# A Heuristic Integrated Scheduling Algorithm via Processing Characteristics of Various Machines

^{1}

^{2}

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Analysis and Description

- (1)
- Each operation has three elements: the operation number, the machine number, and the processing time;
- (2)
- Each occupied machine has two properties: the time certainty and the processing continuity;
- (3)
- Except for leaf node operations, the sufficient and necessary condition of each operation is that all of its immediate predecessor operations are completed;
- (4)
- The end time of the last operation is the total processing time of the product.

_{ij}to indicate whether operation i is allocated earlier to the same machine as operation j. For example, y

_{ij}= 1 and y

_{ji}= 0 mean that operation i processes before j on the same machine. For each operation, if its starting time is S

_{i}, then after processing time P

_{i}on its corresponding machine, its completion time is C

_{i}. The scheduling goal is to shorten the completion time of the final operation, as shown in Formula (1). Constraint (2) says that each operation j must be stated after its immediate predecessor operation i finishes, and the relation set R = {(i, j) |if i is the immediate predecessor of j} is used to store the constraint relationships. It represents the precedence constraints. Constraint (3) guarantees that operations on the same machine cannot be processed at the same time, and L is a large positive constant. Formulas (3) and (4) give the boundary of y

_{ij}, S

_{i}, and P

_{i}to make them significant in practice.

_{MOM}of all the operations in MOM, and the subscript means the schedule number of each operation in the sequence. For example, i

_{11}is the 11th operation, which will be scheduled on multi-operation machines. Formula (7) says the reason for operation i

_{1}is scheduled earlier than i

_{2}. As the operations in MOM are scheduled by the strategy of layer priority and the strategy of leaf node first, if operation i

_{1}is scheduled earlier than i

_{2}, the layer priority ${L}_{{i}_{1}}$ must be larger than ${L}_{{i}_{2}}$. If the layer priorities are the same, the binary leaf node judge parameter ${\alpha}_{{i}_{1}}$ should be the larger one. If ${\alpha}_{{i}_{1}}=1$, operation i

_{1}is a leaf node. Formula (8) represents the scheduling sequence Q

_{MTM}of all the t operations in MTM. Additionally, the operations in MTM are scheduled by the strategy of layer priority and the strategy of the highest constraint degree first. Formula (9) explains why operation i

_{1}is scheduled earlier than i

_{2}. Similar to Formula (7), when the layer priorities are the same, the constraint degree ${D}_{{i}_{1}}$ must be larger than ${D}_{{i}_{2}}$. Formula (10) represents the scheduling sequence Q

_{GM}of all the g operations in GM. Additionally, the operations in GM are scheduled by the strategy of large path value, the strategy of layer priority, and the strategy of the highest constraint degree first. Additionally, Formula (11) explains why operation i

_{1}is scheduled earlier than i

_{2}. If operation i

_{1}has a larger path value ${P}_{{i}_{1}}$, higher layer priority ${L}_{{i}_{1}}$, or larger constraint degree ${D}_{{i}_{1}}$, then operation i

_{1}is more preferred. Formula (12) says that the total number of these three operation sets must be equal to the total number of the operations of the product.

## 3. Algorithm Design Ideas

#### 3.1. Relevant Definitions

**Definition**

**1**

**([33]).**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

#### 3.2. Description of the Algorithm

#### 3.3. Algorithm Complexity Analysis

^{2}). To find the best time in the idle time period, the operation needs to be compared with the end time of its immediate predecessors (sum), and the worst situation is that it not only needs to be compared with all the operations on the same layer with it but also with all the operations on the same machine, so the time complexity is O(n

^{2}).

^{2})} = O(n

^{2}).

## 4. Example Analysis

#### 4.1. Complex Product Scheduling Demonstration

_{2}is the multi-operation machine, M

_{1}is the more-time machine, M

_{3}and M

_{4}are general machines.

_{2}, the seamless scheduling operations are {A27, A25, A21, A20, A15, A10, A6, A3}, as shown in Figure 3.

_{1}in turn on the basis of the constraints between operations, as shown in Figure 4.

#### 4.2. Comparison and Analysis of Asymmetric Complex Product Scheduling

- (1)
- ANCOG is adopted to prioritize operation scheduling in the closely connecting operation groups, which ignores the influence of the relative position of operations with a low constraint degree on the scheduling results and leads to the idle time periods in the operations during the serial scheduling process. A comparative analysis of Figure 8 and Figure 9 shows that the multi-operation machine M
_{2}in Figure 9 has a long idle time from t = 17 to t = 24 with a total of 10 working hours. Due to the priority scheduling of A20 of the special machine M_{2}, in Figure 8, A16, A12, A9, A7, and A4, are 5, 4, 2, 2, and 4 working hours ahead of those in Figure 9. - (2)
- TSACCSP does not consider the processing and utilization of multi-operation machines during the determination of the starting time of operations, which affects the overall scheduling effect. A comparative analysis of Figure 8 and Figure 10 shows that the more-time machine M
_{2}is idle from t = 0 to t = 2, t = 5 to t = 7, t = 14 to t = 18, and t = 19 to t = 24 in Figure 10. In Figure 8, the starting times of A27, A25, A15, A6, and A3 are 7, 8, 17, and 5 working hours, earlier than those in Figure 10, respectively. This only increases the tightness of continuous processing on M_{2}, but it also increases the more-time machine utilization rate by 9.9%. - (3)
- In ACHSO, the focus of optimization is operations at the same layer. The strategies of “Layer first” and “Leaf node process of the same layer first” are both horizontal optimization in nature, and the problem of “emphasizing the horizontal while neglecting the vertical” appears. A comparative analysis of Figure 8 and Figure 11 shows that, due to the priority scheduling of process A21 in Figure 8, its subsequent operations {A16, A12, A9, A7, A4} are 7, 7, 4, 4, and 6 working hours earlier than those shown in Figure 11. This realizes the close processing of the operations.
- (4)
- In HIS-PCVM, the multi-operation machines and the more-time machines are added to the integrated scheduling mechanism as special factors. First, the layer priority strategy and leaf node operation priority strategy ensure the parallel processing effect of the special machine. Then, for the general machine, according to the scheduling principles of the path value and the constraint degree from large to small, the subsequent operations of the operations on special equipment can be processed as soon as possible, vertically.

#### 4.3. Comparison and Analysis of Symmetric Complex Product Scheduling

_{3}. It is not only a multi-operation machine but also a more-time machine. The general machines are M

_{1}, M

_{2}, and M

_{4}.

_{3}is {B19, B13, B11, B6, B9, B4}, and on the general machine, it is {B18, B15, B14, B12, B17, B8, B7, B16, B5, B3, B2, B10, B1}, as shown in Figure 13. The makespan is 185 working hours.

#### 4.4. Comparison and Analysis of Five Scheduling Algorithms

- (1)
- HIS-PCVM adopts the optimization strategy of “special equipment”. It takes the particularity of machine resources as the research perspective. Additionally, it internalizes the overall optimization effect of integrated scheduling into the optimization of special equipment, so as to drive further optimization of the other machines.
- (2)
- The strategies of “the layer priority” and “the path value” fully compensate for the disadvantages of “attaching importance to horizontal optimization and discarding vertical optimization“ and “attaching importance to vertical optimization and discarding horizontal optimization“ in integrated scheduling. HIS-PCVM not only considers the leaf node operations with low layer priority but also considers the scheduling problem on the long path.
- (3)
- HIS-PCVM adopts the strategy of “the earliest scheduling time”, which effectively uses the scheduling gap between the serial operations caused by inserting the relevant operations into the idle time of the machine.
- (4)
- HIS-PCVM adopts the strategy of “the constraint degree”, which is based on the structure properties of the product itself, to comprehensively consider the various constraint relations between the processing operations. It solves the problem of processing gaps on machines due to a weak tight cohesion between the operations.

## 5. Conclusions and Future Research

- (1)
- This paper takes “the special equipment processing characteristics” as the important optimization factors. It considers the variety of machine processing characteristics and the influence of complex products on scheduling results. It achieves the effect of optimizing integrated scheduling by scheduling the operations corresponding to the special equipment.
- (2)
- The proposed algorithm realizes both horizontal optimization and vertical optimization by the layer priority strategy, the earliest scheduling time strategy, and the path value strategy. The layer priority strategy realizes parallel optimization in landscape orientation. The other two strategies realize optimization in the longitudinal direction.
- (3)
- It reduces the serial gap between operations, improves machine utilization, and shortens the makespan of complex products. Thus, it provides a new method to solve the integrated scheduling problem and expands the ideas on solving the problem. Therefore, the proposed algorithm has a certain theoretical and practical significance.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**Gantt chart of asymmetric complex product A after the adjustment at the earliest scheduling time of 26 working hours.

**Figure 10.**Gantt chart of asymmetric complex product A in the reverse sequence by TSACCSP with 31 working hours.

**Figure 13.**Gantt chart of symmetric complex product B by HIS-PCVM with 185 working hours. (

*****Indicates the special equipment is unique.)

**Figure 14.**Gantt chart of symmetric complex product B by RSVHPSA with 205 working hours. (

*****Indicates the special equipment is unique).

**Figure 15.**Results of ANCOG, TSACCSP, ACHSO, RSVHPSA, and HIS-PCVM for integrated scheduling with asymmetric structures: (

**a**) 20 operations, (

**b**) 50 operations, (

**c**) 100 operations, and (

**d**) 200 operations.

**Figure 16.**Results of ANCOG, TSACCSP, ACHSO, RSVHPSA, and HIS-PCVM for integrated scheduling with symmetric structures: (

**a**) 20 operations, (

**b**) 50 operations, (

**c**) 100 operations, and (

**d**) 200 operations.

**Figure 17.**Comparison results of ANCOG, TSACCSP, ACHSO, RSVHPSA, and HIS-PCVM for integrated scheduling with asymmetric structures: (

**a**) optimal solution ratio, (

**b**) variance curves, and (

**c**) CPU time (the points in the small square are the outliers for TSACCSP).

**Figure 18.**Comparison results of ANCOG, TSACCSP, ACHSO, RSVHPSA, and HIS-PCVM for integrated scheduling with symmetric structures: (

**a**) optimal solution ratio, (

**b**) variance curves, and (

**c**) CPU time (the points in the small square are the outliers for TSACCSP).

**Table 1.**Statistics of complex product A: the layer priorities, the leaf nodes, and the constraint degree.

Operation | Layer Priority | Constraint Degree | Leaf Node |
---|---|---|---|

A1 | 1 | 1 | No |

A2 | 2 | 3 | No |

A3 | 3 | 3 | No |

A4 | 3 | 2 | No |

A5 | 4 | 2 | No |

A6 | 4 | 1 | Yes |

A7 | 4 | 2 | No |

A8 | 5 | 2 | No |

A9 | 5 | 3 | No |

A10 | 6 | 3 | No |

A11 | 6 | 2 | No |

A12 | 6 | 2 | No |

A13 | 7 | 2 | No |

A14 | 7 | 1 | Yes |

A15 | 7 | 3 | No |

A16 | 7 | 3 | No |

A17 | 8 | 1 | Yes |

A18 | 8 | 3 | No |

A19 | 8 | 1 | Yes |

A20 | 8 | 2 | No |

A21 | 8 | 1 | Yes |

A22 | 9 | 1 | Yes |

A23 | 9 | 2 | No |

A24 | 9 | 1 | Yes |

A25 | 10 | 3 | No |

A26 | 11 | 1 | Yes |

A27 | 11 | 1 | Yes |

**Table 2.**Comparison and analysis of machine utilization of four algorithms of asymmetric complex product A.

Multi-Operation Machine Utilization Ratio | More-Time Machine Utilization Ratio | Overall Utilization Rate of Machine | Relative Improvement Ratio of Overall Machine Utilization | |
---|---|---|---|---|

ANCOG | 60% | 57.1% | 57.7% | 5.2% |

TSACCSP | 53.6% | 51.6% | 48.2% | 14.7% |

ACHSO | 65.2% | 57.1% | 56.7% | 6.2% |

HIS-PCVM | 65.2% | 61.5% | 62.9% | ------ |

**Table 3.**Comparison and analysis of scheduling results of two algorithms of symmetric complex product B.

HIS-PCVM | RSVHPSA | |
---|---|---|

Algorithm idea | 1. On the special machine: using the strategies of “the layer priority”, “the leaf node operation first”, and “the long processing time first” to establish the scheduling sequence; 2. On the general machine: using the strategies of “the highest path value priority”, “the layer priority”, and “the highest constraint degree priority” to establish the scheduling sequence. | 1. Split symmetric process tree into several sub-trees; 2. Establish the pre-scheduling sequence according to the descending order of processing time of sub-trees; 3. On the same machine: establish the scheduling sequence by the machine process pre-start time. |

Schedule sequence | {B19, B16, B10, B12, B8, B4, B18, B13, B11, B17, B6, B5, B15, B14, B3, B2, B9, B1} | M_{1}: {B17, B6, B2};M _{2}: {B18, B7, B5, B9, B3};M _{3}: {B12, B19, B16, B10, B4, B8};M _{4}: {B15, B11, B13, B14}. |

Total processing time | 185 | 205 |

Overall utilization rate of machine | 75.4% | 67.8% |

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## Share and Cite

**MDPI and ACS Style**

Zhou, W.; Zhou, P.; Zheng, Y.; Xie, Z.
A Heuristic Integrated Scheduling Algorithm via Processing Characteristics of Various Machines. *Symmetry* **2022**, *14*, 2150.
https://doi.org/10.3390/sym14102150

**AMA Style**

Zhou W, Zhou P, Zheng Y, Xie Z.
A Heuristic Integrated Scheduling Algorithm via Processing Characteristics of Various Machines. *Symmetry*. 2022; 14(10):2150.
https://doi.org/10.3390/sym14102150

**Chicago/Turabian Style**

Zhou, Wei, Pengwei Zhou, Ying Zheng, and Zhiqiang Xie.
2022. "A Heuristic Integrated Scheduling Algorithm via Processing Characteristics of Various Machines" *Symmetry* 14, no. 10: 2150.
https://doi.org/10.3390/sym14102150