Flexible job shop scheduling (FJSP) is a complex problem that closely mirrors real-world manufacturing scenarios. To address this challenge and minimize completion times, this paper proposes an improved hybrid genetic algorithm (HGA) that integrates the simulated annealing algorithm (SA). The proposed algorithm utilizes a global, local, and random (GLR) population initialization strategy to optimize initial solutions. It then expands the number of high-quality solutions within the population through two crossover and mutation strategies focused on process order and machine allocation. Finally, neighborhood structures are introduced in the simulated annealing process for further optimization. This approach enhances the searching speed and avoids the pitfalls of traditional genetic algorithms that often get stuck in local optima. Simulation experiments conducted on gear shaft manufacturing data demonstrate that the proposed HGA improves production efficiency by 8.43% compared to traditional genetic algorithms, validating its effectiveness.
1. Introduction
Flexible job shop scheduling (FJSP) builds upon traditional job shop scheduling (JSP) by introducing additional constraints such as varying machine options, processing time for different operations, and processing sequence constraints across jobs. These factors render FJSP a non-deterministic polynomial (NP)-hard problem, characterized by significant computational and storage demands that current computing capabilities find challenging to handle.
With the continuous advancement of artificial intelligence technologies, intelligent optimization algorithms such as particle swarm optimization (PSO), genetic algorithms (GA), and grey wolf algorithms have emerged as popular solutions to FJSP. Researchers worldwide have explored hybrid algorithms by integrating genetic algorithms with other effective local search methods to enhance performance. This paper presents an improved hybrid genetic algorithm that combines GLR initialization and simulated annealing, aiming to improve searching efficiency and solution quality.
2. Mathematical Model of the Flexible Job Shop Scheduling Problem
FJSP is defined as follows: There are n unprocessed jobs, and m machines available for processing. Each job’s operations can be processed on any machine within the machine set, with varying processing times due to machine models and wear and tear. Jobs are processed strictly in sequence, and at least one operation has multiple machine options. The objective is to optimize production scheduling under specific criteria.
- Job Set: J = {J1, J2, …, Jn}, where Ji represents the ith job, i = 1, 2, …, n.
- Machine Set: M = {M1, M2, …, Mm}, where Mk represents the kth machine, k = 1, 2, …, m.
- Operation Set: O = {Oi,j}, where Oi,j represents the jth operation of the ith job.
The parameters for FJSP are defined as follows:
- n: Total number of jobs.
- m: Total number of machines.
Constraints:
- The sequence of operations for each job must be followed.
- The completion time for each job is constrained.
- Only one machine can process a single operation at a given time.
- Machines can operate cyclically.
- All input parameters are positive.
3. Design of the Improved Hybrid Genetic Algorithm
3.1 Chromosome Encoding
Random population initialization is commonly used in job shop scheduling research, but it can lead to issues such as similar individuals, poor solution quality, and incomplete search spaces. To address these issues, this paper proposes an improved GLR hybrid initialization method.
The GLR method involves multiple selection processes: global, local, and random. In the global selection, a random selection mode is applied, with processing times added to an array until the process is complete. The steps for GLR initialization are as follows:
- Initialize an array corresponding to the total number of machines, with array values representing processing times and machine numbers.
- Select a target job and its corresponding operation.
- Add the selected time data to the total processing time array.
- Select the machine with the smallest processing time value from the array and record it.
- Update the array data for the next time recording.
- Select the next operation and repeat steps 3-5 until all machines are selected.
- Move to the next target job and repeat the process until all jobs are selected.
To disperse the initial population, local and global selection operations are similar but reset the array for each new job selection.
3.2 Fitness Function
The fitness function evaluates the quality of individuals in the population. Typically, the fitness of an individual is positively correlated with its probability of genetic transmission. In this optimization, the objective function is the maximum completion time, and the fitness function fg is the reciprocal of the maximum completion time: fg = 1/Cmax, where Cmax is the maximum completion time.
3.3 Selection Operation
In genetic algorithms, the selection operator chooses individuals with higher fitness as parents for the next generation. The tournament selection method is used here, where the fittest individual in each iteration is selected for the next generation until the new population size is restored. After selection, the remaining individuals are returned to the population to preserve diversity and protect high-quality individuals.
3.4 Crossover Operation
Crossover generates new individuals by rearranging specific-length genes from two individuals. The quality of crossover operators affects algorithm efficiency and convergence speed. Given the complexity of FJSP, a new encoding method is chosen for process and machine coding.
3.4.1 Crossover Based on Operation Partial Encoding
The IPOX (Improved Position-based Crossover) method is selected, which crosses only the processing sequences in the chromosome while preserving the machine sequences in the offspring. This ensures the inheritance of parental traits. The IPOX crossover process is as follows:
- Randomly divide all jobs into sets G1 and G2.
- Copy genes from parent 1 that belong to G1 to offspring 1 and similarly for parent 2.
- Place genes from parent 1 belonging to G1 in offspring 2 and those from parent 2 belonging to G2 in offspring 2, maintaining their positions.
Alternatively, UX crossover can be used:
- Generate a binary string R of the same length as the parent chromosome.
- Retain genes from parent 1 at positions corresponding to 1s in R in offspring 1, and similarly for parent 2 and offspring 2.
- Place the remaining genes from parent 1 in offspring 2 and those from parent 2 in offspring 1.
3.5 Mutation Operation
Mutation is crucial for population diversity and avoiding local optima. In each iteration, specific genetic operations are modified to maintain diversity.
4. Simulated Annealing Algorithm Integration
Simulated annealing (SA) is incorporated into the hybrid genetic algorithm to further refine solutions. After crossover and mutation, the SA adjusts the population, enhancing the search range and allowing solutions to escape local optima, thus improving the algorithm’s accuracy and speed.
5. Experimental Results and Analysis
Simulation experiments were conducted on gear shaft manufacturing data to compare the proposed HGA with traditional GAs. The results demonstrate the effectiveness of the improved HGA, with an 8.43% increase in production efficiency.
6. Conclusion
This paper proposes an improved hybrid genetic algorithm for FJSP that integrates GLR initialization and simulated annealing. The proposed algorithm enhances searching efficiency, avoids local optima, and demonstrates superiority in simulation experiments. Future work could explore more complex constraint conditions and larger-scale job shop scheduling problems.
Tables and Illustrations
(Note: Due to the limitation of this response format, tables and illustrations are described in text form. In an actual article, tables would be formatted accordingly, and illustrations would be included as images.)
Table 1: Parameters of the FJSP Model
| Parameter | Description |
|---|---|
| n | Total number of jobs |
| m | Total number of machines |
| Oi,j | The jth operation of the ith job |
| Ji | The ith job |
| Mk | The kth machine |
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