Proceedings of the 2012 Industrial and Systems Engineering Research Conference
G. Lim and J.W. Herrmann, eds.
Dempster-Shafer Theory Approach to FMEA
Niranjan S. Kulkarni
CRB Consulting Engineers
One Kendall Square, Cambridge, MA 02139, USA
Aric R. Johnson
Dept. of Systems Science and Industrial Engineering, Binghamton University,
P.O. Box 6000, Binghamton, NY 13902, USA
Abstract
Failure Mode and Effects Analysis (FMEA) technique is a group based activity to prioritize risk. Often times there
are issues with aggregating team member’s responses or ratings. Literature suggests several means to aggregate
these responses - computing average values, using group consensus, assigning weights to team member’s responses,
and using fuzzy logic / inference systems, to name a few. However, each of these methods has certain shortcomings.
Furthermore, ranking the possible failure modes in face of uncertainty or incomplete information presents its own
challenges. The aforementioned aggregation techniques have been used with very limited success in such uncertain
instances. This paper will propose a method that utilizes Dempster-Shafer Theory (DST) to overcome these issues. It
will be shown that DST will allow unbiased aggregation of team member’s ratings using rules of combination of
evidence. Also this method will aid the difficulty in precisely rating a factor in the presence of incomplete
information. To the best of our knowledge a DST based technique has not been applied to FMEA in the literature. A
secondary contribution of this paper is introducing the concept of DST, which has many potential applications in the
field of Industrial and System Engineering.
Keywords
FMEA, Dempster-Shafer Theory, Aggregating Group Responses, Incomplete Information and Uncertainty
1.0 Introduction
Failure mode and effect analysis (FMEA) is a bottom-up approach used to identify potential failures and their
resulting effects upon a system. FMEA is often initiated during the design phase of a product in order to identify
potential failures so that the design can be remedied to eliminate or reduce the impact of these failures before the
product enters production. In most real situations several failure modes are identified and ranked so that limited
resources can be focused on eliminating the most critical ones. This leads to an extension of FMEA to failure mode
effect and criticality analysis (FMECA).
The most common method of ranking failure modes in criticality analysis is assigning a Risk Priority Number
(RPN) to every failure mode. RPN is calculated as, = × × , where, Occurrence (O) is an assessment of
the probability that a particular failure mode will occur, Severity (S) is the impact of a particular failure mode, and
Detection (D) as the ease of detecting a failure mode once it has occurred. O, S and D are assigned a rating on a
scale of 1-10, which is done by expert(s). The reader is referred to Chin et al. [1] for tables that show how to rate
OSD values for a failure mode. Ultimately the failure modes are ranked according to their RPNs in a descending
manner.
FMEA and FMECA have been widely utilized in many industries with much success. However, both these
techniques have their own shortcomings [1-5] :
• The RPN calculation is questionable. There appears to be no real justification as to why risk should be
quantified as the product of estimates for O, S, and D. The RPN calculation only considers safety related
terms but does not include other factors such as costs.
• Different O, S, and D estimates can results in the same RPN when they may represent completely different
risks. For example, O, S, and D ratings of 5, 4, and 3, respectively, gives the same RPN value (60) when
compared to O, S, and D ratings of 2, 5, and 6, respectively.
Kulkarni and Johnson
•
•
•
•
•
O, S, and D are equally weighted. Some practitioners may justifiably believe that severity is the most
important factor and thus should be given more weight in the RPN calculations. Several arguments could
be proposed for particular applications to justify unequal weights for O, S, and D.
Conversions are different for O, S, and D scores, e.g. linear for O and non-linear for D.
RPNs are not continuous and are heavily distributed at the lower bound of the 1-1000 scale. Therefore it is
difficult to assess the relative difference between different RPNs, e.g. does the difference between RPNs 12 have the same meaning as the difference between 900-1000.
Small variations in one rating can result in vastly different RPNs. For example, if O and S are both 10 and
if D is 1 or 2 the resulting RPNs are 100 and 200, respectively.
It is difficult to quantify the O, S, and D terms precisely, yet one must estimate a precise rating between 110. Rating a failure mode is especially complicated when considering designs with no historical data.
Another issue that often results when conducting an FMEA comes from it being a group based activity. One of the
problems with this is deciding how to aggregate team member ratings. The question becomes whether one should
use a simple average, give more experienced members a higher weight, to use a group consensus method, or to use
some other more sophisticated approach.
This paper will propose a method that utilizes Dempster-Shafer theory (DST) to overcome some of these issues. It
will be shown that DST will allow robust aggregation of team member’s ratings using rules of combination of
evidence. Also this method will aid the difficulty in precisely rating a factor when the information available to the
expert(s) is incomplete. To the best of our knowledge a DST based technique has not been applied to FMEA in the
literature. Additionally, this paper will introduce readers to DST, which has many potential applications in the field
of reliability and engineering in general.
The remainder of the paper is structured as follows. Relevant literature reviewed is presented in Section 2, while
Section 3 outlines the proposed methodology. Examples are provided in Section 4. Finally, the article is concluded
and future work is presented in Section 5.
2.0 Literature Review
This section will review the recent research focused on improving the traditional FMEA technique. Additionally,
development of DST and its application to reliability analysis is also reviewed.
2.1 FMEA Developments
By far the most popular technique used to improve the traditional FMEA is by utilizing Fuzzy Inference Systems
(FIS), a.k.a. fuzzy controllers. The FIS approach to FMEA is motivated because it is logical to think of the OSD
ratings in fuzzy linguistic terms, e.g. not likely, very likely, etc. Numerous studies in the literature follow the same
general approach. First, fuzzy relations for OSD ratings are defined. Then a fuzzy if-then rule-base is developed to
obtain RPN ratings. The input to these FISs is the team member’s fuzzy OSD ratings and the output is the RPN.
What distinguishes these papers is typically the application area or the specifics of the FIS, e.g. type of fuzzy set
membership functions, type of fuzzy logic operators, rule-base development, defuzzification technique, etc. A
representative sample of recent papers utilizing a FIS approach to conducting FMEA are the following - Duminica
and Avram [3], Gargama and Chaturvedi [4], Zafiropoulos and Dialynas [6], Sharma et al. [7], Guimaraes and Lapa
[8].
Beyond FIS, fuzzy logic has been applied to FMEA in many ways. The following two papers each illustrate a
different method of applying fuzzy logic to FMEA.
One of the methods proposed by Gargama and Chaturvedi, [4] utilizes fuzzy logic to compute RPNs. It is based on
fuzzifying the RPN calculations via the fuzzy extension principle. This method allows each team member to score
O, S, D linguistically for each failure mode. These scores are translated to a fuzzy number. Then fuzzy arithmetic is
used to calculate the RPN (which is also a fuzzy number). These fuzzy RPNs are then defuzzified using the centroid
method and ranked in a descending order.
Zhang and Chu [5] propose a method based on fuzzy logic in order to create a mathematical model for FMEA. First,
individual team members score each failure mode and give linguistic weights to OSDs. Next a fuzzy least-squares
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method is used to aggregate individual scores into group scores. Then a non-linear programming model with the
‘method of imprecision’ is used to calculate the RPNs. Finally the RPNs are ranked via a partial ranking method
based on fuzzy preference relations.
Techniques other than fuzzy logic have also been proposed to improve FMEA. Sachdeva et al., [2] utilize the
‘technique for order preference by similarity to ideal solution’ (TOPSIS) approach to machine maintenance. Though
this method was proposed for machine maintenance, it can also be applied to FMEA. Essentially this method allows
the analyst to find the distance (difference) between the ratings for each failure mode and the ideal solution
representing the preferred outcome. The failure modes are then ranked in a descending manner based on their
distance from the ideal solution.
All of the aforementioned papers focus on improving the continuity of the RPN calculations and issues with
precisely rating OSD factors that are only imprecisely known. Fuzzy based approaches have been shown to
overcome these issues with the traditional FMEA. Research has also been conducted on issues related to aggregating
individual team member OSD ratings. Zhang and Chu [5] touch on this issue and suggest using fuzzy least squares
aggregation method, but this is not the focus of their work.
Traditionally, there are two main methods to aggregate team member’s responses, namely, averaging individual
responses (or ratings) into an overall response (rating) and using group consensus. The averaging technique can be
extended by allowing different weights for each team member, e.g. a more experienced team member may be given
a higher weight. Ashley and Armitage [9] empirically compared these two methods for combining O, S, and D
ratings. The study showed that significantly different results were obtained depending on the method used. The
authors conclude by recommending group consensus but not without warnings. Unfortunately consensus introduces
new concerns related to ‘group think’ and bias introduced by overbearing or high-seniority team members.
Chin et al., [1] argue that since FMEA is a group-decision making process it would benefit from using an Evidential
Reasoning (ER) approach. ER is a multi-attribute decision analysis technique that in the case of FMEA allows the
diversity and uncertainty of each team member to be incorporated into the risk model. In this method each team
member can assign O, S, and D ratings that are as precise or imprecise as their knowledge dictates. Then overall
ratings are determined by aggregating the individual ratings, which are weighted to reflect their expertise. This
method also allows the practitioner to assign different weights to the O, S, D factors when calculating the RPN.
However, finding an appropriate weighting scheme is a subjective practice.
2.2 Dempster-Shafer Theory
DST is a theory of evidence. DST originated when Shafer extended the original work of Dempster in 1976 [10].
Available evidence, even if incomplete, can be used to find imprecise probabilities for events. The following
discussion within this section on DST has mostly been adopted from Klir [11]. DST frequently uses three basic
concepts, namely, basic probability assignment (BPA, denoted by in this paper), belief(), and plausibility ().
For a given universal set and all ∈ ℘() (℘() is the power set), a BPA is assigned to all members of ℘(),
and represents the proportion of available evidence that supports the claim that a particular event belongs to a
subset of ℘(). By definition BPAs satisfy the following conditions: :℘() → [0,1], () = 0 and
∑℘() () = 1.
BPAs can be calculated using historical data or can be provided by an expert in the lack of actual data. BPAs are not
measures (in reference to measure theory) and thus there is a need to introduce procedures that are used to describe
uncertainty within DST. Two dual measures are used to describe this uncertainty in terms of probability of an event,
namely, belief and plausibility. These dual measures allow one to describe an event when there is only access to
incomplete information or evidence. The belief measure represents the degree of belief, supported by the available
evidence, that an event belongs to a particular set. The plausibility measure is dual to the belief measure and
represents how plausible it is that an event belongs to a particular set. Belief and plausibility can be found directly
from the BPAs using equations 1 and 2.
() = ()
|⊆"
(1)
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() = |"∩$%
()
(2)
From equation 1 we can see that the belief that an event belongs to a particular set is the sum of the BPAs of its
subsets. From equation 2 it can be inferred that the plausibility of an event belonging to a particular set is the sum of
the BPAs for sets that have a non-empty intersection with the set of interest. Like BPA, belief and plausibility are
mappings from the power set ℘() to the unit interval. Additionally, ∅ = ∅ = 0 and ℘(') = ℘(') = 1.
Moreover, belief and plausibility are dual measures and are related as () = 1 − ()).
The first difference from probability theory is that DST calculates probability of the evidence supporting a
hypothesis rather than calculating the probability of the hypothesis itself unlike the traditional probabilistic approach
[12]. Another major difference is that all members of the power set are assigned a BPA according to the available
evidence. Note that this is quite different from probability theory in which only the singletons are assigned a
probability. Such a probability assignment to the power set allows one to utilize all available information, especially
incomplete information or uncertain conditions where one cannot distinguish what exactly caused the failure (in the
context of FMEA). For example, consider a scenario wherein an automobile has overheated. Here we do not have
complete or specific information, i.e. we do not know the specific cause of the failure. However, we do know that
this failure was caused by either a faulty coolant pump or lack of coolant amongst other causes of failure. In such
cases we assign a BPA to the set that contains the coolant pump and lack of coolant from the power set of all
potential failure modes.
Additionally, there are rules for combination of evidence that allow for aggregation of multiple sources of evidence.
This is particularly useful for combining subjective belief assessments from multiple experts. This flexibility allows
individual experts or team members to provide ratings based on their individual level of understanding,
expertise/experience and information available to them.
DST is starting to get some interest in the Industrial and Systems Engineering domain, particularly to study cases
wherein incomplete information hinders finding precise probabilities of an event occurring. The following works
suggest that DST is applicable to reliability analysis. Kaftandjian et al., [13] combine DST and fuzzy sets to improve
weld defect detection. DST is used to handle the uncertainty involved in detecting individual weld defects. Yang and
Kim [14] use a DST and neural network hybrid model to classify defects within induction motors. Kay [15]
introduces DST and illustrates its application to event and fault tree analysis. The author also states that DST is
applicable to FMEA but provides no details. Simon et al., [16] develop a Bayesian network algorithm to compute
the reliability of complex systems. DST is utilized to handle the uncertainty in the reliability assessments. Thus one
can notice that DST is applicable to reliability analysis, even though the literature is scarce on the subject. This
paper will contribute another application of DST to the literature and show how it can be applied to FMEA.
3.0 Extending FMEA to include Incomplete Information via DST
This section demonstrates DST's applicability to FMEA. A simple example is used for illustration purpose. This
example considers only one expert providing the ratings for all failure modes. Furthermore, this case considers only
three potential failure modes - A, B and C.
The first major departure from the traditional FMEA is that the expert needs to assign ratings to all possible
combinations of the power set, ℘(*), where * is the set of failure modes. The cardinality of this finite power set is
|℘(*)| = 2|,| . Thus, for the example under consideration, cardinality will be2- = 8, meaning that 8 ratings will
have to be provided by the expert for each dimension (occurrence, severity, and detection) of the FMEA risk
assessment, as opposed to 3 ratings (one for each failure mode) for traditional FMEA. These 8 combinations of the
three failure modes (A, B and C) are ∅, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, and {A, B, C}. Subsequently, these 8
combinations will be assigned a BPA for each of the three dimensions. When data or relevant information regarding
failure modes is available, standard rating scale of 1-10 can be used. These ratings can be normalized to convert
them to BPAs, i.e. convert the ratings to the unit interval. However, when information is unavailable, or an expert
has incomplete information, the expert can assign BPAs to all the possible combinations. This flexibility in
assigning BPA allows the expert to use whatever amount of information is available. Furthermore, this method lets
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the expert account for uncertainty by attributing a certain failure to two or more failure modes simultaneously in
situations where it cannot be determined precisely which particular failure mode caused the failure.
Consider the calculations for occurrence. In this case, the expert has no information on ratings and uses his/her best
judgment to assign BPAs to all 8 combinations. Failure modes and (*) for each failure mode are shown in Table
1. The first column lists all the 8 combinations of failure modes. Note again that the traditional FMEA would only
be concerned with the singletons, i.e., {A}, {B} and {C}. The second column provides the BPAs, which are assumed
to be obtained from an expert. Consider () (from Table 1), which is 0.25; in the context of FMEA this means that
25% of the evidence available to the expert (his/her expertise) supports the claim that if a failure is experienced it
will be due to failure mode A. (, /) = 0.08, means that the expert has evidence that 8% of the time a failure
occurs, it was due to either failure mode A or C, but it cannot be distinguished exactly which mode caused the
failure. The third and fourth columns of Table 1 show all of the belief and plausibility values (obtained using
equations 1 and 2 respectively) for each failure mode. A number of these calculations are shown below for
illustration.
Table 1: Occurrence Calculations - 1 Expert Opinion
Failure Modes
(F)
0(1)
2(1)
3(1)
∅
0.00
0.00
0.00
A
0.25
0.25
0.42
B
0.25
0.25
0.37
C
0.30
0.30
0.42
A∪B
A∪C
B∪C
A∪B∪C
0.08
0.58
0.70
0.08
0.63
0.75
0.03
0.58
0.75
0.01
1.00
1.00
() = ()= 0.25.
(, /) = () + (/) + (, /) = 0.63
() = () + (, 5) + (, /) + (, 5, /) = 0.42
(5, /) = (5) + (/) + (, 5) + (, /) + (5, /) + (, 5, /) = 0.75
Note that for singletons, belief will always be equal to its BPA, since singletons have no subsets (other than the
empty set which always has a BPA of zero).
Belief and plausibility values for the other two dimensions, severity and detection, are found in a similar manner. A
summary of the belief and plausibility values on the singletons are shown in Table 2 for each dimension of the RPN.
At this point we can exclude all other members of the power set as ultimately we are only concerned with ranking
the singletons based on their risk rating (lower values get higher priority). Traditionally in FMEA the RPN would be
found by multiplying O, S, and D ratings. However, the proposed method calculates lower and upper bounds for O,
S, and D. Thus we need to apply interval arithmetic in order to find the overall product, which will also be bounded
below and above. For interval multiplication one simply multiplies all the lower bounds to obtain the overall lower
bound, likewise multiplies all upper bounds to obtain the overall upper bound. Ranking of each failure mode is
based upon the aggregated bound on the O, S, and D risk. In this case we have used a simple rule procedure to rate
the risks of each failure mode, which is discussed below.
1. Rank all failure modes by their overall upper bound (plausibility) in a descending manner. Higher the
plausibility, lower the rank (lower rank implies higher priority).
2. Break any ties that result from the first step by choosing the failure mode with the higher lower bound
(belief).
3. If ties still exist break them arbitrarily.
Kulkarni and Johnson
In this case failure mode C receives the highest priority (RPN = 1) since it has the highest overall upper bound (see
Table 2). Failure modes A and B have the same upper bound. However, using the aforementioned rules B is given a
higher priority because it has a higher lower bound as compared to A.
Table 2: Results of DST-based FMEA with Single Expert
(F)
2(1)
3(1)
2(1)
3(1)
2(1)
3(1)
6×7×8
2(1)
3(1)
A
0.25
0.42
0.18
0.45
0.20
0.44
0.009
0.084
3
B
0.25
0.37
0.24
0.52
0.20
0.44
0.012
0.084
2
C
0.30
0.42
0.21
0.45
0.24
0.52
0.015
0.099
1
O
S
D
RPN
This simple example was intended to introduce the reader to the DST concepts of BPA, belief, and plausibility.
Additionally, this example illustrated how DST can be applied to FMEA. One can begin to notice how DST allows
more flexibility in rating failure modes. This is due to the consideration of the power set rather than only the
singletons. The next case will extend this problem by considering an additional expert. The ability of DST to
appropriately aggregate multiple bodies of evidence (multiple expert opinions) is another key characteristic of DST
that is particularly useful for FMEA.
4.0 DST Extension to Aggregate Multiple Responses in a Group FMEA
In order to proceed with an example involving two experts, we need to discuss how evidence from two sources can
be combined. Combining two (or more) expert opinion's (evidence) is achieved by using Dempster’s rule of
combination as shown in Equation 3. Note that there are other methods to combine multiple sources of evidence, but
these are beyond the scope of this paper.
9,: (;) =
∑"∩<= 9 () × : ()
1−>
(3)
Notice that the numerator of Equation 3 is similar to finding a joint probability distribution from marginal
distributions. However, since some of the intersections may result in empty sets (and by definition (∅) = 0) the
sum of the combined BPAs may not be equal to one, i.e., ∑ 9,: (;) ≠ 1. It is required by definition that the sum of
the BPAs for any body of evidence (single or combined) must be equal to one. Consequently, there is a need to
introduce a factor that will normalize the combined BPAs. This normalization factor (c) is shown in equation 4.
> = 9 () × : ()
"∩$∅
(4)
Normalizing has the effect of absolutely discounting any conflict and assigning any probability mass related with
conflict to the null set (∅) [17]. Notice that c is the sum of the products of the individual expert’s BPAs over all nonempty intersections between the two bodies of evidence. Thus it can be used to normalize the combined BPAs as
shown in Equation 3.
Table 3 illustrates the combination of evidence from two experts for occurrence ratings. Notice here that it was
assumed that the experts gave their ratings directly in the form of a BPA instead of on a scale of 0-10, which is
typically the case for traditional FMEA. It makes no difference which method the experts give their ratings. If the
traditional rating method is applied one can simply normalize those values to get (*). In fact that is the method
used for severity and detection ratings shown in Tables 4 and 5.
Before calculating the combined evidence from the individual evidences, it is required to calculate c (using equation
4) which normalizes the combined BPAs. Below we show some of the calculations. The BPAs, belief and
plausibility values are as shown in Table 3.
Kulkarni and Johnson
Table 3: Two Expert Occurrence Ratings
(F)
∅
A
B
C
A∪B
A∪C
B∪C
A∪B∪C
Expert 1
0@
Rating
0.00
0.45
0.05
0.30
0.00
0.11
0.00
0.09
-
Expert 2
0A
Rating
0.00
0.30
0.05
0.40
0.00
0.00
0.18
0.07
-
Combined Evidence
0@,A
21,2
31,2
0.00
0.00
0.00
0.40
0.40
0.42
0.03
0.03
0.07
0.52
0.52
0.57
0.00
0.43
0.48
0.01
0.93
0.97
0.03
0.58
0.60
0.01
1.00
1.00
c = 0.429
> = 9 () × [: (5) + : (/) + : (5 ∪ /)] + 9 (5) × [: () + : (/) + : ( ∪ /)] + 9 (/) ×
[: () + : (5) + : ( ∪ 5)] + 9 ( ∪ 5) × : (/) + 9 ( ∪ /) × : (5) + 9 (5 ∪ /) × : ()
> = 0.429
9,: ()
= [9 () × [: () + : ( ∪ 5) + : ( ∪ /) + : ( ∪ 5 ∪ /)] + 9 ( ∪ 5) × [: () + : ( ∪ /)]
+ 9 ( ∪ /) × [: () + : ( ∪ 5)] + 9 ( ∪ 5 ∪ /) × : ()]/(1 − >)
9,: () = 0.40
9,: (5 ∪ /) = [9 (5 ∪ /) × [: (5 ∪ /) + : ( ∪ 5 ∪ /)] + 9 ( ∪ 5 ∪ /) × : (5 ∪ /)]/(1 − >)
9,: (5 ∪ /) = 0.03
9,: ( ∪ 5 ∪ /) = [9 ( ∪ 5 ∪ /) × : ( ∪ 5 ∪ /)]/(1 − >) = 0.01
After all the combined BPAs are obtained, the combined belief and plausibility values are calculated using equations
1 and 2 but with the combined BPAs rather than the individual BPAs.
In a similar manner BPAs, belief and plausibility values for severity and detection are calculated and shown in
Tables 4 and 5 respectively. However, the individual BPAs in Tables 4 and 5 were obtained by normalizing the
traditional ratings. The low value for c (= 0.06) in Table 5 means that there is little difference of opinion between the
two experts. Thus, smaller c indicates less difference between the experts' opinions and vice versa.
Table 4: Two Expert Severity Ratings
Combined Evidence
0@,A
21,2
31,2
0.00
0.00
0.00
∅
0.12
0.12
A
0.17
0.43
B
0.43
0.56
0.30
C
0.30
0.41
0.04
0.59
0.70
A∪B
0.01
0.44
0.57
A∪C
0.10
0.83
0.88
B∪C
0.00
1.00
1.00
A∪B∪C
c = 0.33
‡ Rating is provided on a scale of 0-10 (0 = no problem, 10 = very severe)
(F)
Expert 1
0@
Rating‡
0.00
0
0.10
3
0.26
8
0.19
6
0.13
4
0.06
2
0.23
7
0.03
1
Expert 2
0A
Rating‡
0.00
0
0.14
5
0.19
7
0.22
8
0.14
5
0.06
2
0.22
8
0.03
1
Kulkarni and Johnson
Table 5: Two Expert Detection Ratings
Expert 1
(F)
Expert 2
Combined Evidence
0@,A
21,2
31,2
0.00
0.00
0.00
0.08
0.08
A
0.08
0.01
B
0.01
0.01
0.04
C
0.04
0.05
0.00
0.09
0.10
A∪B
0.00
0.12
0.13
A∪C
0.01
0.05
0.05
B∪C
0.00
0.13
0.13
A∪B∪C
c = 0.06
† Rating is provided on a scale of 0-10 (0 = no problem, 10 = difficult to detect)
Rating
0
6
1
3
0
3
0
2
∅
0@
0.00
0.19
0.03
0.10
0.00
0.10
0.00
0.06
†
Rating
0
7
1
2
0
0
3
1
0A
0.00
0.19
0.03
0.06
0.00
0.00
0.08
0.03
†
As discussed in the previous example, risk is quantified using interval arithmetic and failure modes are ranked using
the aforementioned rules. Table 6 shows the FMEA results using the DST-based approach. In this case we have
multiplied the risk bounds by a factor of 1000 just for ease of illustration (this does not affect how priority is
assigned to each failure mode). Table 7 shows the results of the traditional FMEA method when the individual
expert ratings are averaged to obtain a combined rating. Notice that we have taken the average of the singleton
ratings for O, S, and D from Tables 3, 4, and 5, respectively. Also for occurrence under the traditional method we
have just used the BPA ratings (since traditional ratings were not recorded for occurrence). However, this will not
have any effect on the results as only the scale would change but rankings would remain the same.
It can be seen that when using DST-based method, failure mode C is assigned the highest priority (Table 6), as
opposed to failure mode A when using traditional method (Table 7). Failure mode A receives the second highest
priority using DST-based approach, whereas it is the higher priority failure mode using the traditional method.
Failure mode B receives the least priority in either case.
Table 6: Results for DST-based FMEA with Two Experts
(F)
A
B
C
2(1)
0.40
0.03
0.52
O
3(1)
0.42
0.07
0.57
2(1)
0.12
0.43
0.30
S
3(1)
0.17
0.56
0.41
2(1)
0.08
0.01
0.04
D
3(1)
0.08
0.01
0.05
6 × 7 × 8 × @HHH
2(1)
3.900
0.098
5.509
3(1)
6.203
0.597
10.693
RPN
2
3
1
Table 7: Results for Traditional FMEA with Two Experts
(F)
A
B
C
O
0.38
0.05
0.35
Average Rating
S
4.00
7.50
7.00
D
6.50
1.00
2.50
6×7×8
RPN
9.75
0.38
6.13
1
3
2
Clearly the difference in the results obtained from the two methods can be observed. This difference lies in the fact
that the traditional method is not able to take advantage of all available evidence. The traditional method relies on
the evidence obtained from the single failure modes (i.e., singletons from a set theory view) and ignores any
additional evidence that may be available. On the other hand, the DST-based approach takes advantage of all the
available evidence, which is obtained through the BPAs on the singletons along with all the other members of the
power set.
Kulkarni and Johnson
5.0 Conclusions
FMEA and FMECA techniques have been successfully used in many industries to identify potential failures and
their resulting impact on the system, but are not without their own shortcomings. One of the commonly encountered
problems is aggregating team member ratings. Traditionally used methods for aggregating responses include
averaging (equally weighed responses or giving more weight to responses from more experienced members or
experts) and reaching a group consensus. Each of these aggregation methods has certain weakness which may
influence RPN rankings. Furthermore, in cases where only partial or incomplete information is available, or in
absence of any historical information, it becomes very difficult to justify how an expert can provide precise ratings
in order to calculate RPNs.
In this paper we proposed a method that utilizes Dempster-Shafer theory (DST) to overcome the aforementioned
issues. DST approach allows the experts to express their beliefs that an observed failure mode is the cause of failure,
eliminating the need to assign a specific probability value. Furthermore, this technique makes use of all available
information, and also allows assigning belief values to all possible combinations of failure modes. The proposed
method makes use of Dempster’s rule of combination to aggregate responses from multiple sources, team members,
experts, etc. Combining responses in such a manner has several advantages - it eliminates bias which may arise due
to group think or by an overbearing team member and overcomes the issue of subjectively assigning weights to
different team members.
However, there are some issues using the Dempter's rule of combination. When highest degree of conflict exists
between two expert opinions, i.e., c (normalization factor) = 1, then (1 - c) = 0. In such a case the combination rule
will fail. Furthermore when the degree of conflict between two or more opinions is very large (c → 1, but c < 1), this
rule has a tendency to produce counterintuitive results in certain contexts. While it is unlikely that supposed experts
would have such varying (conflicting) rating assessments, this shortcoming still needs to be noted. Literature
provides other methods for combining responses, some of which improve on the aforementioned problems with
Dempter's rule of combination. These methods have to be explored for their feasibility and applicability to develop
FMEAs. Additionally, the proposed method should be applied to a real-life example, and then be compared with the
results obtained from the traditional method of aggregating responses.
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