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

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

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.
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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[12]

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration,, Dover Publications, (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. doi: 10.1007/BF02591992.

[14]

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

[15]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems,, North-Holland Pub. Co., (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. 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, (1968).

[18]

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

[19]

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

[21]

M. R. Reynolds, Evaluating properties of variable sampling interval control charts,, Sequentional Analysis, 14 (1995), 59. 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. doi: 10.2307/1270164.

[23]

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

[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.

[25]

G. Taguchi, S. Chowdhury and Y. Wu, Taguchi's Quality Engineering Handbook,, John Wiley and Sons Inc., (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. 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.

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[12]

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration,, Dover Publications, (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. doi: 10.1007/BF02591992.

[14]

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

[15]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems,, North-Holland Pub. Co., (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. 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, (1968).

[18]

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

[19]

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

[21]

M. R. Reynolds, Evaluating properties of variable sampling interval control charts,, Sequentional Analysis, 14 (1995), 59. 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. doi: 10.2307/1270164.

[23]

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

[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.

[25]

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

[1]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[2]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial & Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585

[3]

Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323

[4]

Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019

[5]

Bavo Langerock. Optimal control problems with variable endpoints. Conference Publications, 2003, 2003 (Special) : 507-516. doi: 10.3934/proc.2003.2003.507

[6]

Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1

[7]

Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519

[8]

Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275

[9]

Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100

[10]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

[11]

Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial & Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73

[12]

Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027

[13]

Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1067-1094. doi: 10.3934/mbe.2013.10.1067

[14]

Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764

[15]

Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control & Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83

[16]

Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1685-1700. doi: 10.3934/jimo.2018027

[17]

Andrzej Just, Zdzislaw Stempień. Optimal control problem for a viscoelastic beam and its galerkin approximation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 263-274. doi: 10.3934/dcdsb.2018018

[18]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791

[19]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105

[20]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

[Back to Top]