科技和产业

系统	G₁	G₂	G₃	G₄
A₁	(0.53,0.33,0.09)	(0.89,0.08,0.03)	(0.42,0.35,0.18)	(0.08,0.89,0.02)
A₂	(0.73,0.12,0.08)	(0.13,0.64,0.21)	(0.03,0.82,0.13)	(0.73,0.15,0.08)
A₃	(0.91,0.03,0.02)	(0.07,0.79,0.05)	(0.04,0.85,0.10)	(0.68,0.26,0.06)
A₄	(0.85,0.09,0.05)	(0.74,0.16,0.10)	(0.02,0.89,0.05)	(0.08,0.84,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.53,0.33,0.09)	(0.89,0.08,0.03)	(0.42,0.35,0.18)	(0.08,0.89,0.02)
A₂	(0.73,0.12,0.08)	(0.13,0.64,0.21)	(0.03,0.82,0.13)	(0.73,0.15,0.08)
A₃	(0.91,0.03,0.02)	(0.07,0.79,0.05)	(0.04,0.85,0.10)	(0.68,0.26,0.06)
A₄	(0.85,0.09,0.05)	(0.74,0.16,0.10)	(0.02,0.89,0.05)	(0.08,0.84,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.53,0.33,0.09)	(0.73,0.12,0.08)	(0.91,0.03,0.02)	(0.85,0.09,0.05)
A₂	(0.89,0.08,0.03)	(0.13,0.64,0.21)	(0.77,0.09,0.05)	(0.74,0.16,0.10)
A₃	(0.42,0.35,0.18)	(0.03,0.82,0.13)	(0.04,0.85,0.10)	(0.02,0.89,0.05)
A₄	(0.33,0.51,0.12)	(0.53,0.31,0.16)	(0.68,0.26,0.06)	(0.08,0.84,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.53,0.33,0.09)	(0.73,0.12,0.08)	(0.91,0.03,0.02)	(0.85,0.09,0.05)
A₂	(0.89,0.08,0.03)	(0.13,0.64,0.21)	(0.77,0.09,0.05)	(0.74,0.16,0.10)
A₃	(0.42,0.35,0.18)	(0.03,0.82,0.13)	(0.04,0.85,0.10)	(0.02,0.89,0.05)
A₄	(0.33,0.51,0.12)	(0.53,0.31,0.16)	(0.68,0.26,0.06)	(0.08,0.84,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.33,0.52,0.12)	(0.52,0.31,0.16)	(0.15,0.76,0.07)	(0.16,0.71,0.05)
A₂	(0.17,0.53,0.13)	(0.51,0.24,0.21)	(0.31,0.39,0.25)	(0.64,0.16,0.10)
A₃	(0.90,0.05,0.02)	(0.68,0.08,0.21)	(0.05,0.87,0.06)	(0.13,0.75,0.09)
A₄	(0.15,0.73,0.08)	(0.70,0.20,0.10)	(0.91,0.03,0.05)	(0.18,0.64,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.33,0.52,0.12)	(0.52,0.31,0.16)	(0.15,0.76,0.07)	(0.16,0.71,0.05)
A₂	(0.17,0.53,0.13)	(0.51,0.24,0.21)	(0.31,0.39,0.25)	(0.64,0.16,0.10)
A₃	(0.90,0.05,0.02)	(0.68,0.08,0.21)	(0.05,0.87,0.06)	(0.13,0.75,0.09)
A₄	(0.15,0.73,0.08)	(0.70,0.20,0.10)	(0.91,0.03,0.05)	(0.18,0.64,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.90,0.05,0.02)	(0.68,0.08,0.21)	(0.05,0.87,0.06)	(0.13,0.75,0.09)
A₂	(0.77,0.13,0.10)	(0.62,0.24,0.11)	(0.10,0.75,0.10)	(0.64,0.16,0.10)
A₃	(0.80,0.15,0.02)	(0.68,0.18,0.05)	(0.05,0.87,0.06)	(0.12,0.65,0.20)
A₄	(0.15,0.73,0.08)	(0.61,0.25,0.10)	(0.91,0.03,0.05)	(0.28,0.44,0.16)

系统	G₁	G₂	G₃	G₄
A₁	(0.90,0.05,0.02)	(0.68,0.08,0.21)	(0.05,0.87,0.06)	(0.13,0.75,0.09)
A₂	(0.77,0.13,0.10)	(0.62,0.24,0.11)	(0.10,0.75,0.10)	(0.64,0.16,0.10)
A₃	(0.80,0.15,0.02)	(0.68,0.18,0.05)	(0.05,0.87,0.06)	(0.12,0.65,0.20)
A₄	(0.15,0.73,0.08)	(0.61,0.25,0.10)	(0.91,0.03,0.05)	(0.28,0.44,0.16)

基于投影和交叉熵的Picture模糊多属性群决策方法

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李亚娟 ¹ , 范建平 ² , 吴美琴 ²

科技和产业 | 科技创新 2025,25(10): 18-24

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科技和产业 | 科技创新 2025, 25(10): 18-24

基于投影和交叉熵的Picture模糊多属性群决策方法

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李亚娟¹, 范建平², 吴美琴²

作者信息

1. 山西电子科技学院经济与管理学院, 山西临汾 041000

2. 山西大学经济与管理学院, 太原 030006

李亚娟(1995—),女,山西临汾人,硕士,助教,研究方向为决策科学与技术;

范建平(1975—),男,山西武乡人,博士,教授,研究方向为决策科学与技术;

吴美琴(1980—),女,山西太原人,博士,副教授,研究方向为决策科学与技术。

Projection and Cross Entropy for Picture Fuzzy Multiple Attributes Group Decision Making

Yajuan LI¹, Jianping FAN², Meiqin WU²

Affiliations

1. School of Economics and Management, Shanxi Electronic Science and Technology Institute, Linfen 041000, Shanxi, China

2. School of Economics and Management, Shanxi University, Taiyuan 030006, China

出版时间: 2025-05-25

文章导航

摘要

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针对方案属性值为Picture模糊数(Picture fuzzy numbers,PFN)的多属性群决策(multiple attributes group decision making,MAGDM)问题,提出一种基于投影法和交叉熵的Picture模糊多属性群决策方法。将投影法引入Picture模糊多属性群决策中求出专家的权重,利用交叉熵求出各专家对应的各方案与理想方案的加权对称差异信息测度值,然后根据专家权重对其进行集结求出各方案与理想方案的加权对称差异信息测度值进而排序得出最优方案。最后通过企业资源计划(ERP)系统的选择表明该方法的可行性和有效性。

关键词

Picture模糊集 / 多属性群决策 / 投影法 / 交叉熵

Abstract

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According to the problem of multiple attributes group decision making, in which the attribute values are given in terms of picture fuzzy numbers, a picture fuzzy multiple attributes group decision making model based on projection and cross entropy was established. The projection was introduced into the picture fuzzy multiple attributes group decision making to find the weights of the experts. The weighted symmetric discrimination information measure for PFSs of each alternative corresponding to each expert and the ideal alternative were obtained by using cross entropy, then according to the weight of the experts, the weighted symmetric discrimination information measure values of each alternative and the ideal alternative were assembled. The optimal alternative is obtained. Finally, an example is given to illustrate the practicality and effectiveness of the proposed method.

Key words

picture fuzzy set / multiple attributes group decision making / projection / cross entropy

引用本文

李亚娟, 范建平, 吴美琴. 基于投影和交叉熵的Picture模糊多属性群决策方法. 科技和产业, 2025 , 25 (10) : 18 -24 .

Yajuan LI, Jianping FAN, Meiqin WU. Projection and Cross Entropy for Picture Fuzzy Multiple Attributes Group Decision Making[J]. Science Technology and Industry, 2025 , 25 (10) : 18 -24 .

正文

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Zadeh^[1]于1965年提出模糊集(fuzzy set,FS)理论,用隶属度来表达决策信息的不确定性和模糊性。然而,仅用隶属度来表达模糊性是不全面的,因此Atanassov^[2]将其扩展到直觉模糊集(intuitionistic fuzzy set,IFS),提出同时用隶属度、非隶属度和犹豫度来表达模糊信息。尽管如此,在现实生活中,由于决策环境变得越来越复杂,仅用隶属度、非隶属度及犹豫度来表达决策信息是不够的,这样可能会造成一部分信息的丢失。比如,在一次投票选举中,选民既可以给候选者投出支持的一票,也可以放弃给候选者投票,还可以给候选者投出反对票,同时还可以拒绝给其投票。这时如果仅用直觉模糊集中的隶属度、非隶属度及犹豫度来表达这部分信息,就会造成一部分信息没有得到充分利用。因此,寻找一种更准确、更全面地表达认知信息的方式就显得尤为重要。Guong^[3]研究了Picture模糊集(Picture fuzzy sets,PFS)及其一些运算和属性。PFS能够同时表达正隶属度、中性隶属度、负隶属度及拒绝隶属度信息,能更好地表达涉及支持、弃权、反对及拒绝四种类型的人类认知行为,在处理不确定性和模糊性方面更具灵活性和实用性。

针对方案属性值为Picture模糊数的多属性决策问题,Wei等^[4]引进Picture模糊理想点的概念并提出一种投影模型,根据各方案与理想方案的相似度对方案排序从而选出最优方案,在后续又提出Picture模糊加权交叉熵对方案进行排序^[5]。Liu和Zhang^[6]提出一种新的Picture模糊语言聚合算子并在群决策中得到很好的应用。龙慧丰和罗敏霞^[7]提出图片模糊Frank加权平均聚合算子和图片模糊Frank加权几何聚合算子用于解决多属性决策问题。韩二东^[8]提出一种基于Picture模糊熵和Picture模糊加权对称交叉熵的多属性决策方法。王磊和姚星娜^[9]针对现有Picture模糊距离的不足,构建一种带有反映决策者态度偏好参数的Picture模糊距离。Yuan等^[10]在Picture模糊环境下,基于Jensen-Shannon散度提出一种新的距离测度,并基于新的距离测度和逼近理想解妥协排序法(CRADIS)提出一种新的图片模糊环境下的多属性决策(MADM)方法。吴孝宇等^[11]通过构建得分函数来对多属性群决策问题进行研究。

在群决策中,专家权重的确定一直是学者们研究的重点内容。李泽欣等^[12]同时考虑犹豫度和共识度,基于概率分布的模糊语言环境,提出一种专家权重的确定方法。张毅等^[13]通过数据势理论来确定专家的权重。王志平等^[14]提出基于累积前景理论和多准则妥协解排序(VIKOR)的多属性群决策方法,利用群体一致性原则来确定决策者的权重。投影法也可以用来确定专家的权重,其不仅考虑到两个元素间的距离还考虑到两个元素间的夹角,因此在模糊环境中得到广泛的应用。Yue和Jia^[15]提出一种基于新的标准化投影的群决策模型。Liang等^[16]提出一种基于几何Bonferroni均值Pythagorean模糊多准则群决策投影模型。林原等^[17]提出一种基于加权双向投影的专家权重确定方法。吴波等^[18]将博弈论组合赋权与灰色关联投影法进行耦合并将其用于山岭隧道突涌水风险分析中。

熵是一种衡量不确定信息的有用工具^[19]。自Luca和Termini^[20]引入模糊熵衡量模糊集间的差异度。Bhandari和Pal^[21]定义了一种新的衡量模糊集间差异度的方法——交叉熵。此后,交叉熵在Vague集^[22]、犹豫模糊集^[23]、直觉模糊集^[24]、区间直觉模糊集^[25]中得到广泛应用。但其在PFS中的应用较少,另外在当前的研究趋势下,对Picture聚合算子以及属性权重已知条件下的决策问题研究已经取得了显著进展,相比之下,对群体决策问题研究却显得相对匮乏。群体决策是一个复杂而多维的过程,其涉及多个决策者的意见、偏好和判断。在现实中,许多重要决策都需要通过群体讨论和协商来达成,因此,对群体决策问题的深入研究具有重要的理论和实践意义。鉴于此,本文提出一种基于投影法和交叉熵的Picture模糊多属性群决策方法,利用各专家的正隶属度矩阵、中性隶属度矩阵及负隶属度矩阵分别在群体的正隶属度矩阵、中性隶属度矩阵及负隶属度矩阵上的投影来确定专家的权重,充分考虑并利用专家和群体的正隶属度矩阵、中性隶属度矩阵及负隶属度矩阵信息,提高专家权重的科学性与可靠性。其次,利用交叉熵在求出各专家对应的各方案与理想方案的加权对称差异信息测度值基础上,根据投影法求得的专家权重对不同专家对应的同一方案与理想方案的加权对称差异信息测度值进行集结。最后,根据集结后的各方案与理想方案的加权对称差异信息测度值排序从而求出最优的方案。

1 相关概念

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1.1 Picture模糊集

定义1^[3]:设X是给定的论域,则

$A=\{x,{\mu }_{A}(x),{\eta }_{A}(x),{v}_{A}(x\left)\left|x\in X\right.\right\}$

称为PFS。其中,

${\mu }_{{\rm A}}\left(x\right)\in \left[\mathrm{0,1}\right],$

表示X中的元素x属于X的正隶属度;

${\eta }_{A}\left(x\right)\in \left[\mathrm{0,1}\right],$

表示X中的元素x属于X的中性隶属度,

${v}_{A}\left(x\right)\in \left[\mathrm{0,1}\right],$

表示X中的元素x属于X的负隶属度,并且满足

$0\le {\mu }_{{\rm A}}\left(x\right)+{\eta }_{{\rm A}}\left(x\right)+{v}_{A}\left(x\right)\le 1,\forall x\in X。$

此外,

$\pi_{A}(x)=1-\left[\mu_{A}(x)+\right. \left.\eta_{A}(x)+v_{A}(x)\right]$

,表示X中的元素x属于X的拒绝隶属度。

为了简便,称

$\gamma =({\mu }_{\gamma },{\eta }_{\gamma },{v}_{\gamma })$

为Picture模糊数^[4],其中,

${\mu }_{\gamma }\in \left[\mathrm{0,1}\right],{\eta }_{\gamma }\in \left[\mathrm{0,1}\right],{v}_{\gamma }\in \left[\mathrm{0,1}\right],0\le {\mu }_{\gamma }+{\eta }_{\gamma }+{v}_{\gamma }\le 1。$

1.2 投影法

定义2^[26]:设

$\alpha =({\alpha }_{1},{\alpha }_{2},\dots,{\alpha }_{n})$

和

$\beta =({\beta }_{1},{\beta }_{2},\dots,{\beta }_{n})$

是n维向量,设定

(1)

$Pr{j}_{\beta }\left(\alpha \right)=\left|\alpha \right|cos(\alpha,\beta )=\left|\alpha \right|\frac{\alpha \beta }{\left|\alpha \right|\left|\beta \right|}=\frac{\alpha \beta }{\left|\beta \right|}$

式中:

$Pr{j}_{\beta }\left(\alpha \right)为\alpha 在\beta 上的投影。其中,\left|\alpha \right|=\sqrt{\sum _{j=1}^{n}{\alpha }_{j}^{2}},\left|\beta \right|=\sqrt{\sum _{j=1}^{n}{\beta }_{j}^{2}},\alpha \beta =\sum _{j=1}^{n}{\alpha }_{j}{\beta }_{j}。一般来说,Pr{j}_{\beta }\left(\alpha \right)越大,表示向量\alpha 越接近于向量\beta 。$

与向量间的投影类似,Yue^[27]给出了矩阵间的投影如下。

定义3^[27]:设

$A=({a}_{ij}{)}_{m\times n}和B=({b}_{ij}{)}_{m\times n}$

是两个m×n的实矩阵,设定

(2)

$Pr{j}_{B}\left(A\right)=\frac{\sum _{i=1}^{m}\sum _{j=1}^{n}{a}_{ij}{b}_{ij}}{\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}{b}_{ij}^{2}}}$

式中:

$Pr{j}_{B}\left(A\right)为矩阵A在矩阵B上的投影。类似的,Pr{j}_{B}\left(A\right)越大,表示矩阵A越接近于矩阵B。$

1.3 Picture模糊集之间的交叉熵

定义4^[28]:假设

$\alpha \left[\alpha \right({x}_{1}),\alpha ({x}_{2}),\dots,\alpha ({x}_{n}\left)\right]$

和

$\beta \left[\beta \right({x}_{1}),\beta ({x}_{2}),\dots,\beta ({x}_{n}\left)\right]$

是论域

$({x}_{1},{x}_{2},\dots,{x}_{n})$

上的模糊集,

$\alpha $

和

$\beta $

的模糊交叉熵定义为

(3)

$\begin{array}{c}h(\alpha, \beta)=\sum_{j=1}^{n}\left\{\alpha\left(x_{j}\right) \ln \frac{\alpha\left(x_{j}\right)}{\frac{1}{2}\left[\alpha\left(x_{j}\right)+\beta\left(x_{j}\right)\right]}+\right. \left.\left[1-\alpha\left(x_{j}\right)\right] \ln \frac{1-\alpha\left(x_{j}\right)}{1-\frac{1}{2}\left[\alpha\left(x_{j}\right)+\beta\left(x_{j}\right)\right]}\right\}\end{array}$

式中:

$h(\alpha,\beta )为\alpha 和\beta $

的差异度。

然而,由于

$h(\alpha,\beta )$

不具有对称性。所以Shang和Jiang^[28]提出一种对称差异信息测度为

(4)

$I(\alpha,\beta )=h(\alpha,\beta )+h(\beta,\alpha )$

式中:

$I(\alpha,\beta )\ge 0,$

当且仅当

$\alpha =\beta $

时

$I(\alpha,\beta )=0。$

于是,模糊集上的交叉熵以及对称差异信息测度扩展到PFS。假设

$\alpha =({\mu }_{{\alpha }_{j}},{\eta }_{{\alpha }_{j}},{v}_{{\alpha }_{j}})和\beta =({\mu }_{{\beta }_{j}},{\eta }_{{\beta }_{j}},{v}_{{\beta }_{j}})(j=\mathrm{1,2},\dots,n)$

为两个PFS,Wei^[5]定义了PFS之间的交叉熵为

(5)

$\begin{array}{c}C(\alpha, \beta)=\sum_{j=1}^{n}\left[\mu_{\alpha_{j}} \ln \frac{\mu_{\alpha_{j}}}{\frac{1}{2}\left(\mu_{\alpha_{j}}+\mu_{\beta_{j}}\right)}+\right. \left.\left(1-\mu_{\alpha_{j}}\right) \ln \frac{1-\mu_{\alpha_{j}}}{1-\frac{1}{2}\left(\mu_{\alpha_{j}}+\mu_{\beta_{j}}\right)}\right]+ \sum_{j=1}^{n}\left[\eta_{\alpha_{j}} \ln \frac{\eta_{\alpha_{j}}}{\frac{1}{2}\left(\eta_{\alpha_{j}}+\eta_{\beta_{j}}\right)}+\right. \left.\left(1-\eta_{\alpha_{j}}\right) \ln \frac{1-\eta_{\alpha_{j}}}{1-\frac{1}{2}\left(\eta_{\alpha_{j}}+\eta_{\beta_{j}}\right)}\right]+ \sum_{j=1}^{n}\left[v_{\alpha_{j}} \ln \frac{v_{\alpha_{j}}}{\frac{1}{2}\left(v_{\alpha_{j}}+v_{\beta_{j}}\right)}+\right.\left.\left(1-v_{\alpha_{j}}\right) \ln \frac{1-v_{\alpha_{j}}}{1-\frac{1}{2}\left(v_{\alpha_{j}}+v_{\beta_{j}}\right)}\right]\end{array}$

式中:

$C(\alpha,\beta )为\alpha $

和

$\beta $

的差异度;i=1,2,…,m。由于

$C(\alpha,\beta )$

不具有对称性,所以Wei^[5]将其修改为如下的PFS对称差异信息测度。

(6)

$D(\alpha,\beta )=C(\alpha,\beta )+C(\beta,\alpha )$

式中:

$\alpha $

和

$\beta $

的差异越大,

$D(\alpha,\beta )$

就越大。

如果考虑到

$\alpha $

和

$\beta $

的权重,Wei^[5]定义的

$\alpha $

和

$\beta $

的Picture模糊加权交叉熵测度如式(7)所示。

(7)

$\begin{array}{c}C_{\omega}(\alpha, \beta)=\sum_{j=1}^{n} \omega_{j}\left[\mu_{\alpha_{j}} \ln \frac{\mu_{\alpha_{j}}}{\frac{1}{2}\left(\mu_{\alpha_{j}}+\mu_{\beta_{j}}\right)}+\right. \left.\left(1-\mu_{\alpha_{j}}\right) \ln \frac{1-\mu_{\alpha_{j}}}{1-\frac{1}{2}\left(\mu_{\alpha_{j}}+\mu_{\beta_{j}}\right)}\right]+ \sum_{j=1}^{n} \omega_{j}\left[\eta_{\alpha_{j}} \ln \frac{\eta_{\alpha_{j}}}{\frac{1}{2}\left(\eta_{\alpha_{j}}+\eta_{\beta_{j}}\right)}+\right. \left.\left(1-\eta_{\alpha_{j}}\right) \ln \frac{1-\eta_{\alpha_{j}}}{1-\frac{1}{2}\left(\eta_{\alpha_{j}}+\eta_{\beta_{j}}\right)}\right]+ \sum_{j=1}^{n} \omega_{j}\left[v_{\alpha_{j}} \ln \frac{v_{\alpha_{j}}}{\frac{1}{2}\left(v_{\alpha_{j}}+v_{\beta_{j}}\right)}+\right. \left.\left(1-v_{\alpha_{j}}\right) \ln \frac{1-v_{\alpha_{j}}}{1-\frac{1}{2}\left(v_{\alpha_{j}}+v_{\beta_{j}}\right)}\right]\end{array}$

式中:

$i=\mathrm{1,2},\dots,m;\omega =({\omega }_{1},{\omega }_{2},\dots,{\omega }_{n}{)}^{T}是\alpha 、\beta $

的权重向量,满足

${\omega }_{j}\in \left[\mathrm{0,1}\right],j=\mathrm{1,2},\dots,n,\sum _{j=1}^{n}{\omega }_{j}=1;{C}_{\omega }(\alpha,\beta )为\alpha $

和

$\beta $

的差异度,由于其不具有对称性,所以Wei^[5]将修改为如下的Picture模糊加权对称差异信息测度。

(8)

${D}_{\omega }(\alpha,\beta )={C}_{\omega }(\alpha,\beta )+{C}_{\omega }(\beta,\alpha )$

式中:

$\alpha $

和

$\beta $

的差异越大,

${D}_{\omega }(\alpha,\beta )$

就越大。

2 基于投影和交叉熵的Picture模糊多属性群决策

收起

假设有m个可行的方案$A_{1}, A_{2}, \cdots, A_{m}, n$个评价属性$G_{1}, G_{2}, \cdots, G_{n}, \boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \cdots, \omega_{n}\right)$是属性$G_{j}(j=1,2, \cdots, n)$的权重。其中,$\omega_{j} \in[0,1], j=1 ， 2, \cdots, n, \sum_{j=1}^{n} \omega_{j}=1, t$个决策专家${E}_{1},{E}_{2},\dots,{E}_{t}$。决策专家${E}_{k}(k=\mathrm{1,2},\dots,t)$对方案${A}_{i}(i=\mathrm{1,2},\dots,m)$在评价属性${G}_{j}(j=\mathrm{1,2},\dots,n)$下的决策值为PFN$({\mu }_{ij}^{k},{\eta }_{ij}^{k},{v}_{ij}^{k})$。则专家${E}_{k}(k=\mathrm{1,2},\dots,t)$的Picture模糊决策矩阵为

(9)

${X}^{k}=({\mu }_{ij}^{k},{\eta }_{ij}^{k},{v}_{ij}^{k}{)}_{m\times n}=\left[\begin{array}{llll}({\mu }_{11}^{k},{\eta }_{11}^{k},{v}_{11}^{k})& ({\mu }_{12}^{k},{\eta }_{12}^{k},{v}_{12}^{k})& \dots & ({\mu }_{1n}^{k},{\eta }_{1n}^{k},{v}_{1n}^{k})\\ ({\mu }_{21}^{k},{\eta }_{21}^{k},{v}_{21}^{k})& ({\mu }_{21}^{k},{\eta }_{21}^{k},{v}_{21}^{k})& \dots & ({\mu }_{2n}^{k},{\eta }_{2n}^{k},{v}_{2n}^{k})\\ ︙& ︙& & ︙\\ ({\mu }_{m1}^{k},{\eta }_{m1}^{k},{v}_{m1}^{k})& ({\mu }_{m2}^{k},{\eta }_{m2}^{k},{v}_{m2}^{k})& \dots & ({\mu }_{mn}^{k},{\eta }_{mn}^{k},{v}_{mn}^{k})\end{array}\right]$

接下来给出基于投影和交叉熵的Picture模糊多属性群决策过程。

步骤1:基于投影的Picture模糊多属性群决策专家权重的确定。

(1)将专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

的Picture模糊决策矩阵X^k分为正隶属度矩阵

${X}_{\mu }^{k}、$

中性隶属度矩阵

${X}_{\eta }^{k}$

和负隶属度矩阵

${X}_{v}^{k}。$

(10)

${X}_{\mu }^{k}=\left[\begin{array}{llll}{\mu }_{11}^{k}& {\mu }_{12}^{k}& \dots & {\mu }_{1n}^{k}\\ {\mu }_{21}^{k}& {\mu }_{21}^{k}& \dots & {\mu }_{2n}^{k}\\ ︙& ︙& & ︙\\ {\mu }_{m1}^{k}& {\mu }_{m2}^{k}& \dots & {\mu }_{mn}^{k}\end{array}\right]$

(11)

${X}_{\eta }^{k}=\left[\begin{array}{llll}{\eta }_{11}^{k}& {\eta }_{12}^{k}& \dots & {\eta }_{1n}^{k}\\ {\eta }_{21}^{k}& {\eta }_{21}^{k}& \dots & {\eta }_{2n}^{k}\\ ︙& ︙& & ︙\\ {\eta }_{m1}^{k}& {\eta }_{m2}^{k}& \dots & {\eta }_{mn}^{k}\end{array}\right]$

(12)

${X}_{v}^{k}=\left[\begin{array}{llll}{v}_{11}^{k}& {v}_{12}^{k}& \dots & {v}_{1n}^{k}\\ {v}_{21}^{k}& {v}_{21}^{k}& \dots & {v}_{2n}^{k}\\ ︙& ︙& & ︙\\ {v}_{m1}^{k}& {v}_{m2}^{k}& \dots & {v}_{mn}^{k}\end{array}\right]$

(2)确定所有专家的群体决策矩阵X^*。

(13)

${X}^{*}=({\mu }_{ij}^{*},{\eta }_{ij}^{*},{v}_{ij}^{*}{)}_{m\times n}=\left[\begin{array}{llll}({\mu }_{11}^{*},{\eta }_{11}^{*},{v}_{11}^{*})& ({\mu }_{12}^{*},{\eta }_{12}^{*},{v}_{12}^{*})& \dots & ({\mu }_{1n}^{*},{\eta }_{1n}^{*},{v}_{1n}^{*})\\ ({\mu }_{21}^{*},{\eta }_{21}^{*},{v}_{21}^{*})& ({\mu }_{21}^{*},{\eta }_{21}^{*},{v}_{21}^{*})& \dots & ({\mu }_{2n}^{*},{\eta }_{2n}^{*},{v}_{2n}^{*})\\ ︙& ︙& & ︙\\ ({\mu }_{m1}^{*},{\eta }_{m1}^{*},{v}_{m1}^{*})& ({\mu }_{m2}^{*},{\eta }_{m2}^{*},{v}_{m2}^{*})& \dots & ({\mu }_{mn}^{*},{\eta }_{mn}^{*},{v}_{mn}^{*})\end{array}\right]$

式中:

${\mu }_{ij}^{*}=1-\sum _{k=1}^{t}(1-{\mu }_{ij}^{k}{)}^{\frac{1}{t}};{\eta }_{ij}^{*}=\sum _{k=1}^{t}({\eta }_{ij}^{k}{)}^{\frac{1}{t}};{v}_{ij}^{*}=\sum _{k=1}^{t}({v}_{ij}^{k}{)}^{\frac{1}{t}};i=\mathrm{1,2},\dots,m;j=\mathrm{1,2},\dots,n。$

(3)将群体决策矩阵X^*分为正隶属度矩阵

${X}_{\mu }^{*}、$

中性隶属度矩阵

${X}_{\eta }^{*}$

和负隶属度矩阵

${X}_{v}^{*}。$

(14)

${X}_{\mu }^{*}=\left[\begin{array}{llll}{\mu }_{11}^{*}& {\mu }_{12}^{*}& \dots & {\mu }_{1n}^{*}\\ {\mu }_{21}^{*}& {\mu }_{21}^{*}& \dots & {\mu }_{2n}^{*}\\ ︙& ︙& & ︙\\ {\mu }_{m1}^{*}& {\mu }_{m2}^{*}& \dots & {\mu }_{mn}^{*}\end{array}\right]$

(15)

${X}_{\eta }^{*}=\left[\begin{array}{llll}{\eta }_{11}^{*}& {\eta }_{12}^{*}& \dots & {\eta }_{1n}^{*}\\ {\eta }_{21}^{*}& {\eta }_{21}^{*}& \dots & {\eta }_{2n}^{*}\\ ︙& ︙& & ︙\\ {\eta }_{m1}^{*}& {\eta }_{m2}^{*}& \dots & {\eta }_{mn}^{*}\end{array}\right]$

(16)

${X}_{v}^{*}=\left[\begin{array}{llll}{v}_{11}^{*}& {v}_{12}^{*}& \dots & {v}_{1n}^{*}\\ {v}_{21}^{*}& {v}_{21}^{*}& \dots & {v}_{2n}^{*}\\ ︙& ︙& & ︙\\ {v}_{m1}^{*}& {v}_{m2}^{*}& \dots & {v}_{mn}^{*}\end{array}\right]$

(4)计算专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

的正隶属度矩阵

${X}_{\mu }^{k}$

在群体正隶属度矩阵

${X}_{\mu }^{*}$

上的投影、中性隶属度矩阵

${X}_{\eta }^{k}$

在群体中性隶属度矩阵

${X}_{\eta }^{*}$

上的投影以及负隶属度矩阵

${X}_{v}^{k}$

在群体负隶属度矩阵

${X}_{v}^{*}$

上的投影。

由式(2)可分别计算出:

①专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

的正隶属度矩阵

${X}_{\mu }^{k}$

在群体正隶属度矩阵

${X}_{\mu }^{*}$

上的投影为

(17)

$Pr{j}_{{X}_{\mu }^{*}}\left({X}_{\mu }^{k}\right)=\frac{\sum _{i=1}^{m}\sum _{j=1}^{n}{\mu }_{ij}^{k}{\mu }_{ij}^{*}}{\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}{\mu }_{ij}^{*2}}}$

②专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

的中性隶属度矩阵

${X}_{\eta }^{k}$

在群体中性隶属度矩阵

${X}_{\eta }^{*}$

上的投影为

(18)

$Pr{j}_{{X}_{\eta }^{*}}\left({X}_{\eta }^{k}\right)=\frac{\sum _{i=1}^{m}\sum _{j=1}^{n}{\eta }_{ij}^{k}{\eta }_{ij}^{*}}{\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}{\eta }_{ij}^{*2}}}$

③专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

的负隶属度矩阵

${X}_{v}^{k}$

在群体负隶属度矩阵

${X}_{v}^{*}$

上的投影为

(19)

$Pr{j}_{{X}_{v}^{*}}\left({X}_{v}^{k}\right)=\frac{\sum _{i=1}^{m}\sum _{j=1}^{n}{v}_{ij}^{k}{v}_{ij}^{*}}{\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}{v}_{ij}^{*2}}}$

(5)计算专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

的决策矩阵X^k与群体决策矩阵X^*的相似度为

一个专家的决策矩阵越相似于群体决策矩阵,那么这个专家的评价信息是重要的,应被赋予一个较大的权重。反之,一个专家的决策矩阵与群体决策矩阵差异较大,那么这个专家的评价信息是不重要的,应被赋予一个较小的权重。设定

(20)

${E}_{k}(\theta,\lambda )=\theta Pr{j}_{{X}_{\mu }^{*}}\left({X}_{\mu }^{k}\right)+\lambda Pr{j}_{{X}_{\eta }^{*}}\left({X}_{\eta }^{k}\right)+(1-\theta -\lambda )Pr{j}_{{X}_{v}^{*}}\left({X}_{v}^{k}\right)$

式中:

${E}_{k}(\theta,\lambda )$

为专家E_k的决策矩阵X^k与群体决策矩阵X^*的相似度。其中,

$\theta $

和

$\lambda $

为偏好度,是一个给定的值,表示领导者对正隶属度矩阵信息以及中性隶属度矩阵信息的偏好度。

(6)计算专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

的权重。

(21)

$E(\theta,\lambda )=\sum _{k=1}^{t}{E}_{k}(\theta,\lambda )$

那么专家E_k的权重为

(22)

${\epsilon }_{k}=\frac{{E}_{k}(\theta,\lambda )}{E(\theta,\lambda )}$

步骤2:定义理想方案A⁺。

(23)

${A}^{+}=\left[\right({\mu }_{1}^{+},{\eta }_{1}^{+},{v}_{1}^{+}),({\mu }_{2}^{+},{\eta }_{2}^{+},{v}_{2}^{+}),\dots,({\mu }_{n}^{+},{\eta }_{n}^{+},{v}_{n}^{+}\left)\right]$

式中:

${\mu }_{j}^{+}=\underset{i}{max}\left\{{\mu }_{ij}^{k}\right\},{\eta }_{j}^{+}=\underset{i}{min}\left\{{\eta }_{ij}^{k}\right\},{v}_{j}^{+}=\underset{i}{min}\left\{{v}_{ij}^{k}\right\},j=\mathrm{1,2},\dots,n;k=\mathrm{1,2},\dots,t。$

步骤3:求出每个专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

对应的每个方案

${A}_{i}(i=\mathrm{1,2},\dots,m)$

与理想方案A⁺的Picture模糊加权对称差异信息测度值

${D}_{\omega }^{k}({A}_{i},{A}^{+})。$

(24)

${D}_{\omega }^{k}({A}_{i},{A}^{+})={C}_{\omega }^{k}({A}_{i},{A}^{+})+{C}_{\omega }^{k}({A}^{+},{A}_{i})$

式中:

(25)

\[ {C}_{\omega }^{k}\left( {{A}_{i},{A}^{ + }}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{\omega }_{j}\left\lbrack {{\mu }_{ij}^{k}\ln \frac{{\mu }_{ij}^{k}}{\frac{1}{2}\left( {{\mu }_{ij}^{k} + {\mu }_{j}^{ + }}\right) } + }\right. \\ \left. {\left( {1 - {\mu }_{ij}^{ + }}\right) \ln \frac{1 - {\mu }_{ij}^{k}}{1 - \frac{1}{2}\left( {{\mu }_{ij}^{k} + {\mu }_{j}^{ + }}\right) }}\right\rbrack + \\ \mathop{\sum }\limits_{{j = 1}}^{n}{\omega }_{j}\left\lbrack {{\eta }_{ij}^{k}\ln \frac{{\eta }_{ij}^{k}}{\frac{1}{2}\left( {{\eta }_{ij}^{k} + {\eta }_{j}^{ + }}\right) } + }\right. \\ \left. {\left( {1 - {\eta }_{ij}^{k}}\right) \ln \frac{1 - {\eta }_{ij}^{k}}{1 - \frac{1}{2}\left( {{\eta }_{ij}^{k} + {\eta }_{j}^{ + }}\right) }}\right\rbrack + \\ \mathop{\sum }\limits_{{j = 1}}^{n}{\omega }_{j}\left\lbrack {{v}_{ij}^{k}\ln \frac{{v}_{ij}^{k}}{\frac{1}{2}\left( {{v}_{ij}^{k} + {v}_{j}^{ + }}\right) } + }\right. \\ \left. {\left( {1 - {v}_{ij}^{k}}\right) \ln \frac{1 - {v}_{ij}^{k}}{1 - \frac{1}{2}\left( {{v}_{ij}^{k} + {v}_{j}^{ + }}\right) }}\right\rbrack \]

式中:i=1,2,…,m。

步骤4:根据专家权重

${\epsilon }_{k}(k=\mathrm{1,2},\dots,t)$

对每个专家对应的每个方案

${A}_{i}(i=\mathrm{1,2},\dots,m)$

与理想方案A⁺的Picture模糊加权对称差异信息测度值

${D}_{\omega }^{k}({A}_{i},{A}^{+})$

进行集结。

(26)

${D}_{\omega }({A}_{i},{A}^{+})=\sum _{k=1}^{t}{\epsilon }_{k}{D}_{\omega }^{k}({A}_{i},{A}^{+}), i=\mathrm{1,2},\dots,m$

步骤5:根据每个方案与理想方案的加权对称差异信息测度值

${D}_{\omega }({A}_{i},{A}^{+})$

进行排序。

${D}_{\omega }({A}_{i},{A}^{+})$

越小,方案A_i越好。

3 应用

收起

企业资源计划(ERP)系统在现代企业管理中扮演着至关重要的角色,选择合适的ERP系统对企业来说是一个重要的决策过程。现有一家公司计划实施ERP系统,有不同公司提供的4个系统

$\{{A}_{1},{A}_{2},{A}_{3},{A}_{4}\}$

作为备选方案,邀请4位专家

$\{{E}_{1},{E}_{2},{E}_{3},{E}_{4}\}$

分别在4个属性

$\{{G}_{1},{G}_{2},{G}_{3},{G}_{4}\}$

上做出决策,G₁为技术,G₂为战略适应性,G₃为供应商能力,G₄为供应商声誉,其权重分别为

${\left(\mathrm{0.25,0.3,0.25,0.2}\right)}^{T}。$

专家

${E}_{k}(k=\mathrm{1,2},\mathrm{3,4})$

利用图片模糊数评估ERP系统

${A}_{i}(i=\mathrm{1,2},\mathrm{3,4})$

在属性

${G}_{j}(j=\mathrm{1,2},\mathrm{3,4})$

上的特征。专家

${E}_{k}(k=\mathrm{1,2},\mathrm{3,4})$

的评估结果如表1~表4所示的Picture模糊决策矩阵X^k。

步骤1:基于投影的Picture模糊多属性群决策专家权重的确定。

根据前边的步骤1可求出四位专家的权重分别为(这里取领导者对正隶属度矩阵信息以及中性隶属度矩阵信息的偏好度分别为0.5、0.3):

${\epsilon }_{1}=0.239,{\epsilon }_{2}=0.255,{\epsilon }_{3}=0.260,{\epsilon }_{4}=0.246。$

步骤2:定义理想方案A⁺。A⁺=[(0.91,0.03,0.02),(0.89,0.08,0.03),(0.91,0.03,0.02),(0.85,0.09,0.02)]。

步骤3:求出每个专家

${E}_{k}(k=\mathrm{1,2},\dots,t)$

对应的每个方案

${A}_{i}(i=\mathrm{1,2},\dots,m)$

与理想方案A⁺的Picture模糊加权对称差异信息测度值

${D}_{\omega }^{k}({A}_{i},{A}^{+})$

。

${D}_{\omega }^{1}({A}_{1},{A}^{+})=0.522,{D}_{\omega }^{1}({A}_{2},{A}^{+})=0.809$

${D}_{\omega }^{1}({A}_{3},{A}^{+})=0.870,{D}_{\omega }^{1}({A}_{4},{A}^{+})=0.768$

;

${D}_{\omega }^{2}({A}_{1},{A}^{+})=0.117,{D}_{\omega }^{2}({A}_{2},{A}^{+})=0.364,{D}_{\omega }^{2}({A}_{3},{A}^{+})=1.367,{D}_{\omega }^{2}({A}_{4},{A}^{+})=0.606$

;

${D}_{\omega }^{3}({A}_{1},{A}^{+})=0.817,{D}_{\omega }^{3}({A}_{2},{A}^{+})=0.566,{D}_{\omega }^{3}({A}_{3},{A}^{+})=0.704,{D}_{\omega }^{3}({A}_{4},{A}^{+})=0.528$

;

${D}_{\omega }^{4}({A}_{1},{A}^{+})=0.703,{D}_{\omega }^{4}({A}_{2},{A}^{+})=0.457,{D}_{\omega }^{4}({A}_{3},{A}^{+})=0.695,{D}_{\omega }^{4}({A}_{4},{A}^{+})=0.497。$

步骤4:根据专家权重

${\epsilon }_{k}(k=\mathrm{1,2},\dots,t)$

对每个专家对应的每个方案

${A}_{i}(i=\mathrm{1,2},\dots,m)$

与理想方案A⁺的Picture模糊加权对称差异信息测度值

${D}_{\omega }^{k}({A}_{i},{A}^{+})$

进行集结。

${D}_{\omega }({A}_{1},{A}^{+})=0.540,{D}_{\omega }({A}_{2},{A}^{+})=0.546,{D}_{\omega }({A}_{1},{A}^{+})=0.911,{D}_{\omega }({A}_{1},{A}^{+})=0.598。$

步骤5:根据每个方案与理性方案的加权对称差异信息测度值

${D}_{\omega }({A}_{i},{A}^{+})$

进行排序。

根据各方案的加权对称差异信息测度值

${D}_{\omega }({A}_{i},{A}^{+}),$

可得到各方案的排序为

$ A_{1}>A_{2}> A_{4} > A_{3} $

因此,最优方案为方案1。

4 结论

收起

对于Picture模糊环境下的多属性群决策问题,本文提出基于投影法和交叉熵的Picture模糊多属性群决策模型。将投影法引入Picture模糊多属性群决策问题中,通过投影法给出的专家权重,充分考虑各专家决策矩阵的正隶属度矩阵、中性隶属度矩阵及负隶属度矩阵分别在群体决策矩阵上的正隶属度矩阵、中性隶属度矩阵及负隶属度矩阵上的投影,使得所求得的权重更科学。在交叉熵的基础上,考虑交叉熵的非对称性,使用加权对称差异信息测度来求得各方案与理想方案间的差异度使得所求结果更全面。最后,通过ERP系统的选择证明该方法的有效性和可行性。

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接收时间：2024-11-14
首发时间：2025-07-09
出版时间：2025-05-25

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收稿日期：2024-11-14

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1. 山西电子科技学院经济与管理学院, 山西临汾 041000

2. 山西大学经济与管理学院, 太原 030006

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2种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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系统	G₁	G₂	G₃	G₄
A₁	(0.53,0.33,0.09)	(0.89,0.08,0.03)	(0.42,0.35,0.18)	(0.08,0.89,0.02)
A₂	(0.73,0.12,0.08)	(0.13,0.64,0.21)	(0.03,0.82,0.13)	(0.73,0.15,0.08)
A₃	(0.91,0.03,0.02)	(0.07,0.79,0.05)	(0.04,0.85,0.10)	(0.68,0.26,0.06)
A₄	(0.85,0.09,0.05)	(0.74,0.16,0.10)	(0.02,0.89,0.05)	(0.08,0.84,0.06)

系统

G₁

G₂

G₃

G₄

A₁

(0.53,0.33,0.09)

(0.89,0.08,0.03)

(0.42,0.35,0.18)

(0.08,0.89,0.02)

A₂

(0.73,0.12,0.08)

(0.13,0.64,0.21)

(0.03,0.82,0.13)

(0.73,0.15,0.08)

A₃

(0.91,0.03,0.02)

(0.07,0.79,0.05)

(0.04,0.85,0.10)

(0.68,0.26,0.06)

A₄

(0.85,0.09,0.05)

(0.74,0.16,0.10)

(0.02,0.89,0.05)

(0.08,0.84,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.53,0.33,0.09)	(0.73,0.12,0.08)	(0.91,0.03,0.02)	(0.85,0.09,0.05)
A₂	(0.89,0.08,0.03)	(0.13,0.64,0.21)	(0.77,0.09,0.05)	(0.74,0.16,0.10)
A₃	(0.42,0.35,0.18)	(0.03,0.82,0.13)	(0.04,0.85,0.10)	(0.02,0.89,0.05)
A₄	(0.33,0.51,0.12)	(0.53,0.31,0.16)	(0.68,0.26,0.06)	(0.08,0.84,0.06)

系统

G₁

G₂

G₃

G₄

A₁

(0.53,0.33,0.09)

(0.73,0.12,0.08)

(0.91,0.03,0.02)

(0.85,0.09,0.05)

A₂

(0.89,0.08,0.03)

(0.13,0.64,0.21)

(0.77,0.09,0.05)

(0.74,0.16,0.10)

A₃

(0.42,0.35,0.18)

(0.03,0.82,0.13)

(0.04,0.85,0.10)

(0.02,0.89,0.05)

A₄

(0.33,0.51,0.12)

(0.53,0.31,0.16)

(0.68,0.26,0.06)

(0.08,0.84,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.33,0.52,0.12)	(0.52,0.31,0.16)	(0.15,0.76,0.07)	(0.16,0.71,0.05)
A₂	(0.17,0.53,0.13)	(0.51,0.24,0.21)	(0.31,0.39,0.25)	(0.64,0.16,0.10)
A₃	(0.90,0.05,0.02)	(0.68,0.08,0.21)	(0.05,0.87,0.06)	(0.13,0.75,0.09)
A₄	(0.15,0.73,0.08)	(0.70,0.20,0.10)	(0.91,0.03,0.05)	(0.18,0.64,0.06)

系统

G₁

G₂

G₃

G₄

A₁

(0.33,0.52,0.12)

(0.52,0.31,0.16)

(0.15,0.76,0.07)

(0.16,0.71,0.05)

A₂

(0.17,0.53,0.13)

(0.51,0.24,0.21)

(0.31,0.39,0.25)

(0.64,0.16,0.10)

A₃

(0.90,0.05,0.02)

(0.68,0.08,0.21)

(0.05,0.87,0.06)

(0.13,0.75,0.09)

A₄

(0.15,0.73,0.08)

(0.70,0.20,0.10)

(0.91,0.03,0.05)

(0.18,0.64,0.06)

系统	G₁	G₂	G₃	G₄
A₁	(0.90,0.05,0.02)	(0.68,0.08,0.21)	(0.05,0.87,0.06)	(0.13,0.75,0.09)
A₂	(0.77,0.13,0.10)	(0.62,0.24,0.11)	(0.10,0.75,0.10)	(0.64,0.16,0.10)
A₃	(0.80,0.15,0.02)	(0.68,0.18,0.05)	(0.05,0.87,0.06)	(0.12,0.65,0.20)
A₄	(0.15,0.73,0.08)	(0.61,0.25,0.10)	(0.91,0.03,0.05)	(0.28,0.44,0.16)

系统

G₁

G₂

G₃

G₄

A₁

(0.90,0.05,0.02)

(0.68,0.08,0.21)

(0.05,0.87,0.06)

(0.13,0.75,0.09)

A₂

(0.77,0.13,0.10)

(0.62,0.24,0.11)

(0.10,0.75,0.10)

(0.64,0.16,0.10)

A₃

(0.80,0.15,0.02)

(0.68,0.18,0.05)

(0.05,0.87,0.06)

(0.12,0.65,0.20)

A₄

(0.15,0.73,0.08)

(0.61,0.25,0.10)

(0.91,0.03,0.05)

(0.28,0.44,0.16)