1. Introduction
As an important branch of reliability analysis, failure mode and effects analysis (FMEA) is a methodical way to examine a proposed design in which failure is possible [
1,
2,
3,
4]. Failure mode refers to a form of system failure or system malfunction and effects analysis is used to research the impact on the total system when a local system is unable to work [
5]. The main purpose of FMEA is to define, identify, and eliminate potential failure or problems in different products, designs, systems, and services [
6,
7]. FMEA not only can provide a basis to help designers improve or upgrade their scheme to reduce the loss but also can help decision makers to formulate corresponding preventive measures or improve the emergency response capability [
8,
9,
10]. Nowadays FEMA has been widely used in many industry fields including aerospace, chemical, engineering, design mechanical, medical, and so on [
11,
12,
13,
14,
15,
16,
17,
18].
Risk priority number (RPN) is a popular way to evaluate risk priority in traditional FMEA [
19,
20,
21]. RPN is usually expressed as :
O
S
D, where O, S, and D are three main risk factors which denote the occurrence (O) of a failure mode, the severity (S) of a failure effect, and the probability of not being detected (D), respectively [
22]. A failure mode should be paid more attention and be more important if the value of RPN is higher than others. However, RPN has some shortcomings especially in transforming linguistic variable and considering the difference of weight about risk factors. Therefore, lots of methods are proposed to improve FMEA, such as evidence theory [
23,
24,
25], expert system [
26,
27], uncertainty measure [
28], hybrid approaches [
29], fuzzy set theory [
30,
31], and so on [
32].
The issue with linguistic variable transforming is how to precisely evaluate the three risk factors by RPN. In many cases, the risk factors given by experts are expressed as a linguistic variable rather than the exact numbers [
33]. Fuzzy set theory, proposed by Zadeh in 1965 [
34,
35], makes use of membership to measure the degree of fuzzy linguistic variables and is a precise way to solve the uncertainty of information [
36,
37,
38,
39]. For the issue of setting the weight of risk factors by RPN, the three risk factors are multiplied, which means occurrence, severity, and detection have the same weight. It can be thought that the three factors are independent and equally important. However, in many practical situations the difference in weight must be considered. For example, to some extent the failure mode with a high frequency is easily detected, so the weights of risk factors are related; in some systems decision makers pay more attention to the severity of a failure effect. Therefore, the difference and relevance of risk factors’ weights should be considered.
In this paper, a method is proposed to improve risk priority number based on a fuzzy measure and fuzzy integral, which can effectively reflect the weights’ difference and relevance of risk factors. The weights of risk factors given by domain experts are regarded as fuzzy densities to generate a -fuzzy measure that can take the difference and relevance of risk factors’ weights into consideration. Then, the Choquet integral is applied to fuse every value of the risk factors in order to obtain the comprehensive evaluation result. When the proposed method is applied to FMEA, it has achieved desired results. The proposed method provides a more reasonable and effective method for FMEA.
The rest of this paper is organized as follows. We give a literature review of the traditional FMEA method, fuzzy set theory, fuzzy measure, and fuzzy integral in
Section 2.
Section 3 is about the proposed method of FMEA under a fuzzy environment, using a fuzzy measure and fuzzy integral. An illustrative example and the comparison with another approach are given to show the effectiveness of the proposed approach in
Section 4.
Section 5 provides a brief conclusion.
3. The Proposed Model
As we known, RPN is used to determine the risk priorities in FMEA. However, the RPN approach is difficult in accurately evaluating the relevance of the three risk factors and does not consider the difference of risk factors weights. We need obtain the weights’ values but the weight of risk factors are often expressed as vague linguistic variables such as important, very high, and so on. So we need to transform them into values. Fuzzy set theory is a useful approach to get specific values from vague linguistic variables. Then we should consider the relevance and weights’ difference of risk factors.
This paper proposes a method to improve RPN approach based on a fuzzy measure and fuzzy integral. Because fuzzy integrals do not need to assume the indicators are independent of each other, so a fuzzy integral is widely used in subjective evaluation problems where evaluation indexs are associated. In order to precisely and reasonably simulate the weights’ difference of the risk factors in FMEA, this paper uses a fuzzy measure to build a model for the weights of the occurrence, severity, and detection. The weights of risk factors given by domain experts are seen as fuzzy densities to generate a
-fuzzy measure which can take the relevance and weights’ difference of risk factors into consideration. Then, the Choquet integral is used to fuse the evaluation results of the three factors and we can get the final comprehensive evaluation value of the failure mode. The main process is shown in
Figure 1.
As shown in
Figure 1, the method proposed in this paper mainly includes four key steps.
Evaluate the three risk factors by domain experts including occurrence, severity, and detection and their weights. The results of the assessment include two aspects: one is the importance of each factor, which are defined as g(O), g(S), and g(D) and they are in the range of 0–1; the other is the value of every risk factor in failure modes, which are defined as , , and .
Generate a fuzzy measure and use the fuzzy measure to generate a
-fuzzy measure. The g(O), g(S), and g(D) are regarded as the fuzzy density of the three risk factors. Then generate the
of the
-fuzzy measure according to Equation (
4).
Fuse evaluation values using the Choquet integral and rank the comprehensive evaluation value. The Choquet integral, by Equation (
5), is applied to fuse
,
, and
based on
and the evaluation value by domain experts. Then we can obtain the comprehensive evaluation value
, which reflects the comprehensive risk level of the failure mode. Next, rank the failure mode by
to find the high-risk failure mode.
Prevent and improve failure mode. Domain expert and decision makers can formulate corresponding preventive measures by the ranking. Repeating above steps, the system will be optimized continuously and the reliability will be improved.
4. An Illustrative Example
In this section, the proposed method is applied to FMEA to prove its effectiveness. The example and data are from the literature [
59]. The example is about a grade A class three hospital that uses FMEA for its medical risk management in order to reduce medical accident and iatrogenic disease. Literature [
59] used an extended VIKOR method under a fuzzy environment to get the matrix and weight of every risk factors. The steps of risk evaluation in FMEA are shown as follows:
Step 1: Identify the risk assessment objective. The hospital wants to identify some important failure modes in the general anaesthesia process. After discussion and screening, it identified six possible failure modes which are denoted as FM 1, FM 2, FM 3, FM 4, FM 5, and FM 6.
Step 2: Organize five experts as a team. The five experts are denoted as DM 1, DM 2, DM 3, DM 4, and DM 5 to evaluate the values and weights of occurrence, severity, and detection by linguistic variable. The results are shown in
Table 4 and
Table 5.
Step 3: Transform linguistic variables into detailed numerical values by trapezoidal fuzzy numbers. Then weights of risk factors and fuzzy rating of failure modes are aggregated to get the fuzzy decision matrix and fuzzy weight of risk factors, as in
Table 6.
Step 4: Aggregate these evaluations given by five experts to obtain an integrated evaluation of the three risk factors. In this paper, we directly use the results of literature [
59], as shown in
Table 7.
Step 5: Define the weight of three risk factors as g(O), g(S), and g(D), in the range of 0–1.
g(O) = 0.768
g(S) = 0.787
g(D) = 0.650
Step 6: Generate the fuzzy measure. The g(O), g(S), and g(D) are regarded as fuzzy density of the three risk factors:
(O) = g(O) = 0.768
(S) = g(S) = 0.787
(O) = g(D) = 0.650
Then generate the
of
-fuzzy measure by Equation (
4). According to Equation (
4):
1 + = (1 + 0.768)(1 + 0.878)(1 + 0.650)
Solve the formula: = {−2.911, −0.989, 0}
Because > −1 and ≠ 0, so the value of is −0.989. Therefore we obtain the following -fuzzy measure
(∅) = 0,
({O}) = 0.768,
({S}) = 0.878,
({D}) = 0.650,
({O,S}) = 0.979,
({O,D}) = 0.924,
({S,D}) = 0.964,
({O,S,D}) = 1.
Step 7: Use the Choquet integral to fuse the evaluations of all risk factors and rank the six failure modes. The comprehensive evaluation value
of every failure mode can be found with Equation (
5). The comprehensive evaluation values are shown in
Table 8. It can be seen that the comprehensive evaluation value of every failure mode is still between 1 and 10, which reflects the specific risk level of the failure mode (between 1 and 10) and the fuzzy integral value has a clear physical significance.
Step 8: Prevent and improve the failure mode. Domain experts or decision makers can give priority to making the effective precautionary measures and emergency response schemes.
As a comparison, the literature [
59] gives the failure model risk evaluation results and ranking based on the VIKOR method, as shown in
Table 9 and
Table 10. Among them, S, R, and Q are represented the three ranking indexes of VIKOR method, respectively. It can be seen in the ranking of failure modes based on VIKOR method, the ranking of S index is FM 3, FM 2, FM 6, FM 1, FM 5, and FM 4; the ranking of R index is FM 6, FM 3, FM 2, FM 5, FM 1, and FM 4; and the ranking of Q index is FM 3, FM 6, FM 2, FM 5, FM 1, and FM 4. The ranking result by the Choquet integral is as same as the ranking result by R index and is only different with the first and second ranking by Q index. Every method can draw a conclusion that FM4 is the lowest risk mode of failure. It can prove the method we propose is effective by comparison with the VIKOR method. What is more important, the comprehensive evaluation value by fuzzy integral fusion has a more definite physical significance and can be regarded as the risk degree of failure mode. The risk of the six failure modes is ranked by FM 6, FM 3, FM 2, FM 5, FM 1, and FM 4, using the proposed method. It is no doubt that FM 4 has the lowest degree of risk, because occurrence and detection both are the lowest, and severity is only slightly higher than the lowest one. The highest occurrence of the occurrence failure mode is FM 3, the highest degree of severity is FM 6, and the most difficult of detection is FM 3. Although FM 3 has the highest value in both occurrence and detection, FM 6 has the highest value only in severity but the weight of the severity factor is the highest. Therefore, the final ranking of FM 3 and FM 6 depends on both the weight of the risk factor and the crisp evaluation value. Using the fuzzy integral fusion method given in this paper, we find that FM 6 has the highest risk degree, which shows the effectiveness of the proposed method.
5. Conclusions
FMEA is a useful and important approach to examine potential failure by the risk priority number. The traditional RPN approach is criticized in many aspects, especially in transforming linguistic variable and considering no difference among risk factors’ weights. Fuzzy set theory is a classical method to transform linguistic variables into crisp values. Also, a fuzzy measure and fuzzy integral are appropriate methods to take the difference and relevance of weights into consideration. In this paper, a new method based on fuzzy integral fusion is proposed to solve the issue of the weights’ difference and relevance about risk factors. In the proposed method, the -fuzzy measure is generated according to the weights of the risk factors given by the domain expert and then the Choquet integral is used to fuse the crisp evaluation values to obtain the comprehensive evaluation results. Finally, the effectiveness of the proposed method is shown by comparing the result of a medical risk management system with different methods. By using the proposed method, the comprehensive result can reflect the specific risk level of the failure mode because in the model the comprehensive evaluation value of the fuzzy integral is in the range of 1–10. So the result has a definite physical significance and is easy to be understood and applied. In addition, the proposed method can be effectively used in a group decision question, uncertain decision making environment, and so on.
Further research will focus on the following directions. Firstly, we need future research to perform a comparative study with the obtained results. Secondly, the uncertain information should be considered in FMEA. A fuzzy measure and fuzzy integral can be used when we can obtain the evaluation value and weights of the three risk factors. However, in some situations, diversity and uncertainty of the risk factors can not be ignored. We need to develop a new method to obtain the evaluation value and weights of uncertain information and fuse them by a fuzzy measure and fuzzy integral.