# A Hybrid Metaheuristic for the Unrelated Parallel Machine Scheduling Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Single Operation

#### 2.1.1. Single Machine Scheduling

#### 2.1.2. Parallel Machine Scheduling

#### Identical Parallel Machines

#### 2.2. Multiple Operation

#### 2.2.1. Flow Shop scheduling Problem

#### 2.2.2. Job Shop Scheduling Problem

#### 2.2.3. Open Shop Scheduling Problem

#### 2.3. Summary

## 3. Mathematical Formulation

- Each job contains only one kind of product. The products that need to be assembled in each job are provided by MPS.
- If two consecutive jobs processed in one production line are of different products, setup time is required.
- Each production line contains no job at the beginning of each day.
- The process time of each job on each production line is given and fixed.
- If a job begins in a production line, it will be completed without interruption.

- Each production line has its own maximum number of daily production hours.
- Each product and its corresponding job have some production requirements and can only be processed in the predetermined/specified production lines.
- Each job has its own earliest starting time and can only start after that prespecified time.
- Each job requires a specific level of B/I equipment.
- There is an upper bound of the total production hours for all the production lines considered together.

**Notations**

**Sets**

N | set of jobs |

M | set of production lines |

BI | set of burn-in levels |

T | set of the planning horizon |

**Parameters**

${P}_{ij}$ | processing time of job $j$ on production line$i$ |

${S}_{ijk}$ | setup time between job $j$ and job $k$ on production line $i$. ${S}_{ijk}$= setup time if two consecutive jobs processed on production line $i$ belong to different job types; ${S}_{ijk}=0$ if two consecutive jobs processed on production line $i$ belong to identical job types. |

$ca{p}_{it}$ | maximum daily processing hours for production line$i$ on day $t$. |

$T{P}_{t}$ | maximum daily processing hours for all the production lines considered together. |

$Q$ | an extremely large number |

${U}_{t}$ | penalty parameter for burn/in ratio violations on day$t$ |

$B{I}_{1}$,$B{I}_{2}$,$B{I}_{3}$ | target ratio of jobs assigned to B/I levels 1, 2 and 3, respectively. Suppose that a company hopes to maintain three B/I levels of 5:4:1; we can set $B{I}_{1}=5$, $B{I}_{2}=4$, and $B{I}_{3}=1$. |

$M{N}_{j}$ | number of products that need to be assembled in each job $j$ As the B/I level violations are calculated based on $M{N}_{j}$, this parameter is introduced. |

**Decision Variables**

${X}_{ijkt}$ | ${X}_{ijkt}=1$ if job $j$ is processed immediately before job $k$ on production line $i$ on day $t;{X}_{ijkt}=0$ otherwise. Note that if job$j$ or $k$ cannot be processed on production line $i$, then ${X}_{ijkt}=0$. |

${B}_{jbt}$ | ${B}_{jbt}=1$, if job $j$ has a B/I level of $b$ on day $t$; ${B}_{jbt}=0$ otherwise. |

${C}_{ijt}$ | completion time of job $j$ on production line $i$ on day$t$ |

${F}_{it}$ | complete time for each production line $i$ on day $t$ |

${C}_{max,t}$ | maximum completion time on day$t$ |

**Mathematical Formulation**

## 4. Solution Approach

#### 4.1. Initial Solution

#### 4.2. Algorithm Steps

#### 4.2.1. Initialization

#### 4.2.2. Neighborhood Search

#### 4.2.3. Incumbent Solution Updating

#### 4.2.4. Termination

#### 4.3. Summary

## 5. Solution Approach

#### 5.1. Parameter Calibration

#### 5.2. Validation

#### 5.3. Practical Application Scenarios

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

B/I | burn-in |

PBSA | population-based simulated annealing |

VND | variable neighborhood descent |

GA | genetic algorithm |

SA | simulated annealing |

LS | local search |

ERD | earlier release date first |

AIS | artificial immune system |

PSO | particle swarm optimization |

DWPSA | delay window-time parallel saving algorithm |

DWGSA | delay window-time generalized saving algorithm |

FJSP | flexible job shop scheduling problem |

GDS | general dense scheduling |

ACO | colony optimization |

SPT | shortest processing time |

MPS | master production schedule |

DPS | daily production scheduling problem |

NSJ | number of scheduled jobs |

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Study | Objective | Primary Constraints | Solution Approach |
---|---|---|---|

[10] | Minimizing the total completion time | Setup time and production availability | GA |

[21] | Minimizing the makespan | Resource constraints, sequence-dependent setup times, different release dates, machine eligibility and precedence constraints | GA and AIS |

[22] | Minimizing the total machine load | Past sequence-dependent setup times, release dates, deteriorating jobs and learning effects | Integrated PSO and GA |

[19] | Minimizing the makespan | Nonzero arbitrary release dates, limited additional resources, and non-anticipatory sequence-dependent setup times | Integrated SA and VNS |

[16] | Minimizing the total weighted tardiness | Random rework and due dates | GA and SA |

[20] | Minimizing the total weighted tardiness | Sequence- and machine- dependent setup times | VND |

Current Study | Minimizing the makespan and B/I violations | Sequence-dependent setup times, different work starting times, machine eligibility, burn-in eligibility and work time limits | Integrated PBSA and VND |

$\mathit{\alpha}$$:\mathit{\beta}$ | Cmax | B/I Penalty |
---|---|---|

1:0.0001 | 43.26 | 1437.65 |

1:0.001 | 43.63 | 1357.95 |

1:0.01 | 44.48 | 1308.20 |

1:0.1 | 50.67 | 1303.20 |

1:1 | 57.93 | 1298.45 |

L/T/N ^{1} | Gurobi | PBSA with 3VND ^{4} | PBSA with 2VND ^{5} | |||||
---|---|---|---|---|---|---|---|---|

Objective Value | CPU ^{2} (s) | Objective Value | Cmax | B/I Penalty | Objective Value | Cmax | B/I Penalty | |

3/2/10 | 23.59 | 1.47 | 23.59 | 14.44 | 915.50 | 23.59 | 14.44 | 915.50 |

3/2/15 | 31.74 | 3614.93 | 31.74 | 20.60 | 1113.75 | 31.74 | 20.60 | 1113.75 |

4/2/10 | 19.12 | 0.14 | 19.12 | 9.97 | 915.50 | 19.12 | 9.97 | 915.50 |

4/2/15 | 27.95 | 10,790.65 | 27.95 | 16.81 | 1113.75 | 27.95 | 16.81 | 1113.75 |

4/2/20 | *^{3} | * | 30.01 | 20.22 | 978.75 | 30.00 | 20.22 | 978.75 |

4/2/50 | * | * | 45.86 | 25.34 | 2052.25 | 45.60 | 25.23 | 2036.75 |

4/2/100 | * | * | 58.62 | 36.52 | 2209.50 | 59.69 | 36.86 | 2283.75 |

4/3/100 | * | * | 54.14 | 40.00 | 1414.00 | 54.05 | 39.91 | 1414.00 |

5/3/100 | * | * | 48.34 | 34.18 | 1416.00 | 48.07 | 33.88 | 1418.50 |

6/3/150 | * | * | 71.38 | 50.50 | 2088.00 | 70.57 | 50.24 | 2032.75 |

6/4/150 | * | * | 79.33 | 54.67 | 2466.00 | 79.34 | 54.68 | 2466.00 |

6/3/200 | * | * | 74.06 | 51.69 | 2236.50 | 68.49 | 49.70 | 1878.75 |

6/4/200 | * | * | 90.95 | 65.84 | 2511.00 | 88.81 | 63.70 | 2511.00 |

6/5/200 | * | * | 121.60 | 78.51 | 4308.75 | 122.23 | 78.62 | 4361.25 |

^{1}L: number of production lines; T: number of scheduling days in a week; N: number of jobs.

^{2}CPU: computational time.

^{3}*: fails to determine solutions within 8 h.

^{4}3VND: all three neighborhood structures explained in Figure 2, Figure 3 and Figure 4.

^{5}2VND: the neighborhood structures explained in Figure 2 and Figure 3.

5:4:1 | 6:3:1 | 5:5:0 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

L/T/N | Obj. Value | Cmax | B/I Penalty | NSJ ^{1} | Obj. Value | Cmax | B/I Penalty | NSJ | Obj. Value | Cmax | B/I Penalty | NSJ |

3/2/100 | 59.70 | 39.45 | 2025.00 | 75 | 66.89 | 39.44 | 2745.00 | 75 | 39.54 | 39.51 | 3.00 | 76 |

3/3/100 | 62.62 | 42.37 | 2025.00 | 76 | 69.88 | 42.43 | 2745.00 | 76 | 42.63 | 42.52 | 11.00 | 77 |

4/2/100 | 59.69 | 36.86 | 2283.75 | 63 | 62.28 | 37.23 | 2505.00 | 65 | 38.51 | 37.96 | 55.00 | 68 |

4/3/100 | 54.05 | 39.91 | 1414.00 | 99 | 62.11 | 39.19 | 2292.00 | 99 | 39.86 | 39.78 | 8.00 | 100 |

4/4/100 | 54.78 | 40.64 | 1414.00 | 100 | 62.47 | 39.55 | 2292.00 | 99 | 39.82 | 39.80 | 2.00 | 100 |

5/2/100 | 56.55 | 36.82 | 1973.25 | 64 | 65.94 | 38.12 | 2782.00 | 64 | 38.31 | 37.73 | 58.00 | 66 |

5/3/100 | 48.07 | 33.88 | 1418.50 | 100 | 55.28 | 32.36 | 2292.00 | 99 | 33.34 | 33.06 | 28.00 | 100 |

5/4/100 | 47.82 | 33.68 | 1414.00 | 100 | 55.35 | 32.43 | 2292.00 | 99 | 33.60 | 33.32 | 28.00 | 100 |

6/2/100 | 57.84 | 35.75 | 2209.25 | 62 | 63.05 | 35.87 | 2718.00 | 65 | 37.00 | 36.64 | 36.00 | 66 |

6/3/100 | 42.61 | 28.46 | 1415.50 | 100 | 51.47 | 28.55 | 2292.00 | 100 | 29.06 | 29.06 | 0.00 | 100 |

^{1}NSJ: number of scheduled jobs.

Original Problem | Problem with an Additional Constraint | |||||||
---|---|---|---|---|---|---|---|---|

L/T/N | Obj. Value | Cmax | B/I Penalty | NSJ | Obj. Value | Cmax | B/I Penalty | NSJ |

3/2/100 | 59.70 | 39.45 | 2025.00 | 75 | 55.79 | 38.47 | 1732.50 | 57 |

3/3/100 | 62.62 | 42.37 | 2025.00 | 76 | 55.79 | 38.47 | 1732.50 | 57 |

4/2/100 | 59.69 | 36.86 | 2283.75 | 63 | 55.23 | 37.12 | 1811.25 | 58 |

4/3/100 | 54.05 | 39.91 | 1414.00 | 99 | 51.55 | 38.95 | 1260.00 | 88 |

4/4/100 | 54.78 | 40.64 | 1414.00 | 100 | 50.34 | 37.74 | 1260.00 | 88 |

5/2/100 | 56.55 | 36.82 | 1973.25 | 64 | 58.38 | 34.03 | 2434.25 | 59 |

5/3/100 | 48.07 | 33.88 | 1418.50 | 100 | 46.44 | 32.81 | 1363.00 | 97 |

5/4/100 | 47.82 | 33.68 | 1414.00 | 100 | 47.21 | 33.58 | 1363.50 | 97 |

6/2/100 | 57.84 | 35.75 | 2209.25 | 62 | 51.85 | 30.23 | 2161.50 | 60 |

6/3/100 | 42.61 | 28.46 | 1415.50 | 100 | 44.28 | 30.20 | 1408.50 | 99 |

Original Order | Original Order with Superhot Runs | |||||||
---|---|---|---|---|---|---|---|---|

L/T/N | Obj. Value | Cmax | B/I Penalty | NSJ ^{1} | Obj. Value | Cmax | B/I Penalty | NSJ |

6/4/200 | 88.81 | 63.70 | 2511.00 | 164 | 89.16 | 62.16 | 2699.50 | 160 |

6/5/200 | 122.23 | 78.62 | 4361.25 | 184 | 137.09 | 83.96 | 5312.50 | 179 |

^{1}NSJ: number of scheduled jobs.

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**MDPI and ACS Style**

Lin, D.-Y.; Huang, T.-Y.
A Hybrid Metaheuristic for the Unrelated Parallel Machine Scheduling Problem. *Mathematics* **2021**, *9*, 768.
https://doi.org/10.3390/math9070768

**AMA Style**

Lin D-Y, Huang T-Y.
A Hybrid Metaheuristic for the Unrelated Parallel Machine Scheduling Problem. *Mathematics*. 2021; 9(7):768.
https://doi.org/10.3390/math9070768

**Chicago/Turabian Style**

Lin, Dung-Ying, and Tzu-Yun Huang.
2021. "A Hybrid Metaheuristic for the Unrelated Parallel Machine Scheduling Problem" *Mathematics* 9, no. 7: 768.
https://doi.org/10.3390/math9070768