基于双向投影法的概率语言多属性群决策

doi:10.3969/j.issn.1001-506X.2020.09.21

摘要/Abstract

摘要：

针对概率语言信息下的多属性群决策问题,首先基于概率语言术语集(probabilistic linguistic term sets, PLTS)的调整规则和语言尺度函数,定义了PLTS的新运算,并证明其满足代数运算的基本性质。然后,为解决现有向量模公式在下标对称语言下计算结果与实际不符的问题,提出一种改进的概率语言信息下的向量模公式。在此基础上定义方案与正、负理想解形成的向量表示式,并建立针对概率语言信息的双向投影法,由此提出新的贴近度公式。最后,通过应用算例的对比分析验证了本文所提方法的有效性和可行性。该方法充分考虑了PLTS的数据特征,在简化计算的同时避免了信息损失,为解决概率语言信息中数值差异较小的问题提供了新的方法和思路。

关键词: 多属性群决策, 概率语言术语集, 双向投影法

Abstract:

Aiming at the multi-attribute group decision-making problem with the probabilistic linguistic information, a new operation of the probabilistic linguistic term sets (PLTS) is defined based on the adjustment rules and linguistic scale function of the PLTS, which is proved to satisfy the basic properties of algebraic operations. Then, to solve the inconsistency between the calculation results of the existing vector module formula in subscript symmetric language and the actual situation, an improved vector module formula in the probabilistic linguistic information is proposed. On this basis, the vector expressions formed by the scheme, positive and negative ideal solutions are defined, and the bidirectional projection method for the probabilistic linguistic information is established, thus a new closeness degree formula is presented. Finally, the effectiveness and feasibility of the proposed method are verified by the comparative analysis of the application examples. This method fully considers the data characteristics of the PLTS, and avoids the information loss while simplifying the calculation, which provides a new method and idea to solve the problem of small numerical differences in the probabilistic linguistic information.

Key words: multi-attribute group decision-making, probabilistic linguistic term sets (PLTS), bidirectional projection method

中图分类号:

C934

胡悦, 江登英, 李贺. 基于双向投影法的概率语言多属性群决策[J]. 系统工程与电子技术, 2020, 42(9): 2052-2059.

Yue HU, Dengying JIANG, He LI. Probabilistic linguistic multi-attribute group decision-making based on bidirectional projection method[J]. Systems Engineering and Electronics, 2020, 42(9): 2052-2059.

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参考文献 22

1	HWANG C L , LIN M J . Group decision making under multiple criteria: methods and applications[M]. Berlin: Springer-Verlag, 1988: 132.
2	ZADEH L A . The concept of a linguistic variable and its application to approximate reasoning-I[J]. Information Sciences, 1975, 8 (3): 199- 249.
3	PANG Q , WANG H , XU Z S . Probabilistic linguistic term sets in multi-attribute group decision making[J]. Information Sciences, 2016, 369, 128- 143. doi: 10.1016/j.ins.2016.06.021
4	GUO X J , XU Z S . Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term set[J]. Information Sciences, 2016, 372, 407- 427. doi: 10.1016/j.ins.2016.08.034
5	BAI C Z , ZHANG R , QIAN L X , et al. Comparisons of probabilistic linguistic term sets for multi-criteria decision making[J]. Knowledge-Based Systems, 2017, 119, 284- 291. doi: 10.1016/j.knosys.2016.12.020
6	ZHANG Y X , XU Z S , LIAO H C . A consensus process for group decision making with probabilistic linguistic preference relations[J]. Information Sciences, 2017, 414, 260- 275. doi: 10.1016/j.ins.2017.06.006
7	ZHANG Y X , XU Z S , WANG H , et al. Consistency-based risk assessment with probabilistic linguistic preference relation[J]. Applied Soft Computing, 2016, 49, 817- 833. doi: 10.1016/j.asoc.2016.08.045
8	LIU H B , JIANG L , XU Z S . Entropy measures of probabilistic linguistic term sets[J]. International Journal of Computational Intelligence Systems, 2018, 11 (1): 45- 57. doi: 10.2991/ijcis.11.1.4
9	XIE W Y , XU Z S , REN Z L , et al. Probabilistic linguistic analytic hierarchy process and its application on the performance assessment of Xiongan new area[J]. International Journal of Information Technology & Decision Making, 2017, 16, 1693- 1724.
10	WU X L , LIAO H C . A consensus-based probabilistic linguistic gained and lost dominance score method[J]. European Journal of Operational Research, 2019, 272 (3): 1017- 1027.
11	WU X L , LIAO H C , XU Z S , et al. Probabilistic linguistic multimoora: a multi-criteria decision making method based on the probabilistic linguistic expectation function and the improved Borda rule[J]. IEEE Trans.on Fuzzy Systems, 2018, 26 (6): 3688- 3702. doi: 10.1109/TFUZZ.2018.2843330
12	LIAO H C , JIANG L S , XU Z S , et al. A linear programming method for multiple criteria decision making with probabilistic linguistic information[J]. Information Sciences, 2017, 415, 341- 355.
13	LIU P D , YOU X L . Probabilistic linguistic TODIM approach for multiple attribute decision-making[J]. Granular Computing, 2017, 2 (4): 333- 342. doi: 10.1007/s41066-017-0047-4
14	ZHANG X L , XING X M . Probabilistic linguistic VIKOR method to evaluate green supply chain initiatives[J]. Sustainability, 2017, 9 (7): 1231. doi: 10.3390/su9071231
15	ZHANG P H , SHE S X . Assessment of service quality in wireless sensor networks with probabilistic linguistic term sets[J]. International Journal of Online and Biomedical Engineering, 2017, 13 (3): 125- 135.
16	ZHANG X F , GUO X J , XU Z S , et al. A projection method for multiple attribute group decision making with probabilistic linguistic term sets[J]. International Journal of Machine Learning and Cybernetics, 2019, 10 (9): 2515- 2528. doi: 10.1007/s13042-018-0886-6
17	CHENG X , GU J , XU Z S . Venture capital group decision-making with interaction under probabilistic linguistic environment[J]. Knowledge-Based Systems, 2018, 140, 82- 91. doi: 10.1016/j.knosys.2017.10.030
18	刘小弟, 朱建军, 刘思峰. 犹豫模糊信息下的双向投影决策方法[J]. 系统工程理论与实践, 2014, 34 (10): 2637- 2644. doi: 10.12011/1000-6788(2014)10-2637
	LIU X D , ZHU J J , LIU S F . Bidirectional projection method with hesitant fuzzy information[J]. Systems Engineering Theory & Practice, 2014, 34 (10): 2637- 2644. doi: 10.12011/1000-6788(2014)10-2637
19	孟一鸣.基于一致性度量的概率模糊语言信息多属性群决策方法[D].沈阳:东北大学, 2017.
	MENG Y M. Consensus measurement method for probabilistic fuzzy linguistic information in multi-attribute group decision making[D]. Shenyang: Northeastern University, 2017.
20	LAI Y J , LIU T Y , HWANG C L . TOPSIS for MODM[J]. European Journal of Operational Research, 1994, 76 (3): 486- 500.
21	陈可嘉, 陈萍. 基于三参数区间灰数的TOPSIS决策方法[J]. 系统工程与电子技术, 2019, 41 (1): 124- 130.
	CHEN K J , CHEN P . Decision making method of TOPSIS based on three-parameter interval grey numbers[J]. Systems Engineering and Electronics, 2019, 41 (1): 124- 130.
22	王应明. 多指标决策与评价的新方法—投影法[J]. 系统工程与电子技术, 1999, 21 (3): 1- 4. doi: 10.3321/j.issn:1001-506X.1999.03.001
	WANG Y M . A new method for multi-indices decision and evaluation-a projection method[J]. Systems Engineering and Electronics, 1999, 21 (3): 1- 4. doi: 10.3321/j.issn:1001-506X.1999.03.001

文献	公式
文献[3]	$L_{1}(p) \oplus L_{2}(p)=\bigcup\limits_{L_{1}^{(k)} \in L_{1}(p), L_{2}^{(k)} \in L_{2}(p)}\left\{p_{1}^{(k)} L_{1}^{(k)} \oplus p_{2}^{(k)} L_{2}^{(k)}\right\}\left(k=1, 2, \cdots, \max \left(\# L_{1}(p), \# L_{2}(p)\right)\right) $ $\lambda L(p)=\bigcup\limits_{L^{(k)} \in L(p)} \lambda p^{(k)} L^{(k)}(k=1, 2, \cdots, \# L(p))$
文献[4]	$ L_{1}(p) \oplus L_{2}(p)=g^{-1}\left(\bigcup\limits_{\eta_{1}^{(i)} \in g\left(L_{1}\right), \eta_{2}^{(j)} \in g\left(L_{2}\right)}\left\{\left(\eta_{1}^{(i)}+\eta_{2}^{(j)}-\eta_{1}^{(i)} \eta_{2}^{(j)}\right)\left(p_{1}^{(i)} p_{2}^{(j)}\right)\right\}\right)\left(i=1, 2, \cdots, \# L_{1}(p) ; j=1, 2, \cdots, \# L_{2}(p)\right)$ $\lambda L(p)=g^{-1}(\bigcup\limits_{\eta^{(i)} \in g(L)}\left\{\left(1-\left(1-\eta^{(i)}\right)^{\lambda}\right)\left(p^{(i)}\right)\right\})(i=1, 2, \cdots, \# L(p)) $
文献[5]	$ L_{1}(p) \oplus L_{2}(p)=\bigcup\limits_{L_{1}^{(k)} \in L_{1}(p), L_{2}^{(k)} \in L_{2}(p)}\left\{L_{3}^{(k)}\left(p_{3}^{(k)}\right)\right\}\left(L_{3}^{(k)}=L_{1}^{(k)} \oplus L_{2}^{(k)}, p_{3}^{(k)}=p_{1}^{(k)} \cdot p_{2}^{(k)}, k=1, 2, \cdots, \max \left(\# L_{1}(p), \# L_{2}(p)\right)\right)$ $\lambda L(p)=\bigcup\limits_{L^{(k)} \in L(p)}\left\{\lambda L^{(k)}\left(p^{(k)}\right)\right\}(k=1, 2, \cdots, \# L(p)) $
文献[19]	$L_{1}(p) \oplus L_{2}(p)=\bigcup\limits_{\eta \in\left(L_{1}(p) L_{2}(p)\right)}{\cup}\left\{\eta^{(w)}\left(p_{1}^{(w)}+p_{2}^{(w)}\right)\right\}\left(w=\# L_{1}(p)+\# L_{2}(p)-\#\left(L_{1}(p) \cap L_{2}(p)\right)\right) $ $ \lambda L(p)=\bigcup\limits_{\eta^{(k)} \in L(p)}\left\{\eta^{(k)}\left(\lambda p^{(k)}\right)\right\}(k=1, 2, \cdots, \# L(p))$

文献	L₁(p)⊕L₂(p)
文献[3]	L₁(p)⊕L₂(p)={s_-0.4, s_0.3, s_1.3}
文献[4]	L₁(p)⊕L₂(p)={s_0.5(0.08), s₁(0.06), s_1.5(0.15), s_1.25(0.12), s_1.75(0.09), s₂(0.5)}
文献[5]	L₁(p)⊕L₂(p)={s_-1(0.08), s₀(0.18), s₁(0.35), s₂(0.24), s₃(0.15)}
文献[19]	L₁(p)⊕L₂(p)={s_-1(0.4), s₀(0.5), s₁(0.3), s₂(0.5)}
本文	L₁(p)⊕L₂(p)={s_-0.6(0.2), s_-0.28(0.2), s_0.44(0.1), s_0.84(0.2), s_1.48(0.3)}

文献	λ₁L₁(p)⊕λ₂L₂(p), λ₁=0.3, λ₂=0.7
文献[3]	λ₁L₁(p)⊕λ₂L₂(p)={s_-0.28, s_0.09, s_0.51}
文献[4]	λ₁L₁(p)⊕λ₂L₂(p)={s_-0.66(0.08), s_-0.16(0.12), s₀(0.16), s_0.38(0.09), s_0.77(0.06), s₁(0.09), s₂(0.5)}
文献[5]	λ₁L₁(p)⊕λ₂L₂(p)={s_-0.7(0.08), s_-0.4(0.12), s_-0.1(0.2), s₀(0.06), s_0.3(0.09), s_0.6(0.15), s_0.7(0.06), s₁(0.09), s_1.3(0.15)}
文献[19]	λ₁L₁(p)⊕λ₂L₂(p)={s_-1(0.28), s₀(0.27), s₁(0.3), s₂(0.15)}
本文	λ₁L₁(p)⊕λ₂L₂(p)={s_-0.6(0.2), s_-0.28(0.2), s_0.44(0.1), s_0.84(0.2), s_1.48(0.3)}

模公式	贴近度结果	排序结果
文献[16]	C₁=1＞C₂=0.933 9＞C₄=0.932 1＞C₃=0.877 4＞C₅=0.868 5	A₁?A₂?A₄?A₃?A₅
本文	C₁=1＞C₂=0.896 8＞C₄=0.865 7＞C₃=0.791 3＞C₅=0.773 3	A₁?A₂?A₄?A₃?A₅

	G₁	G₂	G₃	G₄	G₅
A₁	{s₂(0.4), s₁(0.4), s₀(0.2)}	{s₂(0.6), s₁(0.4)}	{s₂(0.6)}	{s₁(0.4)}	{s₂(0.2), s₁(0.4), s₀(0.2)}
A₂	{s_-2(0.4), s₀(0.2), s₁(0.2)}	{s₂(0.6), s₁(0.4)}	{s₂(0.4), s₁(0.6)}	{s₁(0.4), s₀(0.4)}	{s_-1(0.6), s₀(0.2), s₂(0.2)}
A₃	{s₁(0.6), s₀(0.4)}	{s₁(0.2), s₀(0.4), s_-1(0.2)}	{s_-2(0.6), s₁(0.2), s₀(0.2)}	{s₁(0.4), s₀(0.2), s_-2(0.2)}	{s₂(0.2), s₁(0.4), s₀(0.4)}
A₄	{s_-2(0.4), s₀(0.4), s_-1(0.2)}	{s₀(0.6), s₁(0.2)}	{s₀(0.2), s₁(0.2), s₂(0.4)}	{s₂(0.4), s₁(0.4), s₀(0.2)}	{s₁(0.2), s₂(0.4), s_-2(0.2)}
A₅	{s₀(0.4), s₁(0.6)}	{s₁(0.2), s₀(0.2), s_-2(0.2)}	{s_-1(0.4), s_-2(0.4), s₀(0.2)}	{s₂(0.8)}	{s_-2(0.4), s₀(0.6)}