Statistical Design of an Adaptive Synthetic X Control Chart with Run Rule on Service and Management Operation

Copyright © 2016 Shucheng Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An improved syntheticX control chart based onhybrid adaptive scheme and run rule scheme is introduced to enhance the statistical performance of traditional syntheticX control chart on service andmanagement operation.The proposed scientific hybrid adaptive schemes consider both variable sampling interval and variable sample size scheme.Theproperties of the proposed chart are obtained using Markov chain approach. An extensive set of numerical results is presented to test the effectiveness of the proposed model in detecting small and moderate shifts in the process mean. The results show that the proposed chart is quicker than the standard syntheticX chart and CUSUM chart in detecting small and moderate shifts in the process of service and management operation.


Introduction
Since control chart was introduced by Walter A. Shewhart in 1924, it has been treated as an important tool to detect the process shifts that may occur in services and managements operation process.In today's service and management operation practice, processes have obtained a low level of nonconformities or defects as a result of technological advancement and automation.Although the traditional Shewhart charts have the advantage of easy setup, they also have the high false alarm rates and inability to detect further process improvement under a low-defect environment [1].
To deal with this situation, different types of control charts have been proposed to obtain good performance for effective detection with a wide range of shift sizes, such as cumulative sum (CUSUM) chart, synthetic  chart, and time-between-events (TBE) chart.Among them, the synthetic  chart which was introduced by Wu and Spedding (2000) outperformed the traditional  chart in terms of smaller average run length under the small shift in process mean.It is the combination of Shewhart  chart and conforming run length (CRL) chart used to detect shifts in the process mean.The difference between Shewhart  charts and synthetic  charts is that the synthetic  charts do not send an alarm instantly when a sample falls outside of the limits but inspect the number of the samples taken since the last time that a point fell outside the limits, or since the first sample if there have been no previous points outside the limits.If that number of samples is sufficiently small, then a signal is triggered.The synthetic  chart provides a significantly better detection power than Shewhart  chart for all levels of mean shifts [2].Numerous studies and extensions have been performed on the synthetic chart.Among the more recent ones are Chen et al. [3], Zhang et al. [4], Yen et al. [5], and Zhen [6].
On the other hand, to enhance further detection of the power of control charts for better process control, the adaptive control charts in which at least one of input parameters (the sampling interval, the sample size, and the control limits) is allowed to be changed based on the current state of the process are proposed.Some common adaptive control charts are often concerned in SPC studies, such as the variable sample size (VSS) control charts (see [7][8][9][10]), the variable sampling interval (VSI) control charts (see [11,12]), and the variable sample size and sampling interval (VSSI) control charts (see [7,8,13,14]).It is found that totally adaptive 2 Scientific Programming control charts have been shown to detect the process change faster than the corresponding standard Shewhart chart [15].Qu and Meng [16], Qu et al. [17], Zhen [18], and Zhen [19] used the fundamental diagrams for extreme-scenario analysis on transportation.Among these different types of adaptive charts, the VSSI  chart is even quicker than the VSI or VSS  charts in detecting moderate shifts in the process [20].
Some researchers tried to combine together the standard synthetic  chart with adaptive schemes.Khoo et al. [21] proposed a synthetic double sampling chart that integrated the double sampling (DS)  chart and CRL chart and they concluded that the synthetic DS chart is superior to the synthetic or even the DS  chart.Chen and Huang [22] considered the variable sampling interval scheme as an enhancement to their proposed synthetic chart in order to further improve the chart's performance.Lee and Lim [23] proposed a VSSI-CRL synthetic control chart and they concluded that it has better detection power than the CRL synthetic chart or the VSSI chart in general.Costa and Rahim [24] considered the synthetic chart based on the statistic  to monitor both the mean and variance.They found that their proposed chart always detected process disturbances faster than the joint  and  charts.Machado et al. [25] proposed a synthetic control chart based on two sample variances for monitoring the covariance matrix.The proposed chart was thought to be more efficient than the chart based on the generalized variance ||.In this work, the schemes that consider both variable sampling interval and variable sample size combined with run rules are applied to the standard synthetic  chart for obtaining better detection capacity.
In this paper, an adaptive synthetic  chart with a joint sampling strategy combining variable sampling interval and variable sample size is developed.A redefined running rule scheme is adopted to further improve the performance of the control chart.Compared with the work of Lee and Lim (2005), different running rules are embedded in the proposed synthetic chart.In our model, the chart sends an alarm not only when the conforming run length is sufficiently small but also when the measuring index outsides the control limits.Undoubtedly the detection capability is enhanced, but the problem becomes even more complicated.In this work, we present a Markov chain model and use it to evaluate the zero-state and steady-state average time to signal (ATS) performance of the proposed chart.The numerical results show that the proposed chart has achieved better detection power than the traditional synthetic  chart and CUSUM chart in detecting small and moderate shifts in the process.
The rest of the paper is organized as follows: in the next Section, the formulation of the proposed chart is developed; then the design model is presented.In Section 3, the genetic algorithm is used to solve the statistical design model; and the obtained results are reported and discussed.Finally Section 4 concludes the paper.

Description of the Developed Chart
Assuming that the production process starts in in-control (healthy) state, with the in-control mean  0 and the in-control standard deviation  0 , the one key quality characteristic  is assumed to follow an identical and independent normal distribution ( 0 ,  2 0 ).When process shift occurs, the mean  will change: that is,  =  0 +  0 ( ≥ 0), where  is the magnitude of the process shift.Since quality shifts are not directly observable yet undesirable, the process is monitored by a control chart.At each sampling instance, a sample is taken and the sample statistic, , is computed and plotted on the control chart.Then the control chart gives an indication for the actual process condition.In this work, an adaptive synthetic  chart is proposed to monitor the process and two alternative combinations of the control chart parameters are considered: a relaxed one and a tightened one.The relaxed scheme uses sampling interval ℎ  , sample size   , while the respective parameter values for the tightened scheme are ℎ  ,   , where ℎ  > ℎ  > 0,   >   > 0.
If we set  as the lower limit of the CRL subchart and set the constant  1 >  2 > 0 as the parameters of the  subchart, then the lower control limit, upper control limit, and the warning control limit of the adaptive synthetic  are given by where   is the in-control standard deviation of the sample mean.
By referring to the graphical view of the chart in Figure 1, the operation of the proposed chart is as follows: (1) Set the control limits of the charts.
(2) A sample is taken and its mean () is measured at each inspection instance.
(3) If a sample produces a value between in the central region, that is, (UWL, LWL), the process is in control.The relaxed scheme used for next sampling is (  , ℎ  ).
(4) If the sample mean lies outside the limits, that is, (UCL, +∞) or (−∞, LCL), the control chart gives a signal.The process is declared as out-of-control and an investigation and possible restoration take place.
(5) If the sample mean falls in the warning region, that is, (UWL, UCL) or (LCL, LWL), then the CRL is checked.In this work, CRL is defined as the number of samples since the most recent previous sample mean fell in the warning region or since sampling began if no point fell in the warning region.It is obvious that CRL is a larger-the-better characteristic.
(a) If CRL is larger than the lower control limit , where  is a specified positive integer, the process is still considered as in-control, but the tightened sampling scheme (  , ℎ  ) is used in next sampling.(b) If CRL is smaller than the lower control limit , then the process is signaled out-of-control, and an investigation and possible restoration take place.After that the control flow goes back to Step (2) and the relaxed sampling plan is used.
It is worth noting that as  increases, the adaptive synthetic  control chart behaves more and more like an ordinary VSSI  chart.The most important performance metrics of control charts are average run length (ARL) and average time to signal (ATS) in a long-run process [26].ARL is commonly studied under two cases in literatures: one when process is in control (denoted by ARL 0 ) and the other when the process is out-of-control (denoted by ARL 1 ).Usually, the ARL 0 of the chart is preferred to be long to avoid high frequency of false alarms and ARL 1 to be short to reduce the number of produced nonconforming units.ATS is presented by Tagaras [27], which is defined as the expected value of the time from the start of the process to the time when the chart indicates an out-of-control signal.Correspondingly, denote ATS 0 and ATS 1 as the average time to signal when the process is in-control and out-of-control, respectively.

Computation of ARL and ATS of the Proposed Chart.
Assuming that a single assignable cause occurs at a random time and results in a shift in the process mean of a known magnitude  ( > 0) so that the out-of-control mean value is  1 =  0 +  0 , the occurrence of the assignable cause indicates that the process has gone out of control.The Markov chain approach suggested by Davis and Woodall [28] is used to compute the in-control and out-of-control average run lengths in the process.
Denote  0 1 ( 1 1 ) as the probability that a sample falls beyond the control limits of  subchart when the last sample falls in the central (warning) region.Then the detecting power,  0 1 and  1 1 , of  subchart can be calculated as where Φ(⋅) is the cumulative distribution function of a standard normal distribution function.
Denote  0 2 ( 1 2 ) as the probability that a sample falls in warning region of  subchart when the last sample falls in the central (warning) region. 0 2 and  1 2 are given by Denote  0 3 ( 1 3 ) as the probability that a sample falls in central region of  subchart when the last sample falls in the central (warning) region. 0 3 and  1 3 can be calculated as A Markov chain {(),  ≥ 1} is constructed, where () is the number of samples which fall in the central region between the th and the ( − 1)th sample which falls in the warning region of the  subchart.Then the state spaces of {(),  ≥ 1} are {0, 1, 2, . . .,  − 1, , }, where state  represents that the control chart sends out out-of-control signal.Therefore, we can model the adaptive synthetic  chart using a { + 2,  + 2} transition probability matrix  having the following structure: where 0 = (0, 0, . . ., 0)  and  is the ( + 1,  + 1) matrix of transient probabilities.The ( + 1, 1) vector  = 1 −  ⋅ 1 (i.e., the row probabilities must sum to 1) with 1 = (1, 1, . . ., 1)  .The expected average run length is given by where  is a ( + 1, 1) vector of initial probabilities associated with the transient states, with 1 for the initial state and 0 elsewhere, that is,  = (1, 0, . . ., 0)  , and  is a ( + 1,  + 1) identity matrix.The steady-state probability of the process is required due to the uncertainty of the instantaneous probability of the process in each state. = { 0 ,  1 , . . .,   } is represented as the corresponding steady-state probability of the state space.According to the Markov theory, spreading the above matrix, we obtain the following results: Then when the magnitude of the process shift is  (≥ 0), the expected average sampling interval  ℎ [] and the expected average sample size   [] are calculated as The total expected average sample size of the process can be expressed as where   is the probability of the process in control at arbitrary time.
The out-of-control average run length, ARL 1 , is calculated by substituting  > 0, while the in-control average run length, ARL 0 , is computed by substituting  = 0.The ATS of the adaptive synthetic  chart is given by Similarly, ATS 1 is calculated by substituting  > 0, while ATS 0 is computed by substituting  = 0.

Design Model.
The statistical design of the proposed synthetic  chart can be conducted using the following optimization model: Objective function is Design variable is where  and  max are, respectively, the allowed minimum incontrol average run length to signal and the maximum sample size.The value of  is usually decided by the quality assurance (QA) engineer with regard to the false alarm rate.

Numerical Analysis
3.1.Optimization Algorithm.The established design model for the control chart is a nonlinear programming model with mixed continuous discrete variables, which is too complex to be solved in optimality.Therefore, metaheuristic methods, especially genetic algorithms (GA), were commonly used to solve the problem.GA has been commonly used for its adaptiveness and effectiveness.Successful applications of GA in the designs of control charts can be found in Aparisi and García-Díaz [29] and He and Grigoryan [30].In this study, GA toolbox of the University of Sheffield is developed to solve the optimal statistic designs of the proposed chart.

The Statistical Performance.
In this section, we evaluate the statistical performances of the proposed chart.Table 1 shows the optimal ATS 1 of the proposed synthetic  chart; moreover, each row in the table shows the optimal design parameters.
From Table 1, it is found that the value of ATS 1 of the proposed chart decreases along with an increase in the magnitude of the shift ().A significant change occurs when the value of  changes from small to moderate.That is to say, the proposed chart is sensitive to the change of mean shift when the process is out-of-control.On the other hand, when the mean shift changes from moderate to large, the detection  In actual production, 100% sampling is not possible and the assignable causes are not self-announced; therefore, the average sample size,   , is usually an important matter to QA engineer.In Table 2, we compare the optimal value of ATS 1 between the proposed synthetic  chart and traditional synthetic  chart in three cases (  = {3, 5, 9}).It is reasonable that the optimal value of ATS 1 decreases along with an increase in the average sample size (  ).At the same time, it can be noted that the proposed chart generally achieved shorter ATS 1 than the traditional synthetic chart, showing that the variable sample size and sampling interval scheme is helpful in improving the statistical performance of the chart.

Comparison to CUSUM Chart.
In this section, the proposed chart is compared with the cumulative sum (CUSUM) chart in terms of the average time to signal when process is out-of-control.Woodall and Adams [31] recommended use of the ARL approximation given by Siegmund for designing a CUSUM chart.For a one-sided CUSUM with parameters  and , Siegmund gives for Δ ̸ = 0, where Δ = ( 1 −  0 )/ −  for the upper one-side CUSUM, and  =  + 1.166.Similarly, ARL  can be obtained for the lower one-sided CUSUM.Then we can achieve the ATS 1 of a two-sided CUSUM chart as follows: where ℎ is the sampling interval.Table 3 displays the comparison between the proposed chart and CUSUM chart, and we can see that the proposed chart always has the shorter ATS 1 when the mean shift is small; see  = 0.1∼0.6.However, the opposite result is achieved when the mean shift is large enough and simultaneously the sample size is great; see  = 0.8∼1.5 and   = 9.In other words, the proposed chart is the better choice for QA engineer if the production process is fragile and samples are difficult to obtain; conversely, the CUSUM chart is superior to the proposed chart.
A specific example to display the result can be found in Figure 2. We can see that the traditional synthetic  chart is always the worst of the three to QA engineer.The gap between the three is getting smaller and smaller with the increase of .When the value of  exceeds a certain value, the CUSUM chart is the better choice than the adaptive synthetic  chart.The results mean that, when the mean shift of process is small, the detection power of the proposed chart is always superior to traditional synthetic chart and the CUSUM chart; however, when the mean shift of process changes from moderate to large, CUSUM chart would be the better one.

Conclusion
In this research, adaptive synthetic  charts which integrated variable sample size and sampling interval  charts and CRL charts have been developed to control the state of statistical control in service and management operation process.The performances of these charts were evaluated by determining their optimal statistical design and comparing it with tradition synthetic  chart and CUSUM chart schemes commonly used in the literature.The optimal design was obtained by genetic algorithm, which works to determine the minimum ATS 1 under the set of selected constraints.The obtained results show that the proposed charts work better than the tradition synthetic  chart for all levels of mean shifts and better than CUSUM chart when small to moderate shifts in the mean of the controlled parameter are expected.

Table 1 :
The optimal parameters and the values of ATS 1 .

Table 2 :
Comparison among the optimal ATS 1 corresponding to the two charts.Syn and Trad-Syn represent the proposed adaptive synthetic  chart and the traditional synthetic  chart, respectively.

Table 3 :
Results of the proposed chart and CUSUM chart for comparison.