ROBUST DESIGN INCORPORATING COST MODEL FOR SYSTEM ...

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.

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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.

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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

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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

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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

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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

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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

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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

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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.

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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).

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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

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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

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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

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(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

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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.

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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

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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

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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.

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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.

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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

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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.

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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

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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.

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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.

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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

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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.


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


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



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


Manufactured in Process C-1

41


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



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


Manufactured in Process D-1

42


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

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.


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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