CN112115561B - Improved FMEA method based on interval triangular fuzzy number and fuzzy VIKOR method - Google Patents

Improved FMEA method based on interval triangular fuzzy number and fuzzy VIKOR method Download PDF

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CN112115561B
CN112115561B CN202010990123.XA CN202010990123A CN112115561B CN 112115561 B CN112115561 B CN 112115561B CN 202010990123 A CN202010990123 A CN 202010990123A CN 112115561 B CN112115561 B CN 112115561B
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李国发
张微
何佳龙
霍津海
杨海吉
韩良晟
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Jilin University
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Abstract

The invention discloses an improved FMEA method based on an interval triangular fuzzy number and fuzzy VIKOR method. Firstly, the relative importance of the failure mode grade and the risk factor is evaluated by using linguistic variables, and the linguistic variables are expressed by interval triangular fuzzy numbers. Secondly, subjective weight of the risk factors is calculated by using a fuzzy analytic hierarchy process, objective weight of the risk factors is calculated by using an expanded VIKOR process, and comprehensive weight of the risk factors is calculated according to an improved game theory combination weighting process. And finally, sequencing the risk priority of the failure modes by using a fuzzy VIKOR method. The method is used for evaluating the workpiece box system of the numerical control gear milling machine, and the result verifies the applicability and the effectiveness of the method.

Description

Improved FMEA method based on interval triangular fuzzy number and fuzzy VIKOR method
Technical Field
The invention belongs to the technical field of reliability analysis, and particularly relates to an improved FMEA (Fuzzy analysis) method based on an Interval Triangular Fuzzy Number and a Fuzzy VIKOR (Fuzzy Number, IVF) method.
Prior Art
Failure Mode and Effect Analysis (FMEA) is a systematic activity of analyzing subsystems constituting a product or various processes constituting a process one by one in a product design stage or a process design stage, finding out all potential Failure modes, and taking necessary measures in advance, and was originally proposed by the national space agency (NASA) in the 20 th century and the 60 th century. Because of the characteristics of simplicity, convenience, high efficiency and the like, the method is widely applied to risk analysis in the fields of automobiles, aerospace, medical care and the like.
FMEA requires a cross-job team of FMEA members with different specialties who need to evaluate the degree of occurrence (O), severity (S), and degree of detection (D) for each failure mode based on past experience and judgment. In the traditional FMEA method, the score of each Risk factor is an integer of 1-10, the Risk Priority of a failure mode is determined according to the Risk Priority Number (RPN), the Risk Priority Number is obtained by multiplying the three Risk factors, and the larger the value of the RPN is, the more serious the influence of the failure mode is. Although the effectiveness of conventional FMEA methods based on RPN ranking has been variously demonstrated, there are still some drawbacks and shortcomings: (1) many fuzzy, complex or uncertain information can be generated in the FMEA evaluation process, and the FMEA member can hardly give accurate values of the three risk factors; (2) when the failure modes are sorted, the weights of the three default risk indexes are the same. However, when different subjects are evaluated, the weights of the three risk factors may have a large difference; (3) the score combinations of different risk factors may result in the same RPN value, and the potential risk levels of these failure modes may be completely different, which may result in the failure to accurately find the most important failure mode;
(4) when calculating the RPN, only 120 numbers out of 1 to 1000 can be multiplied by three risk factors. This will result in a strong discontinuity in the RPN; (5) the calculation of the RPN is not scientifically based and is very sensitive to changes in the risk factors. Small changes in a certain risk factor may result in large differences in RPN.
The interval triangular fuzzy number is used for evaluating the relative importance of the grade of the failure mode and the risk factor, and is not used for a preset linguistic variable. The diversified view of the FMEA members can be flexibly and accurately expressed through the interval triangular fuzzy number.
The fuzzy VIKOR method is a multi-attribute decision method proposed by professor Opricovic of south Slave for complex systems. The fuzzy VIKOR method has the core content that on the basis of determining the positive and negative ideal solutions, the advantages and the disadvantages are ranked according to the proximity degree of each alternative scheme and the ideal scheme. The method considers the maximization of group benefit and the minimization of individual regret, and simultaneously also considers the subjective preference of a decision maker. And the VIKOR method obtains a compromise scheme with priority, and the compromise scheme is closer to an ideal scheme.
Disclosure of Invention
In order to solve the defects of the traditional FMEA, the invention provides an improved FMEA method based on an interval triangular fuzzy number and fuzzy VIKOR method. The method comprises the steps of firstly, evaluating the relative importance of the failure mode grade and the risk factor by using a linguistic variable, wherein the linguistic variable is represented by an interval triangular fuzzy number. Secondly, subjective weight of the risk factors is calculated by using a fuzzy analytic hierarchy process, objective weight of the risk factors is calculated by using an expanded VIKOR process, and comprehensive weight of the risk factors is calculated according to an improved game theory combination weighting process. And finally, sequencing the risk priority of the failure modes by using a fuzzy VIKOR method. The method is used for evaluating the workpiece box system of the numerical control gear milling machine, and the result verifies the applicability and the effectiveness of the method.
The invention is realized by the following technical scheme:
the invention provides an improved FMEA method based on an interval triangular fuzzy number and fuzzy VIKOR method. Setting of l evaluators in an FMEA evaluation team TMk(k 1, 2.. times.l) for m failure modes FMi(i 1, 2.. m.) n risk factors RFj(j ═ 1, 2.., n) was evaluated. The weight vector of the evaluators in the FMEA evaluation team is (lambda)1，λ2，……λl) And λk＞0，
Figure GDA0002746306600000021
λkIndicating the relative importance of each evaluator in the evaluation team. Setting evaluator TMkGiven the evaluation result, n fuzzy decision matrixes are obtained
Figure GDA0002746306600000031
And n fuzzy decision vectors
Figure GDA0002746306600000032
i＝1,2,…,n；k＝1,2,3。
Figure GDA0002746306600000033
And
Figure GDA0002746306600000034
a matrix that evaluates the level of each failure mode and the relative importance of each risk factor using linguistic variables for FMEA members. Wherein
Figure GDA0002746306600000035
Based on the setting, the method comprises the following steps:
step 1: and determining an evaluation target.
Step 2: potential failure modes are determined.
And step 3: the FMEA member evaluates the level of each failure mode and the relative importance of each risk factor using the linguistic variables in tables 1 and 2.
TABLE 1 linguistic variables for the level of failure modes
Figure GDA0002746306600000036
TABLE 2 linguistic variables for relative importance of risk factors
Figure GDA0002746306600000037
And 4, step 4: summarizing the evaluation results of each FMEA member to obtain n fuzzy decision matrixes
Figure GDA0002746306600000038
And n fuzzy decision vectors
Figure GDA0002746306600000039
Figure GDA0002746306600000041
Figure GDA0002746306600000042
And 5: determining a decision matrix
Figure GDA0002746306600000043
And a decision vector Wk。
Figure GDA0002746306600000044
Figure GDA0002746306600000045
Wherein
Figure GDA0002746306600000046
Step 6: computing a normalized decision matrix
Figure GDA0002746306600000047
Figure GDA0002746306600000048
And 7: calculating subjective weight of risk factor using fuzzy analytic hierarchy process
Figure GDA0002746306600000049
The Fuzzy Analytic Hierarchy Process (FAHP) has the principle that a fuzzy consistent matrix and the analytic hierarchy process are fused, so that the fuzziness of a judgment matrix is reserved, and the consistency of the judgment matrix is guaranteed. The subjective weighting steps for calculating the risk factors by using the fuzzy analytic hierarchy process are as follows:
(1) will be provided with
Figure GDA00027463066000000410
Expressed by 2 triangular fuzzy numbers, i.e.
Figure GDA00027463066000000411
Figure GDA00027463066000000412
k＝1,2,3。
(2) Defined in terms of triangular blur numbers, i.e. assuming any two triangular blur numbers
Figure GDA00027463066000000413
And
Figure GDA00027463066000000414
the degree of probability of (A) is shown in formula (6).
Figure GDA00027463066000000415
Will be provided with
Figure GDA00027463066000000416
Comparing the importance degrees of every two to construct a fuzzy complementary judgment matrix B1＝{Puv}3×3U is 1,2, 3; v is 1,2, 3. Wherein
Figure GDA00027463066000000417
And when u is equal to v,
Figure GDA00027463066000000418
the fuzzy complementary judging matrix contains the possibility degree information of mutual comparison of all elements, and the proportion of each element is determined by calculating the sorting vector of each element.
Figure GDA0002746306600000051
(3) According to the formula (8), calculating
Figure GDA0002746306600000052
Figure GDA0002746306600000053
(4) Repeating the steps (2) to (3) and calculating
Figure GDA0002746306600000054
(5) Calculating subjective weight of risk factor according to formula (9)
Figure GDA0002746306600000055
Figure GDA0002746306600000056
And 8: calculating objective weights for risk factors using extended VIKOR
Figure GDA0002746306600000057
(1) Determining an optimum value
Figure GDA0002746306600000058
Sum worst value
Figure GDA0002746306600000059
Calculating an optimum value
Figure GDA00027463066000000510
To the worst value
Figure GDA00027463066000000511
The distance of (c).
Figure GDA00027463066000000512
(2) According to the formula (10), calculating
Figure GDA00027463066000000513
To
Figure GDA00027463066000000514
Distance.
Figure GDA00027463066000000515
(3) According to equation (11), the standard blur distance is calculated.
Figure GDA00027463066000000516
(4) Calculating subjective weight of risk factor according to equation (12)
Figure GDA00027463066000000517
Figure GDA00027463066000000518
And step 9: comprehensive weight for calculating risk factor by using improved game theory combined weighting method
Figure GDA00027463066000000519
SaidThe game theory combined weighting method aims to find the maximized common points of interests among the subjective and objective weights, so that the deviation between the comprehensive weights and the subjective and objective weights is minimized. Assume that the weight vector obtained by the alpha methods is
Figure GDA00027463066000000520
α is 1,2, …, L. Let an arbitrary linear combination of L weight vectors be:
Figure GDA00027463066000000521
in the formula: epsilonαIs a linear combination coefficient ofα> 0, W represents all possible weight vectors. With W and WαIs the target, for the L linear combination coefficients ε in equation (13)αAnd (6) optimizing. Constructing a strategy model:
Figure GDA0002746306600000061
the optimization condition for equation (14) is derived from the differential nature of the matrix:
Figure GDA0002746306600000062
the equivalent of equation (15) is:
Figure GDA0002746306600000063
solving equation (16) yields the linear combination coefficient εαBut cannot guarantee ∈α> 0, which is clearly contrary to the assumption of equation (13).
By using the constraint conditions of the dispersion maximization objective weighting method for reference, the improved optimization model can be determined as follows:
Figure GDA0002746306600000064
establishing a Lagrange function to solve the model:
Figure GDA0002746306600000065
determining a linear combination coefficient epsilonαAnd to epsilonαAnd (3) carrying out normalization processing, namely determining a combination weight coefficient:
Figure GDA0002746306600000066
Figure GDA0002746306600000067
will be provided with
Figure GDA0002746306600000068
Substituting equation (13) results in the total weight:
Figure GDA0002746306600000069
step 10: s for each protocol was calculated using the fuzzy VIKOR methodj，RjAnd QjThe value of (c).
Calculating the group benefit optimal solution S according to the formula (22), the formula (23) and the formula (24)jRespectively regret worst solution RjAnd the comprehensive index QjSelecting
Figure GDA00027463066000000610
Figure GDA0002746306600000071
Figure GDA0002746306600000072
Figure GDA0002746306600000073
Step 11: according to Sj，RjAnd QjAnd sequencing the risk priority of each failure mode.
Step 12: a compromise is proposed.
Advantageous effects
(1) It is more advantageous in handling uncertain information. The diversified view of the FMEA members can be flexibly and accurately expressed through the interval triangular fuzzy number. (2) The method comprises the steps of calculating subjective weight of the risk factors by using a fuzzy analytic hierarchy process, calculating objective weight of the risk factors by using an extended VIKOR process, and obtaining comprehensive weight of the risk factors by using an improved game theory combination weighting process. The method can take the advantages of subjective and objective weights into consideration, and avoids information loss to a certain extent. (3) The failure modes are sorted using the fuzzy VIKOR method. The method fully considers the maximization of group benefit and the minimization of individual regret, and simultaneously also considers the subjective preference of a decision maker.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Detailed Description
The invention is further illustrated with reference to the following figures and examples. It is to be understood that the described embodiments are merely exemplary of the invention, and not restrictive of the full scope of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The embodiment evaluates a workpiece box system of the numerical control gear milling machine.
In this embodiment, the FMEA team consists of 1 machine tool designer, 1 assembly engineer, 1 quality inspector, and 1 operation engineer, and the number and weight thereof are TM1(λ 1)1＝0.2)、TM2(λ2＝0.35)、TM3(λ30.15) and TM4(λ)40.3). The risk factors are RF 1: degree of occurrence (O); RF 2: severity (S); RF 3: a detected degree (D).
Step 1: and determining an evaluation target. The evaluation target is a numerical control gear milling machine workpiece box system. The inside of the gear box body is provided with a worm gear transmission mechanism which adopts oil-immersed lubrication. The clamping mode of the workpiece is hydraulic clamping. One servo motor drives the workpiece shaft to rotate, and the other servo motor drives the workpiece box system to do arc reciprocating motion. During machining, the nozzle sprays the cutting fluid to a machining area.
Step 2: potential failure modes are determined. The 10 potential failure modes were identified, FM 1: oil leaks from the gear box body; FM 2: the hydraulic cylinder leaks oil; FM 3: insufficient lubrication of the bearing; FM 4: insufficient clamping force; FM 5: gear wear; FM 6: insufficient lubrication of the gears; FM 7: the motor temperature is too high; FM 8: the precision of the workpiece shaft is reduced; FM 9: the flow of cutting fluid is insufficient; FM 10: the bearing wears.
And step 3: the FMEA member evaluates the relative importance of the level of failure mode and risk factor using the linguistic variables in tables 1 and 2.
And 4, step 4: summarizing the evaluation results of each FMEA member to obtain 4 fuzzy decision matrixes
Figure GDA0002746306600000081
And 4 fuzzy decision vectors
Figure GDA0002746306600000082
The results are shown in tables 3 and 4.
TABLE 3 evaluation results of failure mode grades
Figure GDA0002746306600000083
TABLE 4 evaluation results of the relative importance of the Risk factors
Figure GDA0002746306600000091
And 5: determining a decision matrix
Figure GDA0002746306600000092
And decision vector
Figure GDA0002746306600000093
The results are shown in Table 5.
TABLE 5 comprehensive evaluation of failure mode grade and relative importance of risk factors
Figure GDA0002746306600000094
Step 6: computing a normalized decision matrix
Figure GDA0002746306600000095
The results are shown in Table 6.
TABLE 6 normalized decision matrix
Figure GDA0002746306600000096
Figure GDA0002746306600000101
And 7: calculating subjective weight of risk factor using fuzzy analytic hierarchy process
Figure GDA0002746306600000102
(1) Will be provided with
Figure GDA0002746306600000103
Represented by 2 triangular blur numbers.
Figure GDA0002746306600000104
Figure GDA0002746306600000105
Figure GDA0002746306600000106
(2) Constructing a fuzzy complementary judgment matrix B1And B2。
Figure GDA0002746306600000107
Figure GDA0002746306600000108
(3) Computing
Figure GDA0002746306600000109
And
Figure GDA00027463066000001010
the value of (c).
Figure GDA00027463066000001011
Figure GDA00027463066000001012
(4) Calculating subjective weights for risk factors
Figure GDA00027463066000001013
Figure GDA00027463066000001014
And 8: calculating objective weights for risk factors using extended VIKOR
Figure GDA00027463066000001015
(1) Computing
Figure GDA00027463066000001016
To
Figure GDA00027463066000001017
The results are shown in Table 7.
TABLE 7
Figure GDA00027463066000001018
To
Figure GDA00027463066000001019
Is a distance of
Figure GDA00027463066000001020
Figure GDA0002746306600000111
And
Figure GDA0002746306600000112
the value of (c).
Figure GDA0002746306600000113
Figure GDA0002746306600000114
(3) Calculating objective weights for risk factors
Figure GDA0002746306600000115
Figure GDA0002746306600000116
And step 9: comprehensive weight for calculating risk factor by using improved game theory combined weighting method
Figure GDA0002746306600000117
(1) And calculating a combination weight coefficient.
Figure GDA0002746306600000118
(2) Calculating composite weights for risk factors
Figure GDA0002746306600000119
Figure GDA00027463066000001110
Figure GDA00027463066000001111
Figure GDA00027463066000001112
Step 10: s for each protocol was calculated using the fuzzy VIKOR methodj，RjAnd QjThe results are shown in Table 8, Table 9 and Table 10.
TABLE 8 failure mode SjValue of
Figure GDA00027463066000001113
TABLE 9 failure mode RjValue of
Figure GDA00027463066000001114
TABLE 10 failure mode QjValue of
Figure GDA00027463066000001115
Step 11: according to Sj，RjAnd QjAnd sequencing the risk priority of each failure mode.
(1) As can be seen from Table 11, according to Sj，RjAnd QjThe risk priorities of the failure modes are ranked, FM2 is the most severe failure mode and should be given the highest risk priority, and the risk priority order of 10 failure modes is FM2>FM4>FM8>FM3>FM1>FM10>FM6>FM5>FM7>FM9。
TABLE 11 order of failure mode Risk prioritization
Figure GDA0002746306600000121
Step 12: a compromise is proposed.
The compromise herein corresponds to the lowest risk priority scheme. According to the rule described in the fuzzy VIKOR method,
Figure GDA0002746306600000122
FM9 as Sj、RjAnd QjThe ranking is the scheme with the lowest risk priority, and the condition 1 and the condition 2 are met. Thus. Compromise FM9 for failure mode. The present invention is not limited to the above embodiments, and any person skilled in the art can substitute or change the technical solution and the inventive concept of the present invention within the technical scope of the present invention.

Claims (1)

1. The improved FMEA method based on the interval triangular fuzzy number and fuzzy VIKOR method is characterized by comprising the following steps of:
step 1: determining an evaluation target; the numerical control gear milling machine workpiece box system comprises a worm gear transmission mechanism arranged in a gear box body, and oil-immersed lubrication is adopted; the clamping mode of the workpiece is hydraulic clamping; one servo motor drives the workpiece shaft to rotate, and the other servo motor drives the workpiece box system to do arc reciprocating motion; during machining, the nozzle sprays cutting fluid to a machining area; the risk factors are RF 1: degree of occurrence (O); RF 2: severity (S); RF 3: a detected degree (D);
step 2: determining a potential failure mode; for a numerically controlled gear milling machine work box system, 10 potential failure modes were determined, FM 1: oil leaks from the gear box body; FM 2: the hydraulic cylinder leaks oil; FM 3: insufficient lubrication of the bearing; FM 4: insufficient clamping force; FM 5: gear wear; FM 6: insufficient lubrication of the gears; FM 7: the motor temperature is too high; FM 8: the precision of the workpiece shaft is reduced; FM 9: the flow of cutting fluid is insufficient; FM 10: bearing wear;
and step 3: the FMEA member evaluates the level of each failure mode and the relative importance of each risk factor using the linguistic variables in tables 1 and 2:
TABLE 1 linguistic variables for the level of failure modes
Figure FDA0003535580730000011
Figure FDA0003535580730000021
TABLE 2 linguistic variables for relative importance of risk factors
Figure FDA0003535580730000022
And 4, step 4: summarizing the evaluation results of each FMEA member to obtain n fuzzy decision matrixes
Figure FDA0003535580730000023
And n fuzzy decision vectors
Figure FDA0003535580730000024
Figure FDA0003535580730000025
Figure FDA0003535580730000026
And 5: determining a decision matrix
Figure FDA0003535580730000027
And a decision vector Wk；
Figure FDA0003535580730000028
Figure FDA0003535580730000029
Wherein the weight vector of evaluators in the FMEA evaluation team is (lambda)1，λ2，……λw) And λk>0，
Figure FDA00035355807300000210
λkRepresenting the relative importance of each evaluator in the evaluation team;
Figure FDA0003535580730000031
step 6: computing a normalized decision matrix
Figure FDA0003535580730000032
Figure FDA0003535580730000033
And 7: calculating the subjective weight of the risk factor by using a fuzzy analytic hierarchy process:
(1) will be provided with
Figure FDA0003535580730000034
Expressed by 2 triangular fuzzy numbers, i.e.
Figure FDA0003535580730000035
Figure FDA0003535580730000036
k＝1,2,3；
(2) Defined in terms of triangular blur numbers, i.e. assuming any two triangular blur numbers
Figure FDA0003535580730000037
And
Figure FDA0003535580730000038
Figure FDA0003535580730000039
the degree of possibility of (A) is shown in formula (6);
Figure FDA00035355807300000310
will be provided with
Figure FDA00035355807300000311
Comparing the importance degrees of every two to construct a fuzzy complementary judgment matrix B1＝{Puv}3×3U is 1,2, 3; v is 1,2, 3; wherein
Figure FDA00035355807300000312
And when u ═When the value is v, the number of the grooves is,
Figure FDA00035355807300000313
the fuzzy complementary judgment matrix comprises the possibility degree information of mutual comparison of all elements, and the proportion of each element is determined by calculating the sequencing vector of each element;
Figure FDA00035355807300000314
(3) according to the formula (8), calculating
Figure FDA00035355807300000315
Figure FDA00035355807300000316
(4) Repeating the steps (2) to (3) and calculating
Figure FDA00035355807300000317
(5) Calculating subjective weight of risk factor according to formula (9)
Figure FDA00035355807300000318
Figure FDA00035355807300000319
And 8: calculating objective weights for risk factors using extended VIKOR
Figure FDA00035355807300000320
(1) Determining an optimum value
Figure FDA0003535580730000041
Sum worst value
Figure FDA0003535580730000042
Calculating an optimum value
Figure FDA0003535580730000043
To the worst value
Figure FDA0003535580730000044
The distance of (d);
Figure FDA0003535580730000045
(2) according to the formula (10), calculating
Figure FDA0003535580730000046
To
Figure FDA0003535580730000047
A distance;
Figure FDA0003535580730000048
(3) calculating a standard fuzzy distance according to the formula (11);
Figure FDA0003535580730000049
(4) calculating subjective weight of risk factor according to equation (12)
Figure FDA00035355807300000410
Figure FDA00035355807300000411
And step 9: combined weighting method using improved game theoryCalculating the composite weight of the risk factor
Figure FDA00035355807300000412
The game theory combined weighting method aims at searching a maximized common interest point between the subjective and objective weights, so that the deviation between the comprehensive weight and the subjective and objective weights is minimized; assume that the weight vector obtained by the alpha methods is
Figure FDA00035355807300000413
α ═ 1,2, …, L; let an arbitrary linear combination of L weight vectors be:
Figure FDA00035355807300000414
in the formula: epsilonαIs a linear combination coefficient ofα>0, W represents all possible weight vectors; with W and WαIs the target, for the L linear combination coefficients ε in equation (13)αOptimizing; constructing a strategy model:
Figure FDA00035355807300000415
the optimization condition for equation (14) is derived from the differential nature of the matrix:
Figure FDA00035355807300000416
the equivalent of equation (15) is:
Figure FDA00035355807300000417
solving equation (16) yields the linear combination coefficient εαBut cannot guarantee ∈α>0, this showsHowever, it is contrary to the assumption of equation (13);
by using the constraint conditions of the dispersion maximization objective weighting method for reference, the improved optimization model can be determined as follows:
Figure FDA0003535580730000051
establishing a Lagrange function to solve the model:
Figure FDA0003535580730000052
determining a linear combination coefficient epsilonαAnd to epsilonαAnd (3) carrying out normalization processing, namely determining a combination weight coefficient:
Figure FDA0003535580730000053
Figure FDA0003535580730000054
will be provided with
Figure FDA0003535580730000055
Substituting equation (13) results in the total weight:
Figure FDA0003535580730000056
step 10: s for each protocol was calculated using the fuzzy VIKOR methodj，RjAnd QjA value of (d);
calculating the group benefit optimal solution S according to the formula (22), the formula (23) and the formula (24)jRespectively regret worst solution RjAnd the comprehensive index QjSelecting
Figure FDA0003535580730000057
Figure FDA0003535580730000058
Figure FDA0003535580730000059
Figure FDA00035355807300000510
Step 11: according to Sj，RjAnd QjRanking the risk priority of each failure mode;
step 12: a compromise is proposed, which corresponds to the failure mode with the lowest priority risk.
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