Main Article Content
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 Hotelling’s T2 control chart and it is shown that this approach is more desirable statistically and economically.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
- 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 uniﬁed 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. İSTATİSTİK, 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.