A Multiple Dependent State Control Chart Based on Double Control Limits

This study is to propose and design a new X-bar control chart utilizing previous subgroup information called a multiple dependent state control chart based on double control limits. The in-control Average Run Length (ARL) and the out-of-control ARLs according to mean shifts are derived. The performance of the proposed control chart is compared with the traditional X-bar chart in terms of ARLs, which shows the outperformance of the proposed control chart. This idea can be easily extended to other types of control charts.


INTRODUCTION
In a manufacturing process, the variation in the process may be caused due to several reasons for example, the external temperature, improper adjustment of manufacturing machines, human errors such as the operator error in recording or analyzing the data and defective raw material (He et al., 2002).The process is favorable when it is in control because in this state the process is manufacturing the acceptable products.On the other hand, if the process has shifted, then the process may produce products beyond the specification limits and so the industrial engineer wants the quick indication about this situation so as to search for the cause of the variation.The control chart is one of the important tools of statistical process control.The control chart helps industrial engineers to monitor the manufacturer process and point out when the process is going to be out of control.
Researchers are putting efforts to design efficient control schemes to improve the monitoring capability.Variable control charts are used when the variable of interest in the process is measurable.Among them, the Shewhart X-bar control charts have been widely used for the measurement process.Many authors have discussed the advantages and application of X-bar control charts including for example, Rahim (1989), Saniga (1989), Chiu (1995), Ben-Daya andRahim (2000), Saniga and Davis (2001), Mehrafrooz and Noorossana (2011), Zhang et al. (2011), Chen et al. (2011), Fallah Nezhad and Niaki (2010) and Caballero Morales (2013).
By exploring the literature, we note that the control charts utilizing the single, double and triple sampling are available in the literature (He et al., 2002).It is known that the control chart based on double sampling or triple sampling is more efficient than the single sampling.One of the important sampling schemes called the Multiple Dependent State (MDS) sampling has been widely used in the area of acceptance sampling plans.Wortham and Baker (1976) introduced the MDS sampling scheme, where the decision on the deposition of a lot is taken on the basis of the current sample information and the previous lot information.In fact, MDS sampling does not require additional sampling but uses previous sample data.Readers may refer to Balamurali and Jun (2007) and Aslam et al. (2013).
In this study, we will introduce the MDS sampling in the area of control charts.The proposed control chart is based on double control limits and utilizes the sample information from the previous subgroups in addition to the current subgroup.So, it is expected that the proposed control chart will be more efficient than the traditional Shewhart control chart in term of average run length.

METHODOLOGY MDS X-bar chart using double control limits:
Assume that the quality characteristic of interest follows a normal distribution with mean µ and variance σ 2 .It is also assumed that the process mean is m when the process is in control.We propose the MDS X-bar control chart using the double control limits.The operational procedure of the proposed chart is given in the following three steps: Step 1: Select a random sample of size n from the current subgroup and compute ܺ ഥ .
Step 2: Declare the process to be in control if Otherwise, go to Step 3.
Step 3: Declare the process to be in control if ݅ proceeding subgroups shows that the process is in control that is Otherwise, declare the process to be out-ofcontrol.
The double control limits of the proposed control chart are composed of the outer control limits: and the inner control limits given by: The proposed chart involves two control constants k 1 and k 2 as well as the parameter i.The proposed chart is reduced to the traditional Shewhart X-bar control chart when ݇ ଵ = ݇ ଶ = ݇ and ݅ = 1.
The probability that the process is in control for the proposed control chart is given as follows: The probability in Eq. ( 1) is obtained by: So, the Average Run Length (ARL) when the process is in control or the in-control ARL is given as: Suppose that the process is shifted from ݉ to ݉ + ܿߪ, where ܿ is the shift constant in the process.Then, the probability that the process is declared as being in control can be given as follows: The ARL when the process has shifted to ݉ + ܿߪ or the out-of-control ARL is given as: To construct the tables, we used various values of sample size ݊, some specified in-control ARL and various shifts in the process.We wrote the R codes for the proposed control chart to complete the tables.These R codes are available from the authors upon request.
The values of ‫ܮܴܣ‬ ଵ are placed in Table 1 to 3. Table 1 is presented when ‫ܮܴܣ‬ = 200, Table 2 for ‫ܮܴܣ‬ = 300 and Table 3 for ‫ܮܴܣ‬ = 370.The control constants k 1 and k 2 as well as the number of the preceding subgroups (i) are determined to yield the specified ‫ܮܴܣ‬ .
It is observed that the ARL decreases rapidly as the shift constant c increases and that the decreasing rate becomes faster as the sample size gets larger.There seems to be no trends in k 1 , k 2 and i according to the sample size.
Comparative study: In this section, we compare the performance of the proposed control chart with the traditional X-bar control chart in terms of the ARL.As we have mentioned above the proposed control chart is the generalization of the Shewhart X-bar control chart.When k 1 = k 2 = k and ݅ = 0, the proposed control chart becomes the Shewhart X-bar control chart.For the comparison purpose, we selected ‫ܮܴܣ‬ = 370 and considered various values of sample size.We placed the ‫ܮܴܣ‬ ଵ values for the both control charts in Table 4.   From Table 4, we note that the for the same values of ܿ and ݊, the ARLs of the proposed chart are much smaller than those of Shewhart X-bar control chart.The proposed control chart is also more efficient than the existing control chart for small shifts.For example, when ܿ = 0.1 and n = 50, the value of ‫ܮܴܣ‬ ଵ from the existing X-bar control chart is 90.56 and 72.42.The difference between the values of ‫ܮܴܣ‬ ଵ increases as ‫ݎ‬ decreasing or shifts increasing.

CONCLUSION
The MDS X-bar control chart using double control limits is introduced in this study.The proposed control chart not only uses the current information from the sample but also uses the pervious subgroup information.The use of the pervious information increases the efficiency of the control chart in detection of the process mean shift.The efficiency of the proposed control is discussed with the Shewhart X-bar control chart in terms of ARL for the shifted process.From the comparison, we concluded that the proposed chart provides the smaller values of ARL as compared to the existing control chart.The use of the control chart in the industries improves the production process and quality of the product as quick indication will minimize the non-conforming products.The idea of MDS and double control limits can be applied to other types of control charts, which will be pursued in a future study.