doi: 10.3934/jimo.2021096
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Optimal decision in a Statistical Process Control with cubic loss

Department of Applied Mathematics, ORT Braude College of Engineering, 51 Snunit Str., P.O.B. 51, Karmiel, 2161002, Israel

* Corresponding author

Received  September 2020 Revised  February 2021 Early access May 2021

We consider the problem of time-sampling optimization for a Statistical Process Control (SPC). The aim of this optimization is to minimize the expected loss, caused by a delay in the detection of an undesirable process change. The expected loss is chosen as a cubic polynomial function of this delay. Such a form of the expected loss is justified by some real-life problems. The SPC optimization problem is modeled by a nonlinear calculus of variations problem where the functional is minimized by a proper choice of the sampling time-interval. Theoretical results are illustrated by several academic and real-life examples.

In the previous works of the authors, the SPC optimization problem was solved for linear, pure quadratic and quadratic polynomial criteria.

Citation: Vladimir Turetsky, Valery Y. Glizer. Optimal decision in a Statistical Process Control with cubic loss. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021096
References:
[1]

R. W. Amin and R. Hemasinha, The switching behavior of $\overline{X}$ charts with variable sampling intervals, Communication in Statistics - Theory and Methods, 22 (1993), 2081-2102.  doi: 10.1080/03610929308831136.  Google Scholar

[2]

R. W. Amin and R. W. Miller, A robustness study of charts with variable sampling intervals, Journal of Quality Technology, 25 (1993), 36-44.  doi: 10.1080/00224065.1993.11979414.  Google Scholar

[3]

V. Babrauskas, Heat release rates, in SFPE Handbook of Fire Protection Engineering, National Fire Protection Association, 2008, 1-59. 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), 91-107.   Google Scholar

[5]

M. A. K. Bulelzai and J. L. A. Dubbeldam, Long time evolution of atherosclerotic plaques, Journal of Theoretical Biology, 297 (2012), 1-10.  doi: 10.1016/j.jtbi.2011.11.023.  Google Scholar

[6]

M. G. Bulmer, Principles of Statistics, Dover Books on Mathematics Series, Dover Publications, 1979.  Google Scholar

[7]

T. E. CarpenterJ. M. O'BrienA. D. Hagerman and B. A. 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.  Google Scholar

[8]

X. Y. ChewM. B. C. KhooS. Y. Teh and P. Castagliola, The variable sampling interval run sum $\overline{X}$ control chart, Computers & Industrial Engineering, 90 (2015), 25-38.  doi: 10.1016/j.cie.2015.08.015.  Google Scholar

[9]

A. F. B. Costa, $\overline{X}$ charts with variable sample size, Journal of Quality Technology, 26 (1994), 155-163.  doi: 10.1080/00224065.1994.11979523.  Google Scholar

[10]

A. F. B. Costa, $\overline{X}$ charts with variable sample size and sampling intervals, Journal of Quality Technology, 29 (1997), 197-204.  doi: 10.1080/00224065.1997.11979750.  Google Scholar

[11]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963.  Google Scholar

[12]

V. Y. Glizer and V. Turetsky, Optimal time-sampling in a statistical process control with a polynomial expected loss, in Informatics in Control, Automation and Robotics, 15th International Conference ICINCO 2018, Porto, Portugal, July 29-31, 2018, Revised Selected Papers (eds. O. Gusikhin and K. Madani), vol. 613 of Lecture Notes in Electrical Engineering, Springer Nature, Switzerland, 2020, chapter 2, 26-50. Google Scholar

[13]

V. Y. GlizerV. Turetsky and E. Bashkansky, Statistical process control optimization with variable sampling interval and nonlinear expected loss, Journal of Industrial and Management Optimization, 11 (2015), 105-133.  doi: 10.3934/jimo.2015.11.105.  Google Scholar

[14]

C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 2012. Google Scholar

[15]

A. T. Hatjimihail, Estimation of the optimal statistical quality control sampling time intervals using a residual risk measure, PLOS ONE, 4 (2009), e5770. doi: 10.1371/journal.pone.0005770.  Google Scholar

[16]

A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Pub. Co., Amsterdam-New York, 1979.  Google Scholar

[17]

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.  Google Scholar

[18]

Z. Li and P. Qiu, Statistical process control using dynamic sampling scheme, Technometrics, 56 (2014), 325-335.  doi: 10.1080/00401706.2013.844731.  Google Scholar

[19]

D. M. Packwood, Moments of sums of independent and identically distributed random variables, arXiv: 1105.6283 (2011). Google Scholar

[20]

Y. PengL. Xu and M. R. Reynolds, The design of the variable sampling interval generalized likelihood ratio chart for monitoring the process mean, Quality and Reliability Engineering International, 31 (2015), 291-296.  doi: 10.1002/qre.1587.  Google Scholar

[21]

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

[22] P. Qiu, Introduction to Statistical Process Control, CRC Press, Boca Raton, FL, 2013.  doi: 10.1201/b15016.  Google Scholar
[23]

M. R. ReynoldsR. W. AminJ. C. Arnold and J. A. Nachlas, $\bar{X}$ charts with variable sampling intervals, Technometrics, 30 (1988), 181-192.  doi: 10.2307/1270164.  Google Scholar

[24]

S. Ross, A First Course in Probability, Prentice Hall, Upper Saddle River, NJ, 2009. Google Scholar

[25]

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.  Google Scholar

[26]

I. SultanaI. AhmedA. H. Chowdhury and S. K. Paul, Economic design of $\overline{X}$ control chart using genetic algorithm and simulated annealing algorithm, International Journal of Productivity and Quality Management, 14 (2014), 352-372.  doi: 10.1504/IJPQM.2014.064810.  Google Scholar

[27]

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

show all references

References:
[1]

R. W. Amin and R. Hemasinha, The switching behavior of $\overline{X}$ charts with variable sampling intervals, Communication in Statistics - Theory and Methods, 22 (1993), 2081-2102.  doi: 10.1080/03610929308831136.  Google Scholar

[2]

R. W. Amin and R. W. Miller, A robustness study of charts with variable sampling intervals, Journal of Quality Technology, 25 (1993), 36-44.  doi: 10.1080/00224065.1993.11979414.  Google Scholar

[3]

V. Babrauskas, Heat release rates, in SFPE Handbook of Fire Protection Engineering, National Fire Protection Association, 2008, 1-59. 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), 91-107.   Google Scholar

[5]

M. A. K. Bulelzai and J. L. A. Dubbeldam, Long time evolution of atherosclerotic plaques, Journal of Theoretical Biology, 297 (2012), 1-10.  doi: 10.1016/j.jtbi.2011.11.023.  Google Scholar

[6]

M. G. Bulmer, Principles of Statistics, Dover Books on Mathematics Series, Dover Publications, 1979.  Google Scholar

[7]

T. E. CarpenterJ. M. O'BrienA. D. Hagerman and B. A. 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.  Google Scholar

[8]

X. Y. ChewM. B. C. KhooS. Y. Teh and P. Castagliola, The variable sampling interval run sum $\overline{X}$ control chart, Computers & Industrial Engineering, 90 (2015), 25-38.  doi: 10.1016/j.cie.2015.08.015.  Google Scholar

[9]

A. F. B. Costa, $\overline{X}$ charts with variable sample size, Journal of Quality Technology, 26 (1994), 155-163.  doi: 10.1080/00224065.1994.11979523.  Google Scholar

[10]

A. F. B. Costa, $\overline{X}$ charts with variable sample size and sampling intervals, Journal of Quality Technology, 29 (1997), 197-204.  doi: 10.1080/00224065.1997.11979750.  Google Scholar

[11]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963.  Google Scholar

[12]

V. Y. Glizer and V. Turetsky, Optimal time-sampling in a statistical process control with a polynomial expected loss, in Informatics in Control, Automation and Robotics, 15th International Conference ICINCO 2018, Porto, Portugal, July 29-31, 2018, Revised Selected Papers (eds. O. Gusikhin and K. Madani), vol. 613 of Lecture Notes in Electrical Engineering, Springer Nature, Switzerland, 2020, chapter 2, 26-50. Google Scholar

[13]

V. Y. GlizerV. Turetsky and E. Bashkansky, Statistical process control optimization with variable sampling interval and nonlinear expected loss, Journal of Industrial and Management Optimization, 11 (2015), 105-133.  doi: 10.3934/jimo.2015.11.105.  Google Scholar

[14]

C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 2012. Google Scholar

[15]

A. T. Hatjimihail, Estimation of the optimal statistical quality control sampling time intervals using a residual risk measure, PLOS ONE, 4 (2009), e5770. doi: 10.1371/journal.pone.0005770.  Google Scholar

[16]

A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Pub. Co., Amsterdam-New York, 1979.  Google Scholar

[17]

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.  Google Scholar

[18]

Z. Li and P. Qiu, Statistical process control using dynamic sampling scheme, Technometrics, 56 (2014), 325-335.  doi: 10.1080/00401706.2013.844731.  Google Scholar

[19]

D. M. Packwood, Moments of sums of independent and identically distributed random variables, arXiv: 1105.6283 (2011). Google Scholar

[20]

Y. PengL. Xu and M. R. Reynolds, The design of the variable sampling interval generalized likelihood ratio chart for monitoring the process mean, Quality and Reliability Engineering International, 31 (2015), 291-296.  doi: 10.1002/qre.1587.  Google Scholar

[21]

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

[22] P. Qiu, Introduction to Statistical Process Control, CRC Press, Boca Raton, FL, 2013.  doi: 10.1201/b15016.  Google Scholar
[23]

M. R. ReynoldsR. W. AminJ. C. Arnold and J. A. Nachlas, $\bar{X}$ charts with variable sampling intervals, Technometrics, 30 (1988), 181-192.  doi: 10.2307/1270164.  Google Scholar

[24]

S. Ross, A First Course in Probability, Prentice Hall, Upper Saddle River, NJ, 2009. Google Scholar

[25]

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.  Google Scholar

[26]

I. SultanaI. AhmedA. H. Chowdhury and S. K. Paul, Economic design of $\overline{X}$ control chart using genetic algorithm and simulated annealing algorithm, International Journal of Productivity and Quality Management, 14 (2014), 352-372.  doi: 10.1504/IJPQM.2014.064810.  Google Scholar

[27]

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

Figure 1.  Optimal control: $ \delta = 0.5 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $
Figure 2.  Parameter $ \gamma_3 $: $ \delta = 0.5 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $
Figure 3.  Optimal control: $ \delta = 1 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $
Figure 4.  Parameter $ \gamma_3 $: $ \delta = 1 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $
Figure 5.  Optimal control: $ \delta = 0.18 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $
Figure 6.  Parameter $ \gamma_3 $: $ \delta = 0.18 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $
Figure 7.  Optimal sampling time in epidemic control, $ \Delta = 0.025 $
Figure 8.  Optimal sampling time in epidemic control, $ \Delta = 0.05 $
Figure 9.  Subcase I.1
Figure 10.  Subcase I.2
Figure 11.  Subcase I.3
Figure 12.  Case II
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