# A Study on the Optimal Flexible Job-Shop Scheduling with Sequence-Dependent Setup Time Based on a Hybrid Algorithm of Improved Quantum Cat Swarm Optimization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description and Formulation

#### 2.1. Problem Assumption

_{1}, J

_{2}, ⋯, J

_{n}} need to be processed by m machines {M

_{1}, M

_{2}, ⋯, M

_{m}}. Each job contains O

_{i}operations and its own processing route. O

_{ij}represents the jth operation of the ith job. Each operation O

_{ij}can be processed on any machine selected from a compatible machine set. The processing time of each operation is determined by the processing capacity of the assigned machine. The setup time of each machine depends on the two consecutive operations on it. In this study, we chose an eligible machine for each operation, then sequenced the operations on each machine in order to minimize the makespan, i.e., $\mathrm{min}C=\mathrm{min}\left\{\mathrm{max}{C}_{k}|1\le k\le m\right\}$, where C

_{k}represents the completion time of the last job on machine k. For this problem, the following assumptions help to simplify the problem:

- (1)
- Machines and jobs are available at time zero.
- (2)
- There exist precedence constraints among different operations of the same job, i.e., each operation can only be processed after its predecessor is completed.
- (3)
- There are no precedence constraints among different jobs, i.e., jobs are independent of each other.
- (4)
- Preemption is not allowed, i.e., the processing of each operation must not be interrupted once it starts.
- (5)
- Each machine can only process one operation at a given time.
- (6)
- Job transportation and machine breakdown are not considered.

#### 2.2. Description of Parameters and Variables

#### 2.3. Problem Formulation

## 3. Implementation of Proposed QCSO

#### 3.1. Encoding Approach

#### 3.2. Decoding Mechanism

_{i}(t) = 1, else let x

_{i}(t) = 0 (i = 1, 2, …, n). For every P(t), we can get a binary $X\left(t\right)=\left({x}_{1}^{\prime},{x}_{2}^{\prime},\cdots ,{x}_{n}^{\prime}\right)$ of length n.

#### 3.3. Seeking Mode

_{k}and place them in the memory pool; the size of the memory pool is j, and j = SMP. If the value of SPC is true, then j = (SMP − 1), and leave the current position as a candidate solution.

_{k}.

_{b}= FS

_{max}, otherwise FS

_{b}= FS

_{min}.

#### 3.4. Tracing Mode

_{1}, α

_{2}, …, |α

_{i}, …, α

_{j}|, …, α

_{l}

_{1}, β

_{2}, …, |β

_{i}, …, β

_{j}|, …, β

_{l}

_{1}, α

_{2}, …, |β

_{i}, …, β

_{j}|, …, α

_{l}

_{i},

_{d}) of each dimension direction. The best position update that the entire cat group has experienced is the current optimal solution, and it is denoted as x

_{best}. The speed of each cat is denoted as v

_{i}= {v

_{i}

_{1}, v

_{i}

_{2}, …, v

_{id}}, and each cat updates its speed according to Equation (11):

_{i,d}= v

_{i,d}+r

_{1*}c

_{1*}(x

_{best},

_{d}− x

_{i,d}), d = 1,2,…,M

_{best}

_{,d}is the position of the cat with the best fitness value; x

_{i,d}is the position of c

_{k}, c

_{1}is a constant, and r

_{1}is a random value in the range [0, 1]; v

_{i,d}is the updated speed of cat i in dimension d, and M is the dimension size; x

_{best}

_{,d}(t) represents the position of the cat with the best fitness value in the current swarm.

_{i,d}= x

_{i,d}+ v

_{i,d}, d = 1, 2, …, M

#### 3.5. Updating Quantum Rotation Angle

_{i}is updated as follows:

_{1}and c

_{2}are two constants and r

_{1}is a random value in the range [0, 1].

_{max}− (MR

_{max}− MR

_{min})’× L/n

_{max}

_{max}is the maximum iterations and L is the current run time.

#### 3.6. Fitness Function

_{t}(x) − M

_{B}(min)

_{t}(x) and M

_{B}(min) indicate the completion time of the current individual and the current minimum makespan in generation t. In other words, it is the current optimal solution.

#### 3.7. Flowchart of QCSO

_{0}), and randomly create n chromosomes that encoded by qubit.

_{0}), and get a definite solution P(t

_{0}).

## 4. Algorithm Validation

#### 4.1. Data Generation

_{i}refers to job I, O

_{j}refers to operation j, J

_{ij}refers to operation j of job i, and M

_{i}refers to the machine number.

#### 4.2. Calculation Result

_{1}= 2, r

_{1}is a random number in the range [0, 1]. The QCSO algorithm was compared to the parallel genetic algorithm (PGA) [8], and the comparison of calculation results included the following aspects: target minimum value (Min.sol), target average value (Avg.sol), target maximum value (Max.sol), times for optimal solution, average relative percentage error (RPE), and their standard deviation (SD). The percentage, which represents the absolute deviation of a measurement value to the mean value, is called RPE, and it is used to measure the deviation of a single measurement result from the mean value. SD is the square root of the sum of the squared deviation from the mean, and it is also the arithmetic square root of variance. It can reflect the discrete degree of the dataset, which is represented by σ. The smaller the value of σ, the better the stability of the algorithm. The formulas for RPE and σ are as follows:

_{i}are real numbers, and μ is the arithmetic mean value of x

_{i}.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**QCSO and PGA used to solve 6 × 4 problem working sketch. (

**a**) QCSO used to solve 6 × 4 problem working sketch. (

**b**) PGA used to solve 6 × 4 problem working sketch.

**Figure 3.**QCSO and PGA to solve 8 × 4 problem working sketch. (

**a**) QCSO used to solve 8 × 4 problem working sketch (

**b**) PGA used to solve 8 × 4 problem working sketch.

**Figure 4.**QCSO and PGA to solve 10 × 4 problem working sketch. (

**a**) QCSO used to solve 10 × 4 problem working sketch (

**b**) PGA used to solve 10 × 4 problem working sketch.

Literature | Objective Function | Algorithms |
---|---|---|

Mousakhani [2] | Minimize total tardiness | Iterated local search |

Shen et al. [3] | Minimize makespan | Tabu search with specific neighborhood search function |

Bagheri and Zandieh [4] | Minimize makespan and mean tardiness | Variable neighborhood search |

Abdelmaguid [5] | Minimize makespan | Tabu search with specific neighborhood functions |

Naderi et al. [6] | Minimize makespan | Genetic algorithm |

Li and Lei [7] | Minimize makespan, total tardiness, and total energy consumption | Imperialist competitive algorithm with feedback |

Defersha and Chen [8] | Minimize makespan | Parallel genetic algorithm |

Azzouz et al. [9] | Minimize makespan and bi-criteria objective function | Hybrid genetic algorithm and variable neighborhood search |

Wang and Zhu [10] | Minimize makespan | Hybrid genetic algorithm and tabu search |

Li et al. [11] | Minimize makespan and total setup costs | Elitist nondominated sorting hybrid algorithm |

Azzouz et al. [12] | Minimize makespan | Hybrid genetic algorithm and iterated local search |

Azzouz et al. [13] | Minimize makespan | Adaptive genetic algorithm |

Abderrabi et al. [14] | Minimize total flow time | Genetic algorithm and iterated local search |

Parjapati and Ajai [15] | Minimize makespan | Genetic algorithm |

Sadrzadeh [16] | Minimize makespan and mean tardiness | Artificial immune system and particle swarm optimization |

Tayebi Araghi et al. [17] | Minimize makespan | Genetic variable neighborhood search with affinity function |

Sun et al. [18] | Minimize makespan, total workload, workload of critical machine, and penalties of earliness/tardiness | Hybrid many-objective evolutionary algorithm |

Li et al. [19] | Minimize energy consumption and makespan | Improved Jaya algorithm |

Müller et al. [20] | Minimize makespan | Decision trees and deep neural networks |

Wei et al. [21] | Minimize the makespan and total energy consumption | Energy-aware estimation model |

Li et al. [22] | Minimize the makespan and the total workload | Hybrid self-adaptive multi-objective evolutionary algorithm |

Türkyılmaz et al. [23] | Minimize makespan | Hybrid Genetic Algorithm-hypervolume contribution measure |

Jiang et al. [24] | Handle the issues of low production efficiency, high energy consumption and processing cost | A novel improved crossover artificial bee colony algorithm |

Index | Explanation |
---|---|

J | Job set |

O | Operation set |

M | Machine set |

${C}_{\mathrm{max}}$ | Final completion time (makespan) |

${C}_{k}$ | Completion time of machine k |

c_{o}_{, j,k} | Completion time of operation o of job j on machine k |

p_{o}_{, j,k} | Processing time of operation o of job j on machine k |

s_{o}_{, j,k,o’,j’} | Setup time of two adjacent operations arranged on the same machine |

st_{o}_{, j, k} | Start time of operation o of job j on machine k |

R_{m} | Maximum number of processing tasks on machine m |

r | Position index of processing tasks on each machine, r = 1, 2, ⋯, R_{m} |

L | A large positive number |

c_{r}_{,k} | Completion time of task r on machine k |

x_{o,j,k} | x_{o,j,k} = 1 if operation o of job j is processed on machine k, otherwise x_{o, j,k} = 0 |

y_{r,k,o,j} | y_{r,k,o,j} = 1 if the task on position r of machine k is just operation o of job j, otherwise y_{r,k,o,j} = 0 |

Job | Number of Machine | ||
---|---|---|---|

1 | 1 | 2 | 3 |

2 | 3 | 1 | 2 |

3 | 1 | 3 | 2 |

4 | 2 | 3 | 1 |

Job | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|

O_{1} | 1, 2 | 2 | 1, 2, 3 | 4 |

O_{2} | 2, 3 | 1, 4 | 3 | 1, 2 |

O_{3} | 3 | 1, 3 | 2, 3, 4 | 1, 2 |

O_{4} | 1, 2, 3 | 3 | 1 | 2, 4 |

Job | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|

O_{1} | 87, 140 | 200, 220, 200 | 165, 150 | 1102.5, 1347.5, 1125 |

O_{2} | 210, 192 | 280, 260 | 165, 135, 165 | 1102.5, 1125, 1125 |

O_{3} | 245, 280, 262 | 240, 200 | 150, 180 | 1100, 1200 |

O_{4} | 245, 262 | 230, 270 | 140, 160 | 1000, 1050 |

Job | M_{1} | M_{2} | M_{3} | M_{4} |
---|---|---|---|---|

J_{1} | 220 | 90 | 80 | 45 |

J_{2} | 120 | 85 | 60 | 85 |

J_{3} | 235 | 75 | 65 | 127 |

J_{4} | 167 | 129 | 109 | 68 |

Job | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|

J_{1} | 0 | 250 | 176 | 155 |

J_{2} | 260 | 0 | 248 | 165 |

J_{3} | 210 | 50 | 0 | 218 |

J_{4} | 220 | 60 | 205 | 0 |

Job | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|

J_{1} | 0 | 190 | 161 | 292 |

J_{2} | 220 | 0 | 146 | 224 |

J_{3} | 260 | 122 | 0 | 158 |

J_{4} | 215 | 114 | 171 | 0 |

Job | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|

J_{1} | 0 | 235 | 231 | 285 |

J_{2} | 260 | 0 | 162 | 159 |

J_{3} | 290 | 193 | 0 | 202 |

J_{4} | 228 | 213 | 152 | 0 |

Job | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|

J_{1} | 0 | 252 | 203 | 252 |

J_{2} | 65 | 0 | 146 | 156 |

J_{3} | 154 | 68 | 0 | 159 |

J_{4} | 121 | 154 | 154 | 0 |

Job | J_{1} | J_{2} | J_{3} | J_{4} | J_{5} | J_{6} | J_{7} | J_{8} |
---|---|---|---|---|---|---|---|---|

O_{1} | 2, 3 | 1, 2, 4 | 2, 4 | 1, 2, 4 | 1, 2 | 1, 4 | 2, 3 | 4 |

O_{2} | 1, 3 | 1, 3 | 1, 2, 4 | 1, 2, 4 | 2, 4 | 2, 3 | 2, 3, 4 | 1, 2 |

O_{3} | 1, 2, 3 | 3, 4 | 1, 2 | 2, 4 | 1, 3, 4 | 1, 2, 3 | 1, 3 | 1, 2 |

O_{4} | 2, 4 | 1, 4 | 2, 3 | 2, 3 | 3,4 | 1 | 2, 4 | 1, 4 |

J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|

87, 140 | 200, 220, 200 | 165, 150 | 1102.5, 1347.5, 1125 |

210, 192.5 | 280, 260 | 165, 135, 165 | 1102.5, 1125, 1125 |

245, 280, 262.5 | 240, 200 | 150, 180 | 1100, 1200 |

245, 262.5 | 230, 270 | 140, 160 | 1000, 1050 |

J_{5} | J_{6} | J_{7} | J_{8} |

220, 200 | 210, 240 | 180, 200 | 120 |

140, 120 | 260, 300 | 210, 235, 265 | 110, 160 |

180, 200, 220 | 200, 220, 260 | 250, 280 | 220, 260 |

130, 160 | 270 | 150, 180 | 200, 240 |

Job | M_{1} | M_{2} | M_{3} | M_{4} |
---|---|---|---|---|

J_{1} | 220 | 90 | 80 | 45 |

J_{2} | 120 | 85 | 60 | 85 |

J_{3} | 235 | 75 | 65 | 127 |

J_{4} | 167 | 129 | 109 | 68 |

J_{5} | 216 | 143 | 123 | 145 |

J_{6} | 134 | 110 | 95 | 187 |

J_{7} | 146 | 225 | 88 | 122 |

J_{8} | 221 | 219 | 75 | 157 |

Job | J_{1} | J_{2} | J_{3} | J_{4} | J_{5} | J_{6} | J_{7} | J_{8} |
---|---|---|---|---|---|---|---|---|

J_{1} | 0 | 250 | 176 | 155 | 215 | 255 | 190 | 212 |

J_{2} | 260 | 0 | 248 | 165 | 223 | 157 | 154 | 214 |

J_{3} | 210 | 50 | 0 | 218 | 213 | 258 | 259 | 215 |

J_{4} | 220 | 60 | 205 | 0 | 119 | 159 | 164 | 227 |

J_{5} | 150 | 110 | 117 | 178 | 0 | 30 | 116 | 215 |

J_{6} | 130 | 125 | 129 | 132 | 137 | 0 | 40 | 203 |

J_{7} | 120 | 215 | 238 | 181 | 147 | 121 | 0 | 209 |

J_{8} | 150 | 225 | 159 | 169 | 116 | 212 | 113 | 0 |

Job | J_{1} | J_{2} | J_{3} | J_{4} | J_{5} | J_{6} | J_{7} | J_{8} |
---|---|---|---|---|---|---|---|---|

J_{1} | 0 | 190 | 161 | 292 | 201 | 255 | 269 | 248 |

J_{2} | 220 | 0 | 146 | 224 | 209 | 157 | 254 | 218 |

J_{3} | 260 | 122 | 0 | 158 | 213 | 251 | 214 | 148 |

J_{4} | 215 | 114 | 171 | 0 | 151 | 220 | 207 | 214 |

J_{5} | 210 | 152 | 149 | 153 | 0 | 80 | 85 | 161 |

J_{6} | 159 | 155 | 219 | 159 | 156 | 0 | 90 | 142 |

J_{7} | 151 | 121 | 153 | 117 | 112 | 80 | 0 | 217 |

J_{8} | 154 | 216 | 165 | 152 | 119 | 210 | 159 | 0 |

Job | J_{1} | J_{2} | J_{3} | J_{4} | J_{5} | J_{6} | J_{7} | J_{8} |
---|---|---|---|---|---|---|---|---|

J_{1} | 0 | 235 | 231 | 285 | 294 | 240 | 200 | 90 |

J_{2} | 260 | 0 | 162 | 159 | 221 | 100 | 200 | 60 |

J_{3} | 290 | 193 | 0 | 202 | 219 | 150 | 150 | 150 |

J_{4} | 228 | 213 | 152 | 0 | 55 | 118 | 159 | 208 |

J_{5} | 173 | 159 | 158 | 75 | 0 | 106 | 148 | 158 |

J_{6} | 119 | 156 | 159 | 206 | 149 | 0 | 157 | 204 |

J_{7} | 138 | 184 | 215 | 233 | 258 | 126 | 0 | 106 |

J_{8} | 176 | 217 | 208 | 259 | 239 | 137 | 125 | 0 |

Job | J_{1} | J_{2} | J_{3} | J_{4} | J_{5} | J_{6} | J_{7} | J_{8} |
---|---|---|---|---|---|---|---|---|

J_{1} | 0 | 252 | 203 | 252 | 216 | 158 | 206 | 153 |

J_{2} | 65 | 0 | 146 | 156 | 101 | 212 | 103 | 157 |

J_{3} | 154 | 68 | 0 | 159 | 154 | 111 | 155 | 206 |

J_{4} | 121 | 154 | 154 | 0 | 35 | 108 | 203 | 108 |

J_{5} | 206 | 124 | 150 | 85 | 0 | 155 | 159 | 212 |

J_{6} | 151 | 158 | 104 | 104 | 203 | 0 | 95 | 203 |

J_{7} | 159 | 153 | 109 | 152 | 159 | 123 | 0 | 149 |

J_{8} | 107 | 152 | 112 | 101 | 206 | 109 | 45 | 0 |

Problem scale: 2 × 4 | |||||
---|---|---|---|---|---|

Max.sol (min) | Min.sol (min) | Avg.sol (min) | σ | Times for optimal solution | |

QCSO | 1359 | 1359 | 1359 | 0 | 20 |

PGA | 1359 | 1359 | 1359 | 0 | 20 |

Problem scale: 4 × 4 | |||||

Max.sol (min) | Min.sol (min) | Avg.sol (min) | σ | Times for optimal solution | |

QCSO | 4773 | 4744 | 4761.4 | 10.58 | 18 |

PGA | 4773 | 4744 | 4771.55 | 48 | 10 |

Problem scale: 6 × 4 | |||||

Max.sol (min) | Min.sol (min) | Avg.sol (min) | σ | Times for optimal solution | |

QCSO | 4924 | 4764 | 47945 | 20.57 | 16 |

PGA | 4854 | 4764 | 4784.35 | 25.93 | 14 |

Problem scale: 8 × 4 | |||||

Max.sol (min) | Min.sol (min) | Avg.sol (min) | σ | Times for optimal solution | |

QCSO | 5118 | 4853 | 4941.2 | 62.48 | 16 |

PGA | 5184 | 4826 | 4962.8 | 1103 | 12 |

Problem scale: 10 × 4 | |||||

Max.sol (min) | Min.sol (min) | Avg.sol (min) | σ | Times for optimal solution | |

QCSO | 5590 | 5234 | 5444.3 | 91.81 | 15 |

PGA | 5954 | 5164 | 5607 | 192.51 | 8 |

n | m | Instance | RPE | n | m | Instance | RPE | n | m | Instance | RPE |
---|---|---|---|---|---|---|---|---|---|---|---|

6 | 4 | 1 | −0.13 | 8 | 4 | 1 | 2.09 | 10 | 4 | 1 | 1.03 |

2 | 0 | 2 | 0.35 | 2 | −4.55 | ||||||

3 | 0.69 | 3 | −1.37 | 3 | 50 | ||||||

4 | −0.10 | 4 | 4.38 | 4 | 5.72 | ||||||

5 | 0 | 5 | −1.45 | 5 | 1.17 | ||||||

6 | 2.64 | 6 | 1.39 | 6 | 2.52 | ||||||

7 | 0.92 | 7 | 5.83 | 7 | −3.08 | ||||||

8 | −1.45 | 8 | 3.62 | 8 | 2.78 | ||||||

9 | 0.19 | 9 | 3.77 | 9 | 9.37 | ||||||

10 | 1.55 | 10 | 1.94 | 10 | −1.16 | ||||||

11 | 0.02 | 11 | 5.67 | 11 | 5.51 | ||||||

12 | 0.11 | 12 | 0.66 | 12 | 0.78 | ||||||

13 | −0.12 | 13 | 0.16 | 13 | 3.86 | ||||||

14 | −1.12 | 14 | 7.09 | 14 | 4.31 | ||||||

15 | −0.21 | 15 | 1.42 | 15 | 7.44 | ||||||

16 | 3.27 | 16 | −0.80 | 16 | 80 | ||||||

17 | −0.13 | 17 | 4.33 | 17 | −0.71 | ||||||

18 | −1.02 | 18 | 0.37 | 18 | 2.46 | ||||||

19 | −0.10 | 19 | 1.45 | 19 | 0.82 | ||||||

20 | 0.10 | 20 | 0.16 | 20 | 2.36 | ||||||

Mean | 0.26 | Mean | 2.05 | Mean | 2.70 |

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## Share and Cite

**MDPI and ACS Style**

Song, H.; Liu, P.
A Study on the Optimal Flexible Job-Shop Scheduling with Sequence-Dependent Setup Time Based on a Hybrid Algorithm of Improved Quantum Cat Swarm Optimization. *Sustainability* **2022**, *14*, 9547.
https://doi.org/10.3390/su14159547

**AMA Style**

Song H, Liu P.
A Study on the Optimal Flexible Job-Shop Scheduling with Sequence-Dependent Setup Time Based on a Hybrid Algorithm of Improved Quantum Cat Swarm Optimization. *Sustainability*. 2022; 14(15):9547.
https://doi.org/10.3390/su14159547

**Chicago/Turabian Style**

Song, Haicao, and Pan Liu.
2022. "A Study on the Optimal Flexible Job-Shop Scheduling with Sequence-Dependent Setup Time Based on a Hybrid Algorithm of Improved Quantum Cat Swarm Optimization" *Sustainability* 14, no. 15: 9547.
https://doi.org/10.3390/su14159547