### 中 華 大 學 博 士 論 文

題目: 三種失效模式與效應分析之改善考量 Three improvements on Failure Mode and

Effects Analysis

系 所 別：科 技 管 理 研 究 所 學號姓名：D09403009 陳 日 光 指導教授：李 友 錚 博 士

中華民國九十七年八月

**ABSTRACT **

Traditionally, the decision of improvement is based on Risk Priority Number (RPN) in FMEA, but many scholars questioned the RPN method, and proposed some new methods to improve. However, these methods are only measuring from the risks viewpoint while ignoring the utility of corrective actions. This study propose three improvements on FMEA, firstly aims to evaluate the structure of hierarchy and interdependence of corrective action by Interpretive Structural Model (ISM), then to calculate the weight of corrective action through the Analytic Network Process (ANP), furthermore, combine the utility of corrective actions, to made decision on improvement priority order of FMEA by Utility Priority Number (UPN). It verifies the feasibility and effectiveness of this method by case study. Secondly then propose a Fuzzy ISM method to taking account the Fuzzy linguistic consideration to fit in with real complicated situation, and then compare difference of the order of corrective action obtained; The last adopt a new method to analyze the causal relationship and influence strength of corrective actions with Decision Making Trial and Evaluation Laboratory (DEMATEL), then to effectively integrate ANP to get a more complete weight of corrective action considering influence strength. Finally compare the hierarchical structure between hierarchical relation consideration and causal relationship consideration to verify its’ efficient.

**Keywords: RPN; FMEA; ISM; ANP; DEMATEL **

**摘 要 **

傳統上失效模式與效應分析之改善優先順序是依風險優先數之大小來做決 策，但已有許多學者提出以風險優先數為決策考量之一些缺點，並提出一些新 的方法來改善其缺失。但這些方法仍是從風險之角度來衡量，而並未將改善對 策之效果度納入衡量。本研究提出三個失效模式與效應分析之改善方法，首先 以階層明示法將改善對策之階層關係架構求出，再以網路分析法得到各改善對 策之權重，然後進一步結合改善對策之效果度，以效用優先數來做為改善優先 順序之決策。經實例之驗證，此方法確實具可行性與有效性。其次，提出以模 糊階層明示法來納入模糊語意之考量以更貼近人類之思維方式，然後比較此方 法所得到之改善對策優先順序與先前方法之差異性。最後，提出以決策實驗室 法納入改善對策之因果關係與影響強度為考量，並進一步結合網路分析法以得 到較完整考量之權重。最後並比較新方法與先前方法所得結果之差異性，以驗 證其有效性。

關鍵詞：風險優先數、失效模式與效應分析、階層明示法、網路分析法、決策 實驗室法。

**CONTENT **

Abstract ... i

Abstract (in Chinese) ... ii

Content ... iii

List of Tables... v

List of Figures ... vi

Chapter 1 Introduction ... 1

Section 1 Research Background ... 1

Section 2 Research motive and purpose ... 2

Section 3 Research procedure ... 3

Chapter 2 Literaturure review ... 5

Section 1 FMEA basis ... 5

Section 2 Evaluation of risk priority number ... 6

1. Conventional priority evaluation ... 6

2. Argument of risk priority number ... 8

3. Others methods for risk evaluation ... 9

Chapter 3 Hierarchical relation consideration ………...….13

Section 1 Research methodology ... 13

Section 2 Case study ... 16

1. Conventional FMEA implementation ... 16

2. Evaluation of untility priority number ... 18

3. Comparison ... 23

Chapter 4 Fuzzy linguistic consideration ... 24

Section 1 Background and research purpose ... 24

Section 2 Fuzzy ISM approach ... 24

Section 3 Case study ... 28

Section 4 Comparison ... 32

Chapter 5 Causal relationships consideration ... 33

Section 1 Background and research purpose ... 33

Section 2 Brief of DEMATEL ... 33

Section 3 Methodology ... 37

Section 4 Case study ... 38

Section 5 Comparison ... 41

Chapter 6 Conclusion ... 43

Section 1 Result and research summarized ... 43

Section 2 Research limitations ... 44

Reference ... 46

**LIST OF TABLES **

Table 1. Suggested Evaluation Criteria for Severity... .7

Table 2. Suggested Evaluation Criteria for Occurrence... 7

Table 3. Suggested Evaluation Criteria for Detection... 8

Table 4. FMEA report ... 17

Table 5. Priority and Corrective actions ... 18

Table 6. Comparison of Priority Order ... 23

Table 7. Fuzzy linguistic translation ... 27

Table 8. New UPN for *causal relationship consideration ... 41 *

Table 9. Priority comparison between methods ………...41

**LIST OF FIGURES **

Figure 1. The decision tree for S-A-G-W ... .10

Figure 2. Binary relation ... .14

Figure 3. The relation of orders of the corrective action ...19

Figure 4. Hierarchy structure ………...21

Figure 5. Supermatrix structure ………...21

Figure 6. Triangular membership function ………... 26

Figure 7. Fuzzy linguistic ………....26

Figure 8. Hierarchy Structure of Fuzzy linguistic consideration ...31

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**CHAPTER 1 INTRODUCTION ** **Section 1 Research background **

Failure Mode and Effects Analysis (FMEA) was first used in the early 1950’s for the design and development of flight control systems (Dhillon, 1992). An early FMEA implementation took place in the National Aeronautics and Space Administration (NASA) Apollo-Saturn program (Loch, 1998). FMEA was used by NASA to assure that hardware built for space applications has the desired reliability characteristics (Jordan, 1972). In the 1960’s, FMEA became a widely used reliability and safety technique in the aerospace, automotive, electronic and nuclear industries (Russomanno et al, 1994). FMEA became better known when Ford implemented it in approximately 1977. FMEA has been employed in software development since the late 1970’s (Reifer, 1979). Today, FMEA became standard practice in Japan, American and European manufacturing companies (Huang et al, 1999).

FMEA is a kind of design and analysis technology of failure and prevention, which is a structured systematic identifying the potential failure mode in design or manufacturing, then studying the impact of failure to the system and providing qualitative evaluation, thus taking necessary correction while aiming at the problems lying in the systematic reliability. Stamatis (1995) pointed out that FMEA is used in the electromechanical, semiconductor and medical device industries and for computer hardware and software. Onodera (1997) investigated about 100 FMEA applications in various industries in Japan, and he found that the FMEA is used in the areas of electronics, automobiles, consumer products, electrical generating power

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plant, building and road construction, telecommunications, etc. In addition, it is widely used in service industry, for example, Linton (2003) applied to E-Commerce;

Reiling et al. (2002) applied to the cure for medical errors; Shahin (2005) integrated the Kano’s mode and was applied to travel agent; Even some special fields, such as preventing medical accidents (Reliey, 2002); environment management systems (Vandenbrande, 2003) and supplier development (Sawhney & Padiyar, 2004). It indicates that FMEA is the popularization of application.

**Section 2 Research motive and purpose**

Traditionally, using FMEA to improve is in the order from bigger Risk Priority Number (RPN) to smaller ones. However, several drawbacks have been pointed out by researchers concerning the rationality of this approach, especially the risk priority number as determined by multiplying the converted scores of the three predefined factors without considering their relative importance (Gilchrist, 1993). There are some different calculated methods of RPN proposed, such as Criticality Score Evaluate, Level of Risk, Critical Analysis and Matrix Method, etc. Several new methods were proposed in recent years, however, these methods still measures from risks viewpoint, while ignoring the implementation utility of corrective actions.

Corrective actions may be not independent from each other, and may have the relationship of hierarchy or even interdependence. For example, to carry out the corrective action of the Item 1 should start from Item 2; after Item 2, to implement Item 3 and then Item 1 may be relatively more convenient; the implementation of Item 3 may affect the previous setting of Item 2, especially when adjusting the

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equipment parameter setting. Further, the implementation of the corrective actions is in proper order, it not only maximizes the improvement effect, but also brings favorable result in the shortest time at the lowest costs. Therefore, when perform FMEA, besides the measurement of risks, to involve the utility of corrective actions is also an important subject.

**Section 3 Research procedure**

Against these backgrounds, this study firstly aims to evaluate the structure of hierarchy and interdependence structure by Interpretive Structural Model (ISM), and then calculate the weight of corrective action through Analytic Network Process (ANP). Furthermore, combine the utility of corrective actions, and to determine the improvement priority order of FMEA by Utility Priority Number (UPN). Finally, it verifies the feasibility and effectiveness of the method through the case of Surface Mounting Technology (SMT) process improvement of an electronic assembly plant.

Secondly this study further involve the Fuzzy linguistic consideration because crisp values may inadequate to present because people would have difficulties in understanding the difference and uncertainties in human’s semantic expression, due to intangible and subjective information often appearing in the investigation or evaluation process. Hence Fuzzy ISM method is proposed and then to compare the hierarchical structure between hierarchical relation consideration and Fuzzy linguistic consideration via the same illustration case.

Finally this study continually take account the causal relationships and influence strength of corrective action consideration. We adopt a new method to analyze the

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causal relationship and influence strength of corrective actions with Decision Making Trial and Evaluation Laboratory (DEMATEL), then to effectively integrate ANP to get a more complete weight of corrective action considering influence strength. At last, to compare difference priority order of corrective action obtained between conventional method, hierarchical relation consideration and causal relationships consideration via the same illustration case.

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**CHAPTER 2 LITERATURE REVIEW ** **Section 1 FMEA basis**

The FMEA Process columns normally include Item and Function; Potential Failure Mode; Potential Effects of Failure; Severity; Potential Causes of Failure;

Occurrence; Current Controls; Detection; Risk Priority Number; Recommended Action; and Responsibility and Target Completion Date. The Action Results columns include Action Taken, Severity, Occurrence, Detection and Risk Priority Number.

There are four types of FMEA: the system FMEA, The design FMEA, the process FMEA, and the service FMEA (Stamatis, 1995). All types of FMEA produce a list of failure modes ranked by the risk priority number and a list of recommended actions to reduce failures or to improve failure detection.

Loch (1988) identified six purposes of the FMEA method. First, the FMEA provides early recognition of product failures and the consequences to the customer.

Second, the FMEA helps determine potential cause for failure. Third, the FMEA points out preventive methods to decrease a failure’s occurrence and increase its detection. Fourth, the FMEA provides a prioritized list of potential failures. Fifth, the FMEA method helps determine both the reasons for and the consequences of potential failures. Sixth, the FMEA points out countermeasures and requirements changes to be introduced.

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**Section 2 Evaluation of risk priority number **

**1. Conventional priority evaluation **

Traditionally in FMEA, uses the RPN to conduct the risk assessment, which is still widely adopted by all industries. Take the Potential Failure Mode and Effects Analysis Manuel (AIAG, 1995) of QS-9000 as an example, the risk factors include:

(1) Severity (S): Result generated from failure (2) Occurrence (O): Opportunity brought by failure.

(3) Detection (D): Opportunity brought by unidentified failure or the difficulty of detection.

The three factors are to be scored from 1 to 10 on the basis of degree (refer to Table 1; 2 and 3). Risk Priority Number (RPN) is the product of occurrence, detection and severity, which can be expressed as follows:

RPN = S × O × D

RPN is the selection foundation of improvement, the larger RPN stands for more prior improvement. Corrective action is taken by the relevant department beginning with the riskiest, as shown by the RPN. RPN should be recalculated after a certain time lapse to see if they have gone down, and to check the efficiency of the corrective action for each failure cause.

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Table 1

*Suggested Evaluation Criteria for Severity *

Table 2

*Suggested Evaluation Criteria for Occurrence *

8

Table 3

*Suggested Evaluation Criteria for Detection *

**2. Argument of risk priority number **

Ben-Daya and Raouf (1993) advocated giving the occurrence factor the most weight in a risk priority number formula. They argued that the occurrence factor is the most important because it affects the likelihood of a fault reaching the customer.

They advocated calculating the risk priority number using an occurrence value is 1 to 0. They also recommended using expected costs in conjunction with FMEA. Kara- Zaitri and Fleming (1997) argued that the risk priority number method of FMEA is intrinsically subjective because guidelines for rating severity, occurrence, and detection vary from one institution to another. They pointed out that the same risk priority number cab be obtained using a number of different combinations of severity, occurrence, and detection factors.

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Sankar and Prabhu (2001) pointed the distortion is compounded by the non-linear nature of the individual ranking scales. Subjectivity in the ranking scales adds inherent inconsistency. Franceschini and Galetto (2001) pointed out the RNP is the original ordinal scale which is transformed in a new cardinal scale characterized by a metric and by the integer number composition properties. This arbitrary `promotion’

of the scale properties brings about a series of problems in the RPN interpretation.

Bowles (2003) compared four types of measurement scales, Nominal Measurements;

Ordinal Measurements; Equal Interval Measurements and Ratio Measurements, and suggested the RPN should be dropped and an entirely different prioritization technique should be used.

**3. Other methods for risk priority **

A different formulation for evaluating the risk priority of FMEA is used in
Japan, it called Criticality Score Evaluate Method, in which five factors are
*combined to assess the risk of failure. Where C*1 is the degree of importance affecting
*the functional failure; C*2* is the affected the range of the system; C*3 is the frequency
*of the failure; C*4* is the possibility of prevention; and C*5 is the difficulty of changing
the design. The risk priority number is obtained by using the following equation:

5 *C*1 *C*2 *C*3 *C*4 *C*5

*C** _{s}* = × × × ×

Where:

* C**s** : 7 ≦ C**s *≦ 10 : Risk level I
* 4 ≦ C**s* < 7 : Risk level II
* 2 ≦ C**s* < 4 : Risk level III
* C**s* < 2 : Risk level IV

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Another formulation recommended by DINV 19250 (KEMA, 1996), in which the four factors of S-A-G-W are used to evaluate the risk level. Where S is the four- level severity of the possible harm; A is the two-level possibility of avoidance; and G is the three-level probability of occurrence of an event that can cause harm. The decision for the level of risk could be made according to the following decision tree:

* Figure 1. The decision tree for S-A-G-W *

Gilchrist (1993) suggested the use of expected costs in prioritizing failures. He noted that for hundreds of years, it has been generally agreed that the way to express severity has been in financial terms. The expected cost helps justify remedial actions.

He pointed out that ballpark figures are all that are usually needed. Expected costs can be summed to show the impact of all failure modes for a root cause. Barsky and Cutta (1997) also proposed the used of expected costs instead of risk priority number in FMEA.

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Onodera (1997) noted that criticality can be computed as an alternative to risk priority number by omitting the detection rating from the calculation. He suggested the use of criticality in the design and development stages, and stated that the risk priority number should be used at the manufacturing and inspection stages because failure detection rate is an important factor in manufacturing and inspection processes.

Bowles (1998) described multi-criteria Pareto ranking as an alternative method to the classical risk priority number calculation. In this method, failure modes are plotted on a matrix where the horizontal axis is severity and the vertical axis is the probability of occurrence. The vertical access uses a log scale. Failure modes are of increasing criticality the nearer they are to the upper right corner of the matrix. All failure modes with no other failure modes rated higher on both severity and occurrence are labeled as rank 1 item failure modes to be attacked first.

Chang et al. (2001) used the Grey model to rank the priorities, first, they established comparative series, then, established standard series; Obtained difference between comparative series and standard series, next, compute the grey relational coefficient; Determine the degree of relation, finally, rank the priority of risk.

Puente et al. (2002) proposed a RPC index which an intuitive decision system beads on risk priority class instead of RPN. The categories involved to calculate the risk based on a new estimated rule which allows weighting of any input variables seen as being more important, A asymmetric RPC matrix with greater risk implications be considered, thus, priority can be obtained even a new input variable

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be considered, such as cost.

Bowles (2003) recommended the detection ranking should be dropped, and the rankings for the severity categories also should be scrapped. Instead these categories should be considered as nominal classifications for safety, operational, and cosmetic related failure mode effects.

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**CHAPTER 3 HIERARCHICAL RELATION ** **CONSIDERATION **

**Section 1 Research methodology **

Interpretive Structural Model (ISM) is a complicated system structural method proposed by Warifield (1973). ISM is the frameworks of hierarchy of a group of factors in a known order, which are adopt to FMEA in the following steps:

1. Analyze the binary relation of corrective actions:

*If there are n corrective actions, its set is expressed by S, therefore, *
(1) *S* =

### {

*s*

_{1},

*s*

_{2},L,

*s*

*n*

### }

;(2) (*s** _{i}*,

*s*

*) is the ordered pair of*

_{j}*s and*

_{i}*s ;*

_{j}(3) Cartesian product is：* ^{S}*×

*=*

^{S}## {

(

^{s}*i*,

^{s}*j*)|

^{s}*i*,

^{s}*j*∈

*;∀*

^{S}*,*

^{i}

^{j}## }

;*(4) The relationship of factors in S is defined as binary relation;*

*(5) R refers to the set of the ordered pair *(*s** _{i}*,

*s*

*) with the binary relation in product set*

_{j}*S*×

*S, i.e. R is a subset of*

*S*×

*S*;

*R*⊂

*S*×

*S*;

(6) If the ordered pair(*s** _{i}*,

*s*

*)∈*

_{j}*R*shows that

*s is related to*

_{i}*s ; s*

_{j}

_{i}*R s*

*,*

_{j }*S*

*s*
*s*_{i}* _{j}*∈

∀ , ;

If the ordered pair (*s** _{i}*,

*s*

*)∉*

_{j}*R*shows that the factor

*s is irrelevant to*

_{i}*s ;*

_{j}*s*

*i*

*R s*

*j*, ∀ ,

*s*

_{i}*s*

*∈*

_{j}*S*;

*(7) Each factor in the set S can be regarded as a node, which can be found a *
solution by graph theory. If two nodes are related, they can be linked up by
direct line as shown in Figure 2:

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* Figure 2. Binary relation *

2. Adjacency matrix

*After comparison of binary relation R in S with all ordered pairs, the binary *
matrix: *n*×*n matrix A may be evaluated, A is also called as the adjacency *
*matrix of binary relation R, where: *

[ ]*a**ij* _{n}_{n}

*A*= _{×}

Where: *a** _{ij}* =1 ,

*if s*

_{i}*R s*

*;*

_{j}*a*

*=0 ,*

_{ij}*if s*

_{i}*R s*

_{j}3. Reachability matrix

*(1) Since each node can reach its own node, add adjacency matrix A to identity *
*matrix ( I ):N* = *A*+*I* ;

*(2) N is called as element connection matrix, the path length 0 or 1 can be used to *
*express its reachability. Then evaluate the power matrix of N by Boolean *
operation, until it meets*N* ≠ *N*^{2} ≠*N*^{3} ≠L≠ *N*^{r}^{−1} =*N** ^{r}* =

*M*;

*(3) M is called as reachability matrix, which is of transitive relation with *
*element connection matrix N. If * *M*(*s** _{i}*,

*s*

*)=1, it means there is path existing*

_{j}s1

s4

s2

s3

15

from node *s to *_{i}*s ; If *_{j}*M*(*s** _{i}*,

*s*

*)=0, it means there is no path existing from node*

_{j}*s to*

_{i}*s .*

_{j}4. Expression of hierarchy

*In all the figures got from the reachability matrix M, all the nodes can be *
concluded as:

*(1) Adjacency reachiability set ( R ):R*^{=}

## {

*s*

*j*

^{|}

*s*

*j*

^{∈}

*S*

^{,}

*M*

^{(}

*s*

*i*

^{,}

*s*

*j*

^{)}

^{=}

^{1}

## }

*(2) Adjacency antecedent set ( A ):A*^{=}

## {

*s*

*j*

^{|}

*s*

*j*

^{∈}

*S*

^{,}

*M*

^{(}

*s*

*j*

^{,}

*s*

*i*

^{)}

^{=}

^{1}

## }

(3) If n factors in S meets the condition:*R*(*s** _{i}*)I

*A*(

*s*

*)=*

_{i}*R*(

*s*

*),∀*

_{i}*s*

*∈*

_{i}*S*. If so, draw out and list

*s in the same hierarchy, eliminate it from the reachable*

_{i}*matrix M, and repeatedly evaluate adjacency reachiability set R and*

*adjacency antecedent set A until all the factors are drawn out. Finally*establish the hierarchy structure according to it.

Base on the hierarchy structure, thus calculate the weights of corrective action by ANP method. ANP was put forward by Saaty (1996), whose difference from AHP (1980), it allows the inner dependence within cluster and outer dependence among clusters. It provides a complete structure, thus the researchers may find the interactions between elements and clusters from problems, then deduce the priority and proportion of each scheme. ANP method includes two parts:

1. The first part is control hierarchy, which refers to the network relationship of guideline and sub-guideline, influencing the internal relationship of systems.

2. The second part refers to the network relationship between elements and clusters.

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Hence, at first establish pair-wise comparison matrix of corrective action, then the eigenvector of each pair-wise comparison matrix shall be calculated, it is usually by NGM method (Normalization of the Geometric Mean of the Rows). Then group the eigenvectors of each pair comparison matrix into a supermatrix, but it may be the un- weighted supermatrix, it should be transformed into weighted supermatrix, that is, the sum of any column shall be 1. Several times of multiplication until the extreme value is fixed, i.e. limiting supermatrix, then stop. Finally all the weights would be obtained, and there stand for the utility of corrective actions. Further, use the utility of corrective actions to make decision on improvement priority order of FMEA by UPN, which is calculated by following formula:

UPNi = RPNi × Ci = Si × Oi × Di × Ci (1) Where: Ci is the weight of the utility of corrective actions.

**Section 2 Case study **

**1. Conventional FMEA implementation **

In electronic assembly industry, the main operation of SMT manufacturing is to place the parts of main board on Printed Circuit Board (PCB) by Part Mounter Machine, then fix them on PCB by Reflow Oven and Flow Soldering Machine. But the open joint and distorted welding spot are common problems in SMT manufacturing process, the research team tries to improve its quality and implement FMEA in Table 4:

17

18

Upon analysis, there are 10 possible causes of open joint and 12 causes of distorted welding spot. Upon assessment of RPN, the priority order is shown as the column of priority in Table 5. By UPN, the research team choose those with RPN higher than 100 as improvement priority items upon study first, totaling at 9. Then it is to propose corrective actions upon discussion (as shown in Table 5):

Table 5

*Priority and Corrective actions *

**2. Evaluation of utility priority number **

After the assessment of the ordered pair of binary relation, the binary relation of the corrective action in item 9 is shown as Figure. 3:

19

* Figure 3. The relation of orders of the corrective action *

*The adjacency matrix A can be obtained from the binary relation in Figure 3, *
then it can be transformed into element connection matrix N. Finally, the reachability
*matrix M can be obtained by Boolean operation: *

*A = *

*N = A + I = *

20

*= M*

In this way, we can get reachability matrix*M* = *N*^{4}, then further calculate the
*Adjacency reachiability (R ) and Adjacency antecedent set (A) of M, and draw out * *s ** _{i}*
if they meet

*R*(

*s*

*)I*

_{i}*A*(

*s*

*)=*

_{i}*R*(

*s*

*), list them in the same layer. Then, eliminate the*

_{i}*drawn factor from the reachable matrix M, repeatedly evaluate adjacency*

*reachiability set (R) and adjacency antecedent set (A) until all the factors are drawn*out.

=

=

≠ ^{3} ^{4}

2 N N

N

21

Finally, establish the hierarchy structure as Figure 4:

* Figure 4. Hierarchy Structure *

From the hierarchy structure, S1 and S9 are control level, whose weights are given as 0.6 and 0.4 respectively. The lower layer is regarded as network level, by which to evaluate the pair-wise comparison matrix, and then to evaluate its eigenvectors to group into supermatrix. Its structure is showed as Figure 5:

* Figure 5. Supermatrix Structure *

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To group the supermatrix shall first transform them into weighted supermatrix as shown in the following figure:

Get constringent supermatrix after multiplying the supermatrix until its constringency, it can show the weight of each corrective action as follows:

To sum up, the weights of corrective actions are shown in Table 6. Bring them into the Formula (1) and get each UPN, the difference of priority order is found from traditional RPN with UPN as the Table 6:

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Table 6

*Comparison of Priority Order*

**3.3 Comparison **

To compare the improvement priority order by conventional RPN and hierarchy
relation consideration with UPN evaluation, the finding is the 8^{th} in priority order by
conventional RPN but ranks 1^{st} by hierarchy relation consideration with UPN; the 2^{nd}
in priority order by conventional RPN but ranks 7^{th} by hierarchy relation
consideration with UPN; the 6^{th} in priority order by conventional RPN but ranks 3^{rd}
by hierarchy relation consideration with UPN. In other items, there is no big
difference. But follow the new priority order, it may not only involving the utility of
corrective actions, but also maximizes the improvement effect, even brings the
favorable result in the shortest time at lowest cost. The method is indeed feasible and
efficient through this case study.

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**CHAPTER 4** **FUZZY LINGUISTIC ** **CONSIDERATION**

**Section 1 Background and research purpose **

In general, researchers implement the investigation or evaluation with crisp values to present people’s feelings and subjective perceptions. In fact crisp values may inadequate to present because people would have difficulties in understanding the difference and uncertainties in human’s semantic expression, due to intangible and subjective information often appearing in the investigation or evaluation process.

Several researches have proved the Fuzzy linguistic works better than the crisp values in terms of reliance and effectiveness. But until now, very less research has explored Fuzzy linguistic instead of crisp values to ISM. Since the adjacency matrix of ISM is from binary evaluation, it may be 0 or 1, which may be led to errors for subjective and extreme evaluation results. Therefore, in this chapter we propose a Fuzzy ISM method to taking account the Fuzzy linguistic consideration to fit in with real complicated situation, and then compare difference of the order of corrective action obtained from different way via the same illustration case.

**Section 2 Fuzzy ISM Approach **

Since people’s knowledge and language turn on a Fuzzy nature under different conditions are as their subjective consciousness, different time, environmental change and the perspective in judging an issue, which impairs the researcher from clearly knowing the true nature of the object, then to appropriately set the presumed

25

mathematical model. Therefore, the idea of Fuzzy theory occurred, which had referred to the Fuzzy measurement and classification theory used to environment in people’s way of thinking, providing a relatively stable description to define pluralistic and complicated ambiguous and uncertain phenomena. Zadeh (1965) imitated the way of characteristic function to express classical set and established membership function to describe Fuzzy set. In Fuzzy theory, membership function is used to express the degree of universe element belonging to the Fuzzy concept.

Membership function can be divided into discrete type and continuous type. Discrete
membership function is the membership which is directly given to each element of
limited Fuzzy set, and is expressed in vector style. Continuous membership function
has several commonly used function forms (Triangular function, Trapezoidal
function and Normal function) to describe Fuzzy set. Assume *X* =

### [

*x*

_{1},

*x*

_{2},...,

*x*

*n*

### ]

is a discrete set, the membership of Fuzzy subset can be expressed as:### ∑

= +

+ +

=

i i n

n 2

2 1

1 ( ) ( )

) ....

( ) (

*x*
*x*
*x*

*x*
*x*

*x*
*x*

*A* μ^{A}*x* μ* ^{A}* μ

*μ*

^{A}

^{A}

As for the continuous function, if *x*∈

### [ ]

*a*,

*b*is a continuous set, a and b are numbers, the membership of Fuzzy subset can be expressed as:

*x* *dx*
*A* ^{b}*x*

*a*

### ∫

*A*

= μ ( )

Take the commonly used trigonometric membership function as an example, its membership function is showed as Figure 6:

26

*Figure 6. Triangular membership function. *

If the semantic being expressed by Fuzzy number, and the mental feeling and attitude being expressed by discrete membership function, e.g. the five agrees include “Strongly Disagree”; “Disagree”; “Nature”; “Agree” and “Strongly Agree”.

Thus the set *L：L ={Strongly Disagree, Disagree, Nature, Agree, Strongly Agree} *

is expressed by *l*1, *l*2, *l*3, *l*4, *l*5* in order. If F is used to score the Fuzzy linguistics, *
then the corresponding function is:

*i*
*l*

*F*( * _{i}*)

*= , i =1,2,3,4,5, in which*~ ~

*i*is a Fuzzy number.

If symmetric triangular Fuzzy number is used as the linguistic variable, then its measurement can be expressed as Figure 7:

**Figure 7. Fuzzy linguistic **

27

*Hence, obtain the Fuzzy number from the answers after comparing the i** ^{th}* and

*the j*

*corrective actions by the above method, then transform the Fuzzy number into a triangular Fuzzy number by Table 7: A~ ( , , )*

^{th}*j*
*i*
*j*
*i*
*j*
*i*
*j*

*i* = *a* *b* *c* .

Table 7

*Fuzzy linguistic translation. *

*The e**i j p *is ordered to be the valuation matrix, being used to express the decision
*value of corrective action in Item i and j of people number p. Then the judgment *
results are to be integrated. The integrated Fuzzy number matrix is ordered to be

) , , ( A~

*j*
*i*
*j*
*i*
*j*
*i*

***

*i j* = *A* *B* *C* , thus:

} { min

A _{i}_{ p}_{j}

*ij* = *p* *e*

*p*
*i j p*
*p*

*p*

*ij* *e*

*B*

1

1

⎥⎦

⎢ ⎤

⎣

=⎡

### ∏

=

(2)

}
{
max _{i}_{ p}_{j}

*ij* *p* *e*

*C* =

28

Then, use the graded mean integration proposed by Hsien & Chen (1998) to calculate the property average (PA) of the Fuzzy set, by which to stand for the characteristics of the whole set, this method is shown by the following Formula (3):

### ∫

### ∫

^{−}

^{+}

^{−}

= _{w}

0 0

1

1( ) ( ))2 ] [ (

*hdh*
*x* *dh*
*R*
*x*
*h* *L*

*PA*

*w*

*ij*

Compare each property average with the central Fuzzy number 0.5, order the superior central Fuzzy number as 1, and the inferior Fuzzy number as 0, we can get the adjacency matrix, then we can get the diagram of hierarchy structure for all the corrective actions by ISM.

**Section 3 Case study **

We implement further exploration via same empire case in chapter 3. According to Table 4, 10 possible causes of open joint and 12 causes of distorted welding spot are base to explore. Three person of research team also choose 9 items with RPN higher than 100 as Table 5 upon study. Then, evaluate the adjacency of corrective actions by Fuzzy linguistics, transform them into Fuzzy numbers according to Table 7, get ~

*j*

*A by integrating all the members’ advices with Formula (2), then get the **i*

property average matrix by Formula (3). Finally, the adjacency matrix is obtained after comparison by the central Fuzzy number of 0.5 as follows:

= 6

) 4

(*A*_{i}* _{j}* +

*B*

_{i}*+*

_{j}*C*

_{i}*(3)*

_{j}29

*Then can be transformed into element connection matrix N. Finally, the *
*reachability matrix M can be obtained by Boolean operation: *

30

In this way, we can get reachability matrix*M* = *N*^{4}, then further calculate the
*Adjacency reachiability (R ) and Adjacency antecedent set (A) of M, and draw out * *s ** _{i}*
if they meet

*R*(

*s*

*)I*

_{i}*A*(

*s*

*)=*

_{i}*R*(

*s*

*), list them in the same layer. Then, eliminate the*

_{i}*drawn factor from the reachable matrix M, repeatedly evaluate adjacency*

*reachiability set (R) and adjacency antecedent set (A) until all the factors are drawn*out.

*N = A + I = *

=

=

≠N3 N4 N2

31

Finally, establish the hierarchy structure as Figure 8.

*Figure 8. Hierarchy Structure of fuzzy linguistic consideration*

32

**Section 4 Comparison**

According to the hierarchy structure of Figure 8, S1 and S9 are the first hierarchy items to be improved; S2, S6, S7 and S8 are the second hierarchy items to be improved; S3, S4 and S5 are the third hierarchy items to be improved. The result of hierarchy structure is same as the hierarchy relation consideration. Hence, it indicates the Fuzzy ISM indeed can be used, and in order to eliminating people’s intangible and subjective judgment in the investigation or evaluation process, involve Fuzzy linguistic consideration might obtain more accurate priority of corrective action, then to maximizes the improvement effect.

33

**CHAPTER 5 CAUSAL RELATIONSHIP ** **CONSIDERATION **

**Section 1 Background and research purpose **

From deeper perspective, ISM can clarify the relationship between different hierarchies and the corresponding effects of corrective action, but it cannot explain their causal relationships and influence strength. Therefore, to analyze its weight directly with ANP method may lose its accuracy. This section adopt a new method to analyze the causal relationship and influence strength of corrective actions with DEMATEL, then to correct the original DEMATEL by referring to relevant scholars’

researches, finally to effectively integrate ANP to get a more complete weight of corrective action considering influence strength. At last, this chapter compare the hierarchical structure between hierarchical relation consideration and causal relationship consideration to verify its’ efficient.

**Section 2 Brief of DEMATEL **

The Decision Making Traial and Evaluation Laboratory (DEMATEL) method studied the disjointed and antagonistic phenomena of world and investigated integrated solutions from the Geneva Research Centre of the Battelle Memorial Institute (Gabus & Fontela, 1973; Fontela & Gabus, 1976). In recent years, DEMATEL method has been widely used to extract a problem structure of a complex problematique (Fontela & Gabus, 1974), it is especially practical and useful for visualizing the structure of complicated causal relationships with matrices or

34

digraphs, which portray the contextual relations between the elements of a system, where a numeral represents the strength of influence. Therefore, the DEMATEL method can convert the relationship between the causes and effects of criteria into an intelligible structural model of the system.

The DEMATEL method has been successfully applied in many fields. For example, Tamura et al. (2003) try to decrease anxiety of people by extracting and analyzing various uneasy factors in order to create future safe, secure and reliable (SSR) society. More recently, Chiu et al. (2006) adopted the method to study marketing strategy based on customer behavior related to LCD-TVs. Also Hori &

Shimizu (1999) employed it to design and evaluate the software of a display screen structure for analyzing a supervisory control system.

The essentials of the crisp DEMATEL method will be reviewed below. Suppose that a system contains a set of criteria C = {C1, C2, ……, Cn}, and the particular pair- wise relations are determined for modeling with respect to a mathematical relation.

1. Definition 1

The pair-wise comparison scale may be designated into four levels, where the scores of 0, 1, 2, and 3 represent “No influence”, “Low influence”, “High influence”, and “Very high influence”, respectively.

2. Definition 2

*The initial direct-relation matrix Z is a n × n matrix obtained by pair-wise *
*comparisons in terms of influences and directions between criteria, in which Z*ij is
*denoted as the degree to which the criterion C*i* affects criterion C*j. Accordingly, all

35

*principal diagonal elements Z*ii* of matrix Z are set to be zero. *

3. Definition 3

Let λ = _{⎟⎟}

⎠

⎜⎜ ⎞

⎝

⎛

### ∑

≤ =

≤
*n*

*j*
*n* *ij*

*i* *z*

1 1

max , *then the normalized direct-relation matrix X can be *

*obtained through X=*

λ*Z . The DEMATEL method further assumes that at least one *
i such that

### ∑

=
*n* <

*j*

*ij* *s*

*z*

1

. This assumption is satisfied in almost all practical cases.

*Hence, matrix X just resembles the sub-stochastic matrix obtained from an *
absorbing Markov chain matrix by deleting all rows and columns associated with
the absorbing states. It was proved that:

*O*
*X*^{w}

*w* =

∞

limit→ , and limit ( ^{2} ... ) ( )^{−}^{1}

∞

→ *I* +*X* + *X* + +*X** ^{w}* =

*I*−

*X*

ω

*Where O is the null matrix and I is the identity matrix. *

Eg.

36

4. Definition 4

*The total-relation matrix T can be acquired by calculating *

1 2

w ( ... ) ( )

limit ^{−}

∞

→ + + + = −

= *X* *X* *X* *X* *I* *X*

*T* ^{w}

The total-indirect-relation matrix H can be obtained through the following formula.

1 2

3 2

w ( ... ) ( )

limit ^{−}

∞

→ + + + = −

= *X* *X* *X* *X* *I* *X*

*H* ^{w}

5. Definition 5

*Let t**ij** (i, j = 1,2,. . .,n) be the elements of the total-relation matrix T, then the sum *
*of rows and the sum of columns, denoted as D**i** and R**j*, respectively, can be
obtained through the following two formulas:

), ,...., 2 , 1 (

1

*n*
*i*

*t*
*D*

*n*

*j*
*ij*

*i* =

### ∑

==

), ,...., 2 , 1 (

1

*n*
*j*

*t*
*R*

*n*

*j*
*ij*

*i* =

### ∑

==

6. Definition 6

*A causal diagram can be acquired by mapping the ordered pairs of (D**k**+R**k** , D**k**−R**k*),
*where the horizontal axis (D+R), named “Prominence”, is made by adding R**k* to
*D**k** , and the vertical axis (D−R), named “Relation”, is made by subtracting R**k* from
*D**k*. The horizontal axis “Prominence” of the causal diagram shows how important
the criterion is, whereas the vertical axis “Relation” may divide the criteria into the
*cause group and effect group. Generally, when the value (D*_{k}*−R** _{k}*) is positive, the

*criterion belongs to the cause group. If the value (D*

*k*

*−R*

*k*) is negative, the criterion belongs to the effect group. Hence, causal diagrams can visualize the complicated

37

causal relationships between criteria into a visible structural model, providing valuable insight for problem solving. Further, with the help of a causal diagram, we may make proper decisions by recognizing the difference between cause and effect criteria.

**Section 3 Methodology **

This section aims to consider the causal relationship and influence strength of corrective action and to effectively integrate ANP, so that to obtain a more complete weight of corrective action by DEMATEL method which uses the composite importance in order to improve the limit of the original DEMATEL. According to Tamura et al. (2005), the specific procedures are as follows:

1. Conduct pair-wise comparison designated into four levels for corrective action, where the scores of 0, 1, 2, and 3 represent “No influence”, “Low influence”,

“High influence”, and “Very high influence”, respectively. Thus, obtain direct-
*relation matrix Z. *

*2. Normalized direct-relation matrix Z to be matrix X *
*3. Calculating the total-relation matrix T byX*(*I− X*)^{−}^{1}
4. Set composite importance

*Z’=W + T × W (4) *
*where: T is the direct/indirect matrix calculated by DEMATEL; *

* Each element of Z is the influence strength of the weighted “influence *
index” of index.

* W is the vector consisting of the weights calculated by ANP. *

38

**Section 4 Case study **

We implement another further exploration via same illustration case again. It’s also from Table 4 and Table 5, and 10 possible causes of open joint and 12 causes of distorted welding spot, and Table 6, through ANP method to obtain the weights of corrective actions to be base, then implementing further exploration. The specific procedures are as follows:

1. Assess the relationship of corrective actions:

Reorganize the previous assessment team to assess the inter-relationship of corrective actions by discussion and brain storms.

2. Create the direct-relation matrix:

Gather the assessment results and create the matrix, name it as “direct-relation
*matrix” and represent it as Z, as well as set its diaphragm logic element as “0.” Since *
*there are totally 9 corrective actions, we get 9×9 matrices, the figures of Z*ij in the
matrices are the influence strengths of the i^{th} corrective action to the j^{th }corrective
action. The assessment result is listed as follows:

3. Directly relate to the matrix normalization:

39

Sum up each lines of figures, let λ represent the reciprocal of the biggest summation, i.e.:

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

### ∑

≤ =

≤
*n*

*j*
*n* *ij*

*i* *z*

1 1

λ max

Thusλ=7, divided the elements of all the matrix with λ, then work out the normalized direct-relation matrix X, the calculation is as follows:

4. Then calculate direct/indirect matrix with the formula: *T*=*X*(*I*−*X*)^{−}^{1}, the
calculation procedure and result are as follows:

40

5. Calculate the matrix of mixed weight relationships

Then calculate the matrix of mixed weight relationships with Formula (4):

*Z = W + T × W, the specific procedures and results are as follows: *

Normalize the results to obtain the final weights and work out the new UPN with formula (1) as shown in Table 8:

41

Table 8

*New UPN* *for* *causal relationship consideration *

**Section 5 Comparison **

To compare the improvement priority order by conventional RPN method;

hierarchy relation consideration with UPN and causal relationship consideration with UPN, the result showed as Table 9:

Table 9

*Priority comparison between methods *

42

The finding is there are large difference of priority, such as S2 of the 7^{th} priority
in hierarchy relation consideration with UPN but ranks 2^{nd} by causal relationship
consideration with UPN; S5 of the 2^{nd} in hierarchy relation consideration with UPN
but ranks 5^{th} by causal relationship consideration with UPN; and S7 of the 5^{th} in
hierarchy relation consideration with UPN but ranks 1^{st} by causal relationship
consideration with UPN. And there are smaller difference of priority, such as S1 of
the 1^{st} priority in hierarchy relation consideration with UPN but ranks 3^{rd} by causal
relationship consideration with UPN; S4 of the 4^{th} priority in hierarchy relation
consideration with UPN but ranks 6^{th} by causal relationship consideration with UPN.

In other items, there is no significant difference.

To compare with conventional RPN method, still also found large difference of
priority, such as S1 of the 8^{th} priority in conventional RPN but ranks 3^{rd} by causal
relationship consideration with UPN; S4 of the 3^{rd} priority in conventional RPN but
ranks 6^{th} by causal relationship consideration with UPN; S5 of the 1^{st} priority in
conventional RPN but ranks 5^{th} by causal relationship consideration with UPN; S7 of
the 4^{th} priority in conventional RPN but ranks 1^{st} by causal relationship consideration
with UPN.

43

**CHAPTER 6 CONCLUSIONS **

In this chapter, the results and advantages of research will be integrated results and into conclusions. First, the results of research will be summarized. Then advantages of research will be drawn. Finally, some suggestions and research limitations for future studies will be offered.

**Section 1** ** Results of research summarized **

In this study, we firstly evaluate the structure of hierarchy and interdependence structure by Interpretive Structural Model (ISM), and then calculate the weight of corrective action through Analytic Network Process (ANP). Furthermore, combine the utility of corrective actions, and to determine the improvement priority order of FMEA by Utility Priority Number (UPN). Via an empire case study, the finding of comparison is indeed obtained a different priority order of corrective actions. Follow the new priority order, it is not only involving the utility of corrective actions, but also may maximizes the improvement effect, even may brings the favorable result in the shortest time at lowest cost.

Secondly, we implemented a further exploration. In general, researchers evaluate with crisp values to present their feelings and subjective perceptions. In fact, due to intangible and subjective information often appearing in the evaluation process, crisp values are inadequate to present the evaluation ratings of customers, so people would have difficulties in understanding the difference and uncertainties in human’s semantic expression. Therefore, we proposed a Fuzzy ISM method to not only taking account of the relationship of hierarchy and even interdependence of the

44

corrective action to fit in with real complicated situation, and then to compare the hierarchical structure between hierarchical relation consideration and Fuzzy linguistic consideration via the same illustration case. The result of hierarchy structure is same as the hierarchy relation consideration, that is ISM none Fuzzy linguistic consideration. Hence, it indicates the Fuzzy ISM indeed can be used, and involve Fuzzy linguistic consideration might obtain more accurate priority of corrective action, then to maximizes the improvement effect.

Finally, we implement another further exploration. From deeper perspective, though ISM can clarify the relationship between different hierarchies and the corresponding effects of corrective action, it cannot explain their causal relationships and influence strength. We used DEMATEL method to analyze the causal relationship and influence strength of corrective actions, then to effectively integrate ANP to get a more complete weight of corrective action considering influence strength. At last, the finding of comparison is indeed obtained a different priority order of corrective actions.

**Section 2** ** Research Limitations **

Although this study provides theoretical substantive explanations, it still suffers from several limitations, as following:

1. We use ISM to evaluate the structure of hierarchy and interdependence structure, and then calculate the weight of corrective action through ANP. Furthermore, combine the utility of corrective actions, and to determine the improvement priority order of FMEA by UPN instead of RPN. This method can maximizes the

45

improvement effect, and brings the favorable result in the shortest time should be implicit, but whether it can at lowest cost need to be further explored from cost study.

2. We continually to further explore the Fuzzy linguistic consideration to compare the hierarchy structure of corrective actions and causal relationship consideration, to obtain and to compare the different priority order of corrective action. However, we’ve not further combine these two considerations to be a completed concept, that is, to develop Fuzzy DEMATEL method is possible future research direction.

46

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