final version economic design of vsi x-bar control charts for correlated non-normal case (1)
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economic X bar chartTRANSCRIPT
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Economic Design of Variable Sampling Interval X-Bar Control Charts for Monitoring Correlated Non-Normal Samples
Seyed Taghi Akhavan Niaki, Ph.D.1 Professor of Industrial Engineering, Sharif University of Technology
P.O. Box 11155-9414 Azadi Ave., Tehran 1458889694 Iran Phone: (+9821) 66165740, Fax: (+9821) 66022702, E-Mail: [email protected]
Fazlollah Masoumi Gazaneh, M.Sc.
Department of Industrial Engineering, Islamic Azad University (South-Tehran Branch), Tehran, Iran, E-Mail: [email protected]
Moslem Toosheghanian, M.Sc.
Department of Industrial Engineering, Iran University of Science and Technology [email protected]
Abstract
Recent studies have shown the X-barcontrol chart with variable sampling interval detects shifts in the process mean faster than the traditional X-barchart.These studies are usually based on the assumption that the process data are independently and normally distributed. However, many situations in practice violate these assumptions. In this study, a methodology is developed to
economically design a variable sampling interval X-barcontrol chart that takes into consideration correlated non-normal sample data. An example is provided to illustrate the solution procedure.A sensitivity analysis on the input parameters (i.e., the cost and the process parameters) is
performed taking into account the non-normality and the correlation on the optimal design of the
chart.
Keywords: X-bar control chart; Economic design; Variable sampling interval; Non-normality;
Correlated process data; Genetic algorithms
1 Corresponding author
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1. Introduction and literature survey
Control charts are the most important tools in statistical quality control environments for
continuous improvement of the quality of the items being produced and the X chart is the first
that was originally introduced by Shewhart in 1924 (Shewhart 1931). The design of the Shewhart
X control charts requires the determination of three designparameters: the sample size ( n ), the
sampling interval ( h ), and the control limits width ( k ). Shewhart considered these parameters
fixed, however recent studies have shown that varying the sampling interval (VSI), the sample
size (VSS), the sampling interval jointly with the sample size (VSSI), or all design parameters at
the same time, including k , (VP) reduce the detection time of process changes (see for example,
Reynolds et al., 1988; Baxley, 1996; Prabhu et al., 1993,1994; Costa, 1994, 1997, 1999). Further,
a survey of studies on VSR charts is presented in Tagaras (1998).
Duncan (1956) proposed the first economic model to determine the optimal values of n ,
h , and k of an X chart. Since then, the economical design of control charts has received
increasing attention in the literature (e.g., Montgomery, 1980; Vance, 1983; Woodall, 1986;
Pignatiello and Tsai, 1988; Niaki et al. 2010). The usual approach in an economical design is to
develop a cost model for a special type of industrial process, and then derive the optimal
parameters by minimizing the expected cost per unit of time.While Duncans (1956) and Chius
(1975) models for the economic design of control charts have been widely studied in practice, in
the former the process remains in operation during the period of search for the assignable cause
and in the latter the process is ceased.
In the design process of the Shewhart X control chart, one assumes the measurements
(observations) within the subgroups are independently and normally distributed. However, these
assumptions may not hold in many practices. For example, when a production process consists of
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multiple but similar units on a single part, such as several cavities on a single casting, multiple
pins on an integrated circuit chip, or multiple contact pads on a single machine mount, the
collected measurements within a subgroup are usually correlated (Grant and Leavenworth 1988).
Neuhardt (1987) considered the effect of correlation within a subgroup on the performances of
control charts. Yang and Hancock (1990) employed Neuhardts idea to conduct simulation
studies to specify the effect of correlated data on the performances of X , R, S and S2 charts. Liu
et al. (2002) used the correlation model of Yang and Hancock (1990) to develop a minimum-loss
design of fixed sample size and sampling interval (FSI) charts for correlated data. Chen and
Chiou (2005) combined Yang and Hancocks correlation model (1990) with the cost model of
Bai and Lee (1998) to make the economic design of the VSI charts for correlated data. Recently,
Chen et al. (2007) incorporated the Yang Hancocks correlation model (1990) with the cost
model of Costa (2001) to develop an economic design of the VSSI charts for correlated data.
If the size of subgroups (sample size) is large enough, the statistic X is approximately
normal based on the central limit theorem. However, when a control chart is applied to monitor
the process, unfortunately the sample size is usually not sufficiently large due to the sampling
cost. As a result, due to non-normality, the traditional method of designing the control chart may
decrease the ability of detecting assignable causes. Yourstone and Zimmer (1992) considered the
Burr distribution to represent the non-normal distributions and statistically designed the X chart.
Chen (2004) presented the economic design of VSI X control charts for non-normal data. Lin
and Chou (2005) employed the Burr distribution to design both symmetric and asymmetric-limit
VSSI X chart under non-normality. Recently, Lin and Chou (2007) studied the effect of non-
normality on the performances of adaptive X control charts (i.e., variable sample size, sampling
intervals, and and/or action limit coefficients) and concluded the VP X control chart is more
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effective than the other adaptive charts in detecting small process mean shifts under non-normal
process data.
In this research, the economic model of the VSI X chart for correlated non-normal
process data is developed using Burr distribution and the correlation model presented in Yang
and Hancock (1990). The FSI and VSI schemes are compared in terms of the long-run expected
cost per hour and statistical performances for various levels of correlation.
The remainder of the paper are as follows. In the next section, the concept of the VSI
control charts is briefly discussed. The Burr distribution representing the various types of non-
normal distributions is introduced in section 3. The economic model based on the Burr
distribution that works under correlated process data is developed in section 4. A solution
method to find the optimal values of the design parameters of an asymmetric VSI X chart is
presented in section 5. Sensitivity analysis on the process parameters and correlation coefficients
are conducted in section 6. Finally, conclusions make up the last section.
2. The VSI X control chart
Consider a process involving a single quality characteristic following a normal
distribution with in-control mean 0 and in-control known variance 2 . To monitor the process
mean using an X control chart, traditionally at a given sample point a sample of n independent
observations ( ; 1, 2,..., )ix i n are taken and the sample mean 1
1n
ii
x n x
is calculated accordingly. If the plotted x falls within the interval 0 k n , ( 0 is the centerline and k is the control limit width) the process is considered in-control and the next sample is taken in a
fixed interval of 1h time. Otherwise, a signal is sent to inform the operator to search for an
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assignable cause. This control chart is said to operate in a fixed sample-size ( )n and fixed
sampling interval 1( )h (FSI) condition.
When the VSI scheme is in use, the waiting time h until the next sampling point is a
function of the current x value. In other words, if ; 1, 2ix I i , then ih h where
' '1 0 1 0 2 0 2 0 1
'2 0 2 0 2
, ,
,
I k k k kn n n n
I k kn n
(1)
with 2 10 k k , ' '2 10 k k , and 2 1 0h h . According to Chen (2004), the slight cost savings
offered by the VSI schemes with more than two sampling interval sizes does not justify their use.
As an example, Fig. 1 depicts a VSI X chart using two interval lengths of 1h and 2h .
0
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The sample means in Fig.1 are plotted against the time on the horizontal axis. The first
sample mean falls within 2I , so the next sampling interval is 2h . The second one falls within 1I ,
which is close to the control limits. Hence, a shorter sampling interval is adopted to take the third
sample and so on.
The way the VSI X control charts work can improve the detection ability of FSI charts
by shortening the length of time until a signal is occurred. Nonetheless, the complexity will
increase when the number of divided areas becomes large. Further, traditional symmetric VSI
charts can be easily obtained by setting '1 1k k .
3. The Burr distribution
In this study, the Burr distribution is used to model non-normal process data. Burr (1942)
proposed a simple probability distribution function that was capable to model various types of
non-normal data. The cumulative distribution function of the two-parameter Burr distribution is
11 y 0
1
0 y
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Burr (1942) tabulated the first two moments (given in a Table numbered 2) and the
coefficients of skewness and kurtosis (given in another Table numbered 3) for the family of the
Burr distribution. These tables allow users to make a standardized transformation between the
variable of the Burr distribution (Y ) and any other random variable ( X ) when they have the
same coefficients of skewness and kurtosis. The standardized transformation between Y and
X is defined as follows:
x
X X Y MS S (3)
Where S and M are the sample mean and standard deviation of the Burr distribution,
respectively and X and xS are the sample mean and standard deviation of the data set (as a
random variable).
Different combinations of the parameters 1c and 1k can cover an extensive range of skewness and kurtosis coefficients of various probability density functions (e.g., Normal,
Gamma, Beta, etc.). For instance, when c = 4.8621 and k = 6.3412, the Burr distribution
approximates the normal distribution (Yourstone and Zimmer 1992). Further, the first four
moments of the empirical distribution can be used to determine c and k . Burr (1967) used this
distribution to study the impact of non-normality on constant coefficients of the X and R control
charts. Tsai (1990) employed the Burr distribution to design the probabilistic tolerance for a
subsystem. Yourstone and Zimmer (1992) used the Burr distribution to design the control limits
of X control charts for non-normal data. Chou et al. (2000) employed the Burr distribution for
the economic design of X charts under the presence of non-normal process data.
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4. The economic model of the VSI X control chart
The economic design of the VSI X control chart is developed employing a cost function
and searching for the optimal design parameters that minimizes the cost function during a
production cycle. The derivation of the cost function comes in the next subsection.
4.1. The cost function
In the derivation of the cost function, the process is initially assumed in-control 0 . The process will then be disturbed by a single assignable cause that makes a shift of in the mean (i.e., the out-of-control process mean becomes 0 ),where is the magnitude of the shift. After the shift, the process remains out-of-control until the assignable cause is removed.
The inter-arrival time of the assignable cause is assumed to follow an exponential distribution
with a mean of 1 . For monitoring purposes, a sample of size n is taken at each sampling point. If the
calculated sample mean falls inside the two control limits, its position inside the control region
will be used to specify the next sampling point; making sampling interval variable. If the sample
mean goes beyond the two control limits, the process is ceased and a search starts to find a
possible assignable cause and if necessary to adjust the process.
As in Chen (2004), the length of a production cycle is the time between the instant the
process starts in control to the time the process is stopped for adjustments. Fig. 2 shows a
production cycle ( )T that is divided into four intervals including the in-control period 1( )T , the
out-of-control period 2( )T , the searching period due to a false alarm 3( )T , and the period for
identifying and correcting the assignable cause 4( )T .
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Fig 2. Production cycle considered in the cost model
The individual periods are now illustrated before they are grouped together.
1( )T : Based on the assumptions the expected length of the in-control period is 1 .
2( )T : Let A be the length of the sampling interval in which an assignable cause occurs,
and Y be the time interval between sample points just prior to the occurrence of the assignable
cause and the occurrence itself (see Fig. 2). Then, Reynolds et al. (1988) showed that
2 1 11
1m
j jJ
E T E A E Y S h P
(4) Where 1S represents the expected number of samples required to detect the assignable cause and
1 jP represents the conditional probability that X falls within jI , given that X falls inside the
two control limits when 0 . Moreover, the number of samples in detecting the
assignable cause can be modeled by a geometric distribution with parameter 1q , (i.e., 1 11S q ),
Last Sample before process mean shift
Cycle ends
Process mean
Shift
First Sampleafterprocessmeanshift
Assignable
Cause
Assignable
Cause Out-of-control
Detected
(T1) in control period +
(T3) Searching period due to false alarm
(T2) out- of-control period
(T4) Time period for identifying and correcting assignable cause
Y
A
Cycle
Start
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where 1q is the probability of detecting an assignable cause in a sample. Reynolds et al. (1988)
also assumed that the probability of the length of A being jh is
0 01
Pr( )m
j j j j jj
A h h P h P
(5) Where 0 jP is the conditional probability that X falls within jI , given that X falls inside the
control limits when 0 . Then from the result of Duncan (1956), the conditional expected value of Y given jA h is
1 1
1
j
j
hj
j h
h eE Y A h
e
(6)
Therefore, the expected length of the out-of control period is reached using (4)(6) as:
0
2 1 11 101
11 11
j
j
hm mj j j
j jm hj jj jj
P h h eE T S h P
h P e
(7)
3( )T : Let 0S be the expected number of samples in the in-control state. Then, Bai and
Lee (1998) showed that
01
0 021
01
11
j
j
j
m hmj hj
jm h jjj
P eS P e
P e
(8)
The expected searching period due to false alarm becomes
1 0 0t q S (9)
where 0q is the false alarm rate and that an average of 1t hours is spent if the signal is a false
alarm.
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4( )T : The time to identify and correct the assignable cause following an action signal is
assumed a constant 2t .
Thus, the expected length of a production cycle can be aggregately represented by
1 2 3 4
01 1 1 0 0 2
1 101
( ) ( )
11 1 11
j
j
hm mj j j
j jm hj jj jj
E T E T T T T
P h h eS h P t q S t
h P e
(10)
The expected cost of a production cycle includes (a) the cost of false alarms; (b) the cost
of detecting and eliminating the assignable cause; (c) the cost associated with production in the
out-of-control condition; and (d) the cost of sampling and testing. Defining 1c the average search
cost of a false signal, 2c the average cost to discover the assignable cause and adjust the process
to in-control state, 3c the hourly cost associated with production in an out-of-control state, 4c the
fixed sampling cost for each sample, and 5c the variable cost of sampling and testing in each
sample, the expected cost of a production cycle, ( )E C , becomes:
1 0 0 2 3 2 4 5 0 1( )E C c q S c c E T c c n S S (11) The aim of the economic design of a VSI X control chart with generalized control limits
is to determine the appropriate values of design parameters ' '1 2 1 2 1 2, , , , , ,n k k k k h h such that the expected cost per hour, ECT , given in (12) is minimized
1 0 0 2 3 2 4 5 0 1
2 1 0 0 2
( )1( )
c q S c c E T c c n S SE CECTE T E T t q S t
(12)
Note that ECT is a function of both the process and the cost parameters.
Since the objective of this research is to develop an economical design of the VSI X
control chart under correlated non-normal process data, in the next section, correlated data is first
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considered. Then we take advantage of the Burr distribution to model non-normal data in section
4.3. These modeling can help determine the values of 0 jP and 1 jP used in (12).
4.2. The case of correlated process data
Yang and Hancock (1990) assumed each subgroup a realization of the random vector
1 2[ , ,..., ]T
nX X XX following a multivariate normal distribution ( )N ,V , where
1[ ( ),..., ( )]T
nE X E X is the mean vector and { ( , ); 1,..., , 1,... }ij i jv Cov X X i n j n V is
the covariance matrix. Assuming e , where e is a vector of all 1s, i.e., ( )iE X , and 2V R , where { ; 1,..., , 1,..., }ijr i n j n R is the correlation matrix, the sample mean X
can be shown to be normally distributed with mean and variance as follows:
E( ) X (13)
2V( ) 1 1X nn (14)
Where
#
1
iji j
r
n n
(15)
The mean and variance of X are still valid even if the measurements are not normally
distributed.
Now, let 0O and 1O be the conditional probabilities that any sample mean X falls outside
the control limits, given that the process is in control ( 0 ) and out of control ( 0 ), respectively. The conditional probabilities 0O and 1O are also called the false alarm rate and the
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failure-detection power of a control chart, respectively. Then by denoting the upper and the
lower control limits of the X chart by UCL and LCL , respectively, we have
0 1
'0 1
UCL k n
LCL k n
(16)
As a result 0O can be obtained as
0 0
0 0
1 Pr
Pr Pr
O LCL X UCL
X UCL X LCL
Then by replacing UCL and LCL given in (16), we have
1
0 10 0
'0
0
Pr 1 1 1 1
Pr 1 1 1 1
X kOn n n
kXn n n
(17)
Similarly 1O becomes
1
1 0
'0 1
=1 Pr
1 Pr1 1 1 1
O LCL X UCL
k n X k nnn n
(18)
4.3. The case on non-normal process data
The skewness and kurtosis coefficients of the sample mean X (based on a sample of size
n ) according to Dodge and Rousson (1999) are, respectively,
3 43 4 3 , = +3X X nn (19)
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In which 3 and 4 denote the skewness and kurtosis coefficients of the random variable
X (modeling the population). Using the values of 3 4( ) and ( )X X and the tables of the Burr distribution (1942), one can obtain the values of , , , and M S c k for the distribution with
skewness and kurtosis values close to the 3 4( ) and ( )X X through the interpolation. Now, using the Burr transformation given in (2) and (3), expressions for 0O , 1O , 0 jP ,
0mP , 1 jP , and 1mP are derived in the equations A.1 to A.6 of the appendix, respectively.
5. An example and a solution procedure
An examination of the probability components in (12) reveals that finding the optimal
values of the parameters * * * '* ' * * *1 2 1 2 1 2, , , , , ,n k k k k h h of the VSI X chart using a conventional optimization method is not simple. As a result, to describe the nature of the solutions obtained
for the economic design of the VSI X control chart under correlated non-normal process data, a
numerical example is provided and a parameter-tuned genetic algorithm (GA) is developed to
search for a near optimal solution. The numerical example is borrowed from Chou et al. (2002)
and is modified to demonstrate the methodology. In this example, the input based on the
historical data is:
0.01 , 1 10c , 2 30c , 3 100c , 4 0.5c , 5 0.1c , 1 , 1 0.1t , and 2 0.3t . Previous data also indicate the skewness and kurtosis coefficients of the tang dimension are
approximately 3 1.4322 and 4 7.3558 , respectively, which may be approximated by the Burr distribution with 2c and 4k .
The recent 60 successive parts are viewed as a random sample from a multivariate non-
normal distribution. The sample average is 0.66, 0.59, 0.69, 0.54 TT and the corresponding sample covariance matrix is
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1.05 05 8.40 06 7.80 06 6.90 068.40 06 1.07 05 9.90 06 7.70 067.80 06 9.90 06 1.07 05 9.00 066.90 06 7.70 06 9.00 06 1.10 05
E E E EE E E EE E E EE E E E
V
that results in an average correlation coefficient of 0.77 . Now, suppose a set of data is collected and the sample mean and sample standard
deviation are computed 0.5x and 0.001xS , respectively. For a given sample size, in order to calculate the probabilities ijP and ultimately ECT we first find the skewness and kurtosis
coefficients of the process data (3 and 4 ). Then, we obtain 3 X and 4 X using Equation (19). Next, using the Burr (1942) Tables II and III the values of M , S , c , and k that correspond
to 3 X and 4 X are obtained using interpolation. Finally, equations (A.1)-(A.6) are used to determine the probabilities 0 jP , 0mP , 1 jP , and 1mP that are required to evaluate ECT in (12). The
solution procedure is carried out using a parameter-tuned GA to obtain the near optimal values of
* * * '* ' * * *1 2 1 2 1 2, , , , , ,n k k k k h h that minimize ECT . The proposed GA was coded in MATLAB (version 7.8) environment, in which except
the integer sample size n , decimal coding is applied. The initial population contains 40 feasible
chromosomes that are randomly generated. The fitness value of each solution in the initial
population is evaluated by ECT in (12) and the chromosome with the lowest cost replaces the
one with the highest cost. The best 20% of the chromosomes are the survivors for the next
generation. For the crossover operation, among the second 20% of the chromosomes, the parents
are selected using the Rolette-Wheel method, in which the crossover point is randomly selected
and the crossover operation on the parents are performed with the probability of 0.5. In this step,
if a gene does not satisfy its corresponding constraints, its value is transformed to an acceptable
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value within the range uniformly. Moreover, the mutation is performed with a probability of 0.3.
These steps are continued until the stopping condition, 1500 generations, is achieved.
By employing the proposed GA on the economic designs of both the FSI and VSI X-bar
charts, the near optimal solutions along with the minimum costs are obtained as given in Table
(1). As shown in this table, the cost function ETC of Equation (12) gives a minimum of $4.748
if the sample size, the sampling interval, and the control limit coefficient are fixed (FSI).
However, a minimum cost of $3.335 is obtained when the sample size and the control limit
coefficient are fixed and the sampling intervals change (VSI), resulting in 29.76% cost savings.
Table (1): The optimal designs of the FSI and VSI X-Bar control chart
X-Bar n 1h 2h 1k 2k '1k '2k ETC % decrease
FSI 2 0.520 2.457 2.796 4.748
VSI 2 0.010 1.361 2.456 0.0004 2.622 2.622 3.335 29.76
In the operations of the GA,the quality of the solution obtained depends on the setting of
its control parameters: the population size (PS), the crossover probability (CP), the mutation rate
(MR), and the generation number (GN) (Chen 2006). In order to find the optimal setting of these
parameters, an orthogonal array experiment is performed on three different levels of the
parameters. Based on three replicates and the smaller-the-better criterion on the signal-to-noise
ratio, the optimal combination level of the control parameters is obtained as PS=40, CP=0.5,
MR=0.3, GN=1500.
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6. Sensitivity analyses
In this section, a sensitivity analysis is conducted to understand how typical the given
numerical example is and to explore the effect of the model parameters on the solution of the
proposed economic design of the VSI X control chart under correlated non-normal process data.
Since the chart is designed without knowing the shift direction, the optimization should
consider the overall better performance taking into account process mean increases and process
mean decreases. As a result, different cost parameter settings are considered for each of which
the near optimal solution of the economic model of both the FSI and VSI X-Bar control chart
along with its expected cost per hour is obtained for positive and negative mean shifts in Tables
(2) and (3), respectively. In these tables, is the false alarm risk. Further, similar to the tables given in Chen (2004), we set 1 2 0
' 'k k ; for (and 1 2 0k k ; for ) and added them to the constraints in the optimization model. By this way, the chart is designed without lower
(upper) limits in case of increase (decrease) in the process mean. Moreover, a constraint in the
optimization model was added to avoid false alarm risks of bigger that 0.05, and another
constraint was added to avoid 1h smaller than 0.01.
While previous research works (Bai and Lee 1998, Chen 2004, Chen and Chiou 2005,
and Chen et al. 2007) only considered positive mean shifts, a careful examination of the results
in Table (3) reveals that the optimal sample size of "one" has been obtained in all cases in which
negative mean shifts are considered. Based on Equations (A.2)-(A.6), this means the correlation
should not have any effect on the probabilities obtained to reach the near optimal solution.
Moreover, Table (4) summarizes the performances of the economic design of X control charts
for process data with different levels of correlation coefficients and Table (5) shows the
percentage reductions in cost obtained using the VSI X-Bar chart.
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The results given in Tables (2) and (5) reveal the following findings:
1. For positive mean shifts, the expected cost per hour of the VSI control schemes are
consistently smaller than that of the FSI control scheme over the 15 parameter
combination sets.
2. Compared with the FSI schemes, in most cases the corresponding VSI scheme tends to
operate with a larger upper control limit ( 1k ) and the same (or sometimes smaller)
sample size.
3. The percent reduction in ECT is reduced when small or intermediate mean shifts ( = 0.5 or 1) are encountered.
4. The percentage reduction in ECT is significantly increased due to higher searching cost
(involving 1c and 2c ), lower sampling cost (involving 4c and 5c ), when the production
cost in the out-of-control state is large and when is small. Other parameters such as 1t and 2t do not significantly affect the ECT.
5. In most cases, false alarm rate () of the X control chart for the VSI scheme is smaller than the FSI scheme.
Further, the results in Tables (3) and (5) indicate the following
1- For negative mean shifts, the expected cost per hour of the VSI control schemes are
consistently smaller than that of the FSI control scheme over the 15 parameter
combination sets.
2- The percent reduction in ECT is significantly increased when the mean shifts are = -0.5 or -1.
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3- The percentage reduction in ECT is increased due to lower searching cost (involving 1c
and 2c ), lower sampling cost (involving 4c and 5c ), and when is small. Other
parameters such as 1t and 2t do not significantly affect the ECT
From Table (4), several findings can be spelled out as follows:
1. As the measurements in the sample are positively or negatively correlated, both VSI and
FSI charts for highly correlated data require less frequent sampling with small sample
size and narrow width of the control limits.
2. Monitoring the process data by the VSI chart results in an evident improvement in
ECT when a high positive correlation is considered. On the contrary, when the
measurements in the sample are negatively correlated, the obvious improvement only
sounds to be the case for intermediatehighly correlated data .8.05.0 3. The sample sizes taken in VSI schemes are slightly smaller than the corresponding one in
the FSI scheme.
4. As compared with the FSI charts, the VSI chart takes the next sample more slowly for
positively correlated data and more quickly for the most of negatively correlated data.
5. In this study, the percentages of the cost reduction with 2c , 4k , and different values of (Table 4) are consistently more significant than the corresponding percentages of the cost reduction (Table 3) obtained by Chen and Chiou (2005), where
the process data is distributed normality.
It should be noted that when 0 the expected cost per hour obtained for the FSI and VSI charts are 2.398 and 2.163, respectively. These figures are similar to the ones in Chen (2004), in
which he obtained them as 2.440 and 2.145, respectively. The difference is due to different
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computer codes written to solve the problem. In fact, while Chen (2004) employed the
EVOLVER coding to solve the problem, we solved the problem in MATLAB environment.
7. Conclusions and recommendation for future research
In this paper, we first took advantages of the Bai and Lees (1998 ) cost model, the Yang
and Hancocks (1990 ) correlation model, along with the Burr distribution modeling of non-
normal data to develop the economic design of the VSI X-Bar control chart for monitoring the
mean of correlated non-normal process data. An illustrative example to demonstrate the
application of the proposed methodology was then provided and comparisons were made
between the FSI and VSI schemes in terms of the expected cost per hour. To do this a parameter-
tuned genetic algorithm was developed to search for the near optimal solution of the economic
designs. Sensitivity analyses were next carried to determine the influences of the input
parameters on the solutions of the economic design. Based on the results in the sensitivity
analyses, we have the following findings:
1. Over the 15 different combination sets of the model parameters indicated for positive
mean shifts given in Chen and Chiou (2005) the average percentage reduction in the
expected cost per hour ( ECT ) of the VSI scheme under correlated non-normal process
data is about 26.76%, which is more than that for correlated normal process data.
2. For positive mean shifts, the improvements in the VSI scheme in terms of ECT become
evident when the average wasted costs associated with false alarms and the detection and
elimination of the assignable cause after a true alarm are large. Moreover, when the
sampling cost is low, the improvement is also important for both positive and negative
mean shifts.
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3. When the sample measurements are correlated, the improvement in ECT is apparent for
process data with a high degree of positive correlation or with an intermediatehigh
degree of negative correlation.
4. As compared with the FSI charts, the VSI chart takes the next sample more slowly for
positively correlated data and more quickly for the most of negatively correlated data.
5. For monitoring the correlated data, the FSI and VSI chart use a smaller sample than that
for monitoring uncorrelated data.
6. In most cases, false alarm rate () of the X control chart for the VSI scheme is smaller than the FSI scheme.
While a single assignable cause was assumed throughout this research, it would be
interesting to conduct a research on the optimal design of VSI X control charts to monitor
correlated non-normal process data under the consideration of multiple assignable causes in the
future.
8. Acknowledgement
The authors are thankful for constructive comments of the reviewers that helped better
presentation of this paper.
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22
Table 2. The optimum design of FSI and VSI X charts for positive mean shifts under the Burr distributions ( 2 and 4c k ) FSI VSI
No t1 t2 c1 c2 c3 c4 c5 n h1 k1 k'1 ECT n h1 h2 k1 k2 k'1 k'2 ECT 1 0.01 2 0.1 0.3 10 30 100 0.5 0.1 2 0.9438 2.4560 2.8659 0.0500 2.6952 1 0.0103 1.3600 2.9815 0.3796 1.8453 1.8453 0.0120 1.8038
2 0.05 2 0.1 0.3 10 30 100 0.5 0.1 2 0.4644 2.4624 2.8732 0.0496 6.6443 1 0.0101 0.6327 2.6656 0.3989 2.9600 2.9600 0.0179 4.9277
3 0.01 2 0.1 0.3 10 30 10 0.5 0.1 2 3.2542 2.4561 2.6753 0.0500 1.0331 1 0.0101 4.7090 2.8331 0.3844 2.0326 2.0326 0.0145 0.7752
4 0.01 2 0.1 0.3 10 30 1000 0.5 0.1 2 0.3115 2.4560 2.6526 0.0500 7.8436 1 0.0100 0.4641 2.6538 0.3879 2.2658 2.2658 0.0182 5.2736
5 0.01 2 0.1 0.3 25 75 100 0.5 0.1 2 1.2166 2.4596 2.6683 0.0498 3.7817 1 0.0103 1.5817 3.0000 0.3704 2.8698 2.8698 0.0117 2.4090
6 0.01 2 0.5 1.5 10 30 100 0.5 0.1 2 0.9626 2.4569 2.8347 0.0499 2.6102 1 0.0101 1.4578 2.9281 0.3749 2.7379 2.7379 0.0128 1.7727
7 0.01 2 0.1 2.1 10 210 100 0.5 0.1 2 1.0957 2.4559 2.7048 0.0500 4.3896 1 0.0100 1.4379 2.9699 0.3717 2.6024 2.6024 0.0122 3.5173
8 0.01 2 0.1 0.3 1 3 100 0.5 0.1 2 0.7160 2.4584 2.8515 0.0499 1.9395 1 0.0118 1.2941 1.8632 0.4692 1.7030 1.7030 0.0499 1.3475
9 0.01 2 0.1 0.3 10 30 100 5 0.1 2 2.1940 2.4562 2.8601 0.0500 5.5069 1 0.0101 3.7610 1.8617 0.4794 2.0574 2.0574 0.0500 4.1687
10 0.01 2 0.1 0.3 10 30 100 0.5 1 1 1.1107 1.8614 2.7378 0.0500 3.7594 1 0.0110 2.1792 2.5370 0.4040 2.0408 2.0408 0.0211 2.5344
11 0.01 1 0.1 0.3 10 30 100 0.5 0.1 2 0.5202 2.4572 2.7956 0.0499 4.7479 2 0.0102 1.3613 2.4563 0.0004 2.6219 2.6219 0.0500 3.3351
12 0.01 1 0.1 0.3 10 30 20 0.5 0.1 2 1.2372 2.4560 2.7481 0.0500 2.2451 2 0.0102 3.1640 2.4570 0.0018 2.9970 2.9970 0.0499 1.6415
13 0.05 1 0.1 0.3 10 30 100 0.5 0.1 2 0.2405 2.4561 2.8830 0.0500 10.9347 2 0.0100 0.7009 2.4565 0.0009 2.9828 2.9828 0.0500 8.1413
14 0.01 0.5 0.1 0.3 10 30 100 0.5 0.1 1 0.3430 1.8616 2.1894 0.0500 6.6988 2 0.0106 0.7823 2.4581 0.0001 2.8878 2.8878 0.0499 5.6334
15 0.01 0.5 0.1 0.3 10 30 5 0.5 0.1 2 2.0035 2.4560 2.6434 0.0500 1.5727 2 0.0105 3.9988 2.4559 0.0009 2.9479 2.9479 0.0500 1.3989
0.77
-
23
Table 3. The optimum design of FSI and VSI X charts for negative mean shifts under the Burr distributions ( 2 and 4c k ) FSI VSI
No t1 t2 c1 c2 c3 c4 c5 n h1 k1 k'1 ECT n h1 h2 k1 k2 k'1 k'2 ECT 1 0.01 2 0.1 0.3 10 30 100 0.5 0.1 1 0.9000 2.9988 1.5996 0.0137 1.9121 1 0.0100 1.0260 2.9997 2.9997 1.6585 1.1122 0.0118 1.8202
2 0.05 2 0.1 0.3 10 30 100 0.5 0.1 1 0.3933 2.9998 1.5963 0.0139 4.9941 1 0.0100 0.4954 2.9957 2.9957 1.6559 1.0701 0.0119 4.8011
3 0.01 2 0.1 0.3 10 30 10 0.5 0.1 1 2.9803 2.9995 1.6009 0.0137 0.7974 1 0.0104 3.3745 2.9995 2.9995 1.6585 1.1139 0.0118 0.7701
4 0.01 2 0.1 0.3 10 30 1000 0.5 0.1 1 0.2558 2.9994 1.6137 0.0131 5.4336 1 0.0100 0.3256 2.9975 2.9975 1.6606 1.0979 0.0118 5.1187
5 0.01 2 0.1 0.3 25 75 100 0.5 0.1 1 1.0320 2.9997 1.6352 0.0123 2.5528 1 0.0107 1.1484 2.9989 2.9989 1.6688 1.1110 0.0118 2.4391
6 0.01 2 0.5 1.5 10 30 100 0.5 0.1 1 0.9013 3.0000 1.5963 0.0139 1.8783 1 0.0137 1.0344 2.9971 2.9971 1.6546 1.1041 0.0119 1.7904
7 0.01 2 0.1 2.1 10 210 100 0.5 0.1 1 0.9144 2.9995 1.5990 0.0138 3.6248 1 0.0116 1.0439 2.9994 2.9994 1.6579 1.1055 0.0119 3.5358
8 0.01 2 0.1 0.3 1 3 100 0.5 0.1 1 0.9599 2.9996 1.3325 0.0499 1.4305 1 0.5122 1.0320 2.9983 2.9983 1.3328 0.8591 0.0499 1.4126
9 0.01 2 0.1 0.3 10 30 100 5 0.1 1 2.7436 2.9999 1.3322 0.0500 4.3864 1 1.5009 2.9897 2.9937 2.9937 1.3326 0.9011 0.0500 4.3394
10 0.01 2 0.1 0.3 10 30 100 0.5 1 1 1.4055 2.9986 1.5178 0.0201 2.6823 1 0.0114 1.5800 2.9999 2.9999 1.6303 1.1062 0.0125 2.5792
11 0.01 1 0.1 0.3 10 30 100 0.5 0.1 1 0.6331 2.9996 1.4686 0.0260 2.9988 1 0.0105 0.9782 2.9993 2.9993 1.5558 0.5244 0.0166 2.7255
12 0.01 1 0.1 0.3 10 30 20 0.5 0.1 1 1.4326 2.9996 1.4688 0.0259 1.4881 1 0.0105 2.0278 2.9986 2.9986 1.5594 0.6150 0.0163 1.3705
13 0.05 1 0.1 0.3 10 30 100 0.5 0.1 1 0.3142 2.9999 1.4364 0.0306 7.3119 1 0.0101 0.3615 2.9997 2.9997 1.5470 0.6871 0.0173 6.7761
14 0.01 0.5 0.1 0.3 10 30 100 0.5 0.1 1 0.5109 2.9984 1.3325 0.0499 4.4881 1 0.0109 0.9183 2.9988 2.9988 1.3555 0.3117 0.0451 4.2459
15 0.01 0.5 0.1 0.3 10 30 5 0.5 0.1 1 2.5710 2.9983 1.3331 0.0498 1.1636 1 0.0113 4.0903 2.9918 2.9918 1.3354 0.4842 0.0494 1.1204
0.77
-
24
Table 4. Effect of correlation coefficients on the optimal design of the FSI and VSI X charts under the Burr distributions ( 2 and 4c k ) FSI VSI
n h1 k1 k'1 ECT n h1 h2 k1 k2 k'1 k'2 ECT % 0.9 2 0.42696 2.54506 2.96068 0.04997 4.93567 1 0.01000 1.14443 1.86145 0.00106 1.68205 1.68205 0.04997 3.36725 31.77724
0.8 2 0.53618 2.47706 2.81296 0.04997 4.77219 1 0.01068 1.18143 1.86118 0.00113 1.98587 1.98587 0.04999 3.36713 29.44267
0.7 2 0.54953 2.41360 2.68994 0.04963 4.69388 2 0.01023 1.12432 2.48025 0.00245 2.93573 2.93573 0.04609 3.34272 28.78554
0.6 3 0.62383 2.71990 2.99961 0.04997 4.53213 2 0.01057 1.49458 2.34295 0.00492 2.92305 2.92305 0.04955 3.18791 29.65975
0.5 3 0.62306 2.58229 2.99907 0.04999 4.36052 3 0.01026 1.65146 2.58918 0.00052 2.99608 2.99608 0.04963 3.07956 29.37639
0.4 4 0.69670 2.72072 2.99993 0.04982 4.19957 3 0.01101 1.65987 2.56365 0.00098 2.92944 2.92944 0.04388 2.94890 29.78085
0.3 4 0.70525 2.48929 2.99993 0.04998 3.91435 3 0.01047 1.60831 2.58609 0.00396 2.79143 2.79143 0.03563 2.80881 28.24321
0.2 5 0.91768 2.39362 2.99997 0.04985 3.54721 4 0.01022 1.77758 2.84457 0.00388 2.99874 2.99874 0.02411 2.62885 25.88951
0.1 9 1.45073 2.38232 2.99533 0.04986 3.01286 5 0.01013 1.70928 2.78880 0.31280 2.84947 2.84947 0.01873 2.42983 19.35140
0 11 1.86872 1.91101 2.80768 0.03639 2.39795 7 0.01222 1.72507 2.66818 0.85559 2.99931 2.99931 0.00947 2.16310 9.79352
-0.1 8 1.65198 1.67679 1.92691 0.00423 1.93348 7 0.01001 1.57608 2.24190 1.27358 1.90720 1.90720 0.00173 1.88589 2.46151
-0.2 5 1.43798 1.37049 1.08825 0.00521 1.74940 5 0.01005 1.41908 1.82256 1.23673 2.66754 2.32316 0.00079 1.71984 1.68937
-0.3 4 1.36257 1.31588 2.31656 0.00080 1.63784 4 0.01116 1.35429 1.51731 1.29000 1.72271 1.72271 0.00025 1.63485 0.18278
-0.4 3 1.36355 1.08033 1.38650 0.01966 1.75917 3 0.01073 1.29067 1.64592 0.85457 2.52335 2.52335 0.00235 1.61942 7.94401
-0.5 2 0.92253 1.30626 2.57480 0.04990 2.85462 2 0.01018 1.57398 1.99290 0.18527 2.55298 2.55298 0.01172 1.98012 30.63450
-0.6 2 1.03912 1.16751 2.51285 0.04999 2.56525 2 0.01061 1.43037 1.87324 0.30548 1.53344 1.53344 0.00941 1.86062 27.46810
-0.7 2 1.21410 1.01146 2.99077 0.04995 2.24150 2 0.01016 1.37357 1.72249 0.43533 2.10534 2.10534 0.00711 1.73396 22.64305
-0.8 2 1.36156 0.88854 2.17417 0.04076 1.91192 2 0.01023 1.28667 1.53854 0.59497 2.44292 2.44292 0.00453 1.61081 15.74925
-0.9 2 1.25857 0.90606 1.36369 0.01091 1.58033 2 0.01007 1.19620 1.29985 0.81036 1.48247 1.48247 0.00164 1.50857 4.54048
Fixed cost parameters: =0.01, =1, t1=0.1, t2= 0.3, c1= 10, c2=30, c3=100, c4=0.5, c5=0.1
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25
Table (5): Cost Reductions when instead of FSI, the VSI X-Bar chart is employed
Percentages of Cost Reduction
No. Positive Mean Shifts Negative Mean Shifts
1 33.074 4.806
2 25.836 3.863
3 24.962 3.419
4 32.766 5.797
5 36.298 4.456
6 32.086 4.682
7 19.872 2.454
8 30.522 1.251
9 24.300 1.073
10 32.585 3.844
11 29.758 9.112
12 26.883 7.906
13 25.546 7.329
14 15.905 5.396
15 11.050 3.714
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26
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30
Appendix
Using the Burr transformation given in (2) and (3), equation (17) becomes
1
1
'1
0
'1
1
= Pr Pr1 1 1 1
Pr Pr (A.1)1 1 1 1
1 1 1
1 0, 1 01 1
kc
kkY M Y MOS Sn n
kkY M S Y M Sn n
kMax M S Maxn
1
'
,1 1
kck
M Sn
Likewise, 1O in equation (18) can be represented by
1
1
'1
1
'1
1
O 1 Pr1 1 1 1
1 Pr (A.2)1 1 1 1
1 1 1
1 0, 11 1
kc
k n k nY MSn n
k n k nM S Y M Sn n
k nMax M S Maxn
1
'
0,1 1
kck n
M Sn
As a result, the conditional probabilities 0 jP , 0mP , 1 jP , and 1mP are obtained as
0 0 1 0 0
' '0 0 1 0
Pr ,
+ Pr ,
j j j
j j
P k X k LCL X UCLn n
k X k LCL X UCLn n
-
31
0
1
1 1 1 = 1
1+Max 0,M 1+Max 0,M1 1 1 1
k kc c
j j
Ok k
S Sn n
1
' '
1 1
1 0, 1 0,1 1 1 1
for 1, 2, ..., -1 (A.3)
j j
k kc ck k
Max M S Max M Sn n
j m
'0 0 0 0
0 '
Pr ,
1 1 1 (A.4) 1
1 0, 1 0,1 1 1 1
m m m
k kc c
m m
P k X k LCL X UCLn n
Ok kMax M S Max M Sn n
1 0 1 0 0
' '0 0 1 0
Pr ,
Pr ,
j j j
j j
P k X k LCL X UCLn n
k X k LCL X UCLn n
1
1
1 1 1 1
1 0, 1 0,1 1 1 1
k kc c
j j
Ok n k n
Max M S Max M Sn n
-
32
1
' '
1 1
1 0, 1 0,1 1 1 1
for 1, 2, ..., -1
j j
k kc ck n k n
Max M S Max M Sn n
j m
(A.5)
'1 0 0 0
1 '
Pr , (A.6)
1 1 1 1
1 0, 1 0,1 1 1 1
m m m
k kc c
m m
P k X k LCL X UCLn n
Ok n k nMax M S Max M S
n n