# Machine Downtime Effect on the Warm-Up Period in an Economic Production Quantity Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Definition

#### 3.1. Scrap Model

#### 3.2. Solution Algorithm for Scrap Model

- Forward-solving the model is required. Starting with the lowest warm-up period, calculate the values of ${t}_{jL}$, ${t}_{jU}$, and ${T}_{j}$ from Equations (18), (19), and (21), respectively.
- If ${T}_{j}$ is obtained from ${w}_{j}$, stop at the desired distance, i.e., the allowed distance, ${t}_{j}$ (${t}_{jL},{t}_{jU}$). Otherwise, go to the next time, that is, ${t}_{\left(j+1\right)}$, which is greater than ${t}_{j}$.
- Whenever ${T}_{j}$ is at the allowed distance, determine the total cost. Then, calculate the exact cost for the left breakpoint at that point, i.e., for the time ${t}_{\left(j-1\right)}$. The lowest cost is the problem’s optimal answer.

#### 3.3. Rework Model

#### 3.4. Solution Algorithm for Rework Model

- Forward-solving the model is required. Starting with the lowest warm-up period, calculate the values of ${t}_{jL}$, ${t}_{jU}$, and ${T}_{j}$ from Equations (37), (38), and (40), respectively.
- If ${T}_{j}$ is obtained from ${w}_{j}$, stop at the desired distance, i.e., the allowed distance, ${t}_{j}$ (${t}_{jL},{t}_{jU}$). Otherwise, go to the next time, that is, ${t}_{\left(j+1\right)}$, which is greater than ${t}_{j}$.
- Whenever ${T}_{j}$ is at the allowed distance, determine the total cost, calculate the same cost for the left breakpoint at that point, i.e., for the time ${t}_{\left(j-1\right)}$. The lowest cost is the problem’s optimal solution.

## 4. Numerical Examples

#### 4.1. Numerical Example for Scrap Model

#### 4.2. Numerical Example for Rework Model

## 5. Sensitivity Analysis

## 6. Managerial and Environmental Insights

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Partial Derivations of the Total Cost Equation (14)

## Appendix B. Partial Derivations of the Total Cost Equation (35)

## References

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**Figure 3.**The scrap model total cost versus cycle length for each breakpoint. Bold line: permissible limits of each range, *: the answers for each range are obtained from Equation (21).

**Figure 4.**The rework model’s total cost versus the cycle length for each breakpoint. Bold line: permissible limits of each range, *: the answers for each range are obtained from Equation (40).

**Figure 6.**The effects of the system parameters of the numerical example on the total cost of the first model.

**Figure 8.**The effects of the system parameters of the numerical example on the total cost of the second model.

**Figure 9.**The effects of the cost of the numerical example on the difference between the two models.

**Figure 10.**The effects of the system parameters of the numerical example on the difference between the two models.

Paper | Objective Function | Warm-Up Period | Defective Items | Solution Method | Warm-Up-Dependent Machine Downtime | |||||
---|---|---|---|---|---|---|---|---|---|---|

Cost | Profit | Fixed | Variable | Scrap | Rework | Closed-Form | Analytical | Meta-Heuristic | ||

Nobil et al. [24] | * | * | * | * | ||||||

Nobil et al. [30] | * | * | * | |||||||

Ganesan and Uthayakumar [31] | * | * | * | * | * | |||||

Ganesan and Uthayakumar [32] | * | * | * | * | * | |||||

Nobil et al. [33] | * | * | * | * | ||||||

This study | * | * | * | * | * | * |

Range Number | Downtime Range | Warm-Up Time |
---|---|---|

0 | ${l}_{0}\le {t}_{d}<{l}_{1}$ | ${w}_{0}$ |

1 | ${l}_{1}\le {t}_{d}<{l}_{2}$ | ${w}_{1}$ |

2 | ${l}_{2}\le {t}_{d}<{l}_{3}$ | ${w}_{2}$ |

$\vdots $ | $\vdots $ | $\vdots $ |

$j$ | ${l}_{j}\le {t}_{d}<{l}_{j+1}$ | ${w}_{j}$ |

$\vdots $ | $\vdots $ | $\vdots $ |

$n$ | ${l}_{n}\le {t}_{d}$ | ${w}_{n}$ |

Range Number | Downtime Range | Cycle Length Range | Warm-Up Time |
---|---|---|---|

0 | ${l}_{0}\le {t}_{d}\le {l}_{1}$ | ${t}_{0L}\le T<{t}_{0U}$ | ${w}_{0}$ |

1 | ${l}_{1}\le {t}_{d}<{l}_{2}$ | ${t}_{1L}\le T<{t}_{1U}$ | ${w}_{1}$ |

2 | ${l}_{2}\le {t}_{d}<{l}_{3}$ | ${t}_{2L}\le T<{t}_{2U}$ | ${w}_{2}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

$j$ | ${l}_{j}\le {t}_{d}<{l}_{j+1}$ | ${t}_{jL}\le T<{t}_{jU}$ | ${w}_{j}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

$n$ | ${l}_{n}\le {t}_{d}$ | ${t}_{nL}\le T$ | ${w}_{n}$ |

Range Number | Downtime Range | Warm-Up Time |
---|---|---|

0 | ${t}_{d}<0.28$ | ${w}_{0}=0.010$ |

1 | $0.28\le {t}_{d}<0.38$ | ${w}_{1}=0.020$ |

2 | $0.38\le {t}_{d}<0.50$ | ${w}_{2}=0.030$ |

3 | ${t}_{d}\ge 0.50$ | ${w}_{3}=0.035$ |

Range Number | Downtime Range | Cycle Time Range | Warm-Up Time |
---|---|---|---|

0 | ${t}_{d}<0.28$ | $T<0.45$ | ${w}_{0}=0.010$ |

1 | $0.28\le {t}_{d}<0.38$ | $0.45\le T<0.62$ | ${w}_{1}=0.020$ |

2 | $0.38\le {t}_{d}<0.50$ | $0.62\le T<0.81$ | ${w}_{2}=0.030$ |

3 | ${t}_{d}\ge 0.50$ | $T\ge 0.81$ | ${w}_{3}=0.035$ |

Range Number | Downtime Range | Cycle Time Range | Warm-Up Time |
---|---|---|---|

0 | ${t}_{d}<0.28$ | $T<0.44$ | ${w}_{0}=0.010$ |

1 | $0.28\le {t}_{d}<0.38$ | $0.44\le T<0.60$ | ${w}_{1}=0.020$ |

2 | $0.38\le {t}_{d}<0.50$ | $0.61\le T<0.80$ | ${w}_{2}=0.030$ |

4 | ${t}_{d}\ge 0.50$ | $T\ge 0.80$ | ${w}_{3}=0.035$ |

Change in % | Model I with Scrap Items | Model II with Rework | Z(1)* − Z(2)* | |||||
---|---|---|---|---|---|---|---|---|

Breakpoint of ${\mathit{w}}_{\mathit{j}}$ | $\mathit{T}{\left(1\right)}^{\ast}(\%)$ | $\mathit{Z}{\left(1\right)}^{\ast}(\%)$ | Breakpoint of ${\mathit{w}}_{\mathit{j}}$ | $\mathit{T}{\left(2\right)}^{\ast}(\%)$ | $\mathit{Z}{\left(2\right)}^{\ast}(\%)$ | |||

Initial | 2 | 0 | 0 | 2 | 0 | 0 | 3072.317 | |

$A$ | 50 | 3 | 31.965 | 1.030 | 4 | 32.437 | 1.059 | 3095.999 |

25 | 2 | 0 | 0.528 | 2 | 0 | 0.599 | 3069.011 | |

−25 | 1 | −26.812 | −0.641 | 2 | −2.194 | −0.600 | 3041.354 | |

−50 | 1 | −26.812 | −1.363 | 1 | −26.784 | −1.299 | 3012.737 | |

$m$ | 50 | 2 | 0 | 0.264 | 2 | 0 | 0.299 | 3070.664 |

25 | 2 | 0 | 0.132 | 2 | 0 | 0.149 | 3071.491 | |

−25 | 2 | 0 | −0.132 | 2 | 0 | −0.149 | 3073.144 | |

−50 | 1 | −26.812 | −0.280 | 2 | 0 | −0.299 | 3069.049 | |

$c$ | 50 | 1 | −26.812 | 45.526 | 2 | 0 | 45.249 | 4547.566 |

25 | 2 | 0 | 22.769 | 2 | 0 | 22.624 | 3811.821 | |

−25 | 2 | 0 | −22.769 | 2 | 0 | −22.624 | 2332.813 | |

−50 | 2 | 0 | −45.538 | 2 | 0 | −45.249 | 1593.310 | |

$v$ | 50 | 2 | 0 | 1.927 | 2 | 0 | 1.929 | 3130.816 |

25 | 2 | 0 | 0.963 | 2 | 0 | 0.964 | 3101.567 | |

−25 | 2 | 0 | −0.963 | 2 | 0 | −0.964 | 3043.068 | |

−50 | 2 | 0 | −1.927 | 2 | 0 | −1.929 | 3013.818 | |

$h$ | 50 | 1 | −26.812 | 0.980 | 2 | −11.519 | 1.267 | 3022.997 |

25 | 1 | −26.812 | 0.530 | 2 | −3.086 | 0.658 | 3053.375 | |

−25 | 2 | 0 | −0.606 | 4 | 32.437 | −0.709 | 3082.212 | |

−50 | 4 | 79.738 | −1.437 | 4 | 57.448 | −1.638 | 3083.884 |

***:**Refers to the optimal value.

Change in % | Model I with Scrap Items | Model II with Rework | Z(1)* − Z(2)* | |||||
---|---|---|---|---|---|---|---|---|

$\mathbf{Breakpoint}\mathbf{of}{\mathit{w}}_{\mathit{j}}$ | $\mathit{T}{\left(1\right)}^{\ast}(\%)$ | $\mathit{Z}{\left(1\right)}^{\ast}(\%)$ | $\mathbf{Breakpoint}\mathbf{of}{\mathit{w}}_{\mathit{j}}$ | $\mathit{T}{\left(2\right)}^{\ast}(\%)$ | $\mathit{Z}{\left(2\right)}^{\ast}(\%)$ | |||

Initial | 2 | 0 | 0 | 2 | 0 | 0 | 3072.317 | |

${p}_{1}$ | 0.50 | 1 | −27.576 | 0.427 | 2 | −1.590 | 0.214 | 3144.201 |

0.25 | 2 | −0.763 | 0.252 | 2 | −0.795 | 0.106 | 3120.303 | |

−0.25 | 2 | 0.763 | −0.249 | 2 | 0.795 | −0.106 | 3025.045 | |

−0.50 | 2 | 1.526 | −0.495 | 2 | 1.590 | −0.211 | 2978.469 | |

${p}_{2}$ | 0.50 | 2 | −15.542 | 0.613 | 2 | −13.782 | 0.542 | 3110.980 |

0.25 | 2 | −9.979 | 0.369 | 2 | −8.788 | 0.328 | 3095.127 | |

−0.25 | 1 | −9.595 | −0.701 | 2 | 19.555 | −0.572 | 3015.190 | |

−0.50 | infeasible | |||||||

$r$ | 0.50 | 2 | 0 | 0 | 2 | −1.366 | 0.011 | 3069.206 |

0.25 | 2 | 0 | 0 | 2 | −0.824 | 0.006 | 3070.482 | |

−0.25 | 2 | 0 | 0 | 2 | 1.402 | −0.010 | 3075.160 | |

−0.50 | 2 | 0 | 0 | 2 | 4.322 | −0.027 | 3080.003 | |

$d$ | 0.50 | 1 | 3.681 | 46.807 | 1 | 1.578 | 47.133 | 4420.445 |

0.25 | 1 | −14.194 | 23.444 | 2 | 4.876 | 23.678 | 3727.799 | |

−0.25 | 3 | 15.046 | −23.659 | 3 | 15.779 | −23.891 | 2409.420 | |

−0.50 | 4 | 58.080 | −47.864 | 4 | 38.583 | −48.100 | 1666.951 | |

${\alpha}_{1}$ | 0.50 | 1 | −26.622 | 0.625 | 2 | 0.257 | 0.217 | 3204.230 |

0.25 | 1 | −26.717 | 0.353 | 2 | 0.128 | 0.108 | 3150.737 | |

−0.25 | 2 | −0.190 | −0.400 | 2 | −0.128 | −0.109 | 2979.539 | |

−0.50 | 3 | 2.325 | −0.844 | 3 | 4.3115 | −0.266 | 2886.611 | |

${\alpha}_{2}$ | 0.50 | 2 | 3.398 | 6.968 | 2 | 1.855 | 1.606 | 4767.729 |

0.25 | 2 | 1.620 | 3.381 | 2 | 0.918 | 0.803 | 3888.441 | |

−0.25 | 2 | −1.483 | −3.193 | 2 | −0.900 | −0.803 | 2314.082 | |

−0.50 | 1 | −28.932 | −6.220 | 2 | −1.784 | −1.607 | 1606.814 | |

${l}_{j}$ | 0.50 | 1 | 9.255 | −0.384 | 1 | 6.276 | −0.074 | 2974.806 |

0.25 | 1 | −8.778 | −0.288 | 2 | 8.339 | −0.019 | 2989.180 | |

−0.25 | 4 | 27.159 | 0.356 | 4 | 11.411 | 0.062 | 3164.379 | |

−0.50 | 4 | 27.159 | 0.356 | 4 | 11.411 | 0.062 | 3164.379 |

***:**Refers to the optimized value.

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## Share and Cite

**MDPI and ACS Style**

Nobil, E.; Cárdenas-Barrón, L.E.; Garza-Núñez, D.; Treviño-Garza, G.; Céspedes-Mota, A.; Loera-Hernández, I.d.J.; Smith, N.R.; Nobil, A.H.
Machine Downtime Effect on the Warm-Up Period in an Economic Production Quantity Problem. *Mathematics* **2023**, *11*, 1740.
https://doi.org/10.3390/math11071740

**AMA Style**

Nobil E, Cárdenas-Barrón LE, Garza-Núñez D, Treviño-Garza G, Céspedes-Mota A, Loera-Hernández IdJ, Smith NR, Nobil AH.
Machine Downtime Effect on the Warm-Up Period in an Economic Production Quantity Problem. *Mathematics*. 2023; 11(7):1740.
https://doi.org/10.3390/math11071740

**Chicago/Turabian Style**

Nobil, Erfan, Leopoldo Eduardo Cárdenas-Barrón, Dagoberto Garza-Núñez, Gerardo Treviño-Garza, Armando Céspedes-Mota, Imelda de Jesús Loera-Hernández, Neale R. Smith, and Amir Hossein Nobil.
2023. "Machine Downtime Effect on the Warm-Up Period in an Economic Production Quantity Problem" *Mathematics* 11, no. 7: 1740.
https://doi.org/10.3390/math11071740