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# Optimal decision in a Statistical Process Control with cubic loss

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• 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.

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

 Citation: • • 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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