American Institute of Mathematical Sciences

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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

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 & 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.  doi: 10.1080/03610929308831136.  Google Scholar

[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.   Google Scholar

[3]

V. Babrauskas, Heat Release Rates,, in SFPE Handbook of Fire Protection Engineering, (2008), 1.   Google Scholar

[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.   Google Scholar

[5]

M. A. K. Bulelzai and J. L. A. Dubbeldam, Long time evolution of atherosclerotic plaques,, Journal of Theoretical Biology, 297 (2012), 1.  doi: 10.1016/j.jtbi.2011.11.023.  Google Scholar

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[7]

A. F. B. Costa, X control chart with variable sample size,, Journal of Quality Technology, 26 (1994), 155.   Google Scholar

[8]

A. F. B. Costa, X charts with variable sample sizes and sampling intervals,, Journal of Quality Technology, 29 (1997), 197.   Google Scholar

[9]

A. F. B. Costa, Joint X and R control charts with variable parameters,, IIE Transactions, 30 (1998), 505.   Google Scholar

[10]

A. F. B. Costa, X charts with variable parameters,, Journal of Quality Technology, 31 (1999), 408.   Google Scholar

[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.  doi: 10.1002/qre.842.  Google Scholar

[12]

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration,, Dover Publications, (2007).   Google Scholar

[13]

J. P. Dussault, J. A. Ferland and B. Lemaire, Convex quadratic programming with one constraint and bounded variables,, Mathematical Programming, 36 (1986), 90.  doi: 10.1007/BF02591992.  Google Scholar

[14]

I. M. Gelfand and S. V. Fomin, Calculus of Variations,, Prentice-Hall, (1963).   Google Scholar

[15]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems,, North-Holland Pub. Co., (1979).   Google Scholar

[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.  doi: 10.1016/j.ijfatigue.2010.09.018.  Google Scholar

[17]

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers,, McGraw-Hill Book Company, (1968).   Google Scholar

[18]

D. C. Montgomery, Introduction to Statistical Quality Control,, John Wiley and Sons Inc., (2005).   Google Scholar

[19]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Interscience, (1962).   Google Scholar

[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.   Google Scholar

[21]

M. R. Reynolds, Evaluating properties of variable sampling interval control charts,, Sequentional Analysis, 14 (1995), 59.  doi: 10.1080/07474949508836320.  Google Scholar

[22]

M. R. Reynolds, R. W. Amin, J. C. Arnold and J. Nachlas, X charts with variable sampling intervals,, Technometrics, 30 (1988), 181.  doi: 10.2307/1270164.  Google Scholar

[23]

S. Ross, A First Course in Probability,, 9 ed., (2009).   Google Scholar

[24]

P. Sebastião and C. G. Soares, Modeling the fate of oil spills at sea,, Spill Science and Technology Bulletin, 2 (1995), 121.  doi: 10.1016/S1353-2561(96)00009-6.  Google Scholar

[25]

G. Taguchi, S. Chowdhury and Y. Wu, Taguchi's Quality Engineering Handbook,, John Wiley and Sons Inc., (2007).  doi: 10.1002/9780470258354.  Google Scholar

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.  doi: 10.1080/03610929308831136.  Google Scholar

[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.   Google Scholar

[3]

V. Babrauskas, Heat Release Rates,, in SFPE Handbook of Fire Protection Engineering, (2008), 1.   Google Scholar

[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.   Google Scholar

[5]

M. A. K. Bulelzai and J. L. A. Dubbeldam, Long time evolution of atherosclerotic plaques,, Journal of Theoretical Biology, 297 (2012), 1.  doi: 10.1016/j.jtbi.2011.11.023.  Google Scholar

[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.  doi: 10.1177/104063871102300104.  Google Scholar

[7]

A. F. B. Costa, X control chart with variable sample size,, Journal of Quality Technology, 26 (1994), 155.   Google Scholar

[8]

A. F. B. Costa, X charts with variable sample sizes and sampling intervals,, Journal of Quality Technology, 29 (1997), 197.   Google Scholar

[9]

A. F. B. Costa, Joint X and R control charts with variable parameters,, IIE Transactions, 30 (1998), 505.   Google Scholar

[10]

A. F. B. Costa, X charts with variable parameters,, Journal of Quality Technology, 31 (1999), 408.   Google Scholar

[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.  doi: 10.1002/qre.842.  Google Scholar

[12]

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration,, Dover Publications, (2007).   Google Scholar

[13]

J. P. Dussault, J. A. Ferland and B. Lemaire, Convex quadratic programming with one constraint and bounded variables,, Mathematical Programming, 36 (1986), 90.  doi: 10.1007/BF02591992.  Google Scholar

[14]

I. M. Gelfand and S. V. Fomin, Calculus of Variations,, Prentice-Hall, (1963).   Google Scholar

[15]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems,, North-Holland Pub. Co., (1979).   Google Scholar

[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.  doi: 10.1016/j.ijfatigue.2010.09.018.  Google Scholar

[17]

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers,, McGraw-Hill Book Company, (1968).   Google Scholar

[18]

D. C. Montgomery, Introduction to Statistical Quality Control,, John Wiley and Sons Inc., (2005).   Google Scholar

[19]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Interscience, (1962).   Google Scholar

[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.   Google Scholar

[21]

M. R. Reynolds, Evaluating properties of variable sampling interval control charts,, Sequentional Analysis, 14 (1995), 59.  doi: 10.1080/07474949508836320.  Google Scholar

[22]

M. R. Reynolds, R. W. Amin, J. C. Arnold and J. Nachlas, X charts with variable sampling intervals,, Technometrics, 30 (1988), 181.  doi: 10.2307/1270164.  Google Scholar

[23]

S. Ross, A First Course in Probability,, 9 ed., (2009).   Google Scholar

[24]

P. Sebastião and C. G. Soares, Modeling the fate of oil spills at sea,, Spill Science and Technology Bulletin, 2 (1995), 121.  doi: 10.1016/S1353-2561(96)00009-6.  Google Scholar

[25]

G. Taguchi, S. Chowdhury and Y. Wu, Taguchi's Quality Engineering Handbook,, John Wiley and Sons Inc., (2007).  doi: 10.1002/9780470258354.  Google Scholar

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