# Selective Maintenance Optimization for a Multi-State System Considering Human Reliability

^{*}

## Abstract

**:**

## 1. Introduction

- The MSS in this paper consists of multi-state components that are all repairable;
- All the maintenance activities are performed by one maintenance worker, and there is no maintenance activity during a mission; and
- The states of each component at the end of each mission are known.

## 2. Description of Multi-State System and Selective Maintenance Modeling

#### 2.1. Description of Multi-State System

- Do Nothing: for a component under this maintenance option, the state before and after the maintenance is unchanged, i.e., ${X}_{i}={Y}_{i}$.
- Imperfect Maintenance: for a component under this maintenance option, it will not be repaired to their best state after maintenance, although the maintenance worker performs some maintenance actions during the break, i.e., ${X}_{i}<{Y}_{i}<{K}_{i}$.
- Repair: If the target state ${Y}_{i}$ satisfies ${X}_{i}<{Y}_{i}={K}_{i}$, it is considered that this component is to be repaired. Under this option, components function perfectly after the maintenance break (without considering human error).

#### 2.2. Maintenance Time and Costs

#### 2.3. Maintenance Workers

## 3. Selective Maintenance Modeling Considering Human Reliability

#### 3.1. State Determination after Human Error

- If human error occurs in a multi-state component $i$, the true state after maintenance ${Z}_{i}$ is lower than the target state ($0\le {Z}_{i}<{Y}_{i}$). The probability distribution of ${Z}_{i}$ is related to the human reliability level of the worker who performs the task during this maintenance break;
- Experienced maintenance workers not only have lower HEP, but also have lower operational losses after human error occurs, i.e., the component has a higher probability of being in a better state when a human error occurs;
- When ${P}_{m}$ satisfies ${P}_{m}={P}_{f}$, the state of component $i$ after human error satisfies ${Z}_{i}=0$;
- Let ${P}_{b}$ denotes the transition rate between adjacent human reliability levels. The probability distribution of ${Z}_{i}$ is determined by ${P}_{b}$ and ${P}_{m}$.

#### 3.2. Estimation of Component State Degradation and Multi-State System Reliability

#### 3.3. Optimization Model

## 4. Case Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notation

$N$ | total number of components in an MSS |

$i$ | index of components, $i=1,2,\dots ,N$ |

$L$ | number of subsystems in an MSS |

$l$ | index of subsystems, $l=1,2,\dots ,L$ |

${N}_{l}$ | number of components in subsystem $l$ |

${S}_{i}$ | the state set of the component $i$, ${S}_{i}=(0,1,\dots ,{K}_{i}-1,{K}_{i})$ |

$\Phi $ | composition operator for all components |

$X$ | states vector set of all components before maintenance, $X=\{{X}_{1},{X}_{2},\dots ,{X}_{N}\}$ |

$Y$ | states vector set of all components after maintenance, $Y=\{{Y}_{1},{Y}_{2},\dots ,{Y}_{N}\}$ |

$J$ | states vector set of all components at the end of the next mission, $J=\{{j}_{1},{j}_{2},\dots ,{j}_{N}\}$ |

${Z}_{i}$ | state of component $i$ after human error occurs, ${X}_{i}\le {Z}_{i}<{Y}_{i}$ |

${T}_{i}$ | maintenance time of component $i$ during maintenance break |

${C}_{i}$ | maintenance costs of component $i$ during maintenance break |

${T}_{L}$ | maintenance time limit |

${C}_{L}$ | maintenance costs limit |

${g}_{i,j}$ | performance rate of component $i$ in state $j$, $j=0,1,2,\dots {K}_{i}$ |

${g}_{i}(t)$ | performance rate of component $i$ at time $t$ |

${G}_{l}(t)$ | performance rate of subsystem $l$ at time $t$ |

$G(t)$ | performance rate of an MSS at time $t$ |

$M$ | number of maintenance workers available |

$m$ | index of maintenance workers, $m=1,2,\dots ,M$ |

${P}_{m}$ | human error probability of maintenance worker $m$ |

${P}_{{Y}_{i},j}^{i}(t,m)$ | the probability of component $i$ degrading from state ${Y}_{i}$ to state ${j}_{i}$ at time $t$ after maintenance by worker $m$, ${j}_{i}=0,1,\dots ,{Y}_{i}$ |

${u}_{i,{Y}_{i}}(v,t)$ | universal generating function for component $i$ in state ${Y}_{i}$ at time $t$ |

${U}_{Y}(v,t)$ | universal generating function for an MSS in state $Y$ at time $t$ |

$T$ | time required to perform next mission |

$W$ | the requirement of performance rate at the end of next mission |

${R}_{MSS}$ | reliability of MSS to perform the next mission considering human reliability |

${\widehat{R}}_{MSS}$ | reliability of MSS to perform the next mission without considering human reliability |

${R}_{L}$ | minimum reliability required for next mission |

$ME$ | number of years of maintenance experience |

$MQ$ | total number of human errors |

$MT$ | influence of maintenance environment on maintenance time |

$MC$ | the state difference before and after maintenance (${Y}_{i}-{X}_{i}$) |

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Level | Human Error Probability | Probability Ranking |
---|---|---|

0 | $0.4<{P}_{m}\le 0.8$ | $P({Z}_{i}=0)>P({Z}_{i}=1)=P({Z}_{i}=2)=P({Z}_{i}=3)=P({Z}_{i}=4)$ |

1 | $0.2<{P}_{m}\le 0.4$ | $P({Z}_{i}=0)>P({Z}_{i}=1)>P({Z}_{i}=2)=P({Z}_{i}=3)=P({Z}_{i}=4)$ |

2 | $0.1<{P}_{m}\le 0.2$ | $P({Z}_{i}=0)>P({Z}_{i}=1)>P({Z}_{i}=2)>P({Z}_{i}=3)=P({Z}_{i}=4)$ |

3 | $0.05<{P}_{m}\le 0.1$ | $P({Z}_{i}=1)>P({Z}_{i}=0)>P({Z}_{i}=2)>P({Z}_{i}=3)>P({Z}_{i}=4)$ |

4 | $0.025<{P}_{m}\le 0.5$ | $P({Z}_{i}=1)>P({Z}_{i}=2)>P({Z}_{i}=0)>P({Z}_{i}=3)>P({Z}_{i}=4)$ |

5 | $0.0125<{P}_{m}\le 0.025$ | $P({Z}_{i}=1)>P({Z}_{i}=2)>P({Z}_{i}=0)>P({Z}_{i}=3)>P({Z}_{i}=4)$ |

6 | $0.00625<{P}_{m}\le 0.0125$ | $P({Z}_{i}=2)>P({Z}_{i}=1)>P({Z}_{i}=3)>P({Z}_{i}=0)>P({Z}_{i}=4)$ |

7 | $3.125\times {10}^{-3}<{P}_{m}\le 0.00625$ | $P({Z}_{i}=2)>P({Z}_{i}=1)>P({Z}_{i}=3)>P({Z}_{i}=4)>P({Z}_{i}=0)$ |

8 | $1.56\times {10}^{-3}<{P}_{m}\le 3.125\times {10}^{-3}$ | $P({Z}_{i}=2)>P({Z}_{i}=3)>P({Z}_{i}=1)>P({Z}_{i}=4)>P({Z}_{i}=0)$ |

9 | $7.8\times {10}^{-4}<{P}_{m}\le 1.56\times {10}^{-3}$ | $P({Z}_{i}=4)>P({Z}_{i}=3)=P({Z}_{i}=2)>P({Z}_{i}=1)>P({Z}_{i}=0)$ |

10–14 | ${P}_{m}\le 7.8\times {10}^{-4}$ | $P({Z}_{i}=4)>P({Z}_{i}=3)>P({Z}_{i}=2)>P({Z}_{i}=1)>P({Z}_{i}=0)$ |

${\mathit{T}}_{\mathit{L}}\left(\mathbf{Units}\right)$ | ${\mathit{C}}_{\mathit{L}}\left(\mathbf{Units}\right)$ | $\mathit{M}$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{P}}_{\mathit{f}}$ | ${\mathit{P}}_{\mathit{b}}$ | $\mathit{W}$ | $\mathit{\alpha}$ |
---|---|---|---|---|---|---|---|---|

540 | 185 | 3 | 0.50 | 0.50 | 0.50 | 0.30 | 20 | 0.97 |

Component ID | Component Information | ||||||
---|---|---|---|---|---|---|---|

State | Performance Rate | Maintenance Time/Costs (Units) | |||||

0 | 1 | 2 | 3 | 4 | |||

1 | 0 | 0 | 0/0 | 38/7 | 64/13 | 91/27 | 121/40 |

1 | 20 | - | 0/0 | 26/6 | 53/20 | 83/33 | |

2 | 40 | - | - | 0/0 | 27/14 | 57/27 | |

3 | 65 | - | - | - | 0/0 | 30/13 | |

4 | 95 | - | - | - | - | 0/0 | |

2 | 0 | 0 | 0/0 | 32/9 | 56/20 | 76/31 | - |

1 | 30 | - | 0/0 | 24/11 | 44/22 | - | |

2 | 50 | - | - | 0/0 | 20/11 | - | |

3 | 70 | 0/0 | - | ||||

3 | 0 | 0 | 0/0 | 25/8 | 48/16 | 83/20 | - |

1 | 25 | - | 0/0 | 23/8 | 58/4 | - | |

2 | 45 | - | - | 0/0 | 35/4 | - | |

3 | 70 | - | - | - | 0/0 | - | |

4 | 0 | 0 | 0/0 | 33/12 | 68/25 | 108/40 | 140/51 |

1 | 40 | - | 0/0 | 35/13 | 75/28 | 107/39 | |

2 | 75 | - | - | 0/0 | 40/15 | 72/26 | |

3 | 90 | - | - | - | 0/0 | 32/11 | |

4 | 125 | - | - | - | - | 0/0 | |

5 | 0 | 0 | 0/0 | 19/6 | 36/10 | 55/15 | - |

1 | 20 | - | 0/0 | 17/4 | 36/9 | - | |

2 | 35 | - | - | 0/0 | 19/5 | - | |

3 | 50 | - | - | - | 0/0 | - | |

6 | 0 | 0 | 0/0 | 22/8 | 44/17 | 67/26 | - |

1 | 25 | - | 0/0 | 22/9 | 45/18 | - | |

2 | 35 | - | - | 0/0 | 23/9 | - | |

3 | 55 | - | - | - | 0/0 | - | |

7 | 0 | 0 | 0/0 | 15/3 | 31/7 | 45/11 | - |

1 | 15 | - | 0/0 | 16/4 | 30/8 | - | |

2 | 25 | - | - | 0/0 | 14/4 | - | |

3 | 40 | - | - | - | 0/0 | - | |

8 | 0 | 0 | 0/0 | 23/7 | 44/15 | 63/23 | - |

1 | 30 | - | 0/0 | 21/8 | 40/16 | - | |

2 | 50 | - | - | 0/0 | 19/8 | - | |

3 | 75 | - | - | - | 0/0 | - | |

9 | 0 | 0 | 0/0 | 31/10 | 60/19 | 94/30 | - |

1 | 25 | - | 0/0 | 29/9 | 63/20 | - | |

2 | 40 | - | - | 0/0 | 34/11 | - | |

3 | 55 | - | - | 0/0 | - | ||

10 | 0 | 0 | 0/0 | 32/12 | 65/23 | 99/33 | 140/45 |

1 | 35 | - | 0/0 | 33/11 | 69/21 | 110/33 | |

2 | 60 | - | - | 0/0 | 36/10 | 77/22 | |

3 | 95 | - | - | - | 0/0 | 41/12 | |

4 | 115 | - | - | - | - | 0/0 |

Com ID | Initial State | State after Maintenance | Degradation Probability | ||||
---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | |||

1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

1 | 0.20 | 0.80 | 0 | 0 | 0 | ||

2 | 0.15 | 0.24 | 0.61 | 0 | 0 | ||

3 | 0.05 | 0.15 | 0.28 | 0.52 | 0 | ||

4 | 0.02 | 0.09 | 0.14 | 0.26 | 0.49 | ||

2 | 1 | 0 | 1 | 0 | 0 | 0 | - |

1 | 0.30 | 0.70 | 0 | 0 | - | ||

2 | 0.12 | 0.22 | 0.66 | 0 | - | ||

3 | 0.05 | 0.11 | 0.27 | 0.57 | |||

3 | 1 | 0 | 1 | 0 | 0 | 0 | - |

1 | 0.13 | 0.87 | 0 | 0 | - | ||

2 | 0.08 | 0.32 | 0.60 | 0 | - | ||

3 | 0.06 | 0.24 | 0.34 | 0.36 | |||

4 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |

1 | 0.17 | 0.83 | 0 | 0 | 0 | ||

2 | 0.09 | 0.16 | 0.75 | 0 | 0 | ||

3 | 0.05 | 0.11 | 0.21 | 0.63 | 0 | ||

4 | 0.01 | 0.04 | 0.11 | 0.24 | 0.60 | ||

5 | 0 | 0 | 1 | 0 | 0 | 0 | - |

1 | 0.42 | 0.48 | 0 | 0 | - | ||

2 | 0.27 | 0.35 | 0.38 | 0 | - | ||

3 | 0.16 | 0.22 | 0.29 | 0.33 | - | ||

6 | 1 | 0 | 1 | 0 | 0 | 0 | - |

1 | 0.30 | 0.70 | 0 | 0 | - | ||

2 | 0.16 | 0.24 | 0.60 | 0 | - | ||

3 | 0.08 | 0.12 | 0.35 | 0.55 | - | ||

7 | 2 | 0 | 1 | 0 | 0 | 0 | - |

1 | 0.35 | 0.65 | 0 | 0 | - | ||

2 | 0.22 | 0.31 | 0.47 | 0 | - | ||

3 | 0.14 | 0.20 | 0.29 | 0.37 | - | ||

8 | 1 | 0 | 1 | 0 | 0 | 0 | - |

1 | 0.44 | 0.56 | 0 | 0 | - | ||

2 | 0.18 | 0.38 | 0.44 | 0 | - | ||

3 | 0.03 | 0.09 | 0.30 | 0.58 | - | ||

9 | 0 | 0 | 1 | 0 | 0 | 0 | - |

1 | 0.14 | 0.86 | 0 | 0 | - | ||

2 | 0.12 | 0.25 | 0.63 | 0 | - | ||

3 | 0.08 | 0.14 | 0.27 | 0.51 | - | ||

10 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |

1 | 0.27 | 0.73 | 0 | 0 | 0 | ||

2 | 0.15 | 0.23 | 0.62 | 0 | 0 | ||

3 | 0.06 | 0.12 | 0.20 | 0.62 | 0 | ||

4 | 0.01 | 0.03 | 0.13 | 0.18 | 0.65 |

Worker ID | Parameter | |||||
---|---|---|---|---|---|---|

$\mathit{M}\mathit{E}$ | $\mathit{M}\mathit{Q}$ | $\mathit{V}$ | $\mathit{M}\mathit{T}$ | ${\mathit{E}}_{\mathit{d}}$ | ${\mathit{P}}_{\mathit{m}}$ | |

A | 25 | 1 | 120 | 0.95 | 4 | 0.0166 |

B | 15 | 5 | 60 | 0.90 | 3 | 0.0544 |

C | 5 | 5 | 35 | 0.80 | 3 | 0.1174 |

Population Size | Number of Iterations | Mutation Probability | Crossover Probability | Generation Gap |
---|---|---|---|---|

100 | 30 | 0.01 | 0.7 | 0.9 |

Component ID | Target State | Option | Time (Units) | Costs (Units) | ${\widehat{\mathit{R}}}_{\mathit{M}\mathit{S}\mathit{S}}$ | ${\mathit{R}}_{\mathit{L}}$ | Reliability Considering Human Reliability | ||
---|---|---|---|---|---|---|---|---|---|

A | B | C | |||||||

1 | 4 | Repair | 533 | 182 | 0.9316 | 0.9037 | 0.9239 | 0.8948 | 0.8391 |

2 | 2 | Imperfect Maintenance | |||||||

3 | 1 | Do Nothing | |||||||

4 | 4 | Repair | |||||||

5 | 3 | Repair | |||||||

6 | 3 | Repair | |||||||

7 | 2 | Imperfect Maintenance | |||||||

8 | 3 | Repair | |||||||

9 | 1 | Imperfect Maintenance | |||||||

10 | 4 | Repair |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Z.; Xiao, B.; Wang, N.; Yan, X.; Ma, L.
Selective Maintenance Optimization for a Multi-State System Considering Human Reliability. *Symmetry* **2019**, *11*, 652.
https://doi.org/10.3390/sym11050652

**AMA Style**

Zhao Z, Xiao B, Wang N, Yan X, Ma L.
Selective Maintenance Optimization for a Multi-State System Considering Human Reliability. *Symmetry*. 2019; 11(5):652.
https://doi.org/10.3390/sym11050652

**Chicago/Turabian Style**

Zhao, Zhonghao, Boping Xiao, Naichao Wang, Xiaoyuan Yan, and Lin Ma.
2019. "Selective Maintenance Optimization for a Multi-State System Considering Human Reliability" *Symmetry* 11, no. 5: 652.
https://doi.org/10.3390/sym11050652