A Parameter-Tuned Genetic Algorithm for Economic-Statistical Design of Variable Sampling Interval X-Bar Control Charts for Non-Normal Correlated 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, Tehran, Iran E-Mail:
[email protected]
Abstract Among innovations and improvements occurred in the past two decades on the techniques and tools used for statistical process control (SPC), adaptive control charts have shown to substantially improve the statistical and/or economical performances. Variable sampling intervals (VSI) control charts are one of the most applied types of the adaptive control charts and have shown to be faster than traditional Shewhart control charts in identifying small changes of concerned quality characteristics. While in the designing procedure of the VSI control charts the data or measurements are assumed independent normal observations, in real situations, the validity of these assumptions is under question in many processes. This paper develops an economic-statistical design of a VSI X-bar control chart under non-normality and correlation. Since the proposed design consists of a complex nonlinear cost model that cannot be solved using a classical optimization method, a genetic algorithm (GA) is employed to solve it. Moreover, to improve the performances, response surface methodology (RSM) is employed to calibrate GA parameters. The solution procedure, efficiency, and sensitivity analysis of the proposed design are demonstrated through a numerical illustration at the end. Keywords: SPC; Economic-statistical design; Variable sampling interval; Non-normality; Correlation; Genetic algorithm; Response surface methodology 1
Corresponding author
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1. Introduction and Literature Review Control charts, as the most important graphical tools of statistical process control, play a prominent role in monitoring processes and improving firms’ quality and productivity. Before implementing a control chart, three design parameters, namely the sample size ( n ) , the sampling interval ( h ) , and the control-limits ( k ) , should be first determined. Duncan (1956) was the first who introduced the economic model of X control charts to determine the design parameters, in which a single assignable cause was present. In this model, it is assumed the process is not stopped during the search and repair time. Another popular economic model that was proposed by Chiu (1975) assumes that when a control chart signals, the production process is stopped to search for assignable causes. In another comprehensive and easy-to-use economic model that was introduced by Lorenzen & Vance (1986), the complexity of the model analysis is simplified. For a comprehensive review on economic designs, readers are referred to Montgomery (2005a) and Celano (2011). A usual and important assumption in the designing procedures of control charts is the normality of the process data under consideration. However, in many real-world environments the process data or measurements may violate this assumption. This violation specially occurs when the sample size is small and the central limit theorem is not applicable to make non-normal observations almost normal. Ignoring this violation causes increase in the false alarm rate. Many researchers employed the Burr distribution to model non-normal process data (e.g., Burr 1967; Yourstone & Zimmer 1992; Chou et al. 2002; Chen 2003, 2004; Chen & Cheng 2007; Chen & Yeh 2010). Chen & Cheng (2007) stated that although there are many methodologies for developing a control chart under non-normality, the main advantage of applying the Burr distribution for non-normal process data is that it has a closed-form cumulative distribution
2
function, which leads to the simplification of the calculations related to the Type-I and Type-II error probabilities. While another ordinary and important assumption in the design process of control charts is independency of the measurements, in many real-world environments this assumption fails to hold. As an example, collected measurements from multiple contact pads on a single machine are highly correlated. Grant & Leavenworth (1988) presented some of such practical cases. Neuhardt (1987) investigated the effects of within-sample correlation on the performance of the control charts. Yang & Hancock (1990) studied the effects of correlated data on the performances of ( X , R ), ( X , S ), and ( X , S 2 ) charts and showed that correlation increases Type-I error probabilities of the charts. Liu et al. (2002) used the correlation model of Yang and Hancock (1990) and developed a minimum-loss design of X control charts for correlated data. Saniga (1989) introduced economic-statistical designs of control charts in which statistical constraints were added to the economic model. He showed that although the optimal costs of the economic-statistical designs are slightly higher than their corresponding economic models, their statistical performances are as well as the ones of statistically designed control charts. The idea of varying parameters was presented under the title of variable or adaptive control charts to make traditional charts act faster in detecting smaller mean shifts (see for example Baxley 1996, Prabhu et al. 1993, and Costa 1997, 1999). One of these charts is the variable sampling interval (VSI) control chart that has been suggested to improve the conventional fixed sampling interval (FSI) policy (e.g., Reynolds et al., 1988; Reynolds & Arnold, 1989; Baxley, 1996). Researchers have shown that VSI X control charts are
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substantially faster than their FSI versions in detecting small shifts (e.g., Reynolds et al., 1988; Park & Reynolds, 1999). Following Chiu’s (1975) cost model, Bai & Lee (1998) presented the economic design of the VSI X control charts. Chen (2003) used Bai & Lee’s model (1998) and employed the Burr distribution for developing an economic-statistical design of VSI X control charts with symmetric control limits. Chen (2004) presented the economic model of VSI X control charts with asymmetric control limits. For correlated samples, Chen and Chiou (2005) developed the economic model of the VSI X control charts with a combination of the multivariate normal distribution model given by Yang & Hancock (1990) and the cost model of Bai & Lee (1998). Recently Torng et al. (2010) evaluated the performances of the double sampling VSI X control charts under non-normality. Finally, Chen & Yeh (2010) developed an economic design of the VSI X control charts for non-normal data with Burr distribution under gamma failure models. In this research, an economic-statistical design for VSI X control charts in presence of correlation and non-normality is proposed to not only be applicable to closer to reality environments, but also to have desire economic and statistical performances as well. Since the obtained cost function is a complex nonlinear model, a genetic algorithm (GA) is used to solve it and obtain the optimal values of the design parameters. As the quality of a solution obtained by GA usually depends on its control parameters, the response surface methodology (RSM) is used for calibration. The remainder of the paper is organized as follows. In the next section, the concept of the VSI control charts is briefly discussed. In Section 3, the Burr distribution that is used to model non-normal process data is briefly introduced. The model assumptions along with model development come in Section 4. The effects of correlation and non-normality are discussed in 4
Sections 5 and 6, respectively. The solution procedure and the application of the proposed methodology are given in Section 7. The performances of the proposed method are evaluated in Section 8. Section 9 contains the results of sensitivity analyses on the model parameters. Conclusions are made in Section 10.
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 a X control chart, at a given sample point a sample of n independent observations n
( X i ; i 1, 2,..., n) are taken and the sample mean X 1 n X i is calculated accordingly. If the i 1
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 time interval of h1 . Otherwise, a signal is sent to inform the operator to search for an assignable cause. This control chart is said to operate in a fixed sample-size ( n ) and fixed sampling interval (h1 ) (FSI) condition. When the VSI scheme is in use, the time h until the next sampling point is a function of the current X value. In other words, if X I i ; i 1, 2 , then h hi where
I 1 0 k 1 I 2 0 k 2
n , 0 k 2 n , 0 k 2
n
n 0 k 2
n , 0 k 1
n
with 0 k 2 k 1 , 0 k 2' k 1' , and h2 h1 0 . Fig. 1 depicts a VSI X chart using two interval lengths of h1 and h2 . 5
(1)
0
Fig.1. The asymmetrical VSI control chart when m 2
The sample means in Fig.1 are plotted against the time on the horizontal axis. The first sample mean falls within I 2 , so the next sampling interval is h2 . The second one falls within I 1 , which is close to the control limits. Hence, a shorter sampling interval is adopted to take the third sample and so on. Further, traditional symmetric VSI charts can be easily obtained by setting k 1 k '1 .
3. A Brief Review of the Burr Distribution
Burr (1942) introduced the Burr distribution with the probability density function given in (2). c k y c 1 ; y 0 f y (1 y c ) k 1 0 ; y 0
(2)
6
In which c 1 and k 1 are the skewness and the kurtosis coefficients and can be obtained using the first four moments (or the 3rd and 4th moments) of an empirical distribution. Moreover, the cumulative density function of the Burr distribution is obtained as
1 1 c k F y 1 y 0 ; y 0
; y 0
(3)
which can also be expressed in the following form F y 1
1
1 Max 0, y
c
(4)
k
The properties of the Burr distribution have been studied in depth by Burr (1942), where the mean, the standard deviation, the skewness, and the kurtosis coefficients for a set of Burr distributions were given in two Tables (Tables II and III.) These properties enable one to make a standardized transformation between any random variable (such as X ) and a random variable of the Burr distribution (such as Y ) using the following equation, if they share identical skewness and kurtosis coefficients. X X Y M SX S
(5)
In Equation (5), X and S X are the sample mean and sample standard deviation of the data set, and M and S are the sample mean and sample standard deviation of the corresponding random variable following the Burr distribution, respectively.
4. Model Development
The economic-statistical design of the VSI X control chart under correlation and nonnormality of the process data works under the assumptions given in Subsection 4.1.
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4.1. Model Assumptions
The following assumptions are made to simplify the development: 1. Observations within a sample are correlated and the sample mean ( X ) follows a nonnormal distribution with E( X ) µ and V( X ) 1 n 1 n , where 2
r
ij
i#j
n n 1
and rij is an element of the correlation matrix (Yang and Hancock 1990). 2. The production process starts with an in-control state 0 . 3. When an assignable cause occurs, the process mean shifts to 0 where is the shift magnitude related to a assignable cause and is the standard deviation of the incontrol process. 4. The occurrence of an assignable cause is associated with an exponential distribution with parameter . 5. The production process is ceased for searching, identification, and repairing assignable causes when an assignable cause occurs.
4.2. The Expected Cycle Length
A production cycle is defined as the time duration from the start of the process in the incontrol state until the identification and correction of the assignable cause. To simplify the cycle length calculations, the production cycle ( T ) is divided into four intervals including the incontrol period ( T1 ) , the out-of-control period ( T 2 ) , the searching period due to the false alarm ( T 3 ) , and the period of identifying and correcting the assignable cause ( T 4 ) which are individually obtained as follows: 8
( T1 ) : Based on the assumptions, the process starts with the in-control state and remains in-control until the assignable cause occurs. Since is the failure rate and the time until the occurrence of the assignable cause follows an exponential distribution, the mean time in which the process remains in the in-control state is 1 . ( T 2 ) : Let A be the length of the sampling interval in which the assignable cause occurs, and B be the time interval between the sample points just prior to the occurrence of the assignable cause and the occurrence itself. Reynolds et al. (1988) showed that m
E ( T 2 ) E ( A ) E ( B ) ( S 1 1) h j P1 j
(6)
j 1
where m stands for the number of subintervals within the control limits, S 1 represents the expected number of the samples required to detect the assignable cause, and P1 j denotes the conditional probability that X falls within I j , 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 * , (i.e., S1 1 * ), where * is the probability of detecting an assignable cause in a sample. Reynolds et al. (1988) assumed that the probability of the length of A being h j is
Pr( A h j ) h j P0 j
m
h P j 1
j
(7)
0j
where P0 j is the conditional probability that X falls within I j , given that X falls inside the control limits when 0 . Then, from the result of Duncan (1956), the conditional expected value of B given A h j is
9
E ( B A hj )
1 h j 1 e
1e
h j
h j
(8)
Therefore, the expected length of the out-of control period is obtained using Equations (6)–(8) as
E (T2 )
1
h P j
P0 j h j h j 1 e
j 1
1e
m
j 1
m
0j
h j
h j
S
m
1 1 h j P1 j
(9)
j 1
( T 3 ) : Let S 0 be the expected number of the samples in the in-control state. Bai and Lee (1998) showed that m
S0
P j 1
0j
e
h j
P 1 e m
m h j 1 P0 j e j 1
2
j 1
h j
0j
(10)
As a result, the expected length of the searching period due to a false alarm becomes E ( T 3 ) t 1 S 0
(11)
where is the false alarm rate and t 1 is the searching time when the false alarm occurs. ( T 4 ) : The time to identify and correct the assignable cause following a signal is assumed a constant t 2 . Thus, the expected length of a production cycle can be aggregately represented by
E ( T ) E ( T1 ) E ( T 2 ) E ( T 3 ) E ( T 4 ) 1
m
1 m
h P j 1
j
0j
P0 j h j h j 1 e
j 1
1e
h j
h j
S
m
1 1 h j P1 j t 1 S 0 t 2 j 1
4.3. The Expected Cycle Cost
The expected cost of a production cycle includes the following cost components 10
(12)
1. A penalty cost due to the occurrence of a false alarm 2. A penalty cost due to identifying and fixing the assignable cause 3. A penalty cost due to being in an out-of-control conditions (like poor quality) 4. The costs spent due to the sampling, inspecting, evaluating, and plotting. Let C 1 be the average search cost of a false signal, C 2 the average cost to discover the assignable cause and adjust the process to the in-control state, C 3 the hourly cost associated with production in an out-of-control state, C 4 the fixed sampling cost for each sample, and C 5 the variable cost of sampling and testing in each sample. Then, the expected cost of a production cycle, E ( C ) , becomes E (C ) C 1 S 0 C 2 C 3E T 2 C 4 C 5 n S 0 S 1
(13)
Thus, using Equations (12) and (13), the expected cost per unit time ( ECT ) is obtained as ECT
E ( C ) C 1 S 0 C 2 C 3 E T 2 C 4 C 5 n S 0 S 1 E (T ) 1 E ( T 2 ) t1 S 0 t 2
(14)
4.4. Economic-Statistical Model
By adding two constraints on the Types-I and Type-II error probabilities to the economic model, the economic-statistical model becomes Min ECT ( s ) s.t . ( s ) U
* ( s ) * L design s ( n , h1 , h2 , k 1 , k 2 , k 1' , k 2' ) n is a positive integer h1 , h2 , k 1 , k 2 , k 1' , k 2' 0
11
(15)
In this model, ECT ( s ) represents the expected cost per unit time associated with the design s (that is the function of the implementation characteristics of the chart,) (s ) is the false alarm rate of the control chart based on design s , *( s) is the test power of the control chart in design s , n is the sample size, h1 and h2 are the first and the second sampling intervals, k 1 and k1 are the upper and the lower warning limits, k 2 and k 2 are the upper and the lower control
limits of the VSI X control charts, and U and L* are predefined upper and lower bounds that limit the false alarm rate and the failure-detection power of the control chart. Based on the model given in (15), the optimal values of the design parameters (n ,h ,h ,k ,k ,k ,k ) that 1
2
1
2
1
2
minimize the expected cost per unit time function are determined. Since the objective of this research is to develop an economic-statistical design of the VSI X control chart under correlated non-normal process data, in the next section, correlated data is first modeled. Then we take advantage of the Burr distribution to model non-normal data in Section 6. These two sections are provided to show how P0 j and P1 j used in Equations (6) to (12) are determined.
5. Modeling Correlated Observations
Yang & Hancock (1990) assumed each subgroup a realization of the random vector
X { X 1 , X 2 ,..., X n } following a multivariate normal distribution N ( , V ) in which is the mean vector and V is the covariance matrix as
i E (X i ) i 1, 2,..., n
V v ij Cov (X i , X j ) i 1, 2,..., n , j 1, 2,..., n
12
(16)
V 2 R , where
Assuming
is
standard
deviation
of
the
process and
R {rij | i 1,..., n, j 1,..., n} is the correlation matrix, and that all observations within a subgroup share a mean , the sample mean X can be shown to be normally distributed with mean and variance of E ( X ) ; V ( X ) 2 1 ( n 1) n
(17)
where
rij n n 1
(18)
i#j
The mean and variance of X are still valid even if the measurements are not normally distributed. Now, let and * 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 and * are also called the false alarm rate and the 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
UCL 0 k 1
n
LCL 0 k 1
n
(19)
As a result can be obtained by
1 Pr LCL X UCL 0
Pr X UCL 0 Pr X LCL 0
Then, by replacing UCL and LCL given in Equation (19), we have
13
(20)
X 0
Pr
n 1 n 1 X 0 Pr n 1 n 1
1 n 1 k 1 1 n 1 k1
(21)
Similarly * becomes
* 1 Pr LCL X UCL 0 k ' n X ( 0 ) k1 n 1 1 Pr < 1 n 1 n 1 n 1 1 n 1
(22)
6. Modeling Non-normal Observations
According to Dodge and Rousson (1999), the skewness and kurtosis coefficients of the sample mean X (based on a sample of size n ) are, respectively 3 (X ) 3
n
, 4 ( X )= 4 3 n 3
(23)
In which 3 and 4 denote the skewness and kurtosis coefficients of the random variable X modeling the population. Using the values of 3 ( X ) , 4 ( X ) and tables of the Burr
distribution (1942), one can obtain the values of M , S , c , and k for the Burr distribution with skewness and kurtosis values close to the 3 ( X ) and 4 ( X ) by interpolation. Now, using the CDF of the Burr distribution and standard transformation given in (4) and (4), expressions for and * are derived as
Y M k1 S 1 n 1
Pr
Y M k 1' Pr S 1 n 1
14
Sk 1 Sk 1' Pr Y M Pr Y M 1 1 1 1 n n 1 1 1 c k 1 Max 0 , M Sk 1 1 n 1 1 Max 0 , M Sk 1'
k 1' n k1 n Y M 1 Pr 1 ( n 1) S 1 ( n 1) *
1 n 1
c k
M S k 1' n S k1 n 1 Pr Y M 1 ( n 1) 1 ( n 1) 1 1 c k k1 n 1 Max 0 , M S 1 Max n 1 1
(24)
1
k 1' n 0 , M S 1 n 1
c
k
(25)
As a result, the conditional probability P0 j ( j 1, 2,..., m 1 ) becomes
+ Pr k
P0 j Pr 0 k j 1 0
j
1 = 1 1+Max
n X 0 k j n X 0 k j 1
1 Max
n 0
1 c
k j 1 0, M S 1 n 1
1 c
k j M S 0, 1 n 1
k
, LCL X UCL
n 0 , LCL X UCL
k
1
1+Max
c
kj 0, M S 1 n 1
1
1 Max
15
c
k j 1 M S 0, 1 n 1
k
k
(26)
Moreover, P0 m is obtained as
P0 m Pr 0 k m'
1 1 1+Max
n X 0 k m
n 0 , LCL X UCL
1
c
k m' 0 , M S 1 n 1
k
1 1+Max
c
km 0 , M S 1 n 1
k
c
(27)
In a similar way, the conditional probability P1 j ( j 1, 2,..., m 1 ) is
+ Pr k
P1 j Pr 0 k j 1 0
j
1 1 * 1 Max
n , LCL X UCL
n X 0 k j
n 0 , LCL X UCL
n X 0 k j 1
0
1 c
k j 1 n 0, M S 1 n 1
1 1 Max
c
k j n 0, M S 1 n 1
k
k
1 1 Max
k j n 0, M S 1 n 1
1 1 Max
c
k j 1 n 0, M S 1 n 1
In this situation, P1m is calculated as
P1m Pr 0 k m
n X 0 k m
n 0 , LCL X UCL
16
k
k
(28)
1 1 * 1 Max
1 0, M S
k m n 1 n 1 c
k
1 1 Max
0, M S
c k m n 1 n 1
k
(29)
Since the optimization problem in (15) is hard to solve analytically, in the next section, a meta-heuristic solution procedure is proposed to obtain a near optimum solution.
7. A Solution Procedure and Application
In general, the purpose of the proposed economic-statistical model is to determine optimal values of the design parameters (n ,h1 ,h2 ,k1 ,k2 ,k1 ,k2 ) that lead to the minimization of ECT given in Equation (14) subject to all constraints. An examination of the probability components of the model reveals that finding the optimal values of the design parameters for the VSI X chart using a conventional optimization method is not simple. To achieve this goal, in this research, a genetic algorithms (GA) is employed, where it is explained in detail in the next section through a numerical example.
7.1. A Numerical Example
In this section, a numerical example provided in Chou et al. (2002) is borrowed and modified to demonstrate the application of the proposed methodology. Historical data on a manufacturing process reveal that the shifts occur randomly with a frequency of about four every 100 hours of operation (i.e. 0.04 ). Based on the analysis of operators and quality control engineers and the cost of the testing equipment, the fixed cost of taking a sample is determined 17
$0.5 (i.e., C 4 0.5 ), while the variable cost is $0.1 per bottle (i.e., C 5 0.1 ). On the average,
when the process becomes out-of-control, the magnitude of the mean shift is about three standard deviation (i.e., 3 ). The average time to investigate an out-of-control signal that results in the elimination of an assignable cause is 0.3 hour (i.e., t 2 0.3 ), while the time spent to investigate a false alarm is 0.1 hour ( t 1 0.1 ). Besides, on the average a search cost of $10 (i.e., C 1 10 ) is spent if the assignable cause does not exist. However, if an assignable cause exists, it takes an average of $30 ( C 2 30 ) to discover the cause and correct the process to an in-control state. The estimated cost associated with production in an out-of-control period is $100 per hour ( C 3 100 ). Moreover, let the upper bound on the Type-I error probability be 0.05 ( U 0.05 ) and the lower bound on the test power be 0.90 ( *L 0.9 ) . In short, we have
0.04 , C1 10 , C2 30 , C3 100 , C4 0.5 , C5 0.1 , 3.0 , t1 0.1 , t2 0.3 , U 0.05 , and L* 0.9 Previous data also indicate the skewness and kurtosis coefficients of the concerned quality index can be estimated as 3 1.4322 and 4 7.3558 , respectively. These values approximately correspond to the Burr distribution with c 2 and k 4 . The recent 60 successive parts are viewed as a random sample from a multivariate nonnormal distribution. The sample average is
0.66, 0.59, 0.69, 0.54
sample covariance matrix is 1.05E 05 8.40 E 06 V 7.80 E 06 6.90 E 06
8.40 E 06 7.80 E 06 6.90 E 06 1.07 E 05 9.90 E 06 7.70 E 06 9.90 E 06 1.07 E 05 9.00 E 06 7.70 E 06 9.00 E 06 1.10 E 05
18
and the corresponding
that results in an average correlation coefficient of 0.77 . Now, suppose a set of data is collected and the sample mean and the sample standard deviation are computed as x 0.5 and S x 0.001 , respectively. In order to calculate the probabilities Pij and ultimately ECT , we first find the skewness and kurtosis coefficients of the process data ( ˆ 3 and ˆ 4 ). Then, 3 ( X ) and 4 ( X ) are obtained using Equation (23). Next, using Tables II and III of Burr (1942), the values of M ,S , c , and k corresponding to 3 ( X ) and 4 ( X ) , are obtained using interpolation. Finally, the equations (26)-(29) are used to determine the probabilities P0 j , P0m , P1 j , and P1m that are required to evaluate ECT in Equation (14). Moreover, since there is no significant difference in the expected costs resulted from the use of two or more sampling intervals (Reynolds and Arnold 1989, Chen 2004, Yu and Hou 2006 and Yu et al. 2007), in this paper, a two sampling interval control chart ( 0 h1 h2 ) is employed as well. The GA is coded in MATLAB (version 7.8) environment in which real coding is used for the design parameters (n ,h1 ,h2 ,k1 ,k2 ,k1 ,k2 ) at hand, with the exception that the value of n is integer. The steps involved in the solution procedure of this numerical example are described in detail as follows. Step 1. Initialization: The first population including 59 feasible chromosomes that satisfy the model constraints is randomly generated. (The number ‘59’ is derived based on the methodology described in Section 7.2.) Step 2. Evaluation: The fitness value of each chromosome in the population is evaluated by calculating its objective value (fitness function). The fitness function of the model is ECT shown in Equation (14). 19
Step 3. Selection: The best 18.5% of the chromosomes (the chromosomes with the best fitness function values) are the survivors for the next generation. In the initial generation, the chromosome with the highest cost is replaced with the chromosome with the lowest cost. Step 4. Crossover: From the second 18.5% of the chromosomes, the parent chromosomes are first selected using the Rollet Wheel method. Then, the crossover operation point is randomly selected. After applying the crossover operation on parent chromosomes with the crossover probability of 0.45 (based on the methodology described in Section 7.2.), the children chromosomes are produced. Then, the constraints of the model are checked on offspring for feasibility. If a gene does not satisfy its corresponding constraints, its value is transformed to an acceptable value within the range using a uniform function. Step 5. Mutation: In this step, small changes of the genes are allowed, where the survivors are selected for the next generation using the mutation operator. To do this, a random number representing the index of the gene considered for the mutation is first generated. Then, a random number regarding the constraint of the selected gene is produced and replaced. The mutation rate is assumed 0.365 (based on the experiment described in Section 7.2.). Step 6 . Steps 1 through 5 are repeated until the stopping condition is achieved. In this example, stopping condition is ‘500 generations’. By employing the proposed GA on the economic-statistical design of the VSI X control chart, the near optimal solution along with the false alarm rate, test power, and minimum ECT are obtained in Table 1.
Insert Table 1 about here
20
7.2. Determining the GA Control Parameters
The quality of a solution obtained by a GA usually depends on the setting of its control parameters. These parameters are the crossover probability (CP), the population size (PS), the generation number (GN), and the mutation rate (MR). Niaki et al. (2010) used the statistically efficient method of response surface methodology (RSM) to calibrate the control parameters of GA for economic and economic-statistical designs of MEWMA control charts. In this paper, RSM is also conducted to tune these parameters. RSM is a set of mathematical and statistical techniques that are used to model and analyze the problems in which a response variable (output) is influenced by a function of several predictive variables (inputs) and the objective is to find the best combination values for predictive variables that optimize the response. The first step in RSM is to find a suitable approximation for the relationship that exists between the response and predictive variables. Usually, a first order model is the simplest one and considered first where the response variable is fitted by a simple linear combination of predictive variables. However, when there is curvature on the response surface; i.e. either the interaction or the second order of the variables or both have significant effects on the response, then a second-order model is usually used. The RSM guides the designer to an improving path that along with it the general vicinity of the optimum can be found rapidly and efficiently. After finding the optimum region, an elaborating model is then employed to locate the optimum point (Montgomery, 2005a). As mentioned before, four parameters in GA determine the quality of a solution. We first define two levels (low and high) with a center point for each parameter (as shown in Table 2).
Insert Table 2 about here
21
A 24 central composite design with four central points is utilized for experiments. This design involves twenty experiments; sixteen on the factorial points and four on the central point. The responses ( ECT ) based on the factorial points are given in Table 3. In this table, the low and the high values of the parameters are shown with "−1" and "+1", respectively. The response on the center point are 4.1287, 4.1489, 4.1252, and 4.1275 with the mean of yC 4.1326 . The mean value of the response at factorial points is yF 4.1384 . The analysis of variance results is given in Table 4, where the sum of square of the pure quadratic effect (PQ) is calculated as SSPQ
nC n F ( y C y F ) 2 nC n F
4 16 (4.1326 4.1384) 2 4 16
0.000109
Insert Table 3 about here Insert Table 4 about here
Since the upper critical point of the F-distribution is f 0.05,1,1 10.13 , the results shown in Table 4 state that not only there is no significant curvature on the response surface, but also none of the parameters and their interactions, except the main effect of the parameter GN and interaction effect of CP*GN, are statistically significant at 0.95-confidence-level. Hence, a better response value can be achieved by estimating the response function using the least square method. The estimated response function is Response = 4.14 - 0.00825 PS - 0.00473 CP - 0.00628 MR - 0.00960 GN
(30)
Now, with a proper direction, the optimum values of the four parameters can be found. The proper direction using the steepest descends method (Montgomery, 2005b) is Δ = (9, 0.5, 0.065, 100). 22
The steps and corresponding response values for the economic-statistical design of the VSI X control chart are recorded in Table 5.
Insert Table 5 about here
As shown in Table 5, the minimum expected cost per unit time of the economic design is related to the parameters (PS, CP, MR, GN) = (59, 0.45, 0.365, 500) with the value of $4.08934.
8. Comparisons and Analysis
In this section, the results obtained using the economic-statistical design of VSI X control chart are compared to their corresponding economic-statistical design of FSI X chart to show the efficiency of the proposed model. Based on a similar RSM approach taken for the FSI chart, the optimal GA’s parameter setting is (PS, CP, MR, GN) = (62, 0.55, 0.42, 600). Table 6 shows the optimal designs along with their expected cost and statistical properties.
Insert Table 6 about here
As shown in Table 6, while the false alarm rates and the test powers are approximately the same for both FSI and VSI charts where they are in predetermined desired bounds, an improved expected cost per unit time ( ECT ) of 10.17% and an improved average time to signal (ATS) of 97.24% is achieved by implementing the economic-statistical design of the VSI X control chart. Therefore, using a VSI chart for correlated non-normal process data seems a
better alternative. 23
In order to evaluate the efficiency of the proposed model, a comparison between the economic–statistical and economic designs is also made in Table 7 in which the GA’s optimal parameter setting of the economic design is obtained as (PS, CP, MR, GN) = (60, 0.56, 0.38, 600) using a similar methodology explained in Section 7.2. As shown in Table 7, improvement percentages on the test power and the average time to signal (ATS) of the economic-statistical design of the VSI X control chart are 52.18% and 79.98%, respectively. These improvements are obtained at a higher expected cost per unit time of only 2.47%.
Insert Table 7 about here
9. Sensitivity Analysis
In this section, a sensitivity analysis is provided to investigate the effects of the model parameters (process and cost parameters), and correlation coefficients on the solution of the model. In Table 8, the model parameters , , t 1 , t 2 , C 1 , C 2 , C 3 , C 4 , and C 5 are changed by ±10%, ±25% and ±50%, where the effect of each alternative on the performances of the economic-statistical design of the VSI and FSI X charts is obtained. Moreover, Table 9 shows the minimum, the maximum, and the range of the ECT for each input parameter of the VSI X control chart. Table 10 illustrates the performances of the proposed design in presence of different correlation coefficients. The following points can be inferred based on the observations given in Tables 8, and 9.
24
In all cases the VSI X control chart consistently has lower expected cost per unit time ( ECT ) and substantially lower average time to signal (ATS) than the corresponding FSI X control chart.
As expected, ECT increases when , C 1 , C 2 , C 3 , C 4 and C 5 increase; somehow showing the credibility of the obtained results.
ECT decreases when increases.
Except , all the other parameters have no significant impact on the test power of the VSI X control chart. This conclusion can almost be made for the false alarm rate of the VSI X chart as well.
A bad estimation of parameter leads to large deviations of the statistical characteristics of both the FSI and VSI charts from the target. In other words, reducing by 10%, 25% and 50%, force the model in a loop such that it is unable to find a solution based on ≤ 0.05 and β ≤ 0.1. As a result, we have to increase both Type-I and Type-II error probabilities until minimum α and β are found that minimize ECT . In this case, the maximum value of ECT in VSI X control chart is obtained ($6.46906) when is decreased by 50% with 0.18248 and 0.24631 and the maximum value of ECT in FSI X control chart is obtained ($6.9435) when is decreased by 50% with
0.18128 and 0.2365 .
A minimum ECT in FSI and VSI control chart is obtained $3.30342 and $2.70313, respectively, when the parameter is decreased by 50%.
On the one hand, the effects of parameters t 1 and t 2 on ECT of the VSI are lower than those effects corresponding to other parameters (maximum changes of $0.11937 and
25
$0.10225, respectively). On the other hand, the largest effect is due to the changes in ($2.94272).
For all cases, the VSI X control chart tends to operate with asymmetric control and warning limits. A similar behavior can be seen for FSI X control chart as well. Thus, it can be concluded that asymmetric control and warning limits are superior to their corresponding symmetric limits, an important fact which has been often ignored in the literatures.
Insert Table 8 about here Insert Table 9 about here Insert Table 10 about here
From Table 10, several findings can be spelled out as follow:
In all cases, the VSI X control chart consistently has lower expected cost per unit time ( ECT ) and substantially lower average time to signal (ATS) than those of the corresponding FSI X control chart.
In all cases, the VSI X control chart tends to operate with asymmetric control and warning limits. A similar behavior can be seen for the FSI X control chart as well. Hence, asymmetric control and warning limits are superior to their corresponding symmetric limits again.
ECT and false alarm rate of the VSI control chart decrease when the correlation
coefficient decreases from 0.9 to -0.6. This trend can also be seen for the FSI chart when the correlation coefficient decreases from 0.9 to -0.5. 26
For highly negative correlated observations (i.e. 0.7, 0.8, 0.9 ), the VSI X control chart has approximately a fixed ECT (about $4.09) and almost a unique Type-I error probability (about 0.048). A similar behavior can also be seen for the FSI control chart.
While positively correlated data has no significant impact on the test power of the VSI X control chart, the impact of negatively correlated data on the test power is significant. The range of the changes of the test power for negatively correlated data is from 0.90 to 0.99.
For highly negative correlated data, during the solution process of the FSI X control chart, the model traps in a loop and cannot obtain a solution with ≤ 0.05 and β ≤ 0.1, simultaneously. As a result, we have to increase at least one of the type I and II error probabilities until the minimum amounts of α and β are found that minimize ECT.
The required sample size of the VSI X control chart for highly correlated data is smaller than that of the VSI X control chart for lowly correlated data. This may be due to the fact that higher correlation increases the homogeneity of the data and therefore, there is no need for additional extra observations. This result coincides with the conclusion made by Chou et al. (2002).
10. Conclusions
In this paper, an economic-statistical model of the variable sampling interval X control charts was first developed under correlation and non-normality of process data. A RSM-based parameter tuned GA was then proposed to solve the model and obtain the near-optimum design. Next, a numerical example was brought to demonstrate the applicability of the proposed methodology and to evaluate its performances under different scenarios, where the efficiency of 27
the developed method was proved by performance comparisons with both FSI and economic design. The results of the numerical example showed that not only the proposed VSI design has a good statistical performance, but also it has better (lower) expected cost per unit time (ECT) than its counterpart FSI control chart in all cases. Further, the presented VSI model considerably has better (lower) average time to signal (ATS) than FSI. Moreover, the economic-statistical design of the chart in comparison to the economic design can obtain solutions with better statistical properties without increasing the cost very much. The results of sensitivity analyses showed that the change direction of the expected cost per unit time is the same as the ones of the cost parameters and . However, opposite change direction of ECT was observed when change direction. In addition, false alarm rate, test power, and ECT of the VSI X control chart increase when correlation coefficient decreases (except highly negative correlated observations). Finally, it has been seen that for all cases, FSI and VSI X control charts tend to operate with asymmetric control and warning limits. Due to the importance of controlling the variance of a process as well as its mean, a joint economic-statistical model of VSI X and R chart under correlation and non-normality would be an interesting subject for further research. It would also be an interesting subject to compare the performances of different schemes such as VSI X-Bar, EWMA, and CUSUM in a future work.
References Bai, D.S., Lee, K.T., 1998. An economic design of variable sampling interval X control charts.
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Saniga, E.M., 1989. Economic statistical control chart designs with an application to X and R charts.
Technometrics 31, 313–320. Torng, C.C., Tseng, C.C., Lee, P.H., 2010. Non-normality and combined double sampling and variable sampling interval X control charts. Journal of Applied Statistics 37, 955–967. Woodall, W.H., 1986. Weaknesses of the economical design of control charts. Technometrics 28, 408409. Yang, K., Hancock, W.M., 1990. Statistical quality control for correlated samples. International Journal
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31
Table 1: Near-optimal solution of the economic-statistical design of the VSI X control chart n
h1
h2
k1
k2
k'1
k'2
Power
ATS
ECT
1
0.00266
0.76251
1.89189
1.40937
2.72028
1.69728
0.04807
0.90007
0.00240
4.08934
Table 2: Level plan for GA’s control parameters Low
Center
High
PS
40
50
60
CP
0.3
0.4
0.5
MR
0.2
0.3
0.4
GN
300
400
500
Table 3: Experimental results of the economic-statistical model Experiment
PS
CP
MR
GN
Response
1
-1
-1
-1
-1
4.17535
2
-1
-1
-1
1
4.14673
3
-1
-1
1
-1
4.17290
4
-1
-1
1
1
4.10053
5
-1
1
-1
-1
4.14747
6
-1
1
-1
1
4.13953
7
-1
1
1
-1
4.14456
8
-1
1
1
1
4.14626
9
1
-1
-1
-1
4.16367
10
1
-1
-1
1
4.12444
11
1
-1
1
-1
4.13416
12
1
-1
1
1
4.12733
13
1
1
-1
-1
4.14080
14
1
1
-1
1
4.11957
15
1
1
1
-1
4.10519
16
1
1
1
1
4.12613
32
Table 4: The ANOVA for the experimental results of Table 3 Source of variation
Sum of square
df
Mean square
F
PS
0.00109
1
0.00109
9.05066
CP
0.00036
1
0.00036
2.96678
MR
0.00063
1
0.00063
5.24086
GN
0.00147
1
0.00147
12.24502
PS*CP
0.00010
1
0.00010
0.83970
PS*MR
0.00001
1
0.00001
0.06063
PS*GN
0.00023
1
0.00023
1.92525
CP*MR
0.00016
1
0.00016
1.29983
CP*GN
0.00123
1
0.00123
10.24834
MR*GN
0.00010
1
0.00010
0.85050
PS*CP*MR
0.00019
1
0.00019
1.56312
PS*CP*GN
0.00015
1
0.00015
1.24585
PS*MR*GN
0.00074
1
0.00074
6.12957
CP*MR*GN
0.00025
1
0.00025
2.07060
PS*CP*MR*GN
0.00012
1
0.00012
0.98837
PQ
0.00011
1
0.00011
0.90449
Error
0.00036
3
0.00012
Table 5: The steps and response values of the steepest descends method for the economic-statistical design Step
PS
CP
MR
GN
Response
1
Central point
50
0.40
0.300
400
4.13778
2
Δ
9
0.05
0.065
100
-
3
Central point + Δ
59
0.45
0.365
500
4.08934
4
Central point + 2Δ
68
0.50
0.430
600
4.09393
5
Central point + 3Δ
77
0.55
0.495
700
4.12290
33
Table 6: A comparison between the FSI and VSI X control charts n
h1
FSI
2
1.05333
VSI
1
0.00266
h2
k1
k2
2. 61575 0.76251
1.89189
k'1 2.81557
1.40937
Power
ATS
ECT
0.04161
0.92380
0.08689
4.55213
0.04807
0.90007
0.00240
4.08934
97.2379
10.1662
k'2
2.72028
1.69728
Improvement (%)
Table 7: Comparison between the economic and economic-statistical responses Model
n
h1
h2
k1
k2
k'1
k'2
Power
ATS
ECT
Economic
1
0.00848
0.77855
2.99622
1.35807
1.43908
1.43820
0.03026
0.43041
0.01199
3.98820
Economic-Statistical
1
0.00266
0.76251
1.89189
1.40937
2.72028
1.69728
0.04807
0.90007
0.00240
4.08934
52.1800
79.9814
-2.4734
Improvement (%)
34
Table 8. Effect of process and cost parameters on the optimal design of the FSI and VSI X control charts for correlated data (ρ=0.77) under the Burr distribution ( c=2 and k=4 ) FSI Parameter
t1
t2
C1
VSI
%
Rate
n
h1
k1
k'1
Power
ATS
ECT
n
h1
h2
k1
k2
k'1
k'2
Power
ATS
ECT
ATS
ECT
-50
2
0.5184
2.6031
2.7016
0.0422
0.9261
0.0414
3.3034
2
0.0020
1.0967
2.7021
1.9320
2.9148
2.9026
0.0382
0.9015
0.0071
2.7031
82.833
18.172 12.000
-25
2
1.4876
2.6076
2.8085
0.0420
0.9253
0.1201
4.1115
1
0.7846
1.6161
1.8634
0.0038
2.9525
0.0069
0.0498
0.9093
0.0783
3.6181
34.810
-10
2
0.4717
2.7358
2.9620
0.0366
0.9001
0.0523
4.2659
2
0.0063
0.8174
2.7028
1.9405
2.7654
2.7108
0.0382
0.9014
0.0061
3.8635
88.343
9.435
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
10.166
+10
2
0.4655
2.6254
2.9668
0.0412
0.9220
0.0394
4.7569
2
0.0021
0.7563
2.7080
1.9262
2.8454
2.6926
0.0380
0.9003
0.0047
4.3757
88.062
8.015
+25
2
1.2960
2.4427
2.6291
0.0499
0.9522
0.0650
5.7196
2
0.0040
0.5234
2.6931
2.0768
2.9938
2.8667
0.0386
0.9033
0.0083
4.8484
87.234
15.231
+50
2
0.9493
2.6389
2.6462
0.0406
0.9195
0.0831
6.0155
2
0.0008
0.7163
2.7067
1.8916
2.7596
2.7543
0.0380
0.9006
0.0033
5.3528
96.031
11.017
-50
2
1.3561
1.1436
2.6198
0.1813
0.7635
0.4201
6.9744
2
0.8861
0.4438
1.1702
1.1489
2.8402
0.0005
0.1825
0.7537
0.1586
6.4691
62.245
7.245
-25
2
0.7537
1.9321
2.9373
0.0846
0.8397
0.1439
5.1192
1
0.0002
0.8305
1.4052
0.7811
1.7811
1.6972
0.0888
0.8006
0.0148
4.7501
89.716
7.211
-10
3
1.2084
2.7304
2.8111
0.0661
0.9141
0.1135
5.0711
3
0.0007
0.9372
2.9939
1.8957
2.9949
2.9918
0.0502
0.8671
0.0124
4.4971
89.078
11.320
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
10.166
+10
1
0.4902
2.0443
2.6847
0.0499
0.9265
0.0389
4.4371
1
0.0001
0.6880
2.1913
1.6749
2.7731
2.5741
0.0328
0.9002
0.0007
3.8678
98.168
12.832
+25
1
1.0972
2.4806
2.9116
0.0292
0.9309
0.0814
4.3821
1
0.0016
0.6135
2.6344
2.1079
2.1324
2.0612
0.0186
0.9025
0.0005
3.6501
99.441
16.705
+50
1
1.0775
2.9089
1.5518
0.0204
0.9998
0.0003
3.9838
1
0.0015
0.5701
2.9993
2.8369
1.8940
1.7147
0.0117
0.9898
0.0001
3.5263
78.577
11.482
-50
2
0.4366
2.6038
2.6422
0.0421
0.9260
0.0349
4.6660
2
0.0012
0.7811
2.7009
1.9330
2.6380
2.6214
0.0383
0.9017
0.0050
4.1348
85.675
11.383
-25
2
1.0075
2.6690
2.7571
0.0393
0.9137
0.0952
4.5269
1
0.0036
0.7780
1.8883
1.3718
2.6861
2.0399
0.0483
0.9012
0.0011
4.1005
98.844
9.419
-10
2
1.1017
2.7020
2.6465
0.0380
0.9071
0.1128
4.6655
2
0.0060
0.9296
2.6692
1.8671
2.9851
2.8822
0.0396
0.9080
0.0039
4.1977
96.542
10.026 10.166
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
+10
2
0.4689
2.7266
2.2586
0.0424
0.9021
0.0509
4.5968
1
0.0030
0.5953
1.8787
1.4483
1.8194
1.7905
0.0489
0.9044
0.0037
4.1756
92.733
9.163
+25
3
1.0757
2.9547
2.9999
0.0499
0.9518
0.0545
4.6090
1
0.0018
0.7031
1.8852
1.4193
1.8101
1.7272
0.0485
0.9023
0.0026
4.0937
95.230
11.181
+50
2
0.4365
2.6459
2.7581
0.0403
0.9181
0.0389
4.6007
1
0.0072
0.7423
1.8904
1.4119
2.3317
2.1301
0.0482
0.9005
0.0030
4.0784
92.292
11.353
-50
2
1.1716
2.6004
2.6768
0.0423
0.9266
0.0928
4.7249
2
0.0013
0.9012
2.6990
1.8633
2.8886
2.7182
0.0383
0.9021
0.0033
4.1916
96.444
11.288
-25
2
1.0219
2.7051
2.9719
0.0378
0.9065
0.1054
4.5731
1
0.0004
0.7145
1.8918
1.4162
1.9983
1.7365
0.0481
0.9001
0.0023
4.1047
97.818
10.243
-10
2
1.0855
2.4428
2.9339
0.0499
0.9522
0.0545
4.5366
1
0.0254
0.7297
1.8649
1.3953
2.4586
1.9472
0.0498
0.9088
0.0039
4.1192
92.842
9.201 10.166
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
+10
2
0.4329
2.6698
2.3120
0.0427
0.9135
0.0410
4.6869
1
0.0014
0.8692
1.8872
1.3858
1.9739
1.8519
0.0484
0.9016
0.0014
4.1151
96.584
12.199
+25
2
1.2046
2.5009
2.5757
0.0470
0.9434
0.0722
4.6799
2
0.0084
0.7712
2.7062
1.9363
2.9754
2.9659
0.0380
0.9007
0.0059
4.1107
91.833
12.162
+50
2
1.1967
2.6586
2.7210
0.0398
0.9157
0.1102
4.7430
2
0.0003
0.8259
2.7057
1.9147
2.8526
2.7825
0.0381
0.9008
0.0046
4.1080
95.824
13.388
-50
2
0.9735
2.5346
2.7401
0.0453
0.9380
0.0644
4.2172
2
0.6674
0.5673
2.4562
2.4491
2.7011
0.0023
0.0500
0.9447
0.0333
3.9923
48.276
5.331
-25
2
0.4320
2.7303
2.8825
0.0368
0.9013
0.0473
4.3912
2
0.0222
0.7362
2.7092
1.8881
2.9007
2.6217
0.0379
0.9001
0.0056
3.9954
88.164
9.013
35
C2
C3
C4
C5
-10
2
0.4544
2.7175
2.7101
0.0374
0.9039
0.0483
4.4563
2
0.0020
0.7085
2.7075
1.9363
2.7198
2.6249
0.0380
0.9004
0.0048
4.0723
90.059
8.618
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
10.166 12.184
+10
2
1.2050
2.5886
2.7906
0.0428
0.9287
0.0925
4.7708
2
0.0193
0.8298
2.6917
1.8909
2.7175
2.7151
0.0387
0.9036
0.0057
4.1895
93.839
+25
2
1.1974
2.5573
2.5605
0.0443
0.9342
0.0844
4.7964
2
0.0038
1.0168
2.7081
1.8998
2.8384
2.6859
0.0380
0.9003
0.0053
4.3394
93.720
9.527
+50
2
1.3131
2.5909
2.9343
0.0427
0.9283
0.1014
5.0488
2
0.0043
0.8337
2.7057
1.9318
2.7539
2.7220
0.0381
0.9008
0.0057
4.3644
94.381
13.555
-50
2
0.4256
2.7037
2.7875
0.0379
0.9068
0.0438
4.0473
1
0.0033
0.8847
1.8655
1.3953
2.8479
2.6042
0.0497
0.9086
0.0020
3.5611
95.429
12.014
-25
2
0.4600
2.5341
2.7901
0.0454
0.9381
0.0304
4.3197
2
0.0209
0.8199
2.7056
1.9090
2.9007
2.6834
0.0381
0.9008
0.0065
3.8439
78.599
11.015
-10
2
1.0237
2.7260
2.8688
0.0370
0.9022
0.1110
4.4594
2
0.0033
0.7572
2.7076
1.9470
2.6277
2.6275
0.0380
0.9004
0.0056
4.0042
94.954
10.208
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
10.166 13.197
+10
2
1.2468
2.5507
2.8408
0.0446
0.9353
0.0863
4.8901
2
0.0005
0.8197
2.7091
1.9150
2.9276
2.7169
0.0379
0.9001
0.0046
4.2448
94.669
+25
2
1.1743
2.5243
2.5807
0.0458
0.9397
0.0754
4.9513
2
0.4586
0.8151
2.6388
1.0966
2.6982
2.1321
0.0410
0.9138
0.0433
4.5612
42.625
7.880
+50
2
0.8433
2.7302
2.6685
0.0369
0.9013
0.0923
4.9591
2
0.0007
0.7061
2.7045
1.9771
2.7800
2.7605
0.0381
0.9010
0.0061
4.7091
93.394
5.041
-50
2
1.4722
2.7351
2.8481
0.0367
0.9003
0.1630
3.5665
2
0.0037
1.1263
2.7077
1.9392
2.8333
2.6555
0.0380
0.9004
0.0078
3.2453
95.216
9.005
-25
2
1.6464
2.4598
2.6640
0.0490
0.9497
0.0872
4.4570
1
0.0009
0.8099
1.8847
1.4228
2.7713
1.8515
0.0485
0.9024
0.0031
3.7046
96.443
16.880
-10
2
1.1572
2.6897
2.5995
0.0385
0.9096
0.1150
4.4669
1
0.0034
0.7802
1.8889
1.4113
2.9764
2.6357
0.0483
0.9011
0.0026
3.9403
97.739
11.787
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
10.166
+10
2
0.9171
2.6712
2.8766
0.0392
0.9133
0.0871
4.6364
1
0.0002
0.7531
1.8892
1.4085
1.8665
1.6782
0.0482
0.9009
0.0021
4.2376
97.589
8.602
+25
2
1.0873
2.5070
2.8690
0.0467
0.9425
0.0664
5.1400
2
0.0005
0.6815
2.7074
1.9415
2.9476
2.8066
0.0380
0.9004
0.0046
4.4716
93.070
13.003
+50
2
0.7287
2.7048
2.9438
0.0379
0.9065
0.0751
5.1577
2
0.0016
0.5087
2.7080
2.0441
2.6359
2.6235
0.0380
0.9003
0.0067
4.8482
91.081
6.001
-50
2
0.8891
2.4490
2.6336
0.0496
0.9513
0.0455
4.0831
2
0.0020
0.5962
2.7041
1.9900
2.7520
2.6267
0.0381
0.9011
0.0058
3.7686
87.254
7.702
-25
2
0.4661
2.6354
2.6232
0.0408
0.9201
0.0405
4.2750
2
0.0022
0.6077
2.7066
2.0091
2.6587
2.6538
0.0380
0.9006
0.0066
3.9747
83.686
7.026
-10
2
0.4855
2.4774
2.8566
0.0481
0.9471
0.0271
4.4720
2
0.0009
0.8168
2.7087
1.9066
2.8870
2.7852
0.0379
0.9002
0.0043
4.0643
84.155
9.116 10.166
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
+10
2
1.2277
2.5403
2.9458
0.0451
0.9370
0.0825
4.7830
2
0.0009
0.6994
2.7002
1.9746
2.8104
2.6381
0.0383
0.9019
0.0060
4.2057
92.727
12.070
+25
2
1.3381
2.4613
2.9652
0.0490
0.9495
0.0712
4.9511
2
0.0071
0.8589
2.7046
1.9287
2.7028
2.6542
0.0381
0.9010
0.0060
4.2973
91.571
13.206
+50
2
0.4461
2.6962
2.7071
0.0382
0.9083
0.0450
5.1164
1
0.0010
0.6881
1.8840
1.4369
2.8929
2.2367
0.0486
0.9026
0.0034
4.4784
92.452
12.469
-50
2
0.6358
2.6998
2.8279
0.0381
0.9076
0.0648
4.1473
2
0.0038
0.7385
2.7035
1.9334
2.7510
2.6600
0.0382
0.9012
0.0051
3.9827
92.124
3.969
-25
2
0.6806
2.6635
2.3467
0.0419
0.9148
0.0634
4.2530
2
0.0016
0.7477
2.7002
1.9476
2.9890
2.8961
0.0383
0.9019
0.0054
4.0543
91.484
4.671
-10
2
0.5839
2.4950
2.9520
0.0473
0.9444
0.0344
4.3461
2
0.0121
0.6480
2.7032
1.9808
2.7760
2.6495
0.0382
0.9013
0.0069
4.0772
79.942
6.186 10.166
0
2
1.0533
2.6157
2.8156
0.0416
0.9238
0.0869
4.5521
1
0.0027
0.7625
1.8919
1.4094
2.7203
1.6973
0.0481
0.9001
0.0024
4.0893
97.238
+10
2
1.0605
2.6339
2.6529
0.0408
0.9204
0.0917
4.5882
1
0.0002
0.6345
1.8743
1.4422
2.7113
1.8938
0.0492
0.9058
0.0033
4.1608
96.401
9.316
+25
2
0.4599
2.5919
2.5963
0.0427
0.9281
0.0356
4.6834
2
0.0028
0.6278
2.7011
2.0312
2.8944
2.6869
0.0383
0.9017
0.0077
4.2530
78.382
9.191
+50
2
1.2265
2.4490
2.9785
0.0496
0.9513
0.0628
4.7823
2
0.0016
0.8508
2.6946
1.9230
2.7097
2.6386
0.0385
0.9030
0.0051
4.2677
91.877
10.761
36
Table 9: Minimum, maximum, and range of ECT for model parameters of the VSI X control chart Lower bound
Upper bound
Minimum cost
Changing level (%)
Maximum loss cost
Changing level (%)
Range
2.70313
-50
5.35277
+50
2.64965
3.52634
+50
6.46906
-50
2.94272
t1
4.07838
+50
4.19775
-10
0.11937
t2
4.08934
0
4.19159
-50
0.10225
C1
3.99233
-50
4.36444
+50
0.37211
C2 C3
3.56109
-50
4.70906
+50
1.14798
3.24534
-50
4.84818
+50
1.60284
C4
3.76857
-50
4.47840
+50
0.70985
C5
3.98268
-50
4.26769
+50
0.28501
Parameter
37
Table 10: Effect of correlation coefficients on the optimal design of the FSI and VSI X control chart under the Burr distribution (c=2 and k=4) FSI
VSI
%
ρ
n
h1
k1
k'1
Power
ATS
ECT
n
h1
h2
k1
k2
k'1
k'2
Power
ATS
ECT
ATS
ECT
0.9
2
1.2826
2.6129
2.6883
0.0459
0.9136
0.1213
4.9551
1
0.0132
1.0235
1.8622
1.3656
2.7205
1.9497
0.0499
0.9096
0.0020
4.2383
98.352
14.466
0.8
2
1.0481
2.6987
2.7430
0.0390
0.9052
0.1098
4.6100
1
0.0018
0.8001
1.8909
1.3992
2.2917
2.0788
0.0481
0.9004
0.0019
4.0944
98.270
11.183
0.7
2
1.0231
2.7507
2.6472
0.0340
0.9034
0.1094
4.5418
2
0.0002
0.7111
2.7379
1.9955
2.8752
2.7384
0.0345
0.9005
0.0051
4.0717
95.340
10.351
0.6
2
1.0666
2.6148
2.4499
0.0361
0.9387
0.0697
4.4541
2
0.0004
0.8242
2.7837
1.9800
2.5433
2.5352
0.0297
0.9003
0.0029
4.0195
95.839
9.757
0.5
2
1.0630
2.7228
2.2452
0.0295
0.9282
0.0823
4.4413
2
0.0005
0.7442
2.8291
2.0521
2.9193
2.8823
0.0253
0.9005
0.0026
3.9320
96.840
11.468
0.4
2
1.1213
2.6899
2.5995
0.0269
0.9441
0.0664
4.4245
2
0.0049
0.7242
2.8724
2.1070
2.4686
2.4027
0.0212
0.9015
0.0025
3.8679
96.235
12.579
0.3
2
1.0723
2.8824
2.9014
0.0187
0.9163
0.0979
4.4184
2
0.0030
0.6630
2.9275
2.1997
2.5027
2.3235
0.0172
0.9003
0.0025
3.8012
97.447
13.969
0.2
2
0.9784
2.8815
1.9721
0.0174
0.9285
0.0753
4.2228
2
0.0011
0.7055
2.9794
2.2176
2.4847
2.1922
0.0136
0.9002
0.0011
3.7472
98.539
11.263
0.1
2
0.9468
2.9538
2.5847
0.0124
0.9253
0.0765
4.1485
2
0.0002
0.7245
2.9891
2.2412
2.4420
2.3175
0.0112
0.9112
0.0002
3.7126
99.718
10.507
0
2
0.9491
2.9047
1.9289
0.0109
0.9493
0.0507
4.0403
2
0.0103
0.6973
2.9637
2.3506
2.6667
2.2700
0.0094
0.9312
0.0011
3.6740
97.830
9.068
-0.1
2
0.9803
2.9435
1.9316
0.0082
0.9553
0.0458
4.0297
2
0.0012
0.6476
2.9950
2.4479
2.2224
2.0905
0.0070
0.9390
0.0004
3.6230
99.170
10.093
-0.2
2
0.3756
2.9396
2.3005
0.0063
0.9700
0.0116
3.9596
2
0.0005
0.5429
2.9958
2.5396
2.0491
1.7653
0.0052
0.9544
0.0002
3.6126
98.343
8.765
-0.3
2
0.9355
2.9122
1.4081
0.0111
0.9864
0.0129
3.8871
2
0.6206
0.4134
2.7958
0.0104
2.3456
0.0010
0.0053
0.9939
0.0038
3.5966
70.615
7.472
-0.4
2
0.9469
2.8186
2.5339
0.0039
0.9995
0.0005
3.7818
2
0.0002
0.6005
2.9998
2.7370
2.6507
1.7696
0.0023
0.9840
0.0001
3.5401
94.765
6.392
-0.5
2
0.4098
2.9910
1.4604
0.0017
0.9979
0.0009
3.7121
2
0.0009
0.6471
2.9997
2.8603
2.1624
1.8086
0.0013
0.9955
0.0001
3.5279
87.352
4.964
-0.6
1
0.7699
1.7610
2.9305
0.0699
0.9210
0.0660
4.5225
2
0.6025
0.7171
2.9975
0.0009
1.5098
0.0052
0.0006
0.9999
0.0001
3.5115
99.849
22.356
-0.7
1
0.6887
1.8918
1.7287
0.0599
0.8743
0.0990
4.5498
1
0.0020
0.8056
1.8916
1.3979
2.2986
2.0004
0.0481
0.9002
0.0019
4.0953
98.081
9.990
-0.8
1
0.6477
1.8913
1.7438
0.0599
0.8745
0.0930
4.5558
1
0.0030
0.7222
1.8895
1.4138
2.7457
1.9653
0.0482
0.9008
0.0025
4.0935
97.311
10.146
-0.9
1
0.6853
1.8918
2.8936
0.0599
0.8743
0.0985
4.5498
1
0.0015
0.8097
1.8910
1.4002
2.7877
2.5031
0.0481
0.9004
0.0020
4.0965
97.970
9.963
Parameters: 0.04 , C 1 10 , C 2 30 , C 3 100 , C 4 0.5 , C 5 0.1 , 3.0 , t 1 0.1 , t 2 0.3 , U 0.05 , and 0.9 * L
38
