Vol.:(0123456789) 1 3 Life Cycle Reliability and Safety Engineering https://doi.org/10.1007/s41872-020-00147-8 ORIGINAL RESEARCH Effect of inspection error on CUSUM control charts for the Erlang‑truncated exponential distribution Mujahida Sayyed 1  · R. M. Sharma 1  · Farkhunda Sayyed 2 Received: 18 July 2020 / Accepted: 30 August 2020 © Society for Reliability and Safety (SRESA) 2020 Abstract The aim of this paper is to study the effect of inspection error on cumulative sum (CUSUM) control charts for controlling the parameters of a random variable under Erlang-truncated exponential distribution. Expression for the parameter of the CUSUM chart also derived. Keywords SPRT · CUSUM · ARL · ETE distribution 1 Introduction Page (1954, 1961) suggested the CUSUM charts which are more effective than Shewhart control chart. The CUSUM chart is widely used in the examination of the mean of a pro- cess based, on samples taken from the process at given times. The measurements of the samples in a given time comprises a subgroup rather examining the mean of every subgroup independently, the CUSUM chart illustrates the accumulated information of existing and earlier samples. This is the reason why CUSUM chart is usually better than the  X− chart for detecting small shifts in the mean of a process. The cumu- lative sum (CUSUM) chart is commonly used for detecting small or moderate shifts in the fraction of defective manufac- tured items. However, its construction relies on the error-free inspection assumption, which can seldom be met in practice. The traditional control chart methods assume that inspection process have no mistake, but in actually inspection error is very difficult to avoid whatever using visual or mechanical detection. The single sampling plan (SSP) for attribute qual- ity characteristics is one of the fields for statistical quality control. A basic assumption in the construction of acceptance sampling plans is that the inspection is perfect without error. However, an inspection error may exist when the product is inspected by an inspector. Generally, two types of errors are possible in attributes sampling. An item that is good may be classified as defective (this is called type I error) or an item that is defective may be classified as good (this is called type II error). Collins et al. (1973) inspected the effect of inspection error on single sampling plan. Chakraborty (1994) studied sampling plan with inspection error. Chen and Chou (2003) and Chen et al. (2008) worked on inspection error under sam- pling plan. Wu et al. (2009) discuss the construction of an upward CUSUM chart in the presence of inspection error. Luceno and Puig-pey (2000) evaluated the run-length prob- ability distribution for CUSUM Charts. Sayyed and Singh (2015) considered CSCC for binomial parameters under the effect of inspection error where the underlying distribution is Poisson. Singh and Mishra (2017) have considered the effect of inspection error on singly truncated Binomial distribution. Erlang-truncated exponential (ETE) distribution was originally introduced by El-Alosey (2007). Mohsin (2009); Mohsin et al. (2010) studied the recurrence relation of Erlang-truncated exponential distribution. Nasiru et al. (2016) constructed the generalized Erlang-truncated expo- nential distribution. One-sided cumulative sum (CUSUM) control charts for controlling the parameters of a random variable with Erlang-truncated exponential distribution stud- ied by the Rao (2013). Jimoh et al. (2019) worked on Erlang- truncated exponential distribution. This paper studies about CUSUM control charts for the Erlang-truncated exponential distribution under inspection error to detect the shift of the process parameters. Also examined about the lead distance, angle of the mask and average run length. * Mujahida Sayyed mujahida.sayyed@gmail.com 1 College of Agriculture, Jawahar Lal Nehru Krishi Vishwa Vidyalaya, GanjBasoda, Madhya Pradesh, India 2 Department of Applied Science, SAGE, University, Indore, India