Journal of Applied Science and Engineering, Vol. 19, No. 2, pp. 177-186 (2016)

DOI: 10.6180/jase.2016.19.2.08

A Modified Method for Risk Evaluation in Failure Mode and Effects Analysis Jun-Li Shi 1,2*, Ya-Jun Wang1, Hai-Hua Jin1, Shuang-Jiao Fan1, Qin-Yi Ma1 and Mao-Jun Zhou1 1

School of Mechanical Engineering and Automation, Dalian Polytechnic University, Dalian 116034, P.R. China 2 Institute of Sustainable Design and Manufacturing, Dalian University of Technology, Dalian 116024, P.R. China

Abstract This study proposes a modified failure mode and effects analysis (FMEA) method based on fuzzy set theory and fuzzy analytic hierarchy process (FAHP) by analyzing the limitations of the traditional FMEA. First, the fuzzy language set of severity, occurrence, and detection is set up in this method. Second, the failure mode is evaluated by a triangular fuzzy number based on the fuzzy language set. Then, the weights of severity, occurrence, and detection are determined by the FAHP. Finally, the risk priority of the failure modes is determined by the modified risk priority number (RPN). The efficiency and feasibility of the modified FMEA method are verified by using it to deal with risk evaluation of the failure modes for a compressor crankshaft. Key Words: Failure Mode and Effects Analysis, Fuzzy Language Set, Triangular Fuzzy Number, Fuzzy Analytical Hierarchy Process, Risk Priority Number

1. Introduction The failure mode and effects analysis (FMEA), which was first developed as a formal design methodology in the 1960s, is an extensively used risk assessment tool to define and identify potential failures in products, processes, designs, and services [1]. The FMEA technique has been extensively used in a wide range of industries, such as in the aerospace, automotive, electronics, medical and mechanical technology industries [2-5]. In FMEA, prioritization of the failure modes is generally determined through the risk priority number (RPN), which provides an effective method of ranking the failure modes. The traditional RPN is obtained by multiplying the occurrence (O), severity (S), and detection (D) of a failure mode, as expressed in Eq. (1): *Corresponding author. E-mail: [email protected]

RPN = S ´ O ´ D

(1)

where, S is the severity of the failure mode, O is the occurrence of the failure mode, and D is the probability of not detecting the failure mode. The higher the RPN value of a failure mode, the greater the risk for the product/ system. Three risk factors are evaluated by a numeric scale (rating) from 1 to 10 to obtain the RPN of a potential failure mode. Table 1 shows the proposed criteria of the rating for S of a failure mode in the FMEA. The numeric scales for O and D follow the same criteria, because of the length limitation no more tautology here. However, the RPN is criticized in most cases as a crisp value because S, O, and D are crisp numbers [6]. The three main reasons for this are the following: First, experts encounter difficulties in giving a precise number for the three risk parameters in the crisp model because FMEA experts’ opinions are mainly subjective and qual-

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Table 1. Suggested rating criteria of failure severity in FMEA Rating

Effect

Severity of effect

1 2 3 4 5 6 7 8 9

None Very minor Minor Low Moderate Significant Major Extreme Hazardous with warning

10

Hazardous without warning

No effect System performance and satisfaction with slight effect System performance and satisfaction with minor effect System performance is small affected, maintenance may not be needed Performance of system or product is affected seriously, maintenance is needed Operation of system or product is continued and performance is degraded Operation of system or product may be continued but performance is affected Operation of system or product is broken down without compromising safe Higher severity ranking of a failure mode, occurring with warning, consequence is hazardous Highest severity ranking of a failure mode, occurring without warning and consequence is hazardous

itative descriptions [7,8]. Second, the three risk factors are considered to be of equal importance, and the relative importance of the risk parameters is not considered while calculating the RPN value [9]. Third, different combinations of O, S, and D may result in exactly the same RPN value, which in reality may be a different risk implication altogether [10]. The fuzzy concepts of the FMEA have been applied in many attempts to reduce the aforementioned drawbacks. Liu et al. [11] proposed a intuitionistic fuzzy hybrid technique for order preference by similarity to an ideal solution (TOPSIS), which is a new modified method to determine the risk priorities of the identified failure modes. Mandal and Maiti [12] introduced fuzzy numerical technique to remove the drawbacks of the crisp FMEA. This technique integrated the concepts of the similarity value measure of fuzzy numbers and possibility theory. The methodology is more robust because it does not require arbitrary precise operations (e.g., defuzzification) to prioritize the failure modes. Wang et al. [13] dealt with the problem of the crisp RPN not realistically determining the risk priority of the failure modes by proposing fuzzy RPNs (FRPNs). The FRPNs in their method were defined as the fuzzy weighted geometric means of the fuzzy grades for O, S, and D. These means were computed using alpha-level sets and linear programming models. Hu et al. [14] applied the fuzzy analytic hierarchy process (FAHP) to determine the relative weights of

four factors when analyzing the risks of green components in the incoming quality control stage in Taiwan. A green component RPN is used to calculate each of the components in this method. Xu et al. [15] presented a fuzzy logic-based FMEA method to address the issue of uncertain failure modes. They also showed a platform for a fuzzy expert assessment, which was integrated with the proposed method. The aforementioned literature review shows significant achievements in fuzzy FMEA research. This study proposes a new risk assessment model by incorporating the traditional FMEA and FAHP theory. The new model could not only evaluate expert knowledge and experiences more reasonably but could also consider the relative importance of the risk parameters (i.e., S, O, and D) while calculating the RPN value.

2. Modified FMEA Method Based on Fuzzy Theory and the FAHP Figure 1 shows the broad framework of this method. This approach is developed to determine the functional process and possible failure modes of the products, analyze the causes and effects, and establish the fuzzy language set and triangular fuzzy number (TFN) for S, O, and D. Then, a clear number is calculated using defuzzification mathematical operations. Subsequently, a paired comparison matrix is built using FAHP theory to deter-

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set is defined as a language fuzzy set, as: S/O/D = {very low (VL), low (L), medium (M), high (H), very high (VH)}, the meaning of the fuzzy language set is shown in Table 2.

2.2 Defuzzification Algorithm of the TFN Quantification of the fuzzy language set can be achieved using TFN. TFN is one of the major components of fuzzy set theory, which is designed to deal with the extraction of the primary possible outcome from a multiplicity of information vaguely and imprecisely [16]. According to Laarhoven and Pedrycz [17], a TFN should possess the following features: ~ Assuming that the TFN is A = (l, m, u), the membership function is defined in Eq. (2):

Figure 1. Broad framework of modified FMEA.

mine the weights of S, O, and D. Finally, the modified RPN of each failure mode is calculated and corresponding improvement measures are implemented accordingly.

2.1 Fuzzy Language Set of S, O and D In traditional FMEA, since the assessment information for risk factors mainly based on experts’ subjective judgments, there is a high level of uncertainty involved. In this paper, we choose linguistic terms for the assessment of risk factors and the individual evaluation grade

(2)

where l and u represent the lower and upper bounds of

the TFN, and m is the median value. TFN can be determined on the basis of the experts’ knowledge and experiences. Assuming the presence of n experts, if the weight of the ith expert is ai, then the fuzzy evaluation variable of a failure model for this expert is xi, xi Î (1, 10) and xi = (li, mi, ui). Then, the weighted average TFN of this variable is obtained using Eqs. (3) to (6):

Table 2. Meaning of fuzzy language set for S, O, D Evaluation degree Very low (VL) Low (L) Medium (M) High (H) Very high (VH)

Meaning of the fuzzy language set Severity (S) System can basically or even cannot meet the requirements, but few customers could find defective System can run, but the performance of comfort or convenience decreased, customer feel not satisfied System can run, but the components of comfort or convenience cannot work, customers feel not satisfied System can run, but the performance drops, the customer feel not satisfied System cannot run, the basic functions are lost

Occurrence (O)

Detection (D)

Failure is unlikely occurs Failure rarely occurs Failures sometimes occur Failures often occur Failures occur

Probability of failure be detected out is very high Probability of failure be detected out is high Failure cannot be detected out occasionally Probability of failure be detected out is relatively low Probability of failure be detected out is very low

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(3)

(4)

(5) ai can be expressed as follows:

(6) ~ The operational laws of the two TFNs (i.e., A1 (l1, ~ m1, u1) and A2 (l2, m2, u2)) comply with the following rules [18] shown in Eqs. (7) to (10): Addition of the fuzzy number:

(7) Multiplication of the fuzzy number:

(8)

where j(x) is the clear number of xi. M and N are determined by the degree of the deviation of l, m, and u, which indicates that the possibility of m may be M times of u and N times of l.

2.3 Weight Determination of S, O and D Based on the FAHP The AHP process, which was first introduced by Saaty [20], is one of the extensively used multi-criteria decision-making methods. However, the AHP is frequently criticized because of its inability to overcome fuzziness deficiency during decision making [21]. Therefore, the FAHP, which is a fuzzy extension of the AHP, is developed to solve vague problems. Laarhoven and Pedrycz [17] incorporated Saaty’s AHP into fuzzy theory. The FAHP procedure of determining the weights of S, O, and D is described as follows: Step 1: Construction of a hierarchical structure. The goal of the desired problem is placed on the top layer of the hierarchical structure, the evaluation criteria and the alternatives are placed on the second and bottom layer. ~ Step 2: Construction of the fuzzy judgment matrix C . ~ The fuzzy judgment matrix C is a pair wise comparison matrix of each alternative and evaluation criterion, which is expressed in Eqs. (12) and (13), as follows:

Division of the fuzzy number: (12) (9)

Reciprocal of the fuzzy number: (10) With regard to the defuzzification algorithm of the TFN, this study selects the non-fuzzy method presented by Xiao and Lee [19]. The calculation is presented in Eq. (11):

(11)

(13)

Linguistic terms are assigned to the pair wise comparisons by investigating which of the two criteria is more important [22]. Table 3 is the membership function of linguistic scale

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3. Case Study

Step 3: Calculation of the TFN weights of each criterion. The calculation rules of the TFN weights are taken from Buckley [23]. The computations are shown in Eq. (14):

(14) where ~ cij is the fuzzy comparison value of criteria i to j, ~ r is the geometric mean of the fuzzy comparison value ~ is the fuzzy weight of criterion i to each criterion, and w i of the ith criterion (i.e., i is S, O, and D). Step 4: Calculation of the normalized clear weights of each criterion. ~ of each criterion can be transThe TFN weight w i formed to the clear weight w' t using Eq. (11). The clear weight would be transformed to a normalized clear weight, as expressed in Eq. (15): (15) where w¢S, w¢O, and w¢D are the clear weights of S, O, and D, respectively, and wi is the normalized clear weight of S, O, and D.

Compressor is the core component of air conditioning systems and refrigeration products. Its quality performance directly affects the quality of these refrigeration products. A compressor company plans to develop a new type of scroll compressor and set up an FMEA project team. The first task for this team is to identify and predict the potential failure modes for six large parts of the compressor. The “crankshaft” plays an important role in the compressor and is usually used in power transmission, which demands higher quality and reliability. The process using the modified FMEA method to identify and evaluate the potential failure modes for this component is discussed in the subsequent sections.

3.1 Potential Failure Mode Analysis of the Compressor Crankshaft Five cross-functional members in the FMEA team decide to evaluate the modes using linguistic terms. The five members are assigned the relative weights of 0.2, 0.3, 0.1, 0.2 and 0.2. Table 4 defines the fuzzy language sets for five main potential failure modes. 3.2 Determining the TFN and Clear Number The assessment information of the TFNs for the five failure modes is presented in Table 5. The average TFN of xi is calculated using Eqs. (3) to (6). The corresponding clear number is obtained through Eq. (11). When taking the calculation process “VL” of “bad hardness” as an example, the average weight TFN becomes (l, m, u) = (1.3, 2.3, 2.7). Accordingly, M = m/u = 0.85, N = m/l = 1.77. Therefore, the clear number is obtained as follows:

2.4 RPN Calculation This calculation process aims to obtain the value of the modified RPN based on the clear numbers and weights of the three risk factors determined by Eqs. (11) and (15). The modified RPN is calculated, as shown in Eq. (16), for each failure mode:

(17)

(16)

The clear numbers of L, M, H, and VH for all the

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failure modes identified in the FMEA are obtained using the same method. Table 6 shows the results.

3.3 Determining the Weights of S, O and D The fuzzy judgment matrix of S, O, and D that corre-

Table 4. Linguistic terms and fuzzy language sets of five failure modes for compressor crankshaft Potential failure mode analysis Potential failure mode

Consequences of failures

Causes of failures

S

O

D

Bad hardness Coaxiality tolerance Interleaving burr Super size difference Cylindricity error

Unstable working performance Unable to install and connect Unstable working performance Unable to install the connection Cause the device to the cutter

Serious abrasion of Jigs and fixtures Bad clamping and positioning Worker’s weak quality awareness Error compensation value of tools Top Seriously wear

H L M VH VH

M VL L H L

L M VL M M

Table 5. Triangular fuzzy number of compressor crankshaft Potential failure mode Bad hardness

Triangular fuzzy number (TFN) Expert No

a

VL

L

M

H

VH

1 2 3 4 5

0.2 0.3 0.1 0.2 0.2

1 2 3 4 5

0.2 0.3 0.1 0.2 0.2

1 2 3 4 5

0.2 0.3 0.1 0.2 0.2

1 2 3 4 5

0.2 0.3 0.1 0.2 0.2

1 2 3 4 5

0.2 0.3 0.1 0.2 0.2

(1.0, 2.1, 2.6) (1.4, 2.3, 2.9) (1.2, 2.4, 2.7) (1.6, 2.2, 2.8) (1.3, 2.3, 2.6) (1.3, 2.3, 2.7) (1.0, 2.4, 3.1) (1.3, 2.1, 3.3) (1.4, 2.3, 3.5) (1.3, 2.2, 3.4) (1.2, 2.5, 3.2) (1.2, 2.3, 3.3) (1.1, 2.4, 3.1) (1.3, 2.6, 3.8) (1.4, 2.8, 3.8) (1.3, 2.5, 3.9) (1.2, 2.6, 3.2) (1.3, 2.6, 3.6) (1.4, 2.4, 3.4) (1.3, 2.1, 2.8) (1.8, 2.8, 3.7) (1.5, 2.5, 2.9) (1.4, 2.2, 3.3) (1.4, 2.3, 3.1) (1.1, 2.4, 3.1) (1.3, 2.6, 2.8) (1.4, 2.8, 2.8) (1.3, 2.5, 2.9) (1.2, 2.6, 3.2) (1.3, 2.6, 3.0)

(2.2, 3.0, 4.2) (2.4, 3.5, 4.6) (2.6, 3.2, 4.5) (2.0, 3.3, 4.8) (2.4, 3.4, 4.7) (2.3, 3.3, 4.6) (2.6, 3.5, 4.5) (2.7, 3.6, 4.4) (2.5, 3.4, 4.3) (2.6, 3.3, 4.2) (2.9, 3.6, 5.1) (2.7, 3.5, 4.5) (3.1, 4.1, 4.9) (3.2, 4.2, 5.1) (2.9, 4.3, 5.3) (3.1, 4.1, 4.8) (3.2, 4.2, 5.1) (3.2, 4.2, 5.0) (2.9, 4.2, 5.4) (3.4, 4.5, 5.3) (2.8, 3.9, 4.9) (3.3, 4.3, 4.8) (3.8, 4.6, 5.5) (3.3, 4.4, 5.2) (3.6, 4.1, 4.6) (3.7, 4.2, 5.1) (3.5, 4.4, 5.3) (3.6, 4.3, 5.2) (3.9, 4.6, 5.1) (3.7, 4.3, 5.0)

(3.3, 4.6, 6.5) (3.9, 4.7, 6.3) (3.4, 4.3, 6.4) (3.5, 4.5, 6.2) (3.6, 4.4, 6.0) (3.6, 4.5, 6.3) (4.7, 5.8, 6.4) (4.7, 5.7, 6.5) (4.6, 5.8, 6.4) (4.8, 5.6, 6.2) (5.1, 5.8, 6.3) (4.7, 5.7, 6.4) (4.7, 5.8, 6.4) (4.7, 5.7, 6.1) (4.6, 5.7, 6.3) (4.8, 5.6, 6.2) (5.0, 5.8, 6.3) (4.7, 5.7, 6.4) (4.8, 5.8, 6.8) (4.7, 5.7, 6.7) (5.0, 5.8, 6.7) (4.6, 5.6, 6.6) (4.9, 5.9, 6.6) (4.8, 5.7, 6.7) (4.7, 5.8, 6.4) (4.7, 5.7, 6.5) (4.6, 5.8, 6.4) (4.8, 5.6, 6.2) (5.0, 5.8, 6.3) (4.8, 5.7, 6.4)

(5.3, 7.6, 8.6) (5.9, 7.4, 8.8) (5.8, 7.8, 8.5) (5.7, 7.5, 8.7) (5.5, 7.6, 8.4) (5.7, 7.5, 8.6) (6.8, 7.3, 7.9) (6.7, 7.4, 8.3) (6.9, 7.8, 8.5) (6.7, 7.5, 8.3) (6.6, 7.6, 8.4) (6.7, 7.5, 8.3) (6.6, 7.1, 7.8) (6.9, 7.4, 8.4) (6.6, 7.2, 7.7) (6.7, 7.4, 8.2) (6.7, 7.6, 8.3) (6.7, 7.4, 8.2) (6.8, 7.8, 8.8) (6.7, 7.6, 8.5) (5.9, 6.9, 7.6) (6.7, 7.7, 8.3) (6.6, 7.6, 8.6) (6.7, 7.4, 8.5) (6.8, 7.6, 8.4) (6.7, 7.4, 8.3) (6.6, 7.5, 8.5) (6.7, 7.5, 8.3) (6.6, 7.6, 8.4) (6.7, 7.5, 8.4)

(8.3, 9.6, 10) (7.9, 9.8, 10) (8.2, 9.5, 10) (7.8, 9.7, 10) (8.5, 9.4, 10) (8.1, 9.6, 10) (8.6, 9.5, 10) (7.9, 9.1, 10) (8.8, 9.4, 10) (8.7, 9.6, 10) (7.8, 9.3, 10) (8.3, 9.3, 10) (8.7, 9.7, 10) (8.8, 9.6, 10) (7.9, 9.5, 10) (7.7, 9.2, 10) (8.6, 9.5, 10) (8.3, 9.5, 10) (8.7, 9.7, 10) (7.8, 8.8, 10) (8.6, 9.5, 10) (7.8, 8.9, 10) (8.7, 9.7, 10) (8.2, 9.3, 10) (8.6, 9.5, 10) (7.9, 9.1, 10) (7.8, 9.4, 10) (8.7, 9.6, 10) (8.1, 9.1, 10) (8.2, 9.3, 10)

Weighted average

Coaxiality tolerance

Weighted average

Interleaving burr

Weighted average

Supersize difference

Weighted average

Cylindricity error

Weighted average

The TFN here is the measurement of failure mode getting from experts’ experiences and knowledge, for example, (1.0, 2.1, 2.6) is the TFN of ‘VL’ for ‘Bad hardness’ getting from NO. 1 expert, in his opinion, the smallest value is 1.0, the biggest value is 2.6, and the median value is 2.1. The other TFNs are obtained as the same way.

A Modified Method for Risk Evaluation in Failure Mode and Effects Analysis

tained using Eqs. (11) and (15), respectively (Table 8).

Table 6. Clear number of failure modes Potential failure mode Bad hardness Coaxiality tolerance Interleaving burr Super size difference Cylindricity error

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Clear number VL

L

M

H

VH

2.2 2.4 2.7 2.4 2.4

3.5 3.6 4.2 4.4 4.3

4.8 5.7 5.7 5.8 5.7

7.4 7.5 7.4 7.5 7.5

9.4 9.2 9.3 9.2 9.2

sponds to the relative importance of the RPN is determined using Eq. (12) (Table 7). The weights of S, O, and D are calculated using the method proposed in section 2.3. Table 8 shows the results. The weights of wS, wO, and wD for “bad hardness” are calculated as follows:

(18)

The clear weight numbers of w¢S, w¢O, and w¢D and the normalized clear weights of wS, wO, and wD are ob-

3.4 Calculating the Modified RPN Value and Determining the Risk Ranking Finally, the modified RPN value is calculated using Eq. (16). The first failure mode “bad hardness” is taken as an example, as follows: Modified RPN = swC ´ owo ´ dwD =7.4 ´ 0.54 ´ 4.8 ´ 0.28 ´ 3.5 ´ 0.18 = 3.38 The modified RPN value for all the failure modes is calculated using the same method (Table 9). The clear numbers of the modified S, O, and D are obtained from Tables 4 and 6. The weights are obtained from Table 8. As is shown in Table 9, the RPN values of “super size difference” and “bad hardness” are ranked as first and second, respectively. Therefore, they have the highest risk and should be well controlled.

3.5 Traditional RPN Value and Risk Ranking Table 10 shows the risk ranking of the failure modes according to the traditional FMEA method. The traditional values of S, O, and D are obtained from the previous FMEA team of this compressor company. The value selection criteria are obtained from Table 1, and the RPN value is calculated using Eq. (1). Table 10 shows that “super size difference” and “cylindricity error” are the first and second risk potential failure modes to be controlled. 3.6 Comparison and Discussion Figure 2 shows the percentage comparison of the two RPN alternatives for five failure modes. It is clearly that

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Table 9. Modified RPN risk ranking Potential failure mode Bad hardness Coaxiality tolerance Interleaving burr Super size difference Cylindricity error

Clear number of S, O, D and RPN risk ranking s

o

d

RPN

Percent (%)

Risk ranking

7.4 3.6 5.7 9.2 9.2

4.8 2.4 4.2 7.5 2.4

3.5 5.7 2.7 5.8 5.7

3.38 1.03 0.27 6.95 1.45

25.84 07.87 02.06 53.14 11.09

2 4 5 1 3

Table 10. Traditional RPN risk ranking Potential failure mode Bad hardness Coaxiality tolerance Interleaving burr Super size difference Cylindricity error

Traditional number of S, O, D and RPN risk ranking s

o

d

RPN

Percent (%)

Risk ranking

8 4 5 9 9

5 3 3 8 4

4 6 2 6 6

160 072 030 432 216

17.58 07.91 03.30 47.47 23.74

3 4 5 1 2

Figure 2. Modified and traditional RPN percentage of five failure modes.

the two RPN results are similar with each other. The most serious failure mode in the two methods is “super

size difference,” which is 53.14% and 47.47% in the modified and traditional RPNs, respectively. The two least serious failure modes are “interleaving burr” and “coaxiality tolerance” in both methods and the percentage results are similar also. The “interleaving burr” of all the failure modes is only 2.06% and 3.30% in the modified and traditional RPNs, whereas the “coaxiality tolerance” is 7.87% and 7.91%, respectively. However, the severity and detection of the failure mode “cylindricity error” are very high in the traditional FMEA, the RPN value is therefore also higher than the others and ranks second in the modes to be controlled. By contrast, the occurrence of “bad hardness” is higher than “cylindricity error” in the modified FMEA, the RPN value is also higher, which takes this mode to the second

A Modified Method for Risk Evaluation in Failure Mode and Effects Analysis

place, as a result, the risk priority is changed. This is because that the clear numbers of S, O, and D for the failure mode are obtained by the TFN operation, and the weight is fully considered. Furthermore, expert knowledge and experience are more reasonably processed, which could enable the compressor FMEA team to control the measures for the failure modes more objectively. Consequently, the modified FMEA method could make more comprehensive and accurate judgments on the risk priority for the failure modes, which overcome the limitation of crisp RPN, and can be practically used in industrial production.

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number is not limited to the case presented in this study. This method could certainly be used for other products or systems to determine potentially high-risk failure modes.

Acknowledge The authors gratefully acknowledge the support of Liaoning Province Natural Science Foundation (20140 26006) and Dalian Sanyo Co., LTD. The authors would like to thank the editor and reviewers for their constructive suggestions of the paper.

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The FMEA, which has been extensively used in industries, plays an important role in analyzing safety and reliability. This study develops and applies a modified FMEA method to determine the risk priority of the failure modes considering the difficulty in acquiring precise assessment information on failure modes. Accordingly, expert knowledge and experiences are fully considered. This method could provide a qualitative evaluation of the failure modes by establishing a fuzzy language set and TFNs. The weights of severity, occurrence, and detection can also be determined through the FAHP by comprehensively considering the importance of each variable and the decision maker’s risk preference. In this way, the limitations associated with the traditional crisp RPN-based FMEA in risk and failure analysis can be overcome to a significant extent. The potential failure modes of the “compressor crankshaft” are evaluated to test and verify the feasibility and validity of the proposed method. This evaluation is conducted by calculating and comparing the modified and traditional RPNs and determining the risk priority ranking of each failure mode. Notably, the TFNs determining for each potential failure mode is based on expert investigation when using this modified FMEA method. Therefore, the experts selected must be familiar with product design and production. The TFN algorithm employed to derive the clear

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Manuscript Received: Nov. 27, 2015 Accepted: Apr. 21, 2016