# Optimal Control of Degrading Units through Threshold-Based Control Policies

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Degrading Unit with an Instantaneous Failure

#### 2.1. Mathematical Model

#### 2.2. Regenerative Process with Costs

**Proposition 1.**

**Proof.**

**Proposition 2.**

**Proof.**

#### 2.3. Markov Degradation Model

**Proposition 3.**

**Corollary 1.**

**Proof.**

**Proposition 4.**

**Proof.**

#### 2.4. Numerical Examples

**Example 1.**

**Example 2.**

**Example 3.**

## 3. Degrading Unit with a Partial Preventive Repair

#### 3.1. Regenerative Process with Costs

**Proposition 5.**

**Proof.**

#### 3.2. Mean Time to Failure

**Proposition 6.**

**Proof.**

**Proposition 7.**

**Proposition 8.**

**Proof.**

#### 3.3. Numerical Examples

**Example 4.**

**Example 5.**

**Example 6.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Esary, J.D.; Marshall, A.W.; Proshan, F. Shock models and wear processes. Ann. Probab.
**1973**, 1, 627–649. [Google Scholar] [CrossRef] - Kopnov, V.A.; Timashev, S.A. Optimal death process control in two-level policies. In Proceedings of the 4th Vilnius Conference on Probability Theory and Statistics, Vilnius, Lithuania, 24–29 June 1985; Volume 4, pp. 308–309. [Google Scholar]
- Murphy, D.N.P.; Iskandar, B.P. A new shock damage model: Part II—Optimal maintenance policies. Reliab. Eng. Syst. Saf.
**1991**, 31, 211–231. [Google Scholar] [CrossRef] - Singpurwalla, N.D. The hazard potential: Introduction and overview. J. Am. Stat. Assoc.
**2006**, 101, 1705–1717. [Google Scholar] [CrossRef] - Singpurwalla, N.D. Reliability and Risk; Wiley & Sons Ltd.: Chichester, UK, 2006. [Google Scholar]
- Lisnuansky, A.; Levitin, G. Multi-State System Reliability. Assessment, Optimization and Application; World Scientific: Hackensack, NJ, USA; London, UK; Singapore; Hong Kong, China, 2003. [Google Scholar]
- Rykov, V.; Dimitrov, B. On multi-state reliability systems. In Proceedings of the Seminar Applied Stochastic Models and Information Processes, Barcelona, Spain, 8–13 September 2002; pp. 128–135. [Google Scholar]
- Efrosinin, D.; Rykov, V. On reliability control of Fault Tolerance Units. In Proceedings of the Forth International Conference on Mathematical Methods in Reliability, Santa Fe, NM, USA, 21–25 June 2004. [Google Scholar]
- Rykov, V. Generalized birth and death processes and their application to aging models. Autom. Remote Control
**2006**, 3, 103–120. [Google Scholar] - Rykov, V.; Efrosinin, D. Degradation models with random life resources. Commun. Stat. Theory Methods
**2010**, 39, 398–407. [Google Scholar] [CrossRef] - Giorgio, M.; Guida, M.; Pulcini, G. An age- and state-dependent Markov model for degradation processes. IIE Trans.
**2011**, 43, 621–632. [Google Scholar] - Borodina, A.; Efrosinin, D.; Morozov, E. Application of Splitting to Failure Estimation in Controllable Degradation System. In DCCN 2017: Distributed Computer and Communication Networks; Communications in Computer and Information Science; Vishnevskiy, V., Samouylov, K., Kozyrev, D., Eds.; Springer: Cham, Switzerland, 2017; Volume 700. [Google Scholar]
- Kopnov, V.A. Optimal degradation process control by two-level policies. Reliab. Eng. Syst. Saf.
**1999**, 66, 1–11. [Google Scholar] [CrossRef] - Kopnov, V.A.; Kanajev, E.I. Optimal control limit for degradation process of a unit modeled as a Markov chain. Reliab. Eng. Syst. Saf.
**1994**, 43, 29–35. [Google Scholar] [CrossRef] - Gnedenko, B.V. The Questions of the Mathematical Theory of Reliability; Radio and Communications: Moscow, Russia, 1983. (In Russian) [Google Scholar]
- Solovyev, A.D. Asymptotic behaviour of the time to the first occurrence of a rare event. Eng. Cybern.
**1971**, 9, 1038–1048. [Google Scholar]

**Figure 4.**$({m}^{*},{n}^{*})$ versus ${c}_{i}$ (

**a**) and ${c}_{ei}$ (

**b**) for the model with an instantaneous failure.

**Figure 5.**$({m}^{*},{n}^{*})$ versus ${c}_{r}$ (

**a**) and ${c}_{R}$ (

**b**) for the model with an instantaneous failure.

**Figure 8.**$({m}^{*},{n}^{*})$ versus ${c}_{i}$ (

**a**) and ${c}_{ei}$ (

**b**) for the model with a preventive repair.

**Figure 9.**$({m}^{*},{n}^{*})$ versus ${c}_{r}$ (

**a**) and ${c}_{R}$ (

**b**) for the model with a preventive repair.

**Figure 10.**Real and asymptotic reliability functions $R\left(t\right)$ for $\nu =0.05$ (

**a**) and $\nu =0.10$ (

**b**).

Variable | Description |
---|---|

${\left\{X\left(t\right)\right\}}_{t\ge 0}$ | controllable degradation process |

$(m,n)$ | two-threshold policy with a signal state m and a number of degradation states n |

E | set of states of the degradation process |

${E}_{Y}$ | set of states of an auxiliary Markov death process |

${T}_{i}$ | random sojourn time in state i |

${S}_{ij}$ | time to reach state j from state i |

${U}_{i}$ | random repair time in state i |

V | random time to instantaneous failure (in Model 1) or to a preventive repair (in Model 2) |

${Y}_{m,n}$ | random length of the regenerative cycle given that the control policy is $(m,n)$ |

${Z}_{m,n}$ | random cost in a regenerative cycle for the policy $(m,n)$ |

${H}_{i}$ | random time from ${Y}_{m,n}$ when $X\left(t\right)=i$ |

${c}_{i}$ | repair cost per unit of time is state i |

${c}_{ei}$ | operational cost per unit of time in state i |

${c}_{r}$ | fixed cost due to a complete failure |

${c}_{R}$ | fixed cost due to an instantaneous failure (in Model 1) or due to a maintenance repair (in Model 2) |

Case N | Opt. Problem | ${\mathit{n}}^{*}$ | $\mathit{g}(4,\xb7)$ | ${\mathit{p}}_{\mathit{F}}(4,\xb7)$ | ${\mathit{\pi}}_{\mathit{F}}(4,\xb7)$ | $\mathit{r}(4,\xb7)$ |
---|---|---|---|---|---|---|

1 | $g(4,n)\Rightarrow \underset{n}{max}$ | 8 | 0.1148 | 0.1484 | 0.0016 | 442.83 |

2 | ${p}_{F}(4,n)\Rightarrow \underset{n}{min}$ | 5 | 0.1157 | 0.0972 | 0.0015 | 391.67 |

3 | ${\pi}_{F}(4,n)\Rightarrow \underset{n}{min}$ | 6 | 0.1163 | 0.1204 | 0.0015 | 414.82 |

4 | $g(4,n)\Rightarrow \underset{n}{max}$ | 13 | 0.1191 | 0.1765 | 0.0040 | 470.94 |

$r(4,n)>470$ | ||||||

5 | $r(4,n)\Rightarrow \underset{n}{max}$ | 14 | 0.1549 | 0.1814 | 0.0221 | 475.87 |

Case Nr. | Opt. Problem | $({\mathit{m}}^{*},{\mathit{n}}^{*})$ | $\mathit{g}(\xb7,\xb7)$ | ${\mathit{p}}_{\mathit{F}}(\xb7,\xb7)$ | ${\mathit{\pi}}_{\mathit{F}}(\xb7,\xb7)$ | $\mathit{r}(\xb7,\xb7)$ |
---|---|---|---|---|---|---|

1 | $g(m,n)\Rightarrow \underset{m,n}{min}$ | (3, 6) | 0.1279 | 0.1871 | 0.0030 | 848.44 |

2 | ${p}_{F}(m,n)\Rightarrow \underset{m,n}{min}$ | (2, 4) | 0.1280 | 0.1636 | 0.0037 | 626.16 |

3 | ${\pi}_{F}(m,n)\Rightarrow \underset{m,n}{min}$ | (5, 6) | 0.2426 | 0.5718 | 0.0020 | 436.15 |

4 | $g(m,n)\Rightarrow \underset{m,n}{min}$ | (4, 8) | 0.1317 | 0.2535 | 0.0031 | 948.61 |

$r(m,n)>900$ | ||||||

5 | $r(m,n)\Rightarrow \underset{m,n}{max}$ | (4, 8) | 0.1317 | 0.2535 | 0.0031 | 948.61 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Efrosinin, D.; Stepanova, N.
Optimal Control of Degrading Units through Threshold-Based Control Policies. *Mathematics* **2022**, *10*, 4098.
https://doi.org/10.3390/math10214098

**AMA Style**

Efrosinin D, Stepanova N.
Optimal Control of Degrading Units through Threshold-Based Control Policies. *Mathematics*. 2022; 10(21):4098.
https://doi.org/10.3390/math10214098

**Chicago/Turabian Style**

Efrosinin, Dmitry, and Natalia Stepanova.
2022. "Optimal Control of Degrading Units through Threshold-Based Control Policies" *Mathematics* 10, no. 21: 4098.
https://doi.org/10.3390/math10214098