## Main Article Content

## Abstract

Due to the proper performance of Bayesian control chart in detecting process shifts, it recently has become the subject of interest. It has been proved that on Bayesian and traditional control charts, the economic and statistical performances of the variable sampling interval (VSI) scheme are superior to those of the fixed ratio sampling (FRS) strategy in detecting small to moderate shifts. This paper studies the VSI multivariate Bayesian control chart based on economic and economic-statistical designs. Since finding the distribution of Bayesian statistic is t complicated, we apply Monte Carlo method and we employ artificial bee colony (ABC) algorithm to obtain the optimal design parameters (sample size, sampling intervals, warning limit and control limit). In the end, this case study is compared with VSI Hotellingâ€™s T2 control chart and it is shown that this approach is more desirable statistically and economically.

## Keywords

## Article Details

This work is licensed under a Creative Commons Attribution 4.0 International License.

**Authors who publish with this journal agree to the following License**

**CC BY: **This license allows reusers to distribute, remix, adapt, and build upon the material in any medium or format, so long as attribution is given to the creator. The license allows for commercial use.

*How to Cite*

*Pakistan Journal of Statistics and Operation Research*,

*16*(4), 737-750. https://doi.org/10.18187/pjsor.v16i4.3417

* * References

- Banerjee, P. K. and Rahim, M. A. (1988). Economic design of X-bar control charts under Weibull shock models. Technometrics, 30, 407-414.
- Calabrese, J. M. (1995). Bayesian Process Control for Attributes, Management Science, 41, 637-645.
- Chen, Y. K. (2006-b). Economic design of T2 control charts with the VSSI sampling scheme, Quality and Quantity. DOI: 10.1007/s11135-007-9101-7.
- Costa, A. F. B. and Rahim, M. A. (2001). Economic design of X-bar charts with variable parameters: the Markov chain approach. Journal of Applied Statistics, 287, 875-885.
- Duncan, A. J. (1956). The economic design of X-bar charts used to maintain current control of a process. Journal of the American Statistical Association, 51, 228-242.
- Duncan, A. J. (1971). The economic design of 2 charts where there is a multiplicity of assignable causes. J. Am. Statist. Assoc, 66, 107-121.
- Faraz, A. Saniga, E. and Kazemzadeh, R. B. (2009). Economic and Economic Statistical Design of T2 Control Chart with two-adaptive Sample Sizes. Journal of Statistical Computation and Simulation, 80, 1299-1316.
- Girshick, M. A. and Rubin, H. (1952). A Bayes approach to a quality control model. Annals of Mathematical Statistics, 23, 114-125.
- Hotelling, H. (1947). Multivariate quality control-Illustrated by the air testing of sample bombsights. Techniques of Statistical Analysis, Eisenhart, C., Hastay, M.W., Wallis,W.A. (eds), New York: MacGraw- Hill.
- Karaboga, D. (2005). An Idea Based On Honey Bee Swarm for Numerical Optimization, TR-06, October 2005.
- Karaboga, D. and Akay, B. (2009). A comparative study of Artificial Bee Colony algorithm. Applied Mathematics and Computation, 214, 108-132.
- Karaboga, D. and Basturk, B. (2007). A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Glob Optim, 39, 459-471.
- Karaboga, D. and Basturk, B. (2008). On the performance of artificial bee colony (ABC) algorithm. Applied Soft Computing, 8, 687-697.
- Lorenzen, T. J. and Vance, L. C. (1986). The economic design of control charts: a uniï¬ed approach. Technometrics, 28, 3-10.
- Maikis, V. (2009). Multivariate Bayesian process control for a finite production run. European Journal of Operational Research, 194, 795-806.
- Prabhu, S. S. Mongomery, D. C. and Runger, G. C. (1997). Economic-statistical design of an adaptive X-bar chart. Int. J. Production Economics, 49, l-l 5.
- Reynolds, Jr. M. R., Amin, R. W., Arnolds J. C. and Nachlas, J. A. (1988). X-bar chart with variable sampling intervals, Technometrics, 30, 181-192.
- Seif, A. Sadeghifar M. (2015). Non-dominated sorting genetic algorithm (nSGA-II) approach to the multi-objective economic statistical design of variable sampling interval T2 control charts. Hacet J Math Stat. 44, 203-214
- Saniga, E. M. (1989). Economic statistical control chart designs with an application to X-bar and R charts. Technometrics, 31, 313-320.
- Tagaras, G. and Nikolaidis, Y. (2002). Comparing the Effectiveness of Various Bayesian X-bar Control Charts. Operations Research, 50, 878-888.
- Tavakoli, M., Pourtaheri, R. and Moghadam, M. B. (2014). Economic-Statistical design of FRS Bayesian control chart using Monte Carlo method and artificial bee colony (ABC) algorithm. Ä°STATÄ°STÄ°K, 8, 74-81.
- Tavakoli, M., Pourtaheri, R. and Moghadam, M. B. (2015). Economic and Economic-Statistical designs of VSI Bayesian control chart Using Monte Carlo method and ABC algorithm. Submitted.
- Tavakoli, M., Pourtaheri, R., and Moghadam, M. B. Economic-Statistical Design of VSI Hotelling's T2 Control Chart Using ABC algorithm. 2nd National Industrial Mathematics Conference, Tabriz, Iran, May, 2015.
- Taylor, H. M. (1965). Markovian sequential replacement processes. Annals of Mathematical Statistics, 36, 13-21.
- Taylor, H.M. (1967). Statistical Control of a Gaussian Process. Technometrics, 9, 29-41.
- Torabian, M., Moghadam, M. B. and Faraz, A. (2010). Economically Designed Hotelling's T2control chart using VSICL scheme. The Arabian Journal for Science and Engineering, 35, 251-263.
- Woodall, W. H. (1986). Weaknesses of the economical design of control charts, Technometrics 28, 408-409.
- Yang, S. F. and Rahim, M. A. (2005). Economic statistical process control for multivariate quality characteristics under Weibull shock model. Int. J. Production Economics, 98, 215-226.
- Yeong, W. Khoo, MBC. Yanjing, O. (2015). Economic-statistical design of the synthetic X-bar chart with estimated process parameters. Qual Reliab Eng Int. 31, 863-876.
- Zhang, G. and Berardi, V. (1997). Economic statistical design of X-bar control charts for systems with Weibull in-control times. Computers and Industrial Engineering, 32, 575-586.