# Optimal Stopping Rules for Preventing Overloading of Multicomponent Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Array of Pillars under Progressive Loading

#### 2.1. Pillars

#### 2.2. Load Transfer Rule

#### 2.3. Arrays’ Strengths

## 3. Blackjack-Type Optimal Stopping Problems

**Proposition 1.**

## 4. Optimal Stopping Rules for a Pillar-Array Loading

#### 4.1. Load Limit and Payoff Function

#### 4.2. Uniform Distribution of Load Steps

#### 4.3. Solution of Optimal Stopping Problem

`1`.`Set Load=0``2`.`Set ActualLimit = RandomNumber[LimitDistribution]``3`.`While (Load<Q*)AND(Load<ActualLimit) set``Load=Load+RandomNumber[StepDistribution]``4`.`If Load < ActualLimit set Payoff = Load else set Payoff = -Load``5`.`Return Payoff`

- After simulating ${10}^{5}$ runs of the above procedure, we found the approximate value of the problem, which equals 2944.18.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Chekurov, N.; Grigoras, K.; Peltonen, A.; Franssila, S.; Tittonen, I. The fabrication of silicon nanostructures by local gallium implantation and cryogenic deep reactive ion etching. Nanotechnology
**2009**, 20, 65307. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Park, J.E.; Won, S.; Cho, W.; Kim, J.G.; Jhang, S.; Lee, J.G.; Wie, J.J. Fabrication and applications of stimuli-responsive micro/nanopillar arrays. J. Polym. Sci.
**2021**, 59, 1491–1517. [Google Scholar] [CrossRef] - Harding, F.J.; Surdo, S.; Delalat, B.; Cozzi, C.; Elnathan, R.; Gronthos, S.; Voelcker, N.H.; Barillaro, G. Ordered Silicon Pillar Arrays Prepared by Electrochemical Micromachining: Substrates for High-Efficiency Cell Transfection. ACS Appl. Mater. Interfaces
**2016**, 8, 29197–29202. [Google Scholar] [CrossRef] [PubMed] - Schoen, I.; Hu, W.; Klotzsch, E.; Vogel, V. Probing Cellular Traction Forces by Micropillar Arrays: Contribution of Substrate Warping to Pillar Deflection. Nano Lett.
**2010**, 10, 1823–1830. [Google Scholar] [CrossRef] [Green Version] - Qiu, X.; Lo, J.C.C.; Lee, S.W.R.; Liou, Y.-H.; Chiu, P. Evaluation and Benchmarking of Cu Pillar Micro-bumps with Printed Polymer Core. In Proceedings of the 2019 International Conference on Electronics Packaging (ICEP), Niigata, Japan, 17–20 April 2019; pp. 24–27. [Google Scholar] [CrossRef]
- Chen, X.; Shao, J.; Tian, H.; Li, X.; Tian, Y.; Wang, C. Flexible three-axial tactile sensors with microstructure-enhanced piezoelectric effect and specially-arranged piezoelectric arrays. Smart Mater. Struct.
**2018**, 27, 025018. [Google Scholar] [CrossRef] - Chen, X.; Li, X.; Shao, J.; An, N.; Tian, H.; Wang, C.; Han, T.; Wang, L.; Lu, B. High-Performance Piezoelectric Nanogenerators with Imprinted P(VDF-TrFE)/BaTiO3 Nanocomposite Micropillars for Self-Powered Flexible Sensors. Small
**2017**, 13, 1604245. [Google Scholar] [CrossRef] - Choi, Y.-Y.; Yun, T.G.; Qaiser, N.; Paik, H.; Roh, H.S.; Hong, J.; Hong, S.; Han, S.M.; No, K. Vertically aligned P(VDF-TrFE) core-shell structures on flexible pillar arrays. Sci. Rep.
**2015**, 5, 10728. [Google Scholar] [CrossRef] [Green Version] - Mervat, I.; Jinxing, J.; Zhen., W.; Xuhui, S. Surface Engineering for Enhanced Triboelectric Nanogenerator. Nanoenergy Adv.
**2021**, 1, 4. [Google Scholar] [CrossRef] - Rakotondrabe, M.; Yang, R.; Wang, L.Z. Editorial for the Special Issue on Piezoelectric Nanogenerators for Micro-Energy and Self-Powered Sensors. Micromachines
**2022**, 13, 1443. [Google Scholar] [CrossRef] - Greer, J.R.; Jang, D.; Kim, J.-Y.; Burek, J. Emergence of New Mechanical Functionality in Materials via Size Reduction. Adv. Funct. Mater.
**2009**, 19, 2880–2886. [Google Scholar] [CrossRef] - Jang, D.; Greer, J.R. Transition from a strong-yet-brittle to a stronger-and-ductile state by size reduction of metallic glasses. Nat. Mater.
**2010**, 9, 215–219. [Google Scholar] [CrossRef] - Derda, T.; Domanski, Z. Enhanced strength of cyclically preloaded arrays of pillars. Acta Mech.
**2020**, 231, 3145–3155. [Google Scholar] [CrossRef] - Derda, T.; Domanski, Z. Survivability of Suddenly Loaded Arrays of Micropillars. Materials
**2021**, 14, 7173. [Google Scholar] [CrossRef] [PubMed] - Zhu, Y.; Yang, B.; Liu, J.; Wnag, X.; Wang, L.; Yang, C. A flexible and biocompatible triboelectric nanogenerator with tunable internal resistance for powering wearable devices. Sci. Rep.
**2016**, 6, 22233. [Google Scholar] [CrossRef] [Green Version] - Shin, S.-H.; Choi, S.-Y.; Lee, M.H.; Nah, J. High-Performance Piezoelectric Nanogenerators via Imprinted Sol–Gel BaTiO
_{3}Nanopillar Array. ACS Appl. Mater. Interfaces**2017**, 9, 41099–41103. [Google Scholar] [CrossRef] [PubMed] - Hidalgo, R.C.; Moreno, J.; Kun, F.; Herrmann, H.J. Fracture model with variable range of interaction. Phys. Rev. E
**2002**, 65, 046148. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Roy, S.; Biswas, S.; Ray, P. Modes of failure in disordered solids. Phys. Rev. E
**2017**, 96, 063003. [Google Scholar] [CrossRef] [Green Version] - Arnold, T.; Emerson, J. Nonparametric goodness-of-fit tests for discrete null distributions. R J.
**2011**, 3, 34–39. [Google Scholar] [CrossRef] [Green Version] - Wu, Y. Optimal Stopping and Loading Rules Considering Multiple Attempts and Task Success Criteria. Mathematics
**2023**, 11, 1065. [Google Scholar] [CrossRef] - Qiu, Q.; Cui, L. Reliability evaluation based on a dependent two-stage failure process with competing failures. Appl. Math. Model.
**2018**, 64, 699–712. [Google Scholar] [CrossRef] - Oosterom, C.D.; Elwany, A.H.; Çelebi, D.; Houtum, G.J. Optimal policies for a delay time model with postponed replacement. Eur. J. Oper. Res.
**2014**, 232, 186–197. [Google Scholar] [CrossRef] - Liu, X.; Wang, W.; Peng, R.; Zhao, F. A delay-time-based inspection model for parallel systems. J. Risk Reliab.
**2015**, 229, 556–567. [Google Scholar] [CrossRef] - Sun, Y.T.; Liu, C.; Zhang, Q.; Qin, X.R. Multiple Failure Modes Reliability Modeling and Analysis in Crack Growth Life Based on JC Method. Math. Probl. Eng.
**2017**, 2017, 2068620. [Google Scholar] [CrossRef] [Green Version] - Grzybowski, A.Z. Optimal Stopping Rules For Some Blackjack Type Problem. In Current Themes in Engineering Science; Korsunsky, A., Ed.; American Institute of Physics: Melville, NY, USA, 2010; pp. 91–100. [Google Scholar] [CrossRef]
- Grzybowski, A.Z. Monte Carlo Analysis of Risk Measures for Blackjack Type Optimal Stopping Problems. Eng. Lett.
**2011**, 19, 147–154. [Google Scholar] - Chow, Y.S.; Robbins, H.E.; Siegmund, D. Great Expectations: The Theory of Optimal Stopping; Houghton Mifflin: Boston, MA, USA, 1971. [Google Scholar]
- Shiryaev, A.N. Optimal Stopping Rules; Springer: New York, NY, USA, 2008. [Google Scholar] [CrossRef]
- Cha, J.M.; Mi, J. Study of a stochastic failure model in a random environment. J. Appl. Probab.
**2007**, 44, 151–163. [Google Scholar] [CrossRef] [Green Version] - Yang, L.; Peng, R.; Zhao, Y. Hybrid preventive maintenance of competing failures under random environment. Reliab. Eng. Sys. Saf.
**2018**, 174, 130–140. [Google Scholar] [CrossRef]

**Figure 1.**Schematic view of array of pillars: before loading (

**left**panel) and under a load (

**right**panel). High columns represent intact pillars and low ones refer to crushed pillars.

**Figure 2.**Empirical distribution of ${Q}_{c}$ obtained from ${10}^{4}$ arrays of $100\times 100$ pillars (

**left**panel) and the resulting probability density of the Weibull distribution, see Equation (7), with parameters estimated from the simulations (

**right**panel).

**Figure 4.**Sketch of considered load steps distributions $f\left(q\right)$, as drawn in the case $\overline{q}=250$.

**Left**panel: PDFs with the finite support $\left[0,\overline{q}\right]$.

**Right**panel: PDFs with the semi-infinite support $\left[0,\infty \right]$.

**Figure 5.**The function ${V}_{1}$ and the payoff function Z in the case of uniformly distributed loading steps with $\overline{q}=250$.

Step Distribution | Mean Load Step $\overline{\mathit{q}}$ | Step Standard Deviation | Optimal Threshold ${\mathit{Q}}^{*}$ | Value of the Problem V |
---|---|---|---|---|

Bates | 250 | 83.33 | 2927.98 | 2975.70 |

truncated normal | 250 | 109.95 | 2905.02 | 2963.97 |

uniform | 250 | 144.34 | 2872.45 | 2944.18 |

half-normal | 250 | 188.88 | 2760.99 | 2827.05 |

exponential | 250 | 250.00 | 2600.77 | 2600.55 |

Bates | 225 | 75.00 | 2950.35 | 2984.43 |

truncated normal | 225 | 98.95 | 2930.82 | 2973.76 |

uniform | 225 | 129.90 | 2902.56 | 2960.94 |

half-normal | 225 | 169.99 | 2805.83 | 2861.57 |

exponential | 225 | 225.00 | 2657.74 | 2658.53 |

Bates | 200 | 66.67 | 2971.88 | 2992.69 |

truncated normal | 200 | 87.96 | 2955.56 | 2983.30 |

uniform | 200 | 115.47 | 2931.45 | 2973.25 |

half-normal | 200 | 151.10 | 2849.89 | 2895.03 |

exponential | 200 | 200.00 | 2715.57 | 2714.77 |

Bates | 175 | 58.33 | 2992.52 | 3001.67 |

truncated normal | 175 | 76.97 | 2979.17 | 2994.49 |

uniform | 175 | 101.04 | 2959.05 | 2985.46 |

half-normal | 175 | 132.21 | 2892.68 | 2926.08 |

exponential | 175 | 175.00 | 2773.93 | 2771.59 |

Bates | 150 | 50.00 | 3012.26 | 3007.89 |

truncated normal | 150 | 65.97 | 3001.62 | 2999.43 |

uniform | 150 | 86.60 | 2985.26 | 2997.42 |

half-normal | 150 | 113.33 | 2933.63 | 2952.24 |

exponential | 150 | 150.00 | 2832.40 | 2831.59 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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**

Grzybowski, A.Z.; Domański, Z.; Derda, T.
Optimal Stopping Rules for Preventing Overloading of Multicomponent Systems. *Materials* **2023**, *16*, 2817.
https://doi.org/10.3390/ma16072817

**AMA Style**

Grzybowski AZ, Domański Z, Derda T.
Optimal Stopping Rules for Preventing Overloading of Multicomponent Systems. *Materials*. 2023; 16(7):2817.
https://doi.org/10.3390/ma16072817

**Chicago/Turabian Style**

Grzybowski, Andrzej Z., Zbigniew Domański, and Tomasz Derda.
2023. "Optimal Stopping Rules for Preventing Overloading of Multicomponent Systems" *Materials* 16, no. 7: 2817.
https://doi.org/10.3390/ma16072817