Statistical process control optimization with variable sampling interval and nonlinear expected loss

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January 2015, 11(1): 105-133. doi: 10.3934/jimo.2015.11.105

Statistical process control optimization with variable sampling interval and nonlinear expected loss

Valery Y. Glizer ^1, , Vladimir Turetsky ^1, and Emil Bashkansky ^2,

1.	Department of Applied Mathematics, Ort Braude College of Engineering, 51 Snunit Str., P.O.B. 78, Karmiel 2161002, Israel, Israel
2.	Department of Industrial Engineering and Management, Ort Braude College of Engineering, 51 Snunit Str., P.O.B. 78, Karmiel 2161002, Israel

Received April 2013 Revised December 2013 Published May 2014

Abstract
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The optimization of a statistical process control with a variable sampling interval is studied, aiming in minimization of the expected loss. This loss is caused by delay in detecting process change and depends nonlinearly on the sampling interval. An approximate solution of this optimization problem is obtained by its decomposition into two simpler subproblems: linear and quadratic. Two approaches to the solution of the quadratic subproblem are proposed. The first approach is based on the Pontryagin's Maximum Principle, leading to an exact analytical solution. The second approach is based on a discretization of the problem and using proper mathematical programming tools, providing an approximate numerical solution. Composite solution of the original problem is constructed. Illustrative examples are presented.

Keywords: variable sampling interval, nonlinear loss, optimal control, approximation, quadratic programming., Statistical process control, optimization.

Mathematics Subject Classification: Primary: 49N90, 49M25; Secondary: 90C2.

Citation: Valery Y. Glizer, Vladimir Turetsky, Emil Bashkansky. Statistical process control optimization with variable sampling interval and nonlinear expected loss. Journal of Industrial and Management Optimization, 2015, 11 (1) : 105-133. doi: 10.3934/jimo.2015.11.105

References:

[1]	R. W. Amin and R. Hemasinha, The switching behavior of X charts with variable sampling intervals, Communication in Statistics - Theory and Methods, 22 (1993), 2081-2102. doi: 10.1080/03610929308831136.
[2]	R. W. Amin and R. W. Miller, A robustness study of X charts with variable sampling intervals, Journal of Quality Technology, 25 (1993), 36-44.
[3]	V. Babrauskas, Heat Release Rates, in SFPE Handbook of Fire Protection Engineering, (Ed. P.J. DiNenno), National Fire Protection Association, (2008), 1-59.
[4]	E. Bashkansky and V. Y. Glizer, Novel approach to adaptive statistical process control optimization with variable sampling interval and minimum expected loss, International Journal of Quality Engineering and Technology, 3 (2012), 91-107.
[5]	M. A. K. Bulelzai and J. L. A. Dubbeldam, Long time evolution of atherosclerotic plaques, Journal of Theoretical Biology, 297 (2012), 1-10. doi: 10.1016/j.jtbi.2011.11.023.
[6]	T. E. Carpenter, J. M. O'Brien, A. Hagerman and B. McCarl, Epidemic and economic impacts of delayed detection of foot-and-mouth disease: A case study of a simulated outbreak in California, Journal of Veterinary Diagnostic Investigation, 23 (2011), 26-33. doi: 10.1177/104063871102300104.
[7]	A. F. B. Costa, X control chart with variable sample size, Journal of Quality Technology, 26 (1994), 155-163.
[8]	A. F. B. Costa, X charts with variable sample sizes and sampling intervals, Journal of Quality Technology, 29 (1997), 197-204.
[9]	A. F. B. Costa, Joint X and R control charts with variable parameters, IIE Transactions, 30 (1998), 505-514.
[10]	A. F. B. Costa, X charts with variable parameters, Journal of Quality Technology, 31 (1999), 408-416.
[11]	A. F. B. Costa and M. S. De Magalhães, An adaptive chart for monitoring the process mean and variance, Quality and Reliability Engineering International, 23 (2007), 821-831. doi: 10.1002/qre.842.
[12]	P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Dover Publications, Inc., Mineola, NY, 2007.
[13]	J. P. Dussault, J. A. Ferland and B. Lemaire, Convex quadratic programming with one constraint and bounded variables, Mathematical Programming, 36 (1986), 90-104. doi: 10.1007/BF02591992.
[14]	I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963.
[15]	A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holland Pub. Co., Amsterdam, Netherlands, 1979.
[16]	S. Kim and D. M. Frangopol, Optimum inspection planning for minimizing fatigue damage detection delay of ship hull structures, International Journal of Fatigue, 33 (2011), 448-459. doi: 10.1016/j.ijfatigue.2010.09.018.
[17]	G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, Inc., New York, NY, 1968.
[18]	D. C. Montgomery, Introduction to Statistical Quality Control, John Wiley and Sons Inc., New York, NY, 2005.
[19]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, NY, 1962.
[20]	S. S. Prabhu, D. C. Montgomery and G. C. Runger, A combined adaptive sample size and sampling interval X control scheme, Journal of Quality Technology, 26 (1994), 164-176.
[21]	M. R. Reynolds, Evaluating properties of variable sampling interval control charts, Sequentional Analysis, 14 (1995), 59-97. doi: 10.1080/07474949508836320.
[22]	M. R. Reynolds, R. W. Amin, J. C. Arnold and J. Nachlas, X charts with variable sampling intervals, Technometrics, 30 (1988), 181-192. doi: 10.2307/1270164.
[23]	S. Ross, A First Course in Probability, 9 ed., Prentice Hall, Upper Saddle River, NJ, 2009.
[24]	P. Sebastião and C. G. Soares, Modeling the fate of oil spills at sea, Spill Science and Technology Bulletin, 2 (1995), 121-131. doi: 10.1016/S1353-2561(96)00009-6.
[25]	G. Taguchi, S. Chowdhury and Y. Wu, Taguchi's Quality Engineering Handbook, John Wiley and Sons Inc., Hoboken, NJ, 2007. doi: 10.1002/9780470258354.

show all references

References:

[1]	R. W. Amin and R. Hemasinha, The switching behavior of X charts with variable sampling intervals, Communication in Statistics - Theory and Methods, 22 (1993), 2081-2102. doi: 10.1080/03610929308831136.
[2]	R. W. Amin and R. W. Miller, A robustness study of X charts with variable sampling intervals, Journal of Quality Technology, 25 (1993), 36-44.
[3]	V. Babrauskas, Heat Release Rates, in SFPE Handbook of Fire Protection Engineering, (Ed. P.J. DiNenno), National Fire Protection Association, (2008), 1-59.
[4]	E. Bashkansky and V. Y. Glizer, Novel approach to adaptive statistical process control optimization with variable sampling interval and minimum expected loss, International Journal of Quality Engineering and Technology, 3 (2012), 91-107.
[5]	M. A. K. Bulelzai and J. L. A. Dubbeldam, Long time evolution of atherosclerotic plaques, Journal of Theoretical Biology, 297 (2012), 1-10. doi: 10.1016/j.jtbi.2011.11.023.
[6]	T. E. Carpenter, J. M. O'Brien, A. Hagerman and B. McCarl, Epidemic and economic impacts of delayed detection of foot-and-mouth disease: A case study of a simulated outbreak in California, Journal of Veterinary Diagnostic Investigation, 23 (2011), 26-33. doi: 10.1177/104063871102300104.
[7]	A. F. B. Costa, X control chart with variable sample size, Journal of Quality Technology, 26 (1994), 155-163.
[8]	A. F. B. Costa, X charts with variable sample sizes and sampling intervals, Journal of Quality Technology, 29 (1997), 197-204.
[9]	A. F. B. Costa, Joint X and R control charts with variable parameters, IIE Transactions, 30 (1998), 505-514.
[10]	A. F. B. Costa, X charts with variable parameters, Journal of Quality Technology, 31 (1999), 408-416.
[11]	A. F. B. Costa and M. S. De Magalhães, An adaptive chart for monitoring the process mean and variance, Quality and Reliability Engineering International, 23 (2007), 821-831. doi: 10.1002/qre.842.
[12]	P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Dover Publications, Inc., Mineola, NY, 2007.
[13]	J. P. Dussault, J. A. Ferland and B. Lemaire, Convex quadratic programming with one constraint and bounded variables, Mathematical Programming, 36 (1986), 90-104. doi: 10.1007/BF02591992.
[14]	I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963.
[15]	A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holland Pub. Co., Amsterdam, Netherlands, 1979.
[16]	S. Kim and D. M. Frangopol, Optimum inspection planning for minimizing fatigue damage detection delay of ship hull structures, International Journal of Fatigue, 33 (2011), 448-459. doi: 10.1016/j.ijfatigue.2010.09.018.
[17]	G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, Inc., New York, NY, 1968.
[18]	D. C. Montgomery, Introduction to Statistical Quality Control, John Wiley and Sons Inc., New York, NY, 2005.
[19]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, NY, 1962.
[20]	S. S. Prabhu, D. C. Montgomery and G. C. Runger, A combined adaptive sample size and sampling interval X control scheme, Journal of Quality Technology, 26 (1994), 164-176.
[21]	M. R. Reynolds, Evaluating properties of variable sampling interval control charts, Sequentional Analysis, 14 (1995), 59-97. doi: 10.1080/07474949508836320.
[22]	M. R. Reynolds, R. W. Amin, J. C. Arnold and J. Nachlas, X charts with variable sampling intervals, Technometrics, 30 (1988), 181-192. doi: 10.2307/1270164.
[23]	S. Ross, A First Course in Probability, 9 ed., Prentice Hall, Upper Saddle River, NJ, 2009.
[24]	P. Sebastião and C. G. Soares, Modeling the fate of oil spills at sea, Spill Science and Technology Bulletin, 2 (1995), 121-131. doi: 10.1016/S1353-2561(96)00009-6.
[25]	G. Taguchi, S. Chowdhury and Y. Wu, Taguchi's Quality Engineering Handbook, John Wiley and Sons Inc., Hoboken, NJ, 2007. doi: 10.1002/9780470258354.

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