UNIVERZITET U BEOGRADU
MATEMATICˇKI FAKULTET
Halima Elfaghihe
Konstrukcija i osobine kontrolnih kartica za
stacionarne i nekorelisane podatke
DOKTORSKA DISERTACIJA
BEOGRAD, 2016
UNIVERSITY OF BELGRADE
FACULTY OF MATHEMATICS
Halima Elfaghihe
Design and performances of control charts
for stationary and uncorrelated data
DOCTORAL DISSERTATION
BELGRADE, 2016
Podaci o mentoru i cˇlanovima komisije:
MENTOR:
redovni profesor dr Slobodanka Jankovic´
Univerzitet u Beogradu,
Matematicˇki fakultet
CˇLANOVI KOMISIJE :
redovni profesor dr Slobodanka Jankovic´
Univerzitet u Beogradu,
Matematicˇki fakultet
redovni profesor dr Pavle Mladenovic´
Univerzitet u Beogradu,
Matematicˇki fakultet
vanredni profesor u penziji dr Vesna Jevremovic´
Univerzitet u Beogradu,
Matematicˇki fakultet
Datum odbrane:
Abstract
The subject of this thesis belongs the area of quality control, which represents the
practical usage of statistics in following and improving the production process. In
1930 Walter Shewhart started studying quality control, based on control charts, and
using statistical principles. Later on, after World War II, Edward Deming took this
discipline to Japan, where it flourished. The design and the performance of control
charts are the most important problems in this area. The purpose of studying the
characteristics of control charts in this thesis is to compare the existing and the
suggested control charts.
The thesis is divided into four chapters. The first chapter is introductory and con-
tains motivation and basic definitions related to this subject. In this study it is
always assumed that the data are normally distributed, and that the in-control pro-
cess data are stationary and uncorrelated.
Shewhart control charts and the corresponding limits are constructed in order to
meet the given specifications for the quality characteristic that we investigate. Qual-
ity control that is applied to a production process always has costs related to the
control. The important parameters connected to the cost of quality control are:
width of control limits k, the sample size n and the interval between the samples h.
In Chapter 2 a new loss function is given, which is connected to the production pro-
cess and to X−bar quality control chart. Using Matlab program for optimization,
values of kˆ, nˆ and hˆ are found, which minimize the loss function for given costs. For
given values of cost, a non-linear regression model is built using a package Sigma plot
and the obtained values are compared to those obtained by numerical optimization.
In Chapter 3, the time series model Yi = λXi + (1 − λ)Yi−1 is investigated, where
0 < λ ≤ 1 is a constant, Xi are N(µ,σ2) distributed. Exponentially Weighted Mov-
ing Average (EWMA) control charts for this model are presented, and type I and
type II errors are calculated in the case when i is large. For different sample sizes,
the new comparison between the optimal design of the X-bar and EWMA control
charts for Normally distributed quality characteristic is given, comparing the corre-
sponding cost-loss functions, power functions and average run lengths.
i
The process of calibration is one of the methods in statistical process control, in-
troduced for improving the quality of the products and for reducing the production
costs. In Chapter 4, two new models of non-symmetrical loss function are intro-
duced. Here, the loss function is connected to one product under control (not to
the whole sample). Using our program, written in statistical software R, the value
which minimizes the expected loss for Shewhart X¯ control chart is found. This
value is used as the new central target value of the quality characteristic, that is,
the production process is calibrated with this new value.
The thesis ends with Conclusions, where the results of the thesis are summarized,
and with some open problems to be investigated.
ii
Apstrakt
Tema ove doktorske disertacije pripada oblasti kontrole kvaliteta, koja predstavlja
praktic`no koriˇsc´enje statistike prilikom prac´enja i poboljˇsavanja proizvodnog procesa.
1930. godine Volter Sˇuhart je pocˇeo da proucˇava kontrolu kvaliteta, baziranu na
kontrolnim karticama, koristec´i statisticˇke principe. Kasnije, posle Drugog svetskog
rata, Edvard Deming odneo je ovu disciplinu u Japan, gde je ona dozˇivela proc-
vat. Konstrukcija i osobine kontrolnih karti su najvazˇnija pitanja u ovoj oblasti.
Cilj izucˇavanja osobina kontrolnih karti u ovoj tezi je da se uporede postojec´e i
predlozˇene kontrolne karte.
Teza je podeljena na cˇetiri glave. Prva glava je uvodna i sadrzˇi motivaciju i osnovne
definicije koje se odnose na ovu temu. U ovoj tezi uvek se pretpostavlja da su podaci
normalno raspodeljeni i da su podaci, kada je proces pod kontrolom, stacionarni i
nekorelirani.
Sˇuhartove kontrolne karte i odgovarajuc´e granice konstruiˇsu se tako da odgovaraju
zadatim specifikacijama za karakteristiku kvaliteta koju ispitujemo. Kontrola kvaliteta
koja se primenjuje na proces proizvodnje uvek ima trosˇkove koji su povezani sa kon-
trolom. Vazˇni parametri koji su povezani troskovima kontrole kvaliteta su: sˇirina
kontrolnih granica k, obim uzorka n i interval izmedu uzoraka h. U Glavi 2 data je
nova funkcija gubitka, koja je povezana sa proizvodnim procesom i sa kontrolnom
karticom. Koristec´i program Matlab za optimizaciju, nadene su vrednosti kˆ, nˆ i
hˆ, koje minimizuju funkciju gubitaka pri zadatim trosˇkovima. Za date vrednosti
trosˇkova, napravljen je i nelinearan regresioni model koriˇsc´enjem paketa Sigma plot
i dobijene vrednosti su uporedene sa onima dobijenim numericˇkom optimizacijom.
U Glavi 3 izucˇavan je model vremenskih serija Yi = λXi+(1−λ)Yi−1, gde je 0 < λ ≤ 1
konstanta, a Xi imaju N(µ,σ2) raspodelu. Kontrolne karte za vremenske serije
sa pokretnim sredinama i eksponencijalnim tezˇinama (EWMA) za ovaj model su
predlozˇene i izracˇunate su gresˇke I i II vrste u slucˇaju kada je i veliko. Za razlicˇite
obime uzorka, dato je novo poredenje izmedju optimalnog dizajna za EWMA kon-
trolne karte, kod kojih karakteristika kvaliteta ima Normalnu raspodelu, i uporedene
su odgovarajuc´e funkcije trosˇka-gubitka, funkcije moc´i i srednje duzˇine ciklusa.
Proces kalibracije je jedan od metoda u statisticˇkoj kontroli kvaliteta, koji se uvodi
da bi se poboljˇsao kvalitet proizvoda i da bi se smanjili trosˇkovi proizvodnje. U Glavi
4 predlozˇena su dva nova modela nesimetricˇne funkcije gubitka. Ovde se funkcija
gubitka odnosi na jedan proizvod koji se kontroliˇse (a ne na ceo uzorak). Koristec´i
nasˇ program napisan za statisticˇki softver R, nadena je za oba slucˇaja vrednost koja
minimizuje ocˇekivanje funkcije gubitka kod Sˇuhartove kontrolne karte. Ta vrednost
se onda koristi kao nova centralna ciljna vrednost karakteristike kvaliteta, tj. proces
se kalibrira sa ovom novom vrednosˇc´u.
Teza se zavrsˇava sa Zakljucˇkom, u kome su sumirani rezultati iz teze i u kome su
izlozˇeni neki otvoreni problemi koji c´e se izucˇavati.
Acknowledgements
I would like to express the deepest appreciation to my adviser and committee chair,
dr Slobodanka Jankovic´ who helped me with her advice to improve my thesis,I am
deeply indebted to her for everything she have done for me. I would like to express
my sincere gratitude to the committee members, dr Pavle Mladenovic´ and dr Vesna
Jevremovic´, I also owe thanks to my colleague Kristina Veljkovic´ for reviewing the
papers. I am indebted to all of you. Thanks again to dr Vesna Jevremovic´ who
supported me throughout my stay in Serbia.
I would like to thank my loved ones, who have supported me throughout the entire
process, both by keeping me harmonious and helping me to shape my today and
tomorrow. I am grateful to my mother and all my family for giving me the opportu-
nity to have an education from the best institutions and for helping me throughout
my life. This thesis would not have been possible without my husband, the person
who is always with me, Osama Alsaaeh and my children, Ahmed, Anas and Hala.
This thesis is dedicated to my son Ahmed who deserves the best in his life.
Contents
Abstract i
1 Introduction 1
1.1 Quality characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Causes of Quality Variation . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Quality control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Quality control chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Phase I and Phase II of Control Chart Application . . . . . . . . . . . 8
1.6 Time series analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Design of control charts . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Shewhart-type control chart for the mean 14
2.1 Construction of the X-bar control chart . . . . . . . . . . . . . . . . . . 14
2.2 Production process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Loss Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 EWMA-type control charts for the mean 35
3.1 Construction of EWMA control chart . . . . . . . . . . . . . . . . . . . 35
3.2 Basic assumptions and parameters for the design of control charts . . 41
3.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Control Chart Performance . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Optimal Process Calibration for Non-symmetric Loss Functions 66
4.1 Symmetric loss function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Non-symmetric loss function . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Implementation in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Conclusions 75
Bibliography 76
Chapter 1
Introduction
1.1 Quality characteristics
Quality has become the most important criteria for making a distinction between
products and services. Consumers use quality nowadays as the most relevant pa-
rameter for choosing which products and services they need. In order to improve
competitiveness on the market, companies highlight quality in business strategies
and in this way they also secure growth and success.
Quality characteristic represents the key feature of the product that can be mea-
sured. It is a random variable that can be continuous (length, diameter, weight,
thickness of a product etc.) or discrete (the number of errors or mistakes made
in completing a loan application, the number of medical errors made in a hospi-
tal etc). The desired value of quality characteristic is called the nominal or target
value, which is usually bounded by interval of values called specification limits. It
is assumed that specification limits will be sufficiently close to the target value so
as to not impact the function or performance of the product if the value of quality
characteristic is in that range. The largest allowable value for a quality characteris-
tic is called the upper specification limit (USL), and the smallest allowable value for
a quality characteristic is called the lower specification limit (LSL). Some quality
characteristics have specification limits on only one side of the target. A product is
called nonconforming if the value of quality characteristic lies outside the interval
of specification limits [LSL, USL].
1.2 Causes of Quality Variation
Some variability of products always exists, because there are no two products that
are exactly the same. For example, the weight of each chocolate bar ”Milka” is not
exactly 100 gr, some will weight more, others less.
There are two general causes of variability:
1
ˆ Chance (or common cause): Variability happens randomly and it isn’t possible
to control it or to remove it.
ˆ Assignable causes (or special causes): This variability in the observed quality
characteristic usually arises from three sources: improperly adjusted or con-
trolled machines, operator errors or defective raw material. Variability caused
by assignable causes can be identified and we can control it. After doing some
research and investigation we can find the cause and eliminate it.
Production process that operates with only chance causes of variability is said to be
in statistical control (or in-control state). A process that operates with the presence
of assignable causes is said to be an out-of-control process. When a process shifts
to the out-of-control state, larger proportion of nonconforming products will be
produced.
Quality improvement is used to decrease the variability in processes and products.
Statistics and engineering are useful for improving the quality. Moreover, there are
techniques, including statistical and analytical tools which are helpful in the analysis
of quality problems and for the improvement of the performance of processes. These
techniques belongs to time series analysis and depend on the type of the input and
output data. For detailed discussions of time series modeling see for example [44],
[11]. Important techniques which are used to improve the quality (statistical and
engineering quality and important modeling of time series) are to be discussed in
what follows.
1.3 Quality control
Quality control is a set of techniques used to secure a required level of quality in
a product or service. Statistical quality control (SQC) uses statistical methods
to determine when product quality deviates from the specifications. The primary
goal of statistical quality control is to maintain and improve the production process
through techniques (see [27]) such as
ˆ Inspection Inspection is used to examine and test the quality of the product
and classify it into a conforming or a nonconforming product using statistical
sampling.
ˆ Statistical Process Control Statistical process control (SPC) is a primary
tool for monitoring and improving quality. The following figure (1.1), (see
[35]) shows the steps of the statistical process control procedure
2
Figure 1.1: Statistical process control procedure.
SPC is one of the greatest technological developments of the twentieth century
because it is built on firm basic statistical principles, it is user-friendly, has important
influence on quality improvement, and can be implemented on any process.
The statistical control process it usually accompanied by Engineering Quality Con-
trol. Engineering Quality Control is made up of operations, management, and en-
gineering actions, which any company uses in order to make sure that the quality
of a product meets the necessary target goals and that the variability is kept at the
minimum. Engineering process control (EPC) attempts to compensate the causes
of deviations by using process adjustment which is usually computed and applied
automatically. This type of control is sometimes called Automatic process control
(APC). If the process is out of control, the engineer process control (EPC) looks
for an assignable cause by following the out-of-control action plan (OCAP), which
is associated with statistical process control (SPC) by using the control chart. The
following figure (1.2),(see [35]) shows the out-of-control action plan.
3
Figure 1.2: Engineering process control.
The product specifications are usually the result of the process design, and the
specification limits are usually set by the design engineer. Once we have the spec-
ifications, we can assess the quality characteristics. The specifications tell us what
kind of quality we want for the components of the product. It should be emphasized
that there is no connection between the control limits of the control chart and the
specification limits of the process. The process should be both in control and should
meet specification limits. As it was explained previously statistical process control
and engineering process control have the same objective and that is to ensure that
the final result of the process is closest to its target. But this is achieved in dif-
ferent ways, by using different techniques. The different approaches of SPC and
EPC are not in competition, but they just have their own points of view on how a
manufacturing process works (for example see [44], [11], [49]):
1. SPC highlights process monitoring and detecting faults. This is in contrast
with process adjustment when the process is in the statistically in-control
state. In this situation changing the process can increase the process variance
when we need to decrease it.
2. EPC pushes forward many process adjustment strategies in order to keep the
4
process on the way to its target.
Reasons for using the process control are the following ones:
ˆ We use process control in order to lower the production costs, to reduce the
losses and avoid unnecessary actions (see [27]).
ˆ Process control keeps within boundaries the production errors and deviation,
as well as the reasons for causing them.
ˆ It is very beneficial to reveal the errors at an early stage of the production.
1.4 Quality control chart
Control charts are tools used to determine whether a manufacturing or business
process is in a state of statistical control. Walter Shewhart invented control charts
for statistical process control eighty years ago. He suggested a general model for
control charts and they are now called Shewhart charts or process-behavior charts.
The quality control charts have been used a lot, because they help reduce the cost
and they improve the quality of the products. They use statistical methods to con-
trol and monitor the production process, find the deviations and find the ways how
to fix them. The typical control chart is a graphical display of the measured quality
characteristic. We present now a general model of the control chart.
General model for control chart
Every control chart has three lines: central line which represents target value of
quality characteristic and two control limits called upper control limit (UCL) and
lower control limit (LCL). These control limits help us find whether the data indi-
cates a state of statistical control or not, (see [44], [27]).
Let M be a sample statistic which is a measure of quality. Suppose that M has
a normal distribution with mean µM and the standard deviation σM . The mean of
M is equal to the target value µM = µ0, when the process is in control. Then the
central line is CL = µM , the upper control limit is UCL = µM + σM , and the lower
control limit is LSL = µM − σM . Shewhart control chart is shown in Figure (1.3).
The quantity kσM represents the distance of the control limits from the central line.
Parameters of the control chart are the following: the sample size n and the control
limit width k. It should be noticed that Shewhart’s control charts actually represent
tests corresponding to hypothesis testing. Null hypothesis H0(µM = µ0) corresponds
5
Figure 1.3: Shewhart control chart
to the situation when the production process is in control, while the alternative hy-
pothesis H1(µM ≠ µ0) corresponds to the situation when the production process is
out of control.
For each sample i, i = 1,2, ..m we calculate the value of sample statistic M and
we denote it by Mi. If the point Mi is within limits UCL and LCL, then the process
is in-control state and this leads to the acceptance of the hypothesis (H0 ∶ µM = µ0).
But if the point Mi is out of the interval (UCL, LCL), this means that the process
is out of control, and we reject the hypothesis (H0 ∶ µM ≠ µ0). The result is an
acceptance of the alternative hypothesis and it becomes necessary to discover the
causes that led to this out-of-control state, and to correct them.
Control charts are classified into two general types: if the quality characteristic can
be measured (quantified), then we use the variable control charts. If the quality
characteristic can not be measured (it is not quantitative), then we use attribute
control charts.
Variable control charts
The X-bar chart is used for monitoring the process mean. On this chart, we plot the
6
sample mean. For monitoring the process variability, we use S-chart and R-chart.
If we have a sample (x1, x2, ..., xn) of the size n, then on S-chart we represent the
sample standard deviation,
S = √∑ni=1(Xi − X¯)2
n − 1 .
On R-chart, we represent the sample range, R =Xmax −Xmin. R-chart is used when
the sample size is small (n < 10).
When we want to detect smaller process shifts we usually use the time-weighted
charts (CUSUM, EWMA and MA). On the cumulative sum chart (CUSUM)the
target level deviations are accumulated. The CUSUM will not increase if there are
just the chance causes. If the cumulative sum increases too much, this tells us that
the process level changed because of the assignable causes. Therefore, a decision
limit is assigned to the cumulative sums, which are similar to the control limits.
MA chart is based on a simple, unweighted moving average and may be of interest
for some purposes. The exponentially weighted moving average (EWMA) control
chart is very efficient in detecting small shifts in the mean of a process. We will
explain this further in Chapter 3.
Attribute control charts
The most frequently used attribute charts are: np-chart, c-chart, u-chart and p-
chart. We use np-chart to plot the number of defectives (per batch, per day, per
machine). The control limits in this chart are calculated from binominal distribu-
tion. On the c-chart, the number of defects (per batch, per day, per machine, per
100 feet of pipe, etc.) is plotted. It is presumed that the quality defects are not
frequent and the control limits are computed using Poisson distribution. Defects per
unit chart (u-chart) is used to plot the rate of defects. We divide the number of de-
fects by the number of inspected units (the n; e.g., feet of pipe, number of batches).
The differences between u-chart and c-chart is that u-chart does not need a constant
number of units, and we can use it, for example, when the batches (samples) are not
of the same size. We use proportion defective chart (p-chart) in order to plot the
fraction of defectives (per any unit of measure, per day, per machine, etc.) just like
in the u-chart. The control limits are based on binominal distribution of proportions.
Control chart for Individual Values
Sometimes, the sample size used for process monitoring is n = 1, e.g. the sample
consists of an individual unit. There are many situations in which control chart for
7
individual values is appropriate. For example when production rate is low, (mean-
ing data become available slowly) it is inconvenient to allow sample size of n > 1 to
accumulate before analysis.
1.5 Phase I and Phase II of Control Chart Appli-
cation
When we use standard control chart, then there are two phases: Phase I and Phase
II of applications. They have two different objectives which can be used in many ap-
plications of control charting, and it is useful to know the difference between them.
(see [44] [23]).
Phase I: A set of data is collected and analyzed all at the same time in a ret-
rospective analysis. Trial control limits are constructed so that we can determine if
the process has been in control during the period of data collection, and to check if
we can determine reliable control limits for monitoring future production. Usually
we do this first when we apply control limits to any process. The primary function of
control charts in this phase is to assist operating personnel with putting the process
into a statistical control state.
Phase II: This phase can start when we have a ”clean” set of process data which
are gathered under stable conditions. In Phase II, the control chart is used to moni-
tor the process. The control chart compares the sample statistic for each successive
sample, as it is drawn from the process, to the control limits.
For example, in Phase I, we are comparing a collection of m points to a set of
control limits computed from these points. Typically m = 20 or 25 subgroups are
used in phase. Typically in a Phase I it is assumed that the process is initially out
of control, so the goal of the analyst is to bring the process into a state of statistical
control. Control limits are calculated based on the m subgroups and the data plot-
ted on the control charts. Points outside the control limits are investigated in search
of potential assignable causes. If some assignable causes are found, engineering and
operating personnel work on eliminating them. Then those points are excluded and
a new set of revised control limits are calculated. Next step is to collect new data
and compare them to these revised limits. Under the SPC framework, Phase I meth-
ods are compared by assessing the probability of the process being out of control,
i.e. unstable. In Phase II, the most important thing is to detect quickly the process
trends or shifts.
8
The shifts are usually measured by parameters of the run-length distribution, where
the run length is the number of samples taken before an out-of-control signal is
given. The average run-length (ARL) is frequently used for comparison of compet-
ing control chart methods in Phase II. It is usually assumed that there are trends
or sustained step shifts in the parameters of the model which was estimated from
Phase I. Under the SPC framework, we mainly want to control the false-alarm rate
in Phase I and the in-control ARL in Phase II, and to use methods which are proved
to be effective in detecting specified trends or shifts in the distribution of the quality
variable over time.
1.6 Time series analysis
In order to understand better the types of variability exhibited by the process, we
introduce some definitions and properties of time series.
A time series is a sequence of observations measured through time. We can make
these measurements continuously through time or take them at a discrete set of
time points. Conventionally, these two types of series are called continuous and
discrete time series, although the measured variable may be discrete or continuous
in both cases. Many sets of data appear as time series: a monthly sequence of the
quantity of goods which are shipped from a factory, a weekly series of the number
of the road accidents, hourly observations made on the yield of a chemical process,
etc. There are many examples of time series in: economics, business, engineering,
natural sciences (geophysics, meteorology, etc.), as well as in the social sciences.
For a continuous time series, the observed variable is typically a continuous variable
recorded continuously, such as a measure of brain activity recorded from an EEG
machine. The analysis of such a series starts by sampling (or digitizing) the series
at equal intervals of time, in order to obtain a discrete time series. By using this
process we lose little or no information as long as the sampling interval is small
enough. The models of time series show the relation between the series {Zt, t ∈ T},
which is to be modeled, and a white noise generating series {at}. The elements
at, at−1, at−2, ... may be regarded as a series of shocks, which govern the system. It
consists of a sequence of uncorrelated random variables with mean zero and constant
variance, that is
E(at) = 0,
D(at) = σ2a.
9
Since the random variables {at} are uncorrelated, it follows that their autocovariance
function is
E(atas) = ⎧⎪⎪⎨⎪⎪⎩ σ
2
a, k = s
0, k ≠ s, (1.1)
where {at, t ∈ T} is called white noise. We use the notation at ⊥ (0, σ2a) in order to
write down all three characteristics of at as time series.
Time series could be stationary or non stationary and we give the definition of
stationarity (see [10], [36]). Time series {Zt, t ∈ T} is said to be weakly stationary if
1. The expectation of Zt is constant over the time i.e.
E(Zt) = constant, t ∈ T.
2. The variance of Zt has to be finite, and it is enough that:
E(Z2t ) <∞,
3. The autocovariance function is a function of the difference of arguments, i.e.
E[(Zt −E(Zt))(Zs −E(Zs))] = B(t − s).
Time series is strict stationary if its probability distribution does not change when
shifted in time.
Process behavior
One of the important consideration in control chart usage is the type of variability
exhibited by the process. Figure 1.4. represents data from three different processes
(see [44]): (a) Stationary and uncorrelated (white noise). (b) Stationary and auto-
correlated. (c) Nonstationary.
Figures 1.4.a and 1.4.b illustrate the stationary behavior. By this we mean that the
process data vary around a fixed mean in a stable or predictable manner. This is
the type of behavior that is produced by an in-control process. The data in Figure
1.4.a are uncorrelated-this type of data represents white noise.
Figure 1.4.b illustrates stationary but autocorrelated process data. Successive obser-
vations in these data are dependent - a value above the mean tends to be followed by
10
Figure 1.4: Data from three different processes
another value above the mean, whereas a value below the mean is usually followed
by another such value. This produces a data series that has a tendency to move in
moderately long ”runs” on both sides of the mean.
Figure 1.4.c illustrates nonstationary variation. This type of process data occurs
frequently in the chemical and process industries. Note that the process is very
unstable in that it drifts or ”wanders about” without any sense of a stable or fixed
mean. In many industrial settings, this type of behavior is stabilized by using engi-
neering process control.
Shewhart control charts are the most effective when the in-control process data
look like those given in Figure 1.4.a. In the study of the control charts we always
assume that the in-control process data are stationary and uncorrelated.
1.7 Design of control charts
In order to use a control chart it is necessary to determine the value of its parame-
ters. For example X-bar control chart has three parameters : sample size n, sample
frequency h and control limits width k. EWMA chart has two parameters: control
limits width L and λ. The design of the control chart is defined by the selection
of these parameters. We use statistical design and economic design and economic
statistical design as criteria for the design of control charts (see [27], [34], [1]).
Originally, only the first criteria (statistical criteria) have been used to design con-
trol charts. Based on the effects on the statistical properties of the charts, statistical
design decides about the parameters. For example we can have pre-selected values
of the probabilities of errors, because the control charts are usually designed to min-
imize the errors. There are two types of errors in the control chart:
Type I error is the situation when a false alarm occurs: although the process is
11
in-control. the out-of-control state is proclaimed (a point falls outside the control
limits and this occurs only because of the natural causes). This makes us look for
an assignable cause, when none exists and adds extra cost to the process. The prob-
ability of type I error is denoted by α.
Type II error is the opposite situation when the process seems to be in control,
although it is out of control. This happens because of the shift in the process mean
(a point falls inside the control limits, although some assignable causes are present
in the process). If this is the case we do not take any corrective action, while the
process is really out of control. That is why the use of control chart to detect process
shift is not good enough. Therefore, we need to measure the efficiency of the control
chart and for this purpose we will also use the probability of type II error, usually
denoted by β, the average run length (ARL), (the average number of samples which
we take before the shift is detected) and the average time to signal ATS (the average
time necessary until a signal is obtained).
There are also some economic consequences when we use and design the control
chart and this is second criteria. These economic consequences are, for example,
the cost of sampling and checking out-of-control signals, and the cost of eliminat-
ing assignable cause and cost of non-conforming units which the consumer will get.
Therefore, it is strongly advisable to take into account these consequences when we
design a control chart, since the above mentioned qualities affect parameters selec-
tion. We need to establish a mathematical model which relates the chart parameter
to their economic consequence, and then to select the values of the parameters which
optimize a specific objective function. This approach is criticized because it does
not take into consideration the statistical properties of the chart.
The third criteria for designing control charts are economic-statistical design of
the control chart. In order to satisfy specific statistical economic properties of the
design, this approach puts statistical constraints on economic models.
In an economic model, we aim to find the optimal control charts parameters to
minimize the process cost by using the economic design of the control charts. This
approach does not include the statistical properties, so the combined economic-
statistical design should be used as a better approach (see [64]).
The design of control charts was investigated a lot by many authors because it offers
material for learning about the performance of control charts. For example, Duncan
[26] first suggested an economic model for the optimum economic design of X-bar
12
control chart. His research dealt with a completely economic model of a Shewhart
type control chart, and it included formal optimization methodology in establishing
the control chart parameters. Duncan’s finding inspired a lot of further research in
this area. Lorenzen and Vance [40] used Duncan’s original model and turned it into
general model which can be applied directly to most types of control charts. Woodall
[68] criticized the economic approach for, among other reasons, its possibility of pro-
viding a plan that generates an excessive number of false alarms (i.e., looking for
assignable causes when none exists). Goel, Jain and Wu [31] came to a practical
solution and specified the values of parameters (n, k, h). Serel and Moskowitz [56]
proposed an economic model to EWMA (exponentially weighted moving average)
control charts which is based on economic or both economic and statistical criteria.
Hamoda [34] discussed the optimal economic design for x¯ − 3σ chart with warning
limits. Alarfi [1] discussed the optimal economic design for x¯−kσ chart with warning
limits. Elfaghihe [27] discussed the optimal economic model of x¯-kσ chart without
warning limits. When we compare the economic and the economic-statistical mod-
els, the results showed better statistical performances of the economic-statistical
design with non important rise in cost.
The method of economic-statistical design of control charts was developed by Saniga
[52]. He proposed economic-statistical model which combines properties of economic
and statistical design: it is a method for economic design of control charts that have
bounds on type I and type II error probabilities and the average time to signal (ATS)
of an expected shift. Economic-statistical design considers economic factors while
achieving desirable statistical properties and, therefore, represents the improvement
of economic or statistical design.
13
Chapter 2
Shewhart-type control chart for
the mean
In this chapter we construct the X-bar control chart which is designed in order
to achieve minimal quality control costs. We create models which can be used for
determining the parameters (n, k, h) for different values of variables that rely on the
objective function by:
1. Determination of objective function (function of loss by unit of the product)
that connects the production process and its characteristics with the X-bar
chart, its attributes and the economical consequences of its usage.
2. Defining the values of parameters (n, k, h) which create the objective function
as small as possible for different values of incomes.
3. Use of the results obtained in previous step in order to create mathemati-
cal methods which will be used to determine the values of the parameters
depending on the values of reliable economical variables.
2.1 Construction of the X-bar control chart
The X-bar chart is the most popular control chart. It is used to monitor the average
of the process characteristic we are interested in, and we obtain it from the above
mentioned principles. The sample statistic is X¯ (the sample mean) and the control
chart consists of:
1. Measurements of a quality characteristic in samples which are taken from the
process at different times.
2. The mean x¯ of this characteristic is calculated by using all the samples (e.g.,
the mean of the means). A central line (CL) of X-bar control chart represents
the value of the mean of the quality characteristic. If X is normally distributed
14
quality characteristic with known mean µ0 (target value) when the process is
in-control, and known variance σ2, then CL = µx¯ = µ0 and σ2x¯ = σ2n .
3. Upper and lower control limits (UCL,LCL), which indicate the threshold at
which the process output is considered to be statistically ’unlikely’, are typi-
cally drawn at k standard deviations away from the central line. As a result
we have
UCL = µ0 + k σ√
n
, LCL = µ0 − k σ√
n
.
This chart is called k-sigma X-bar chart. When k = 2, the probability that a given
sample lies within the control limits is 95.44%. When k = 3, the probability is
99.74%. The X-bar chart has three parameters (n, k, h) where n is sample size, k
is the width of control limits, h is the sample interval (sampling frequency).
Suppose that a quality characteristic is normally distributed with mean µ and stan-
dard deviation σ. There are many statistical tests to check out the normality of
distribution - χ2 -test and Kolmogorov-Smirnov, to mention only the two mostly
used. In practice for the Normal distribution of a quality characteristic, we usually
will not know µ and σ. Therefore, they must be estimated from preliminary samples
or subgroups taken when the process is thought to be in control. These estimates
should usually be based on at least 20 to 25 samples. Suppose that m samples are
available, each containing n observations of the quality characteristic. Typically, n
will be small, often either 4, 5, or 6. These small sample sizes usually result from
the fact that the sampling and inspection costs, associated with variables measure-
ments, are usually relatively large.
Let x¯1, x¯2, ..., x¯m be the average of each sample. Then the best estimator of m,
the process average, is the overall average
x¯ = x¯1 + x¯2 + ... + x¯m
m
. (2.1)
Thus, x¯ would be used as the center line on the chart.
To construct the control limits, we need an estimate of the standard deviation σ. We
may estimate σ from either the standard deviations or the ranges of the m samples.
We will explain both methods, following [44].
15
Range method
If x1, x2, ..., xn is a sample of size n, then the range of the sample is the difference
between the largest and the smallest observations, that is
R = xmax − xmin.
Let R1,R2, ...,Rm be the ranges of m samples. The average range is
R¯ = R1 +R2 + ... +Rm
m
.
An unbiased estimator of the standard deviation σ of the Normal distribution is
σˆ = R¯
d2
, (2.2)
where d2 is a function of the sample size n defined by
d2 = d2(n) = E(Z(n) −Z(1)), (2.3)
with Z(n) and Z(1) the largest term and the smallest term, respectively, in a random
N(0,1) sample Z1, . . . , Zn.
We may now give the formulas for constructing the control limits on the X-bar
chart. They are as follows (where σˆ is as above):
UCL = x¯ + k√
n
σˆ
CL = x¯
LCL = x¯ − k√
n
σˆ. (2.4)
Standard deviations method
Suppose that m preliminary samples are available (xi1, xi2, ..., xin, i = 1, ...m) each
of the size n, and let Si be the standard deviation of the ith sample. That is
Sj = √∑ni=1(xij − x¯i)2
n − 1 .
Let S1, S2, ..., Sn be sample deviations of m samples. The average of standard devi-
ations is
S¯ = S1 + S2 + ... + Sm
m
,
16
where Si is the standard deviation of the ith subgroup and m is the number of
subgroups. The standard deviation is then estimated from the following equation:
σˆ = S¯
c4
, (2.5)
where c4 is a function of the sample size n defined by
c4 = c4(n) = ( 2
n − 1)
1
n Γ(n2 )
Γ[n−12 ] . (2.6)
We define the control limits of the corresponding X-bar control chart as in (2.4),
but now σˆ = S¯c4 :
UCL = x¯ + k√
n
σˆ
CL = x¯
LCL = x¯ − k√
n
σˆ. (2.7)
Phase I Application of X-bar control chart
In phase I control chart usage, we usually treat the control limits obtained from
(2.4) and (2.7) as trial control limits, when preliminary samples are used to con-
struct X-bar control chart. They let us determine if the process was in control
during the selection of m initial samples. In order to determine whether the process
was in control when the preliminary samples were collected, we need to plot the
mean values from each sample on the chart and analyze the results.
If all points are inside the control limits, we infer that the process was in con-
trol in the past, and we can conclude that the trial control limits are appropriate
for controlling the present or future production. It is good to have 20 to 25 samples
or subgroups of size n (typically n is between 3 and 5) to compute the trial control
limits. It is possible to work with less data but then the control limits are less reliable.
Let us now suppose the values of X-bar plot are out of control when we com-
pare them to the trial control limits. It is obvious that if want the control limits for
present or future production to mean something, then they have to be based on data
from an in control process. If this is not the case, then we have to revise the trial
control limits. Therefore, we examine each of the out-of-control points, in order to
find an assignable cause. If we find an assignable cause, then we discard the point
and recalculate the trial control limits with only the remaining points. Then these
17
remaining points are reexamined for control. This process continues until all points
are in control and when this happens we adopt the trial control limits for current use.
Sometimes we cannot find an assignable cause for a point that is out-of-control.
Then we have two options. One is to eliminate the point, pretending that the
assignable cause had been found. The other option is to keep the point (or points)
assuming the trial control limits are appropriate for current control. If the point
really represents an out-of-control condition, the resulting control limits will be too
wide. However, if it is just one or two points of the kind, then it will not do any
significant damage to the control chart. If future samples still indicate in-control
state, then it is probably safe to drop the unexplained points. Sometimes, when the
initial sample values are plotted against the trial control limits, many points will be
out-of-control.
Clearly, if we arbitrarily drop the out-of-control points, we will not have a satisfying
situation, because this approach could ignore a lot of useful information. However,
it is not always possible to have success in searching for an assignable cause for each
out-of-control point.
As an example of phase I application of the X-bar chart, we use the data of diameter
measurements (mm) for automobile engine piston rings (given in Table 2.1, taken
from [45]). In the Table 2.1, 25 samples of diameter measurements (mm) for auto-
mobile engine piston rings are given, where each sample or subgroup consists of five
piston rings. As the variance is unknown, we estimated it with mean sample range.
First, we calculate the sample mean and sample range for each of the 25 samples. We
get x¯ = 74.00118 ,σˆ = 0.009171,where we used sample range to estimate σˆ (see (2.2)).
Then, the upper control limit, central line and lower control limit of X-bar control
chart are equal, respectively
UCL = x¯ + k√
n
σˆ = 74.00118 + 3√
5
(0.009171) = 74.0135
CL = x¯ = 74.00118
LCL = x¯ − k√
n
σˆ = 74.00118 − 3√
5
(0.009171) = 73.9889.
The X-bar control chart is shown in the Figure (2.1).
18
Sample
Number,i Observations x¯i R¯i
1 74.030 74.002 74.019 73.992 74.008 74.010 0.038
2 73.995 73.992 74.001 74.011 74.004 74.001 0.019
3 73.988 74.024 74.021 74.005 74.002 74.008 0.036
4 74.002 73.996 73.993 74.015 74.009 74.003 0.022
5 73.992 74.007 74.015 73.989 74.014 74.003 0.026
6 74.009 73.994 73.997 73.985 73.993 73.996 0.024
7 73.995 74.006 73.994 74.000 74.005 74.000 0.012
8 73.985 74.003 73.993 74.015 73.988 73.997 0.030
9 74.008 73.995 74.009 74.005 74.004 74.004 0.014
10 73.998 74.000 73.990 74.007 73.995 73.9987 0.017
11 73.994 73.998 73.994 73.995 73.990 73.994 0.008
12 74.004 74.000 74.007 74.000 73.996 74.001 0.011
13 73.983 74.002 73.998 73.997 74.012 73.998 0.029
14 74.006 73.967 73.994 74.000 73.984 73.990 0.039
15 74.012 74.014 73.998 73.999 74.007 74.006 0.016
16 74.000 73.984 74.005 73.998 73.996 73.997 0.021
17 73.994 74.012 73.986 74.005 74.007 74.001 0.026
18 74.006 74.010 74.018 74.003 74.000 74.007 0.018
19 73.984 74.002 74.003 74.005 73.997 73.998 0.021
20 74.000 74.010 74.013 74.020 74.003 74.0097 0.020
21 73.982 74.001 74.015 74.005 73.996 74.000 0.033
22 74.004 73.999 73.990 74.006 74.009 74.002 0.019
23 74.010 73.989 73.990 74.009 74.014 74.0027 0.025
24 74.015 74.008 73.993 74.000 74.010 74.005 0.022
25 73.982 73.984 73.995 74.017 74.013 73.998 0.035∑ = 1850.029 ∑ = 0.581
x¯ = 74.00118 R¯ = 0.02324
Table 2.1: The data of diameter in phase I
19
Figure 2.1: X-bar control chart in phase I
As we can see on Figure 2.1, no indication of an out-of-control condition is ob-
served (all sample points fall between the control limits). Therefore, we conclude
that the process is in control, and adopt the trial control limits for use in phase II.
Phase II Application of the X-bar chart
When we establish a set of reliable control limits, we use the control chart for
monitoring future production. This is the phase II of control chart usage. We use
additional samples (given in Table 2.2 taken from [44]) for the piston ring process,
taken after the initial control charts were established. The X-bar control chart
for monitoring process mean is shown in Figure (2.2). We can see that there is
indications that the process is out-of-control state (some sample points after the
11th sample are out of control limits).
2.2 Production process
The design of control charts has some statistical and economic consequences, which
must be taken into account during the determination of the control chart parame-
ters. The economical design of charts demands formulation of the objective function,
which connects economic consequences with parameters from the charts. To deter-
mine the objective function (loss function), first we need to make certain assumptions
about the behavior of the production process (see [27]).
A production cycle is defined as the interval of time from the start (in the in-control
20
Figure 2.2: X-bar control chart in phase II
state) to the detection and correction of the assignable cause. The cycle consists of
four periods:
1. In-control period
2. Out-of-control period
3. The time taken for sampling and interpreting the results
4. The time taken to find the assignable cause (see [5]).
We need to make some assumptions about the process behavior in order to be able
to formulate a model for the design of a control chart. These assumptions are given
below and they are standard in most models. Process behavior, statistical properties
of the chart, control procedure, and economic factors are discussed in the frame of
these assumptions. The characteristics and relations for the investigated process are
as follows:
I Production has mean of υ items per hour.
II The sample size is n and the sampling interval is h.
III The quality characteristic of interest X has Normal distribution with mean µ
and variance σ2.
IV The production is thought to consist of a series of cycles. Each cycle starts
in state I i.e. under control (or in-control state), with mean µ0 (target level),
and ends in state II (out-of-control state) with µ ≠ µ0. After we detected and
corrected the assignable cause, the process returns to the state I again and a
21
Sample
Number,i Observations x¯i
26 74.012 74.015 74.030 73.986 74.000 74.009
27 73.995 74.010 73.990 74.015 74.001 74.002
28 73.987 73.999 73.985 74.000 73.990 73.992
29 74.008 74.010 74.003 73.991 74.006 74.007
30 74.003 74.000 74.001 73.986 73.997 73.997
31 73.994 74.003 74.015 74.020 74.004 74.007
32 74.008 74.002 74.018 73.995 74.005 74.006
33 74.001 74.004 73.990 73.996 73.998 73.998
34 74.015 74.000 74.016 74.025 74.000 74.011
35 74.030 74.005 74.000 74.016 74.012 74.013
36 74.001 73.990 73.995 74.010 74.024 74.004
37 74.015 74.020 74.024 74.005 74.019 74.017
38 74.035 74.010 74.012 74.015 74.026 74.020
39 74.017 74.013 74.036 74.025 74.026 74.0237
40 74.010 74.005 74.029 74.000 74.020 74.013
Table 2.2: The data of diameter in phase II
new cycle starts and so on. If the assignable cause occurs, it shifts the process
to the state II with mean µ1 = µ0−δσ with probability P (µ = µ1) or µ2 = µ0+δσ
with probability P (µ = µ2). Here δ is a known shift in process mean.
V Time T (duration of in-control state) is thought to be a random variable that
has an exponential distribution with parameter λ. Thus the average duration
of the state which is under control in the cycle is E(T ) = 1λ .
VI The states of the process can be recognized by charting only.
Characteristics of the chart
There are two set of characteristics of the chart, statistical and economic (see [27],
[35]) .
Statistical characteristic of the chart We draw samples of size n at the interval
of h hours. The sample point X¯j that falls outside the control limits [LCL,UCL] is
taken as an indication of the out-of-control state. In case we detect an out-of-control
state, then we begin to search for its causes and we need to take suitable steps in
order to bring the process back into state I.
22
The types of errors
There are two types of errors that can happen:
Type-I error (The false alarm): This type of error happens when the chart
signals that the process is in state II and actually it is not. The probability of type
I error of X¯ control chart is given by (see [27]):
α = P (X¯j ∉ [LCL,UCL]∣µ = µo)= 1 − P (X¯j ∈ [LCL,UCL]∣µ = µo)
= 1 − P ((µ0 − kσ√
n
≤ X¯j ≤ µ0 + kσ√
n
)
= 1 − P [−k ≤ Zj ≤ k]= 1 − [Φ(k) −Φ(−k)] = 2Φ(−k). (2.8)
Type-II error (not detecting the real deviations): This error happens when the
chart signals that the process is in state I, while it is in state II. The probability of
type II error of X¯ control chart is given by (see [27]):
β = P (X¯j ∈ [LCL,UCL]∣µ ≠ µo) (2.9)
= P (µ0 − kσ√
n
≤ X¯j ≤ µ0 + kσ√
n
∣µ ≠ µo)
= P (µ0 − kσ√
n
≤ X¯j ≤ µ0 + kσ√
n
∣µ ∈ {µ1, µ2})
= P (µ0 − kσ√
n
≤ X¯j ≤ µ0 + kσ√
n
∣µ = µ1)P (µ = µ1)
+P (0 − kσ√
n
≤ X¯j ≤ µ0 + kσ√
n
∣µ = µ2)P (µ = µ2).
Put B1 = P (µ0 − kσ√n ≤ X¯j ≤ µ0 + kσ√n ∣µ = µ1)P (µ = µ1)
= P (µ0 − kσ√
n
≤ X¯j ≤ µ0 + kσ√
n
)P (µ = µ1).
We have µ1 = µ0 − δσ so B1 becomes:
B1 = P (−k + δ√n ≤ Zj ≤ k + δ√n)P (µ = µ1)
= [Φ(k + δ√n) −Φ(−k + δ√n)]P (µ = µ1)
23
Since Φ(−k + δ√n) = 1 −Φ(k − δ√n) and Φ(k + δ√n) = 1 −Φ(−k − δ√n)
then
B1 = [Φ(k − δ√n) −Φ(−k − δ√n)]P (µ = µ1).
Put B2 = P (µ0 − kσ√n ≤ X¯j ≤ µ0 + kσ√n ∣µ = µ2)P (µ = µ2)
= P (µ0 − kσ√
n
≤ X¯j ≤ µ0 + kσ√
n
)P (µ = µ2).
We have µ2 = µ0 + δσ and P (µ = µ2) = 1 − P (µ = µ1). Then
B2 = P [−k − δ√n ≤ Z ≤ k − δ√n][1 − P (µ = µ1]
= [Φ(k − δ√n) −Φ(−k − δ√n)][1 − P (µ = µ1)].
Because β = B1 +B2, we finally obtain:
β = Φ(k − δ√n) −Φ(−k − δ√n). (2.10)
It follows that the power is:
1 − β = 1 − [Φ(k − δ√n) −Φ(−k − δ√n)]
= Φ(−k + δ√n) +Φ(−k − δ√n). (2.11)
The average run length (ARL)
The average run length (ARL) is another characteristic of the chart which shows the
average number of the samples which are taken before the moment we get the first
out-of-control signal. Let M be a random variable which represents the number
of samples necessary to obtain an out-of-control signal. Then M has geometric
distribution with parameter P (X¯ ∉ [UCL,LCL]∣µ) and the average run is
ARL = E[M ∣µ] = ∞∑
m=1mP (X¯ ∉ [UCL,LCL]∣µ)P (X¯ ∈ [LCL,UCL]∣µ)m−1
= 1
P (X¯ ∉ [LCL,UCL]∣µ) .
There are two types of ARL:
ARL1: In order to obtain the first false alarm, we need to draw an average number
of samples and this is ARL1 - the average run length when the process is in control:
ARL1 = E(M ∣µ = µo) = ∞∑
m=1mα(1 − α)m−1 = 1α, (2.12)
24
where α is the type-I error.
ARL2: The average run length when the process is out-of-control represents the
average number of samples drawn for detecting the shifts:
ARL2 = E(M ∣µ ≠ µo) = ∞∑
m=1mβm−1(1 − β) = 11 − β, (2.13)
where β is the type-II error.
The expected number of samples
To determine the expected loss per hour, we first need to determine the expected
number of samples for in-control and for out-of-control states and the expected num-
ber of false alarms. Below are the definitions of the random variables we use and
the related quantities.
Expected number of samples drawn in the state I.
Let SI be a random variable which represents the number of samples drawn in state I{SI = i}, means that the transition to state II happens in the interval [(ih, (i+1)h)]:
{SI = i} = {(ih < T ≤ (i + 1)h)}.
Probability mass function of Si is
P (SI = i) = P ((ih < T ≤ (i + 1)h)) = ∫ (i+1)h
ih
λe−λtdt
= (1 − e−λh)(e−λh)i, i = 0,1, . . . .
This is the probability mass function for geometric distribution with parameter(1 − e−λh). Therefore, the expected value of the number of samples drawn in the
state I is:
µSI = E(SI) = e−λh1 − e−λh = 1eλh − 1 . (2.14)
Expected number of samples drawn in the state II
Let SII be a random variable which represents the number of samples drawn in
state II. {SII = j} means that the shift in state II is detected at jth sample. The
probability mass function for the variable SII is:
P (SII = j) = (1 − β)βj−1, j = 1,2,3, ...
25
This is the probability mass function for geometric distribution with the parameter(1 − β). Then the expected value of SII is
µsII = E(SII) = 11 − β . (2.15)
The expected number of samples in cycle
If we suppose that S is a random variable which represents the number of samples
in cycle, then S = SI +SII . Let µs be the expected number of samples in cycle. Then
µs = µsI + µsII . (2.16)
The expected number of the false alarms
False alarms occur only from sampling in state I. Let F be a random variable that
represents the number of false alarms in the cycle. In each sample in state I, the
probability of false alarm is α. Therefore {F ∣SI} is a random variable that has
Binomial distribution with parameters (SI , α) and mean
E(F ∣SI) = αSI .
The expected number of false alarms in the cycle is:
µF = E(F ) = E(E(F ∣SI)) = E(αSI) = αE(SI)= α
eλh − 1 . (2.17)
Economic characteristics of the chart
The most important goal of improving the quality of production is to get the great-
est income. When the production process is in in-control state, it is more profitable
than when the process is out-of-control. Therefore, the process must be put under
control by doing appropriate procedures when there is any warning. Two types of
procedures exist and they are executed during the production process, and each of
them has economic consequences see [27], [35].
26
Inspection and sampling procedure
This procedure is done in order to detect a shift. The periodic sampling and testing,
as well as the investigation of false alarm are performed. The economic consequence
of this procedure is described by the cost and it is as follows.
The cost of inspection and sampling:
It is presumed that c∗1 is the fixed cost per sample, and c∗2 is a variable cost per unit
sampled. Then the sampling and inspection costs for sample size n are c∗1 + c∗2n.
The expected number of samples in a cycle is denoted µs. The expected cycle cost
of sampling and inspection is:
Cs = (c∗1 + c∗2n)µs.
The cost of false alarms
It is presumed that c∗3 is the cost of a false alarm, which includes the cost for
suspension of production during the examination. Then the expected cost of false
alarms in the cycle is:
CF = c∗3µF ,
where µF is expected number of false alarms.
The renewal procedures
These procedures are responsible for the transition from state II to state I. When
the cause of the deviation is corrected, the process returns into the controlled stage
again and this is related to the following consequences:
Cost of correction
It is presumed that c∗4 is equivalent to the cost for correcting the whole cycle, which
includes the cost of suspension of the production process, while the process of cor-
rection is in progress.
Revenues from correction
It is presumed that g1 is the profit of a produced unit when the process is in control,
27
g2 is the profit of the produced unit in the out-of-control state. So, g1 − g2 is the
revenue of the transition from a state that is out of control to the state that is in
control by the unit. Hourly penalty cost associated with production in the out-of-
control state is C4 = g1 − g2. The rate of production is υ unit of hours, and the
average length of the period when the process is in control is the 1λ hour. Then the
expected benefit from the transition into state I is
(g1 − g2)υ
λ
.
Assuming that b∗ is the expected net benefit per renewal, then
b∗ = (g1 − g2)υ
λ
−C∗4 . (2.18)
And the average number of units produced in the cycle is N = µshυ by the unit.
2.3 Loss Function
The loss function is L(n, k, h) constructed by combining some characteristics of the
production process: If we suppose that P is the expected profit per cycle then
P = b∗ −Cs −CF , (2.19)
and we define the loss as follows
L = −P = Cs +CF − b∗. (2.20)
Then, the expected loss per unit produced, expressed as a function of the chart
parameters n, k, h is given by:
L(n, k.h) = L
N
L(n, k, h) = c∗3
hυ
[c∗1
c∗3 + c∗2c∗3 n −
b∗
c∗3 − µF
µs
]. (2.21)
To simplify the optimization procedure for the function (2.21) we define the flowing
relative quantities: the relative fixed cost per sample is C1 = c∗1c∗3 ≥ 0, the relative cost
per unit sampled is C2 = c∗2c∗3 ≥ 0, and the relative benefit renewal is b = b∗c∗3 ≥ 0.
Then the loss function can be written as
L(n, k, h) = c∗3
hυ
[C1 +C2n − b − µF
µS
]. (2.22)
28
Since µS = 1(eλh−1) +ARL2 and µF = 1(ehλ−1)ARL1 ,
then
L(n, k, h) = c∗3
hυ
[C1 +C2n − b(ehλ − 1) − 1ARL1
1 +ARL2(ehλ − 1)]. (2.23)
The equation 2.23 is the required loss function which has to be minimized in order
to determine the parameters (n, k, h).
The values (n∗, k∗, h∗) are the optimum design for the X-bar chart iff
L(n∗, k∗, h∗) ≤ L(n, k, h) ∀(n, k, h).
In order to reduce the number of parameters of the objective function and also to
simplify the optimization procedure, we multiply it by υλc∗3 . Put s = δn, x = λh and
c = C2δ2 . Then the standardized loss function is given by:
Ls(s, k, x) = 1
x
[C1 +Cs2 − b(ex − 1) − 1ARL1
1 +ARL2(ex − 1)]. (2.24)
Is is obvious that the loss function L(n, k, h) attains its minimum at (n∗, k∗, h∗) iff
the the standardized loss function Ls(s, k, x) attains its minimum at s∗, k∗, x∗. So
the values (n∗, k∗, h∗) are the optimum design for the X-bar chart iff
L(s∗, k∗, x∗) ≤ Ls(s, k, x) ∀(s, k, x).
Numerical minimization
The minimizing problem of the loss function is solved by using MATLAB Optimiza-
tion: Toolbox for minimization of nonlinear function with constraints for c1 = 0, c
in the interval [0.0001, 1], and b in the interval [10, 860]. A sample of the output of
this program (s∗, k∗, x∗) given in Table (2.3).
Modeling
A non-linear regression models are built by using the a package (Sigma plot) on the
results obtained from the standardized loss function (s∗, k∗, x∗). They are dependent
variables while (c1, c, b) are independent variables. In this case when C ≥ 0.067 and[0.105 − c]b ≤ 0.12638 we obtain (s∗ = k∗ = x∗ = 0) the following models:
ˆ Model s∗
sˆ∗ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩
1.2980b2−1.3342ln(b2)− 115090.6604c2
ln(b2) −40756.0351c2b2+0.0091cln(b)
b2+ 1323037.3265c4
b2
−22616.3265c2b2−790.8770cb2−1.7369ln(b2) − 6.5210bln(b) +3.4296+0.0091ln(b)−17.3746
0
29
ˆ Model k∗
kˆ∗ = ⎧⎪⎪⎨⎪⎪⎩
1.8569b+2.1325cc−0.0177ln(b)+8.4195cb
b(70486cc−58.84825+1.4296c) − 8.6777ln(b)
0
ˆ Model x∗
xˆ∗ = ⎧⎪⎪⎨⎪⎪⎩
0.0001c+25.2567bc2−61.3759c3b+264477ln(b)−0.0366ln(b)−1699cb+0.1659
ln(b)+1.2856c2b2−0.0190cb2+24.9502cb) 77996.3927b4 +0.0217ln(b)
0
In all cases we obtain that R2 = 0.9999 and P-value were for all parameters and
coefficients less than the 0.0001.
b c s∗ k∗ x∗ Ls
10 0.0401 2.3197 1.5193 0.2441 -7.1111
60 0.0606 2.1796 1.3918 0.1047 -51.3540
110 0.0591 2.2194 1.4248 0.0758 -98.1887
160 0.0701 2.1127 1.3423 0.0654 -144.6392
210 0.0321 2.6057 1.2742 0.0589 -196.6382
260 0.0801 2.0335 1.2784 0.0527 -239.7041
310 .01910 2.8967 1.9686 0.0313 -296.5662
360 0.0371 2.5303 1.6744 0.0360 -341.4782
410 0.0301 2.6526 1.7719 0.0316 -391.6524
460 0.0291 2.6730 1.7881 0.0295 -440.7895
510 0.0471 2.3925 1.5637 0.0324 -485.9551
560 0.0291 2.6751 1.7895 0.0267 -538.7770
610 0.0491 2.3694 1.5450 0.0299 -583.2849
660 0.0551 2.2985 1.4885 0.0297 -631.0765
710 0.0471 2.3963 1.6664 0.0274 -681.5591
760 0.0711 2.1350 1.3583 0.0296 -726.1856
810 0.0651 2.1936 1.4056 0.0280 -776.0662
860 0.0621 2.2253 1.4306 0.0268 -825.5621
Table 2.3: Sample of output (s∗, k∗, x∗) and loss function for given values of b, c
The Table 2.4 contains the estimated values (sˆ∗, kˆ∗, xˆ∗) obtained by using the pre-
vious models for some values (c, b).
Note that, by comparing the results in the (Table 2.3) with those in the (Table 2.4) it
is obvious that there are no significant differences between actual values (s∗, k∗, x∗)
and estimated values (sˆ∗, kˆ∗, xˆ∗). However, some differences have been found, which
30
b c sˆ kˆ xˆ Ls
10 0.0401 2.3190 1.5294 0.2415 -7.1111
60 0.0601 2.1867 1.3988 0.1047 -51.3539
110 0.0591 2.2164 1.4262 0.0761 -98.1886
160 0.0701 2.1128 1.3372 0.0658 -144.8491
210 0.0321 2.6021 1.7316 0.0448 -196.6374
260 0.0801 2.0325 1.2706 0.0527 -239.7035
310 0.0191 2.8966 1.9599 0.0305 -296.5597
360 0.0371 2.5267 1.6724 0.0359 -341.4781
410 0.0301 2.6494 1.0661 0.0314 -391.6520
460 0.0291 2.6703 1.7815 0.0294 -440.7891
510 0.0471 2.3907 1.5691 0.0324 -485.9547
560 0.0291 2.6730 1.7830 0.0268 -538.7767
610 0.0491 2.3686 1.5511 0.0299 -583.2845
660 0.0551 2.2994 1.4947 0.0296 -631.0763
710 0.0471 2.3962 1.5726 0.0274 -681.5587
760 0.0711 2.1389 1.3586 0.0295 -726.1854
810 0.0651 2.2918 1.4879 0.0268 -777.7131
860 0.0621 2.2291 1.4351 0.0268 -825.5619
Table 2.4: Estimated the optimal values (s∗, k∗, x∗)
have a slight impact on the standardized loss function.
Algorithms
The next algorithm is used to obtain the approximate optimal sampling interval
h∗, the approximate optimal control limits k∗, as well as the approximate optimal
sample size n∗. The following values are given:
1. The mean of in-control period 1λ
2. The cost parameters:
ˆ b∗ benefit per renewal
ˆ c∗1 fixed cost per sample (in this study c∗1 = 0 )
ˆ c∗2 cost per unit sampled
ˆ c∗3 cost per erroneous inspection
3. Shift parameter δ
31
b c hˆ∗ h∗
λ = 0.05 λ = 0.5 λ = 3 λ = 0.05 λ = 0.5 λ = 3
10 0.0411 4.924 0.1641 0.0821 4.874 0.1625 0.0812
60 0.0391 1.840 0.0613 0.0307 1.846 0.0615 0.0376
110 0.0711 1.596 0.0532 0.0266 1.606 0.0535 0.02677
160 0.0501 1.192 0.0397 0.0199 1.194 0.0398 0.0199
210 0.0351 0.932 0.0311 0.0155 0.924 0.0308 0.0154
260 0.0461 0.906 0.0302 0.0151 0.908 0.0303 0.0151
310 0.0241 0.676 0.0225 0.0113 0.666 0.0222 0.011
360 0.0401 0.736 0.0245 0.0123 0.738 0.0246 0.0123
410 0.0441 0.710 0.0237 0.0118 0.710 0.237 0.0118
460 0.0391 0.606 0.0215 0.0108 0.646 0.0215 0.0108
510 0.0321 0.576 0.0192 0.0996 0.578 0.0193 0.096
560 0.0601 0.662 0.0221 0.0110 0.658 0.0219 0.011
610 0.0311 0.522 0.0174 0.0087 0.526 0.0175 0.088
660 0.0101 0. 342 0.0114 0.0057 0.304 0.0113 0.0057
710 0.0471 0.550 0.0183 0.0092 0.550 0.0183 0.0092
760 0.0611 0.568 0.0189 0.0095 0.568 0.0189 0.0095
810 0.0391 0.484 0.0161 0.0081 0.488 0.0163 0.0081
860 0.0511 0.508 0.0169 0.0085 0.508 0.0169 0.0085
Table 2.5: Approximate values of optimal parameters
4. Relative economical consequences b = b∗c∗3 and c = c∗2c∗3δ will be used as known.
a) hˆ∗ Algorithm
To determine the approximate optimal sample interval xˆ∗ determine the following:
ˆ Determine the optimal values of xˆ∗ from x∗ model at the specified values of
the relative economic consequence using 1-4 above.
ˆ The optimal value (hˆ∗ the approximate samples interval) is obtained through
the relationship hˆ∗ = xˆ∗λ .
The result of applying this algorithm for selected values b, c and λ is given in the
table (2.5).
b) kˆ∗ Algorithm
32
b c kˆ∗ x∗
10 0.0391 1.6187 1.6212
110 0.0711 1.3252 1.3186
160 0.0501 1.5177 1.5220
210 0.0351 1.6942 1.6925
260 0.0461 1.5665 1.5708
310 0.0241 1.8684 1.8592
360 0.0401 1.6377 1.6395
410 0.0441 1.5935 1.5975
460 0.0391 1.6521 1.6524
510 0.0321 1.7444 1.7403
560 0.0071 2.3626 2.3830
610 0.0311 1.7601 1.7553
660 0.0101 2.2300 2.2398
710 0.0471 1.5663 1.5626
760 0.0611 1.4374 1.4423
810 0.0391 1.6565 1.6586
860 0.0511 1.5282 1.5354
Table 2.6: The estimated optimal values of control limits kˆ∗
The optimal values of control limits kˆ∗ are obtained directly from x∗ model at the
values of b and c specified using 1-4 above.
The Table (2.6) presents the result of applying kˆ∗ algorithm for selected value of b
and c.
c) nˆ∗ Algorithm
To determine the approximate optimal sample size nˆ∗, first the optimal value of sˆ∗
is obtained from s∗ model at the values of b and c specified using 1-4 above. Then
the approximate optimal sample size nˆ∗ is given by:
nˆ∗ = [(sˆ∗)2
δ2
].
The result of applying this algorithm for selected values of b, c and δ is presented in
the Table (2.7).
33
b c nˆ∗ n∗
δ = 0.5 δ = 1.5 δ = 2 δ = 0.5 δ = 1.5 δ = 2
10 0.0411 21 2 1 21 2 1
6 0.0391 24 3 2 24 3 2
110 0.0711 17 2 1 18 2 1
160 0.0501 22 2 1 22 2 1
210 0.0351 26 3 2 26 3 2
260 0.0461 23 3 1 23 3 1
310 0.0241 31 3 2 31 3 2
360 0.0401 25 3 2 25 3 2
410 0.0441 24 3 1 24 3 1
460 0.0391 25 3 2 25 3 2
510 0.0321 27 3 2 27 3 2
560 0.0601 20 2 1 20 2 1
610 0.0311 28 3 2 28 3 2
660 0.0101 41 5 3 42 5 3
710 0.0471 23 3 1 23 3 1
760 0.0611 20 2 1 20 2 1
810 0.0391 25 3 3 25 3 2
860 0.0511 22 2 1 22 2 1
Table 2.7: Approximate optimal values of control limits
As can be seen, there are different effect on economical consequences for the optimal
design of X-bar chart:
ˆ Increasing the per unit (relative) sampling cost (the relative c2) c∗2 leads to the
decrease of the optimum sample size nˆ∗ and optimum control limits kˆ∗, but it
leads to the increase of the optimum interval between samples hˆ∗.
ˆ Increasing the benefit per renewal (the relative b) b∗ leads to the increase of
both the optimum sample size nˆ∗ and the optimum control limits kˆ∗, but
decreases the optimum interval between samples hˆ∗.
ˆ Increasing the intensity parameter λ decreases the optimum interval between
samples nˆ∗.
ˆ Increasing of δ decreases the optimum sample size nˆ∗ .
34
Chapter 3
EWMA-type control charts for
the mean
The idea to design the implementation of an the exponential weighted moving aver-
age chart is based on the idea of time series moving average model. In this chapter
we explain how to construct EWMA chart for monitoring the process mean and
we discuss the basic assumptions about the process and method of design of X-bar
and EWMA control charts which we use when the quality characteristic has Normal
distribution. A model of construction of the chart, together with conclusions, and
also sensitivity analysis are discussed.
3.1 Construction of EWMA control chart
Exponentially Weighted Moving Average (EWMA) control chart is one of the con-
trol charts that can be used for detecting shifts in the process. Roberts was the first
to use it in 1959. It consists of three horizontal lines, central line or mean (CL),
upper control limit (UCL) and lower control limit (LCL).
The Shewhart control chart applies statistical test to the process using only the
immediate data, while the EWMA chart uses the previous data with weights. The
EWMA chart can detect small shifts in the process mean. When λ is near unity or
when λ = 1, then EWMA chart is simillar to, or it is equal to the X-bar chart. Mont-
gomery [44] contributed to this subject and specified the Exponentially Weighted
Moving-Average as:
Yi = λXi + (1 − λ)Yi−1. (3.1)
Here 0 < λ ≤ 1 is constant, Xi are observations, and the starting value of the process
is Y0 = µ0.
It can happen sometimes that the mean of preliminary data is used as starting value
of the EWMA control chart, Y0 = X¯. EWMA Yi is then the weighted average of all
previous sample means.
35
We may substitute Yi−1 on the right hand side of equation 3.1. Then
Yi = λXi + (1 − λ)[λXi−1 + (1 − λ)Yi−2]
= λXi + λ(1 − λ)Xi−1 + (1 − λ)2Yi−2.
Continuing to substitute recursively Yi−j, j = 1,2,3, .., i − 1 we obtain
Yi = λ i−1∑
j=0(1 − λ)jXj−1 + (1 − λ)iY0.
If the observations Xi are independent random variables with mean µ0 and variance
σ2, then the mean and the variance of Yi for EWMA control chart are given by:
EYi = µ0,
σ2Yi = σ2( λ2 − λ)[1 − (1 − λ)2i]. (3.2)
In this way, it is possible to construct the EWMA control chart by plotting the Yi
versus the sample number i (or time). Upper Control Limit (UCL), Control Limit
(CL) and Lower Control Limit (LCL) for the EWMA control chart are stated below:
UCL = µ0 +Lσ√ λ
2 − λ[1 − (1 − λ)2i] (3.3)
CL = µ0
LCL = µ0 −Lσ√ λ
2 − λ[1 − (1 − λ)2i], (3.4)
where L is the width of the control limits, usually equal 2 or 3, as in the design of
Shewhart control chart limits.
In case the sample means fall within the control limits, we can say that the process
is in control regarding the mean of process, and we can say that the process is out-
of-control if the sample means are out of the control limits.
The term [1 − (1 − λ)2i] approaches unity, when i gets larger. After the EWMA
control chart’s was used for several time periods, the control limits will get closer to
steady-state values.
UCL = µ0 +Lσ√ λ
2 − λ (3.5)
CL = µ0
36
LCL = µ0 −Lσ√ λ
2 − λ. (3.6)
In a starting period, it is good to use the exact control limits, because this will
improve the detection of value that is far from the process target.
EWMA control chart for the mean
We just need to substitute Xi with X¯i and σ with
σ
n
in the previous equations if
the sample size of subgroups is n, n > 1. So that
Yi = λX¯i + (1 − λ)Yi−1. (3.7)
Then the starting value is the overall average, Y0 = X¯, and the variance of Yi for
EWMA control chart is given by:
σ2Yi = σ2n ( λ2 − λ)[1 − (1 − λ)2i]. (3.8)
Our assumption is that the quality characteristic is normally N(µ0, σ2) distributed,
where µ0 and σ are unknown. Then it is necessary to estimate them from a sample
or a subgroup. In this case the limits of EWMA will be
UCL = X¯ +Lσˆ
n
√
λ
2 − λ[1 − (1 − λ)2i] (3.9)
CL = µ0 = X¯
LCL = X¯ −Lσˆ
n
√
λ
2 − λ[1 − (1 − λ)2i]. (3.10)
Since
lim
i→∞ σ
2
n
( λ
2 − λ)[1 − (1 − λ)2i] = σ2n ( λ2 − λ),
then control limits after several time periods will get closer to steady-state value
values:
UCL = X¯ +Lσˆ
n
√
λ
2 − λ (3.11)
CL = µ0 = X¯
LCL = X¯ −Lσˆ
n
√
λ
2 − λ. (3.12)
37
Phase I and phase II of EWMA chart
We use the same data for application of phase I and phase II of EWMA control
chart as in the previous chapter. We assume that the EWMA control chart has
λ = 0.20 and L = 3. We have Y0 = X¯ = 74.00118 and σˆ = 0.00917. Then the first two
values of EWMA control chart are:
Y1 = λX¯1 + (1 − λ)Y0= 0.2(74.010) + 0.8(74.00118) = 74.00298
and
Y2 = λX¯2 + (1 − λ)Y1= 0.2(74.001) + 0.8(74.00298) = 74.00250
The other values of the EWMA statistic, for both cases are computed similarly and
summarized in the Table 3.1.
The control limits for period i = 1 are found from the formulas
UCL = X¯ +L σˆ√
n
√
λ
2 − λ[1 − (1 − λ)2i]
UCL = 74.00118 + 30.00917√
5
√
0.2
2 − 0.2[1 − (1 − 0.2)2(1)] = 74.00668
and
LCL = X¯ −L σˆ√
n
√
λ
2 − λ[1 − (1 − λ)2i]
LCL = 74.0018 − 30.00917√
5
√
0.2
2 − 0.2[1 − (1 − 0.2)2(1)] = 73.99567.
For period i = 2, the limits are
UCL = 74.00118 + 30.00917√
5
√
0.2
2 − 0.2[1 − (1 − 0.2)2(2)] = 74.00822
LCL = 74.00118 − 30.00917√
5
√
0.2
2 − 0.2[1 − (1 − 0.2)2(2)] = 73.99413.
The control limits vary with i = 1,2, ... until they stabilize at the steady-state value
given by 3.9 and 3.10:
UCL = X¯ +L σˆ√
n
√
λ
2 − λ
= 74.00118 + 30.00917√
5
√
0.2
2 − 0.2 = 74.01034
38
and
LCL = X¯ −L σˆ√
n
√
λ
2 − λ
= 74.00118 − 30.00917√
5
√
0.2
2 − 0.2 = 73.99201.
The corresponding control limits are summarized in the table 3.1.
The EWMA control charts are shown in the Figures 3.1 and 3.2, for the phase
I and phase II, respectively. From the first chart, we can conclude that process
is in-control and retain estimates we got from preliminary samples. The second
EWMA chart, constructed for additional samples, gives indications that the process
is out-of-control.
Figure 3.1: The EWMA control chart in phase I
39
su
b
g
ro
u
p
,
i
X¯
i
E
W
M
A
,
Y
i
L
C
L
Y
i
U
C
L
Y
i
su
b
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ro
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p
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i
X¯
i
E
W
M
A
,
Y
i
L
C
L
Y
i
U
C
L
Y
i
1
74
.0
10
74
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02
98
73
.9
95
67
74
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06
68
21
74
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00
74
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01
62
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92
01
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35
2
74
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02
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13
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22
22
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3
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23
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01
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4
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24
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25
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6
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33
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21
74
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27
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73
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13
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08
22
8
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28
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9
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54
10
73
.9
98
74
.0
00
64
73
.9
92
06
74
.0
10
29
30
73
.9
97
74
.0
00
37
73
.9
92
51
74
.0
09
84
11
73
.9
94
73
.9
99
35
73
.9
92
04
74
.0
10
31
31
74
.0
07
74
.0
01
74
73
.9
92
33
74
.0
10
03
12
74
.0
01
73
.9
99
76
73
.9
92
03
74
.0
10
33
32
74
.0
06
74
.0
02
51
73
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92
21
74
.0
10
14
13
73
.9
98
73
.9
99
49
73
.9
92
02
74
.0
10
33
33
73
.9
98
74
.0
01
57
73
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92
13
74
.0
10
22
14
73
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90
73
.9
97
63
73
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92
01
74
.0
10
34
34
74
.0
11
74
.0
03
49
73
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92
09
74
.0
10
26
15
74
.0
06
73
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99
30
73
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92
01
74
.0
10
34
35
74
.0
13
74
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05
32
73
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92
06
74
.0
10
29
16
73
.9
97
73
.9
98
76
73
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92
01
74
.0
10
34
36
74
.0
04
74
.0
05
05
73
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92
04
74
.0
10
31
17
74
.0
01
73
.9
99
17
73
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92
01
74
.0
10
34
37
74
.0
17
74
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07
36
73
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92
03
74
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10
33
18
74
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07
74
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00
82
73
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92
01
74
.0
10
34
38
74
.0
2
74
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09
81
73
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92
02
74
.0
10
33
19
73
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98
74
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00
29
73
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92
01
74
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10
35
39
74
.0
23
74
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12
53
73
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92
01
74
.0
10
34
20
74
.0
09
74
.0
02
07
73
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92
01
74
.0
10
35
40
74
.0
13
74
.0
12
58
73
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92
01
74
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10
34
T
ab
le
3.
1:
T
h
e
E
W
M
A
va
lu
es
Y
i
40
Figure 3.2: The EWMA control chart in phase II
Shewhart control charts are very effective in phase I, because they are easily con-
structed and interpreted, and because they are effective in detecting both large,
sustained shifts in the process parameters and outliers and measurement errors.
The types of assignable causes that usually occur in phase I result in fairly large
process shifts, which represents exactly the situation in which the Shewhart control
chart is most effective. In phase II, it is usually assumed that the process is rea-
sonably stable. Frequently, the assignable causes that occur in phase II only cause
smaller process shifts, because (it is hoped) most of the sources of greater variability
have been systematically removed during phase I. Now we want to highlight process
monitoring, not bringing the process into control. The EWMA control chart is much
more likely to be effective in phase II, when smaller shifts occur.
3.2 Basic assumptions and parameters for the de-
sign of control charts
n: Sample size
h: Sampling interval
k: Width of control limits.
σ: Standard deviation of observations.
µ0: In-control process mean.
µ1: Out-control process mean.
δ: Shifts in process mean in standard deviation units when assignable cause occurs(δ = ∣µ1−µ0∣σ ).
41
λ: It is assumed that in-control time follows the exponential distribution with mean
1
λ .
τ : Expected time of occurrence of the assignable cause, given it occurs between the
j − th and (j + 1) − st samples is
τ = 1 − (1 + λh)e−λh
λ(1 − e−λh) .
α: Probability of type I error.
β: Probability of type II error.
ATS: Average time to signal an expected shift, ATS = h1−β .
g: The time required to take a sample and interpret the results.
D: The time required to find an assignable cause following an out-of-control signal.
c∗1: Fixed cost per sample. c∗2: Variable cost per unit sampled.
Cs: Cost for searching and repairing the assignable cause.
CF : Cost of investigating a false alarm.
g1: Profit per hour earned by the process operating in-control.
g2: Profit per hour earned by the process operating out-of-control (g1 > g2).
C4: Hourly penalty cost associated with production in the out-of-control state.
µF : Expected number of false alarms, µF = α e−hλ1−e−hλ .
E(T ): Expected length of a production cycle. The production cycle time T is defined
as the interval of time from the start of production (the process is assumed to start
in the in-control state), following an adjustment to the detection and elimination of
the assignable cause.
E(C): Expected cost per cycle. Total cost C consists of the costs of testing and
sampling, the costs associated with investigating an out-of-control signal and with
the repair or correction of any assignable causes found, and the costs associated with
the production of nonconforming items.
µS: Expected number of samples taken within a cycle, µS = E(T )h .
Remark: we derived the errors of type I and II of X-bar chart on the previous
chapter, but for EWMA chart we suppose that [1− [1−λ]2i] = 1 when i is large. So
the probability of type error is:
ˆ Type-I error (The false alarm). This type of error happens when the chart
signals that the process is in state II and actually it is not. Then:
42
α = P (Yi ∉ [LCL,UCL]∣µ = µo)= 1 − P (Yi ∈ [LCL,UCL]∣µ = µo)
= 1 − ⎛⎝µ0 − kσ√n
√
λ
2 − λ ≤ Yi ≤ µ0 − kσ√n
√
λ
2 − λ⎞⎠= 1 − P [−k ≤ Zj ≤ k]= 1 − [Φ(k) −Φ(−k)] = 2Φ(−k). (3.13)
ˆ The probabilities of type II error of EWMA control chart given by:
β = P (Yi ∈ [LCL,UCL]∣µ ≠ µo) (3.14)
= P ⎛⎝µ0 − kσ√n
√
λ
2 − λ ≤ Yi ≤ µ0 + kσ√n
√
λ
2 − λ ∣µ ≠ µo⎞⎠
= P ⎛⎝µ0 − kσ√n
√
λ
2 − λ ≤ Yi ≤ µ0 + kσ√n
√
λ
2 − λ ∣µ ∈ {µ1, µ2}⎞⎠
= P ⎛⎝µ0 − kσ√n
√
λ
2 − λ ≤ Yi ≤ µ0 + kσ√n
√
λ
2 − λ ∣µ = µ1⎞⎠P (µ = µ1)
+P ⎛⎝µ0 − kσ√n
√
λ
2 − λ ≤ Yi ≤ µ0 + kσ√n
√
λ
2 − λ ∣µ = µ2⎞⎠P (µ = µ2).
Put
B1 = P ⎛⎝µ0 − kσ√n
√
λ
2 − λ ≤ Yi ≤ µ0 + kσ√n
√
λ
2 − λ ∣µ = µ1⎞⎠P (µ = µ1).
We have µ1 = µ0 − δσ so B1 becomes:
B1 = P (−k + δ√n√2−λλ ≤ Zi ≤ k + δ√n√2−λλ )P (µ = µ1)
= ⎡⎢⎢⎢⎢⎣Φ
⎛⎝k + δ√n
√
2 − λ
λ
⎞⎠ −Φ⎛⎝−k + δ√n
√
2 − λ
λ
⎞⎠
⎤⎥⎥⎥⎥⎦P (µ = µ1).
Since
Φ(−k + δ√n√2−λλ ) = 1 −Φ(k − δ√n√2−λλ ) and
Φ(k + δ√n√2−λλ ) = 1 −Φ(−k − δ√n√2−λλ ), then
43
B1 = ⎡⎢⎢⎢⎢⎣Φ
⎛⎝k − δ√n
√
2 − λ
λ
⎞⎠ −Φ⎛⎝−k − δ√n
√
2 − λ
λ
⎞⎠
⎤⎥⎥⎥⎥⎦P (µ = µ1).
Put
B2 = P ⎛⎝µ0 − kσ√n
√
λ
2 − λ ≤ Yi ≤ µ0 + kσ√n
√
λ
2 − λ ∣µ = µ2⎞⎠P (µ = µ2).
We have µ2 = µ0 + δσ and P (µ = µ2) = 1 − P (µ = µ1). Then
B2 = P ⎡⎢⎢⎢⎢⎣−k − δ
√
n
√
2 − λ
λ
≤ Zi ≤ k − δ√n√2 − λ
λ
⎤⎥⎥⎥⎥⎦ [1 − P (µ = µ1]
= ⎡⎢⎢⎢⎢⎣Φ
⎛⎝k − δ√n
√
2 − λ
λ
⎞⎠ −Φ⎛⎝−k − δ√n
√
2 − λ
λ
⎞⎠
⎤⎥⎥⎥⎥⎦ [1 − P (µ = µ1)].
Because β = B1 +B2, we finally obtain:
β = Φ⎛⎝k − δ√n
√
2 − λ
λ
⎞⎠ −Φ⎛⎝−k − δ√n
√
2 − λ
λ
⎞⎠ . (3.15)
It follows that the power of determining the real (existing) deviation is:
1 − β = 1 − ⎡⎢⎢⎢⎢⎣Φ
⎛⎝k − δ√n
√
2 − λ
λ
⎞⎠ −Φ⎛⎝−k − δ√n
√
2 − λ
λ
⎞⎠
⎤⎥⎥⎥⎥⎦
= Φ⎛⎝−k + δ√n
√
2 − λ
λ
⎞⎠ +Φ⎛⎝−k − δ√n
√
2 − λ
λ
⎞⎠ . (3.16)
The expected cycle time
Each production cycle begins with the process in the in-control state and continues
until process monitoring via control charts results in an out-of-control signal. Fol-
lowing an adjustment in which the process is returned to the in-control state, a new
cycle begins. As it is shown in Figure 1.3 (see [63]), a cycle is partitioned into five
sub-intervals of time, Ia,Ib,Ic,Id, and Ie.
ˆ Ia: The time from the start of the cycle, until the assignable cause occurs .
ˆ Ib: The time until the next sample is taken from the end of Ia.
ˆ Ic: The time until the chart gives an out-of-control signal from the end of Ib.
ˆ Id: The time to analyze the sample and the chart from the end of Ic
44
Figure 3.3: Cycle Process
ˆ Ie: The time to discover the assignable cause and repair the process from the
end of Id.
Three time points are marked on Figure 3.3: the point 1 represents the last sample
before the occurrence of the assignable cause; the point 2 represents the first sample
taken after the assignable cause and the point 3 is the moment when lack of control
is detected. Process remains in-control state during time interval Ia and it is in the
out-control state during intervals Ib, Ic, Id and Ie. For each part the expected time
is calculated as follows:
E(Ia) = 1
λ
E(Ib) = h − τ
E(Ic) = ATS − h
E(Id) = gn
E(Ie) =D.
Finally by combining all these expected values we get E(T ), which is the expected
length of a production cycle. The production cycle time T is defined as the interval
of time from the start of production (the process is assumed to start in the in-control
state), following an adjustment for detection and elimination of the assignable cause:
E(T ) = E(Ia) +E(Ib) +E(Ic) +E(Id) +E(Ie)
= 1
λ
− τ + gn + h/(1 − β) +D.
Then the expected cost per cycle is
E(C) = g1
λ
+ g2(−τ + gn + h
1 − β +D) − c∗4 −CF − (c∗1 + c∗2n)µs.
45
Loss function
The efficiency of the design of the control chart is presented by the expected loss
per hour (loss function), defined as
E(L) = g1 + E(c)
E(T ) .
Then
E(L) = c∗1 + c∗2n
h
+ C4(−τ + gn + h1−β +D) − c∗4 −CF )( 1λ − τ + gn + h1−β +D) . (3.17)
Method of design
The design of a control chart is defined as a design in which the expected loss
cost function E(L) is minimized, subject to a constrained minimum value of power
and maximum value of the Type I error probability, and ATS of an expected shift.
We wish to find sample size, control-limit width, and sampling frequency to
minimize E(L) (3.18)
subject to
α ≤ α0, p ≥ p0, ATS ≤ ATS0,
where p = 1 − β, α0, p0 and ATS0 are, respectively, the power, the desired bound of
the Type I error probability, the power and ATS of an expected shift (see [53]).
3.3 Numerical example
This example was presented by Montgomery [45]. There is a manufacturer who
produces a nonreturnable glass bottles which are used for packaging a carbonated
soft drink beverage. The thickness of the bottle walls is considered to be an impor-
tant quality characteristic. If the wall isn’t thick enough, then the internal pressure
during filling may cause the bottle to burst. The manufacturer has for some time
applied X¯ control charts to monitor the process. These control charts were designed
in accordance with statistical criteria. But, because the manufacturer wishes to cut
the costs, he wants to design an economically optimum X¯ control chart for the pro-
cess. After analyzing the salaries of quality control technicians and the costs of the
test equipment, the estimate cost of taking a sample is $1, whereas the variable cost
is approximately $0.01 per bottle. On average, 1 minute (0.0167h) is necessary to
measure and record the thickness of the bottle’s wall. The process can be subjected
46
to a couple of types of assignable causes. But when the process goes out-of-control,
the magnitude of the shift is equal to about two standard deviations.
Process shift happens randomly with a frequency of approximately one every 20h.
Therefore, the exponential distribution with the parameter λ = 0.05 is a realistic
model of the run length. One hour is usually necessary to investigate an out-of-
control signal. To investigate an action signal which ends with the elimination of
the assignable cause costs $25. The investigation of a false alarm costs $50.
It is estimated by the manufacturer that when we operate in the out-of-control
state for one hour the penalty cost is $100. It was necessary to design a model
for the X¯ control chart and EWMA chart. (see [26],[45]). Therefore the costs are,
C∗1 = 1,C∗2 = 0.1, C∗4 = 25 ,C∗3 = 50, δ = 2 g = 0.0167, λ = 0.05, C4 = 100, D = 1.
Let X be the measurable characteristic. Constrained minimization of expected loss
function was done using genetic algorithm ga from R package GA (see [55]).
We selected the following values for bounds on type I error probability, power and
ATS an expected shift: α0 = 0.05; p0 = 0.9; ATS0 = 2h. We consider that the quality
characteristics of X-bar and EWMA control charts have normal distribution. By
solving equations α = α(k) = 0.05 and p = p(k) = 0.9 using Brent’s root-finding
method (see [12]), we got values k1 and k2. By the constraints 3.18 we got interval
of values [k1; k2] for parameter k and [0; 2p(k1)] for parameter h. By constrained
optimization of expected loss function, we found the optimal values of parameters
k and h . Results of the minimization of expected loss function is given in Tables
1-4, for different values of the shift, λ and sample sizes from 3 to 10.
By looking at the results in Tables 3.2-3.5, we can make following conclusions:
1. The value of k in X-bar chart is less than the value of k in EWMA chart for
larger shifts.
2. The value of h in X-bar chart is bigger than the value of h in EWMA chart.
3. At the same value of the shift δ we can see the value of ATS in X-bar chart is
bigger than the value of ATS in EWMA chart.
4. Increasing the magnitude of the shift δ increases the power of X-bar chart.
Also, we can see that the power of X-bar chart is smaller than the power of
EWMA chart.
5. For both charts increasing the sample size n increases the power of the chart
at same value of the shift δ.
47
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
k
h
α
1
−β
A
T
S
E
L
k
h
α
1
−β
A
T
S
E
L
n
=3
1.
92
81
4
0.
25
88
9
0.
05
38
4
0.
14
67
9
1.
76
48
26
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97
7
1.
94
74
3
0.
26
75
3
0.
05
14
8
0.
74
23
7
0.
36
03
8
34
.8
86
62
n
=4
1.
91
46
0
0.
32
32
1
0.
05
55
4
0.
18
19
8
1.
77
60
7
23
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44
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1.
95
97
3
0.
32
98
2
0.
05
00
5
0.
85
09
3
0.
38
75
9
33
.0
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71
n
=5
1.
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69
3
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04
7
0.
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27
5
0.
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75
5
1.
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5
0.
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30
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=6
1.
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4
0.
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31
0
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5
1.
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1
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4
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15
7
0.
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53
7
28
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35
66
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=7
1.
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8
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5
0.
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14
7
0.
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66
2
1.
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95
1
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7
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4
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4
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n
=8
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1
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55
9
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7
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2
0.
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4
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n
=9
1.
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14
8
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3.
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5
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05
3
0.
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93
7
28
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1
n
=10
1.
92
34
8
0.
60
59
1
0.
05
44
2
0.
36
62
8
1.
65
42
3
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09
3.
45
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1
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05
4
0.
90
06
0
0.
47
32
3
28
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13
85
T
ab
le
3.
2:
D
es
ig
n
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
w
it
h
δ
=0.5
48
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
k
h
α
1
−β
A
T
S
E
L
k
h
α
1
−β
A
T
S
E
L
n
=3
1.
93
37
3
0.
68
83
8
0.
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31
5
0.
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02
1
1.
63
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9
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41
3.
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6
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5
0.
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01
1
0.
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73
5
0.
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42
7
26
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1
n
=4
1.
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97
8
0.
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32
6
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00
2
0.
51
60
8
1.
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95
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4.
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2
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-0
6
0.
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=7
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9
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1
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0.
42
72
7
4.
39
E
-1
3
0.
96
05
4
0.
44
48
2
27
.9
70
79
n
=10
1.
95
97
6
1.
15
35
4
0.
05
00
2
0.
88
54
2
1.
30
28
2
12
.6
88
86
6.
86
25
0.
42
84
0
6.
77
E
-1
2
0.
99
56
6
0.
43
02
7
28
.1
65
94
T
ab
le
3.
3:
D
es
ig
n
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
δ
=1
49
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
k
h
α
1
−β
A
T
S
E
L
k
h
α
1
−β
A
T
S
E
L
n
=3
1.
95
18
4
1.
00
08
5
0.
05
09
6
0.
74
09
4
1.
35
07
8
13
.2
02
18
6.
01
68
3
0.
42
03
5
1.
78
E
-0
9
0.
96
22
5
0.
43
68
4
26
.7
77
23
n
=4
1.
95
97
3
1.
14
52
8
0.
05
00
3
0.
85
08
9
1.
34
59
8
12
.4
96
09
6.
46
20
9
0.
42
15
4
1.
03
E
-1
0
0.
99
44
2
0.
42
39
0
26
.9
78
93
n
=5
2.
07
19
4
1.
05
58
4
0.
03
82
7
0.
90
01
1
1.
17
30
1
11
.7
91
37
8.
06
85
7
0.
42
27
5
7.
11
E
-1
6
0.
97
69
1
0.
43
27
4
27
.1
79
48
n
=6
2.
38
05
6
0.
86
10
3
0.
01
72
9
0.
90
21
1
0.
95
44
6
10
.8
65
95
8.
29
98
9
0.
42
38
4
1.
04
E
-1
6
0.
99
67
6
0.
42
52
1
27
.3
78
94
n
=7
2.
67
86
3
0.
75
22
9
0.
00
73
9
0.
90
14
7
0.
83
45
1
10
.3
79
09
8.
57
58
5
0.
42
49
8
9.
84
E
-1
8
0.
99
95
7
0.
42
51
6
27
.5
77
3
n
=8
2.
94
96
3
0.
69
50
6
0.
00
31
8
0.
90
2
0.
77
05
8
10
.1
80
74
6.
96
25
7
0.
42
61
2
3.
34
E
-1
2
1
0.
42
61
2
27
.7
74
58
n
=9
3.
05
30
7
0.
69
63
4
0.
00
22
7
0.
92
60
4
0.
75
19
5
10
.1
06
6
7.
39
43
3
0.
42
72
7
1.
42
E
-1
3
1
0.
42
72
7
27
.9
70
79
n
=10
3.
14
15
1
0.
70
23
1
0.
00
16
8
0.
94
54
1
0.
74
28
6
10
.0
73
73
7.
22
11
0.
42
84
1
5.
16
E
-1
3
1
0.
42
84
09
28
.1
65
94
T
ab
le
3.
4:
D
es
ig
n
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
δ
=1.5
50
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
k
h
α
1
−β
A
T
S
E
L
k
h
α
1
−β
A
T
S
E
L
n
=3
2.
18
16
6
0.
99
96
9
0.
02
91
3
0.
90
01
6
1.
11
05
8
11
.2
51
3
7.
96
35
8
0.
42
04
0
1.
67
E
-1
5
0.
99
24
2
0.
42
36
1
26
.7
77
3
n
=4
2.
71
54
4
0.
72
66
4
0.
00
66
2
0.
90
05
3
0.
80
69
1
10
.1
22
86
7.
76
14
2
0.
42
15
3
8.
40
E
-1
5
1
0.
42
15
3
26
.9
78
93
n
=5
3.
04
71
9
0.
69
38
3
0.
00
23
1
0.
92
29
1
0.
75
17
8
9.
83
60
3
8.
86
82
2
0.
42
26
9
7.
43
E
-1
9
1
0.
42
26
9
27
.1
79
48
n
=6
3.
19
79
6
0.
70
46
4
0.
00
13
8
0.
95
55
3
0.
73
74
3
9.
73
64
8
9.
64
40
1
0.
42
37
5
5.
21
E
-2
2
1
0.
42
37
5
27
.3
78
9
n
=7
3.
34
37
7
0.
71
19
7
0.
00
08
3
0.
97
42
8
0.
73
07
7
9.
71
10
8
10
.5
60
75
0.
42
49
8
4.
53
E
-2
6
1
0.
42
49
8
27
.5
77
3
n
=8
3.
48
36
6
0.
71
10
1
0.
00
04
9
0.
98
51
2
0.
72
17
6
9.
72
67
5
11
.9
06
9
9.
11
07
7
1.
09
E
-3
2
1
9.
11
07
8
60
.5
42
19
n
=9
3.
62
15
5
0.
71
26
1
0.
00
02
9
0.
99
13
1
0.
71
88
6
9.
76
58
3
13
.3
71
6
0.
42
84
8.
86
E
-4
1
1
0.
42
84
28
.1
65
94
n
=10
3.
75
54
9
0.
71
41
3
0.
00
01
7
0.
99
49
0.
71
78
9.
81
82
8
13
.3
71
59
0.
42
84
0
8.
86
E
-4
1
1
0.
42
84
0
28
.1
65
94
T
ab
le
3.
5:
D
es
ig
n
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
w
it
h
δ
=2
51
6. At the same value of the shift δ, we can see that the expected value of the
loss function in X-bar chart is less than the expected value of loss function
of EWMA chart. Increasing the magnitude of the shift δ decreases the loss
function in X-bar chart and increasing the magnitude of the shift δ increases
the loss function of EWMA chart.
7. At the same value of the shift δ, we can see that increasing the sample size n
decreases the loss function of X-bar chart and increases the loss function of
EWMA chart.
3.4 Control Chart Performance
There are two commonly used methods for comparing control chart performance.
The first method is to determine the charts operating characteristic (OC) curve.
The second is to determine the average run length (ARL). Those methods are re-
lated with sensitivity analysis of the chart.
This research focuses on increasing the sensitivity of X-bar chart and EWMA chart
in order to detect small shifts. The sensitivity of the charts is measured by their
power 1 − β and the average run length (ARL). The power of the chart represents
the probability of a shift in the process level, and it is described in the process level
and given in the form of an operating characteristic (OC) curve.
The OC Curve: The OC Curve indicates the probability of an observation falling
within the control limits according to the state of the process. For example, for the
X¯ and EWMA charts, we can see the probability of staying within the limits β for
particular values of the process mean, as it is presented in the equations(2.11) and
(3.16), respectively. The curve is generated by calculating β for different values of µ.
The next five tables have the same value of probability of power X-chart and EWMA
chart with different values of the sample size n and shift δ and λ and L = k = 3 (the
usual three-sigma limits). The OC curves are also presented in the following figures.
The figure 3.4 (later used several times for comparison) shows the powers of X-bar
chart with different values of sample size n and k = 3. It is compared with figures
(3.5),(3.7), (3.9), (3.11) and (3.13) which show the EWMA chart with different val-
ues of sample size n for the given values λ and L = 3.
52
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
δ
=0.2
5
δ
=0.5
δ
=0.7
5
δ
=1.5
δ
=2.5
δ
=3
δ
=0.2
5
δ
=0.5
δ
=0.7
5
δ
=1.5
δ
=2.5
δ
=3
n
=3
0.
00
54
3
0.
01
64
8
0.
04
44
9
0.
34
38
8
0.
90
82
7
0.
98
59
6
0.
38
36
9
0.
99
19
9
1
1
1
1
n
=4
0.
00
64
4
0.
02
27
8
0.
06
68
1
0.
50
00
1
0.
97
72
5
0.
99
86
5
0.
54
87
6
0.
99
94
1
1
1
1
1
n
=5
0.
00
75
1
0.
02
99
4
0.
05
27
5
0.
63
83
8
0.
99
52
0.
99
99
0.
68
83
2
0.
99
99
7
1
1
1
1
n
=6
0.
00
86
3
0.
03
79
4
0.
05
31
0
0.
74
99
3
0.
99
91
1
0.
99
99
9
0.
79
51
1
1
1
1
1
1
n
=7
0.
00
98
1
0.
04
67
7
0.
05
14
7
0.
83
36
4
0.
99
98
5
1
0.
87
09
1
1
1
1
1
1
n
=8
0.
01
10
3
0.
05
64
0.
05
43
8
0.
89
3
0.
99
99
8
1
0.
92
16
1
1
1
1
1
n
=9
0.
01
23
1
0.
06
68
1
0.
05
27
9
0.
93
32
1
1
0.
95
38
9
1
1
1
1
1
n
=10
0.
01
36
5
0.
07
79
8
0.
05
44
2
0.
95
93
7
1
1
0.
97
36
4
1
1
1
1
1
n
=11
0.
01
50
4
0.
08
98
5
0.
05
44
2
0.
97
58
6
1
1
0.
98
53
1
1
1
1
1
n
=12
0.
01
64
8
0.
10
24
1
0.
05
44
2
0.
98
59
6
1
1
0.
99
19
9
1
1
1
1
1
n
=13
0.
01
79
7
0.
11
56
1
0.
05
44
2
0.
99
19
9
1
1
0.
99
57
2
1
1
1
1
1
n
=14
0.
01
95
2
0.
12
94
2
0.
05
44
2
0.
99
55
1
1
1
0.
99
77
6
1
1
1
1
1
n
=15
0.
02
11
3
0.
14
37
8
0.
05
44
2
0.
99
75
2
1
1
0.
99
88
4
1
1
1
1
1
T
ab
le
3.
6:
T
h
e
p
ow
er
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
w
it
h
λ
=0.0
5
53
Figure 3.4: The power of X¯ chart
Figure 3.5: The power of EWMA chart with λ = 0.05
54
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
δ
=0.2
5
δ
=0.5
δ
=0.7
5
δ
=1.5
δ
=2.5
δ
=3
δ
=0.2
5
δ
=0.5
δ
=0.7
5
δ
=1.5
δ
=2.5
δ
=3
n
=3
0.
00
54
3
0.
01
64
8
0.
04
44
9
0.
34
38
8
0.
90
82
7
0.
98
59
6
0.
13
29
6
0.
78
08
1
0.
99
61
2
1
1
1
n
=4
0.
00
64
4
0.
02
27
8
0.
06
68
1
0.
50
00
1
0.
97
72
5
0.
99
86
5
0.
20
59
6
0.
91
29
1
0.
99
98
1
1
1
n
=5
0.
00
75
1
0.
02
99
4
0.
09
29
3
0.
63
83
8
0.
99
52
0.
99
99
0.
28
66
2
0.
96
94
9
0.
99
99
9
1
1
1
n
=6
0.
00
86
3
0.
03
79
4
0.
12
24
4
0.
74
99
3
0.
99
91
1
0.
99
99
9
0.
37
04
3
0.
99
03
2
1
1
1
1
n
=7
0.
00
98
1
0.
04
67
7
0.
15
49
0.
83
36
4
0.
99
98
5
1
0.
45
34
9
0.
99
71
7
1
1
1
1
n
=8
0.
01
10
3
0.
05
64
0.
18
97
9
0.
89
3
0.
99
99
8
1
0.
53
27
7
0.
99
92
2
1
1
1
1
n
=9
0.
01
23
1
0.
06
68
1
0.
22
66
3
0.
93
32
1
1
0.
60
61
1
0.
99
98
1
1
1
1
n
=10
0.
01
36
5
0.
07
79
8
0.
26
49
1
0.
95
93
7
1
1
0.
67
22
1
0.
99
99
5
1
1
1
1
n
=11
0.
01
50
4
0.
08
98
5
0.
30
41
5
0.
97
58
6
1
1
0.
73
04
7
0.
99
99
9
1
1
1
1
n
=12
0.
01
64
8
0.
10
24
1
0.
34
38
8
0.
98
59
6
1
1
0.
78
08
1
1
1
1
1
1
n
=13
0.
01
79
7
0.
11
56
1
0.
38
36
9
0.
99
19
9
1
1
0.
82
35
8
1
1
1
1
1
n
=14
0.
01
95
2
0.
12
94
2
0.
42
31
9
0.
99
55
1
1
1
0.
85
93
5
1
1
1
1
1
n
=15
0.
02
11
3
0.
14
37
8
0.
46
20
6
0.
99
75
2
1
1
0.
88
88
6
1
1
1
1
1
T
ab
le
3.
7:
T
h
e
p
ow
er
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
w
it
h
λ
=0.1
55
Figure 3.6: The power of X¯ chart
Figure 3.7: The power of EWMA chart with λ = 0.1
56
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
δ
=0.5
δ
=1.5
δ
=2
δ
=2.5
δ
=3
δ
=3.5
δ
=0.5
δ
=1.5
δ
=2
δ
=2.5
δ
=3
δ
=3.5
n
=3
0.
01
64
8
0.
34
38
8
0.
67
87
2
0.
90
82
7
0.
98
59
6
0.
99
89
0.
34
38
8
1
1
1
1
1
n
=4
0.
02
27
8
0.
50
00
1
0.
84
13
5
0.
97
72
5
0.
99
86
5
0.
99
99
7
0.
50
00
1
1
1
1
1
1
n
=5
0.
02
99
4
0.
63
83
8
0.
92
95
1
0.
99
52
0.
99
99
1
0.
63
83
8
1
1
1
1
1
n
=6
0.
03
79
4
0.
74
99
3
0.
97
12
2
0.
99
91
1
0.
99
99
9
1
0.
74
99
3
1
1
1
1
1
n
=7
0.
04
67
7
0.
83
36
4
0.
98
90
3
0.
99
98
5
1
1
0.
83
36
4
1
1
1
1
1
n
=8
0.
05
64
0.
89
3
0.
99
60
6
0.
99
99
8
1
1
0.
89
3
1
1
1
1
1
n
=9
0.
06
68
1
0.
93
32
0.
99
86
5
1
1
1
0.
93
32
1
1
1
1
1
n
=10
0.
07
79
8
0.
95
93
7
0.
99
95
6
1
1
1
0.
97
58
6
1
1
1
1
1
n
=11
0.
08
98
5
0.
97
58
6
0.
99
98
6
1
1
1
0.
97
58
6
1
1
1
1
1
n
=12
0.
10
24
1
0.
98
59
6
0.
99
99
6
1
1
1
0.
98
59
6
1
1
1
1
1
n
=13
0.
11
56
1
0.
99
19
9
0.
99
99
9
1
1
1
0.
99
19
9
1
1
1
1
1
n
=14
0.
12
94
2
0.
99
55
1
1
1
1
1
0.
99
55
1
1
1
1
1
1
n
=15
0.
14
37
8
0.
99
75
2
1
1
1
1
0.
99
75
2
1
1
1
1
1
T
ab
le
3.
8:
T
h
e
p
ow
er
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
w
it
h
λ
=0.2
57
Figure 3.8: The power of X¯ chart
Figure 3.9: The power of EWMA chart with λ = 0.2
58
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
δ
=0.5
δ
=1.5
δ
=2
δ
=2.5
δ
=3
δ
=3.5
δ
=0.5
δ
=1.5
δ
=2
δ
=2.5
δ
=3
δ
=3.5
n
=3
0.
01
64
8
0.
34
38
8
0.
67
87
2
0.
90
82
7
0.
98
59
6
0.
99
89
0.
23
92
6
0.
99
99
5
1
1
1
1
n
=4
0.
02
27
8
0.
50
00
1
0.
84
13
5
0.
97
72
5
0.
99
86
5
0.
99
99
7
0.
36
15
8
1
1
1
1
1
n
=5
0.
02
99
4
0.
63
83
8
0.
92
95
1
0.
99
52
0.
99
99
1
0.
48
32
7
1
1
1
1
1
n
=6
0.
03
79
4
0.
74
99
3
0.
97
12
2
0.
99
91
1
0.
99
99
9
1
0.
59
49
9
1
1
1
1
1
n
=7
0.
04
67
7
0.
83
36
4
0.
98
90
3
0.
99
98
5
1
1
0.
69
14
7
1
1
1
1
1
n
=8
0.
05
64
0.
89
3
0.
99
60
6
0.
99
99
8
1
1
0.
77
08
6
1
1
1
1
1
n
=9
0.
06
68
1
0.
93
32
0.
99
86
5
1
1
1
0.
83
36
4
1
1
1
1
1
n
=10
0.
07
79
8
0.
95
93
7
0.
99
95
6
1
1
1
0.
88
16
6
1
1
1
1
1
n
=11
0.
08
98
5
0.
97
58
6
0.
99
98
6
1
1
1
0.
91
73
6
1
1
1
1
1
n
=12
0.
10
24
1
0.
98
59
6
0.
99
99
6
1
1
1
0.
94
32
4
1
1
1
1
1
n
=13
0.
11
56
1
0.
99
19
9
0.
99
99
9
1
1
1
0.
96
16
1
1
1
1
1
1
n
=14
0.
12
94
2
0.
99
55
1
1
1
1
1
0.
97
44
1
1
1
1
1
n
=15
0.
14
37
8
0.
99
75
2
1
1
1
1
0.
98
31
4
1
1
1
1
1
T
ab
le
3.
9:
T
h
e
p
ow
er
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
w
it
h
λ
=0.2
5
59
Figure 3.10: The power of X¯ chart
Figure 3.11: The power of EWMA chart with λ = 0.25
60
S
a
m
p
le
X
-b
a
r
E
W
M
A
si
ze
δ
=0.5
δ
=1.5
δ
=2
δ
=2.5
δ
=3
δ
=3.5
δ
=0.5
δ
=1.5
δ
=2
δ
=2.5
δ
=3
δ
=3.5
n
=3
0.
01
64
8
0.
34
38
8
0.
67
87
2
0.
90
82
7
0.
98
59
6
0.
99
89
0.
10
24
1
0.
98
59
6
0.
99
99
6
1
1
1
n
=4
0.
02
27
8
0.
50
00
1
0.
84
13
5
0.
97
72
5
0.
99
86
5
0.
99
99
7
0.
15
86
6
0.
99
86
5
1
1
1
1
n
=5
0.
02
99
4
0.
63
83
8
0.
92
95
1
0.
99
52
0.
99
99
1
0.
22
24
6
0.
99
99
1
1
1
1
n
=6
0.
03
79
4
0.
74
99
3
0.
97
12
2
0.
99
91
1
0.
99
99
9
1
0.
29
09
9
0.
99
99
9
1
1
1
1
n
=7
0.
04
67
7
0.
83
36
4
0.
98
90
3
0.
99
98
5
1
1
0.
36
15
8
1
1
1
1
1
n
=8
0.
05
64
0.
89
3
0.
99
60
6
0.
99
99
8
1
1
0.
43
19
1
1
1
1
1
n
=9
0.
06
68
1
0.
93
32
0.
99
86
5
1
1
1
0.
0.
50
00
1
1
1
1
1
1
n
=10
0.
07
79
8
0.
95
93
7
0.
99
95
6
1
1
1
0.
56
44
7
1
1
1
1
1
n
=11
0.
08
98
5
0.
97
58
6
0.
99
98
6
1
1
1
0.
62
42
4
1
1
1
1
1
n
=12
0.
10
24
1
0.
98
59
6
0.
99
99
6
1
1
1
0.
67
87
2
1
1
1
1
1
n
=13
0.
11
56
1
0.
99
19
9
0.
99
99
9
1
1
1
0.
72
76
1
1
1
1
1
n
=14
0.
12
94
2
0.
99
55
1
1
1
1
1
0.
77
08
6
1
1
1
1
1
n
=15
0.
14
37
8
0.
99
75
2
1
1
1
1
0.
80
86
7
1
1
1
1
1
T
ab
le
3.
10
:
T
h
e
p
ow
er
of
X
-b
ar
an
d
E
W
M
A
ch
ar
ts
w
it
h
λ
=0.4
61
Figure 3.12: The power of X¯ chart
Figure 3.13: The power of EWMA chart with λ = 0.4
62
By looking at the results in the tables and figures, we can make following conclusions.
ˆ The increase power of charts to detect the deviation increases with the large
sample size n. Therefore, detecting of the small variations is done by the use
of large sized samples.
ˆ The power of EWMA chart increases with the decrease of the value of λ.
The Average Run Length: The average run length (ARL) is the average number
of points plotted on the chart until an out-of-control condition is signaled. It is the
expected value of the run length distribution. It is connected to the OC curve as it
can be seen from equation (2.12) and (2.13). The ARL is used more often in control
chart literature than OC curve values, since it’s concept easier to comprehend. The
tables below show the ARL values for both charts with different values of λ and δ
and k = L = 3. We have found that in this case the sample size did not affect ARL1,
but decreasing the sample size leads to increasing ARL2 for both charts and chang-
ing the shift size produces a change in ARL2; decreasing the λ leads to increasing
ARL2 for EWMA chart.
shift in mean X-bar EWMA
δ λ = 1 λ = 0.05 λ = 0.1 λ = 0.2 λ = 0.25 λ = 0.4
0 370.37 370.37 370.37 370.37 370.37 370.37
0.25 184.16 60.68 31.39 22.48 7.52 2.61
0.5 60.68 9.76 4.18 2.91 1.28 1.01
0.75 22.48 2.91 1.49 1.23 1 1
1 9.76 1.47 1.06 1.01 1 1
1.5 2.91 1.01 1 1 1 1
2 1.1 1 1 1 1 1
2.5 1.1 1 1 1 1 1
3 1.01 1 1 1 1 1
4 1 1 1 1 1 1
Table 3.11: ARL of X-bar and EWMA charts with n = 3
63
shift in mean X-bar EWMA
δ λ = 1 λ = 0.05 λ = 0.1 λ = 0.2 λ = 0.25 λ = 0.4
0 370.37 370.37 370.37 370.37 370.37 370.37
0.25 133.16 33.4 15.59 10.76 3.49 1.45
0.5 33.4 4.5 2.07 1.57 1.03 1
0.75 10.76 1.57 1.08 1.02 1 1
1 4.5 1.08 1 1 1 1
1.5 1.57 1 1 1 1 1
2 1.10 1 1 1 1 1
2.5 1.08 1 1 1 1 1
3 1 1 1 1 1 1
4 1 1 1 1 1 1
Table 3.12: ARL of X-bar and EWMA charts with n = 5
shift in mean X-bar EWMA
δ λ = 1 λ = 0.05 λ = 0.1 λ = 0.2 λ = 0.25 λ = 0.4
0 370.37 370.37 370.37 370.37 370.37 370.37
0.25 73.26 12.82 5.5 3.77 1.49 1.03
0.5 12.82 1.77 1.13 1.04 1 1
0.75 3.77 1.04 1 1 1 1
1 1.77 1 1 1 1 1
1.5 1.04 1 1 1 1 1
2 1 1 1 1 1 1
2.5 1 1 1 1 1 1
3 1 1 1 1 1 1
4 1 1 1 1 1 1
Table 3.13: ARL of X-bar and EWMA charts with n = 10
64
shift in mean X-bar EWMA
δ λ = 1 λ = 0.05 λ = 0.1 λ = 0.2 λ = 0.25 λ = 0.4
0 370.37 370.37 370.37 370.37 370.37 370.37
0.25 47.33 6.96 3.02 2.16 1.13 1
0.5 6.96 1.24 1.01 1 1 1
0.75 2.16 1 1 1 1 1
1 1.24 1 1 1 1 1
1.5 1 1 1 1 1 1
2 1 1 1 1 1 1
2.5 1 1 1 1 1 1
3 1 1 1 1 1 1
4 1 1 1 1 1 1
Table 3.14: ARL of X-bar and EWMA charts with n = 15
65
Chapter 4
Optimal Process Calibration for
Non-symmetric Loss Functions
The process calibration is one of the methods in statistical process control for im-
proving the quality of the products. We mention again basic notions of statistical
process control and present loss functions that can be used in this area. This chap-
ter is concerned with the problem of process calibration for some non-symmetric
loss functions. We suggest an optimal calibration policy which can be implemented
using our programs written in statistical software R.
In this chapter we consider the problem of the process calibration, i.e. how to
set-up a manufacturing process in order to meet specification limits, by which we
mean how to minimize the expected loss. In Section 4.1, we introduced traditional
method of calibration, which corresponds to the symmetric loss function. However,
quite often the lower and the upper specification limits cannot be treated in the same
way. For example, it may happen that although one of the specification limits can be
exceeded, we obtain a nonconforming product that could be reworked (improved or
corrected) which requires some costs. Three models of non-symmetric loss functions
and their implementation in statistical software R are considered, respectively, in
Sections 4.2 and 4.3. The R code is given at the end of the chapter.
4.1 Symmetric loss function
Let X denote a quality characteristic of the product. We assume that X is normally
distributed with mean µ and standard deviation σ, i.e. X ∼ N (µ,σ2). We suppose
that the standard deviation σ is known.
The usual practice is to set-up the mean at the middle distance between the speci-
fication limits, i.e. µ0 = LSL +USL
2
and this is legitimate only if the loss does not
depend on which specification limit - the lower or upper - is exceeded. In that case
66
the loss function L(x) is symmetrical,
L(x) =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
w, x < LSL
0, LSL ≤ x ≤ USL
w, x > USL
where w > 0 (see Figure 4.1).
Figure 4.1: Symmetric loss function
This is a very simple method of process calibration. However, very often the as-
sumption of the symmetrical loss function is not fulfilled in practice. Let us consider
the following examples.
Example 4.1.1 The quality characteristic under study is the inner diameter of a
hole. Undersized holes can be reworked at extra costs. But the reduction of the
over-sized hole would be either impossible or would require much higher costs.
Example 4.1.2 The quality characteristic under study is the outside diameter of
the milled item. Over-sized items can be re-grounded at additional costs. But the
undersized item could be sold only for scrap.
Example 4.1.3 The quality characteristic under study is the blood glucose level
in diabetes patients. If measured value of blood sugar is too low or too high, i.e.
below or above allowed limits, then the blood sugar can be stabilized with some
extra costs (injection of insulin, food, tablets, etc.). These costs does not have to
be the same for too low and too high level of blood sugar.
67
These examples show that in situations described above, instead of the symmetrical
loss function, a non-symmetric loss function should be used.
4.2 Non-symmetric loss function
To find the optimal calibration policy for situations as described in the examples
given above we will consider three models of loss functions. First model was pro-
posed by Grzegorzewski and Mrowka [33] and other two models are suggested by
us.
First model
The first model of the loss function is given by the formula
L(x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
w, x < LSL
0, LSL ≤ x ≤ USL
z
x −USL
ULR −USL, USL < x ≤ ULR
z, x > ULR
where w > z and upper limit for rework ULR > USL (see Fig.4.2).
Figure 4.2: The first model of the loss function
This model could be used in cases when over-sized product items can be reworked
with some extra costs (Example 2).
68
In all models we suppose that the loss function is connected to the quality char-
acteristic X, where X has Normal distribution N(µ,σ2) and the corresponding
density is denoted by f(x). The expected loss function is equal to
EL(X) = EL = wP (X < LSL) + zP (X > ULR) + z∫ ULR
USL
x −USL
ULR −USLf(x)dx.
After some transformations we get
EL = wΦ(LSL − µ
σ
) + z + z ( µ −USL
ULR −US − 1)Φ(ULR − µσ ) ++ z µ −USL
ULR −USLΦ(USL − µσ ) −− σ z
ULR −USL (φ(ULR − µσ ) − φ(USL − µσ )) ,
where Φ and φ are, respectively, distribution function and density function of stan-
dard Normal distribution N (0,1). We consider the expected loss as a function of
a process mean µ, EL = EL(µ) and we are searching for such µ in which minimal
expected loss occurs. Thus we need
dEL(µ)
dµ
=
= z
ULR −USL [Φ(ULR − µσ ) −Φ(USL − µσ )] − wσ φ(LSL − µσ ) = 0.
This equation depends on µ and also LSL, USL, ULR, w, z, σ. To simplify it, we
introduce use new parameters:
a = ULR −USL
σ
,
b = USL −LSL
σ
,
K = w
z
.
The preceding equation reduces to
1
a
[Φ(a +∆) −Φ(∆)] −Kφ(−b +∆) = 0. (4.1)
The optimal calibration is given by
µ = USL −∆σ, (4.2)
where ∆ is the correction factor, and it is the solution of the equation 4.1.
69
For cases when oversized items are reworked with some extra costs (Example 4.1.2),
we could consider loss function of this kind.
The following two models could be appropriate in the cases when there is a possi-
bility of rework for both undersized and oversized items (Example 4.1.3).
Second model
Loss function is given by the formula
L(x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
4w, x < LLR
3w, LLR ≤ x < LSL
0, LSL ≤ x < USL
w, USL ≤ x ≤ ULR
2w, x > ULR
,
where w > 0 (see Fig.4.3).
Figure 4.3: The second model of the loss function
In this situation the expected loss is equal to
EL(X) = EL = wP (X < LLR)+3wP (X ≤ LSL)−wP (X ≤ ULR)−wP (X ≤ USL)+2w.
After some transformations we get
EL = wΦ(LLR − µ
σ
) + 3wΦ(LSL − µ
σ
) −wΦ(ULR − µ
σ
) −wΦ(USL − µ
σ
) + 2w.
70
To find the minimal expected loss we need to find µ for which dEL(µ)dµ is equal to
zero. we have
dEL(µ)
dµ
= w
σ
[φ(LLR − µ
σ
) + 3φ(LSL − µ
σ
) − φ(ULR − µ
σ
) − φ(USL − µ
σ
)] = 0.
After dividing this equation with −wσ and using the notation
a = ULR −USL
σ
= LSL −LLR
σ
,
b and ∆ as before, we get
φ(−a − b −∆) + 3φ(−b +∆) − φ(a +∆) − φ(∆) = 0. (4.3)
Third model
This model is given by the formula
L(x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
w2, x < LLR
w1
x −LLR
LSL −LLR −w2 x −LSLLSL −LLR, LLR ≤ x < LSL
0, LSL ≤ x ≤ USL
z
x −USL
ULR −USL, USL < x ≤ ULR
z, x > ULR
where w2 > w1 > z (see Fig.4.4).
In this situation the expected loss is equal to
EL(X) = EL = [w2 + w2 −w1
aσ
µ +w1LLR
aσ
−w2LSL
aσ
]Φ(LLR − µ
σ
) +
+ [w1 −w2
aσ
µ −w1LLR
aσ
+ +w2LSL
aσ
]Φ(LSL − µ
σ
) +
+ [− z
aσ
µ + zUSL
aσ
]Φ(USL − µ
σ
) +
+ [ z
aσ
µ − zUSL
aσ
− z]Φ(ULR − µ
σ
) +
+ w1 −w2
a
φ(LLR − µ
σ
) + w2 −w1
a
φ(LSL − µ
σ
) +
+ z
a
φ(USL − µ
σ
) − z
a
φ(ULR − µ
σ
) + z.
We assumed that
a = ULR −USL
σ
= LSL −LLR
σ
.
71
Figure 4.4: The third model of the loss function
To find the minimal expected loss we need to find µ which minimizes EL(µ) as
function of µ.
Thus
dEL(µ)
dµ
= w2 −w1
aσ
Φ(LLR − µ
σ
) − w2 −w1
aσ
Φ(LSL − µ
σ
) +
+ z
aσ
[Φ(ULR − µ
σ
) −Φ(USL − µ
σ
)] − w1
σ
φ(LSL − µ
σ
) +
+ z
σ
[φ(USL − µ
σ
) − φ(ULR − µ
σ
)] = 0.
Using also the notation b = USL−LSLσ , K1 = w1z and K2 = w2z , we get
K2 −K1
a
[Φ(−a − b +∆) −Φ(−b +∆)] + Φ(a +∆) −Φ(∆)
a
−− K1φ(−b +∆) + φ(∆) − φ(a +∆) = 0. (4.4)
4.3 Implementation in R
To calibrate optimally the process mean, we need to calculate the correction factor
∆. For three considered models, ∆ is calculated as a solution of equations (4.1),(4.3)
and (4.4), respectively. These equations depend on various parameters: equation
(4.1) depends on tree parameters a, b and K; equation (4.3) on two parameters a
and b, and the equation (4.4) depends on four parameters a, b, K1 and K2. We
solve these equations numerically, using Brent’s root- finding method (see [12]) im-
72
plemented in R function uniroot.
Correction factor for these three models of non-symmetric loss functions can be
found calling our R function corr.factor. The code is presented bellow:
corr.factor(delta,a,b,K,K1,K2,par)
by specifying the values of parameters (by choosing non-zero values for parameters
a, b and/or K, K1 and K2). Parameter par represents upper limit for correction
function necessary for Brent’s method to work properly (by default the value of
parameter par = 3).
Code of R-function:
model1<-function(delta,a,b,K){
1/a*(pnorm(a+delta)-pnorm(delta))-K*dnorm(-b+delta)}
model2<-function(delta,a,b){
dnorm(-a-b+delta)+3*dnorm(-b+delta)-dnorm(a+delta)-dnorm(delta) }
model3<-function(delta,a,b,K1,K2){
(K2-K1)/a*(pnorm(-a-b+delta)-pnorm(-b+delta))+1/a*(pnorm(a+delta)-
pnorm(delta))-K1*dnorm(-b+delta)+dnorm(delta)-dnorm(a+delta) }
corr.factor<-function(delta,a=3/0.1,b=3,K=0,K1=0,K2=0,par=3) {
if(K==0&&K1==0&K2==0)
{model.2<-function(delta){model2(delta,a,b)}
solution<-uniroot(model2,lower=0,upper=par)$root
}
else if(K1>0&&K2>0)
{model.3<-function(delta){model3(delta,a,b,K1,K2)}
solution<-uniroot(model3,lower=0,upper=par)$root
}
else if(K>0)
{model.1<-function(delta){model1(delta,a,b,K)}
73
solution<-uniroot(model1,lower=0,upper=par)$root
}
solution }
Now, we can find correction factors for considered non-symmetric loss functions for
given values of parameters.
First model. Parameters of the first model of non-symmetric loss function a, b and
K have the following values: a = 12, b = 6 and K = 2. By calling the function
corr.function(a=12,b=6,K=2)
we get ∆ = 2.308.
Second model. Parameters of the second model of non-symmetric loss function a,
b have the following values: a = 12, b = 6. By calling the function
corr.function(a=12,b=6)
we get ∆ = 2.817.
Third model. Parameters of the third model of non-symmetric loss function a, b
and K1 and K2 have the following values: a = 12, b = 6, K1 = 2, K2 = 3. By calling
the function
corr.function(a=12,b=6,K1=2,K2=3)
we get ∆ = 2.887.
74
Conclusions
The design and the performance of control charts are the most important problems
in Quality Control. In this thesis we studied and compared various characteristics
of control charts, together with related costs. We proposed a loss function, which
is connected to the production process and to X−bar quality control chart. Using
Matlab program for optimization, we found values of k, n and h, minimizing the
loss function for given costs. These values are compared with those obtained from
a non-linear regression model, built using a package Sigma plot, for given values of
cost. Production process can be modeled by Yi = λXi + (1 − λ)Yi−1, where 0 < λ ≤ 1
is a constant, Xi are N(µ,σ2) distributed. Exponentially Weighted Moving Aver-
age (EWMA) charts for this model are presented, and type I and type II errors
are calculated in the case when i is large. For different sample sizes, the new com-
parison between the optimal design of the X-bar and EWMA control charts for
Normally distributed quality characteristic is given, comparing the corresponding
cost-loss functions, power functions and average run lengths. The process of cali-
bration is one of the methods in statistical process control, introduced for improving
the quality of the products and for reducing the production costs. Two new models
of non-symmetrical loss function are introduced, and the value which minimizes the
expected loss is found for both cases. The production process is calibrated with this
new value.
One direction for further investigation in the area of Quality Control is the case
when the quality characteristic does not have symmetrical distribution. The other
direction is to study calibration for non-symmetric loss functions, which are not
piecewise constant, or piecewise linear, but have some parts with polynomial, or
exponential growth.
75
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81
Prijava teme doktorske disertacije
1. Biografija kandidata
ˆ Ime i prezime: Halima Ibrahim M. Elfaghihe
ˆ Datum rodenja: 13. 12. 1981
ˆ Zvanje: asistent na Odseku za statistiku, Univerzitet 7. april, Azave,
Libija
ˆ e-adresa: halimaebrahim80@gmail.com
Obrazovanje: Osnovnu i srednju sˇkolu zavrsˇila u Libiji. Osnovne studije
upisala 1998. a zavrsˇila 2002. u Libiji na Univerzitetu 7. april, Azave. Mag-
istarske studije upisala 2004, a zavrsˇila 2008. na Akademiji za poslediplomske
studije u Tripoliju, Libija, odbranivsˇi magistarski rad pod nazivom Optimal
Economic Design of kσ −X Chart(na arapskom).
Radno iskustvo: 2004-2008 demonstrator u nastavi na Odseku za statistiku
na Univerzitetu 7. april, 2008-2009 asistent na istom Univerzitetu.
Ucˇesˇc´e na projektima: ne
Oblasti naucˇnog istrazˇivanja: matematicˇka statistika, testiranje statisticˇkih
hipoteza, neparametarska statistika, kontrola kvaliteta, planiranje eksperime-
nata.
2. Spisak naucˇnih radova
Radovi u medunarodnim cˇasopisima sa SCI liste:
Economic statistical design of X-bar control chart for non-normal symmetric
distribution of quality characteristic, FILOMAT. Vol. 29, No. 10, pp. 2325-
2338 (2015) (sa K. Veljkovic´ i V. Jevremovic´).
Radovi u medunarodnim cˇasopisima:
One Possible Optimization of Economic design for X-Chart European Inter-
national Journal of Science and Technology, Vol. 2, No. 5, pp. 45-52 (2013).
82
Poglavlje u monografiji: ne
Radovi u Zbornicima:
Process Calibration for Some Examples of Non-symmetric Loss Functions.
Proceedings of Fourth Mathematical Confernce of the Republic of Srpska,
Trebinje, p.141-149. (2014) (sa K. Veljkovic´ i V. Jevremovic´)
Saops˜tenja na kongresima i konferencijama:
13. Srpski matematicˇki kongres, 22-25. maja 2014. Vrnjacˇka Banja sa radom
publikovanim u FILOMAT-u.
Cˇetvrta matematicˇka konferencija Republike Srpske, 6-7. juni 2014, Trebinje
sa radom publikovanim u Zborniku sa konferencije.
83
Prilog 1.
Izjava o autorstvu
Potpisani Halima Elfaghihe
broj indeksa 2048/2009
Izjavljujem
da je doktorska disertacija pod naslovom
”Konstrukcija i osobine kontrolnih kartica za stacionarne i neko-
relisane podatke”
ˆ rezultat sopstvenog istrazˇivacˇkog rada,
ˆ da predlozˇena disertacija u celini ni u delovima nije bila
predlozˇena za dobijanje bilo koje diplome prema studijskim
programima drugih visokosˇkolskih ustanova,
ˆ da su rezultati korektno navedeni i
ˆ da nisam krsˇio autorska prava i koristio intelektualnu svojinu
drugih lica.
Potpis doktoranda
U Beogradu,
84
Prilog 2.
Izjava o istovetnosti sˇtampane i elektronske
verzije doktorskog rada
Ime i prezime autora: Halima Elfaghihe
Broj indeksa: 2048/2009
Studijski program: Matematika-Verovatnoc´a i Statistika
Naslov rada: ”Konstrukcija i osobine kontrolnih kartica za stacionarne i
nekorelisane podatke”
Mentor: redovni Prof. dr.Slobodanka Jankovic´
Potpisani Halima Elfaghihe
Izjavljujem da je sˇtampana verzija mog doktorskog rada istovetna
elektronskoj verziji koju sam predao za objavljivanje na portalu
Digitalnog repozitorijuma Univerziteta u Beogradu.
Dozvoljavam da se objave moji licˇni podaci vezani za dobijanje
akademskog zvanja doktora nauka, kao sˇto su ime i prezime, go-
dina i mesto rodjenja i datum odbrane rada.
Ovi licˇni podaci mogu se objaviti na mrezˇnim stranicama digi-
talne biblioteke, u elektronskom katalogu i u publikacijama Uni-
verziteta u Beogradu.
Potpis doktoranda
U Beogradu,
85
Prilog 3.
Izjava o koriˇsc´enju
Ovlasˇc´ujem Univerzitetsku biblioteku ”Svetozar Markovic´” da u
Digitalni repozitorijum Univerziteta u Beogradu unese moju dok-
torsku disertaciju pod naslovom:
”Konstrukcija i osobine kontrolnih kartica za stacionarne i neko-
relisane podatke”
koja je moje autorsko delo.
Disertaciju sa svim prilozima predao sam u elektronskom formatu
pogodnom za trajno arhiviranje.
Moju doktorsku disertaciju pohranjenu u Digitalni repozitorijum
Univerziteta u Beogradu mogu da koriste svi koji posˇtuju odredbe
sadrzˇane u odabranom tipu licence Kreativne zajednice (Creative
Commons) za koju sam se odlucˇio.
1. Autorstvo
2. Autorstvo - nekomercijalno
3. Autorstvo - nekomercijalno - bez prerade
4. Autorstvo - nekomercijalno - deliti pod istim uslovima
5. Autorstvo - bez prerade
6. Autorstvo - deliti pod istim uslovima
(Molimo da zaokruzˇite samo jednu od sˇest ponudjenih licenci,
kratak opis licenci dat je na poledjini lista).
Potpis doktoranda
U Beogradu,
86
1. Autorstvo - Dozvoljavate umnozˇavanje, distribuciju i javno
saopsˇtavanje dela, i prerade, ako se navede ime autora na nacˇin
odredjen od strane autora ili davaoca licence, cˇak i u komercijalne
svrhe. Ovo je najslobodnija od svih licenci.
2. Autorstvo - nekomercijalno. Dozvoljavate umnozˇavanje, dis-
tribuciju i javno saopsˇtavanje dela, i prerade, ako se navede ime
autora na nacˇin odredjen od strane autora ili davaoca licence.
Ova licenca ne dozvoljava komercijalnu upotrebu dela.
3. Autorstvo - nekomercijalno - bez prerade. Dozvoljavate umno-
zˇavanje, distribuciju i javno saopsˇtavanje dela, bez promena, pre-
oblikovanja ili upotrebe dela u svom delu, ako se navede ime
autora na nacˇin odredjen od strane autora ili davaoca licence.
Ova licenca ne dozvoljava komercijalnu upotrebu dela. U odnosu
na sve ostale licence, ovom licencom se ogranicˇava najvec´i obim
prava koriˇsc´enja dela.
4. Autorstvo - nekomercijalno - deliti pod istim uslovima. Dozvol-
javate umnozˇavanje, distribuciju i javno saopsˇtavanje dela, i pre-
rade, ako se navede ime autora na nacˇin odredjen od strane
autora ili davaoca licence i ako se prerada distribuira pod is-
tom ili slicˇnom licencom. Ova licenca ne dozvoljava komercijalnu
upotrebu dela i prerada.
5. Autorstvo - bez prerade. Dozvoljavate umnozˇavanje, distribu-
ciju i javno saopsˇtavanje dela, bez promena, preoblikovanja ili
upotrebe dela u svom delu, ako se navede ime autora na nacˇin
odredjen od strane autora ili davaoca licence. Ova licenca dozvol-
java komercijalnu upotrebu dela.
6. Autorstvo - deliti pod istim uslovima. Dozvoljavate umnozˇavanje,
87
distribuciju i javno saopsˇtavanje dela, i prerade, ako se navede ime
autora na nacˇin odredjen od strane autora ili davaoca licence i
ako se prerada distribuira pod istom ili slicˇnom licencom. Ova
licenca dozvoljava komercijalnu upotrebu dela i prerada. Slicˇna
je softverskim licencama, odnosno licencama otvorenog koda.
88