﻿ 基于TOPSIS的区间直觉模糊集多属性群决策方法研究综述 The Research Reviewed of the Multi-Attribute Group Decision Making Method for Interval-Valued Intuitionistic Fuzzy Sets Based on TOPSIS

Vol. 07  No. 10 ( 2018 ), Article ID: 27248 , 11 pages
10.12677/AAM.2018.710149

The Research Reviewed of the Multi-Attribute Group Decision Making Method for Interval-Valued Intuitionistic Fuzzy Sets Based on TOPSIS

Min Li, Bianping Su, Qiangqiang Zhang

College of Mathematics, College of Science, Xi’an University of Architecture and Technology, Xi’an Shaanxi

Received: Oct. 1st, 2018; accepted: Oct. 17th, 2018; published: Oct. 24th, 2018

ABSTRACT

In multiple attribute group decision making, because every expert has his own knowledge and expertise, different experts have different weights for different attributes. A new method to determine the expert’s weights is put forward based on the TOPSIS method in interval-triangular fuzzy setting. If the evaluation value is close to the positive idea evaluation value and far away from the negative ideal evaluation value, it will be given a large weight; otherwise, the evaluation value will be given a small weight. Experience shows that the weight of experts determined by this method has a significant effect on solving practical decision-making problems. A new method of multiple attribute interval-valued intuitionistic fuzzy group decision making is presented in this paper, including the attribute weights which are completely known, partly known and completely unknown. Finally, the feasibility and validity of this method are proved by our examples.

Keywords:Multiple Attribute Group Decision Making, Interval-Valued Intuitioniastic Fuzzy Set, Weight, TOPSI

1. 引言

2. 区间直觉模糊集多属性决策方法研究现状

3. 区间直觉模糊集多属性群决策方法研究现状

Wei定义了区间直觉模糊数的得分函数和精确函数和诱导区间直觉模糊有序加权几何(I-IIFOWG)算子，同时给出了该算子的一些重要性质：单调性、幂等性和交换性，进一步提出了直觉模糊多属性决策方法，并且将结果扩展到了区间直觉模糊集群决策方法中 [21]。Wei and Wang定义了一些几何集结算子：区间直觉模糊有序加权几何(IIFOWG)算子和区间直觉模糊混合几何(IIFHG)算子，提出了一种属性权重完全已知的区间直觉模糊集多属性群决策方法 [22]。Liu提出了区间直觉模糊Hamacher混合加权平均(IVIFHHWA)算子、区间直觉模糊Hamacher几何加权平均(IVIFGWA)算子等，并进一步给出了基于各算子的区间直觉模糊集群决策方法 [23]。

4. 多属性群决策问题中确定专家权重方法的研究

5. 基于TOPSIS的确定专家权重的新方法

Hwang和Yoon于1981年提出了TOPSIS方法 [54] ，并且被广泛地研究与应用，在TOPSIS方法中，一个理想方案的选择应该在接近正理想方案的同时远离负理想方案。不少文献已对基于TOPSIS的区间直觉模糊集群决策方法进行了研究 [55] [56] ，如：Park et al.于2011年在区间直觉模糊信息的条件下，用TOPSIS方法解决了属性权重部分已知情形下的多属性群决策方法问题。Tan于2011年通过用基于海明距离的Choquet积分提出了一种基于TOPSIS的区间直觉模糊集群决策方法。Wei Yang，Zhiping Chen，Fang Zhang提出了一种用TOPSIS [43] [57] 确定专家权重的新方法，不过其中决策者给出的评价值是以直觉模糊数的形式给出的，这样对于处理决策过程中的模糊性和不确定性有一定的弊端。基于此，我们对上述方法进行了改进，在决策者的评价值是以区间直觉模糊数给出的情况下，给出了一种用TOPSIS确定专家权重的一种新方法。该方法更有利于考虑到决策过程中的模糊性和不确定性，而且适用于方案集和属性集非常大的情况。为了将不同的决策者给出的评价值集结为一个，首先定义了区间直觉模糊正，负理想矩阵，然后计算每个评价值到区间直觉模糊正，负理想矩阵之间的距离，接着通过计算得到了每个评价值的贴近度，进一步求得了专家权重。通过该方法确定的权重有两个优势：第一如果评价值接近正理想点并且同时远离负理想点就会被赋予一个较高的权重；否则，评价值就会被赋予一个小的权重；第二，降低了过高或过低的评价值对排序结果的影响。并且考虑了不同属性权重信息的情况：属性权重完全已知，部分已知和完全未知。如果属性权重完全已知，在将不同的决策者给出的评价值集结为一个之后，用TOPSIS方法对决策结果进行排序；如果属性权重部分已知，通过求解一个线性规划模型来求得属性权重；如果属性权重完全未知，我同样用TOPSIS方法得到属性权重。并且提出了一种与之相一致的算法。

6. 基于新权重方法的区间直觉模糊集多属性群决策方法

${D}^{\left(k\right)}={\left({\stackrel{˜}{\alpha }}_{ij}^{\left(k\right)}\right)}_{m×n}=\left[\begin{array}{cccc}{\stackrel{˜}{\alpha }}_{11}^{\left(k\right)}& {\stackrel{˜}{\alpha }}_{12}^{\left(k\right)}& \cdots & {\stackrel{˜}{\alpha }}_{1n}^{\left(k\right)}\\ {\stackrel{˜}{\alpha }}_{21}^{\left(k\right)}& {\stackrel{˜}{\alpha }}_{22}^{\left(k\right)}& \cdots & {\stackrel{˜}{\alpha }}_{2n}^{\left(k\right)}\\ ⋮& ⋮& \ddots & ⋮\\ {\stackrel{˜}{\alpha }}_{n1}^{\left(k\right)}& {\stackrel{˜}{\alpha }}_{n2}^{\left(k\right)}& \cdots & {\stackrel{˜}{\alpha }}_{nn}^{\left(k\right)}\end{array}\right]$ (1)

${\stackrel{˜}{D}}^{+}={\left({\stackrel{˜}{\alpha }}_{ij}^{+}\right)}_{m×n}=\left(\begin{array}{cccc}{\stackrel{˜}{\alpha }}_{11}^{+}& {\stackrel{˜}{\alpha }}_{12}^{+}& \cdots & {\stackrel{˜}{\alpha }}_{1n}^{+}\\ {\stackrel{˜}{\alpha }}_{21}^{+}& {\stackrel{˜}{\alpha }}_{22}^{+}& \cdots & {\stackrel{˜}{\alpha }}_{2n}^{+}\\ ⋮& ⋮& \ddots & ⋮\\ {\stackrel{˜}{\alpha }}_{n1}^{+}& {\stackrel{˜}{\alpha }}_{n2}^{+}& \cdots & {\stackrel{˜}{\alpha }}_{nn}^{+}\end{array}\right)$ (2)

${D}_{d}^{-}={\left({\stackrel{˜}{\alpha }}_{ij}^{d}\right)}_{m×n}=\left(\begin{array}{cccc}{\stackrel{˜}{\alpha }}_{11}^{d}& {\stackrel{˜}{\alpha }}_{12}^{d}& \cdots & {\stackrel{˜}{\alpha }}_{1n}^{d}\\ {\stackrel{˜}{\alpha }}_{21}^{d}& {\stackrel{˜}{\alpha }}_{22}^{d}& \cdots & {\stackrel{⌢}{\alpha }}_{2n}^{d}\\ ⋮& ⋮& \ddots & ⋮\\ {\stackrel{˜}{\alpha }}_{n1}^{d}& {\stackrel{˜}{\alpha }}_{n2}^{d}& \cdots & {\stackrel{˜}{\alpha }}_{nn}^{d}\end{array}\right)$ (3)

${D}_{u}^{-}={\left({\stackrel{˜}{\alpha }}_{ij}^{u}\right)}_{m×n}=\left(\begin{array}{cccc}{\stackrel{˜}{\alpha }}_{11}^{u}& {\stackrel{˜}{\alpha }}_{12}^{u}& \cdots & {\stackrel{˜}{\alpha }}_{1n}^{u}\\ {\stackrel{˜}{\alpha }}_{21}^{u}& {\stackrel{˜}{\alpha }}_{22}^{u}& \cdots & {\stackrel{˜}{\alpha }}_{2n}^{u}\\ ⋮& ⋮& \ddots & ⋮\\ {\stackrel{˜}{\alpha }}_{n1}^{u}& {\stackrel{˜}{\alpha }}_{n2}^{u}& \cdots & {\stackrel{˜}{\alpha }}_{nn}^{u}\end{array}\right)$ (4)

${d}_{ij}^{+}=\frac{1}{4}\left(|{a}_{ij}^{\left(k\right)}-{a}_{ij}^{+}|+|{b}_{ij}^{\left(k\right)}-{b}_{ij}^{+}|+|{c}_{ij}^{\left(k\right)}-{c}_{ij}^{+}|+|{d}_{ij}^{\left(k\right)}-{d}_{ij}^{+}|\right)$ (5)

${d}_{ij}^{d}=\frac{1}{4}\left(|{a}_{ij}^{\left(k\right)}-{a}_{ij}^{d}|+|{b}_{ij}^{\left(k\right)}-{b}_{ij}^{d}|+|{c}_{ij}^{\left(k\right)}-{c}_{ij}^{d}|+|{d}_{ij}^{\left(k\right)}-{d}_{ij}^{d}|\right)$ (6)

${d}_{ij}^{u}=\frac{1}{4}\left(|{a}_{ij}^{\left(k\right)}-{a}_{ij}^{u}|+|{b}_{ij}^{\left(k\right)}-{b}_{ij}^{u}|+|{c}_{ij}^{\left(k\right)}-{c}_{ij}^{u}|+|{d}_{ij}^{\left(k\right)}-{d}_{ij}^{u}|\right)$ (7)

${c}_{ij}^{\left(k\right)}=\frac{{d}_{ij}^{u}+{d}_{ij}^{d}}{{d}_{ij}^{u}+{d}_{ij}^{d}+{d}_{ij}^{+}}$ $i=1,2,\cdots ,m,\text{\hspace{0.17em}}j=1,2,\cdots ,n,\text{\hspace{0.17em}}k=1,2,\cdots ,t$ (8)

${\omega }_{ij}^{\left(k\right)}=\frac{{c}_{ij}^{\left(k\right)}}{{\sum }_{k=1}^{t}{c}_{ij}^{\left(k\right)}}$$i=1,2,\cdots ,m,\text{\hspace{0.17em}}j=1,2,\cdots ,n,\text{\hspace{0.17em}}k=1,2,\cdots ,t$ (9)

${\stackrel{˜}{\alpha }}_{ij}={\omega }_{ij}^{\left(1\right)}{\stackrel{˜}{\alpha }}_{ij}^{\left(1\right)}+{\omega }_{ij}^{\left(2\right)}{\stackrel{˜}{\alpha }}_{ij}^{\left(2\right)}+\cdots +{\omega }_{ij}^{\left(t\right)}{\stackrel{˜}{\alpha }}_{ij}^{\left(t\right)}$ (10)

$D={\left({\stackrel{˜}{\alpha }}_{ij}\right)}_{m×n}=\left(\begin{array}{cccc}{\stackrel{˜}{\alpha }}_{11}& {\stackrel{˜}{\alpha }}_{12}& \cdots & {\stackrel{˜}{\alpha }}_{1n}\\ {\stackrel{˜}{\alpha }}_{21}& {\stackrel{˜}{\alpha }}_{22}& \cdots & {\stackrel{˜}{\alpha }}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\stackrel{˜}{\alpha }}_{n1}& {\stackrel{˜}{\alpha }}_{n2}& \cdots & {\stackrel{˜}{\alpha }}_{nn}\end{array}\right)$ (11)

(1) $\left\{{\omega }_{i}\ge {\omega }_{j}\right\},i\ne j$

(2) $\left\{{\omega }_{i}-{\omega }_{j}\ge {\epsilon }_{i}\left(>0\right)\right\},i\ne j$

(3) $\left\{{\omega }_{i}\ge {\alpha }_{i}{\omega }_{i}\right\},0\le {\alpha }_{i}\le 1,i\ne j$

(4) $\left\{{\beta }_{i}\le {\omega }_{j}\le {\beta }_{j}+{\epsilon }_{j}\right\},0\le {\beta }_{j}\le {\beta }_{j}+{\epsilon }_{j}\le 1$

(5) $\left\{\omega i-\omega j\ge \omega k-\omega l\right\},i\ne j\ne k\ne l$

${\stackrel{˜}{A}}^{\text{+}}\text{=}\left({\stackrel{˜}{\alpha }}_{1}^{\text{+}},{\stackrel{˜}{\alpha }}_{2}^{\text{+}},\cdots ,{\stackrel{˜}{\alpha }}_{n}^{\text{+}}\right)\text{=}\left(\underset{i}{\mathrm{max}}{\stackrel{˜}{\alpha }}_{i1},\underset{i}{\mathrm{max}}\stackrel{˜}{\alpha }i2,\cdots ,\underset{i}{\mathrm{max}}\stackrel{˜}{\alpha }in\right)$ (12)

${\stackrel{˜}{A}}^{\text{-}}\text{=}\left({\stackrel{˜}{\alpha }}_{1}^{\text{-}},{\stackrel{˜}{\alpha }}_{2}^{\text{-}},\cdots ,{\stackrel{˜}{\alpha }}_{n}^{\text{-}}\right)\text{=}\left(\underset{i}{\mathrm{min}}{\stackrel{˜}{\alpha }}_{i1},\underset{i}{\mathrm{min}}\stackrel{˜}{\alpha }i2,\cdots ,\underset{i}{\mathrm{min}}\stackrel{˜}{\alpha }in\right)$ (13)

${c}_{i}{}_{j}=\frac{d\left({\stackrel{˜}{\alpha }}_{ij},{\stackrel{˜}{\alpha }}_{j}^{-}\right)}{d\left({\stackrel{˜}{\alpha }}_{ij},{\stackrel{˜}{\alpha }}_{j}^{+}\right)+d\left({\stackrel{˜}{\alpha }}_{ij},{\stackrel{˜}{\alpha }}_{j}^{-}\right)},\text{\hspace{0.17em}}i=1,2,\cdots ,m,\text{\hspace{0.17em}}j=1,2,\cdots ,n.$ (14)

${c}_{i}=\sum _{j=1}^{n}{c}_{ij}{\omega }_{j},i=1,2,\cdots ,m.$ (15)

$\left(M-1\right)\text{ }\mathrm{max}\left\{\sum _{j=1}^{n}{c}_{1j}{\omega }_{j},\sum _{j=1}^{n}{c}_{2j}{\omega }_{j},\cdots ,\sum _{j=1}^{n}{c}_{nj},{\omega }_{j}\right\}$

$s.t\text{ }\text{\hspace{0.17em}}W\in H,$

${\omega }_{j}\ge 0,j=1,2,\cdots ,n,$

${\omega }_{1}\text{+}{\omega }_{2}\text{+}\cdots +{\omega }_{n}\text{=}1.$

$s.t\text{ }\text{\hspace{0.17em}}\text{ }W\in H,$

$\omega j\ge 0,j=1,2,\dots ,n,$

${\omega }_{1}\text{+}\omega {}_{2}\text{+}\cdots +{\omega }_{n}\text{=}1.$

${\omega }_{j}=\frac{{c}_{j}}{{\sum }_{j=1}^{n}{c}_{j}}=\frac{{\sum }_{i=1}^{m}{c}_{ij}}{\sum _{j=1}^{n}\sum _{i=m}^{m}{c}_{ij}},j=1,\cdots ,n,$ (16)

${\omega }_{j}\ge 0,{\sum }_{j=1}^{n}{w}_{j}=1.$

${d}_{i}^{+}=\sum _{j=1}^{n}d\left({\stackrel{˜}{\alpha }}_{ij}^{\text{'}},{\alpha }_{j}^{+}\right),\text{\hspace{0.17em}}i=1,2,\cdots ,m$ (17)

${d}_{i}^{-}=\sum _{j=1}^{n}d\left({\stackrel{˜}{\alpha }}_{ij}^{\text{'}},{\alpha }_{j}^{-}\right),\text{\hspace{0.17em}}i=1,2,\cdots ,m$ (18)

${c}_{i}=\frac{{d}_{i}^{-}}{{d}_{i}^{-}+{d}_{i}^{+}},i=1,2,\cdots ,m$ (19)

7. 结论与展望

1) 在区间直觉模糊集多属性群决策方法中，关于属性权重的确定，虽然不同的学者都提出了较为客观的方法，但是，这些方法不能体现决策者的主观偏好。在一些决策问题中，由于决策者对于方案集的选择存在一定的主观偏好值，这些偏好信息往往会改变决策结果。因此，对于决策者对方案集有偏好的区间直觉模糊集多属性群决策方法的研究是一个值得努力的方向。

2) 由于区间直觉梯形模糊数、区间三角模糊数和三角模糊数更能刻画客观世界的模糊性和不确定性。因此，对于其多属性群决策的方法值得进一步研究。可以把基于TOPSIS的多属性群决策方法拓展到区间梯形模糊集、区间三角模糊集和三角模糊集多属性群决策问题中，这将会是一个不错的研究方向。

The Research Reviewed of the Multi-Attribute Group Decision Making Method for Interval-Valued Intuitionistic Fuzzy Sets Based on TOPSIS[J]. 应用数学进展, 2018, 07(10): 1278-1288. https://doi.org/10.12677/AAM.2018.710149

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