applying some quality control charts to study the strength of building concrete blocks and clay...

7/21/2019 Applying some quality control charts to study the strength of building concrete blocks and clay building brricks.pdf

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QUALITY CONTROL CHARTS / SHERKO HASSAN ABDULRAHMAN

Iraqi Kurdistan Region

Presidency of the Ministerial Council

Ministry of Higher Education and Scientific Research

University of Sulaimani

College of Basic Education

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CERTIFICATION OF THE SUPERVISOR

I certify that (SHERKO HASSAN ABDULRAHMAN) was prepared this

thesis (Applying Some Quality Control Charts to Study the Strength of Building

Concrete Blocks and Clay Building Bricks) under my supervision at the

Department of Statistics (College of Basic Education/University of Sulaimani) in

partial fulfillment of the requirements for the degree of Master of Science in

Statistics.

Signature

SupervisorDDrr..TTAALLIIBBSSHHAARRIIFFJJAALLIILL

AAssssiissttaannttPPrrooffeessssoorr

Date: 9 / 12 / 2007


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LLIINNGGUUIISSTTIICCEEVVAALLUUAATTIIOONNCCEERRTTIIFFIICCAATTIIOONN

I hereby certify that this thesis has been checked by me after indicating all

the grammatical and spelling mistakes; the thesis was given again to the candidate

to make the adequate corrections. After the second reading, I found that the

candidate corrected the indicated mistakes. Therefore, I certify that this thesis is

free from mistakes.

Signature

Name: Talla Abdullah Rashid

Date: / /


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RREECCOOMMMMEENNDDAATTIIOONNOOFFTTHHEECCHHAAIIRRMMAANNOOFFSSCCIIEENNTTIIFFIICCCCOOMMMMIITTTTEEEE::

In view of the available evidence and recommendations, I forward

this thesis for debate by the examining committee.

Signature:

Name : Dr. Jawad N. Hussain

Chairman of the College committee

on post-Graduate studies

Date 3 / 5 / 2008


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MMEEMMBBEERREEXXAAMMIINNIINNGGCCOOMMMMIITTTTEEEE

We certify that we have read this thesis as an Examining Committee,

examined the student (Sherko Hassan Abdualrahman) in its contents, and that in

our opinion it is adequate with (APPLYING SOME QUALITY CONTROL CHARTS TO

STUDY THE STRENGTH OF BUILDING CONCRETE BLOCKS AND CLAY BUILDING

BRICKS) as a thesis for the degree of Master of Science in Statistics.

Signature:

Name: Prof. Assist. Dr. Shawnm A. Muhayddin

Chairman

Date: 21 / 4 / 2008

Signature:

Name: Prof. Assist. Dr. Abdulrahim K. Rahi

Member

Date: / /

Signature:

Name: Prof. Assist. Dr. Samir M. Khdir

Member

Date: / /

Signature:

Name: Prof. Assist. Dr. Talib S. Jalil

Supervisor / Member

Date: / /


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APPROVED BY THE COLLEGE COUNCIL

The council of the (college of Basic Education /University of Sulaimani) approvesthe decision arrived at: / / by ( ) the examining committee.

Signature:

Name: Dr. Ezzadin N. Baban

Assist. Professor

Dean of the college of Basic Education

Date: 4 / 5 / 2008


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--MMyycchhiillddrreenn::


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I would like to present my thanks to my Supervisor, (DDrr..TTaalliibbSShhaarriiffJJaalliill)

who gave me valuable advice and continuous guidance through preparing this

thesis.

I express my respect and appreciation to the council of higher education and

scientific research, the president of the University of Sulaimani, the dean of

College of Basic Education, the chairman of the department of (Mathematic and

Computer / College of Basic Education), and the entire staff and personnel of the

College of Basic Education.

I would like also to express my appreciation to the staff of the College of

Engineering / University of Sulaimani, especially Construction and Building

department, and all the staff of concrete and building material laboratory.

I express my respect and appreciation to all the staff of Constructional

Laboratory of Sulaimani / Ministry of Reconstruction and Housing, specially the

staff of concrete department for their assistance and valuable information about

BCB and CBB.

I would like to express my pure thanks and enormous gratitude to my

family, because really without them I could not prepare this thesis.

SHERKO


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Lists of Abbreviations

SYMBOLS Details

ARL Average Run Length

BCB Building Concrete Blocks

CBB Clay Building Bricks

Cm Centimeter

CUSUM Cumulative Sum Chart

d Distance

E Expectation

eq Equation

EWBA Exponential Weighted Bayesian Average

GMA Geometric Moving Average

k Reference Value of CUSUM

LB Lower Bound

LCL Lower Control Limit

MA Moving Averagem Subgroup Number

M Mean

Max Maximum

Min Minimum

MR-chart Moving Range-chart

QC Quality Control

R Range

R Average of range

SLB Standard Lower Bound

T TargetV Vertical

w Weighted

x Observed value

x Sample average

X Average of sample averages

0 Reference value

B Bayesian estimate

r.s random sample

S Sample standard deviation

xS Standard deviation of subgroup average

S Average of sample standard deviations


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Abstract

A considerable amount of attention is paid to the concern of the quality

control by many researchers and engineers since the quality is considered the main

factor in many fields, such as industry, agriculture, health, education, and some

other scientific fields. Therefore, many companies and industry institutions have

tried to make their product's quality meet the standard specification, and to increase

the consumers for their products.

As it is obvious, any problem in the quality of the product may be resulted

from the quality of the raw materials in the process of production, the technical and

the problem in the machinery and all these affect the quality of the product. That is

why, the statistical quality control is necessary to detect changes in the behavior of

these processes.

As there is a great demand on BCB and CBB, that is why constructed many

of industries, such as BCB and CBB factories, the factories produce a large amount

of products; indeed, this may causes some problems and defects, in their products.

To controlling quality of their products and testing the strength of materials

or quality characteristics, established many of laboratories.

In this thesis, tested the strength of BCBs and CBBs, and use some

classical and new control charts, also the new suggested Exponential Weighted

Bayesian Average control chart (EWBA-chart), to know that these products meet

standard specification, such as International, Iraqi and Local standard

specifications.

We also have made a comparison between control charts, to know which of

them is the most sensitive and adequate in identifying changes or shifts in the

production process and reduce error in decision-making.


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CONTENTS

No. Objects page

Chapter 1: Quality Control and Basic Definitions1.1 Introduction 1

1.2 Historical Background 3

1.3 The problem of the thesis 6

1.4 The aim of the thesis 6

1.5 Some Concepts Related to Quality Control 7

Chapter 2 : Control Charts

2.1 Quality Control Charts 10

2.1.1 Variable Quality Control Charts 112.1.2 Standard Control Charts 11

2.1.2.1 Shewhart Control Charts 11

2.1.2.2 Non-Shewhart Control Charts 15

2.1.3 Bayesian Process Control 19

2.1.4 EWBA-Chart 22

Chapter 3: Application of Some Control Charts on the Real Data

3.1 Introduction 24

3.2 Collection Data 253.3 Quality Control Charts interprets 25

Application 27

3.4 X-bar Charts 27

3.5 R- Charts 30

3.6 Geometric Moving Average Control Chart 35

3.7 Bayesian-Charts 40

3.8 Construction EWBA-Charts 47

3.9 Comparison between x-bar chart and Bayes-chart 52

3.10 Comparison between GMA-chart and EWBA-chart 52

Chapter 4: Conclusions and Recommendations

4.1 Conclusions 54

4.2 Recommendations 56

References 57

Appendix 60


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CCHHAAPPTTEERR11____________________________________________________________________________________________________________________________________

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1.1. Introduction:(1)(3)(4)(8)(24)

Quality control is the use of statistical techniques and activities to achieve,

sustain, and improve the quality of a product or service, and indirectly to improve

the technology to meet consumer requirements.

Quality control has become a major concern in todays competitive

industrial environment. Therefore, quality has become one of the most important

consumer decision factors in the selection among competing products and services.

The quality of product is determined by how a product meets the needs of

consumer and the ideal quality would be zero-defects or defectives. A defect or

nonconformity is the unit of a product that dose not satisfy consumers not conform

with the specification. The production process has variation in the quality of the

output. There are usually two types of variation in the quality of the output of the

process, which are stochastic variation and non-stochastic variation, which is due

to some assignable causes.

The variation can be traced to a particular problem or assignable cause (e.g.

human error, a problem with raw material, or machine failureetc).

Assignable causes occur at unpredictable times because of having a non-

stochastic nature, and the aim of the control chart is to detect them soon and

correction should be made after that.

The use of control chart became widespread after World War II. This was

due to the importance of maintaining quality production during that period.


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Dr. Shewhart and many researchers were attending to control chart, improve,

and modify them to increase the sensitivity of these charts to detect out of control

signal. Quality control is receiving increasing attention as a management tool, by

which important characteristics of a product are observed, assessed, and compared

with some types of standards.

By using quality control charts, scientific valid methods and sticking to

standards specification, the following objectives can be achieved:

1. Improvement of the quality of products

2. Reducing the cost

3. Safety and health for customers and producers.

4. Conservation for consumer and producer from commercial skulduggery.

5. Increasing the chances of producing and marketing.

6. Reducing the defects or defectives.

This thesis begins with an introduction and the basic concepts about quality

control, and a historical background about the brief history of quality control, and

definitions of some concepts related to the quality control, in chapter one.

Chapter two contains theoretical side about some classical control charts,Bayesian control chart and apply a new suggested control chart based on the

exponential weighted Bayesian average (EWBA).

Chapter three contains the application of some control charts on the real

data. Such as classical control charts, Bayesian control chart, and new control chart

(exponential weighted Bayesian average), and contained comparison between

some control chars to help us to obtain a good result in decision-making about the

production process.

Chapter four contains some conclusions, and recommendations for the future

study.


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1.2. Historical Background:

Statistical quality control comparatively new, the science of statistics itself

belongs to two to three centuries ago. Moreover, it is the greatest development has

taken place during the 20th

century.

The notion of using sampling and statistical analysis techniques in a

production started in 1920and applied effectively to quality control as a result of

the development of sampling theory.

The first who applied the newly discovered statistical methods to the

problem of quality control was Walter A. Shewhart (18911963) in the BellTelephone Laboratories. He issued a memorandum on (May 16, 1924) that

featured a sketch of modern quality control chart to control and detect non-

stochastic variation in the process of a production. For the first time statistical

methods applied to the problem of quality control by Walter Shewhart and he kept

improving and working on this scheme, in 19261927 published three papers on

the (Quality Control and Control chart). Therefore, for the first time after

publishing these papers in the Journal of (American statistical society), he used

control chart in Bell Telephone Laboratories. This marked the beginning of

statistical quality control.(3)(12)(19)

In 1931 W.A. Shewhart published a book on statistical quality control under

the title of (Economic Control of Quality of Manufactured Product), published by

(Van Nostrand in New York). He gave some lectures on statistical methods in

production and control charts at the University of London in 1932, and invited

Shewhart by Deming in 1938, to present seminars on control charts at the U.S.

Department of Agriculture Graduate School.(7)(16)

From 1942 to 1946, the training courses on statistical quality control were

given to industry. And more than fifteen quality societies are formed in North


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America, such as (American society for quality control) formed on 16 / 2 /1946.

This organization, through its publication, conference and training session, has

promoted the use of quality control for all types of production and service.(3)(4)

In 1946, the International Standards Organization (ISO) was founded and the

Japanese Union of Scientists and Engineers (JUSE) was formed.(12)(30)

In 1950 W. Edward Deming (19001993), who had worked with Shewhart

at Bell Telephone Laboratories, gave a series of lectures on statistical method to

Japanese Engineers and on quality responsibility to top management.(3)

In addition, Joseph M. Juran was invited by the Japanese to give lecture on

quality management and improvement in 1954.(3)(12)

The British statistician, (E.S.Page) in 1954, when he was in the statistical

laboratory at the Cambirg University, introduced a new control chart in the name

of CUSUM (Cumulative Sum) control chart. This is a draw of Cumulative Sum of

error of observations. In 1959 statistician, Barnard introduced a V-Mask, for

making the decision with CUSUM-chart.(12)(23)

The statistician S.Roberts introduced 3-new control charts. The first was

based on Moving Average (1958), the second was the Moving Range (1958), andthe third was based on Geometric Moving Average (1959). The decision about the

production process for these three charts are depending on the past data as well as

the current data, therefore the decision is not based on a single plotted data

.(11)(18)(20)

Van Dobberede Bruyn (1968), tabulated simulation results of Average Run

Length (ARLs) for (CUSUM) procedures with different decision intervals and

reference values.(14)(25)

Brook and Evans (1972), used Markov chain to approximate the moment

and percentages points of Run Length distribution for integervalued (CUSUM)

procedures.(12)(14)


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Lucas (1982) introduced a new control chart, which was a composite from

shewhart control chart and CUSUM control chart. This chart was sensitive to

detect small and large changes at the same time.(33)

Bourke (1991), proposed the Run-Length control chart, based on the plotting

of the number (RL) of conforming items between successive nonconforming items.

(14)

AL-RASSAM (1996) introduced (Bayesian Two Dimensional Control Chart

for Monitoring Production Process).(33)

Al-Zbedy (1997), constructed two new charts (Bayes chart for fraction

defectives and defects).(29)

Lucas (1998), and Olwell had been studied the steps of CUSUM control

charts to control quality characteristics of output of a process.(12)

Hamad (2003), proposed the use of Beta-content tolerance intervals as the

basis for control limits and more precisely probability control limits.(12)

Yeh (2004), and others suggested the new control charts for (CUSUM

Control Chart) in the name of (CUSUM M-Chart) to detect small changes.(22)

Al- Rawy (2004), used characteristic function and applied quality control onthe birth of premature children.

(35)

Rashid (2006), used multivariate control chart for two characteristics of

cement in Taslujah cement factory in Sulaimani.(33)

Nainawa N. Dawd (2006), studied Process Capability Indices by Simulation,

using process capability in order to asses the real capability of the production

process whose distribution is different from the normal distribution and used

Quality Control Charts to determine specification limits.(27)

Dler Al-Debagay used (Bayesian Decision Technique for Ala Pepsi Soft

Drinks Company in Sulaimani).(28)


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1.3. The problem of thesis:

In undertaking any research, there must be a problem or some problems.

A quality control problem may be an error or mistake, the correction of

which requires effort and expense.

In general, there are two problems in product quality:

- Deviations from target line specifications.

- Excessive variability around target line specifications.

This thesis tries to study the tolerance strength of Building Concrete Blocks

(BCB) and Clay Building Bricks (CBB). Therefore, these two are used more than

any other materials in building houses, schools, hospitals, and in general

construction and reconstruction process.

This is the reason behind increasing the factories of building concrete blocks

and Clay Building Bricks. To control the quality of these products is very hard and

shows the defects in the quality of the product. In all production process, we need

to monitor the extent to which our products meet specifications. This is why, the

study of the variation in the data of a process helps us to analyze the process,

evaluate its effectiveness in producing highquality, low cost, output and adjust,

and improve the process to achieve better results.

1.4. The aim of thesis:

The prime objective of this thesis is to:

Study state of the quality strength of the Building Concrete Blocks (BCBs)

and Clay Building Bricks (CBBs) for some factories by using different control

charts, including a new suggested Exponential Weighted Bayesian Average-chart

(EWBA-chart) to see if these products meet the international, Iraqi and Local

specification standard.


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CCHHAAPPTTEERR22__________________________________________________________________________________________________________________________________

CCoonnttrroollCChhaarrttss

2.1. Quality Control Charts:(3)(12)(13)

Control charts are widely used in industry as a tool to monitor output

process. Different types of control charts may be used depending upon the type of

the data that is measured or computed from samples. A control chart always has a

centerline for the average value of the quality characteristic, and two other

horizontal lines, an upper line called the upper control limit (UCL), and a lower

line for the lower control limit (LCL), are shown on the Fig. 2-1.

The control limits are chosen because if the process in control means that all

of the plotted points will fall between them, and no action is necessary. If a point

falls outside one of the control limits, this would be an indication that the process

is out of control. Then, an action should be taken, and this is to search for

assignable causes (or non-stochastic cause or causes).

Figure 2-1 General control chart

The control limits usually are set at (3 ) from the centerline.

(UCL, LCL) = T 3

Sample Number

Quality

Centerline

Upper Control Limit

Lower Control Limit


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where:

T = target value (centerline)= standard deviation

The quality control charts are divided into two main types, which are

variable quality control chart, and attribute quality control chart. The termsvariable andattributeare associated with the type of data being collected from the

production line of the process.(13)

2.1.1. Variable Quality Control Charts:(12)

A quality characteristic that is measured on a numerical scale is called a

variable; and includes dimensions such as length or width, temperature, time, and

volume (can be measured in fraction or decimals).

When dealing with a quality characteristic that is of variable type, it is

usually preferable to monitor both the mean level of the quality characteristic and

its variability.

The variable quality control charts are divided into two types:

(i). Shewhart variable quality control charts

(ii). Non-Shewhart variable quality control chartsThis thesis has divided control charts into two main parts:

(a) Standard Control Charts

(b) Bayesian process Control Charts

2.1.2. Standard Control Charts:

2.1.2.1. Shewhart Control Charts:

The familiar shewhart control charts are:

. X-chart (Average-chart or Meanchart)

. R-chart (Rangechart)

. -chart (Standard deviation chart)


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1. X-Chart: (9)(12)

X -chart for variables data, (data that is both quantitative and continuous in

measurement, such as a measured dimension or time).

The aim of using the x -chart is to control the mean level of the output of a

process.

The point plots on this chart are the average (x ) of subgroups data, calculate

from:

n

xx

n

1ii

j

=

=

The centerline of the -chart is ( X ) calculate from:

m

x

X

m

1j

j=

= ... (2-1)

The control limits of the -chart are established at three-standard deviation

(3 ) from the target value, and calculate by the formulas:

xx3XUCL += ... (2-2)

xx3XLCL = ... (2-3)

In practice the calculations are simplified by using (for n < 10)

RAXUCL2

+= .. (2-4)

RAXLCL2

==== ... (2-5)

where:

2d

R =

R = average of the subgroup range


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But for (n 10) the control limits are:

1

AXUCL ++++==== ... (2-6)

=1

AXLCL ... (2-7)

where:

= average of the subgroups (or samples) standard deviations

The constants (A1)and (A2) are quality control factors given in the table (9)

in the appendix.

2. R-Chart (Range Chart):(12)(3)

The R-chart is developed from the range of each subgroup data. The aim of

using Range chart is to control the variation in the output of a process. The point

plots on this chart are the range (R) of subgroup data, calculate from:

Rj= Max{x1, x2, ,xn} - Min{x1, x2, ,xn} .... (2-8)

When subgroup sizes are less than (10), both ( R-chart and s-chart) will

graphically portray the same variation, however, as subgroup sizes increase to (10)

or more, extreme values have an undue influence on the R-chart. Therefore, at

larger subgroup sizes the s-chart ( -chart) is used.(3)

If the sample size is relatively small, (n < 10) it is preferable to use R-chart

The centerline for the R-chart is (R ), calculate from:

m

R

R

m

1jj

= =

.... (2-9)

The control limits for the control chart, calculate by the formulas:

R3RUCL += .... (2-10)

R3RLCL = .... (2-11)

In practice, the calculations are simplified by using the formulas:

RDUCL4

==== .. (2-12)


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RDLCL3

==== .. (2-13)

Remark:

In general the (R-chart) is not symmetric around target line, because for (n =

2, 3, ..., 6), the D3= 0.

The constant D3and D4are the quality control factors given in the table (9)

of the appendix.

3. -Chart: (3)(12)

The aim of using (sigmachart) is to control the variation of the output in the

quality of a process.

The point plots on this chart is () or standard deviation of sample.

The centerline of sigma-chart is ( ) and calculated as:

m

j

m

1j

= =

If is unknown, then it would be estimated from the data.

2d

R =

The action lines are:

4

B3UCL ====++++==== ... (2-14)

== 3

B3LCL ... (2-15)

The constants d2, B3and B4 are quality control factors given in the table (9) of

the appendix.

2.1.2.2. Non-Shewhart control charts:

Many other variable control charts (classical) introduced by the statisticians

denoted as non-shewhart charts. The reasons behind introducing these charts

mainly are;

i. The decision in using Shewhart charts is based on a single plotted point.


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ii. Shewhart charts are not very sensitive to small changes in the process

parameters.

Popular non-Shewhart control charts are:

1- Cumulative Sum Control Scheme (CUSUM-scheme).

2- Moving Average Control Chart (MA-chart).

3- Moving Range Control Chart (MR-chart).

4- Geometric Moving Average Control Chart (GMA-chart).

1. Cumulative Sum Control Scheme (CUSUM):(25)The Cumulative Sum Control Scheme was first introduced by Page (1954).

In fact this is not like Shewhart or other control charts in the sense that it has

central line and control limits, this is why, in literature it is called scheme, but it isjust a graph of (n, Sn) as given in fig. (2-2)

Figure 2-2 Cumulative Sum Control-Scheme

The aim of using this chart is to control the mean level of output of a

process. Cumulative Sum Control Scheme is more effective than Shewhart control

charts for detecting small process shifts. Consider a CUSUM chart with reference

value (k), and observations: x1, x2,, xm .

Calculate CUSUM values by the formula:

Sn

Observation

(n,Sn)

(1, S1)


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nkx)kx(Sj

n

1jjn ==

=

... (2-16)

where:

k is constant

Sn= accumulative of differences from the reference value.

For taking the decision in CUSUM-Scheme used:

1-V-Mask.

2- Decision Interval.

If the graph of CUSUM crosses one of the arms of the V. mask, then an

action should be taken, and this would be to search for assignable causes.

Figure 2-3 V-Mask

2. Moving Average Control Chart (MA-chart):(12)

The aim of using this chart is to control the mean level of the output of a

process. It is generally used for detecting small shifts in the process mean. The

formula for calculating moving averages of order (k); for x1,x2,x3,...,xN is:

k

x...xxx

j)2k(j)1k(jk

)1k(j

+++=

, j= k, k+1, ,N

Upper arm

Lower arm

Vertex

Origin

d ((

Observation

Sn


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where:

k 1

The centerline of the moving average chart is:

1kn

x...xxX

)k(

1kN

)k(

2

)k(

1k

1+

+++

=

+

... (2-17)

The variance of the moving average control chart is:

kkk

)x(V)x(V

k

ii

)k(

2

2

2

2

1

=

==

=

The control limits for moving average chart are:

k

2

k

X

k

X

k

2

k

X

k

X

RAX3XLCL

RAX3XUCL

kk

kk

==

+=+=

... (2-18)

where:

kk

X

=

The point plots on this chart are the moving average (kx ).

Observe that the decision is not based on a single plotted point as in

Shewhart charts. Different moving averages have some common observations.

3. Moving Range Control Chart (MR-chart): (3)

The aim of using this chart is to control the variation of the output of a

process (or to control standard deviation of a process). The formula for calculating

moving range of order (k), for x1, x2, x3,...,xN is a group of observations:

}x,...,x,x{Min}x,...,x,x{MaxR i)2k(i)1k(ii)2k(i)1k(ik

)1k(i = , i=k, k+1,, N

The centerline of this chart is:


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1kN

RR

k

j

1kN

1j

++++====

++++

====

The action lines of this chart are:

k

3RR

k

4RR

RD3RLCLRD3RUCL

k

k

==

=+=

... (2-19)

The point plots on this chart is the moving range ( Rk).

One canobserve that here also the decision is not based on a single plotted

point because different moving ranges have some common observations.

4. Geometric Moving Average Control Chart (GMA-chart): (12)(8)(6)

By Roberts (1959) introduced Geometric Moving Average Control Chart.

The aim of using this chart is to control the mean level of the quality of the

output of a process.

The GMA-chart is sensitive to detect a small change of the quality of a

process.

The point plots of this chart are GMA (Zj), which is calculated as:

Zj = ( 1 - )Zj-1 + jx ... (2-20)

where the weight parameter () ( 0 <


28/44


The centerline of GMA-chart is:

T = Z0 = target value (if predetermined).

And

XT ==== = Z0 (if it is not predetermined)

and then:

n)2(Z

= ... (2-21)


z0

z0

3ZLCL

3ZUCL

=

+= ... (2-22)

2.1.3. Bayesian Process Control

Introduction(5)(15)(17)(30)(31)(34)

Bayesian inference method is distinguished from classical statistical

inference methods by the fact that the Bayesian inference method can be applied to

every type of problems and often shapes the optimal method to use. Statisticians,philosophers, and computer scientists have proposed numerous guides to reliable

inference. In addition, Bayesian inference method judges upper most as the most

versatile and often the most effective set of methods to use for a broad range of

inductive inference problems.

Bayesian inference is statistical inference in which evidence or observations

are used to update or to newly infer the probability that a hypothesis may be true.

The name "Bayesian" comes from the frequent use of Bayes'theorem in the

inference process. Bayes' theorem was derived from the work of the Reverend

Thomas Bayes (1702-1761).


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Bayesian Parameter Estimation :(21)

A Bayes estimator combines information from a prior probability )(P ,

defines that the distribution of ( ) before sampling, and a likelihood function;

L( ) = )|x,...,x,x(Pn

21

defines that joint distribution of (n) random

observation (x1,x2,,xn) from population. To arrive at posterior probability

distribution and;

)x,...,x,x|(Pn21

defines that the distribution of ( ) after sampling, and

summarizes all the information about ( ).

where:

)|x,...x,x(p)(LFunctionLikelihoodn21

==

)|x(pi

n

1i=

=

, i = 1,2,,n , (if xs are independent)

and:

The prior distribution of is p() [I( )]1/2

= Jefferys standard non-

informative prior

In the Bayesian parameter estimation formula below, x1,x2,,xnstands as a

random sample got from a population with pdf f(x; ).Posterior probability prior probability likelihood function

where:

is read as proportion to.

)|x,...,x,x(P)(P)x,...,x,x|(Pn21n21

Then Bayesian Estimate is:

)x,...,x,x|( n21B (Under quadratic loss function)

(2-23)

Bayesian Estimation for mean of Normal Distribution:

The normal distribution is central to inference and other fields of statistics.


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To obtain Bayes estimate for the mean of N ( , 2 ) where 2 is known.

Bayesian estimate ( B ):

),(N~)(p2

00

=

n

1i

2

i2})x(

2

1{EXP)(L

)(L)(p)x|(p

=

n

1i

2

i2

2

02

0

})x(2

1{EXP})(

2

1{EXP

+

=

n

1i

2

2

2

02

0

})x(n

)(1

{

2

1EXP

Let:

A =2

0

1

prior precision

B =2

n

likelihood precision

D = A + B posterior precision

a =0

mean of prior distribution

b = x average of sample

And:

BA

)BbAa(c

+

+=

22

0

2

n11

+

=

posterior precision

n0

in00

ww

)xw()w(

+

+= ... (2-24)

where:


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2

0

0

1w

= = prior precision

0 is the mean of the prior distribution.

2

0

is the variance of the prior distribution.

2

n

n

nw

= is sample (or Likelihood)precision

x is the average of sample (or subgroup)

})(1

{EXP)x|(p 22

The centerline and control limits of B-chart are:

Centerline =0

The action limits are:

RA3UCL200B

B

+=+=

RA3LCL200B

B

==

2.1.4. EWBA -Chart:The Exponential Weighted Bayesian Average Chart is very effective for

detecting small process shifts since EWBA Chart can be viewed as a Exponential

Weighted Bayes Average of all past and current data. That is why; EWBA chart is

an effective alternative to the shewhart control charts and Geometric Moving

Average Chart, when we are interested in detecting small shifts.

The aim of using this chart is to control the mean level of the quality of

output of a process (or the mean of a process).

The plotted points on the EWBA Chart are (Zj) which may be calculated

from:

Zj = (1- ) Zj-1+ jB

... (2-25)


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Where is the weight parameter (0 < < 1).

The successive values of EWBA are:

Z1 = (1- )Z0 + 1B

Z2 = (1- )Z1 + 2B

= (1- )2Z0

+ (1- ) 1B + 2B

Therefore at the stage (m) the EWBA is:

Zm =(1- )m

Z0+ (1- )m-1 1B +(1- )

m-2

2B +..... + Bm

to:

Zn =(1- )N

Z0

+ (1- )N-1

1B

+(1- )N-2

2B

+..... + Bm

The centerline of EWBA-chart is:

T = ZEWBA = target value (if predetermined).

And

0BT = = ZEWBA0= x (if it is not predetermined)


EWBA0

EWBA0

3ZLCL

3ZUCL

=

+=

where:

Z0 = ZEWBA0

)x...x|(En1EWBA

= Standard deviation of the posterior

distribution


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CCHHAAPPTTEERR33____________________________________________________________________________________________________________________________________

AApppplliiccaattiioonnooffSSoommeeCCoonnttrroollCChhaarrttssoonntthheeRReeaallDDaattaa

3.1 Introduction:(2) (10)(32)

The use of different materials in building and construction, such as Building

Concrete Blocks (BCB) and Clay Building Bricks (CBB) are gaining popularity

because of high strength of these materials and low prices.

BCB is a hardened building material usually consists of (sand, gravel,

cement, and water). The BCB may be solid or hollow. The hollow type is widely

used in building. The actual dimension of BCB is (20 20 40) cm3

.

CBB is a hardened building material which is constructed from clay (clay

and silt). It has uniform dimension and of different dimension such as

(2411.57.5) cm3.

The different types and various colors of CBB are produced when the CBB

burned by kiln in different grades (750-1000) Centigrade.

The strength of the BCB and CBB are certainly the most important

characteristic that must be tested to see if it conforms with the required

specifications. To achieve the desired strength, manufacturer must carefully control

the production process, which they normally do by using statistical process control

technique. The American Standard, Iraqi Standard, of Testing Materials, and other

organizations have developed a variety of methods for testing the strength. Quality

control charts are widely used by the Engineers of BCBs and CBBs production

processes, and by the Engineers at the site to continually monitor the strength of

these products. Standard test methods have been developed for this purpose as

well.

Engineering Department usually specify the required compressive strength

time of building blocks which is normally 28 days and compressive strength unit


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Kilo-grams per square centimeter (Kg/Cm2) or MEGA PASCALs (MPa) and

pounds per square inch (p/s inch). 28 days is however a long time to wait to

determine the desired hardness that is to be obtained. Therefore, 7 days and 14

days can be useful to predict the ultimate 28 days compressive strength of the

BCB. 25% strength may be gained between (7 and 28) days, and often it is

observed with 100%.

BCB is typically sampled while being placed, with testing acceptance

requiring, that the samples testing be processed under laboratory conditions

(standard processor).

In addition, the test compressive strength for CBB can be done at any age.

3.2. Collection Data:

The Data, which is collected for quality control purposes, is obtained

directly from the production line. The data shown in the tables (1, 2, 3, 4, 5, 6, 7

and 8) in the appendix were obtained from inspection compressive strength of

BCB at different age (7, 14 and 28) days and CBB of different {types A (Iraqi Clay

Building Bricks) and type B (Foreign Clay Building Bricks)} by Hydraulic

Machines directly in:

1. Concrete and Building Material Laboratory / College of Engineering /

University of Sulaimani.

2. Constructional Laboratory of Sulaimani / Ministry of Reconstruction and

Housing.

The purpose of this work is to test the quality strength of BCB and CBB, use

quality control charts and analyze the results. Therefore; we were inspected

compressive strength of those materials and recorded the required data during the

collection of the data.


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3.3. Quality Control Charts Interprets:

To construct control charts for the processes of BCB and CBB products, we

are using the collected data given in the tables (1, 2, 3, 4, 5, 6, 7 and 8) in the

appendix, concerning the strength of BCB at ( age 7, 14, and 28 days ) and CBB

{type (A) and type (B)}. The following control charts are constructed from the

collected data mentioned above, then the data would be plotted on them and

analyze these charts in the quality strength sense.

(a) Standard Control Charts:

The following classical control charts are used for testing compressive

strength of BCB and CBB.

1- X- Bar Chart

2- R-Chart

3- GMA-Chart

(b) Bayesian process Control:

Here, we take B-chart and the new suggested EWBA-chart into

consideration to test the compressive strength of BCB and CBB.

1- B-chart (Bayesian-Chart)

2- EWBA-Chart

Notice:

1. In this thesis, the Minitab statistical software will be used to construct the

control charts.

2. Collected data by inspection compressive strength of BCB during two

different periods. The first one started from 29/8/2006 to 11/1/2007 and the secondone started from 19/4/2007 to 19/8/2007.

3- Construction control charts with only centerline and standard lower bound

(SLB) because of nature of real data were obtained by testing compressive strength

of BCB and CBB directly from laboratories.


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4- Local Standard Lower Bound used to control quality of output of the

process.

5- Constructed control charts, for (age 28 Days), on the strength of Building

Concrete Blocks, and deferent type of Clay Building Bricks.

6- Constructed some control charts, for (ages 7 and 14 days) on the Building

Concrete Blocks, they exist in the appendix.

AApppplliiccaattiioo

..

....

..


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CCHHAAPPTTEERR44

____________________________________________________________________________________________________________________________________

4-1. Conclusions:On the basis of the results that are obtained from analytic real data on the

compressive strength of the BCB and CBB in this thesis, we draw some

conclusions a concerning using quality control charts to monitor these products, as

the following:

4-2. Recommendations:

The following recommendations for future studies are suggested:


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References:

1- Alt, F. B. (1985). Multivariate Quality Control, in Encyclopedia of Statistical

Sciences, Vol. 6, N. L. Johnson and S. Kotz, (eds.) Wiley, New York.

2- Bayan Anwer Ali (2005). Punching Shear Strength of High ReinforcedConcrete Slabs, a thesis submitted to the council of the college of engineering of

university of Sulaimani.

3-Besterfield, Dale H. Besterfield, ph.D., p.E. (1986). Quality Control, II edition.

4- Bisgaard, W. G. Hunter, and L. Pallesen, (1984). Economic Selection of

Quality of Manufactured Product, Technometric, Vol. 26(1), pp. 9-18.

5- Bolstad, William M. (2004). Introduction to Bayesian Statistics, John Wiley

ISBN 0-471-27020-2.

6- Borror, C. M., C. W. champ, and S. E. Rigdon, (1998). Poisson EWMA

Control Charts, Journal of Quality Technology, Vol. 30(4), pp. 352-361.

7- Del Castillo, E., (1998). Statistical Process Adjustment for Quality Control,

Prentice Hall. Inc. New Jersey 07458.

8- Enrique del Castillo (2002). Statistical Adjustment for Quality Control, John

Wiley & Sones. Inc New York, printed in U.S.A.

9- Hamad. M. (2003). Tolerance Interval Control Limits for the x , R, and S

chart, Quality Engineering, Vol. 15(3), pp. 471-487.

10- Harmer E. Davis, Georg Earl Troxell, Clement T. Wiskocil (1964). The

Testing and Inspection Engineering Materials, third edition.

11- MacGregor, J. F., and T. J. Harris (1993). The Exponential Weighted Moving

Average, Journal of Quality Technology, Vol. 25(2), pp. 106-118.12-Montgomery, Doglas C.Montgomery(2005).Statistical Quality Control,5

th

edition

13- Murray R. Spiegel, Larry J. Stephens (1999). Schaums Outlines-Theory and

Problems of Statistics, Chapter 19, Statistical Process Control and Process


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Capability- pp. 470-494 / third edition.

14 - M Xie, TN Goh, V Kuralmani, (2002). Statistical Models and Control Charts

for High-Quality Process, Kluwer Academic Published Boston / Dordrecht /

London.

15 - Peter Congdon, Bayesian Statistical Modeling, Queen Mary, University of

London. UK.

16- Runger, G.C., and M.C Testik (2003). Control Charts for Monitoring Fault

Signatures: Cuscore Versus GLR, Quality and Reliability Engineering

International, Vol. 19(4), pp. 387-396.

17- Samuel A. Schmitt (1969). An Elementary Introduction to Bayesian Statistics

Measuring Uncertainty.

18- Steir, S. H. (1999). EWMA Control Charts with Tim-Varying Control Limits

and Fast Initial Response, Journal Quality Technology Vol. 32(3), pp. 199-208.

19- Walter A. Shewhart, from internet / Deming W. Edwards (1967), Walter A.

Shewhart 1891-1967, American Statistician, Vol. 21(April., 1967), pp 39-40.

20- Testik, M.C.,and C.M.Borror(2004).Design Strategies for the Multivariate

EWMA Control Chart, to appear in Quality and Reliability EngineeringInternational.

21- Winkler, Robert L, Introduction to Bayesian Inference and Decision, 2nd

Edition (2003), probabilistic ISBN 0-9647938-4-9.

22- Yeh, Arthur B, Dennis K.J.Lin and Chandramouliswaran Venkataramani:

(2003),Unified CUSUM chart for Monitoring Process Mean and Variability ,

(Quality Technology and Quantitative Management Vol. 1 No. 1 pp 65-86)


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42/44


APPENDIX:

1-Tables

2-Some Control Charts on the Strength of BCB

3-Material(BCB & CBB) Specifications

4-Abstract in Arabic Language

5-Abstract in Kurdish Language


43/44


TABLE (3) Compressive Strength of BCB age 28 days (First period).*

Measurements (Kg )Subgroup

No.Date Time

x1 x2 x3 x4

Average

Range GMA Bayes

EWB

0113/ 11

200612:30 pm 53 48 46 58 51.3 12 53.1 51.7 53.3

02 14 / 11 01:00 pm 56 59 52 57 56.0 07 53.6 55.953.8

03 15 / 11 02:00 pm 54 57 45 47 50.8 12 53.1 51.353.3

04 16 / 11 02:00 pm 49 45 59 43 49.0 16 52.3 50.352.7

05 18 / 11 09:00 am 50 58 44 48 50.0 14 51.8 50.852.3

06 19 / 11 10:30 am 64 57 52 53 56.5 12 52.7 56.053.1

07 21 / 11 12:30 pm 51 56 48 54 52.3 08 52.6 52.452.9

08 25 / 11 09:00 am 59 66 54 63 60.5 12 54.2 59.354.2

09 26 / 11 10:00 am 61 53 56 61 57.8 08 54.9 57.354.8

10 28 / 11 01:00 pm 50 60 56 48 53.5 12 54.6 53.554.6

11 04 / 12 10:00 am 62 48 49 47 51.5 15 54.0 52.154.1

12 06 / 12 11:30 am 60 60 55 57 58.0 05 54.8 57.854.8

13 09 / 12 09:00 am 59 55 56 52 55.5 07 54.9 55.454.9

14 11 / 12 09:00 am 60 46 51 48 51.3 14 54.2 51.854.3

15 13 / 12 02:00 pm 58 57 60 53 57.0 07 54.8 56.8 54.816 14 / 12 10:00 am 59 60 53 60 58.0 07 55.4 57.6

55.4

17 16 / 12 11:00 am 58 56 52 53 54.8 06 55.3 54.755.2

18 18 / 12 11:00 am 45 39 41 55 45.0 16 53.2 47.553.7

19 20 / 12 01:00 pm 44 53 54 51 50.5 10 52.7 50.953.1

20 21 / 12 09:00 am 50 59 61 57 56.8 11 53.5 56.353.8

21 22 / 12 11:00 am 54 56 53 50 53.3 06 53.5 53.353.7

22 23 / 12 09:00 am 54 47 43 53 49.3 11 52.6 50.0 52.9

23 24/ 12/ 06 11:00 am 48 57 53 59 54.3 11 52.9 54.153.2

24 10/ 01/ 07 02:00 pm 50 50 55 51 51.5 05 52.7 51.652.9

25 11/ 01/ 07 02:00 pm 54 55 52 59 55.0 07 53.1 54.953.3

Sum 1339.5 251 1340.5 1343.3 1345.1


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* inspected from (concrete and building materials laboratory/College of Engineering/Univ. ofSulaimani)

applying some quality control charts to study the strength of building concrete blocks and clay...

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