• Previous Article
    Multi-objective optimization model for planning metro-based underground logistics system network: Nanjing case study
  • JIMO Home
  • This Issue
  • Next Article
    Relaxation schemes for the joint linear chance constraint based on probability inequalities
doi: 10.3934/jimo.2021096
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

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
[1]

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

[2]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial and 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 and Continuous Dynamical Systems, 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 and 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]

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

[7]

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

[8]

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

[9]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1709-1721. doi: 10.3934/jimo.2021040

[10]

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

[11]

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

[12]

Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1663-1683. doi: 10.3934/jimo.2019023

[13]

Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456

[14]

Piermarco Cannarsa, Carlo Sinestrari. On a class of nonlinear time optimal control problems. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 285-300. doi: 10.3934/dcds.1995.1.285

[15]

Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078

[16]

Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial and Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082

[17]

Bin Li, Kok Lay Teo, Cheng-Chew Lim, Guang Ren Duan. An optimal PID controller design for nonlinear constrained optimal control problems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1101-1117. doi: 10.3934/dcdsb.2011.16.1101

[18]

Nasser H. Sweilam, Taghreed A. Assiri, Muner M. Abou Hasan. Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1247-1268. doi: 10.3934/dcdss.2021085

[19]

Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112

[20]

Leszek Gasiński, Nikolaos S. Papageorgiou. Relaxation of optimal control problems driven by nonlinear evolution equations. Evolution Equations and Control Theory, 2020, 9 (4) : 1027-1040. doi: 10.3934/eect.2020050

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (355)
  • HTML views (361)
  • Cited by (0)

Other articles
by authors

[Back to Top]