A fuzzy logic based approach to reliability improvement estimation during product development

A fuzzy logic based approach to reliability improvement estimation

during product development

Om Prakash Yadava, Nanua Singha,*, Ratna Babu Chinnama, Parveen S. Goelb

aDepartment of Industrial and Manufacturing Engineering, Wayne State University, Detroit, MI 48202, USAbTRW Automotive, Chassis System, EAS, Sterling Heights, MI, USA

Received 17 August 2002; accepted 15 November 2002

Abstract

During early stages of product development process, a vast amount of knowledge and information is generated. However, most of it is

subjective (imprecise) in nature and remains unutilized. This paper presents a formal structure for capturing this information and knowledge

and utilizing it in reliability improvement estimation. The information is extracted as improvement indices from various design tools,

experiments, and design review records and treated as fuzzy numbers or linguistic variables. Fuzzy reasoning method is used to combine and

quantify the subjective information to map their impact on product reliability. The crisp output of the fuzzy reasoning process is treated as

new evidence and incorporated into a Bayesian framework to update the reliability estimates. A case example is presented to demonstrate the

proposed approach.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Reliability estimation; Reliability improvement; Product development; Fuzzy logic; Bayesian estimation

1. Introduction

Mastery of the product development (PD) process has

become an increasingly important prerequisite competency

to ensure corporate success in today’s global market

environment. Companies are under severe pressure to

deliver high quality and reliable products under shorter

development lead-times and at lower costs. Over the years,

product durability has become one of the primary

differentiators that dictate the degree of product success in

the market place. While there exists a lot of good literature

that outlines key engineering principles involved in

designing good product [1,2], there is not any literature

that addresses reliability estimation and improvement issues

during the PD process itself. To be able to build reliable

products in the current day competitive environment, it is

not only necessary that companies effectively track and

manage reliability improvement during PD, but also they

should do so beginning at the earliest stages of the PD. This

requires continuous reliability estimation and updating as

PD moves form one stage to another.

During early stages of PD, a vast amount of qualitative

information is available from different sources, such as

physics-of-failure models, results from robust design

experiments, computer aided engineering analysis, failure

modes and effect analysis, and pure expert opinions among

many others. Prior information from existing similar

products and reliability demonstration test results also

provide additional evidence about performance of a new

product under development. However, there is no well-

established methodology or systematic approach to gather

this information and use it for tracking reliability.

Furthermore, the existing reliability prediction methods

are not flexible enough to integrate all these different kinds

of the information, especially the subjective (imprecise)

information. What is really needed is a formal structure for

capturing this available information and knowledge for

timely and accurate assessment of reliability throughout the

PD process.

A lot of efforts have been made to incorporate qualitative

information in reliability estimation. Morris [3], Apostola-

kis and Mosleh [4] and Walls and Quigley [5] have

attempted to capture this information as experts’ opinion

and proposed to treat uncertainty in experts’ opinion as a

probabilistic process to be incorporated into Bayesian

0951-8320/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10 . 10 1 6 /S0 9 51 -8 32 0 (0 2) 00 2 68 -5

Reliability Engineering and System Safety 80 (2003) 63–74

www.elsevier.com/locate/ress

* Corresponding author. Tel.: þ1-313-577-7586; fax: þ1-313-577-8833.

E-mail address: [email protected] (N. Singh).

http://www.elsevier.com/locate/ress

analysis. Much of the work related to uncertainty modeling,

especially in risk analysis, is based on the theory of

subjective probability developed by Savage [6], de Finetti

[7], Lindley [8] and others. However, a more prevalent

belief in the scientific community is that uncertainty in

engineering judgment represents a lack of knowledge due to

imprecise (vague) information available at early stages of

the PD process and not due to randomness [9–14]. Several

suggested the application of fuzzy logic to reliability

engineering and risk analysis to deal with subjective

judgments and vague information. Elsayed [15] has also

emphasized the need for new and efficient methods for

reliability estimation and prediction to improve effective-

ness of the design process. It is, however, not obvious how

one should proceed to capture and quantify the subjective

information generated from various sources during

implementation of design changes and/or corrective actions.

To our knowledge, there exists no integrated framework or

methodology that helps reliability engineers capture and

track the subjective information from different sources at

different stages of the PD process in the form of recursive

reliability estimates.

This paper proposes a comprehensive framework, which

provides a structured way of gathering the qualitative

information from all available sources, processing it, and

integrating it with existing statistical methods to facilitate

continuous updating of reliability estimation in a multistage

PD process. The subjective information is captured as

engineering judgment or improvement indices, which

represent improvement in a particular failure mode or

mechanism. These improvement indices are treated as fuzzy

numbers or linguistic variables, combined and processed

through a fuzzy reasoning method to map their impact on

product reliability. The crisp output of the fuzzy reasoning

method is treated as new evidence. The Bayesian frame-

work is used to integrate the new evidence with other types

of evidence such as quantitative data and qualitative

information having probabilistic uncertainty. Probabilistic

uncertainty is captured in the form of a prior distribution.

The framework allows continuous updating of system

reliability as new evidence is available at any stage of the

PD process. The Weibull distribution is considered as

failure/survival time distribution with the assumption of a

known shape factor value. It is proposed that initial

parameters of the Weibull distribution be estimated from

warranty data of similar prior designs.

The paper is organized as follows. Section 2 introduces,

in brief, the basic components of fuzzy logic system. Section

3 describes a comprehensive methodology of capturing

subjective information, building a rule base, information

processing, quantification and reliability estimation. Section

4 presents a case example to demonstrate the proposed

approach. Section 5 presents some concluding remarks.

2. Fuzzy logic system

Fuzzy logic was originally introduced by Zadeh [16] as a

mathematical way to represent vagueness in everyday life.

Fuzzy logic systems approximate functions rather than

exact reasoning. They are shown to be universal approx-

imators in that fuzzy logic systems can model any

continuous function. The importance of fuzzy logic derives

from the fact that most modes of human reasoning,

especially common sense reasoning, are approximate in

nature. It is for all these reasons that we believe that fuzzy

logic inference systems are ideal for capturing subjective

information during the earliest stages of the PD. The

proposed overall procedure for approximating the impact of

corrective actions on product/system reliability is shown in

Fig. 1, which consists of four components, namely, fuzzy

rule base, fuzzy inference process, fuzzification process, and

defuzzification process.

The basic unit of any fuzzy system is the fuzzy rule base.

All other components of the fuzzy logic system are used to

implement these rules in a reasonable and efficient manner.

The fuzzy inference process combines the rules in the fuzzy

rule base and then carries out a mapping from fuzzy set A in

Fig. 1. An overview of fuzzy logic system for reliability modeling.

O.P. Yadav et al. / Reliability Engineering and System Safety 80 (2003) 63–7464

universe of discourse U to fuzzy set B in universe of

discourse V. Owing to the fact that in most applications the

input and output of the fuzzy system are real-valued

numbers, one must construct interfaces between the fuzzy

inference process and the environment. These interfaces are

the fuzzification and defuzzification processes shown in

Fig. 1.

The fuzzification process can be defined as a process of

mapping a real-valued point x [ U , Rn to a fuzzy set A in

U. The defuzzification process can be defined as a process of

mapping from fuzzy set B in V [ R (which is the output of

the fuzzy inference process) to a crisp value y [ V : For

further readings on fuzzy logic approach, readers are

advised to refer Refs. [17,18].

3. Methodology

The proposed reliability improvement estimation model

based on fuzzy logic includes: (1) the identification of

various input–output attributes, (2) the development of

fuzzy sets or linguistic values to model uncertainty in

improvement indices for different failure modes and

mechanisms, and (3) the development of fuzzy IF-THEN

rules to define the fuzzy relation between inputs and output

variables. The fuzzy inference model then uses these fuzzy

‘IF-THEN’ rules and linguistic input variables to generate a

crisp value representing improvement in product/system

reliability. The basic steps of the methodology can be

grouped into three phases as follows: (1) Knowledge

acquisition phase, (2) Information processing phase, and

(3) Reliability estimation phase.

3.1. Knowledge acquisition phase

In any fuzzy logic system, the quality of the fuzzy

approximation depends predominantly on the quality of

engineering judgment (subjective knowledge) and human

expertise used to build the fuzzy inference database (fuzzy

rules). It is for this reason that the knowledge acquisition

phase is often considered the most important step in

developing a fuzzy logic system. Therefore, the develop-

ment of fuzzy sets for identified input–output variables and

fuzzy IF-THEN rules are assumed as building blocks of the

fuzzy inference system. The knowledge acquisition phase

includes three steps as discussed below.

3.1.1. Identify the input and output variables

Engineering judgments are made throughout the PD

process, right from the concept selection to product launch

and even during field use, to evaluate the effect of design

changes and modifications, improvement in manufacturing

process and management, and environmental and operating

conditions. We consider these engineering judgments as

input variables in the proposed fuzzy model and derive them

by using analytical/computational models, physics-of-fail-

ure models, and/or with help from professional experts.

The output of various tools—such as design of

experiments (DOE), robust design methods, failure modes

and effect analysis (FMEA), statistical process control, and

computer aided engineering (CAE) analysis—can be used

to obtain engineering judgment representing improvements

in the system performance against different failure modes

and mechanisms. In our model, these engineering judg-

ments are defined as improvement indices ðIXXÞ :

IXX ¼ðgainÞ

ðpreviousÞ£ 100 ð1Þ

where the subscript XX represents the source of information

or particular failure mode or mechanism, (gain) represents

net improvement in failure mode or functional characteristic

and (previous) denotes the level of functional characteristic

before corrective action or design change. For example, IFF

represents improvement in fatigue failure (fatigue life)

derived from CAE analysis results and IDOE represents

improvement in a particular failure mode or functional

characteristic obtained from DOE results. These improve-

ment indices are considered as input attributes in the

proposed model. Yadav [19] gives detailed discussion on

various types of improvement indices and their equations to

derive crisp values from available experimental results and

design review records.

The change or improvement in system reliability

parameter due to corrective actions or design changes is

considered as an output variable and termed as reliability

improvement index (RII). In the present study, the RII is

defined as

RII ¼ðlb 2 laÞ

lb

£ 100 ð2Þ

where la is the system failure rate after considering the

impact of corrective actions and design changes and lb is

the system failure rate before any changes or actions were

taken.

3.1.2. Define fuzzy subsets or linguistic variables for each of

the input–output variables

To capture the uncertainty associated with both input and

output attributes and imprecise knowledge about the

relationship between input–output variables, fuzzy set

theory provides a fundamental basis to map the approximate

relationship between fuzzy variables. The input (improve-

ment indices) and output attributes are treated as fuzzy

numbers (sets) and uncertainty is characterized by member-

ship function. In this study, the membership function of

each fuzzy set is assumed to be triangular as shown in Fig. 2.

The triangular membership function has the advantage of

simplicity and is commonly used in reliability analysis [11,

20]. Fuzzy membership functions are generated utilizing the

linguistic categories identified by a human expert to express

his/her assessment. This is the first step in incorporating

O.P. Yadav et al. / Reliability Engineering and System Safety 80 (2003) 63–74 65

human knowledge into engineering systems in a systematic

and efficient manner.

Improvement Indices (IXX), representing the effect of

corrective actions or design changes on various failure

modes and mechanisms such as fatigue, overstress,

corrosion, wear, manufacturing related failure, and any

other failure modes, are used as input attributes or variables

in the current model. To estimate the improvement for any

failure category in linguistic terms, one may choose to use

linguistic variable levels such as negligible, very low, low,

moderate, high, and very high improvement. Fig. 2 shows

the fuzzy set definition for an input variable.

The output variable is defined as RII for improvement in

reliability parameter. To estimate the impact of corrective

actions on system reliability in linguistic terms, one may

choose to use linguistic variable levels such as negligible,

very low, low, moderate, high, and very high improvement

in system reliability. Fig. 3 shows the fuzzy set definition for

an output (RII) variable.

3.1.3. Develop fuzzy rule base to relate the input sets

to the output set

The knowledge required to develop fuzzy rules can be

extracted from several sources. The most widely known

sources are—human knowledge and expertise, historical

data analysis of similar system, and engineering knowledge

from existing literature. The human experts have good

intuitive knowledge of the system behavior or different

failure modes and mechanisms. Since the fuzzy rules are

linguistic rather than numerical, they provide a natural

framework for expressing human knowledge. Thus, experts

often find fuzzy rules to be a convenient way to express their

knowledge about the relationship between input and output

variables. The existing engineering knowledge in the

literature along with historical evidence and human

judgment and expertise could be of great help in developing

a reasonably good fuzzy rule base.

Expert’s knowledge and expertise about the impact of

corrective actions on product reliability can be represented

by fuzzy IF-THEN rules. Such rules are usually more

conveniently formulated in linguistic terms than in numeri-

cal terms and they are often expressed as IF-THEN rules,

which are easily implemented by fuzzy conditional state-

ments. A generic single fuzzy IF-THEN rule assumes the

form

If x is A then y is B ð3Þ

where A and B are linguistic values defined by fuzzy sets on

the universe of discourse U and V, respectively. The if-part

of the rule ‘x is A’ is called the antecedent or premise, while

the then-part of the rule ‘y is B’ is called the consequent or

conclusion.

All the rules that have any truth in their antecedent will

‘fire’ and contribute to the fuzzy conclusion set. If the

antecedent is true to some degree of membership, then the

consequent is also true to that same degree. This point leads

to a natural way to combine multiple qualitative assess-

ments. In practical applications, the fuzziness of the

antecedent eliminates the need for a precise match with

the inputs. With the help of fuzzy rules, an analyst tries to

geometrize knowledge in terms of fuzzy or gray patches in

the input–output space U £ V : These fuzzy patches are a

measure of uncertainty in the rule. The fuzzy system

approximates a function f : X ! Y by covering its graph

into these overlapping fuzzy patches. In essence, the

expression describes a relation between input and output

variables x and y; this suggests that a fuzzy IF-THEN rule

can be defined as a fuzzy relation R on the product space

U £ V

3.2. Information processing phase

In the information-processing phase, the fuzzy system

maps all inputs x ¼ {x1; x2;…; xn}T on to an output f ðxÞ:

This process of mapping inputs on to output is known as

fuzzy inference process or fuzzy reasoning [17,18]. With

crisp inputs and output, a fuzzy inference process

implements a nonlinear mapping from its input space to

output space. This mapping is accomplished by a number of

fuzzy IF-THEN rules, each of which describes the local

behavior of the mapping. The antecedent (premise part) of a

rule defines a fuzzy region in the input space, while the

consequent (conclusion part) specifies the output in the

fuzzy region. This whole process of mapping or function

approximation is carried out in three steps as shown in Fig. 4.

3.2.1. Combining inputs

The first step in fuzzy inference process is to match input

X ¼ {x1; x2;…; xn}T to all the IF-part fuzzy sets in parallel.

This step ‘fires’ or ‘activates’ the rule by how much the

input belongs to each IF-part set A. All the rules that haveFig. 3. Fuzzy set definition for an output variable.

Fig. 2. Fuzzy set definition for improvement index (input variables).


any truth in their antecedent will fire and contribute to the

fuzzy conclusion set. This leads to a natural way of

combining multiple qualitative inputs or engineering

assessments. The fuzzy inference system allows us to pick

arbitrary joint IF-part fuzzy sets. Suppose, if there are n

input variables, the most common way to combine IF-part of

each input variable is with a pair wise minimum or min T-

norm operator as:

mBl ¼ min{mAl1ðxÞ;mAl

2ðxÞ;…;mAl

nðxÞ}

¼ mAl1ðxÞ ^ mAl

2ðxÞ ^ · · · ^ mAl

nðxÞ ð4Þ

where n is number of inputs and l represents rule number.

The min combiner or T-norm operator is a conjunctive or

AND operator. Sometimes people prefer to use another T-

norm operator known as ‘product’ in place of min T-norm

operator to combine IF-part of fuzzy rules:

mBl ¼Yn

i¼1

mAiðxÞ ð5Þ

The only advantage of using the product T-norm operator is

that it does not ignore any information, as does the min

combiner or operator.

3.2.2. Aggregation of THEN-part fuzzy sets

The second step of the fuzzy inference process is to add

all THEN-part fuzzy sets into a final output set. The

‘extension principle’ converts or extends a fuzzy function f

(fuzzy IF-THEN rules) that maps points in X to point in Y to

a function that maps input fuzzy sets A , X to output fuzzy

set B , Y—from y ¼ f ðxÞ to B ¼ FðAÞ [18]. The extension

principle has the form of a supremum or maximum of pair

wise minima (or products). The extension principle asserts

that the fuzzy set B induced by the mapping f is defined as:

B ¼[mj¼1

Bj ð6Þ

The membership function of the output fuzzy set is given

by:

mBðyÞ ¼ max{mB1ðyÞ;mB2ðyÞ;…;mBm ðyÞ} ð7Þ

The system that invokes the max operator or extension

principle or compositional max–min rule is known as a

Mamdani fuzzy inference system [17,21] as shown in Fig. 5,

or fuzzy additive system [18] (Fig. 4). The Mamdani fuzzy

inference system with two inputs and single output in Fig. 5

shows the first two steps of the information processing

phase, i.e. matching inputs and aggregation of THEN part of

fuzzy sets.

3.2.3. Defuzzification

The fuzzy inference system often computes the crisp

value as the centroid or center of gravity of this final output

set. Defuzzification refers to the way a crisp value is derived

from a fuzzy set as a point in the output domain that best

represents fuzzy set B (Fig. 4). In general, five methods of

defuzzification have been discussed in the literature [17],

but centroid of area is the most frequently used method.

With the max–min compositional rule and centroid of area

defuzzification, the fuzzy inference system derives the crisp

Fig. 5. Mamdani fuzzy inference system for two inputs and single output.

Fig. 4. Fuzzy inference system architecture [18].


value of output, (ZCOA) when subjected to crisp inputs, such

as two inputs x and y. In this study we proposed to use the

centroid of area (ZCOA) as a defuzzifier, which is given as

ZCOA ¼

ðZmBðZÞZ dZÐmBðZÞdZ

ð8Þ

where mBðZÞ is the aggregated output membership function.

The calculations needed to carry out the centroid of area

defuzzification operation are slightly time consuming unless

special hardware support is available. However, the quality

of the defuzzification process makes this computational

complexity worthwhile.

3.3. Reliability assessment phase

The output of the fuzzy inference model, i.e. crisp

value of RII, is converted into transformed failure rate

ðlpt ¼ 1=TpÞ using Eq. (2) of RII. The quantified point

estimate failure rate ðlpt Þ or mean time to failure rate

(T p) is treated as new evidence and offered to the

Bayesian framework [22] to update reliability parameters

of the design after corrective actions or design changes

as follows:

f ðlt=TpÞ ¼

LðTp=ltÞf ðltÞðl

LðTp=ltÞf ðltÞdlt

ð9Þ

where f ðlt=TpÞ is the posterior probability density

function; LðTp=ltÞ; the likelihood function; f ðltÞ; the

prior probability density function; T p, the new evidence;

lt is the transformed failure rate of the product or

system.

The Weibull distribution is used to construct the

likelihood function, as this distribution has proven to be

particularly effective in characterizing the time-to-failure

tendencies of mechanical systems. In the proposed analysis,

we assume that the shape factor b of the Weibull

distribution is known. The Weibull distribution with an

assumed value of shape factor is usually a better

representation for the life of most of products than

exponential distribution [23]. With this assumption, the

transformed time t b follows an exponential distribution [22,

24] and allows us to select the gamma distribution as the

prior distribution for Bayesian analysis. The posterior

probability density function f ðlt=t; TpÞ is the analyst’s

modified density function after incorporating new evidence,

derived from the fuzzy inference model. The prior

distribution is selected from warranty database of a similar

product. During the later stages of PD process and

production stage, the posterior distribution (the Bayesian

output) from the previous stage can be considered as prior

distribution for successive stages.

4. The case example

A case of an automotive subsystem, i.e. hydraulic

power steering system, is used to demonstrate the

proposed fuzzy logic based reliability estimation method.

Since the new design is based on the existing hydraulic

power steering system, the major decisions regarding

design changes or modifications are made based on the

analysis of failure data. Failure analysis of warranty data,

or test results during later stages of PD process, provides

valuable information about different failure modes and

mechanisms. It also helps to decide the appropriate

design changes or corrective actions to be undertaken to

eliminate any existing problems and improve upon them.

For that purpose, different failure modes and mechanisms

are classified according to their position and frequency of

occurrence. Since the new steering system is an

evolutionary design, warranty data of a similar existing

system were analyzed and following major problems

identified for further actions: (1) noise, (2) fluid leakage,

and (3) fatigue failure. The appropriate design changes/-

modifications are implemented to mitigate these pro-

blems. The example of the PD process for a steering

system represents a real life scenario, but other details

and actual numbers have been somewhat modified due to

the proprietary nature of the information and to

demonstrate the applicability of the framework.

The hydraulic rack and pinion power steering system

is a high bandwidth servo with stringent performance

requirements on accuracy, reliability, and cost. Design of

such a system can best be achieved by continuous

validation and reliability estimation throughout the

development process.

Fig. 6 shows a schematic for a typical hydraulic rack and

pinion power steering system. The hydraulic rack and

pinion power steering system of an automobile vehicle

consists of the following two main subsystems:

1. Closed-loop hydraulic circuit; including vane pump, flow

control valve with pressure relief, tuned supply and

return lines (hoses), cooler, reservoir, and rotary spool

valve (RSV)

2. Power actuator; including the steering wheel, intermedi-

ate shaft, torsion bar, rack and pinion gear, power piston,

inner and outer tie rods, knuckle, tires and housing. Drive

belt and pulley on the engine are also included in the

same subsystem.

Sensors (torque and wheel speed) measure two primary

inputs—drive torque on the steering shaft and wheel speed.

The torsion bar is connected to the steering wheel

intermediate shaft and the pinion at its top and bottom

ends, respectively. Torsion bar stiffness yields the driver’s

steering efforts, and is designed to let the driver turn the

steering wheel with ease, and at the same time, to give a

memory function for the RSV in order to reduce the error


angle towards zero, after the end of the driver’s steering

demand.

The steering noise and vibration related problems occur

from dynamic interaction between the steering gear and

suspension systems subjected to disturbances, such as

engine torque pulsation, tire shocks, and vibration loads.

These noise and vibration problems include; steering

shudder, steering wheel nibble, mechanical front-end

noise, and steering wheel dither.

The dynamic interaction between the rack and steering

gear housing or excitation of the fundamental frequency

(or oscillatory motion) of the rack and pinion gear

mechanism by a periodic rack force induced by tire

shocks/rough road conditions and braking disk roughness,

during braking at highway speed can induce fatigue

problem and damage the gear teeth. Inner and outer tie

rods, torsion bar, and intermediate shaft are also

subjected to vibrations and shocks and are potential

candidates for fatigue problem.

The closed loop hydraulic circuit suffers from a fluid

leakage problem mainly due to corrosion, environmental

and operational damage of parts, improper fit, etc.

The analysis that follows closely mirrors the basic steps

as described in the methodology.

4.1. Knowledge acquisition

4.1.1. Identification of input–output variables

The input variables are improvement indices represent-

ing the impact of design changes and corrective actions on

identified failure modes and mechanisms. These variables

are derived as follows:

Noise. A measure of improvement in the noise problem

(INOISE) is obtained using signal-to-noise ratios (SN) from

robust design experiments

INOISE ¼ðSNa 2 SNbÞ

SNb

� �£ 100 ð10Þ

where SNa is SN ratio after corrective action and SNb is

signal-to-noise ratio before corrective action and ðSNa 2

SNbÞ represents gain in SN ratios after taking necessary

corrective actions based on DOE output.

Leakage. There is no experimental output for the

corrective actions taken to eliminate or mitigate the leakage

problem. The only source of appropriate information is the

FMEA document which gives revised RPN based on

expert’s opinion. The appropriate corrective action reduces

the probability of failure occurrences and hence RPN in the

revised FMEA document represents reduction in the failure

Fig. 6. Hydraulic rack and pinion power steering system [25].


occurrences or impact of changes. The improvement in the

oil leakage failure mode is derived from a revised system

level FMEA and is given as

ILEAK ¼ðRPNB 2 RPNAÞ

RPNB

£ 100 ð11Þ

where RPNa is re-evaluated RPN after corrective action

taken and RPNB is RPN before corrective actions are taken

and (RPNB 2 RPNA) represents impact of corrective

actions in terms of decrease in risk priority number or in

frequency of failure occurrence.

Fatigue failure. The output of finite element analysis is

used to derive an improvement index for fatigue failure

problem. The fatigue life of a product can be estimated

using a CAE analysis tool before and after corrective action.

There are various fatigue models available in the literature

for estimating the fatigue life of any product, such as, strain-

life relationship, stress-life relationship, cumulative damage

accumulation, and fracture mechanics [26]. The improve-

ment index for fatigue life can be given as

IFF ¼ðNa 2 NbÞ

Nb

£ 100 ð12Þ

where Na and Nb are mean fatigue lives after and before

corrective action, respectively.

The output variable is defined as RII, which represents

improvement in system failure rate as a result of improve-

ment in different failure modes and is expressed as

RII ¼ðlb 2 laÞ

lb

£ 100

where la is the system failure rate after considering the

impact of corrective actions and design changes and lb is

the system failure rate before any changes or actions were

taken.

4.1.2. Define fuzzy sets or linguistic variables for each

input–output variable

Only three input variables, i.e. improvement indices for

noise (INOISE), leakage (ILEAK), and fatigue failure (IFF), are

considered here. The fuzzy sets or the term sets TIXXfor each

input (improvement index IXX) is defined in Eq. (13) and is

graphically represented in Fig. 2

TIXX¼{negligible; very low; low; moderate; high; very high}

ð13Þ

The fuzzy sets or term sets TRII for output (RII) is defined in

Eq. (14) and is graphically represented in Fig. 3

TRII ¼ {negligible; very low; low; moderate; high; very high}

ð14Þ

4.1.3. Developing fuzzy rule base

The fuzzy IF-THEN rules are developed on the basis of

expert opinion and information available from failure

analysis. The warranty data analysis along with expert

opinion could be of great help in building fuzzy IF-THEN

rules because this analysis represents system behavior in

field use specifying the contribution of each failure mode or

mechanisms in system failure. This information, i.e.

proportion of each failure mode in overall system failure,

along with expert opinion is used for building fuzzy IF-

THEN rules.

For each combination of terms of linguistic variables, a

separate rule should be defined. For the present example,

216 (6 £ 6 £ 6) rules are required to consider each

combination of terms of linguistic variables. Some of the

rules are given below:

#1 IF INOISE is negligible and ILEAK is negligible and IFF

if negligible THEN RII is negligible

#10 IF INOISE is negligible and ILEAK is very low and IFF

if moderate THEN RII is low

#37 IF INOISE is very low and ILEAK is negligible and IFF

if negligible THEN RII is negligible

#80 IF INOISE is low and ILEAK is very low and IFF if very

low THEN RII is very low

#120 IF INOISE is moderate and ILEAK is very low and IFF

if very high THEN RII is moderate

#160 IF INOISE is high and ILEAK is low and IFF if

moderate THEN RII is moderate

These fuzzy IF-THEN rules build a fuzzy system that

converts fuzzy inputs into fuzzy output. Fig. 7 shows fuzzy

mapping or functions, defined by fuzzy rules, between two

inputs and output in a three-dimensional input–output

space.

4.2. Information processing

The following input data are considered representative of

improvement in different failure modes attributed to

corrective actions:

Fig. 7. Fuzzy function defined by IF-THEN rules between two inputs and

output.


Improvement in noise failure INOISE ¼ 0:65

Improvement in leakage problem ILEAK ¼ 0:50

Improvement in fatigue failure IFF ¼ 0:35

Since the input is generally a crisp value, fuzzification of

input is the first step in the fuzzy reasoning process. Once

the inputs have been fuzzified, the degree to which each part

of the premise has been satisfied for each rule is to be

determined. If there is more than one input, matching

(combination) of different values of linguistic variables in

the premise is achieved through fuzzy IF-THEN rules using

AND or min T-norm operator as given in Eq. (4). The given

values of input fire the IF-part of all rules in parallel and to

some degree. The fuzzy reasoning was performed using

MATLAB’s fuzzy logic module for all three inputs. Among

all the rules, the following eight rules are fired contributing

to estimation of the system reliability improvement.

#122 IF INOISE is moderate and ILEAK is low and IFF if

very low THEN RII is low

#123 IF INOISE is moderate and ILEAK is low and IFF if

low THEN RII is low

#128 IF INOISE is moderate and ILEAK is moderate and IFF

if very low THEN RII is low.

#129 IF INOISE is moderate and ILEAK is moderate and IFF

if low THEN RII is moderate

#158 IF INOISE is high and ILEAK is low and IFF if very

low THEN RII is low

#159 IF INOISE is high and ILEAK is low and IFF if low

THEN RII is moderate

#164 IF INOISE is high and ILEAK is moderate and IFF if

very low THEN RII is moderate

#165 IF INOISE is high and ILEAK is moderate and IFF if

low THEN RII is moderate

The output of each fired rule is a fuzzy set representing

the RII with certain membership function given by min T-

norm operator. These fuzzy sets are then aggregated into a

single fuzzy set with the help of max S-norm operator as

given in Eq. (7). The single aggregated fuzzy output of all

eight fuzzy sets is shown in Fig. 8.

4.2.1. Defuzzification

The output of the fuzzy reasoning process is also a fuzzy

set and it needs to be converted into a crisp value to

incorporate it into reliability analysis. The centroid of area

(ZCOA) method is used here for the defuzzification process

using Eq. (8) as

ZCOA ¼

ðZmBðZÞZ dZÐmBðZÞdZ

where mBðZÞ is the aggregated output membership function.

The crisp value of RII is 0.501 for the given input values, as

shown in Fig. 8 by a vertical line.

4.3. Reliability analysis

Warranty data of the existing design were used to

estimate the parameters of Weibull distribution and prior

distribution. In the automobile industry, warranty data are

commonly measured in incidents or repairs per a set number

of vehicles. Using a linear regression approach [27], the

Weibull parameters, i.e. shape factor (b) and characteristic

life (u), were estimated as follows:

shape factor ¼ 1:399 ø 1:4

characteristic life ¼ 622 525 miles:

Table 1

Results of defuzzification process and Bayesian analysis

Defuzzification results Posterior gamma distribution

RII Transformed failure rate (lt) Transformed mean time to failure (MTTF) a b

0.501 3:92 £ 1029 2:55 £ 108 4.93 7:54 £ 108

Table 2

Failure rate analysis over product life cycle

Miles 10 000 30 000 50 000 70 000 100 000 120 000 150 000

Prior 4.31 £ 1027 6.69 £ 1027 8.20 £ 1027 9.38 £ 1027 1.08 £ 1026 1.16 £ 1026 1.27 £ 1026

New design 3.70 £ 1027 5.74 £ 1027 7.04 £ 1027 8.05 £ 1027 9.28 £ 1027 9.99 £ 1027 1.09 £ 1026

Fig. 8. Aggregated output fuzzy set.


From these two parameters, we derived the transformed

failure rate lt as:

lt ¼ 1=ub ¼ ð1=622 5251:4Þ ¼ 7:86 £ 1029

Hereafter, we assume that the shape factor is known and

equal to 1.4. This transformed failure rate from warranty

data is considered as initial failure rate (lt0) and is given by

the following equation:

lt0 ¼ a0=b0

According to our assumption lt0 ¼ lt; implies:

lt0 ¼ a0=b0 ¼ 7:86 £ 1029

or

a0 ¼ b0lt ¼ b0 ð7:86 £ 1029Þ

Considering the following relation:

Pð:20lt0 # lt0 # 8lt0Þ ¼ :99

or

ð8l0

:20l0

ba0

0 la021 e2b0l dl ¼ :99

One can easily obtain a0 ¼ 3:93 ø 4 and b0 ¼ 4:99 £ 108:

The prior distribution has now been completely defined and

can be used for the Bayesian analysis.

The Weibull distribution is used to construct the

likelihood function, as this distribution has proven to be

particularly effective in characterizing the time-to-failure

tendencies of mechanical systems. The transformed failure

rate (or MTTF) was calculated from the crisp value of RII

by using Eq. (2). Posterior gamma distribution parameters

are updated by incorporating fuzzy model output, i.e.

transformed MTTF, into Bayesian analysis. The results of

the defuzzification process and updated parameters of

gamma distribution (posterior) are given in Table 1.

The characteristic life (u) for Weibull distribution is

calculated from the Bayesian analysis result. The changes in

failure rate and reliability for both prior and after design

changes/modifications (new design) over the life cycle of

150 000 miles are given in Tables 2 and 3 and shown in

Figs. 9 and 10.

Fig. 11 shows the reliability growth at 50 000 miles at

95% confidence coefficient and shows reduction in

confidence interval after incorporating engineering judg-

ments, which were derived utilizing the outputs of design

tools, experimental results, and design review records.

4.3.1. Sensitivity analysis of the effect of b

on product reliability

In practice, one does not know the actual value of the

shape factor (b). It is always assumed either from historical

data or based on experience. This assumed b value always

differs from the true population value and is not accurate. To

see the effect of this uncertainty, we repeat the calculations

to find out how the estimates and confidence limits change

with different values of shape factor.

For the steering system data, the estimates of character-

istic life (u), Pth percentile tp, and system reliability (R) at

50 000 miles and 95% confidence limits were calculated

using b ¼ 1:0; 1.2, 1.4, and 1.6 as given in Table 4. The

system reliability over the life cycle of 150 000 miles is

calculated using different values of shape factor as shown in

Fig. 12. The figure shows that the shape parameter has great

influence on product reliability and the reliability curve

declines more sharply for b ¼ 1:6 than other values of it.

Fig. 13 shows that shape factor has significant impact on

confidence limits also. The confidence limits become wider

for higher values of shape factor. The results indicate that

the shape factor has significant impact on the system

reliability and confidence interval. It shows that results do

depend on the value of shape factor and wrong value of

shape factor might lead to erroneous conclusions.

4.3.2. Sample size requirement to demonstrate

reliability growth

Here we try to estimate how many more units we need to

test further to demonstrate the reliability growth for the

required reliability target, had we not incorporated sub-

jective information in reliability estimation. The following

formula is used for calculating the required number of tests

Table 3

Reliability analysis over product life cycle

Miles 10 000 30 000 50 000 70 000 100 000 120 000 150 000

Prior 0.996875768 0.985537852 0.970655 0.9534154 0.92441 0.903521 0.870524

New design 0.997363384 0.987784153 0.975184 0.96055 0.935835 0.917961 0.889594

Fig. 9. Instantaneous failure rate at two stages of development process.


to achieve the reliability improvement target

DN ¼logð1 2 CÞ

logðRpÞ

1

L

� �b2

logð1 2 CÞ

logðRÞ

1

L

� �b

¼ logð1 2 CÞ1

logðRpÞ2

1

logðRÞ

� �1

L

� �bð15Þ

where DN is the additional sample size; b, the Weibull

shape factor; C, the confidence level; R, the current

reliability level; R p, the reliability improvement target; L

is the number of lives to be tested.

As shown in Fig. 11, the reliability growth at 50 000

miles with 95% confidence coefficient, after incorporating

subjective (fuzzy) information in reliability estimation

method, is very small, i.e. from only.970 to.975. In

general, to demonstrate the required growth in reliability

one has to do additional testing. The required sample

size for further testing is calculated by Eq. (15). In our

example we have estimated the reliability improvement

attributed to design changes by utilizing the available

subjective evidence without resorting to further testing.

Had we not included the subjective information in our

estimate, we would have required at least eight or more

steering systems to test further for achieving the same

reliability target. It demonstrates the amount of time and

money one can save by using the proposed methodology

of reliability estimation.

5. Conclusions

The paper presents a structured framework for capturing

subjective information as well as quantitative data available

during the PD process, beginning with the earliest stages,

and estimating reliability improvement by employing fuzzy

logic and Bayesian methods. Prior information such as

warranty data of similar product, failure physics knowledge,

and expert opinions are employed in developing a fuzzy rule

base. The framework also makes provision for incorporation

of subjective information derived from employing design

Fig. 10. System reliability behavior after design modification.

Fig. 11. Reliability growth at 50 000 miles.

Table 4

Effect of different b values on estimates and confidence limits

Parameters Beta values

1 1.2 1.4 1.6

uU 4:64 £ 108 1:67 £ 107 1:55 £ 106 2:61 £ 105

u 1:53 £ 108 6:61 £ 106 7:02 £ 105 1:30 £ 105

uL 7:36 £ 107 3:60 £ 106 4:16 £ 105 8:26 £ 104

tp U 4:89 £ 107 1:12 £ 106 6:64 £ 104 7:13 £ 103

tp 1:61 £ 107 4:44 £ 105 3:00 £ 104 3:56 £ 103

tp L 7:76 £ 106 2:42 £ 105 1:78 £ 104 2:26 £ 103

RUðtÞ 99.989% 99.906% 99.187% 93.137%

RðtÞ 99.967% 99.716% 97.553% 80.598%

RLðtÞ 99.932% 99.410% 94.983% 63.885%

Fig. 12. Sensitivity analysis of the effect of b values on system reliability.

Fig. 13. Effect of b values on system reliability confidence limits.


tools and experiments. The Bayesian methods facilitate

recursive estimation of reliability improvements throughout

PD process.

The proposed methodology is a step towards application

of a systems approach to reliability estimation by integrat-

ing physics-of-failure models with classical statistical

methods. The integration of imprecise subjective infor-

mation into the reliability estimation process generally

reduces the requirements of extensive testing. The results

still depend on the quality of information derived from

different sources and expert’s judgments in building the rule

base. As with any modeling framework, one has to exercise

great care to ensure that the data and inputs presented to the

method are of good quality without which the results could

be biased. The proposed method is particularly sensitive to

the fuzzy rule base that maps the impact of corrective

actions on reliability improvement.

While one could perform validation tests to confirm the

model reliability predictions, in general, experience in using

the proposed methods over time tends to address issues of

bias and variance in the model’s predictions. Our experience

has shown that the proposed methods are particularly

appropriate for tracking reliability improvements in the

design and development of electromechanical systems. This

method is most beneficial during the earliest stages of the

PD, where one is generally constrained from collecting

adequate quantitative data to accurately estimate product

reliability parameters.

Acknowledgements

The authors would like to express their sincere

appreciation for the many constructive and helpful com-

ments and suggestions offered by the reviewers.

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A fuzzy logic based approach to reliability improvement estimation during product development