Vol.:(0123456789) Operational Research https://doi.org/10.1007/s12351-018-0424-z 1 3 ORIGINAL PAPER Effects of non‑normal quality data on the integrated model of imperfect maintenance, early replacement, and economic design of   X ‑control charts M. A. Pasha 1  · M. Bameni Moghadam 1  · M. A. Rahim 2 Received: 7 May 2017 / Revised: 23 June 2018 / Accepted: 7 September 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract The quality characteristic of a product for designing a control chart is commonly assumed to be normally distributed. However, this may not be reasonable in many practical applications. This study investigates the effects of non-normality data on the economic design of  X-control charts for systems under imperfect maintenance and the possibility of age-dependent repair before failure. These preventive main- tenance policies increase the reliability of a system by reducing the shift rate to the out-of-control state of the process. A generalized economic model of the joint opti- mization problem is adopted for the Burr distributed quality data to derive the opti- mal design parameters of the integrated model under non-normality; i.e., the main- tenance level, the replacement time, the sample size, the sampling intervals, and the control limits coefficient. Numerical examples reveal that the choice of quality char- acteristic distribution significantly affects the optimal design parameters of the inte- grated model. Furthermore, the higher maintenance levels reduce the quality control costs in non-normal cases similar to normal traditional approach. Keywords Non-normal distributed data · Imperfect maintenance · Truncated production cycle · Deteriorating process · Integrated hazard over sampling intervals * M. A. Pasha m.a.pasha@atu.ac.ir M. Bameni Moghadam bamenimoghadam@atu.ac.ir M. A. Rahim rahim@unb.ca 1 Department of Statistics, Allameh Tabataba’i University, Beheshti-Ghasir Ave., Tehran 1513615411, Iran 2 Faculty of Business Administration, University of New Brunswick, Fredericton, NB E3B 5A3, Canada