The recent industrial revolution is a result of modern technological advancement and industrial improvements require quick detection of assignable causes in a process. This study presents a monitoring scheme for unit interval data assuming beta and unit Nadarajah and Haghighi distributions. To this end, a maximum exponentially weighted moving average (Max-EWMA) chart is introduced to jointly monitor unit interval bounded time and magnitude data. The performance of the proposed chart is evaluated by using average run length and other characteristics of run length distribution using extensive Monte Carlo simulations. Besides a comprehensive simulation study, a real data set is also used to assess the performance of the chart. The results supplementing the proposed chart are efficient for joint monitoring time and magnitude, and simultaneous shifts are detected more quickly than separate shifts in the process parameters.

1. Introduction

In the context of services or goods provided by industry, quality excellence plays an important role that needs to be judged by a metric. If a good or service is termed to be of superior quality, there must be a standard to determine which products or services are of superior quality. Thus, there arises a need to establish specifications, measurement systems, acceptability standards, and methods for the result assessment. To improve monitoring and control, statistical quality control (SQC) tools are widely used. A fundamental tool of SQC is the control chart, which is a graphical presentation of the process history. If a monitoring statistic falls within the boundaries of a control chart, it is termed as in-control (IC); otherwise, it is termed as out-of-control (OOC).

There are two types of control charts. The first one is the memory-less charts that use the information of the current sample only while completely ignoring the process history. These charts work well with a large shift in the process. Attribute charts like p and np using Bernoulli distribution are commonly used to monitor the fraction of the number of nonconforming items [1, 2]. However, due to the memory-less nature of these charts, the historical information of the process is ignored as the new points are plotted [3]. As a result, these charts cannot detect small shifts in process parameters satisfactorily. Consequently, memory-type control charts such as the exponentially weighted moving average chart (EWMA) and the cumulative sum (CUSUM) chart are required. The EWMA chart is more sensitive as compared to the Shewhart chart because it takes into account the historical information of the process. The EWMA chart is a Markov process and simpler than a CUSUM chart. It is reported in the literature that the EWMA chart is not sensitive to the non-normality of data for small smoothing values [4].

Many processes require very high precision, also known as the high quality processes, which cannot be monitored by using ordinary attribute control charts. To this end, time-between events (TBE) control charts are used for a high quality process monitoring, for example, the production process of an armed facility requires utmost precision. The recent popularity of TBE charts is an indication that these charts are suitable for high quality processes such as high yield production lines with extremely small defect rates [5] because these charts overcome the drawbacks of the traditional Shewhart charting schemes [6]. Ali [7] proposed a TBE control chart based on the renewal process assuming a continuous exponentiated family of distributions. Some recent works on TBE can be seen in [8], [9], [10, 11], [12, 13], and [14].

In many cases, the impact of an event occurrence on the process performance is determined by the event magnitude or frequency. Thus, the occurrence of events along their magnitudes needs to be monitored simultaneously with sophisticated tools. Wu et al. [15] developed a rate chart for monitoring time and magnitude simultaneously by using the ratio of magnitude and time. In another study, Wu et al. [16] utilized a Shewhart control chart for monitoring the frequency and magnitude of an event simultaneously. Some recent contributions in this direction can be seen in [17], [18], [19], and references cited therein.

In many practical settings, our interest lies in the monitoring processes bounded in unit interval (0, 1) and existing charts such as based on Bernoulli distribution [20] have poor performance. Thus, there is a dire need to develop control charts assuming other suitable distributions to monitor rates or proportion data. Many practical situations where such data are frequently encountered include unemployment rate, TV rating, and area (in proportion) specified for agricultural purposes [21]. Sant’Anna and Caten [22] used beta distribution to construct a Shewhart control chart for monitoring fractional type data. Lima-Filho et al. [23] introduced an inflated beta control chart. Simplex distribution is somehow similar to exponential dispersion family as well as normal distribution; that is, if the dispersion parameter of simplex distribution is small, this distribution is similar to the normal distribution and becomes flexible to cover many features of data if the dispersion parameter is large [24]. This distribution also has applications in medical science such as mapping the proportion of remaining gas volume relative to its initial volume in the eyes of patients [21, 25]. Zhang and Qiu [26] presented a regression analysis for proportion data using the simplex distribution.

To monitor time and magnitude data bounded in the (0, 1) interval, this article proposes a Max-EWMA chart. The plotting statistic of this chart comprises the maximum the absolute of two independent EWMA statistics. Thus, it is suitable for simultaneously monitoring shifts in magnitude (X) as well as TBE (T) of events. The performance of the chart is assessed by using different average run length (ARL) characteristics. Besides comprehensive Monte Carlo simulations, a real data example is also part of the study to show the application of the proposed method.

The rest of the study is organized as follows. Section 2 presents information regarding beta and unit Nadarajah and Haghighi distributions. The construction of the proposed chart for time and magnitude monitoring is also discussed in the same section. In- and out-of-control run length properties are discussed in Section 3. Real life application of the proposed chart is presented in Section 4. Finally, the concluding remarks are discussed in Section 5.

2. Max-EWMA Control Chart to Monitor Unit Bounded Magnitude and Time Data

To monitor the processes based on unit interval bound variables, a Max-EWMA control chart is proposed. To design the Max-EWMA chart, we used beta and unit Nadarajah and Haghighi distributions.

2.1. Beta Distribution

A random variable is said to be beta distributed with the first shape parameter (shape 1) and the second shape parameter (shape (2) , i.e., , if it has the probability density function as follows:where is the beta function and .

The mean and variance of are and , respectively. Suppose the parameters of beta distribution are transformed as and , then the density of beta distribution can be written as

Now, the transformed has the mean and variance as and .

2.2. Unit Nadarajah and Haghighi Distribution

The UNH distribution is proposed by Shah et al. [27]. The probability density function of Nadarajah and Haghighi (NH) distribution iswhere and are the shape and rate parameters, respectively. Using the transformation , one can obtain the unit NH density as

The mean and variance of this distribution can be computed numerically.

2.3. Max-EWMA Chart

In order to develop the proposed charting scheme, assume that the TBE follows UNH distribution, , and magnitude follows beta distribution, . Furthermore, the process is assumed to start at , and IC parameters are denoted by and . For the sake of simplification, independence between the magnitude and TBE is assumed. The shifts in the parameters of X and T, respectively, are considered as follows. , , , and . Here, , and represent the shift in the shape and rate parameters of UNH distribution and shape 1 and shape 2 parameters of the beta distribution, respectively. The process is IC when both and are equal to one, i.e., . For constructing the Max-EWMA control chart, the standardized statistics are defined as follows:where and represent the mean and variance of the UNH which are computed numerically. Using these statistics, two independent EWMA statistics are constructed aswhere is known as the smoothing constant of the Max-EWMA control chart, and L is the constant multiplier selected to have a desired IC run length. In the literature, a large value of the smoothing parameter is suitable for detecting a large shift and vice versa. Typically, to detect small to medium size shifts, we use [28]. The initial values of the EWMA statistics are fixed at 0, i.e., and .

The charting statistic of the proposed chart is given bywherebecause consists of the maximum of absolute of two EWMA statistics. Moreover, E (M) and V (M) denote the values of mean and variance of the monitoring statistic M when the process is under natural variations. A downward (DW) as well the upward (UW) shift in either or both of the parameters can be detected using the proposed chart, and further, it can be labeled to initiate the diagnostic process listed in Table 1.

2.4. Charting Procedure for the Max-EWMA Control Chart

The following steps are used in the construction of the Max-EWMA control chart.(1)Fix the initial values of both of the EWMA statistics at 0, i.e., .(2)Choose a suitable value of constant multiplier L and smoothing parameter for different values of , , , and in relation to different combinations of and . Different values of L are listed in Tables 2 and 3.(3)Using equation (8), compute the value of UCL.(4)Obtain the kth test observation , where is the kth observation of the magnitude, and represents the TBE between the kth and event’s occurrence.(5)For the test observation, calculate the value of the monitoring statistic .(6)Plot against the counter value . If , the process is IC and go to Step 3. indicates the possibility of the occurrence of an OOC condition and proceed to the next step.(7)Conduct an investigation to know if the process is operating under an assignable cause. If yes, remove the assignable cause by making necessary adjustments to the process and move to Step 3.(8)Using the criteria in Table 1, one can label the OOC points. That is, if or are greater than the UCL, label or to represent shifts in the TBE and magnitude, respectively. On the other hand, if both and are greater than the UCL, label to represent the occurrence of a simultaneous shift in both and .

3. Run Length Properties of the Proposed Chart

The average run length (ARL) is widely used as a performance assessment measure of a control chart. It is defined as the mean value of the number of samples or subgroups that are plotted until an OOC signal occurs. The IC ARL is calculated assuming that the process is operating under natural variations and denoted by , while the OOC ARL is calculated when the process is operating under a shift and denoted by . If there is no shift in the process, no OOC signal should occur, but due to the natural variations in a process, a false alarm signal does occur. On the other hand, when the process moves to an OOC state, an immediate OOC signal should be detected by a chart and adjustments should be made immediately. Thus, it is required that should be large so that unnecessary inspection of the process is avoided and should be small to quickly adjust the process. A smaller value has become a standard of efficiency when comparing the performance of different control charts. To study the properties of the run length (RL) distribution, we computed the standard deviation of RL (SDRL) and several percentiles (q (k), k = 10, 25, 50, 75, 90) of run length distribution. For assessing the performance of the control chart under Case 1 (a), consider that . Table 4, Table 5, and Table S1 are constructed, respectively, to study the shift in the shape1 parameter of the beta distribution, rate parameter of the UNH distribution, and simultaneous shift in both parameters. Under Case 1 (b), Tables S2–S4 are constructed, respectively, to study the shift in shape 2 parameter of beta distribution, shape parameter of UNH distribution, and simultaneous shift in both of the parameters. On the other hand, Table S5 to Table S10 are computed to study similar shifts considering .

Different sizes of shifts, simultaneous as well as pure, are considered to evaluate the performance of the chart. More specifically, we considered UW shifts as (1.40, 1.30, 1.20, 1.10) and DW shifts as (0.50, 0.60, 0.70, 0.80, 0.90), where indicates that the respective parameter is IC. The value indicates a DW shift of a 50% size, and represents an UW shift of size 40%. The IC values of mean and variance of the monitoring statistic, denoted by EM and VM, are also listed in the tables. The cases considered for the analysis are as follows:Case 1: and Case 2: and Case 3: and Case 4: and

is considered for all of the cases given above. The IC values of the parameters of UNH distribution are and . Furthermore, the IC ARL is fixed at (370, 500). The results are obtained using an algorithm written in statistical software R (version 4.0.0 (2020-04-24)).

3.1. Results and Discussion

A detailed discussion on the performance of the proposed Max-EWMA control chart is given in this section. Considering a pure shift in the shape 1 parameter of beta distribution with and in Table 4, the ARL reduces to 20.497, 8.723, 5.714, and 4.322 with respect the value for UW shifts of sizes (1.10, 1.20, 1.30, 1.40) and 19.242, 7.959, 4.993, 3.689, and 2.965 for DW shifts of sizes (0.90, 0.80, 0.70, 0.60, 0.50), with . In terms of percentage, the ARL reduces by 94.47%, 97.65%, 98.46%, and 98.83% for UW shifts and by 94.81%, 97.85%, 98.65%, 99.00%, and 99.20% for DW shifts. For , the ARL reduces to 20.489, 7.612, 4.760, and 3.557 and 19.063, 6.787, 4.133, 2.998, and 2.337, respectively, for UW and DW shifts as mentioned earlier. Finally, for , the ARL reduces to 22.213, 7.266, 4.376, and 3.198, and 21.465, 6.400, 3.727, 2.661, and 2.119 for the aforementioned shift sizes. Thus, small smoothing values are more efficient for shift detection than large values. In the case of , the ARL degenerates a similar pattern. The rest of the table can be interpreted similarly.

For a pure shift in the rate parameter of the UNH distribution, Table 5 shows that under the parameterization given in Case 1 with and , the values for the UW shifts in the rate parameter of T are 275.766, 196.429, 139.191, and 103.370, and for the DW shifts, these are 380.233, 277.300, 168.377, 97.526, and 60.585 assuming that . For , the values for UW and DW shifts are 277.474, 201.030, 152.240, and 113.700 and 458.108, 495.728, 405.167, 255.970, and 138.681, respectively. Therefore, the DW shifts are not detected efficiently as compared to the UW shifts.

Simultaneous shifts in the shape 1 parameter of X and rate parameter of T are discussed in Table S1 assuming that . Unlike a pure shift in the rate parameter of T, we observed a smooth degenerating pattern in the case of simultaneous shifts. For example, in Case 1 when , , (1.10, 1.20, 1.30, 1.40), and , the ARL reduces to 20.404, 8.748, 5.688, and 4.332, respectively. The decrease in the ARL compared to the nominal ARL value is 94.49%, 97.64%, 98.47%, and 98.83%, respectively. On the other hand, if there is a DW shift in the shape 1 parameter of X, i.e., when (0.90, 0.80, 0.70, 0.60, 0.50), the ARL reduces to 19.139, 7.926, 5.009, 3.670, and 2.978, respectively; that is, the ARL gets reduced by 94.84%, 97.86%, 98.65% 99.01%, and 99.20%. Similarly, with , the ARL reduces to 20.114, 8.742, 5.699, and 4.348 for UW shifts and reduces to 18.941, 7.971, 5.001, 3.682, and 2.963 for DW shifts in the rate parameter of T. The ARL gets reduced similarly for the rest cases.

It is noticed that the performance of the control chart under a pure shift in the rate parameter of T is poor as many values are either large or biased . On the other hand, in the case of a shift in the first shape parameter of X, the proposed chart performs much better compared to the former. However, as our purpose lies in the joint monitoring, the proposed chart performs better for joint shifts as compared to a pure shift in the rate parameter of T, especially in Case 1 and Case 2. The SDRL, similar to the ARL, decreased by increasing the shift size. The percentile analysis shows that the ARL has a rightly skewed distribution because the ARL is greater than the 50th percentile (median) and less than 75th percentile.

In addition, the proposed chart shows a high sensitivity for smaller shifts and smoothing parameter values. However, its sensitivity, for moderate to large shifts, increases as increases. For example, in Case 1 of Table 4 with , when and for . Similarly, in Case 2 of Table S1 with , for and for . Thus, it is concluded that the smaller values of are suitable for detecting smaller shifts and vice versa [28].

Tables S2–S4 are tabulated to study the shift in shape 2 parameter of beta distribution, shape parameter of UNH distribution, and simultaneous shift in these two parameters, respectively. The proposed chart behaves very similarly as discussed previously. The chart is efficient in detecting pure shifts in the shape 2 parameter of X and inefficient in detecting pure shifts in the shape parameter of T. The performance of the chart with simultaneous shifts is more or less the same. However, one can notice that a pure UW shift in the shape 1 parameter of X is detected more quickly as compared to a pure UW in the shape 2 parameter of X and vice versa. Similarly, simultaneous UW shifts in the shape 1 parameter of X and rate parameter of T are detected quickly as compared to the simultaneous UW shifts in the shape 2 parameter of X and shape parameter of T and vice versa. The chart performs poorly for pure shifts in either parameter of UNH distribution. The interpretation of the rest of the tables from Table S5 to Table S10, which are constructed assuming , can be done on similar lines.

4. A Real Data Application

This section shows the implementation of the proposed chart on a real-life data set. The data set consists of shooting percentages of 292 players as reported by Simonoff [29]. This data set is considered as the magnitude (X). As the relevant TBE data set is not available, we generated TBE data (T) assuming the UNH distribution with the IC parameters and . The TBE data set represents the proportion of minutes spent on the training by the players (out of 120 minutes) daily. The beta distribution is fitted to X and UNH on T, whereas the estimates of the parameters are obtained by using the maximum likelihood method. These 292 observations of X and T are assumed to be IC and are used as Phase I data, whereas 58 OC observations are generated as the Phase II data by introducing 15% UW shift in the shape 1 parameter of X and shape parameter of T, i.e., . We considered the following parameters to construct the chart , and .

The estimates of the parameters of Phase I data through the maximum likelihood method are listed in Table 6. Besides these estimates, this table also includes standard errors (given in parentheses) of these estimates, i.e., AIC and BIC. In Figure 1, a control chart is constructed for the combined Phase I and Phase II data set. As all of the Phase I data are plotted under the UCL, the chart triggers the OOC signal for the first time at the 305th sample. The reason behind the occurrence of the OOC signal is the presence of an assignable cause in the shooting percentage.

5. Conclusion

Generally, a process deteriorates due to assignable causes and we need efficient monitoring tools to detect such changes. There are very few charts for monitoring unit interval bounded data. To this end, we modified the EWMA chart and introduced a Max-EWMA control chart for monitoring TBE and magnitude data simultaneously. As the charting statistic of the proposed chart is comprised of two independent EWMA statistics, the proposed chart can detect small to medium UW and DW shifts effectively. We have considered different shifts and values of the smoothing parameter to assess the behavior of the control charts. Based on the beta and unit Nadarajah and Haghighi distributions, the efficiency of the proposed chart is evaluated in terms of different run length characteristics. The results indicated that the proposed chart is efficient for small or medium size simultaneous shifts. The performance of the chart is observed to be poor in the case of pure shifts suggesting that charts based on simultaneous shifts are more efficient as compared to pure shifts. Finally, a real data example is also shown to illustrate the implementation the chart in practice. In the future, the effect of estimation on in- and out-of-control ARL can be studied. Also, the use of the smoothing parameter value based on the fact that the smaller value is suitable for smaller shifts and vice versa is not very reliable, since, in real life applications, we cannot predict whether a shift size is large or small. To address this particular issue, an adaptive Max-EWMA chart can be used. Furthermore, other unit interval distributions, including truncated normal, Kumarshwamy, unit logistic, log-Lindley, unit Birnbaum-Saunders, and unit Gompertz, can also be used to construct time and magnitude charts.

Data Availability

The data used in the study can be requested from the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Supplementary Materials

The pdf file contains additional tables for simulation studies. (Supplementary Materials)