﻿ 基于模糊评价和灰色关联度的动力定位系统FMEA方法
 舰船科学技术  2017, Vol. 39 Issue (12): 134-138 PDF

Dynamic positioning FMEA method based on fuzzy evaluation and grey relational grade
HE Zu-jun, DAI San-song, YANG Yi-fei
School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: To aim at the limitations of traditional FMEA method which can’t conduct comprehensive evaluation on failure consequence and lacks quantified evaluation indicators. In this paper, on the basis of traditional FMEA theory, we propose a Dynamic Positioning FMEA Method which based on Fuzzy Evaluation and Grey Relational Grade. Firstly, the fuzzy set theory is applied to establish fuzzy linguistic term and corresponding fuzzy number in failure mode, and according to grey relational grade, we calculate the relevancy of each failure mode. Finally, we determine the risk ranking of all failure mode by the calculations. The results show the method considers the weight of Occurrence Probability Ranking, Effect Severity Ranking and Detection Difficulty Ranking comprehensively, it is more effective and feasibility.
Key words: fuzzy set theory     fuzzy confidence theory     FMEA     risk ranking
0 引　言

1 建立O，S，D的模糊术语集

1.1 建立O，S，D的模糊术语集

3个风险因子O，S，D是模糊语言变量，每个语言变量的评语集为：O/S/D={很低（R）、低（L）、中等（M），高（H），很高（VH），模糊术语的具体含义见表1

1.2 模糊术语对应的模糊数

 ${\lambda _N}(x) = \left\{ \begin{array}{l}0\text{，}\quad \quad \quad \quad \quad \quad x \leqslant a\text{，}\\(x - a)/(b - a)\text{，}\quad \,a < x \leqslant b\text{，}\\(c - x)/(c - b)\text{，}\quad \;b < x \leqslant c\text{，}\\0\text{，} \quad \quad \quad \quad \quad \quad x > c\text{。}\end{array} \right.$ (1)

 $l = \sum\limits_{i = 1}^n {{\alpha _i}{l_i}} \text{，}$ (2)
 $m = \sum\limits_{i = 1}^n {{\alpha _i}{m_i}}\text{，}$ (3)
 $u = \sum\limits_{i = 1}^n {{\alpha _i}{u_i}}\text{。}$ (4)
1.3 三角模糊数去模糊化

 $\begin{split}A(x) = &\displaystyle\frac{1}{{2(1 + N)}}l + \frac{{N + 2NM + M}}{{2(1 + N)(1 + M)}}m+\\ & \frac{1}{{2(1 + M)}}u\text{。}\end{split}$ (5)

2 基于灰色关联度的风险排序

2.1 建立比较判断矩阵

 $\{ {x_j}(i)\} = \left( \begin{array}{l}{x_1}\\{x_2}\\ \vdots \\{x_n}\end{array} \right) = \left( {\begin{array}{*{20}{c}}{{x_1}(1)}&{{x_1}(2)}&{{x_1}(3)}\\{{x_2}(1)}&{{x_2}(2)}&{{x_2}(3)}\\ \vdots & \vdots & \vdots \\{{x_n}(1)}&{{x_n}(2)}&{{x_n}(3)}\end{array}} \right)\text{。}$

 $\{ {x_j}(i)\} = \left\{ {\begin{array}{*{20}{c}}{{x_1}(1)}&{{x_1}(2)}&{{x_1}(3)}\\{{x_2}(1)}&{{x_2}(2)}&{{x_2}(3)}\\{{x_3}(1)}&{{x_3}(2)}&{{x_3}(3)}\\{{x_4}(1)}&{{x_4}(2)}&{{x_4}(3)}\\{{x_5}(1)}&{{x_5}(2)}&{{x_5}(3)}\end{array}} \right\}\text{。}$
2.2 建立参考矩阵

 $\left\{ {\mathit{\boldsymbol{x_0}}\left( {t} \right)} \right\} = \left( {\begin{array}{*{20}{c}}{{h_{11}}}&{{h_{11}}}&{{h_{11}}}\\ \vdots & \vdots & \vdots \\{{h_{11}}}&{{h_{11}}}&{{h_{11}}}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{10}&{10}&{10}\\{10}&{10}&{10}\\{10}&{10}&{10}\\{10}&{10}&{10}\\{10}&{10}&{10}\end{array}} \right)\text{。}$
2.3 计算灰色关联系数

 $\begin{split}{l}&\varepsilon ({x_0}(t),{x_j}(t)) = \\&\displaystyle\frac{{\mathop {\min }\limits_j \;\mathop {\min }\limits_t \;\;\left| {{x_0}(t) \!-\! {x_j}(t)} \right|\! +\! \xi \mathop {\max }\limits_j \;\mathop {\max }\limits_t \;\;\left| {{x_0}(t) - {x_j}(t)} \right|}}{{\left| {{x_0}(t) \!-\! {x_j}(t)} \right| \!+\! \xi \mathop {\max }\limits_j \;\mathop {\max }\limits_t \;\;\left| {{x_0}(t) - {x_j}(t)} \right|}}\end{split}\text{。}\!\!$ (6)

2.4 计算灰色关联度

 $\Gamma ({x_0},{x_{_j}}) = \sum\limits_{t = 1}^3 {{\omega _t}\{ } \varepsilon ({x_0}(t),{x_j}(t))\}\text{。}$ (7)

2.5 风险排序

 图 1 改进的FMEA流程图 Fig. 1 Modified FMEA flow chart
3 实例分析 3.1 确定模糊术语及模糊数

3.2 评价各种故障模式

3.3 应用灰色关联理论计算风险排序

1）根据表3表4建立比较矩阵：

 $\{ {x_j}(i)\} = \left\{ {\begin{array}{*{20}{c}}H&M&M\\M&L&M\\H&L&L\\L&L&M\\L&H&M\end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}}8&{5.6}&{5.6}\\{5.6}&{3.3}&{5.6}\\8&{3.3}&{3.3}\\{3.3}&{3.3}&{5.6}\\{3.3}&8&{5.6}\end{array}} \right\}\text{。}$

2）参考矩阵选择最低水平，则参考矩阵如下：

 $\left\{ {{x_0}\left( {\rm{i}} \right)} \right\} = \left( {\begin{array}{*{20}{c}}{{h_{11}}}&{{h_{11}}}&{{h_{11}}}\\ \vdots & \vdots & \vdots \\{{h_{11}}}&{{h_{11}}}&{{h_{11}}}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{10}&{10}&{10}\\{10}&{10}&{10}\\{10}&{10}&{10}\\{10}&{10}&{10}\\{10}&{10}&{10}\end{array}} \right)\text{。}$

3）计算灰色关联系数

${X_0} = {(10,10,10,10,10)^{\rm T}}\text{，}$

 $\begin{array}{l}{X_0} = {(10,10,10,10,10)^{\rm T}},\\{X_1} = {(8,5.6,8,3.3,3.3)^{\rm T}},\\{X_2} = {(5.6,3.3,3.3,3.3,8)^{\rm T}},\\{X_3} = {(5.6,5.6,3.3,5.6,5.6)^{\rm T}}\text{。}\end{array}$

${\Delta _i}\left( {\rm{t}} \right) = \left| {x_0'\left( {\rm{t}} \right) - x_i'\left( {\rm{t}} \right)} \right|$ i=1，2，3，得 ${X_3} = (5.6,5.6,$ 3.3,5.6,5.6)T

 $\begin{array}{l}{\Delta _1} = \left( {0,0.3,0,0.587\;5,0.587\;5} \right)\text{，}\\{\Delta _2} = \left( {0.3,0.587\;5,0.587\;5,0.587\;5,0} \right)\text{，}\\{\Delta _3} = \left( {0,0,0.41,0,0} \right)\text{。}\end{array}$

 $\begin{array}{l}M = \mathop {\max }\limits_i \mathop {\max }\limits_t {\Delta _i}\left( t \right) = 0.587\;5\text{，}\\m = \mathop {\min }\limits_i \mathop {\min }\limits_t {\Delta _i}\left( t \right) = 0\text{。}\end{array}$

 ${\varepsilon _{1i}}(t) = \frac{{m + \zeta M}}{{{\Delta _i}(t) + \zeta M}} = \frac{{0.5 \times 0.587\;5}}{{{\Delta _i}(t) + 0.5 \times 0.587\;5}}\text{。}$

 $\varepsilon ({x_0}(t),{x_j}(t)) = \left( {\begin{array}{*{20}{c}}1&{0.495}&1\\{0.495}&{0.333}&1\\1&{0.333}&{0.417}\\{0.333}&{0.333}&1\\{0.333}&1&1\end{array}} \right)\text{。}$

4）计算灰色关联度

 $\begin{array}{l}{R} = \varepsilon ({x_0}(t),{x_j}(t)) * \omega = \left( {\begin{array}{*{20}{c}}1&{0.495}&1\\{0.495}&{0.333}&1\\1&{0.333}&{0.417}\\{0.333}&{0.333}&1\\{0.333}&1&1\end{array}} \right)\\ * {(0.4,0.4,0.2)^{\rm T}} = \left( {\begin{array}{*{20}{c}}{0.798}\\{0.531\;2}\\{0.616\;6}\\{0.466\;4}\\{0.733\;2}\end{array}} \right)\text{。}\end{array}$

5）风险排序，见表5

3.4 比较分析

4 结　语

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