# Superior hypergeometric p-chart for efficient process monitoring

### Significance

Control charts are statistical tools widely used in various sectors to determine whether various processes are in a state of control. Commonly, the process fraction of non-conforming objects is monitored by p-charts that are based on the binomial distribution with approximate Shewhart-type control limits (an approach similar to confidence bounds). While binomial p-charts are useful for routine monitoring of binary outcomes, where the binomial probability model is appropriate and the requirements for the usage of approximate control limits are satisfied, they provide unsatisfactory results especially in monitoring processes with a finite horizon. However, in modern manufacturing, we often face finite horizon processes. For instance, frequent setup changeovers in mass production aimed at switching from one product code to another and the growing need to manufacture small lots of customized products are challenging tasks for quality engineers. As a fact, an efficient monitoring of finite horizon processes requires in the first line the consideration of the exact underlying distribution. But, this crucial point regarding the type of production process is mostly neglected or solved using approximations.

To overcome these challenges of traditional p-charts, a joint effort by Dr. Nataliya Chukhrova and Dr. Arne Johannssen from the University of Hamburg is the development of the hypergeometric p-chart. This chart is not only based on the exact underlying distribution of finite horizon processes, i.e. the hypergeometric distribution, but it is also constructed my means of exact probability control limits. Probability control limits help to avoid problems resulting from the asymmetry of the underlying distribution, such as biased upper and lower type I error, undefined upper and lower control limits or inappropriate sample size for efficient monitoring. Moreover, the authors suggest a dynamic modeling approach, where variability in sample and/or population sizes is appropriately incorporated in the calculation of the control limits. In addition, the hypergeometric p-chart can be established for monitoring both finite horizon and continuous processes (such as mass production). The study illustrates a method to accommodate any population proportions, any population sizes, and any sample sizes from very low to large, while retaining a satisfactory false-alarm risk.

The authors provide practical applications using a real data set and conduct comprehensive sensitivity analyses with regard to the control limits and the average run length of the hypergeometric p-chart. Moreover, comparative analyses show the superior performance of the hypergeometric p-chart when monitoring the fraction non-conforming of finite horizon processes. The authors demonstrate that their dynamic approach not only improves the monitoring efficiency but also enables the practitioner to detect the source of variability in the system for appropriate actions. The article has been published in the renowned journal Computers and Industrial Engineering.

In a nutshell, the hypergeometric p-chart with dynamic probability control limits presented by Dr. Chukhrova and Dr. Johannssen meets several current requirements of efficient process monitoring and builds a satisfactory system for measuring, monitoring, and improving finite horizon processes. Considering its ease of implementation in practical applications, Dr. Arne Johannssen in a statement to Advances in Engineering noted that their approach will be of great benefit in improving process monitoring in various fields and particularly in areas where finite horizon processes are most prevalent, such as modern manufacturing, service operations management, health care monitoring, and public health surveillance.

Dr Nataliya Chukhrova holds a Master’s degree in Business Administration and a PhD in Statistics from the University of Hamburg (Germany). She currently works as a postdoctoral researcher at the Institute of Mathematics and Statistics, University of Hamburg. Her research interests include data science, business analytics, soft computing, fuzzy statistics, statistical quality control, reliability engineering, and statistical inference. She authored numerous publications in high-ranked journals, such as Applied Soft Computing, Computers and Industrial Engineering, Fuzzy Sets and Systems, and Journal of Quality Technology.

Her research is funded by the German Research Foundation (DFG) and the Claussen Simon Foundation, among others. Nataliya is a regular member of the American Society for Quality (ASQ), the American Statistical Association (ASA), and the German Statistical Society (DStatG). She serves as a reviewer for various journals including Computers and Industrial Engineering, European Journal of Operational Research, International Journal of Fuzzy Systems, and Soft Computing. Nataliya has given numerous lectures on business mathematics, advanced statistics, statistical quality control, reliability engineering, and lifetime analysis. She has been awarded the Excellence in Higher Education Award.

Dr Arne Johannssen holds a Master’s degree in Economics and a PhD in Applied Statistics/Actuarial Sciences from the University of Hamburg (Germany). He currently works as a postdoctoral researcher at the Institute of Mathematics and Statistics, University of Hamburg.

His research interests include data science, business analytics, soft computing, fuzzy statistics, statistical quality control, quantitative risk management, multiple time series analysis, and statistical inference. He authored numerous publications in high-ranked journals, and especially in the top-tier journals of soft computing (Applied Soft Computing), fuzzy statistics (Fuzzy Sets and Systems), and statistical quality assurance (Journal of Quality Technology). His research is funded by the German Research Foundation (DFG), among others. Arne is a regular member of the American Statistical Association (ASA) and the German Statistical Society (DStatG). He serves as a reviewer for various journals including Applied Soft Computing, ASTIN Bulletin, Computational Statistics, European Journal of Operational Research, Journal of Computational and Applied Mathematics, and Quality Technology & Quantitative Management.

Arne has given numerous lectures on economics, mathematics, and statistics (with up to 1200 students), and has been awarded the Excellence in Higher Education Award twice.

### Reference

Chukhrova, N., & Johannssen, A. (2019). Hypergeometric p-chart with dynamic probability control limits for monitoring processes with variable sample and population sizes. Computers & Industrial Engineering, 136, 681-701.