1

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: Niaki@Sharif.edu

Fazlollah Masoumi Gazaneh, M.Sc.

Department of Industrial Engineering, Islamic Azad University (South-Tehran Branch), Tehran, Iran, E-Mail: masoumi_313@yahoo.com

Moslem Toosheghanian, M.Sc.

Department of Industrial Engineering, Iran University of Science and Technology toosheghan@yahoo.com

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

2

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

3

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

4

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

5

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

6

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

7

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.

8

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 .

9

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

10

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.

11

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

12

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

13

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)

14

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

15

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

16

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.

17

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.

18

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.

19

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

20

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.

21

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.

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

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

26

References

Bai, D.S., Lee, K.T. (1998). An economic design of variable sampling interval X control

chart. International Journal of Production Economics 54: 57-64.

Baxley Jr., R.V. (1996). An application of variable sampling interval control charts. Journal

of Quality Technology 27: 275282.

Burr, I.W. (1942). Cumulative frequency distribution. Annals of Mathematical Statistics 13:

215-232.

Burr, I.W. (1967). The effect of non-normality on constants for X and R charts. Industrial

Quality Control 22: 563569.

Chen, Y.K. (2004). Economic design of X control chart for Non-normal data using variable

sampling policy. International Journal of Production Economics 92: 61-74.

Chen, Y.K., Chiou, K.C. (2005). Optimal design of VSI X control charts for monitoring

correlated sample. Quality and Reliability Engineering International 21: 757768.

Chen, Y.K., Chiou, K.C., Hsieh, K.L., Chang, C.C. (2007). Economic design of the VSSI X

control charts for correlated data. International Journal of Production Economics 107:

528539.

Chiu, W.K. (1975). Economic design of attribute control charts. Technometrics 17: 8187.

Chiu, W.K., Cheung, K.C. (1977). An economic study of X charts with warning limits.

Journal of Quality Technology 9: 166171.

Chou, C.Y., Chang, C.L., Chen, C.H. (2002). Minimum-loss design of X-Bar control

charts for Non-normally correlated data. Journal of the Chinese Institute of

Industrial Engineers 19: 16-24.

Chou, C.Y., Chen, C.H., Liu, H.R. (2000). Economic-statistical design of X charts for non-

normal data by considering quality loss. Journal of Applied Statistics 27: 939951.

27

Costa, A.F.B. (1994). X-BAR chart with run rule and variable sample size. Journal of Quality

Technology 26: 155-163.

Costa, A.F.B. (1997). X-BAR chart with variable sample size and sampling interval. Journal

of Quality Technology 29: 197-204.

Costa, A.F.B. (1999). X charts with variable parameters. Journal of Quality Technology 31:

408416.

Costa, A.F.B. (2001). Economic design of X charts with variable parameter: the Markov

chain approach. Journal of Applied Statistics 28: 875885.

Dodge, Y., Rousson, V. (1999). The complications of the fourth central moment. The

American Statistician 53: 267269

Duncan, A.J. (1956). The economic design of X charts used to maintain current control of a

process. Journal of American Statistical Association 51: 228-242.

Grant, E.L., Leavenworth, R.H. (1988). Statistical Quality Control. McGraw-Hill Book

Company, Singapore.

Lin, Y.C., Chou, C.Y. (2005). On the design of variable sample size and sampling intervals

X charts under non-normality. International Journal of Production Economics 96: 249

261.

Lin, Y.C., Chou, C.Y. (2007). Non-normality and the variable parameters X control charts.

European Journal of Operational Research 176: 361-373.

Liu, H.R., Chou, C.Y., Chen, C.H. (2002). Minimum-loss design of X charts for correlated

data. Journal of Loss Prevention in the Process Industries 15: 405411.

Montgomery, D.C. (1980). The economic design of control charts: A review and literature

survey. Journal of Quality Technology 1: 24-32.

28

Niaki, S.T.A., Ershadi, M.J., Malaki, M. (2010). Economic and economic-statistical designs

of MEWMA control charts, A hybrid Taguchi loss, Markov chain and genetic algorithm

approach. International Journal of Advanced Manufacturing Technology 48: 283-296.

Neuhardt, J.B. (1987). Effect of correlated sub-samples in statistical process control. IIE

Transactions 19: 208-214.

Pignatiello, J.J., Tsai, A. (1988). Optimal economic design of control chart when cost model

parameters are not known. IIE Transactions 20: 103110.

Prabhu, S.S., Montgomery, D.C., Runger, G.C. (1994). A combined adaptive sample size and

sampling interval X control scheme. Journal of Quality Technology 26: 16-76.

Prabhu, S.S., Runger, G.C., Keats, J.B. (1993). An adaptive sample size X chart.

International Journal of Production Research 31: 28952909.

Reynolds Jr., M.R., Arnolds, J.C., Amin, R.W., Nachlas, J.A. (1988). X chart with variable

sampling interval. Technometrics 30: 181-192.

Shewhart, W.A.T. (1931). Economic control of quality of manufactured product. Van

Nostrand, New York, USA.

Tagaras, G. (1998). A Survey of recent developments in the design of adaptive control charts.

Journal of Quality Technology 30:.37-62.

Taguchi, G. (1987). System of experimental design: Engineering method to optimize quality

and minimize costs. UNIPU , New York.

Tsai, H.T. (1990). Probabilistic tolerance design for a subsystem under Burr distribution

using Taguchis quadratic loss function. Communications in Statistics: Theory and

Methods 19: 4679-4696.

Vance, L.C. (1983). A bibliography of statistical quality control chart techniques, 1970

1980. Journal of Quality Technology 15: 5962.

29

Woodall, W.H. (1986). Weakness of the economic design of control charts. Technometrics

28: 408409.

Yang K., Hancock W.M. (1990). Statistical quality control for correlated samples.

International Journal of Production Research 28: 595608.

Yourstone, S.A., Zimmer, W.J. (1992). Non-normality and the design of control charts for

averages. Decision Sciences 23: 1099-1113.

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