The Principle of Mathematical Induction Applied to the Generalized Model for the Economic Design of X-Control Charts

Affiliation(s)

Department of Information Systems, King Faisal University, Al-Ahsa, Saudi Arabia.

Department of Systems Engineering, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia.

Department of Information Systems, King Faisal University, Al-Ahsa, Saudi Arabia.

Department of Systems Engineering, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia.

ABSTRACT

Rahim and Banerjee [1] developed a general model for the optimal design of x-control charts. The model minimizes the expected cost per unit time. The heart of the model is a theorem that derives the expected total cost and the expected cycle length. In this paper an alternative simple proof for the theorem is provided based on mathematical induction.

Rahim and Banerjee [1] developed a general model for the optimal design of x-control charts. The model minimizes the expected cost per unit time. The heart of the model is a theorem that derives the expected total cost and the expected cycle length. In this paper an alternative simple proof for the theorem is provided based on mathematical induction.

Cite this paper

M. Seliaman and S. Duffuaa, "The Principle of Mathematical Induction Applied to the Generalized Model for the Economic Design of X-Control Charts,"*Open Journal of Applied Sciences*, Vol. 2 No. 4, 2012, pp. 236-240. doi: 10.4236/ojapps.2012.24035.

M. Seliaman and S. Duffuaa, "The Principle of Mathematical Induction Applied to the Generalized Model for the Economic Design of X-Control Charts,"

References

[1] M. A. Rahim and P. K. Banerjee, “A Generalized Model for the Economic Design of -Control Charts for Production Systems with Increasing Failure Rate and Early Replacement,” Naval Research Logistics, Vol. 40, No. 6, 1993, pp. 787-809. doi:10.1002/1520-6750(199310)40:6<787::AID-NAV3220400605>3.0.CO;2-4

[2] D. C. Montgomery, “The Economic Design of Control Charts: A Review and Literature Survey,” Journal of Quality Technology, Vol. 12, No. 2, 1980, pp. 75-87.

[3] D. Patel, “Economic Design of Control Chart,” B. Tech Thesis, National Institute of Technology, Rourkela, 2009.

[4] R.-C. Wang and C.-H. Chen, “Economic Statistical NpControl Chart Designs Based on Fuzzy Optimization,” International Journal of Quality & Reliability Management, Vol. 12, No. 1, 1995, pp. 82-92. doi:10.1108/02656719510076276

[5] Y.-S. Chen and Y.-M. Yang, “An Extension of Banerjee and Rahim’s Model for Economic Design of Moving Average Control Chart for a Continuous Flow Process,” European Journal of Operational Research, Vol. 143, No. 3, 2002, pp. 600-610. doi:10.1016/S0377-2217(01)00341-1

[6] I. N. Gibra, “Economically Optimal Determination of the Parameters of X-Control Chart,” Management Science, Vol. 17, No. 9, 1971, pp. 635-646.

[7] E. M. Saniga, “Economic Statistical Control-Chart Designs with an Application to X and R Charts,” Technometrics, Vol. 31, No. 3, 1989, pp. 313-320.

[8] A. J. Duncan, “The Economic Design of -Control Charts Used to Maintain Current Control of a Process,” Journal of the American Statistics Association, Vol. 51, 1956, pp. 228-242.

[9] P. K. Banerjee and M. A. Rahim, “Economic Design of -Control Charts under Weibull Shock Models,” Technometrics, Vol. 30, No. 4, 1998, pp. 407-414.

[1] M. A. Rahim and P. K. Banerjee, “A Generalized Model for the Economic Design of -Control Charts for Production Systems with Increasing Failure Rate and Early Replacement,” Naval Research Logistics, Vol. 40, No. 6, 1993, pp. 787-809. doi:10.1002/1520-6750(199310)40:6<787::AID-NAV3220400605>3.0.CO;2-4

[2] D. C. Montgomery, “The Economic Design of Control Charts: A Review and Literature Survey,” Journal of Quality Technology, Vol. 12, No. 2, 1980, pp. 75-87.

[3] D. Patel, “Economic Design of Control Chart,” B. Tech Thesis, National Institute of Technology, Rourkela, 2009.

[4] R.-C. Wang and C.-H. Chen, “Economic Statistical NpControl Chart Designs Based on Fuzzy Optimization,” International Journal of Quality & Reliability Management, Vol. 12, No. 1, 1995, pp. 82-92. doi:10.1108/02656719510076276

[5] Y.-S. Chen and Y.-M. Yang, “An Extension of Banerjee and Rahim’s Model for Economic Design of Moving Average Control Chart for a Continuous Flow Process,” European Journal of Operational Research, Vol. 143, No. 3, 2002, pp. 600-610. doi:10.1016/S0377-2217(01)00341-1

[6] I. N. Gibra, “Economically Optimal Determination of the Parameters of X-Control Chart,” Management Science, Vol. 17, No. 9, 1971, pp. 635-646.

[7] E. M. Saniga, “Economic Statistical Control-Chart Designs with an Application to X and R Charts,” Technometrics, Vol. 31, No. 3, 1989, pp. 313-320.

[8] A. J. Duncan, “The Economic Design of -Control Charts Used to Maintain Current Control of a Process,” Journal of the American Statistics Association, Vol. 51, 1956, pp. 228-242.

[9] P. K. Banerjee and M. A. Rahim, “Economic Design of -Control Charts under Weibull Shock Models,” Technometrics, Vol. 30, No. 4, 1998, pp. 407-414.