac{{\left| {i + j} \right| - \left| {k + l} \right|}}{{2n - 2}}} \end{array}$
 $\begin{array}{*{20}{c}} {{d_{{\rm{LIF}}}}\left( {\left( {\left( {{h_k},t} \right),\left( {{h_l},f} \right)} \right),\left( {\left( {{h_p},t} \right),\left( {{h_q},f} \right)} \right)} \right) = }\\ {\frac{{\left| {k - p} \right| + \left| {l - q} \right|}}{{2n - 2}} = \frac{{\left| {k + l} \right| - \left| {p + q} \right|}}{{2n - 2}}} \end{array}$

 $r_j^ + = \left\{ \begin{array}{l} \bigcup\limits_i {{r_{ij}}} ,\;\;\;\;\;\;j \in {\mathit{\Omega }_1}\\ \bigcap\limits_i {{r_{ij}}} ,\;\;\;\;\;\;\;j \in {\mathit{\Omega }_2} \end{array} \right.$
 $r_j^ - = \left\{ \begin{array}{l} \bigcap\limits_i {{r_{ij}}} ,\;\;\;\;\;\;j \in {\mathit{\Omega }_1}\\ \bigcup\limits_i {{r_{ij}}} ,\;\;\;\;\;\;\;j \in {\mathit{\Omega }_2} \end{array} \right.$

3 基于TOPSIS的语言真值直觉模糊多属性决策方法

1) 对于某一多属性决策问题，设A={A1, A2, …, Am}为方案集，G={G1, G2, …, Gn}为属性集。w=[w1 w2wn]T为属性权重向量，其中，wj∈[0, 1]，$\sum\limits_{j = 1}^n {{w_j} = 1}$。设方案AiAGjG下的语言评估值(属性值)rij，并得到决策矩阵R=(rij)m×n，其中，rijSi=1, 2, …, mj=1, 2, …, nS为2n元语言真值直觉模糊对集合。

2) 利用定义4确定语言真值直觉模糊正、负理想方案：

① 语言真值直觉模糊正理想方案为

 ${A^ + } = \left( {r_1^ + ,r_2^ + , \cdots ,r_n^ + } \right)$

② 语言真值直觉模糊负理想方案为

 ${A^ - } = \left( {r_1^ - ,r_2^ - , \cdots ,r_n^ - } \right)$

3) 分别计算各方案与语言真值直觉模糊正、负理想方案之间的距离：

① 方案Ai与语言真值直觉模糊正理想方案之间的距离为

 $d\left( {{A_i},{A^ + }} \right) = \sum\limits_{j = 1}^n {{w_j}{d_{{\rm{LIF}}}}\left( {{r_{ij}},r_j^ + } \right)}$

② 方案Ai与语言真值直觉模糊负理想方案之间的距离为

 $d\left( {{A_i},{A^ - }} \right) = \sum\limits_{j = 1}^n {{w_j}{d_{{\rm{LIF}}}}\left( {{r_{ij}},r_j^ - } \right)}$

4) 计算方案Ai与理想方案之间的相对贴近度：

 $C\left( {{A_i}} \right) = \frac{{d\left( {{A_i},{A^ - }} \right)}}{{d\left( {{A_i},{A^ + }} \right) + d\left( {{A_i},{A^ - }} \right)}}$

5) 按照相对贴近度C(Ai)(i=1, 2, …, m)由大到小的顺序对方案Ai(i=1, 2, …, m)进行排序，C(Ai)值越大，则方案Ai越优。

4 案例

G1为销售能力；G2为管理能力；G3为生产能力；G4为技术能力；G5为资金能力；G6为风险承担能力。

1) 将表格1转换成语言真值直觉模糊对决策矩阵R

 $\mathit{\boldsymbol{R}} = \left[ {\begin{array}{*{20}{c}} {\left( {\left( {{h_3},t} \right),\left( {{h_4},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_3},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_5},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_4},f} \right)} \right)}&{\left( {\left( {{h_5},t} \right),\left( {{h_5},f} \right)} \right)}&{\left( {\left( {{h_2},t} \right),\left( {{h_2},f} \right)} \right)}\\ {\left( {\left( {{h_5},t} \right),\left( {{h_5},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_5},f} \right)} \right)}&{\left( {\left( {{h_2},t} \right),\left( {{h_3},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_4},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_3},f} \right)} \right)}&{\left( {\left( {{h_2},t} \right),\left( {{h_5},f} \right)} \right)}\\ {\left( {\left( {{h_4},t} \right),\left( {{h_4},f} \right)} \right)}&{\left( {\left( {{h_2},t} \right),\left( {{h_4},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_4},f} \right)} \right)}&{\left( {\left( {{h_4},t} \right),\left( {{h_5},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_3},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_3},f} \right)} \right)}\\ {\left( {\left( {{h_3},t} \right),\left( {{h_5},f} \right)} \right)}&{\left( {\left( {{h_3},t} \right),\left( {{h_3},f} \right)} \right)}&{\left( {\left( {{h_4},t} \right),\left( {{h_5},f} \right)} \right)}&{\left( {\left( {{h_1},t} \right),\left( {{h_3},f} \right)} \right)}&{\left( {\left( {{h_2},t} \right),\left( {{h_4},f} \right)} \right)}&{\left( {\left( {{h_4},t} \right),\left( {{h_5},f} \right)} \right)} \end{array}} \right]$

2) 该决策中的6个属性均为效益型属性，确定语言真值直觉模糊正、负理想点。

 ${A^ + } = \left( {r_1^ + ,r_2^ + , \cdots ,r_n^ + } \right)$

 $\begin{array}{l} r_1^ + = \left( {\left( {{h_5},t} \right),\left( {{h_5},f} \right)} \right),r_2^ + = \left( {\left( {{h_3},t} \right),\left( {{h_5},f} \right)} \right)\\ r_3^ + = \left( {\left( {{h_4},t} \right),\left( {{h_5},f} \right)} \right),r_4^ + = \left( {\left( {{h_4},t} \right),\left( {{h_5},f} \right)} \right)\\ r_5^ + = \left( {\left( {{h_5},t} \right),\left( {{h_5},f} \right)} \right),r_6^ + = \left( {\left( {{h_4},t} \right),\left( {{h_5},f} \right)} \right) \end{array}$

 ${A^ - } = \left( {r_1^ - ,r_2^ - , \cdots ,r_n^ - } \right)$

 $\begin{array}{l} r_1^ - = \left( {\left( {{h_3},t} \right),\left( {{h_4},f} \right)} \right),r_2^ - = \left( {\left( {{h_2},t} \right),\left( {{h_3},f} \right)} \right)\\ r_3^ - = \left( {\left( {{h_2},t} \right),\left( {{h_3},f} \right)} \right),r_4^ - = \left( {\left( {{h_1},t} \right),\left( {{h_3},f} \right)} \right)\\ r_5^ - = \left( {\left( {{h_2},t} \right),\left( {{h_3},f} \right)} \right),r_6^ - = \left( {\left( {{h_2},t} \right),\left( {{h_2},f} \right)} \right) \end{array}$

3) 取权重w=(0.1, 0.2, 0.15, 0.2, 0.15, 0.2)，分别计算各方案与语言真值直觉模糊正、负理想方案之间的距离：

① 方案Ai与语言真值直觉模糊正理想方案之间的距离为

 $\begin{array}{*{20}{c}} {d\left( {{A_1},{A^ + }} \right) = }\\ {\sum\limits_{j = 1}^n {{w_j}{d_{{\rm{LIF}}}}\left( {{r_{1j}},r_j^ + } \right)} = }\\ {0.1{d_{{\rm{LIF}}}}\left( {{r_{11}},r_1^ + } \right) + 0.2{d_{{\rm{LIF}}}}\left( {{r_{12}},r_2^ + } \right) + }\\ {0.15{d_{{\rm{LIF}}}}\left( {{r_{13}},r_3^ + } \right) + 0.2{d_{{\rm{LIF}}}}\left( {{r_{14}},r_4^ + } \right) + }\\ {0.15{d_{{\rm{LIF}}}}\left( {{r_{15}},r_5^ + } \right) + 0.2{d_{{\rm{LIF}}}}\left( {{r_{16}},r_6^ + } \right) = }\\ {0.1 \times \frac{3}{8} + 0.2 \times \frac{2}{8} + 0.15 \times \frac{1}{8} + 0.2 \times \frac{2}{8} + }\\ {0.15 \times 0 + 0.2 \times \frac{5}{8} = 0.281\;25} \end{array}$

② 方案Ai与语言真值直觉模糊负理想方案之间的距离为

 $\begin{array}{*{20}{c}} {d\left( {{A_1},{A^ - }} \right) = }\\ {\sum\limits_{j = 1}^n {{w_j}{d_{{\rm{LIF}}}}\left( {{r_{1j}},r_j^ - } \right)} = }\\ {0.1{d_{{\rm{LIF}}}}\left( {{r_{11}},r_1^ - } \right) + 0.2{d_{{\rm{LIF}}}}\left( {{r_{12}},r_2^ - } \right) + }\\ {0.15{d_{{\rm{LIF}}}}\left( {{r_{13}},r_3^ - } \right) + 0.2{d_{{\rm{LIF}}}}\left( {{r_{14}},r_4^ - } \right) + }\\ {0.15{d_{{\rm{LIF}}}}\left( {{r_{15}},r_5^ - } \right) + 0.2{d_{{\rm{LIF}}}}\left( {{r_{16}},r_6^ - } \right) = }\\ {0.1 \times 0 + 0.2 \times \frac{1}{8} + 0.15 \times \frac{3}{8} + 0.2 \times \frac{3}{8} + }\\ {0.15 \times \frac{5}{8} + 0.2 \times 0 = 0.25} \end{array}$

4) 计算方案Ai与理想方案的相对贴近度为

 $\begin{array}{*{20}{c}} {\left( {{A_1}} \right) = \frac{{d\left( {{A_1},{A^ - }} \right)}}{{d\left( {{A_1},{A^ + }} \right) + d\left( {{A_1},{A^ - }} \right)}} = }\\ {\frac{{0.25}}{{0.281\;25 + 0.25}} = 0.470\;6} \end{array}$

5) 根据相对贴近度C(Ai)对方案Ai进行排序：

 ${A_2} > {A_3} > {A_4} > {A_1}$

1) 取案例评估值的真值部分，且转换成三角模糊数形式表示，则决策矩阵R′及属性权重如表 4所示。

2) 对决策矩阵进行规范化并加权，得到模糊加权规范化决策矩阵V，如表 5所示。

1) 文献[16]的语言值只是从正面对属性进行评价，而本文使用的语言真值直觉模糊对形式的语言值既可以从正反两方面进行评估，也可同时处理信息的可比性与不可比性；文献[16]将语言值转换成三角模糊数形式进行运算，而本文直接对语言值进行运算。

2) 文献[16]中的模糊正、负理想点为固定值，即(1, 1, 1) 和(0, 0, 0)，而本文中的语言真值直觉模糊正、负理想点需要计算进行确定，该理想点确定方法可以同时确定可比的和不可比的语言值信息的理想点。

3) 相对于文献[16]中的三角模糊数之间的距离公式，本文提出了语言真值直觉模糊对之间的归一化距离算法，利用该距离公式得到相对贴近度，并进行决策排序，使该决策方法具有较高的实用价值。

5 结束语

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