A New Generalized Range Control Chart for the Weibull Distribution

In this study, a generalized range control chart is designed for the Weibull distribution using generally weighted moving average statistics. The proposed chart is based on minimum generally weighted moving average statistic and maximum generally weighted moving average statistics. We utilize the inverse erf function to transform the Weibull data to normal data. The necessary measures are given to assess the performance of the proposed control chart. The comparison study shows that the proposed control chart outperforms the existing control charts based on exponentially weighted moving average statistic in terms of the average run length. A real example is given for applying the proposed chart in the industry.


Introduction
The control charts are an integral part of statistical process control which is used to monitor and improve the manufacturing process in the industry.Among the control charts, the Shewhart control charts are preferred due to ease in assumptions and use in the industry.These charts are usually designed under the assumption that the quality of interest follows the normal distribution.But the data obtained from the chemical, semiconductor, cutting tool wear, and failure time processes are often skewed [1,2].So, monitoring the skewed data using the Shewhart control chart may mislead to observe the manufacturing process.
The Weibull distribution has been widely used to characterize material breaking strength, describe the lifetime of electronic items, and monitor the reliability data obtained from life test [3,4].For the skewed data, it is not a proper approach to monitor the process using Shewhart X-bar, standard deviation, and range charts.According to [5], these charts are not effective to detect a shift in the process in low population percentiles of these distributions and increase the false signal out-of-control.Several people worked on the designing of control chart for the Weibull distribution.
Nelson [6] designed various control charts for the Weibull process.Bai and Choi [1] designed X-bar and range charts to monitor the skewed data.Hawkins and Olwell [7] proposed cumulative sum (CUSUM) control chart to monitor the scale parameter of the Weibull distribution.Ramalhoto and Morais [8] focused on the development of Shewhart for the Weibull distribution with fixed and variable sampling intervals.Chang and Bai [9] designed various control chart including exponentially weighted moving average (EWMA) chart using weighted standard deviations for the skewed distribution.More details about these charts can be seen in [10][11][12].
Transforming the Weibull distributed data to normal data may make ease for industrial engineering.Therefore, several transformation methods including power transformation, inverse erf function, and Box-Cox transformation are available in the literature.Batson et al. [13] designed moving average control chart using the power transformation.Pascual [14] studied moving average chart using natural logarithm of the Weibull distribution.More details about control chart using transformation can be read in [3,[15][16][17] and in [18].
Recently, Wang [19] proposed the MaxEWMA control chart for the Weibull distribution using three different transformations.Wang [19] recommended that inverse erf function performs better than power transformation and Box-Cox transformation when shape parameter of the Weibull distribution is greater than one.Woodall and Montgomery [20] pointed that development of control chart for the Weibull distribution has a potential for future research.
In this study, a new generalized range control chart is designed for the Weibull distribution using inverse erf function which has been commonly used to transform the Weibull distributed data to normal data.Some tables for practical use are also given.The comparison study shows that the proposed control chart outperforms the existing control chart in terms of average run length.A real example is also given to apply the proposed chart in the industry.

Design of Proposed Control Chart
The Weibull distribution with shape parameter β and scale parameter θ has the following probability density function: Usually, industrial engineers have the information about the parameters β and θ.These parameters can be estimated from the monitoring data if unknown in the practice.This density function is skewed, so symmetric type of control limits may not work for a Weibull distribution.But there are several methods of transforming Weibull distributed data to a normal distributed data.Wang [19] discussed and analyzed three transformation methods and recommended the method in [17] when β ≥ 1.
Faraz et al. [17] suggested the use of erf function to convert the Weibull distributed random variable to standard normal variable.When x is a Weibull distributed data, the following transformed data were shown to follow a standard normal data: where erf z = 1/π z −z e −u 2 du.As suggested by Wang [19], we utilized another statistic from a sequence of above transformed data u i 's: Note here that Φ −1 presents the inverse cumulative standard normal distribution and χ 2 1 cumulative chisquare distribution with one degree of freedom.
The generally weighted moving average (GWMA) statistic is the generalization of traditional EWMA statistic.The GWMA statistic empowers the decision about the state of process by utilizing the weight assigned to current and past information; for more details, see [21][22][23][24].Suppose that M denotes the number of samples in the sequence of independent samples until the 1 st occurrence of event A since the previous occurrence of event A [25].Suppose that P M = 1 , P M = 2 , … , P M = i are assigned weight for current and previous sample such that P M > i shows the weight to target of the process.
Two GWMA statistic to monitor the mean and variability of the process are defined.
The GWMA statistics applied to two transformed data are given by where i = 1, 2, 3, … and the starting values of G 0 and H 0 are usually set at 0. The MaxEWMA control chart has the ability to detect a small shift in the mean or variability than the traditional EWMA control charts [25].The range chart (R i ) which is computed from the difference of maximum quantity, say maxGWMA, and minimum quantity, say minGWMA, has been commonly used to monitor the system variability.
We define the maxGWMA statistic as The minGWMA statistic is defined as follows Following [26], Sheu et al. [25] suggested that when U i and V i are independently normally distributed, the quantity P M > i can be evaluated as follows for the design parameter q and adjustment parameter α.
where i = 1, 2, 3, … , 0 ≤ q ≤ 1, and α > 0. Now, the plotting statistic R can be defined as follows The proposed control chart is stated as follows.
Step 1. Obtain the Weibull data X i 's at each subgroup of size n and transform it to the standard normal data using Step 2. Obtain GWMA statistics G i and Step 3. Declare the process out-of-control if The proposed control chart is a generalization of several control charts.The proposed control chart reduces the chart 2 Complexity in [27] when α = 1 and q = 1 − λ, where λ is smoothing constant in EWMA statistic.The proposed control chart becomes the chart in [19] when α = 1 and R = 0.
The operational process of the proposed control chart is based on two control limits which are defined as where R is the average and SD is the standard deviation of statistic R i , respectively.Also, k is the control chart coefficient which can be determined by closed-form solution, the Markov chain approach, and Monte Carlo simulation (MCS).These are alternative methods for obtaining the properties of the run-length distribution.In this paper, the control chart coefficient will be determined through the following MCS.
(1) Specify the shape parameter β and scale parameter θ of the Weibull distribution.Generate a random sample of size n from the Weibull distribution by assuming in-control state (2) Compute statistic R i and plot it on the control limits.Note that the first of out-of-control is called run length (RL).Repeat this process 10,000 times (3) Compute average run length (ARL) and standard deviation (SD) using these RL.Determine k for which ARL 0 ≥ r 0 , where r 0 is specified ARL (4) Repeat Steps 1-3 to find the ARL for the shifted process, say ARL 1 when scale parameter is shifted to θ 1 = dθ 0 , where d is a constant Using the above MCS, the values of ARL 1 and SD are reported for various control chart parameters in Tables 1-5 as well as the mean and standard deviation when q=0.95, n=5, and various values of α, β, and θ.The ARL 1 , k, and SD for various values of shape and scale parameters of the Weibull distribution and shift constant d are reported in Tables 2-4.
From Tables 1-5, we note the following trend in the control chart parameters.
(1) For same values of all other parameters, the values of ARL 1 and SD decrease as α values decrease.For example, when α = 1 10 and d = 0 99, the value of ARL 1 from Table 2 is 347, and it is 346 when α = 1 0 (2) For the same values of all other parameters, the values of ARL 1 and SD decrease for β ≤ 1 5 and decrease when β ≥ 2 (3) For the same values of all other parameters and q = 0 95, α = 0 80, the values of ARL 1 and SD increase when β increases and θ decreases

Comparison and Simulation Study
In this section, we compare the performance of the proposed control chart with chart based on EWMA statistic and the chart in [19].
3.1.Proposed Chart vs EWMA Chart.As mentioned earlier, the proposed control chart is the extensions of several charts.The proposed control chart reduces to control chart based on EWMA statistic when α = 1 and q = 1 − λ.The performance of the proposed control chart will be compared with chart based on EWMA statistic proposed by [27] using the simulation data.In this study, it is assumed that ARL 0 = 370, n = 5, and α = 1 10.Suppose that the process is declared to be θ  6.
The calculations of statistic R is also shown in Table 6.The values of statistic R are plotted when α = 1 10; β = 2 0, α =1.10; β = 1 0 (exponential distribution) and α = 1 0; β = 2 0 ([27] statistic) are plotted on control charts given in Figure 1.The UCL and LCL of the three charts are also shown in Figure 1.From Figure 1, it can be noted that the proposed chart using GWMA statistic detects shift in the process at around 45 th values while the proposed control charts using the statistic in [27] and exponential distributions are unable to detect shift in the process.From Figure 1, it can be concluded that the proposed control chart using GWMA 3 Complexity statistic performs better than the control chart using the statistic in [27] in detecting earlier shift in the process.[19].The performance of the proposed chart in terms of ARL and SD will be compared with the MaxEWMA control chart proposed by [19] for the same values of specified control chart parameters.To save the space, the comparison will be given when q = 0 95, α = 1 10, β = 2, and θ = 50.The ARLs of both control charts are depicted in Figure 2.

Proposed Chart vs the Chart in
From Figure 2, it can be noted that the proposed control chart has lower curve of ARLs as compared to the chart proposed by [19].From Figure 2, it can be observed that the proposed chart has smaller values of ARLs than the chart in [19] for small-shift level that is d ≤ 0 95.For example, when d = 0 99, the ARL and SD from the proposed control charts are 338 and 332, respectively, while the ARL and SD from the chart in [19] are 344 and 336, respectively.The two control charts have the same performance when shift is large that is when d ≥ 0 95.So, by comparison, it is concluded that the proposed control chart is more sensitive to detect small shift in the process.

Case Study
Monitoring the breaking strength fibrous composite is very important in the industries which guarantee the safety of 5 Complexity material used in the aero industry and construction of bridges.The application of the proposed control chart is given on the breaking strength of carbon fiber data which is modeled by the Weibull distribution with β = 4 8 and θ = 3 2.
The data were originally reported by [5] and used by [17] in the area of the control chart.Let ARL 0 = 370 and the control chart coefficient for this real data is 4.27.The data for the breaking strength of carbon fibers for n = 5 are reported  6 Complexity in Table 7 whereas the first 10 samples are recorded when the process is in-control.The values of statistic R i are also reported in Table 7.
The values of R i are also plotted in Figure 3. From Figure 3, it can be noted that 13 th values are out-of-control which lead the experimenter to take action to bring back the process to the in-control state.

Concluding Remarks
In this study, a new generalized range control chart is designed for the Weibull distribution using inverse erf function which has been commonly used to transform the Weibull distributed data to normal data.Some tables for practical use are also given.From comparison study, it is concluded that the proposed control chart is more efficient than the existing charts in detecting quick shift at small-shift levels.The proposed control chart can be applied in the industry for monitoring of the process that follows the Weibull distribution.The proposed control chart using some other sampling scheme and cost models can be considered as future research.
0 = 50 and β = 2 0. The first 20 observations of subgroup size n = 5 are generated from the Weibull distribution with parameters β = 2 0 and θ 0 = 50.The next 30 observations of subgroup size n = 5 are generated from the Weibull distribution with parameters β = 2 0 and shift in scale parameter that is d = 0 75.The simulation data, G i and H i , of the proposed control chart are shown in Table

Figure 1 :
Figure 1: The proposed control chart and existing charts using simulation data.

Figure 2 :
Figure2: The ARL curve between the proposed chart and the chart in[19].

10 Figure 3 :
Figure 3: The proposed chart using the real data.

Table 1 :
Mean and standard deviation at different level.

Table 7 :
The real example data.