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The Pennsylvania State University
The Graduate School
Department of Industrial and Manufacturing Engineering
ROBUST DESIGN INCORPORATING COST MODEL FOR SYSTEM
RELIABILITY: FINDING BALANCE BETWEEN HIGH
QUALITY AND LOW COST
A Thesis in
Industrial Engineering
by
Zhenhua Sun
© 2012 Zhenhua Sun
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
December 2012
The thesis of Zhenhua Sun was reviewed and approved* by the following M. Jeya Chandra Professor of Industrial and Manufacturing Engineering Thesis Advisor Chia-Jung Chang Assistant Professor of Industrial and Manufacturing Engineering Paul Griffin Professor of Industrial and Manufacturing Engineering Head of the Department of Industrial and Manufacturing Engineering
*Signatures are on file in the Graduate School.
iii
ABSTRACT
One of the robust design methods is finding the optimum process parameter
setting that minimizes total cost of manufacturing a product. Dr. Taguchi’s robust
design method provides quality engineers with a means to relate the quality
management to cost management. It can help quality engineers to find the target
values of quality characteristics that minimize external failure cost and thus
theoretically satisfy all customers. However, setting means of the processes equal to
target values may not be the most economical robust method: it incurs a high product
price which is unaffordable for customers. What’s more, Taguchi’s robust design
method fails to give the amount of cost.
What this research does is to take not only quality loss but also production cost
and setup cost into consideration. That is, a more complete cost model is included in
the robust design procedure. It facilitates the finding of optimum process parameter
setting with which the balance between high quality and low manufacturing cost is
achieved.
In this research, the external failure cost, production cost and setup cost are all
presented as functions of optimum means and variances of quality characteristics.
With the variances of quality characteristics known, the optimum mean settings could
be obtained by minimizing the total cost composed of the external failure cost,
production cost and setup cost. The total cost of manufacturing the product can also
be estimated as every item in the total cost model is quantifiable.
iv
TABLE OF CONTENTS
List of Tables vi List of Figures vii Chapter 1: Introduction 1
1.1 Description of the Problem 1 1.2 Literature Review 5 1.2.1 Review of Robust Design Terminology and Methodology 5
1.2.2 Review of the Optimum Mean Setting Methods 8 1.3 Summary and Conclusion 11
Chapter 2: Model Development and Validation 13 2.1 Introduction 13 2.2 Robust Design Using Loss Function 14 2.3 Reliability Model Including Mean and Variance 16 2.4 External Failure Cost Model including Reliability Model 18
2.5 Determining the Reliability Model Parameters Using Maximum Likelihood Estimation (MLE) 20 2.6 Derivation of Total Manufacturing Cost Model for the
Product with Sub-Assemblies 21 2.6.1 Relationship between Product Reliability and
Sub-assemblies’ Reliabilities 22 2.6.2 Total cost model for the product with sub-assemblies 24
2.7 Robust Design Utilizing Total Cost Model 26 2.8 Summary and Conclusion 28
Chapter 3: Numerical Example and Model Analysis 30
3.1 Introduction 30 3.2 Numerical Example 30 3.2.1 Stage 1. Finding the Target Values of Quality Characteristics 34
3.2.2 Stage 2. Estimation of the Reliability Models Using Maximum Likelihood Estimation 36
3.2.3 Stage 3. Finding External Failure Cost Function 42 3.2.4 Stage 4. Creating total cost model 43 3.2.5 Stage 5. Finding Optimum Mean Settings of Processes 44 3.2.6 Stage 6. Obtaining the Optimum Process Combination and
Mean Settings 45
v
3.3 Model Analysis 46 3.3.1 The Effect of Deviation of Mean from the Target Value on Total Cost 47
3.3.2 The Effect of Process Variance on Total Cost 49 3.3.3 The Effect of the Length of Warranty Period on Total Cost 52
3.4 Summary and Conclusion 53 Chapter 4:Conclusion and Future Research 54 4.1 Concluding Remarks 54 4.2 Future Research 55 References 56
vi
LIST OF TABLES
Table 3.1 Processes Available and Associated Variances 31 Table 3.2 Quality Characteristics and Failure Times of Sample Components Manufactured in Process A-1 38 Table 3.3 Quality Characteristics and Failure Times of Sample Components Manufactured in Process B-1 39 Table 3.4 Quality Characteristics and Failure Times of Sample Components Manufactured in Process C-1 40 Table 3.5 Quality Characteristics and Failure Times of Sample Components Manufactured in Process D-1 41 Table 3.6 Table of Results 46
vii
LIST OF FIGURES
Figure 1.1 Goal Post Mentality 2
Figure 1.2 Taguchi Loss Function 4
Figure 2.1 The Reliability Function Under Replacement Policy 19
Figure 2.2 Reliability Function Under Minimal Repair Policy 19
Figure 2.3 Series System 22
Figure 2.4 Parallel System 22
Figure 2.5 Mixed System 23
Figure 3.1 Structure of Product 31
Figure 3.2 Effect of |Xa,A-Xt,A| on Total Cost 47
Figure 3.3 Effect of |Xa,B-Xt,B| on Total Cost 48
Figure 3.4 Effect of |Xa,C-Xt,C| on Total Cost 48
Figure 3.5 Effect of |Xa,D-Xt,D| on Total Cost 49
Figure 3.6 Effect of σA on Total Cost 50
Figure 3.7 Effect of σB on Total Cost 50
Figure 3.8 Effect of σC on Total Cost 51
Figure 3.9 Effect of σD on Total Cost 51
Figure 3.10 Effect of Warranty Period on Total Cost 52
1
Chapter 1
Introduction
1.1 Description of the Problem
As per Dr. Taguchi’s theory, the best and most economical way to eliminate
variances is during the stage of a product’s design and its manufacturing. The product’s
design stage is subdivided by Dr. Taguchi into three stages: system design stage,
parameter design stage and tolerance design stage. System design is a concept formation
step in design stage at which the concept of a new product is made. Starting the
parameter design marks the beginning of engineering process. At this stage, the nominal
sizes of different dimensions of the parts are chosen and the design parameters are
adjusted and decided according to the manufacturing resources available. These values
should be accurately chosen to make the product performance satisfy the requirements
of the customer. The accurately chosen parameters will allow the product to keep its
performance as designed even under the circumstances of varied serving conditions,
manufacturing environment and unexpected fatigue damage. This is called robustness of
a system. Tolerance Design is the process of defining and allocating the product’s
tolerances to the components exactly in order to manufacture high quality products at
the minimum cost. This research focuses on target value setting problem at the
parameter design stage.
Customers always want the product’s characteristics value to be as close as possible
to the ideal value so that the product can keep a high quality and last long. The difficulty
faced by companies in achieving this customer expectation is: the cost of producing a
2
product with ideal quality will be very high and sometimes it is even impossible to get
the ideal quality. For companies, the key to solving this “quality-cost-tradeoff” problem
is to include cost analysis in the parameter design process.
The original theory is that “no cost will incur if the product’s quality characteristic
stays within the specification. This is the well-known “goal post mentality” (Figure 1.1).
It was believed that if the quality characteristics stayed within the specifications, no
quality failure cost would incur and there will not be any quality issue. The other
meaning behind this theory is: even if products’ quality characteristic is located in the
internal: X0-Δ, X0+Δ ( which are the lower specification limit and upper specification
limit of quality characteristic respectively), they will all perform as well as designed. If
the quality characteristics lie outside the specification limits, the same quality loss will
be incurred regardless of the location of the quality characteristic values.
Loss $ No Loss Loss $
C B A
x0-Δ x0 the x0+Δ
Lower Target Upper
Limit Value Limit
Figure 1.1 Goal Post Mentality
But there exist a lot of realities in real industrial practice that are in conflict with
the “Goal Post Mentality”. These cases indicate that the shift from target value will of
course increase the customer’s complaints as well as the company’s quality loss. One big
deficiency in “goal post mentality” theory is its failure to measure the degree of a
Tolerance Range
3
product’s “goodness” and “badness”. Also, “goal post mentality” fails to support the
manufacturer’s expectation to minimize the quality loss.
Dr. Taguchi developed the quadratic loss function to show a more accurate
relationship between the quality characteristic value and quality loss. The core meaning
of Taguchi’s loss function is: as the quality characteristic value deviates from the
nominal value, the quality loss of the product increases. Taguchi referred to the quality
cost due to defects found after the product was delivered to customers as external failure
cost. External failure cost may include warranty cost, repair or replacement cost and
other costs incurred by whole society due to defective products. The Nominal-The-Best
Taguchi Loss Function is:
L(x)=k’(X-X0)2, LSL≤X≤USL,
=0 otherwise (1.1)
In the Loss Function above, X is the quality characteristic value, X0 is the nominal
size of quality character, LSL and USL are specification limits and k’ is a proportional
constant value which converts the quality value into the economic value .The graph of
Taguchi loss function is shown in figure 1.2. Taguchi Loss Function demonstrated that
even if the quality characteristic stays within its specification limits, the external failure
cost increases as the characteristic value deviates from the nominal size. From loss
function, the expected loss per product is obtained as follows:
E[L(x)]=k’[σ2+(µ-X0)2] (1.2)
where σ is the standard deviation of quality characteristic value, µ is the actual mean and
X0 is the nominal size.
4
C cost of rework C cost of rework
Δ Δ
LSL X0 USL
Figure 1.2 Taguchi Loss Function
The importance of this expectation function is that variance and deviation are the
major engineering factors which affect quality loss. It gave a way for the engineers to
effectively control external failure cost, and is widely used in today’s industrial practice.
The methodology which Taguchi loss function provides as a bridge between
engineering indicators and economic indicators makes it a good application in robust
design. The assembly’s quality characteristic is X, the mean and variance of X are μ, σ2
respectively and the target value of X is X0. According to Taguchi’s theory, the best way
to minimize the external failure cost is:
1. Making the mean of X, μ=X0.
2. Minimizing the variance of X, σ2.
Taguchi’s theory is a good method of minimizing the quality loss and making the
product’s parameter design procedure robust. But there are two common criticisms about
Taguchi’s robust design theory: one is that the proportionality constant k’ in Taguchi
Loss Function makes the estimation of the amount of quality loss very difficult; another
is that “producing exactly at target value may be unnecessarily costly and thus increases
the price of the product” (Schneider et al., 1995).
5
To solve the “k’ estimation” problem, a new model to convert the quality
engineering data into economic data is needed. Blue (2002) suggested the use of
reliability model as a bridge to relate the process mean and variance to warranty cost. A
big strength of his model is that every term in it can be quantified. So Blue’s model
provides a clear and definite alternative to the Taguchi Loss Function.
The solution to the problem of “producing exactly at target value may be
unnecessarily costly” problem is a robust design issue. According to Phadke(1989),
robust design is an engineering methodology for improving productivity during research
and development, so that high-quality products can be manufactured quickly and at low
cost. Resit and Edwin (1991) stated that robust design focuses mostly on finding the
near optimum combination of design parameters so that the product is functional,
exhibits a high level of performance and can be manufactured at a low cost.
This research will apply Blue’s model to the solution of “producing exactly at target
value may be unnecessarily costly” problem.
1.2 Literature Review
In this section, review of the evolvement of robust design theory (1.2.1) will be
presented first. Then papers about different kinds of optimum mean setting problems
(1.2.2) will be discussed.
1.2.1 Review of Robust Design Terminology and Methodology
The essence behind robust design was firstly elaborated in the publication by
6
Taguchi and Wu (1979). The fundamental principle of the robust design is minimizing
the effect of the causes of the variation without eliminating the causes (Taguchi and Wu
1979). This can be achieved by performing the off-line quality control in the design
stage which makes the product’s performance insensitive to uncontrollable variances.
But there exists two different understandings of the concept of robust design. One
is that robust design is a specific procedure that optimizes the design process of a
product. This kind of view can be seen from what Tsui (1992) wrote: ‘robust design is an
efficient and systematic methodology that applies statistical experimental design for
improving product and manufacturing process design.’ Lin, et al. (1990) and
Chowdhury(2002) also supported this view of robust design as a specific procedure.
The other understanding of the concept of robust design is that robust design is an
overall approach. Nair (1992) stated that ‘It should be emphasized that robust design is a
problem in product design and manufacturing-process design and that it does not imply
any specific solution methods.’ This viewpoint was also supported by Thornton et
al.(2000), Tennant (2002), and Tatjana et al. (2011). As numerous kinds of methods of
producing robust product have been developed, the view of robust design as a general
guidance of optimizing design stage is accepted by more researchers.
Most of the methodologies of robust design focus on optimizing the parameter
design stage and only a limited number of them deal with system design optimization.
Johansson et al. (2006) developed a method called VMEA (Variation Mode and Effect
Analysis) aiming at variance identification which is applicable in system design
procedure. The VMEA is a statistically based engineering method aimed at guiding the
7
engineer to find critical areas in terms of unwanted variation (Johansson et al., 2006).
Matthiasen (1997) and Andersson (1997) suggested a qualitative method in the form of
design principles to support the robustness of product in the system design procedure.
The trend to perform robust design with emphasize on parameter design partly is
partly due to the work done by Taguchi and Wu (1979). Taguchi and Wu (1979)
suggested the use of orthogonal array and S/N ratio where variation exits in both
controllable factors and noise factor. The level of controllable factors which maximize
the S/N can be selected. This setting can minimize the quality loss without adding to the
manufacturing cost.
After the work by Taguchi and Wu, a large number of papers have been published
on the topic of employing the design of experiment to make the parameter design
process robust. These researches were summarized by Torben et al. (2009) in their
review paper. However, as Li et al. (1997) stated, efforts are needed to develop more
methods than just the use of design of experiments, at the parameter design stage.
Vining and Myers (1990) proposed a response surface method on robust parameter
design. The estimates of mean and variance are presented as two response surfaces for
mean and variance with respect to the operation conditions. The optimum set of
operation conditions can be obtained by setting the mean to target value and minimizing
the variance.
Sometimes, it is too expensive or time consuming to perform the physical
experiment to support the robust design. But with the development of computer based
simulation, the system can be investigated by using virtual simulation. Gijo and Scaria
8
(2012) utilized a computer program to perform the orthogonal array experiment. By
analyzing the result of simulation, optimal combination of factor setting to produce
product was achieved.
1.2.2 Review of the Optimum Mean Setting Methods
The purpose of robust design is to perform the off-line quality control before the
manufacturing process to support the producing of a robust product. The optimum
setting of process parameter is the key component of a good off-line quality controlling.
The following is a review of literatures about target process mean setting.
The first research which focused on finding the target mean was performed by
Springer (1951). Two assumptions he made were: both upper specification limit and
lower specification limit are known; and the losses due to the producing of products
oversized and undersized are specified but not necessarily equal (Springer, 1951). The
target of his research was finding an optimum mean setting which minimizes the total
cost.
After that, a number of researches were performed to solve the target mean setting
problem. Researchers researched this problem under different assumptions of pricing
policies and with focuses on different product quality index.
Bettes (1962) did his research of finding optimum means setting under the
assumption that the minimum content was regulated but the upper specification limit
was arbitrary. The objective is minimizing the total cost composed of cost of
reprocessing and the ingredient loss of producing an oversized product. His procedure
9
was based on trial and error computation. The assumption he made was that oversized
and undersized products would be reprocessed at the same price. But Glohar and Pollock
(1986) indicated in their work that “Bettes’s procedure is based on trial and error and is
computationally tedious”. They replaced Bettes’s trial and error procedure with an
objective optimization procedure which is easy to grasp mathematically. Similar
researches have been done by a lot of scholars, which included controlling lower
specification limit only (Chen and Khoo (2008)), or upper specification limit only (e.g.,
Chen and Huang (2011), or both upper and lower limits (e.g., Chen and Lai (2007)).
Al-Sultan and Pulak (1997) investigated the problem of including variance in the
procedure of finding the optimum mean setting. This kind of problem is defined as
“simultaneously selecting of the most economical target mean and the variance for a
continuous production process” problem. For the single filling operation in which the
lower specification limit is specified, they indicated two ways in which reduction of
variance would affect the cost. The first one is well known by the industry which is the
smaller the variance, the smaller the proportion of rejected products will be. The second
one is that smaller variance allows the process mean to be set closer to lower
specification limit which further leads to the saving of materials.
The function which measured the effect of variance reduction on the cost reduction
was proposed in Al-Sultan and Pulak’s paper. The cost should be compared to the cost of
reducing the variance to see if the variance reducing plan is applicable. The optimum
process mean value can be obtained by utilizing six-sigma mean shifting rule after the
decision on process variance reduction has been made.
10
Determining the optimum mean setting along with the variance in a continuous
process topic was also researched by Mukhopadhyay and Chakraborty (1995), Rahim et
al. (2002), and other researchers. Li (2000) stated that, for a common occurrence where
the deviation of the quality characteristic in one direction is more harmful than in two
directions, manufacturers’ traditional method of setting the process mean at the middle
and choosing smaller tolerance for both sides fails to minimize the expected quality loss.
He suggested that a shift of mean from target value is needed to meet the requirement of
minimizing the quality loss.
In Li’s first paper (2000) about optimum mean setting topic, he discussed two
specific models of quality loss function: when the standard deviation of quality
characteristic remains constant, and when the standard deviation of the quality
characteristic is proportional to the process mean. Despite the different formats of the
two models, Li used the same mathematical method to find the optimal mean setting. He
set the expected quality loss function based on normal distribution assumption as the
objective function. The decision variable is the optimum process mean value that
minimizes the expected quality loss. After that, he used the same optimization method to
determine the optimum mean setting based on truncated asymmetrical quality loss
function (Maghsoodloo and Li, 2000) and asymmetric linear loss function (Li, 2002).
The problem is that these works are based on the assumption that quality losses at
both specifications are equal; but in many situations, they are different. So Li wrote
several papers taking this situation into consideration by modifying his previous works.
He modified the truncated asymmetrical quadratic loss function (Li, 2004) and the
11
truncated asymmetrical linear loss function (Li, 2005) to incorporate difference
at-specification quality loss scenario. The optimization method he had used in his
previous work was still applicable to find the optimum mean minimizing quality loss.
The Optimum mean setting can also be obtained through the process capability.
Veevers and Sparks (2002) defined a capability index as the proportion of products
continuously conforming to the specifications during the service time period. Instead of
paying attention only to conformance of product to the quality requirements in the
process, manufacturers and customers pay more attention on conformance of product
during the service period. Because the process mean is included in the function of
capability index, the target value of mean can be achieved by placing restrictions on the
capability index. The restrictions are the requirements of customers on the conformance
of product.
1.3 Summary and Conclusion
At the first section of this chapter, the description of the problem which this
research attempted to solve was given. This research will apply the model developed by
Blue to the solving of robust parameter design problem. The objective of this research is
to solve the problem of optimum parameter setting and combination of the available
manufacturing resources so that a high quality product can be manufactured quickly and
at low price.
The second section of this chapter reviewed the development of robust design theory
and previous works related to optimum mean setting. From the previous works about
12
robust design, it can be seen that parameter design is the most popular and essential part
in robust design. The reason is that robust parameter design methods provide clear
engineering methods for the manufacturers to produce robust products.
Because robust parameter design is an important area, previous researches about
optimum mean setting were also reviewed. These works dealt with the optimum setting
problem either from the view of pure engineering or based on various kinds of loss
function. The shortcoming with the pure engineering view is that it fails to take the cost
to improve the quality into consideration. The problem with the loss function based
method is that the loss function is difficult to use in real practice.
This research tried to pursue a balance between quality enhancement and cost
reduction at the design stage through the use of a clear and easily applied model. The
method provided in this research gives a good way of setting robust parameter values in
design stage.
The thesis is organized as follows. In Chapter 2, the total cost model will be
developed by incorporating reliability model, and afterwards procedure of utilizing the
total cost model to solve robust design problem will be presented. In Chapter 3, a
numerical example will be provided to show in detail the solution of robust design
problem using total cost model. The analysis of the total cost model will also be given in
Chapter 3 as well. Finally, the conclusion and suggestions for further research will be
made in Chapter 4.
13
Chapter 2
Model Formulation
2.1 Introduction
This chapter focuses on total cost model development and application of total cost model
into the solution of robust design problem. The content of this chapter will be divided into six
sections.
The first section of this chapter will review the Taguchi’s robust design method using loss
function and recommend it as a way to find out the target value of quality characteristic. The
second section of this chapter will review the development of reliability model including mean
and variance. The third section of this chapter will incorporate the reliability model into the
development of external failure cost model. The fourth section of this chapter will describe the
use of maximum likelihood estimation (MLE) to find out the values of proportionality
parameters in the external failure cost model. The fifth section of this chapter will develop a total
cost model incorporating the external failure cost, production cost and setup cost. The sixth
section of this chapter will employ the total cost model to solve a robust design problem with the
objective of finding the optimum manufacturing resource combination and optimum process
parameter setting.
The object of this chapter is to develop an easy-to-use cost model to estimate the total cost
of manufacturing a product and provide a robust design method utilizing this total cost model.
14
2.2 Robust Design Using Loss Function
The expected value of Taguchi’s loss function (Function 1.2) gives a good
indication of how to produce a product with highest quality and minimum external
failure cost: making the mean of assembly quality characteristic equal to its target value
and minimizing the variance of assembly quality characteristic.
Assume that the assembly quality characteristic X is a given function of the
components’ quality characteristics X1, X2, X3, ..., Xk and the function is given as
following (Chandra, 2001):
X=e(X1, X2, X3, …, Xk) (2.1)
As the assembly quality characteristic X can be obtained from the components’
quality characteristics through the function, it is a reasonable inference that the
controlling of mean and variance of X can be achieved by controlling the means and
variances of components’ quality characteristics.
By using Taylor series and statistical operations, the relationship between the mean
of assembly quality characteristic and means of components’ quality characteristics is
given as following function:
X0=e(μ)+ ∑ ∑ h σ (2.2)
The relationship between the variance of assembly quality characteristic and
variances of components’ quality characteristics is:
σ =∑ ∑ g g σ (2.3)
In (2.2) and (2.3), k is the number of components; X0 is the target value of product
quality characteristic; σ is the variance of product quality characteristic; μ is a row
15
vector equal to μ=[ µ1, µ2, µ3, …, µk]; g =∂e(•)/∂Xi and g =∂e(•)/∂Xj, both are
evaluated at μ=[ µ1, µ2, µ3, …, µk]; h = e(•)/XiXj evaluated at μ=[ µ1, µ2, µ3, …, µk];
σ = cov(Xi, Xj) when i≠j and σ =σ when i=j.
According to Taguchi’s theory, the optimum strategy of setting the process
parameter to produce high quality products is:
Minimize:
∑ ∑ g g σ
subject to:
e(μ)+ ∑ ∑ h σ = X0 (2.4)
The values of σ and cov(Xi, Xj) are usually known when the manufacturing
processes are adopted. Therefore, the optimum mean settings of component processes
can be found through solving of this nonlinear program.
This robust design method provides a good way to find the target values of
processes, but the problem with setting process means exactly equal to the respective
target values is that the cost of production will be unnecessarily high. What is more, the
cost of production, which is a major index for the manufacturer to make a series of
manufacturing decisions, is failed to be estimated by using Taguchi’s robust design
method.
Despite of the shortcomings, Taguchi’s robust design method is still an efficient
way to find the target value. The robust design methodology developed in this research
will utilize Taguchi’s robust design method to achieve the target values.
16
2.3 Reliability Model Including Mean and Variance
In his work, Elsayed (1996) illustrated accurately the definition of reliability:
reliability is the probability that a product or service will operate properly for a specified
period of time under the design operation conditions. He also indicated the relationships
among reliability function R(t), probability density function of failure time f(t), and
hazardous rate function h(t), in his work:
f(t) =
(2.5)
h(t) = (2.6)
where t is a variable denoting a random time point during the product’s life period.
A large number of practical cases have proved that a product with smaller variance
and with its quality characteristic closer to the target value will perform longer. Blue
(2001) included this important property in his reliability model development. He
suggested a conditional reliability model as follows:
R(t/X) = exp[(-(λ+β(X-µt)2)tξ] (2.7)
where X is the quality characteristic value; µt is the target value; t is the variable
denoting the random time point during the product’s life period; λ, β, ξ are
proportionality parameters that provide flexibility for the model.
From functions (2.5) and (2.6), the unconditional hazardous rate function and pdf
can be obtained as per following:
f(t/X) = (λ+β(X- µt)2)ξtξ-1exp[-(λ+β(X- µt)
2)tξ] (2.8)
h(t/X) = (λ+β(X- µt)2)ξtξ-1 (2.9)
Manufacturers are more interested in the whole process’s capability of producing
17
high reliability product than in a single product’s reliability. Thus the unconditional
reliability function is needed so that the description of the reliability of whole product
population can be made.
It is assumed that the product’s quality characteristic is normally distributed. Then,
the unconditional reliability function is:
R(t) = exp[-(λ+
)t (2.10)
where t is the variable denoting the random time point during the product’s life period; σ
is the variance of product quality characteristic; µa is the mean of quality characteristic;
µt is the target value; λ, β, ξ are proportionality parameters that provides flexibility for
the model.
If = , the function (2.10) could be written as follows:
R(t) = exp[-(λ+
)t (2.11)
It can be seen from function (2.10) that as σ, µa-µt, and t increase, the value of R(t)
decreases. This is in accordance with the basic understanding of the reliability: larger
variance, larger deviation of mean from target value and longer performance time lead to
lower reliability of the product. This model provides a good estimation of product
reliability.
18
2.4 External Failure Cost Model including Reliability Model
The external failure cost is an important factor in the manufacture’s decision
making process on process parameter setting, warranty period and policies of repair and
replacement. So a rigorous expression of external failure cost is necessary for
manufacturers. An accurate external failure cost model will provide a good estimation of
external failure cost.
According to Elsayed (1996), there are normally two kinds of warranty policies for
failed product. That is, failed product will be subject to two different kinds of warranty
policy. Type one warranty policy is replacing the failed product with a brand new one.
Under this warranty policy, the reliability of failed product after replacement service will
revert to the state of a brand new product. Let the expected number of replacement
services during warranty period [0, w] be M1[w] per unit. The graph of reliability
function under type one warranty policy is shown in figure 2.1. It can be seen from the
graph that reliability curve will continuously decrease until failure, and then a new
reliability curve will begin after replacement service.
Type two warranty policy is called minimal repair policy which is restoring the
product to the state it had just before failure. This minimal repair policy costs less than
replacement policy, so most of the manufacturers use it as their warranty policy. The
expected number of minimal repair services during the warranty period [0, w] is M2(w)
per unit. The graph of reliability function under minimal repair policy is given in figure
2.2. It can be seen from the graph that the product will continue its reliability decreasing
trend after minimal repair.
19
Replaced
1 Product
Failure Time
0
Time
Figure 2.1 The Reliability Function Under Replacement Policy
1
Failure Time Repaired
Product Service
0 Time
Time
Figure 2.2 Reliability Function Under Minimal Repair Policy
Thus, the expected total number of failures during warranty period per unit is:
M(w) = M1[w]+ M2[w] (2.12)
If the proportion of failures under replacement policy is p, and the proportion of
failures under minimal repair policy is 1-p, then the expressions of M1[w] and M2[w]
can be calculated as (Elsayed, 2001):
Rel
iabi
lity
Rel
iabi
lit
20
M1[w] = [R(w)]p+ M w t d R t (2.13)
M2[w] = -(1-p) ln [R(w)] (2.14)
where R(t) is the reliability function given in function (2.10).
In this paper, it is assumed that all failures occurring during the product’s warranty
period will be serviced under minimal repair policy. Thus the total number of failures
during warranty period is:
M[w] = -ln [R(w)] (2.15)
If the unit cost of repairing the product per failure is Cr, then the warranty cost per
unit of product is:
Cw=-Cr ln [R(w)] (2.16)
This external failure cost model provides a good estimation of the external failure
cost as every item in the model can be quantified. It also facilitates the development of
total cost model in this research.
2.5 Determining the Reliability Model Parameters Using Maximum Likelihood
Estimation (MLE)
Before using the reliability model showed in function (2.10), the value of
parameters λ, β, ξ are needed to know. Here, the Maximum Likelihood Estimation (MLE)
is chosen to estimate the values of parameters.
First, a sample of n independent products with quality characteristics Xi (i=1,2, …,
n) should be produced. Then experiments are performed to find the failure times of these
sample products which are ti (i=1,2, …, n). The data set generated is uncensored and
21
complete. Thus, the likelihood function is defined as follows:
L(t, X; λ, β, ξ) = ∏ f t , ; λ, β, ξ)
= f(t1, x1; λ, β, ξ)* f (t2, x2; λ, β, ξ)* ……
* f (tn, xn; λ, β, ξ) (2.17)
where t is the time to failure; X is the product’s quality characteristic; λ, β, ξ are
parameters in reliability model needed to be estimated; f (t, X; λ, β, ξ) is the conditional
probability density function showed in function (2.8) with the value of t, X and X0
known.
The next step is maximizing the likelihood function. The values of λ, β, ξ are
obtained by solving the following nonlinear programing problem:
Maximize
L(t, X; λ, β, ξ) = f (t1, x1; λ, β, ξ)* f (t2, x2; λ, β, ξ)* ……* f (tn, xn; λ, β, ξ) (2.18)
2.6 Derivation of Total Manufacturing Cost Model for the Product with
Sub-Assemblies
In modern manufacturing, it is normal that products are usually composed of
sub-assemblies. But it makes the cost estimation of the product more complex. In this
section, the reliability model will be employed to develop the total cost model of the
product with sub-assemblies.
22
2.6.1 Relationship between Product Reliability and Sub-assemblies’ Reliabilities
The product can be seen as a system composed of several sub-systems
(sub-assemblies). Systems can be classed into series system (figure 2.3), parallel system
(figure 2.4) and mixed system (figure 2.5) according to different connection types of
sub-assemblies. The relationship between product reliability and sub-assemblies’
reliabilities is quite different among these different kinds of systems.
Sub-assembly 1 Sub-assembly 2 Sub-assembly n
Figure 2.3 Series System
Sub-assembly
Figure 2.4 Parallel System
••••••
1
2
n
•••
23
Sub-assembly 1 Sub-assembly 2 Sub-assembly n
Figure 2.5 Mixed System
It is assumed that the paralleled components in each sub-assembly are identical
with the same variance and mean of the associated quality characteristic. The reliabilities
of systems shown in the above figures are as follows:
For series system showed in figure 2.3, the reliability of the system is:
R(t) = R1(t)* R2(t)*……* Rn(t)=∏ R (t) (2.19)
where R(t) is the reliability of the system at time t; R1(t), R2(t), ……, Rn(t) are
reliabilities of sub-assembly 1, 2, ……, n respectively at time t; and n is the number of
the sub-assemblies.
For paralleled system showed in figure (2.4), the reliability of the system is:
R(t) =1-(1-R’(t))n (2.20)
where R(t) is the reliability of the system at time t; R’(t) is the reliability of
1
•••
2
m1
1
•••
2
m2
1
•••
2
mn
•••
24
sub-assembly at time t; n is the number of components of the sub-assembly.
For mixed system showed in figure 2.5, the reliability of the system is:
R(t) = [1- 1 R t ]* [1- 1 R t ]*…… [1- 1 R t ]
= ∏ 1 1 R t (2.21)
where R(t) is the reliability of the system at time t; R1(t), R2(t), ……, Rn(t) are
reliabilities of sub-assembly 1, 2, ……, n respectively at time t; m1, m2, ……, mn are the
number of components of sub-assembly 1, 2, ……n respectively; and n is the number of
the sub-assemblies.
2.6.2 Total Cost Model for the Product with Sub-assemblies
The total cost in this research is composed of three parts: external failure cost,
production cost, and setup cost. External failure cost is influenced by the reliability of
the product: the more reliable the product is, the less number of failures it will have
which leads to less external failure cost. Production cost is influenced by ∂ (∂= ):
the larger ∂ is, the lower the producing cost is. Setup cost is influenced by the variance:
smaller variance in production process requires more sophisticated manufacturing
equipment, and thus increases the setup cost.
The functions of product’s external failure cost, production cost, and set-up cost are
shown as follows:
It is assumed that the relationship between product reliability and sub-assemblies
reliabilities is R(t) = s(R1(t), R2(t), …, Rn(t)), where R(t) is the reliability of the system at
time t; R1(t), R2(t), ……, Rn(t) are reliabilities of sub-assemblies 1, 2, ……, n
respectively at time t. The warranty period is assumed to be w. Then the external failure
25
cost function of the product is:
Cw = -Cr ln [R(w)]
= -Cr ln [s(R1(w), R2(w), …, Rn(w))]; (2.22)
where,
Cr is unit cost of repairing the product per failure;
Rj(w)=
exp[-(λj+
)w (j=1,2, …, n);
= , , (j=1,2, …, n); and
σj is the variance of the quality characteristic of sub-assembly j;
The set-up cost of each sub-assembly is assumed to be as follows:
Csj =ηje (ηj, τ >0 and j=1,2, …, n) (2.23)
where, ηj, τ are proportionality parameters; ∂ = , , ;
The production cost of each sub-assembly is assumed to be as follows:
Cpj =hjσ (hj, α >0 and j=1,2, …, n) (2.24)
where, hj, α are proportionality parameters; σj is the variance of the quality
characteristic sub-assembly j;
The sum of producing cost and setup cost is manufacturing cost Cm, and hence the
total cost could be written as:
TC = Cw+∑ C
= Cw+∑ C C
= -Cr ln [s(R1(w), R2(w), …, Rn(w))]+∑ h σ η e (2.25)
where
26
Rj(w)=
exp[-(λj+
)w (j=1,2, …, n); (2.26)
∂ = , (j=1,2, …, n). (2.27)
2.7 Robust Design Utilizing Total Cost Model
Robust design is an approach that aims at improving the product’s productivity
during research and development stages. As one of the critical robust design problems,
the optimum setting of process parameter problem needs a definite and easily applied
solution. Because every item in the total cost model can be quantified and easily
understood, the total cost model showed in function (2.23) provides a very efficient way
to solve the robust design problem. In this section, the robust design will be performed
using total cost model, and suggested procedures will be provided.
It is assumed that a product has n sub-assemblies, and sub-assembly j (j=1, 2, ……,
n) has xj components arranged in parallel. For components in sub-assembly j (j=1,
2, ……, n) there are yj processes available. The parallel components in each
sub-assembly are identical with the same variance and mean of the associated quality
characteristics. Thus, there are y1*y2*…* yn process combinations available to
manufacture this product. The target value of product’s quality characteristic is known as
X0. The variances of all processes and covariance of any two processes are known.
Warranty period of the product is w. Product’s quality characteristic X is a given
function of the components’ quality characteristics X1, X2, X3, ..., Xn which is X=e(X1,
X2, X3, ..., Xn). The relationship between product reliability and components’ reliabilities
is R(t) = s(R1(t), R2(t), …, Rn(t)). Unit cost of repairing the product per failure is known
27
as Cr. The objective is to find the optimum process combination and optimum mean
setting to support the production of high quality product at low cost.
This is a problem that manufacturers usually meet in the product design stage:
finding the optimum parameter setting and optimum combination of available
manufacturing resource. Here, the total cost model is employed to solve this problem
and the suggested procedure of solution is described as following:
Stage 1: The target values of sub-assemblies’ quality characteristics in a combination
should be estimated using Taguchi’s method described in section 2.2.
Stage 2: First, the maximum likelihood estimation (MLE) described in section 2.5 is
performed to find the values of proportionality parameters λj, βj, ξj in the
reliability functions. Second, the reliability function Rj(t) for components in
subassembly j is obtained by plugging the parameter values into the
reliability model :
Rj(t)=
exp[-(λj+
)t (j=1,2, …, n)
Stage 3: The external failure cost function of product is found using equation:
Cw = -Cr ln [R(w)]
= -Cr ln [s(R1(w), R2(w), …, Rn(w))];
Stage 4: The total cost model including external failure cost, producing cost and setup
cost is:
TC=-Cr ln [s(R1(w), R2(w), …, Rn(w))]+∑ x η e x h σ
28
where,
∂ = , , ;
μ , is the actual mean setting of the process;
Stage 5: As the values of process variance σ and proportionality parameters ηj, τj, hj,
α are known, the optimum mean setting μ , for the processes in this
combination are obtained by:
Minimizing
TC=-Cr ln [s(R1(w), R2(w), …, Rn(w))]+∑ x η e x h σ
Where, ∂ = , , .
Stage 6: Repeat stage 1 to stage 5 for all y1*y2*…* yn combinations, then optimum
process combination and the optimum process mean settings of this process
combination are obtained.
To successfully solve this robust design problem, the softwares such as MATLAB
and LINGO is needed to solve the non-linear programming included in the problem
solving procedure. In this research, MATLAB is employed. A numerical example for
solving this robust design problem will be presented in chapter 3.
2.8 Summary and Conclusion
This chapter defined and developed the reliability model, external failure cost
model and total cost model for a single product. Models defined and developed in this
chapter can satisfy the manufacturer’s requirements of quantifying product’s quality and
estimating the total cost of manufacturing the product. The reason is that every item in
29
these models can be quantified and easily understood.
It was also illustrated in this chapter that the total cost model can be used to solve
the robust design problem aiming at finding the optimum manufacturing process
combination and optimum mean setting of processes. The steps of employing the total
cost model to solve robust design problem was suggested in this chapter. A numerical
example will be presented in the next chapter to describe the method in more detail.
30
Chapter 3
Numerical Example and Model Analysis
3.1 Introduction
This chapter will provide a numerical example to describe the application of the
total cost model to the solving of robust design problem. The analysis of the total cost
model will also be performed in this chapter. The first section of this chapter is related to
a numerical example. The total cost model developed in the previous chapter will be
used to find the optimum manufacturing process combination and the optimum mean
setting of producing a product. The product is composed of several sub-assemblies with
components. The second section of this chapter is concerned with model analysis. The
effects of “shift from target value”, “variance” and “length of warranty period” on total
cost will be studied in this section.
3.2 Numerical Example
It is assumed that the product’s target value of quality characteristic is 10. The
warranty period of this product is 4. This product consists of four sub-assemblies
arranged in series (Figure 3.1). Sub-assembly A has three components A-1, A-2 and A-3
arranged in parallel. Sub-assembly B has two components B-1 and B-2 arranged in
parallel. Sub-assembly C and D have only one component each, which are C-1 and D-1
respectively. For production of components A-1, A-2 and A-3, there are three processes
A1, A2 and A3 available. For production of components B-1 and B-2, there are two
31
processes B1 and B2 available. For production of component C-1, there are two
processes C1 and C2 available. For production of component D-1, there is only one
process D1 available. Here, the parallel components in each sub-assembly are assumed
to be produced in the same process. That is, parallel components in each sub-assembly
are identical with the same variance and mean of the associated quality characteristic.
What’s more, quality characteristics of components are normally distributed. The
variances of all available processes are shown in table 3.1.
Sub-assembly A Sub-assembly B Sub-assembly C Sub-assemblyD
Figure 3.1 Structure of Product
Components Processes Available Variances of Processes A-1, A-2, A-3 A1 A2 A3 0.36 0.57 0.81
B-1, B-2 B1 B2 0.05 0.25 C-1 C1 C2 0.03 0.25 D-1 D1 0.16
Table 3.1 Processes Available and Associated Variances
Therefore, there are 3*2*2=12 combinations of processes available to manufacture
the component in this product. The robust design problem related with this product is to
find the optimum process combination among the 12 combinations and the optimum
A‐1
A‐3
A‐2
B‐2
B‐1
C‐1 D‐1
32
mean settings for the processes in this combination. The solution of this robust design
problem will employ the total cost model developed in the previous chapter and follow
the suggested procedure.
In real practice, no process can produce a product with an unlimited large or small
size. So, there usually exists a numerical range for the process mean setting. The
numeric ranges for mean settings of the processes shown in the above table are all
assumed to be from 1 to 20.
A six sigma company allows the actual mean to shift plus or minus 1.5 standard
deviations (±1.5σ) from the target value. This allowance ensures that even if the process
mean drifts from the target value, as long as it stays within ±1.5σ from target value, the
goal of 6-sigma will be guaranteed.
It is assumed that the relationship between the quality characteristic of product and
the quality characteristics of sub-assemblies A, B, C, D is X=∗ ∗
( where X is the
quality characteristic of the product, XA, XB, XC, XD are the quality characteristics of
components in sub-assembly A, B, C, D, respectively.) Characteristics XA, XB, XC, XD
are independent of each other.
Because parallel components in each sub-assembly are identical with the same
variance and mean of the associated quality characteristic, the reliability of product at
time t is written as:
R(t)=[1-(1-RA(t))3]*[1-(1-RB(t))2]*RC(t)*RD(t) (3.1)
where, RA(t) is the reliability of components A-1, A-2 and A-3 at time t; RB(t) is the
reliability of components B-1 and B-2 at time t; RC(t) and RD(t) are reliabilities of
33
components C-1 and D-1 respectively, at time t.
Rj(t) is the reliability of component of sub-assembly j, j=A, B, C, D at time t, and
according to function (2.26) and (2.27), the function of Rj(t) is written as:
Rj(t)=
exp[-(λj+
)t (j= A, B, C, or D); (3.2)
and
∂ = , , (j = A, B, C or D). (3.3)
where, β , λj,ξ are constants; σ is the variance of characteristics XA, XB, XC or XD;
μ , is the target value of Xj, j=A, B, C, D; μ , is the actual mean setting of
characteristics Xj, j=A, B, C, D; w is the warranty period of the product.
According to function (2.16), the external failure cost of the product is written as
follows:
Cw=-Cr ln[R(w)] (3.4)
where,
R(w)=[1-(1-RA(w))3]*[1-(1-RB(w))2]*RC(w)*RD(w) (3.5)
and Cr is the unit cost of repairing the product per failure.
As per function (2.23)and (2.24), the setup cost and production cost of a component
of sub-assembly j are ηj e and hjσ respectively, where ηj, τ , hj, α are
proportionality parameters. If sub-assembly j has nj components arranged in parallel, the
setup cost Csj and production cost Cpj of sub-assembly j (j=A, B, C, D) are:
Csj = njηje (3.6)
Cpj = njhjσ (3.7)
34
Thus, the total cost model for this product is as per following:
TC= Cw+∑Cp +∑Cs (j =A, B, C, D) (3.8)
Of the 12 different process combinations, combination A1, B1, C1, D1 is
researched first. They are listed in table 3.1 with the variances of process A1, B1, C1and
D1.
3.2.1 Stage 1. Finding the Target Values of Quality Characteristics
Taguchi’s robust design methodology is used to find the target values. This
methodology is based on the strategy that the expected loss of the product quality
characteristic should be minimized: adjusting the product quality characteristic mean to
the target value and minimizing the variance of it.
First, express mean and variance of X with the means and variances of XA, XB, XC,
and XD as it is described in section 2.2:
=∗
; gA(μ)= ∗. =
∗; gB(μ)= ∗
.
=∗
; gC(μ)= ∗
. =∗ ∗
; gD(μ)= ∗ ∗.
= ;hAB(μ)= . = ;hAC(μ)= .
= ∗
;hAD(μ)= ∗. = ;hBC(μ)= .
= ∗
;hBD(μ)= ∗. =
∗;hCD(μ)= ∗
.
=0; hAA(μ)=0. =0; hBB(μ)=0.
=0; hCC(μ)=0. =∗ ∗
; hDD(μ)= ∗ ∗.
μ=e(μ)+ ∑ ∑ h σ
=∗ ∗
+∗ ∗
*σ (3.9)
35
σ =∑ ∑ g g σ
=gA(μ)gA(μ)σ + gB(μ)gB(μ)σ + gC(μ)gC(μ)σ + gD(μ)gD(μ)σ
=∗
*σ +∗
*σ +∗
*σ +∗ ∗
*σ (3.10)
The strategy to get the target values by minimizing the variance of the product quality
characteristic and making the mean of product quality characteristic exactly equal the
target value:
Find µA, µB, µC and µD so as to
minimize
∗
*σ +∗
*σ +∗
*σ +∗ ∗
*σ
subject to
X0=∗ ∗
+∗ ∗
*σ ,
Then incorporate the data into the formulas above:
Find µA, µB, µC and µD so as to
minimize
∗*0.36+
∗*0.05+
∗*0.03+
∗ ∗*0.16
subject to
∗ ∗
+∗ ∗
*0.16=10
and
1≤ µA, µB, µC, µD≤ 20 (3.11)
By using MATLAB to solve the nonlinear program above, the target values of µA,
µB, µC, µD are obtained as: μ , =12.2943, μ , =4.5818, μ , =3.5491 and μ , =20. The
minimum value of product variance is 0.7539. This means that when the means of
36
processes A1, B1, C1, and D1 are set at 12.2943, 4.5818, 3.5491 and 20 respectively, the
product has lowest expected external failure cost.
3.2.2 Stage 2. Estimation of the Reliability Models Using Maximum Likelihood
Estimation
In order to use the reliability function to get the external failure cost, it is essential
to estimate the parameters in the reliability function. In real practice, the model is fitted
to real manufacturing data by using proper parameters. So sample data of a proper size
should be obtained to estimate the parameter. In this research, the method of maximum
likelihood estimation is applied to achieving the values of parameters. In real practice,
the sample data of manufacturing are obtained from the experimental operations of this
manufacturing process in order to estimate the proportionality parameters. The reliability
model needed to be estimated here in order to solve the robust design problem is:
Rj(t)=
exp[-(λj+
)t (j = A, B, C, or D)
where,
∂ = , , .
Parameters β , λj, andξ in above function are to be estimated. In order to estimate
the parameters, sample data of a proper size with the information on the quality
characteristics and failure times are needed. When sample quality characteristics and the
associated failure times are obtained, the method to get the values of β , λj, andξ is to
maximize the likelihood function as follows:
L(t, X; λj, βj, ξj) = f (t1,j, x1,j; λj, βj, ξj)* f (t2,j, x2,j; λj, βj, ξj)* …* f (tn, xn; λj, βj, ξj)
37
= ∏ f t , , x , ; λ , β , ξ (3.12)
where, n is the number of sample; x , is the quality characteristic of sample component
i in sub-assembly j; t , is failure time of sample component with characteristic x , ;
f(ti,j, xi,j; λj, βj, ξj) is the pdf presented in function (2.8) for item with quality
characteristic x , and failure time t , .
Then the procedures of achieving reliability functions of components in
sub-assemblies A, B, C and D are as follows:
1) Estimation of reliability function for the components in sub-assembly A
It is assumed that the means of the processes A-1 are set at 12.5, and the variance of
the process is known as 0.36. So a set of 15 random numbers with mean 12.5 and
variance 0.36 are obtained using MATLAB. These numbers are the quality
characteristics of the component samples. In order to ensure that there is a relationship
between quality characteristic and failure time, the failure times of these parts are
assumed to be generated by dividing random numbers generated by deviation from the
target values and variances. The quality characteristics and failure time values are shown
in the following table:
38
X T 12.8266 2.7613 13.6003 0.6280 11.1447 0.9454 13.0173 2.3879 12.6913 3.9592 11.7154 2.9088 12.2398 165.9793 12.7056 7.2768 14.6470 0.2048 14.1617 0.2302 11.6901 2.2336 14.3210 0.2270 12.9352 2.6120 12.4622 24.4055 12.9288 1.2484
Table 3.2 Quality Characteristics and Failure Times of Sample Components
Manufactured in Process A-1
Thus, the likelihood function of the component in subassembly A is written as
following:
L(t,x; λA,, βA,ξ ) = ∏ f t , , x , ; λ , β , ξ
= (λA+βA(x , - µt,A)2)ξAt , exp[-(λA+βA(x , - µt,A)2)t , ]
=(λA+βA*(12.8266-12.2943)2)*ξA* 2.7613 *exp[-(λA+βA*(12.8266-12.2943)2)*
2.7613 ]*(λA+βA*(13.6003-12.2943)2)*ξA*0.6280 ∗exp[-(λA+βA*(13.6003-12.
2943)2)* 0.6280 ]*……*(λA+βA*(12.9288-12.2943)2)*ξA*1.2484 ∗exp[-(λA+
βA*(12.9288-12.2943)2)* 1.2484 ]
Values of λA,βA,ξ can be obtained by maximizing L(t, x; λA, βA,ξ ). By using
39
MATLAB, it is found that when λA=0, βA=0.717,ξ =1.395, L(t, x; λA, βA,ξ ) is
maximized. Then the reliability function of sub-assembly A is obtained as:
RA(t)=. ∗ . exp[
. ∗ , .
. ∗ . *t . (3.13)
The estimations of reliability functions for the components in sub-assembly B, C
and D will follow the same steps described above.
2) Estimation of reliability function for the components in sub-assembly B
It is assumed that the mean of the process B-1 is set at 4.6, and the variance of the
process is known as 0.05. The quality characteristics and relevant failure time values are
shown in the following table:
X T 4.4873 31.9252 4.6734 54.0012 4.3036 9.6828 4.5626 1831.8685 4.5556 1130.0332 4.8245 10.0741 4.6727 26.7948 4.4595 47.4351 4.6577 131.9953 4.1082 2.7071 4.5329 273.7526 4.7371 26.7807 4.6618 68.0156 4.0234 1.9028 4.7328 13.2757
Table 3.3 Quality Characteristics and Failure Times of Sample Components
Manufactured in Process B-1
40
By using MLE, the proportionality parameters of reliability function are found as
λB=0, βB= 0.735,ξ = 1.274. So the reliability function of components in sub-assembly B
is:
RB(t)=√ . ∗ . exp[
. ∗ , .
. ∗ . *t . (3.14)
3) Estimation of reliability function for the components in sub-assembly C
The mean of the process C-1 is assumed to be set at 3.6, and the variance of the
process is known as 0.03. The quality characteristics and relevant failure time values are
shown in the following table:
X T 3.6296 39.6281 3.5094 233.9333 3.7170 20.0884 3.5373 3712.2953 3.6535 53.5628 3.3291 9.5517 3.6644 16.1199 3.7079 21.4019 3.8327 7.1318 3.4623 63.0588 3.3739 16.3903 3.7940 8.2916 3.5341 1586.2222 3.4660 66.9311 3.2844 3.8321
Table 3.4 Quality Characteristics and Failure Times of Sample Components
Manufactured in Process C-1
41
By using MLE, the proportionality parameters of reliability function are found as
λC=0, βC= 0.769,ξ = 1.244. So the reliability function of components in sub-assembly C
is:
RC(t)=√ . ∗ . exp[
. ∗ , .
. ∗ . *t . (3.15)
4) Estimation of reliability function for the components in sub-assembly D
The mean of the process D-1 is assumed to be set at 20.2, and the variance of the
process is known as 0.16. The quality characteristics and relevant failure time values are
shown in the following table:
X T 20.0898 93.3279 19.5219 2.5698 19.5001 2.0348 20.2730 6.7249 19.7756 17.5810 19.9632 620.1264 20.1410 26.1104 20.9543 0.6190 19.8335 14.0970 19.9446 202.5636 20.6441 1.9698 19.9550 295.3580 20.5603 2.0135 20.6790 1.7638 20.3671 3.8438
Table 3.5 Quality Characteristics and Failure Times of Sample Components
Manufactured in Process D-1
42
By using MLE, the proportionality parameters of reliability function are found as
λD=0, βD= 0.848,ξ = 1.186. So the reliability function of components in sub-assembly C
is:
RD(t)=√ . ∗ . exp[
. ∗ ,
. ∗ . *t . (3.16)
From above results, the R(w) be found as follows:
RA(w)=. ∗ . exp[
. ∗ , .
. ∗ . *w . (3.17)
RB(w)=√ . ∗ . exp[
. ∗ , .
. ∗ . *w . (3.18)
RC(w)=√ . ∗ . exp[
. ∗ , .
. ∗ . *w . (3.19)
RD(w)=√ . ∗ . exp[
. ∗ ,
. ∗ . *w . (3.20)
So the reliability of the product at the end of warranty period w is:
R(w)=[1-(1-RA(w))3]*[1-(1-RB(w))2]*RC(w)*RD(w) (3.21)
where RA(w), RB(w), RC(w), RD(w) are shown in functions (3.17), (3.18), (3.19) and
(3.20)
3.2.3 Stage 3. Finding External Failure Cost Function
It is assumed that the unit cost of repairing the product per failure Cr is 200. Then
according to function (2.22), the external failure cost function is written as follows:
Cw = -Cr ln[R(w)]
= -200 ln[R(w)]
43
= -200ln{[1-(1-RA(w))3]*[1-(1-RB(w))2]*RC(w)*RD(w)} (3.22)
where RA(w), RB(w), RC(w), RD(w) are shown in functions (3.17), (3.18), (3.19) and
(3.20).
3.2.4 Stage 4. Creating total cost model
To set up the total-failure-cost, the producing cost and set-up cost are still needed
besides the external failure cost. As per functions (3.6) and (3.7), the setup cost and
production cost for sub-assemblies A, B, C and D are
Csj = njηje (j = A, B, C or D)
and
Cpj = njhjσ (j = A, B, C or D)
It is assumed that: hA=20/3, α = 3, ηA =66, τA =0.5; hB=5, α = 1, ηB =83.5, τB
=0.6; hC=13, α = 1, ηC =172, τC =0.55; hD=11, α = 2, ηD =156, τD =0.53.
Sub-assembly A has three components, so nA= 3. Similarly, nB= 2, nC= 1 and nD= 1. The
warranty period of this product is assumed to be 4.
Then the total cost of one product manufactured by process combination A1, B1,
C1 and D1 is:
TC= Cw+∑ Cs +∑ Cp (j =A, B, C, D)
= -Cr ln[R(w)]+ ∑ n η e + ∑ n h σ
=-200{[1-(1-RA(4))3]*[1-(1-RB(4))2]*RC(4)*RD(4)}+198*e . ∗ , . / . +
167* e . ∗ , . / . +172*e . ∗ , . / . +156*e . ∗ , / .
+20*0.6-3+10*0.224-1+13*0.173-1+11*0.4-2 (3.23)
44
where,
RA(4)=. ∗ . exp[
. ∗ , .
. ∗ . *4 .
RB(4)=√ . ∗ . exp[
. ∗ , .
. ∗ . *4 .
RC(4)=√ . ∗ . exp[
. ∗ , .
. ∗ . *4 .
RD(4)=√ . ∗ . exp[
. ∗ ,
. ∗ . *4 .
3.2.5 Stage 5. Finding Optimum Mean Settings of Processes
As it is stated in previous chapter, the actual mean is allowed to shift plus or minus
1.5 deviations (±1.5σ) from the target value, and it is assumed at the beginning of this
chapter that the numeric ranges of mean settings of the processes are all assumed to be
(1,20). The limits of the mean settings of process A1, B1, C1 and D1 are:
11.3943≦ μ , ≦ 13.1943; 4.2464≦ μ , ≦4.9172; 3.2893≦ μ , ≦ 3.8089; and
19.40≦μ , ≦20.
The optimum mean settings of the process A1, B1, C1 and D1, which are μ , ,
μ , , μ , and μ , respectively, can be obtained by minimizing the total cost function
TC:
Minimize
TC =-200{[1-(1-RA(4))3]*[1-(1-RB(4))2]*RC(4)*RD(4)}+198*e . ∗ , . / . +
167* e . ∗ , . / . +172*e . ∗ , . / . +156*e . ∗ , / .
+20*0.6-3+10*0.224-1+13*0.173-1+11*0.4-2
45
where,
RA(4)=. ∗ . exp[
. ∗ , .
. ∗ . *4 .
RB(4)=√ . ∗ . exp[
. ∗ , .
. ∗ . *4 .
RC(4)=√ . ∗ . exp[
. ∗ , .
. ∗ . *4 .
RD(4)=√ . ∗ . exp[
. ∗ ,
. ∗ . *4 .
Subject to
11.3943≦μ , ≦ 13.1943;
4.2464≦μ , ≦4.9172;
3.2893≦μ , ≦ 3.8089;
19.40≦μ , ≦20; 3.24
Here, Genetic Algorithm in Matlab is used to solve this non-linear programing
problem. The minimum value of TC is found to be 911.3918,
atμ , =11.635,μ , =4.2466, μ , =3.289, andμ , =19.647. That is, when the mean of
process A1 is set at 11.635, the mean of process B1 is set at 4.2466, the mean of process
C1 is set at 3.289 and the mean of process D1 is set at 19.647, the total cost of
manufacturing this product under process combination A1, B1, C1and D1 is minimized.
3.2.6 Stage 6. Obtaining the Optimum Process Combination and Mean Settings
The optimum mean settings for all the other 11 combinations can be obtained by
repeating stage 1 to 5 for these combinations. The results are summarized in the
46
following table:
Combinations Optimum Mean Settings Minimum Total Cost
A1,B1,C1,D1 µa,A=11.635,µa,B=4.2466, µa,C=3.289, µa,D=19.647 911.3918 A1,B1,C2,D1 µa,A=7.9738,µa,B=2.8828,µa,C=6.971, µa,D=19.674 1101.9 A1,B2,C1,D1 µa,A=8.7161,µa,B=7.4456,µa,C=2.455, µa,D=19.678 1050.5 A1,B2,C2,D1 µa,A=5.9320, µa,B=5.07, µa,C=5.279, µa,D=19.677 1222.2 A2,B1,C1,D1 µa,A=13.6028, µa,B=3.92, µa,C=3.028, µa,D=19.676 914.9699 A2,B1,C2,D1 µa,A=9.324,µa,B=2.6552,µa,C=6.3938, µa,D=19.676 1091.9 A2,B2,C1,D1 µa,A=10.22, µa,B=6.6094, µa,C=2.255, µa,D=19.675 933.2176 A2,B2,C2,D1 µa,A=6.813,µa,B=4.6506,µa,C=4.8446, µa,D=19.672 1142 A3,B1,C1,D1 µa,A=15.286,µa,B=3.668,µa,C=2.8408, µa,D=19.675 916.7199 A3,B1,C2,D1 µa,A=10.581,µa,B=2.4762,µa,C=6.039, µa,D=19.676 1118.9 A3,B2,C1,D1 µa,A=11.505,µa,B=6.4206, µa,C=2.111, µa,D=19.677 1028.9 A3,B2,C2,D1 µa,A=7.791, µa,B=4.344, µa,C=4.1082, µa,D=19.677 1164.9
Table 3.6 Table of Results
It can be seen obviously from the table that when combination A1, B1, C1, D1 is
chosen and the means are set as µa,A=11.635, µa,B=4.917, µa,C=3.289, µa,D=19.647 the
global minimum total cost can be achieved as 911.3918.
The next section of this chapter will employ the data produced in this section to
analyze the effects some important factors have on the total cost.
3.3 Model Analysis
The output of total cost model presented in the previous chapters is affected a lot by
the deviation of mean from the target value, the process variance and the length of
warranty period. So the model analysis is needed to achieve the knowledge of effects
these important factors have on the total cost.
47
In this section, process combination A2, B1, C1, D1 from the previous numerical
example is taken as an example to perform the model analysis. Only the variable
analyzed is changed while all the variables are kept constant.
3.3.1 The Effect of Deviation of Mean from the Target Value on Total Cost
The deviation of mean from the target value affects the quality lost and setting-cost
greatly. That is, the larger the deviation of mean from the target value is, the less reliable
the product is, which leads to higher external failure cost. The larger the deviation of
mean setting from the target value is, the lower the setup cost is. The following figures
show how the deviation of mean from the target value affects the total cost:
|Xa,A-Xt,A|
Figure 3.2 Effect of |Xa,A-Xt,A| on Total Cost
910
915
920
925
930
935
940
945
0 0.2 0.4 0.6 0.8 1 1.2
Total Cost
48
|Xa,B-Xt,B|
Figure 3.3 Effect of |Xa,B-Xt,B| on Total Cost
|Xa,C-Xt,C|
Figure 3.4 Effect of |Xa,C-Xt,C| on Total Cost
910
920
930
940
950
960
970
980
990
0 0.1 0.2 0.3 0.4 0.5 0.6
900
920
940
960
980
1000
1020
1040
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Total Cost
Total Cost
49
|Xa,D-Xt,D|
Figure 3.5 Effect of |Xa,D-Xt,D| on Total Cost
It can be seen from these figures that there exists certain values of mean settings
for processes A2, B1, C1, D1 which can minimize the total cost. These values are called
the optimum mean settings in this research.
3.3.2 The Effect of Process Variance on Total Cost
The increase of process variance leads to a higher warranty cost but results in
lower production cost. The effect of variance on the total cost is reflected in the
following figures:
910
915
920
925
930
935
940
945
950
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Total Cost
50
σA
Figure 3.6 Effect of σA on Total Cost
σB
Figure 3.7 Effect of σB on Total Cost
0
500
1000
1500
2000
2500
3000
3500
4000
0 1 2 3 4 5 6
0
200
400
600
800
1000
1200
1400
0 0.2 0.4 0.6 0.8
Total Cost
Total Cost
51
σC
Figure 3.8 Effect of σC on Total Cost
σD
Figure 3.9 Effect of σD on Total Cost
From the figures above, it can be seen that if the appropriate variances are chosen
while other variables are kept constant, the minimum total cost can be achieved. But in
0
200
400
600
800
1000
1200
1400
1600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 0.5 1 1.5 2 2.5 3 3.5
Total Cost
Total Cost
52
this research, because it is assumed that only limited number processes are available to
produce the components, the global optimum value of variance usually cannot be
obtained.
3.3.3 The Effect of the Length of Warranty Period on Total Cost
Usually, the longer the warranty period is, the higher the warranty cost and total
cost are. The effect of the length of the warranty period on the total cost is illustrated in
the following figure:
Warranty Period
Figure 3.10 Effect of Warranty Period on Total Cost
It can be seen from the above figure that the model follows the rule that the longer
warranty period results in higher cost.
The model analysis above illustrated that process mean setting, variance and length
of warranty period are all essential factors that affect the total cost. So in manufacturing
practice, it is critical to consider these factors. The model analysis also helped to validate
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 5 10 15 20 25 30
Total Cost
53
the model. The model captures the effects of the process mean, variance, and warranty
period on the costs.
3.4 Summary and Conclusion
This object provided further information to support a thorough understanding of the
total cost model. The first section of this chapter provided an example of how to use the
total cost model to solve the robust design problem. The numerical example illustrated
in detail, how the model is used to solve robust design problem and what data is needed
to use it. The second section of this chapter showed that the deviation of mean from
target value, process variance and warranty period have great effects on the total cost.
The study of these effects can help to get more detailed understanding of the total cost
model and validate the model. What’s more, it can facilitate future researches to expand
this model.
54
Chapter 4
Conclusion and Future Research
4.1 Concluding Remarks
This thesis developed a total cost model composed of external failure cost, setup
cost and production cost. The external failure cost function developed in this thesis was
developed from reliability function using the concept of Taguchi’s loss function. The
reliability function could be created with respect to process mean and variance, so the
external failure cost was formulated with respect to process mean and variance. Because
setup cost and production cost are also functions of process means and variances, the
total cost model was developed as a function of process means and variances. In this
thesis, the total cost model was further applied to the solution of robust design problem.
With the process variances known, the optimum setting of process means could be
obtained by minimizing the total cost. If different process combinations are available,
the optimum process combination could be found by comparing the minimum total cost
in each combination.
One of the contributions of this thesis is that it provides a way to find a balance
between high quality and low manufacturing cost. This balance is achieved by finding
the optimum parameter setting that minimizes the total cost composed of external failure
cost, setup cost and production cost. Because every item in the total cost model is
quantifiable, another contribution of this thesis is that it provides an efficient method to
quantify the cost of manufacturing a product.
55
4.2 Future Research
There are several opportunities that this research can be extended. First, only the
case of nominal-the-best quality characteristics was considered in this thesis. So the
problem can be extended to consider the cases of larger-the-better and smaller-the-better
characteristics. Second, when the external failure cost model was developed from the
reliability model, it was assumed in this thesis that all failed products during warranty
period will be serviced under the minimal repair policy. The case of failed products
under replacement policy or under mixed policy should be considered in future research.
56
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