# Joint Optimization of Production Lot Sizing and Preventive Maintenance Threshold Based on Nonlinear Degradation

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## Abstract

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## 1. Introduction

- (1)
- Determining thresholds is a major challenge for maintenance management and this paper considers production lot sizing jointly with PM thresholds for more practicality.
- (2)
- Based on the nonlinear degradation of the system, a joint optimization model of production lot sizing and PM threshold is developed and a solution algorithm is given.

## 2. Problem Description

#### 2.1. System Specification

#### 2.2. Maintenance Scheduling

#### 2.3. Production–Maintenance Interaction

#### 2.4. Assumptions

- (1)
- The demand for all products is fixed and can be divided into small lots for production.
- (2)
- The production system will produce various products sequentially in a predetermined order.
- (3)
- Each product is produced only once in a production cycle and the production cycle is a complete run of all products produced according to their lot sizes.
- (4)
- Inspection time is negligible.
- (5)
- The magnitude of degradation does not change after the set up.
- (6)
- During the production process, the magnitude of degradation recovered by the system due to some protective measures is not considered.
- (7)
- CM results in a fixed cost of loss.
- (8)
- In case of failure, minimal repairs are always performed without changing the failed process and interrupting the production process.

## 3. Integrated Model

#### 3.1. Maintenance Situations

- 1.
- If there is neither CM nor PM at time ${t}_{i}$, it means that $X({t}_{i})<{L}_{p}$, neither PM nor CM is used at this time, so the degradation magnitude does not change either, $X({t}_{mt}^{i})=X({t}_{i})$, ${t}_{mt}^{i}={t}_{i}^{\prime}$. Therefore, the magnitude of degradation at time ${t}_{i+1}$ is $X({t}_{i+1})=X({t}_{mt}^{i})+\Delta X(\Delta {t}_{s})$. At time ${t}_{i+1}$, there are the following three sub-situations.
- (1)
- The system has neither CM nor PM: $X({t}_{i+1})=X({t}_{mt}^{i})+\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})<{L}_{p}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+\u25b3{c}_{i}={C}_{i}+{C}_{m}+{C}_{s}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}$, no change in the number of PM as ${r}_{i+1}={r}_{i}$, no change in the number of CM as ${f}_{i+1}={f}_{i}$.
- (2)
- The system performs PM only: ${L}_{p}\le X({t}_{i+1})=X({t}_{mt}^{i})+\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})<{L}_{f}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+\Delta {c}_{i}={C}_{i}+{C}_{m}+{C}_{s}+{C}_{p}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}+\u25b3{t}_{pm}$, the number of PM changes to ${r}_{i+1}={r}_{i}+1$, no change in the number of CM as ${f}_{i+1}={f}_{i}$.
- (3)
- The system performs CM only: $X({t}_{i+1})=X({t}_{mt}^{i})+\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})\ge {L}_{f}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+{C}_{m}+{C}_{s}+{C}_{f}+{C}_{l}+\u25b3{t}_{s}\cdot {C}_{\mathrm{min}}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}+\u25b3{t}_{cm}$, the number of CM changes to ${f}_{i+1}={f}_{i}+1$, no change in the number of PM as ${r}_{i+1}={r}_{i}$. The minimal repair cost is $\u25b3{t}_{s}\cdot {C}_{\mathrm{min}}$, which ensures that the system can continue to operate from ${t}_{mt}^{i}$ to ${t}_{i+1}$ even though ${L}_{f}$ is reached in $({t}_{mt}^{i},{t}_{i+1})$.

- 2.
- If PM is performed at time ${t}_{i}$, it means that ${L}_{p}\le X({t}_{i})<{L}_{f}$, the magnitude of degradation at time ${t}_{mt}^{i}$ after maintenance becomes $X({t}_{mt}^{i})$, which $X({t}_{mt}^{i})=\gamma $, ${t}_{mt}^{i}={t}_{i}^{\prime}+\u25b3{t}_{pm}$. Here, to ensure the effectiveness of PM, we assume that $\gamma =\Phi (\psi )\cdot Lp$, $\Psi ~N(0,1)$, $-1<\psi <1$. The magnitude of degradation at time ${t}_{i+1}$ is $X({t}_{i+1})=\gamma +\Delta X(\Delta {t}_{s})$. At time ${t}_{i+1}$, there are the following three sub-situations.
- (1)
- The system has neither CM nor PM: ${X}^{\prime}({t}_{i+1})=\gamma +\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})<{L}_{p}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+{C}_{m}+{C}_{s}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}$, no change in the number of PM as ${r}_{i+1}={r}_{i}$, no change in the number of CM as ${f}_{i+1}={f}_{i}$.
- (2)
- The system performs PM only: ${L}_{p}\le {X}^{\prime}({t}_{i+1})=\gamma +\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})<{L}_{f}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+{C}_{m}+{C}_{s}+{C}_{p}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}+\u25b3{t}_{pm}$. The number of PM changes to ${r}_{i+1}={r}_{i}+1$, no change in the number of CM as ${f}_{i+1}={f}_{i}$.
- (3)
- The system performs CM: ${X}^{\prime}({t}_{i+1})=\gamma +\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})\ge {L}_{f}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+{C}_{m}+{C}_{s}+{C}_{f}+{C}_{l}+\u25b3{t}_{s}\cdot {C}_{\mathrm{min}}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}+\u25b3{t}_{cm}$, the minimal repair cost is $\u25b3{t}_{s}\cdot {C}_{\mathrm{min}}$. The number of CM changes to ${f}_{i+1}={f}_{i}+1$, no change in the number of PM as ${r}_{i+1}={r}_{i}$.

- 3.
- If CM is performed at time ${t}_{i}$, it means that $X({t}_{i})\ge {L}_{f}$. After maintenance, the magnitude of degradation at time ${t}_{mt}^{i}$ becomes 0, which is $X({t}_{mt}^{i})=0$. The magnitude of degradation at time ${t}_{i+1}$ is $X({t}_{i+1})=X({t}_{mt}^{i})+\Delta X(\Delta {t}_{s})=\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})$. At time ${t}_{i+1}$, there are the following three sub-situations.
- (1)
- The system has neither CM nor PM: ${X}^{\prime}({t}_{i+1})=\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})<{L}_{p}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+{C}_{m}+{C}_{s}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}$, no change in the number of PM as ${r}_{i+1}={r}_{i}$, no change in the number of CM as ${f}_{i+1}={f}_{i}$.
- (2)
- The system performs PM only: ${L}_{p}\le {X}^{\prime}({t}_{i+1})=\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})<{L}_{f}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+{C}_{m}+{C}_{s}+{C}_{p}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}+\u25b3{t}_{pm}$. The number of PM changes to ${r}_{i+1}={r}_{i}+1$, no change in the number of CM as ${f}_{i+1}={f}_{i}$.
- (3)
- The system performs CM: ${X}^{\prime}({t}_{i+1})=\lambda ({t}_{i+1}{}^{b}-{({t}_{mt}^{i})}^{b})\ge {L}_{f}$. The cost at time ${t}_{i+1}$ is ${C}_{i+1}={C}_{i}+{C}_{m}+{C}_{s}+{C}_{f}+{C}_{l}+\u25b3{t}_{s}\cdot {C}_{\mathrm{min}}$, ${t}_{mt}^{i+1}={t}_{i+1}^{\prime}+\u25b3{t}_{cm}$, the minimal repair cost is $\u25b3{t}_{s}\cdot {C}_{\mathrm{min}}$. The number of CM changes to ${f}_{i+1}={f}_{i}+1$, no change in the number of PM as ${r}_{i+1}={r}_{i}$.

#### 3.2. Computation Algorithm for the Model

Algorithm 1. Computation algorithm for the model. |

1. Give the value range of $N$ and ${L}_{p}$. |

2. Assign $X({t}_{0})=0,t=0,TC=0,r=0,f=0$ |

3. Obtain $\Delta {t}_{s}$ by ${Q}_{s}/(u\cdot N)$, generate the production time of one cycle. |

4. for $j=0:N$ |

5. for $s=0:m$ |

6. Generate ${t}_{i}^{\prime}$ based on the required set-up time for each product. Determine which product should be produced at time ${t}_{mt}^{i}$. |

7. Determine the maintenance status at time ${t}_{i}^{\prime}$ based on the degradation quantity $X({t}_{i}^{\prime})$. |

8. Determine the relationship of $X({t}_{i}^{\prime})$ with ${L}_{p}$ and ${L}_{f}$, if $X({t}_{i}^{\prime})<{L}_{p}$, doing nothing, means no repair time. Generate $X({t}_{i+1})$ from $X({t}_{i}^{\prime})$ and $\Delta X$. Obtain the cost $C\left({t}_{i+1}\right)$, times of PM $r({t}_{i+1})$ and times of CM $f({t}_{i+1})$ at ${t}_{i+1}$. |

9. else if ${L}_{p}\le X({t}_{i}^{\prime})<{L}_{f}$, carry out PM, need to spend the corresponding PM time, then $X({t}_{i}^{\prime})$ into $X({t}_{mt}^{i})$, where $X({t}_{mt}^{i})=\gamma $. Generate $X({t}_{i+1})$ from $X({t}_{mt}^{i})$ and $\Delta X$. Obtain the cost $C\left({t}_{i+1}\right)$, times of PM $r({t}_{i+1})$ and times of CM $f({t}_{i+1})$ at ${t}_{i+1}$. |

10. else $X({t}_{i}^{\prime})\ge {L}_{f}$, carry out CM, need to spend the corresponding CM time, $X({t}_{i}^{\prime})$ into $X({t}_{mt}^{i})$, where $X({t}_{mt}^{i})=0$. Generate $X({t}_{i+1})$ from $\Delta X$. Obtain the cost $C\left({t}_{i+1}\right)$, times of PM $r({t}_{i+1})$ and times of CM $f({t}_{i+1})$ at ${t}_{i+1}$. |

11. end |

12. end |

13. Obtain the total cost $TC$, total number of PM $r$, total number of CM $f$. |

14. Calculate profit per unit of time. |

## 4. Numerical Example

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Abbreviations |

PM: Preventive Maintenance |

CM: Corrective Maintenance |

Notations |

$N$: Production lot sizing |

$m$: The system produces $m$ products in total |

${Q}_{s}$: Demand for the $s-\mathrm{th}$ product ($s=1,2,\cdots ,m$) |

${u}_{s}$: Productivity of the $s-\mathrm{th}$ product |

${d}_{s}$: Consumption rate of the $s-\mathrm{th}$ product |

$t$: Manufacturing system production time |

$\Delta {t}_{s}$: In a production cycle, the time required to produce the $s-\mathrm{th}$ product |

${t}_{i}$: The time when the system starts the $i+1-\mathrm{th}$ set-up $(i=0,1,\cdots m,m+1,\cdots ,n-1$$,{t}_{0}=0$) |

${t}_{i}^{\prime}$: The time when the system starts the $i+1-\mathrm{th}$ maintenance |

$\u25b3{t}_{s}^{\prime}$: The $s-\mathrm{th}$ product requires a system set-up time |

${t}_{mt}^{i}$: The time when the system starts producing the i+1-th product |

$\u25b3{t}_{mt}^{s}$: The $s-\mathrm{th}$ product requires system maintenance time |

${C}_{i}$: Costs incurred in $({t}_{0},{t}_{i})$ |

$\Delta {c}_{i}$: Costs incurred in $({t}_{i-1},{t}_{i})$$,{C}_{1}=\Delta {c}_{1}$ |

$X({t}_{k})$: The magnitude of degradation of the system at the time ${t}_{k}$$(k=0,1,2,\cdots m,m+1,\cdots ,n,X({t}_{0})=0)$ |

$\Delta X$: The increment of the degradation magnitude, which is $X({t}_{i+1})-X({t}_{i})$ |

$\gamma $: Degradation after PM |

${L}_{f}$: CM threshold |

${L}_{p}$: PM threshold |

${r}_{j}$: The number of PMs performed at the time ${t}_{j}$ |

$r$: The total number of PMs in the entire production process |

${f}_{j}$: The number of CMs at time ${t}_{j}$ |

$f$: The total number of CMs in the entire production process |

%: Remainder sign |

${C}_{s}$: The set-up cost of the $s-\mathrm{th}$ product |

${C}_{m}$: Inspection cost |

${C}_{\mathrm{min}}$: Average cost of minimal repair |

${C}_{f}$: Average cost of a CM cost |

${C}_{p}$: Average cost of a PM |

${C}_{l}$: The cost of loss caused by a CM |

${C}_{h}^{s}$: The average inventory holding cost per unit time of the $s-\mathrm{th}$ product |

${C}_{H}$: The inventory holding cost of the entire production and maintenance process |

$TC$: The total cost of the entire production and maintenance process |

${R}_{s}$: The gross profit of each product, which is equal to the unit sales price minus the unit production cost, excluding maintenance, repair and inventory costs |

$ER$: Net profit |

$UR$: Profit per unit time |

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Product | $\mathit{Q}$ | $\mathit{u}$ | ${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{h}}^{\mathit{s}}$ | ${\mathit{R}}_{\mathit{s}}$ |
---|---|---|---|---|---|

1 | 4500 | 50 | 200 | 0.31 | 380 |

2 | 2500 | 50 | 210 | 0.53 | 410 |

3 | 4000 | 80 | 205 | 0.42 | 400 |

4 | 3600 | 60 | 200 | 0.55 | 380 |

5 | 2000 | 50 | 220 | 0.34 | 420 |

6 | 3500 | 50 | 208 | 0.52 | 400 |

${\mathit{C}}_{\mathit{M}}$ | ${\mathit{C}}_{\mathit{p}}$ | ${\mathit{C}}_{\mathit{f}}$ | ${\mathit{C}}_{\mathit{l}}$ | ${\mathit{C}}_{\mathbf{min}}$ |
---|---|---|---|---|

100 | 500 | 5000 | 1000 | 200 |

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## Share and Cite

**MDPI and ACS Style**

Qu, L.; Liao, J.; Gao, K.; Yang, L.
Joint Optimization of Production Lot Sizing and Preventive Maintenance Threshold Based on Nonlinear Degradation. *Appl. Sci.* **2022**, *12*, 8638.
https://doi.org/10.3390/app12178638

**AMA Style**

Qu L, Liao J, Gao K, Yang L.
Joint Optimization of Production Lot Sizing and Preventive Maintenance Threshold Based on Nonlinear Degradation. *Applied Sciences*. 2022; 12(17):8638.
https://doi.org/10.3390/app12178638

**Chicago/Turabian Style**

Qu, Li, Junli Liao, Kaiye Gao, and Li Yang.
2022. "Joint Optimization of Production Lot Sizing and Preventive Maintenance Threshold Based on Nonlinear Degradation" *Applied Sciences* 12, no. 17: 8638.
https://doi.org/10.3390/app12178638