Control charts with random interarrival times between successive samplings

Planning a cost‐efficient monitoring policy of stochastic processes arises from many industrial problems. We formulate a simple discrete‐time monitoring problem of continuous‐time stochastic processes with its applications to several industrial problems. A key in our model is a doubling trick of the variables, with which we can construct an algorithm to solve the problem. The cost‐efficient monitoring policy balancing between the observation cost and information loss is governed by an optimality equation of a fixed point type, which is solvable with an iterative algorithm based on the Feynman‐Kac formula. This is a new linkage between monitoring problems and mathematical sciences. We show regularity results of the optimization problem and present a numerical algorithm for its approximation. A problem having model ambiguity is presented as well. The presented model is applied to problems of environment, ecology, and energy, having qualitatively different target stochastic processes with each other.

Section: Discussionmentioning

confidence: 99%

Cost‐efficient monitoring of continuous‐time stochastic processes based on discrete observations

Yoshioka

Yaegashi

Tsujimura

et al. 2020

“…To measure the detection ability of the control charts, run‐length properties such as average run‐length (ARL), the SD of run‐length (SDRL) and median of run‐length (MDRL) are evaluated. The ARL is a well‐known measure and defined as the average number of points until a point falls outside the limits 44 . Further, it is categorized into IC average run‐length (ARL 0 ) and OOC average run‐length (ARL 1 ).…”

Section: Performance Evaluationsmentioning

confidence: 99%

The generalized linear model‐based exponentially weighted moving average and cumulative sum charts for the monitoring of high‐quality processes

Mahmood

Xie

2021

In this industry 4.0 revolution, most of the manufacturing processes are equipped with the digital devices which are continuously recording the data. To monitor the quality of a manufacturing system, variable about number of conforming or nonconforming items is usually used and statistical analysis based on it is further utilized for developing the policies. In this era of sophisticated and modern technology, most of the manufacturing systems are producing near zero‐defect items. Such processes are known as high‐quality processes, and their dataset consists of excess number of zeros. Generally, the zero excess or near zero‐defect dataset is well fitted by the zero‐inflated distributions, and the zero‐inflated Poisson (ZIP) and zero‐inflated Negative Binomial (ZINB) distributions are the most common models. Most of the time, in high‐quality processes, few covariates are also measured along with defect counts. Hence, to model such processes, generalized linear model (GLM) based on ZIP and ZINB distributions are used to fit the data. In monitoring perspective, data‐based control charts are designed to monitor high‐quality datasets while the GLM‐based control charts based on the residuals of the GLM models are used to monitor a change in the mean of the zero excess datasets. In this study, we have developed memory‐type data‐based and GLM‐based control charts (i.e., exponentially weighted moving average and cumulative sum) to monitor the increasing average defect counts in a high‐quality process. Further, the proposed methods are evaluated using run‐length properties and compared with its competitive charts. Furthermore, to highlight the importance of the study, the proposed methods are implemented on a dataset concerning the number of flight delays between Atlanta and Orlando airports.

“…Phase‐type distributions have been widely used in reliability and quality. See, for example, Neuts and Meier, 10 Montoro‐Cazorla and Perez‐Ocon, 11 Koutras and Rakitsiz 12 . Recently, Alkaff and Qomarudin 13 presented a framework for modeling and analysis of system reliability using phase‐type closure properties.…”

Section: Introductionmentioning

confidence: 99%

Reliability assessment for discrete time shock models via phase‐type distributions

Eryılmaz

Kan

2020

In this paper, particular shock models are studied for the case when the times between successive shocks and the magnitudes of shocks have discrete phase‐type distributions. The well‐known shock models such as delta shock model, extreme shock model, and the mixed shock model which is obtained by combining delta and extreme shock models are considered. The probability generating function and recursive equation for the distribution of the system's lifetime are obtained for the cases when the interarrival times between shocks and the magnitudes of shocks are independent and when they are dependent. System reliability is computed for particular interarrival distributions such as geometric, negative Binomial and generalized geometric distributions.