Optimal decision in a Statistical Process Control with cubic loss

Vladimir Turetsky; Valery Y. Glizer

doi:10.3934/jimo.2021096

Article Contents

2022, Volume 18, Issue 4: 2903-2926. Doi: 10.3934/jimo.2021096

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Optimal decision in a Statistical Process Control with cubic loss

Vladimir Turetsky^, and
Valery Y. Glizer

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

^* Corresponding author
^* Corresponding author

Received: September 2020

Revised: February 2021

Published: July 2022

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 62P30; Secondary: 49B10.

Citation:

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Figure 1. Optimal control: $ \delta = 0.5 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $

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Figure 2. Parameter $ \gamma_3 $: $ \delta = 0.5 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $

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Figure 3. Optimal control: $ \delta = 1 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $

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Figure 4. Parameter $ \gamma_3 $: $ \delta = 1 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $

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Figure 5. Optimal control: $ \delta = 0.18 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $

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Figure 6. Parameter $ \gamma_3 $: $ \delta = 0.18 $, $ u_{\min} = 0.5 $, $ u_{\max} = 1.5 $, $ T = 1 $

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Figure 7. Optimal sampling time in epidemic control, $ \Delta = 0.025 $

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Figure 8. Optimal sampling time in epidemic control, $ \Delta = 0.05 $

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Figure 9. Subcase I.1

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Figure 10. Subcase I.2

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Figure 11. Subcase I.3

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Figure 12. Case II

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