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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 and 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. [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. [3] V. Babrauskas, Heat release rates, in SFPE Handbook of Fire Protection Engineering, 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] M. G. Bulmer, Principles of Statistics, Dover Books on Mathematics Series, Dover Publications, 1979. [7] T. E. Carpenter, J. M. O'Brien, A. 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. [8] X. Y. Chew, M. B. C. Khoo, S. 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. [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. [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. [11] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963. [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. [13] V. Y. Glizer, V. 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. [14] C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 2012. [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. [16] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Pub. Co., Amsterdam-New York, 1979. [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. [18] Z. Li and P. Qiu, Statistical process control using dynamic sampling scheme, Technometrics, 56 (2014), 325-335.  doi: 10.1080/00401706.2013.844731. [19] D. M. Packwood, Moments of sums of independent and identically distributed random variables, arXiv: 1105.6283 (2011). [20] Y. Peng, L. 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. [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. [22] P. Qiu, Introduction to Statistical Process Control, CRC Press, Boca Raton, FL, 2013.  doi: 10.1201/b15016. [23] M. R. Reynolds, R. W. Amin, J. C. Arnold and J. A. Nachlas, $\bar{X}$ charts with variable sampling intervals, Technometrics, 30 (1988), 181-192.  doi: 10.2307/1270164. [24] S. Ross, A First Course in Probability, Prentice Hall, Upper Saddle River, NJ, 2009. [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. [26] I. Sultana, I. Ahmed, A. 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. [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.

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. [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. [3] V. Babrauskas, Heat release rates, in SFPE Handbook of Fire Protection Engineering, 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] M. G. Bulmer, Principles of Statistics, Dover Books on Mathematics Series, Dover Publications, 1979. [7] T. E. Carpenter, J. M. O'Brien, A. 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. [8] X. Y. Chew, M. B. C. Khoo, S. 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. [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. [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. [11] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963. [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. [13] V. Y. Glizer, V. 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. [14] C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 2012. [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. [16] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland Pub. Co., Amsterdam-New York, 1979. [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. [18] Z. Li and P. Qiu, Statistical process control using dynamic sampling scheme, Technometrics, 56 (2014), 325-335.  doi: 10.1080/00401706.2013.844731. [19] D. M. Packwood, Moments of sums of independent and identically distributed random variables, arXiv: 1105.6283 (2011). [20] Y. Peng, L. 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. [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. [22] P. Qiu, Introduction to Statistical Process Control, CRC Press, Boca Raton, FL, 2013.  doi: 10.1201/b15016. [23] M. R. Reynolds, R. W. Amin, J. C. Arnold and J. A. Nachlas, $\bar{X}$ charts with variable sampling intervals, Technometrics, 30 (1988), 181-192.  doi: 10.2307/1270164. [24] S. Ross, A First Course in Probability, Prentice Hall, Upper Saddle River, NJ, 2009. [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. [26] I. Sultana, I. Ahmed, A. 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. [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.
Optimal control: $\delta = 0.5$, $u_{\min} = 0.5$, $u_{\max} = 1.5$, $T = 1$
Parameter $\gamma_3$: $\delta = 0.5$, $u_{\min} = 0.5$, $u_{\max} = 1.5$, $T = 1$
Optimal control: $\delta = 1$, $u_{\min} = 0.5$, $u_{\max} = 1.5$, $T = 1$
Parameter $\gamma_3$: $\delta = 1$, $u_{\min} = 0.5$, $u_{\max} = 1.5$, $T = 1$
Optimal control: $\delta = 0.18$, $u_{\min} = 0.5$, $u_{\max} = 1.5$, $T = 1$
Parameter $\gamma_3$: $\delta = 0.18$, $u_{\min} = 0.5$, $u_{\max} = 1.5$, $T = 1$
Optimal sampling time in epidemic control, $\Delta = 0.025$
Optimal sampling time in epidemic control, $\Delta = 0.05$
Subcase I.1
Subcase I.2
Subcase I.3
Case II
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