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Chapter 39
Research on Hardware Product’s
Reliability Step Change During the Phase
Switch Process Based on Catastrophe
Theory
Weidong Liu, Lilin Jie, Zheng Sun, Shasha Teng and Gen Wen
39.1 Introduction
The manufacturing process of the mass-produced hardware products usually includes
three parts, namely development, trial production and total production phase. The
enterprise will eliminate the weak link of hardware products through continuous
quality improvement activities to promote the product reliability in the three phases.
As a result, the product reliability is increased with time [9,11]. The reliability growth
process of product life cycle is shown in Fig.39.1. When one phase transitions to
another phase, the step change of hardware product’s failure rate will occur during the
reliability growth process of product life cycle. The failure rate λof hardware products
at the development phase gradually decreases with the continuous improvement of
quality and gets the minimum at the point A. When the development phase transitions
to trial production phase, a step change of failure rate λwill occur, and the failure rate
will jump from A to B. Similarly, the failure rate λof hardware products at the trial
production phase gradually decreases with the continuous improvement of quality,
and it obtains the minimum at point C. Moreover, when the trial production phase is
converted to total production phase, the step change of failure rate will occur again,
and the failure rate will jump from C to D.
W. Liu (B)·L. Jie ·S. Teng ·G. Wen
College of Mechatronic Engineering, Nanchang University, Nanchang 330031,
People’s Republic of China
e-mail: liuwd@ncu.edu.cn
W. Liu
College of Economics and Management, Nanchang Hangkong University,
Nanchang 330063, People’s Republic of China
Z. Sun
Chassis Department, BYD Automobile Industry Company, Shenzhen 518118,
People’s Republic of China
© Springer International Publishing AG, part of Springer Nature 2019
J. Xu et al. (eds.), Proceedings of the Twelfth International Conference on Management
Science and Engineering Management, Lecture Notes on Multidisciplinary
Industrial Engineering, https://doi.org/10.1007/978-3-319-93351-1_39
491
492 W. Liu e t a l .
Fig. 39.1 The reliability growth process of hardware product life cycle
In recent years, the study of reliability assessment and the mechanism of hardware
products reliability change is mainly concentrated in each phase of the manufactur-
ing process. In terms of reliability assessment: Robinson and Dietrich [13] proposed
a new nonparametric reliability growth model for the analysis of the failure rate of
a system that is undergoing development test. The study indicated that the model is
generally superior to the popular AMSAA model, regardless of the actual underlying
failure process. The Weibull distribution and the Crow-AMSAA model were applied
for analyzing the cable joint failures by Tang et al. [15]. It was found that the Weibull
distribution provides more reliable results in the analysis of early-failure data. Frenkel
and Khvatskin [5] presented a study on reliability measures assessment for the air
conditioning system with rental equipment working in heavy weather conditions by
using Lz-transform and stochastic processes method. As for the operational relia-
bility, both Algarni et al. [1] and El-Berry and Al-Bossly [3] utilized Weibull and
Gamma distribution to estimate the reliability of the air conditioner system based
on its field data during use. The study revealed that Weibull method has performed
well for decision making. In terms of the mechanism of hardware products reliabil-
ity change: Wu et al. [18] analyzed the characteristics of manufacturing system and
pointed out that many factors can affect the reliability of systems and products, such
as staff, machine equipment, materials, process methods, software and system inte-
gration technologies. Wang et al. [12] put forward the specific measure and scheme
to solve the limitations of the reliability research of military products at the phase of
development and mass production. Jie et al. [7] investigated natural environmental
factors and user habit factors that influence the reliability of air-conditioning systems
during use. Li [8] reported the reliability growth method of product at the use stage
according to the technology principle of the reliability growth. Yang and Su [19] ana-
lyzed the reliability growth process of electronic system equipment according to the
actual project experience. That study briefly described the reliability growth mecha-
nism of hardware product at the development phase, trial production phase and total
production phase. In spite of many obvious advantages of all the studies discussed
above, they still have limitations, such as the little attention paid to the systematic
analysis of influence factors and their influence degree. Furthermore, the study about
hardware products reliability step change during the phase switch process is still
absent.
39 Research on Hardware Product’s Reliability Step Change … 493
In an attempt to reveal the mechanism of hardware product’s reliability step change
during the phase switch process, and find out the significant influence factors and their
influence degree that could not be settled by the above-mentioned methods, we focus
on catastrophe theory in this study. The key reason for choosing catastrophe theory
is that, it can comprehensively consider the relative importance of each evaluation
index and effectively reduce the interference of human factors, which is conducive
to more objective results [2,6,16]. Based on the analysis of influence factors of
reliability step change from the perspective of manufacturing system, an evaluation
model of reliability step change will be established by catastrophe theory. The model
might reveal the rule of product reliability step change and provide the theoretical
basis for taking some relevant measures to control the negative effect of the phase
switch process on product reliability. It can also be used to promote or accelerate the
reliability growth of the hardware product life cycle.
39.2 Analysis and Evaluation of the Phase Transformation
Process
The reliability of hardware product is formed in the manufacturing process, so the
factors that affect the step change of reliability are also derived from its manufacturing
process, which is determined by the manufacturing system and the management
system. Therefore, the factors that affect product reliability can be determined from
the perspective of the manufacturing system and the management system, thereby
analyzing the phase transformation process.
39.2.1 Analysis of the Factors of Reliability Step Change
According to the system engineering theory, the factors that affect the reliability step
change can be divided into two categories, namely structural factors and nonstructural
factors, as shown in Fig. 39.2. The two cooperate with each other and interact with
each other, so that the system can operate normally and utilize fully its function [14].
The structural factors of manufacturing system mainly consist of all the hardware
and the relationship between them, including manufacturing equipment, manufac-
turing process and manufacturing materials. The nonstructural factors mainly refer
to some elements that support and control the operation of the manufacturing sys-
tem, including the manufacturing environment, manufacturing staff, manufacturing
management, quality inspection standards, manufacturing targets and manufacturing
rhythm.
The main factors that affect the reliability step change of hardware product may be
different at different phases. Through enterprise research and analysis of production
systems, we know that when the development phase is converted to trial production
494 W. Liu e t a l .
Manufacturing equipment
Manufact uring process
Manufacturing materials
Nonstructural factors
Structural factors
Manufacturing environme nt
Manufact uring staffs
Manufact uring mana gement
Quality inspection standards
Manufacturing target
Manufact uring rhythm
Reliability step change
Fig. 39.2 The influence factors of reliability step change
phase, the key factors that contribute to the reliability step change are manufacturing
equipment, manufacturing materials, manufacturing staffs, manufacturing manage-
ment level, the manufacturing target and manufacturing rhythm.
(1) Manufacturing equipment factors. For the development phase, the production
line is shorter and can adapt to many types of products, as well as quickly match
the new product. But due to the increase in the number of production at the trial
production phase, the requirement of equipment and technology level become
higher, which leading to a decline in the reliability.
(2) Manufacturing material factors. At the development phase, the R & D department
will provide the bill of materials and trial factory prepare its own material. At
this phase, whether the mechanical structure design of the product is reasonable
or the ability of related functions meets the requirements will be verified. At the
trial production phase, manufacturing materials is to be re-matched. The more
stringent production cost requirements for raw material will lead to a lower
reliability.
(3) Manufacturing staff factors. The manufacturing staffs at the end of the develop-
ment phase are experienced professional trial engineers, while the manufacturing
39 Research on Hardware Product’s Reliability Step Change … 495
staffs at the early trial production phase are ordinary workers in the small-batch
production line, and the related experience is scarce.
(4) Manufacturing management factors. At the development phase, the number of
hardware products and the number of employees is less, which conducive to
quality management work.
(5) Manufacturing target factors. At the development phase, the main purpose is to
verify whether the design method is correct, mechanical institutions and elec-
trical systems are reliable, and the key ability meets the requirements. At the
trial production phase, the main purpose is to verify the feasibility of product
technology. At this phase, the hardware products with complete structure and
function need to install more accessories, and the process becomes longer and
more complex. So the reliability will decline slightly.
(6) Manufacturing rhythm factors. The manufacturing time at the development phase
is relatively low and the trial engineers have a long time to process, install and
debug. During the trial production phase, the hardware products have certain
requirement for the residence time of each process. The more tense rhythm of
production leads to a relative increase in the probability of production errors.
However, the trial production phase transitions to the total production phase, the
factors causing the reliability step change are mainly the manufacturing processes and
manufacturing materials in the structural factors, and the manufacturing staff, quality
inspection standards and manufacturing rhythm and other factors in the nonstructural
factors.
(1) Manufacturing process factors. At the end of the trial production phase, the tech-
nology and process for small-batch production are gradually maturing. However,
in the early phase of total production, the technology level is relatively weak in
the face of the more stringent quality stability requirements of mass production.
(2) Manufacturing material factors. The similar outsourced parts of enterprises gen-
erally have multiple suppliers. The trial production phase uses only purchased
parts supplied by 1–2 similar suppliers due to less production. As the demand
for purchased parts at the total production phase is much higher than that at the
trial production phase, and the supply capacity of different suppliers is differ-
ent, there will be some differences in manufacturing materials between different
phases.
(3) Manufacturing staff factors. At the end of the trial production phase, the pro-
duction staffs are trial factory employees who already have certain experience
in the production. However, the production staffs at the beginning of the total
production phase are the workers just entering the production line. The lack of
production experience leads to the decrease of human reliability and the decrease
of hardware products reliability.
(4) Quality inspection standards. Since the number of production at the trial pro-
duction phase is small, a wide range of quality testing and even 100% inspection
can be carried out at the total production phase, due to a substantial increase in
production, a wide range of quality testing becomes impractical, which makes
it difficult to find the quality problem.
496 W. Liu e t a l .
(5) Manufacturing rhythm factors. At the trial production phase, the product starts
to be produced in small batch and has certain requirements on the manufacturing
time, but mainly tests whether the products have the conditions to enter the mass
production line. At the total production phase, the hardware products are put
into mass production line. Thus, the residence time of each process is very short
and the requirements for manufacturing time are more stringent. As a result, the
probability of operation error in hardware process is relatively large (Table 39.1).
39.2.2 The Evaluation Index System of Phase Transition
State
The system, objective and quantifiable evaluation indexes are the basis of the sci-
entific evaluation for the phase transition state of manufacturing process. According
to the analysis of the influence factors of reliability step change and the general
performance evaluation index of the manufacturing enterprises, the evaluation index
system for the phase transition state of manufacturing process can be established, as
shown in Table 39.2.
The evaluation indexes are quantitatively calculated according to the statistics,
actual measurement or prediction data, so the measured value of evaluation index
Wiis given. By using a standard transformation method, the evaluation indexes are
dimensionless. So that the initial control variable Wiwill be obtained. For the-larger-
the-better indexes such as equipment availability, let
wi=Wi−Wmin
Wmax −Wmin
.(39.1)
For the-smaller-the-better indexes such as equipment failure rate, let
wi=Wmax −Wi
Wmax −Wmin
,(39.2)
where Wmax and Wmin are the maximum and minimum values of each evaluation
index, respectively.
39.3 Evaluation Model of Phase Transition State Based on
Catastrophe Theory
As mentioned above, due to the change of the production system, product’s reliabil-
ity step change occurs when the development phase transitions to trial production
phase, as well as the trial production phase transitions to mass production phase.
The catastrophe theory is studied in a dynamical system. Due to the change in the
39 Research on Hardware Product’s Reliability Step Change … 497
Table 39.1 The evaluation index system for the phase transition state of manufacturing process (1)
First level index Second level index Third level index
Third level index Index measure
Structural factors A Manufacturing equipment
A1
Equipment availability A11 The number of intact equipments
The total number of authorize equipments ×100%
Equipment failure rate A12 The total number of downtime for each device
The total operating time of all equipment ×100%
Equipment update rate A13 Annual equipment update number
Total number of device update plans ×100%
Manufacturing process A2 Process test completion rate
on time A21
Complete process test times on time
Total number of process trials ×100%
Error rate of process tooling
A22
The number of the faulty process documents
Total number of process documents ×100%
Process design completion
rate on time A23
Actual design cycle
Planned design cycle ×100%
Process stability A24 According to 1–5 standard, the greater the number the better
Process self-correction A25 According to 1–5 standard, the greater the number the better
Manufacturing materials A3 Quality passing rate A31 The number of sampling qualified products
The number of sampling product ×100%
Average pass rate A32 The sum of each delivery qualification rate
Delivery times ×100%
Reject rate A33 Retreat batch
Delivery batch ×100%
Exemption rate of materials
A34
The number of products exempted from inspection
Number of products supplied ×100%
498 W. Liu e t a l .
Table 39.2 The evaluation index system for the phase transition state of manufacturing process (2)
First level index Second level index Third level index
Third level index Index measure
Nonstructural factors B Manufacturing environment
B1
Storage environment B11 According to 1–5 standard, the greater the number the better
Internal transport
environment B12
According to 1–5 standard, the greater the number the better
Manufacturing environment
B13
According to 1–5 standard, the greater the number the better
Manufacturing environment
B2
Staff violations B21 Employee’s monthly average violation of command, violation
operation and the number of violation of labor discipline.
Staff training B22 Average monthly training time
Average age of staffs B23 Average working years in the field of employees
Manufacturing management
B3
Related system
establishment and
implementation efficiency
B31
The number of published regimes
The total number of planned releases ×100%
Manufacturing equipment
routine maintenance and
technical support B32
The number of problems solved
The total number of problems ×100%
Production site 5 S
management B33
The number of 5 S management requirements unqualified items
of during operation period
Quality inspection standards
B4
Product random inspection
rate B41
The number of sampling products
Total production ×100%
Standard implementation
B42
According to 1–5 standarde, the greater the number the better
(continued)
39 Research on Hardware Product’s Reliability Step Change … 499
Table 39.2 (continued)
First level index Second level index Third level index
Third level index Index measure
Manufacturing target B5 Difficulty degree of reach
B51
According to 1–5 standard, the greater the number the better
Manufacturing rhythm B6 Monthly output B61 The number of products produced in the current month
Labor productivity B62 output ×Standard working hours
Daily working hours ×the number of labor −Lost workday ×
100%
500 W. Liu e t a l .
control variables, a sudden jump in the system state variables will occur. Therefore,
the catastrophe theory can be used to analyze product’s reliability step change caused
by the phase transition.
Catastrophe theory [4,10] is an important modern nonlinear theory based on
singularity theory and bifurcation theory. Catastrophe means that the system which
is stable or gradual change process suddenly interrupted or changed. As an effective
modeling method, catastrophe theory is often used to analyze complex nonlinear
systems. The change of the control variables in a dynamical system results in a
sudden jump in the state variables. In catastrophe theory, the potential function is
commonly used to describe the state characteristics of a system. It is determined
by the interrelationship and the interaction between each component of the system,
and the relative relationship between the system and the external environment. The
variables of the potential functions are divided into two types, namely state variables
and control variables. The state of the system is the unity of the state variables and
the control variables.
V=V(X,U),(39.3)
where Xis the collection of the state variables, and X={x1,x2,...,xn}.Uis the
collection of the control variables U={u1,u2,...,un}.
At present, there are four common elementary catastrophe systems which are
usually used [20]. They are fold catastrophe, cusp catastrophe, swallowtail catastro-
phe and butterfly catastrophe, respectively, whose potential functions are shown in
Table 39.3.
Through the analysis of four catastrophe models, we can get the general expres-
sion of the potential function and the corresponding normalization formulas of the
catastrophe models (n≥5) [17].
Table 39.3 The elementary catastrophe models
Types Dimensions of control
variables
Potential function Normalization formula
Fold 1V(x)=x3+ux Xu=√u
Cusp 2 V(x)=
x4+ux2+vx
Xu=√u,Xv=3
√v
Swallowtail 3V(x)=
x5+ux3+vx2+ωx
Xu=√u,Xv=
3
√v, Xω=4
√ω
Butterfly 4 V(x)=x6+ux4+
vx3+ωx2+tx
Xu=√u,Xv=3
√v,
Xω=4
√ω, Xt=5
√t
39 Research on Hardware Product’s Reliability Step Change … 501
Vn(x)=1
n+2xn+2+1
na1xn+1
n−1a2xn−1+1
n−2a3xn−2+···+anx,
(39.4)
Xu=√u,Xv=3
√v,..., Xn=n+1
√n,(39.5)
where nis a positive integer, xi(i=1,2,...,n)is the state variable, and ai
(i=1,2,...,n)is the control variable.
According to the characteristics of the evaluation index system, the catastrophe
type is confirmed in every evaluation index layer (see Table 39.2). And the catastrophe
membership function values are calculated recursively according to the following two
criteria.
(1) Non-complementary criteria. Such as the control variables of a system (e.g.,
a,b,c,d), they cannot be replaced by each other and make up for their defi-
ciencies. It can be calculated as:
x=min [xa,xb,xc,xd].(39.6)
(2) Complementary criteria. Such as the control variables of a system (e.g., such as
a,b,c,d), their role can be replaced by each other. It can be calculated as:
x=(xa+xb+xc+xd)/4.(39.7)
According to the normalization formula of catastrophe theory, the total catastrophe
membership function value of the operation state of manufacturing system at each
phase can be calculated and the operation state of the system can be judged. The
multi-objective fuzzy decision theory shows that, for the same multi-layer system,
the greater the total catastrophe membership function value of the different system
state, the more stable the operation state of the system.
39.4 Application Research
In order to verify the validity and feasibility of the evaluation index system and
evaluation model of reliability step change based on catastrophe theory, the four
production lines of the same products of an air enterprise for the period 2008–2012
were taken as the objects to study the phase transition state, i.e., from trial production
phase to total production phase.
502 W. Liu e t a l .
39.4.1 Original Data and Its Processing
Since A2, A3, B2, B4 and B6 are the most significant factors affecting the reliability
step change, the five indexes were extracted in the application study. The original
data for the operation state indexes of different production systems is depicted in
Table 39.4.
The original data in the above table is transformed into the fuzzy membership
function in the range of [0,1]by using the membership function method. Then the
normalization formula of each catastrophe model is gradually calculated upwards to
the upper index layer, and the membership value of the top target can be calculated.
Taking A21 as an example, the calculation process of benefit type index is introduced
according to the Eq. (39.1). And so on, we can obtain the results shown in Table 39.5.
y1=1.424 −1.245
2.698 −1.245 =0.1.y2=1.245 −1.245
2.698 −1.245 =0.
y3=1.782 −1.245
2.698 −1.245 =0.370.y4=2.698 −1.245
2.698 −1.245 =1.
Table 39.4 The original data for the operation state indexes of different production systems
First level
index
Second
level index
Third level
index
Production line
y1 y2 y3 y4
AA2 A21 1.424 1.245 1.782 2.698
A22 2.220 2.179 3.221 2.254
A23 1.332 2.247 1.445 1.147
A24 1.647 3.005 1.245 2.225
A25 1.832 2.782 1.156 3.245
A3 A31 1.356 1.246 1.210 1.101
A32 1.352 2.089 2.803 1.045
A33 2.230 2.121 1.457 1.123
A34 1.087 1.204 1.992 2.101
BB2 B21 2.362 3.136 1.367 1.252
B22 2.017 2.695 1.333 1.823
B23 1.214 2.044 1.928 1.552
B4 B41 3.002 1.926 2.664 3.087
B42 2.025 1.358 2.102 2.391
B6 B61 1.835 2.369 1.567 2.959
B62 1.389 1.864 2.339 2.991
39 Research on Hardware Product’s Reliability Step Change … 503
Table 39.5 The dimensionless values of the evaluation indexes
Third class index Production line
y1 y2 y3 y4
A21 0.123 0.000 0.370 1.000
A22 0.931 1.000 0.000 0.928
A23 0.168 1.000 0.271 0.000
A24 0.228 1.000 0.000 0.557
A25 0.324 0.778 0.000 1.000
A31 1.000 0.568 0.427 0.000
A32 0.307 0.594 1.000 0.000
A33 0.000 0.098 0.098 0.000
A34 0.000 0.115 0.893 1.000
B21 0.411 0.000 0.939 1.000
B22 0.502 1.000 0.000 0.360
B23 0.000 1.000 0.860 0.407
B41 0.927 0.000 0.636 1.000
B42 0.646 0.000 0.720 1.000
B61 0.807 0.424 1.000 0.000
B62 0.000 0.297 0.593 1.000
39.4.2 Analysis of the Total Catastrophe Membership
Function Value
According to the multi-criteria evaluation method of catastrophe theory, the nor-
malization formula of each catastrophe model is used to calculate the catastrophe
membership function value gradually upwards to the upper index layer until that of
the top target is calculated. The indexes A21, A22, A23, A24 and A25 constitute a
fold catastrophe model. The indexes A32, A33 and A34 constitute a butterfly catas-
trophe model. The indexes B21, B22 and B23 constitute a swallowtail catastrophe
model. The indexes B41, B42 and B61 and B62 constitute a cusp catastrophe model.
Taking the manufacturing process A2 of the production system y1 as an example,
the corresponding calculation process is as follows:
XA21 =(0.123 )1
2=0.351.XA22 =(0.931 )1
3=0.976.
XA23 =(0.168 )1
4=0.640.XA24 =(0.228 )1
5=0.744.XA25 =(0.324 )1
6=0.829.
As each control variable can make up for each other, the mean value is taken are as
follows.
A2=(XD4+XD5+XD6+XD7+XD8)
5=0.708.
504 W. Liu e t a l .
Table 39.6 The catastrophe membership function values of the production system y1
Third class index Second class index First class index
C2 0.708 B1 0.419 A1 0.402
C3 0.419
C5 0.385 B2 0.385
C7 0.864
C9 0.466
Table 39.7 Evaluation results of operation state of production systems
Evaluation results Production lines
y1 y2 y3 y4
Structural factors
B1
0.419 0.701 0.266 0.500
Nonstructural
factors B2
0.385 0.000 0.644 0.500
Comprehensive
evaluation results
of operation state
A
0.402 0.351 0.455 0.500
Δλ Large Larger Small Smaller
The indexes A2, A3 constitute a cusp catastrophe model. Hence, we can obtain
A=0.419 according to the non-complementary criterion. In the same way, we can
calculate B=0.385. The indexes A, B constitute a cusp catastrophe. Thus, we can
obtain M=0.402 according to the complementary criterion. And it is equal to
catastrophe membership function value of operation state of the production line y1.
Similarly, the other indexes are normalized, the results are shown in Tables 39.5 and
39.6.
Similarly, the total catastrophe membership function values of the production lines
y2, y3 and y4 can be obtained. They are 0.351, 0.455 and 0.500, respectively. The
above evaluation values of the four production systems are compared with the relative
evaluation values of the reliability change of the four production lines provided by
enterprise. The results are shown in Table 39.7.
As shown in Table39.7, we can see that Ay2<Ay1<Ay3<Ay4, i.e., the catas-
trophe series increases in turn, which indicates that the stability of each manufac-
turing system is increasing in turn. In addition, it is also noticeable that Δλy2>
Δλy1>Δλ
y3>Δλ
y4. It shows that the degree of reliability step change of each
manufacturing system decreases from trial production to total production phase. It
can be concluded that the catastrophic series of the operation state of the manufac-
turing system can effectively reflect the degree of reliability step change between
two adjacent phases, thereby verifying the validity of the proposed method.
39 Research on Hardware Product’s Reliability Step Change … 505
39.5 Conclusion
This study proposed a method based on catastrophe theory to quantify and analyze
the hardware products reliability step change. The evaluation results of operation
state of the manufacturing system in the applied research were compared with the
actual reliability step change. The results showed that both of them have basically
the same trend, which indicates that the comprehensive evaluation method based on
the catastrophe theory is feasible. The quantitative analysis of influence degree of
the control variables on the reliability step change is carried out. The results of this
study might establish the foundation for preventing and controlling the reliability
step change.
Acknowledgements This work is supported by the National Natural Science Foundation of China
(No. 71461020), Production-Study-Research Cooperation Fund of Science and Technology Bureau
of Guangdong Province (No. 2012B091100175), and the Science and Technology Research Fund
of the Ministry of Education of Jiangxi Province (No. 11686).
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