Document Type : Original Article

**Authors**

Department of Industrial Engineering, Faculty of Technology and Engineering, University of Qom, Qom, Iran.

**Abstract**

Control charts are powerful tools to monitor quality characteristics of services or production processes. However, in some processes, the performance of process or product cannot be controlled by monitoring a characteristic; instead, they require to be controlled by a function that usually refers as a profile. This study suggests employing exponentially weighted moving average (EWMA) and range (R) control charts for profile monitoring, simultaneously. For this purpose, the parameters of these control charts should be determined in a way that the expected total cost is minimized. In order to evaluate the statistical performance of the proposed model, the in-control and out-of-control average run lengths are applied. Moreover, the existence of uncertain parameters in many processes is a barrier to attain the best design of control charts in practice. In this paper, the economic-statistical design of control charts for linear profile monitoring under uncertain conditions is investigated. A genetic algorithm is used for solving the proposed robust model, and the Taguchi experimental design is employed for tuning its parameters. Furthermore, the effectiveness of the developed model is illustrated through a numerical example.

**Keywords**

Amiri, A., Willis, A., and Kazemzadeh, RB, (2010). "A Case Study on Monitoring Polynomial Profiles in the Automotive Industry", *Quality and Reliability Engineering International*, Vol. 26, pp. 509–520.

Amiri, A., Moghadam, A.S., and Aghababaee, Z., (2015). "Robust economic-statistical design of multivariate exponentially weighted moving average control chart under uncertainty with interval date", *Scientia Iranica*, Vol. 22, pp. 1189-1202.

Bertsimas, D., and Sim, M., (2004). "The Price of Robustness", *Operations Research*, Vol. 52, pp. 35–53.

Bertsimas, D., Brown, D., and Caramanis, C., (2011). "Theory and Applications of Robust Optimization", *SIAM Review*, Vol. 53, pp. 464–501.

Ben-Tal., A., Nemirovski, A., (2000). "Robust Solutions of Linear Programming Problems Contaminated with uncertain data", *Mathematical Programming*, Vol. 88, pp. 411-424.

Chen, Y., (1995). Economic and Economic-Statistical Designs of Hotelling’s T2 Control Chart, MS thesis, Department of Statistics, National Tsing-Hua University, Hsinchu, Taiwan.

Duncan, A. J., (1956). "The Economic Design of x-bar charts Used to Maintain Current Control of a Process", *Journal of the American Statistica*l Association, Vol. 51, pp. 228–242.

Ershadi, M. J., Noorossana, R., and Niaki, S. T. A., (2015). "Economic-Statistical Design of Simple Linear Profiles with Variable Sampling Interval", *Journal of Applied Statistics*, Vol. 43, pp. 1400-1418.

Fakhrzad, M. B., and Alidoosti, Z., (2018). "A realistic perishability inventory management for location-inventory-routing problem based on Genetic Algorithm", *Journal of Industrial Engineering and Management Studies*, Vol. 5, No. 1, pp. 106-121.

Faraz, A., Chalaki, K., Saniga, E., and Heuchenne, C., (2016). "The robust economic-statistical design of the Hoteling’s T^{2} chart", *Communications in statistics-Theory and methods*, Vol. 45, pp. 6989-7001.

Faraz, A., and Saniga, E., (2013). "Multi-objective Genetic Algorithm Approach to the Economic Statistical Design of Control Charts with an Application to X-bar and S2 Charts", *Quality and Reliability Engineering International*, Vol. 29, pp. 407–415.

Kang, L., and Albin, S. L., (2000). "On-Line Monitoring when the Process Yields a Linear Profile",* Journal of Quality Technology*, Vol. 32, pp. 418–426.

Khedmati, M., and Niaki, S. T. A., (2015). "Phase-II monitoring of general linear profiles in the presence of between profile autocorrelation", *Quality and Reliability Engineering International*, Vol. 32, pp. 443-452.

Khedmati, M., and Niaki, S. T. A., (2016). "A new control scheme for Phase-II monitoring of simple linear profiles in multistage processes", *Quality and Reliability Engineering International*, Vol. 32, pp. 443-452.

Li, C., Mukherjee, A., Su, Q., and Xie, M., (2016). "Robust algorithms for economic designing of a nonparametric control chart for abrupt shift in location", *Journal of Statistical Computation and Simulation*, Vol. 86, pp. 306-323.

Linderman, K., and Choo, A. S., (2002). "Robust Economic Control Chart Design", *IIE Transactions*, Vol. 34, pp. 1069–1078.

Linderman, K., and Love, T. E., (2000). "Economic and Economic Statistical Designs for MEWMA Control Charts", *Journal of Quality Technology*, Vol. 32, pp. 410–417.

Lorenzen, T. J., and Vance, L. C., (1986). "The Economic Design of Control Charts: A Unified Approach", *Technometrics*, Vol. 28, pp. 3–10.

Mahmoud, M., Morgan, J. P., and Woodall W., (2010). "The Monitoring of Simple Linear Regression Profiles with Two Observations per Sample",* Journal of Applied Statistics*, Vol. 37, pp. 1249–1263.

McWilliams, T. P., Saniga, E. M., and Davis, D. J., (2001). "Economic Statistical Design of x-bar and R or x-bar and S x-bar", *Journal of Quality Technology*, Vol. 33, pp. 234–241.

Mestek, O., Pavlik, J., and Suchánek, M., (1994). "Multivariate Control Charts: Control Charts for Calibration Curves", *Journal of Analytical Chemistry*, Vol. 350, pp. 344–351.

Molavi, F., and Rezaee Nik, E., (2018). "A stochastic model for project selection and scheduling problem", *Journal of Industrial Engineering and Management Studies*, Vol. 3, No. 1, pp. 77-88.

Molnau, W. E., Montgomery, D. C., and Runger, G. C., (2001). "Statistically Constrained Designs of the Multivariate Exponentially Weighted Moving Average Control Chart", *Quality and Reliability Engineering international*, Vol. 7, pp. 39–49.

Montgomery, D. C., (2012). Introduction to Statistical Quality Control, 7th edition, *John Wiley & Sons*, Inc., New York.

Montgomery, D. C., Klatt, P. J., (1972). "Economic Design of T^{2} Control Charts to Maintain Current Control of a Process", *Management Science*, Vol. 19, pp. 76–89.

Montgomery, D. C., Torng, J. C. C., Cochran, J. K., and Lowerance, F. P., (1995). "Statistically Constrained Economic Design of the EWMA Control Chart", *Journal of Quality Technology*, Vol. 27, pp. 250–256.

M. S., Saccucci, and J. M., Lucas, (1990). "Average run lengths for exponentially weighted moving average control schemes using the Markov chain approach", *Journal of Quality Technology*, Vol. 22, pp.154-162.

Niaki, S. T. A., and Ershadi, M. J., (2012). "A Hybrid Ant Colony, Markov Chain, and Experimental Design Approach for Statistically Constrained Economic Design of MEWMA Control Charts", *Expert Systems with Applications*, Vol. 39, pp. 3265–3275.

Niaki, S. T. A., Ershadi, M. J., and Malaki, M., (2010). "Economic and economic-statistical designs of MEWMA control charts–a hybrid Taguchi loss, Markov chain, and genetic algorithm approach", *International Journal of Advanced Manufacturing Technology*, Vol. 48, pp. 283-296.

Niaki, S. T. A., Gazaneh, F. M., and Toosheghanian, M., (2013). "A Parameter-tuned Genetic Algorithm for Economic- statistical Design of Variable Sampling Interval X-bar Control Charts for Non-normal Correlated Samples", *Communications in Statistics - Simulation and Computation*, Vol. 43, pp. 1212–1240.

Noorossana, R., Niaki, S. T. A., and Ershadi, M. J., (2014). "Economic and Economic-Statistical Design of Phase II Profile Monitoring", *Quality and Reliability Engineering International*, Vol. 30, pp. 645-655.

Noorossana, R., Saghaei, A., and Amiri, A., (2011). *Statistical Analysis of Profile Monitoring*, John Wiley & Sons, Inc., New York.

Park, C., Lee, J., and Kim, Y., (2004). "Economic Design of a Variable Sampling Rate EWMA Chart", *IIE Transactions*, Vol. 36, pp. 387–399.

Pignatiello, J. J., and Tsai, A., (1988). "Optimal Economic Design of X-control Charts when Cost Model Parameters are not precisely known", *IIE Transactions*, Vol. 20, pp. 103-110.

Lucas, J. M., and Saccucci, M. S., (1990). "Exponentially weighted moving average control schemes: Properties and enhancements", *Technometrics*, Vol. 32, pp. 1-12.

Safaei, A. S., Kazemzadeh, R., and Gan, H. S., (2015). "Robust Economic-Statistical Design f X-bar Control Chart", *International Journal of Production research*, Vol. 14, pp. 4446-4458.

Saghaei, A., Mehrjoo, M., and Amiri, A., (2009). "A CUSUM-based Method for Monitoring Simple Linear Profiles", *The International Journal of Advanced Manufacturing Technology*, Vol. 45, pp. 1252–1260.

Saniga, E. M., (1989). "Economic Statistical Control Chart with an Application to X-bar and R Charts", *Technometrics*, Vol. 31, pp. 313–320.

Serel, D. A., and Moskowitz, H., (2008). "Joint Economic Design of EWMA Control Charts for Mean and Variance", *European Journal of Operational Research*, Vol. 184, pp. 157–168.

Tolley, G. O., and English, J. R., (2001). "Economic Design of Constrained EWMA and Combined EWMA- x-bar Control Schemes", *IIE Transactions*, Vol. 33, pp. 429–436.

Vaghefi, A., (2003). "Phase Two Monitoring of Nonlinear Profiles", *Communications in statistics-Theory and methods*, Vol. 38, pp. 1834-1851.

Vommi, V. B., and Seetala, M. S. N., (2007). "A New Approach to Robust Economic Design of Control Charts", *Applied Soft Computing*, Vol. 7, pp. 211–228.

Williams, J. D., Woodall, W. H., and Birch, J. B., (2003). "Phase I Monitoring of Nonlinear Profiles", Quality and Productivity Research Conference, Yorktown Heights, New York.

Woodall, W. H., (2007). "Current Research on Profile Monitoring", *ProduBaο*, Vol. 17, pp. 420–425.

Wu, C., Yu, M., and Zhuang, F., (2017). "Properties and enhancements of robust likelihood CUSUM control chart", *Computers & Industrial Engineering*, Vol. 114, pp. 80-100.

Zhonghua, L., and Zhaojun, W., (2010). "An Exponentially Weighted Moving Average Scheme with Variable Sampling Intervals for Monitoring Linear Profiles",* Computers and Industrial Engineering*, Vol. 59, pp. 630–637.