# Hybrid Algorithm Based on an Estimation of Distribution Algorithm and Cuckoo Search for the No Idle Permutation Flow Shop Scheduling Problem with the Total Tardiness Criterion Minimization

^{*}

## Abstract

**:**

## 1. Introduction

^{2})) greedy algorithm, which was shown numerically to perform better than other published heuristics. Kalczynski and Kamburowski [13] addressed the problem of determining a job sequence that minimized the makespan in m-machine flow shops under the no-idle condition. Pan and Wang [14] proposed a novel discrete differential evolution algorithm to solve NIPFSP. That study presented two simple approaches to calculate the makespan and a speed-up method to insert the neighborhood and thus improve the efficiency of the entire algorithm. Pan et al. [15] then proposed a new and novel referenced local search procedure hybridized with both algorithms to further improve the solution quality. The referenced local search exploited the space based on reference positions taken from a reference solution to determine better job positions when performing the insertion operation. Researchers [16] also solved the single machine total weighted tardiness problem with sequence dependent setup times by a discrete differential evolution algorithm. He and Wang [17] proposed a hybrid algorithm that combines evolutionary computation and constraint-handling techniques. Li and Wang [18] proposed a hybrid quantum-inspired genetic algorithm for the multi-objective PFSP. A new random-key representation was used to convert the Q-bit representation to job permutation in evaluating the objective values of the schedule solution. Deng and Gu [19] proposed a hybrid discrete differential evolution (HDDE) algorithm for the no-idle permutation flow shop scheduling problem with makespan criterion. Tasgetiren et al. [20] investigated the utilization of a continuous algorithm for the NIPFSP with tardiness criterion. A differential evolution algorithm with variable parameter search was developed to solve the NIPFSP. Researchers [21] also designed a variable iterated greedy algorithm with differential evolution to solve the NIPFSP in recent years. Pan and Ruiz [22] first proposed an effective iterated greedy algorithm for a mixed no idle flow shop, where several machines had the no-idle constraint, whereas others were regular machines. The researchers also proposed a formula set to accelerate the insertion calculation that was utilized in heuristics and in local search procedures. A novel nature-inspired cuckoo search (CS) algorithm was developed by Yang and Deb [23] in 2009. CS has a series of successful engineering examples [24,25,26]. The improved CS algorithm was proposed for hybrid flow shop scheduling problems by Marichelvam et al. [27], and the algorithm was validated with the data from a leading furniture manufacturing company. A discrete version of the inter-species CS algorithm was proposed and applied to solve two significant types of the PFSP [28].

## 2. Problem Formulation

_{1}, J

_{2}, …, J

_{i}, …, J

_{n−}

_{1}, J

_{n}} must be processed on m machines M = {M

_{1}, M

_{2}, …, M

_{j}, …, M

_{m−}

_{1}, M

_{m}} with the same sequence. Consequently, n products {P

_{1}, P

_{2}, …, P

_{i}, …, P

_{n−}

_{1}, P

_{n}} are attained. Therefore, determining the processing sequence of n jobs over m machines in PFSP to satisfy several objectives is widely applied in actual production, especially in one-piece mass production. The NIPFSP is a highly important branch of PFSP and an NP-hard problem. Additional mathematical descriptions are presented in the following subsections.

#### 2.1. Notation

i, j | normally utilized as loop variables (i.e., i represents the job number, and j represents the machine number) |

m | machine number |

n | job number |

Job | {J_{1}, J_{2}, ….., J_{n}}; represents the job set to be processed |

π | scheduling solution that is the processing sequence of the job set {J_{1}, J_{2}, ….., J_{n}} |

T_{i,j} | represents the processing time of the i-th job processed on the j-th machine |

Ts_{i,j} | represents the starting time of the i-th processed job on the j-th machine; |

Given that all of the jobs are prepared to be processed at time zero, then $T{s}_{\pi (1),1}=0$ | |

Te_{i,j} | represents the ending time of the i-th processed job on the j-th machine |

DifT_{i,j} | represents the minimum difference time between the π(i)-th processed job completion time of the j-th machine and (j + 1)-th machine |

d_{i} | represents the due date of the i-th job |

$TTd(\pi )$ | represents the total tardiness of the schedule π |

#### 2.2. Mathematical Model

_{i,j}is employed to ensure the exact delay time. The ending time of the last processed job at each machine can be calculated through the sum DifT

_{i,j}with T

_{i,j}. The ending time of each job at the last machine can then be attained by forward pass calculation. Finally, the total tardiness of the schedule π can be obtained. The objective of solving NIPFSP is to determine a suitable solution (i.e., π) with the total tardiness $TTd(\pi )$ minimization. An example of the NIPFSP problem with five jobs and three machines is shown in Figure 2 in describing the manufacturing process. The five jobs are processed on the three machines with the same job process vector [4 1 5 3 2]. The no idle constraint exists, whether the jobs are processed on the first machine or the following machines. The starting time and end time of each job are also posted in Figure 2.

## 3. HEDA_CS for NIPFSP

#### 3.1. Solution Representation

#### 3.2. Initialization and Probability Model

_{ij}(l) in the probability matrix P represents the probability that job j appears before or in the i-th position at the l-th iteration. The value of p

_{ij}(l) refers to the importance of a job when decoding a solution into a schedule. As the processing matrix T can be calculated to obtain the total processing time of each job, the original probability matrix P is designed considering the knowledge of the longest total processing time job priority principle and roulette wheel selection. The original probability matrix P’s concrete implementation process is illustrated as follows:

#### 3.3. Lévy Flight Strategy in CS

#### 3.4. Updating Mechanism

_{i,j}describes whether job j is located before position i. Considering the operation of the SP better individuals in Section 3.5, the value of $\alpha $ can be set to be slightly large. A sufficient strategy can maintain the exploration in the HEDA_CS in the aforementioned operation. As an operation to obtain better knowledge information in relatively better solutions, the $\alpha $ value must be set as extremely small. This scenario is also an updating mechanism to obtain new individuals during the iteration.

#### 3.5. Knowledge-Based Local Search

#### 3.6. Overall Implementation

^{2}) is also observed. Each new generated individual in the sampling process is generated by the roulette strategy with computational complexity of O(n

^{2}). The aforementioned analysis shows that the computation complexity of the proposed HEDA_CS is not excessively large. It can solve the NIPFFSP with the total tardiness criterion minimization within an acceptable range of calculations.

## 4. Results and Analysis

_{best}is the total tardiness of the best solution obtained by all of the compared algorithms, and avg corresponds to the average value of the total tardiness of the solution obtained by a selected algorithm. So the lower value of the ARPD means that better solutions are achieved.

#### 4.1. Parameter Setting

#### 4.2. Results and Comparison of the Instances

#### 4.3. Discussion of Experimental Results

## 5. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 7.**Factor trends of the tightness factor λ in NIPFSP with the total tardiness criterion minimization.

Tightness Factor $\mathit{\lambda}$ | Main Parameters | Factor Levels |
---|---|---|

1, 2, 3 | Maxgeneration | 100(1), 500(2), 1000(3) |

1, 2, 3 | PopSize | 10(1), 50(2), 100(3) |

1, 2, 3 | SP | 5(1), 8(2), 10(3) |

Experiment Number | Tightness Factor $\mathit{\lambda}$ | Main Parameters | ARV | ||
---|---|---|---|---|---|

Maxgeneration | PopSize | SP | |||

1 | 1 | 100(1) | 10(1) | 5(1) | 0.3664 |

2 | 1 | 500(2) | 50(2) | 8(2) | 0.1719 |

3 | 1 | 100(1) | 100(3) | 10(3) | 0.2064 |

4 | 2 | 100(1) | 50(2) | 10(3) | 0.2021 |

5 | 2 | 500(2) | 100(3) | 5(1) | 0.2449 |

6 | 2 | 1000(3) | 10(1) | 8(2) | 0.1749 |

7 | 3 | 100(1) | 100(3) | 8(2) | 0.2686 |

8 | 3 | 500(2) | 10(1) | 10(3) | 0.1571 |

9 | 3 | 1000(3) | 50(2) | 5(1) | 0.0728 |

Factor Level | Main Parameters | ||
---|---|---|---|

Maxgeneration | PopSize | SP | |

1 | 0.2790 | 0.2330 | 0.2280 |

2 | 0.1915 | 0.1489 | 0.2051 |

3 | 0.1514 | 0.2400 | 0.1887 |

Range | 0.1277 | 0.0910 | 0.0393 |

Rank | 1 | 2 | 3 |

Factors | Levels |
---|---|

Number of jobs | 40, 50, 60, 100 |

Number of machines | 20, 40, 60 |

Processing time on each machine | U(1, 100) |

Problem | GA | IEDA | CS | HEDA_CS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

AVE | MIN | MAX | AVE | MIN | MAX | AVE | MIN | MAX | AVE | MIN | MAX | |

n = 40, m = 20 | 1.13 | 0.33 | 1.98 | 0.86 | 0.03 | 1.89 | 0.85 | 0.09 | 1.79 | 0.83 | 0.00 | 1.82 |

n = 50, m = 20 | 1.31 | 0.45 | 2.05 | 0.94 | 0.23 | 1.87 | 0.89 | 0.10 | 1.77 | 0.77 | 0.00 | 1.70 |

n = 60, m = 20 | 1.93 | 1.03 | 4.26 | 1.31 | 0.35 | 3.06 | 1.38 | 0.21 | 2.89 | 0.73 | 0.00 | 1.52 |

n = 100, m = 20 | 2.36 | 1.31 | 5.18 | 1.53 | 0.36 | 3.86 | 1.21 | 0.18 | 3.41 | 0.43 | 0.00 | 0.93 |

n = 100, m = 40 | 2.53 | 2.13 | 6.31 | 1.43 | 0.61 | 4.19 | 1.10 | 0.53 | 1.95 | 0.85 | 0.00 | 1.76 |

n = 100, m = 60 | 3.76 | 3.35 | 7.51 | 1.51 | 1.03 | 4.51 | 1.43 | 0.99 | 2.97 | 0.94 | 0.00 | 1.92 |

Average | 2.17 | 1.43 | 4.55 | 1.26 | 0.44 | 3.23 | 1.14 | 0.35 | 2.46 | 0.76 | 0.00 | 1.61 |

Problem | GA | IEDA | CS | HEDA_CS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

AVE | MIN | MAX | AVE | MIN | MAX | AVE | MIN | MAX | AVE | MIN | MAX | |

n = 40, m = 20 | 1.14 | 0.29 | 2.07 | 0.89 | 0.01 | 2.13 | 0.91 | 0.06 | 1.93 | 0.89 | 0.00 | 1.84 |

n = 50, m = 20 | 1.19 | 0.36 | 2.84 | 1.18 | 0.17 | 2.52 | 1.09 | 0.15 | 2.21 | 0.75 | 0.00 | 1.88 |

n = 60, m = 20 | 1.67 | 1.07 | 2.65 | 1.51 | 0.68 | 3.15 | 1.38 | 0.23 | 2.97 | 0.62 | 0.00 | 1.42 |

n = 100, m = 20 | 2.13 | 1.79 | 3.09 | 1.87 | 1.01 | 3.24 | 1.57 | 0.29 | 3.31 | 0.41 | 0.00 | 0.92 |

n =100, m = 40 | 2.31 | 1.93 | 4.35 | 2.01 | 0.97 | 4.13 | 1.51 | 0.35 | 3.15 | 0.93 | 0.00 | 1.99 |

n = 100, m = 60 | 2.60 | 2.01 | 4.32 | 1.97 | 1.13 | 5.01 | 1.89 | 0.51 | 3.68 | 1.16 | 0.00 | 2.07 |

Average | 1.84 | 1.24 | 3.22 | 1.57 | 0.66 | 3.36 | 1.39 | 0.27 | 2.88 | 0.79 | 0.00 | 1.69 |

Problem | GA | IEDA | CS | HEDA_CS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

AVE | MIN | MAX | AVE | MIN | MAX | AVE | MIN | MAX | AVE | MIN | MAX | |

n =40, m = 20 | 0.83 | 0.13 | 2.11 | 0.79 | 0.09 | 1.99 | 0.81 | 0.05 | 1.81 | 0.78 | 0.00 | 1.93 |

n = 50, m = 20 | 0.99 | 0.27 | 2.25 | 0.86 | 0.14 | 2.07 | 0.82 | 0.01 | 1.83 | 0.78 | 0.00 | 1.62 |

n = 60, m = 20 | 1.46 | 1.12 | 2.69 | 0.97 | 0.31 | 2.21 | 0.95 | 0.16 | 2.12 | 0.67 | 0.00 | 1.52 |

n = 100, m = 20 | 1.68 | 1.53 | 2.86 | 1.31 | 0.46 | 2.56 | 1.16 | 0.25 | 2.51 | 0.42 | 0.00 | 0.93 |

n = 100, m = 40 | 1.79 | 1.67 | 2.90 | 1.46 | 0.51 | 2.73 | 1.31 | 0.29 | 2.75 | 1.02 | 0.00 | 1.85 |

n = 100, m = 60 | 2.14 | 1.89 | 2.98 | 1.58 | 0.59 | 2.64 | 1.62 | 0.36 | 2.81 | 1.28 | 0.00 | 2.12 |

Average | 1.48 | 1.10 | 2.63 | 1.16 | 0.35 | 2.37 | 1.11 | 0.19 | 2.03 | 0.83 | 0.00 | 1.66 |

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**MDPI and ACS Style**

Sun, Z.; Gu, X.
Hybrid Algorithm Based on an Estimation of Distribution Algorithm and Cuckoo Search for the No Idle Permutation Flow Shop Scheduling Problem with the Total Tardiness Criterion Minimization. *Sustainability* **2017**, *9*, 953.
https://doi.org/10.3390/su9060953

**AMA Style**

Sun Z, Gu X.
Hybrid Algorithm Based on an Estimation of Distribution Algorithm and Cuckoo Search for the No Idle Permutation Flow Shop Scheduling Problem with the Total Tardiness Criterion Minimization. *Sustainability*. 2017; 9(6):953.
https://doi.org/10.3390/su9060953

**Chicago/Turabian Style**

Sun, Zewen, and Xingsheng Gu.
2017. "Hybrid Algorithm Based on an Estimation of Distribution Algorithm and Cuckoo Search for the No Idle Permutation Flow Shop Scheduling Problem with the Total Tardiness Criterion Minimization" *Sustainability* 9, no. 6: 953.
https://doi.org/10.3390/su9060953