# Lot-Sizing and Scheduling for the Plastic Injection Molding Industry—A Hybrid Optimization Approach

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Mathematical Model

- $m:$ Number of molds
- $n:$ Number of jobs
- $r:$ Number of machines
- ${f}_{j}:$ Mold for job $j$
- ${d}_{j}:$ Due date of job $j$
- ${p}_{j}:$ Processing time of job $j$
- ${A}_{j}:$ Time to mount the mold of job $j$ on the machine
- ${D}_{j}:$ Time to dismount the mold of job $j$
- ${M}_{jk}$: The setup time when job $k$ follows job $j$ on a machine. This setup time is equal to the time to dismount the mold used by $j$ (${D}_{j}$) plus the time to mount the mold used by $k$ (${A}_{k}$), if $j$ and $k$ require different molds. Thus:${M}_{jk}={A}_{k}+{D}_{j}$ if ${f}_{j}\ne {f}_{k}$0 otherwise
- $C{F}_{jb}$: Binary variable denoting if job $j$ is compatible with machine $b$:
- $C{F}_{jb}=\{\begin{array}{c}1\mathrm{if}\mathrm{job}j\mathrm{is}\mathrm{compatible}\mathrm{with}\mathrm{machine}b\\ 0\mathrm{otherwise}\end{array}$
- $G$: “Big M,” sufficiently large number to make the problem solvable by linear programming. In our case, $G$ should be much higher than the ending time of any job.

- ${C}_{j}:$ Completion time of job $j$
- ${T}_{j}:$ Tardiness of job $j$
- ${\alpha}_{jb}$: Binary variable denoting if $j$ is the first job to be processed on machine $b$:${\alpha}_{jb}=\{\begin{array}{c}1\mathrm{if}\mathrm{job}j\mathrm{is}\mathrm{the}\mathrm{first}\mathrm{to}\mathrm{be}\mathrm{processed}\mathrm{on}\mathrm{machine}b\\ 0\mathrm{otherwise}\end{array}$
- ${\theta}_{jk}$: Binary variable denoting if job $k$ is scheduled right after job $j$:${\theta}_{jk}=\{\begin{array}{c}1\mathrm{if}\mathrm{job}k\mathrm{is}\mathrm{immediately}\mathrm{processed}\mathrm{after}\mathrm{job}j\\ 0\mathrm{otherwise}\end{array}$
- ${\beta}_{jb}$: Binary variable denoting if job $j$ is processed on machine $b$ but not in the first place:${\beta}_{jb}=\{\begin{array}{c}1\mathrm{if}\mathrm{job}j\mathrm{is}\mathrm{processed}\mathrm{on}\mathrm{machine}b\mathrm{but}\mathrm{not}\mathrm{on}\mathrm{the}\mathrm{first}\mathrm{place}\\ 0\mathrm{otherwise}\end{array}$

## 4. Proposed Solution Approach

#### 4.1. List Algorithm

Algorithm 1 List algorithm for the injection problem |

Data: List of jobs ${({\sigma}_{i})}_{i\in N}$ |

forallJob${\sigma}_{i}$do |

Order the machines according to their release date: $R\left[i\right]$ |

j: = 0 |

while Job ${\sigma}_{i}$ not assigned AND$j<N{b}_{machines}$ do |

if Job ${\sigma}_{i}$ and machine$R\left[j\right]$ compatible with the mold then |

if Needed mold available then |

Release date machine $R\left[j\right]$: = Release date machine $R\left[j\right]$ + processing time of Job ${\sigma}_{i}$ + setup time |

Release date needed mold: = Release date machine $R\left[j\right]$ |

Assign Job ${\sigma}_{i}$ to machine $R\left[j\right]$ |

else |

Find on which machine the needed mold is used: MachineUsing |

If the mold currently used in MachineUsing is the one needed by Job ${\sigma}_{i}$ then |

Release date MachineUsing: = Release date MachineUsing + processing time of Job ${\sigma}_{i}$ |

else |

Release date MachineUsing: = Release date MachineUsing + processing time of Job ${\sigma}_{i}$ + setup time |

Release date needed mold: = Release date machine MachineUsing |

Assign Job ${\sigma}_{i}$ to machine MachineUsing |

else |

Next machine: $j\u2254\left(j+1\right)\%N{b}_{machines}$ |

#### 4.2. Metaheuristic

#### 4.2.1. Tuning

#### 4.2.2. Stochastic Descent

Algorithm 2 Principle algorithm of stochastic descent |

Data: Initial solution $Y$ |

$X\u2254L\left(Y\right)$ |

whilenecessarydo |

choose uniformly and randomly ${Y}^{\prime}\in V\left(Y\right)$ |

${X}^{\prime}\u2254L\left({Y}^{\prime}\right)$ |

if $H\left({X}^{\prime}\right)\le H\left(X\right)$ then |

$Y\u2254{Y}^{\prime}$ |

#### 4.2.3. Simulated Annealing

Algorithm 3 Principle algorithm of simulated annealing |

Data: Initial solution $Y$, Temperature ${T}_{0}$, Decreasing factor $\alpha $ |

$T\u2254{T}_{0}$ |

$X\u2254L\left(Y\right)$ |

whilenecessarydo |

choose uniformly and randomly ${Y}^{\prime}\in V\left(Y\right)$ |

${X}^{\prime}\u2254L\left({Y}^{\prime}\right)$ |

if $H\left({X}^{\prime}\right)\le H\left(X\right)$ then |

$Y\u2254{Y}^{\prime}$ |

else |

if rand[0, 1]$\le {e}^{-\frac{H\left({Y}^{\prime}\right)-H\left(Y\right)}{T}}$then$Y\u2254{Y}^{\prime}$ |

Compute the new temperature $T\u2254\alpha \ast T$ |

- The initial temperature ${T}_{0}$ is chosen to be big enough so that most of the transitions are accepted at the beginning: ${e}^{-\frac{H\left({Y}^{\prime}\right)-H\left(Y\right)}{{T}_{0}}}\approx 1\forall \left(Y,{Y}^{\prime}\right)$.
- The decreasing factor $\alpha $is chosen in such a way that the final temperature would be close to zero.

## 5. Experiments

#### 5.1. Instances Generation

- (1)
- Jobs’ processing times: Exponential with an average of 10.75 h.
- (2)
- Due dates: Uniform with a minimum of 24 and a maximum of 312 h.
- (3)
- Setup times:
- ○
- Time required to dismount a mold: Uniform with a minimum of 15 and a maximum of 45 min
- ○
- Time required to mount a mold: Uniform with a minimum of 20 and a maximum of 60 min.

#### 5.2. Results

## 6. Conclusions and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Decision Level | Strategic | Tactical | Operational |

Planning Horizon | Years | Months | Weeks/Days |

Typical Problems | Determine the number of resources | Plan activities Assign resources | Schedule activities Consider emergencies |

Instance number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

Machines | 2 | 3 | 5 | 10 | |||||||

Molds | 3 | 6 | 12 | 16 | 18 | 20 | 26 | 26 | 25 | 59 | 63 |

Jobs | 10 | 15 | 32 | 47 | 53 | 57 | 79 | 80 | 81 | 177 | 191 |

Solver | Our Method | ||||
---|---|---|---|---|---|

Instance | Number of Jobs | Time | Tardiness | Time | Tardiness |

1 | 10 | 19 s | 38.00 | 19 s | 38.00 |

2 | 15 | 12 min | 18.21 | 7 s | 18.21 |

3 | 32 | 30 h | [37.65, 80.76] | 96 s | 80.76 |

Current Approach | Proposed Approach | |||||||
---|---|---|---|---|---|---|---|---|

Stochastic Descent | Simulated Annealing | Time | ||||||

Instance | Jobs | Tardiness | Setup | Tardiness | Setup | Tardiness | Setup | |

4 | 47 | 8.32 | 16 | 0 | 33 | 0 | 31 | 4 min |

5 | 53 | 37.76 | 26 | 24.37 | 40 | 17.98 | 42 | 37 min |

6 | 57 | 159.6 | 23 | 136.29 | 33 | 66.16 | 27 | 3 min |

7 | 79 | 395 | 37 | 241.26 | 49 | 148.63 | 44 | 6 min |

8 | 80 | 77.8 | 33 | 0.98 | 49 | 0 | 41 | 1 h |

9 | 81 | 165.9 | 28 | 52.56 | 39 | 46.17 | 43 | 1 h |

10 | 177 | 1020.4 | 82 | 963.95 | 114 | 343.82 | 101 | 5 h |

11 | 191 | 580.2 | 83 | 786.18 | 122 | 579.5 | 124 | 4 h |

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**MDPI and ACS Style**

Klement, N.; Abdeljaouad, M.A.; Porto, L.; Silva, C.
Lot-Sizing and Scheduling for the Plastic Injection Molding Industry—A Hybrid Optimization Approach. *Appl. Sci.* **2021**, *11*, 1202.
https://doi.org/10.3390/app11031202

**AMA Style**

Klement N, Abdeljaouad MA, Porto L, Silva C.
Lot-Sizing and Scheduling for the Plastic Injection Molding Industry—A Hybrid Optimization Approach. *Applied Sciences*. 2021; 11(3):1202.
https://doi.org/10.3390/app11031202

**Chicago/Turabian Style**

Klement, Nathalie, Mohamed Amine Abdeljaouad, Leonardo Porto, and Cristóvão Silva.
2021. "Lot-Sizing and Scheduling for the Plastic Injection Molding Industry—A Hybrid Optimization Approach" *Applied Sciences* 11, no. 3: 1202.
https://doi.org/10.3390/app11031202