Enhancing EWMA chart using CRSS for cost models: application to implantable cardioverter-defibrillator monitor

Introduction

Statistical Process Control (SPC) is an important methodology used in manufacturing and industrial processes to monitor process stability and improve product quality. Control charts are one of the most widely used SPC tools for identifying variations in process parameters and detecting the presence of assignable causes. Among the different types of control charts, the Exponentially Weighted Moving Average (EWMA) control chart is particularly effective in detecting small and moderate shifts in the process mean, since it assigns exponentially decreasing weights to past observations and gives more importance to recent data (Lucas et al., 1990) [1] . The statistical performance of control charts is commonly evaluated using measures such as the Average Run Length (ARL). The in-control average run length (𝐴𝑅𝐿 0 ) represents the expected number of samples taken before a false alarm occurs when the process is operating under control, while the out-of-control average run length (𝐴𝑅𝐿 1 ) represents the expected number of samples required to detect a shift in the process mean. Although ARL-based design focuses on the statistical efficiency of control charts, it does not consider the economic consequences associated with process monitoring and quality control (Montgomery, 2009) [2] . In real-time industrial applications, implementing a control chart involves several cost components such as sampling and inspection costs. These costs are associated with investigating false alarms, costs of locating and repairing assignable causes, and the loss incurred due to the production of non-conforming items when the process operates in an out-of-control state. Therefore, considering economic aspects in control chart design is essential for achieving efficient process monitoring and minimizing operational costs. The economic design of control charts aims to determine optimal design parameters such as sample size, sampling interval, and control limits by minimizing the expected cost associated with the monitoring system. To incorporate economic considerations into control chart design, a cost function is introduced to evaluate the expected cost of operating the monitoring system. The unified economic model proposed by Lorenzen et al. (1986) [3] evaluates the expected cost per unit time as the ratio of the expected cycle cost to the expected cycle time, allowing optimal chart parameters to be determined while maintaining acceptable statistical performance. In recent years, many researchers have focused on the economic and statistical design of control charts to improve both statistical detection capability and operational efficiency. For example, Linderman et al. (2000) [4] developed an economic and economic-statistical design framework for the Multivariate Exponentially Weighted Moving Average (MEWMA) control chart, while other studies have applied economic models to optimize sampling intervals and monitoring strategies in SPC systems. Therefore, incorporating economic considerations through a cost function plays a vital role in the design of EWMA control charts. By minimizing the expected cost per unit time while maintaining desirable statistical properties such as ARL performance, economically designed EWMA control charts provide an effective and practical approach for monitoring industrial processes and improving overall production quality.

Economic design of the proposed CRSS-EWMA chart

where the design parameters are: 𝑛 = Sample size ℎ = interval between samples (sampling interval) 𝜆 = Exponential parameter (1/ 𝜆 is the mean time to the occurrence of an assignable cause).

𝐶 0 = the cost of producing non-conformities/hour while the process is in-control. 𝛿 = 𝑠ℎ𝑖𝑓𝑡 in the process average.

The strategy to compute the optimal sampling interval (h) and expected cost/hour for proposed CRSS-EWMA chart is discussed as follows.

Strategy to determine the sampling interval (h) for CRSS-EWMA

Step 1: Construct the CRSS sample using the circular ranking procedure and compute the observations used in the EWMA chart.

Step 2: Choose the set size (𝑚), number of cycles (𝑟) for CRSS, and the EWMA smoothing parameter (𝑤). Fix the in-control average run length (𝐴𝑅𝐿₀).

Step 3: Calculate the out-of-control average run length (𝐴𝑅𝐿 1 ) for a specified shift in the failure rate.

Step 4: Specify the parameters of the economic cost model 𝜆, 𝐶 0 , 𝐶 1 ,𝑎,𝑏,𝑌,𝑊, 𝑇 0 , 𝑇 1 , 𝑇 2 , 𝐸 as given in equation ( 1)

Step 5: Compute the expected cost per hour for a given set of parameters as given in equation ( 1).

Step 6: Check the minimum value of expected cost per hour for a given set of parameters.

Automatic implantable cardioverter-defibrillators (AICDs)

An Automatic Implantable Cardioverter-Defibrillator (AICD) is a small implantable medical device used to monitor and correct lifethreatening abnormal heart rhythms, especially ventricular Tachycardia (VT) and Ventricular Fibrillation (VF). These conditions can lead to sudden cardiac arrest, so continuous monitoring and immediate correction are critical. An AICD is surgically implanted under the skin near the upper chest, with electrode leads connected to the heart. In-control ARL of 370 is maintained to compute the expected cost per hour for a sample size of 4. Both sets of parameters are considered to determine the optimal sampling intervals for the proposed CRSS-EWMA control chart used to monitor the reliability of Automatic Implantable Cardioverter-Defibrillators (AICDs). The lifetime of AICD devices is assumed to follow an exponential distribution. Expected cost is calculated for shifts in the process parameter of 0.25 and 0.5. Sample calculations to determine the optimal sampling interval ℎ are given as follows. Various parameters for the proposed CRSS-EWMA control chart are considered as follows:

Sample size (𝑛) = 4, CRSS set size (𝑚) = 4, number of cycles (𝑟) = 1, EWMA smoothing parameter (𝑤) = 0.2, control limit coefficient (𝐿) = 2.343. In-control average run length (𝐴𝑅𝐿₀) = 370,

Comparison of CRSS-EWMA and conventional EWMA control charts

The economic design of the proposed CRSS-EWMA control chart has been compared with the economic design of the RSS-EWMA control chart for monitoring the reliability of Automatic Implantable Cardioverter-Defibrillators (AICDs). The expected cost per hour and optimal sampling intervals are computed for different sample sizes and shifts in the failure rate, with both charts designed for 𝐴𝑅𝐿 0 = 370. The comparison results are presented in Tables 2 and3. The performance of the CRSS-EWMA control chart is compared with the conventional EWMA control chart using the optimal sampling interval and expected cost per hour. From the results in Tables 2 and3

First set of parameters

Discussion

The performance of the proposed CRSS-EWMA control chart is compared with the conventional EWMA control chart using optimal sampling intervals and expected cost per hour. The results show that the CRSS-EWMA chart provides a lower expected cost per hour than the conventional EWMA chart for all sample sizes and shifts in the failure rate. Under the first set of parameters (Table 2, 𝑛 = 7 to 10), the saving ranges from ₹2.24 to ₹4.43 per hour, a reduction of 9.54% to 16.22%. Under the second set of parameters (Table 3, 𝑛 = 2 𝑡𝑜 5), the saving ranges from ₹5.82 to ₹15.69 per hour, a reduction of 6.31% to 17.73%. For an AICD monitoring system that runs continuously (8,760 hours per year), this is roughly ₹19,600 to ₹38,800 per year under the first set and ₹51,000 to ₹1,37,400 per year under the second set for each monitored stream, and the saving grows in proportion to the number of device lots or production streams being monitored. The savings increase with the sample size, in line with the increase in PRE, and under the first set it is larger for the smaller shift (15.36% to 16.22% at 𝛿 = 0.25 against 9.54% to 10.12% at 𝛿 = 0.5), because small shifts take longer to detect and the out-of-control cost dominates. The CRSS-EWMA chart also needed longer optimal sampling intervals than the conventional EWMA chart (by 0.16 to 0.30 hr under the first set and 0 to 0.20 hr under the second set). Together with its lower cost, this indicates that the more informative CRSS observations allow earlier detection of an increase in the device failure rate with less frequent sampling. In practice, earlier detection means that a batch of devices whose reliability is degrading can be identified and investigated sooner, so that corrective action can start earlier. The optimal sampling intervals became longer as the sample size and shift size increased, in line with the smaller 𝐴𝑅𝐿 1 values for larger samples and larger shifts.

Conclusion

This study developed an economic design of the EWMA control chart based on Circular Ranked Set Sampling (CRSS) for monitoring the reliability of Automatic Implantable Cardioverter-Defibrillators (AICDs), whose lifetimes follow an exponential distribution. The optimal sampling intervals were obtained by minimizing the expected cost per hour under the Lorenzen-Vance unified cost model at a fixed in-control ARL of 370. The proposed CRSS-EWMA chart gave a lower expected cost per hour than the conventional EWMA chart in every combination of sample size (𝑛 = 2 𝑡𝑜 10) and shift (𝛿 = 0.25, 0.5) considered, with savings of 9.54% to 16.22% under the first set of parameters and 6.31% to 17.73% under the second set in Tables 2 and3. In the numerical example (𝑛 = 4, 𝛿 = 0.25), the minimum expected cost is ₹25.55 per hour at an optimal sampling interval of 0.24 hr, against ₹29.17 per hr for the conventional EWMA chart. In summary, the CRSS-EWMA chart is an economically efficient and statistically sound alternative to the RSS-EWMA chart for the reliability monitoring of implantable cardiac devices.

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