In what follows, we use the proposed approach to identify the key failure modes in the product development of the HDD machine. The steps are summarized as follows.
5.1. Implement the Proposed Method
An FMEA team consisting of five cross-functional experts Ek (k = 1,2,3,4,5) from different departments is formed to identify the most important failure modes; we suppose that the relative weights of the five team experts are assigned as 0.15, 0.2, 0.25, 0.1, and 0.3 according to their different professional knowledge and expertise. The FMEA team identifies nine potential failure modes through brainstorming, namely, gear abrasion of dynamic head, action invalidation of force motor, non-normal friction of pedrail, leak of hydraulic system, abrasion of feed mechanism, unexpected halt of engine, cavitation erosion of hydraulic pump, failures of hydraulic system induced by hydraulic oil pollution, and nozzle choking of aiguilles.
Step1: Construct linguistic assessment matrix.
In a practical application, the five experts use the linguistic variables to evaluate the nine failure modes on each risk factor and the relative importance of risk factors; the evaluation results are shown in
Table 3.
Step 2: Construct a consensus-reaching process.
Step 2.1: Generally, the threshold value and the maximum number of cycles are determined by the team experts according to the actual situation. In this case study, let , , and .
Step 2.2: Convert all linguistic variables in each linguistic evaluation matrix
L1,
L2,
L3,
L4, and
L5 into the IVPFNs according to
Table 2 to construct assessment matrices
A1,
A2,
A3,
A4, and
A5.
Step 2.3: The collective matrix
C is aggregated by the IVPFWA operator to compute the consensus degree.
Step 2.4: Calculate the consensus degree.
Based on Equations (11) and (12), the
CDF value is determined as follows.
Then, according to Equation (13), the
CDM value is calculated as follows.
Finally, using Equation (14), the CDT value is obtained as .
Step 2.5: Due to , go to next step.
Step 2.6: Based on Equations (16)–(18), the elements of failure modes that need to be modified are determined to be:
AES = {(2,1,1), (2,1,3), (2,4,1), (2,4,2), (2,4,3), (2,5,1), (2,5,2), (2,5,3), (2,6,3), (2,7,2), (2,8,1), (2,8,2), (2,8,3), (3,2,2), (3,2,3), (3,3,1), (3,3,2), (3,4,1), (3,4,2), (3,4,3), (3,6,1), (3,6,2), (3,7,1), (3,7,2), (3,7,3), (3,8,2), (3,9,1), (3,9,2)}.
Step 2.7: According to Equation (19), the personalized suggestions for experts are obtained. The modified assessment information on the nine failure modes is shown in
Table 4. Then, let
, go back to step 2.2.
Next is the second round of the consensus-reaching process.
Step 2.2: Convert all linguistic variables in each linguistic evaluation matrix
L2 and
L3 into the IVPFNs according to
Table 2 to construct assessment matrices
A2 and
A3.
Step 2.3: Based on the IVPFWA operator, the comprehensive matrix
C is obtained as follows:
Step 2.4 Compute the consensus degree.
Based on Equations (11) and (12), the
CDF value is determined as follows.
Then, according to Equation (13), the CDM value is calculated as follows.
Finally, using Equation (14), the CDT value is obtained as .
Step 2.5: Due to , go to step 2.8.
Step 2.8: Output the adjusted IVPFN assessment matrix.
Step 3: The comprehensive evaluation matrix
D is obtained by considering the weights of experts and applying the IVPFWGBM operator. The result is as follows.
Step 4: Calculate the combination weights of risk factors.
Step 4.1: Determine the subjective weights of risk factors.
The collective evaluation matrix H of risk factors is constructed by aggregating all individual assessments for risk factors. Subsequently, the normalized subjective weights of risk factors based on Equation (21) are determined to be .
Step 4.2: Compute the objective weights of risk factors by the entropy method.
Based on the comprehensive evaluation matrix D, the entropy value of each failure mode with regard to each risk factor can be calculated by applying Equation (22). Then, the objective weights for each risk factor can be derived as by utilizing Equation (23).
Step 4.3: Obtain the combination weights of risk factors.
Based on the subjective weights and objective weights of risk factors, the combination weights of risk factors are calculated as according to Equation (24).
Step 5: Compute the weighted group decision matrix Y.
The weighted decision matrix can be calculated by Equation (25) as follows.
Step 6: Determine the border approximation area vector G.
According to Equation (26), the border approximation area vector
G can be obtained as
Step7: Construct the distance matrix X.
The distance of failure modes from the border approximation area is computed by Equation (27); the distance matrix is determined as
Step 8: Determine the ranking of failure modes.
The closeness coefficient
can be calculated by Equation (28) as follows.
The risk priority of failure modes is ranked as FM7 > FM2 > FM3 > FM8 > FM6 > FM9 > FM1 > FM5 > FM4 by the decreasing order of the closeness coefficient .
5.3. Comparisons and Discussion
To further demonstrate the effectiveness of the proposed method, we use the result to compare some similar approaches, including the RPN method, the fuzzy TOPSIS [
14], the fuzzy VIKOR [
36], the fuzzy MULTIMOOR [
20], and the IFHWED-based FMEA [
54] in this section.
Table 6 shows the ranking results of all nine failure modes obtained by implementing the six approaches.
The ranking order of failure modes obtained by the proposed method is partly different from the ones according to the RPN value, but FM7 possesses the highest risk in the two methods. The FM2 and FM7 failure modes have the same risk in the RPN method, which indicates that different values of S, O, and D may produce the same value of RPN. In this situation, it is difficult to distinguish the risk between FM2 and FM7 for a decision-maker. However, the risk of FM2 is lower than that of FM7 in the proposed method. Therefore, this drawback of the RPN approach can be solved easily by using the proposed method.
The priority of the nine failure modes produced by the fuzzy TOPSIS method is significantly different from the ones determined by the proposed approach. The ranking order obtained by the fuzzy TOPSIS method may be irrational because it does not take into account the interdependent relationships between the experts’ preferences. However, the process of risk assessment based on the FMEA team is considered as a group decision process, and there are many different types of correlations between experts. In addition, each FMEA team expert was considered to have equal importance in the fuzzy TOPSIS method. In reality, FMEA team experts usually come from various departments, and have different professional backgrounds, practical experiences, and knowledge structures. Therefore, they should be assigned different importance during risk assessments.
We can see that there are some differences between the priority of failure modes determined by the proposed method and the fuzzy VIKOR approach. The ranking results determined by the fuzzy VIKOR method may be unreasonable because the consensus-reaching process and the experts’ judgments dependencies were not considered in the risk assessment. For example, FM9 was the least important failure mode using the fuzzy VIKOR method, whereas it ranked sixth using the proposed method. Interestingly, FM6 ranked before FM3 with the fuzzy VIKOR approach, however, the latter is more important in reality. Therefore, FM3 was merited a higher priority in comparison with FM6 in our proposed method. Furthermore, the solution obtained through the fuzzy VIKOR method compared with the proposed approach is a set of compromise solutions.
Apart from FM1, FM4, and FM6, the ranking orders of the failure modes obtained by the fuzzy MULTIMOORA approach are totally different from those determined through the proposed method. These inconsistent ranking results may be expressed by the fact that the subjective weights of risk factors were not taken into account in the fuzzy MULTIMOORA method, which may result in irrational rankings of failure modes. For example, FM8 is given to be a more important failure mode than FM7 according to the fuzzy MULTIMOORA approach. However, in the proposed approach, it ranks only the fourth position; FM7 has the top risk priority, which also can be validated by the IFHWED-based and the fuzzy VIKOR methods. In addition, the consensus-reaching process is not considered in the fuzzy MULTIMOORA approach, which may be another reason that leads to the biased ranking results.
As shown in
Table 5, the failure modes for
FM3,
FM5,
FM6,
FM8, and
FM9 have different ranking orders between the proposed approach and IFHWED-based method. Interestingly, the sorting positions of
FM5 and
FM9 have been interchanged in the proposed method comparing with the IFHWED-based approach. There are many different types of interdependent relationships between the experts’ preferences in the real world because of the mutual influences of FMEA team experts. However, the interdependent relationships among experts’ preferences are not considered in the IFHWED-based method, which may be the reason that results in the different ranking results. In addition, the IFHWED-based approach also fails to take into account the consensus-reaching process.
Through the comparative analysis above, we can conclude that the risk evaluation results determined by the proposed approach are more reasonable and accurate than those obtained by the fuzzy TOPSIS, fuzzy VIKOR, fuzzy MULTIMOORA, and IFHWED-based methods. Compared with the listed approaches, the advantages of the proposed method are summarized as follows:
- (1)
The IVPFWGBM operator was used to aggregate the experts’ preferences into group assessments, which sufficiently reflect the interdependent relationships between the experts’ preferences.
- (2)
Compared with the other improved FMEA approach, the ranking results obtained by the proposed method are more acceptable because the level of agreement between decision-maker and group is considered through introducing a consensus-reaching process into the risk assessment process of FMEA.
- (3)
The ranking results of failure modes obtained by the proposed approach are more reasonable when compared with the other improved FMEA methods; the reason is that the improved MABAC method adopted the IVPFGBM operator to construct the border approximation area matrix, which considers the direct and indirect relationships among failure modes.