The job shop scheduling problem is a classical combinatorial optimization challenge frequently encountered in manufacturing systems. It involves determining the optimal execution sequences for a set of jobs on various machines to maximize production efficiency and meet multiple objectives. The Non-dominated Sorting Genetic Algorithm III (NSGA-III) is an effective approach for solving the multi-objective job shop scheduling problem. Nevertheless, it has some limitations in solving scheduling problems, including inadequate global search capability, susceptibility to premature convergence, and challenges in balancing convergence and diversity. To enhance its performance, this paper introduces a strengthened dominance relation NSGA-III algorithm based on differential evolution (NSGA-III-SD). By incorporating constrained differential evolution and simulated binary crossover genetic operators, this algorithm effectively improves NSGA-III’s global search capability while mitigating premature convergence issues. Furthermore, it introduces a reinforced dominance relation to address the trade-off between convergence and diversity in NSGA-III. Additionally, effective encoding and decoding methods for discrete job shop scheduling are proposed, which can improve the overall performance of the algorithm without complex computation. To validate the algorithm’s effectiveness, NSGA-III-SD is extensively compared with other advanced multi-objective optimization algorithms using 20 job shop scheduling test instances. The experimental results demonstrate that NSGA-III-SD achieves better solution quality and diversity, proving its effectiveness in solving the multi-objective job shop scheduling problem.

The Job Shop Scheduling Problem (JSSP) refers to a class of problems where the objective is to optimize scheduling by arranging the processing sequence of operations within a set of tasks, while adhering to task-specific constraints and precedence relations among operations [

JSSP, a prominent focus in intelligent manufacturing research, exemplifies a classic NP-hard combinatorial optimization challenge, rendering its resolution exceedingly arduous [

In classical JSSP models, the primary optimization objective has traditionally focused on minimizing the single criterion of the makespan. However, with the advancement of factory automation, a single scheduling criterion can no longer meet the diverse production requirements. Therefore, in this paper, we consider a total of five objectives to solve the Multi-Objective Job Shop Scheduling Problem (MO-JSSP). These objectives encompass total flow time, total tardiness time, average machine idle time, and implementing a Just-in-Time (JIT) production mode, in addition to the traditional objective of makespan. When there are too many objectives, traditional algorithms struggle to find suitable trade-offs and equilibrium points among multiple objectives. Moreover, the solution space of MO-JSSP is vast, which may lead to computational complexity and long computation time for traditional optimization algorithms.

The NSGA-III algorithm, proposed by Deb et al. [

The main contributions of this paper are as follows:

Proposed hybrid constraint differential evolution and simulated binary crossover to enhance the searchability of NSGA-III and avoid premature convergence.

Introduced a new reinforced dominance relation to improve the convergence and diversity of the algorithm.

Designed effective encoding and decoding methods for MO-JSSP.

The rest of the paper is organized as follows:

JSSP has remained a focal point of research for numerous years, and scholars have undertaken extensive investigations to find efficient solutions. For the single-objective JSSP, the primary methods include mathematical programming and metaheuristics. For instance, Meng et al. [

With the rapid development of the manufacturing industry and the increasing complexity of customer requirements, traditional single-objective JSSP struggles to accommodate today’s diverse production requirements. Consequently, more and more scholars have embarked on research into the multi-objective JSSP. For instance, Liu et al. [

NSGA-III is considered a favorite algorithm for solving multi-objective problems due to its efficient crossover and mutation operations. These operations help overcome the inefficiency of simple crossover and mutation patterns found in many algorithms. Many scholars have dedicated their efforts to improving the NSGA-III algorithm for efficient resolution of MO-JSSP. Wu et al. [

In summary, the methods mentioned above have made notable advancements. However, they still possess certain limitations. As a result, researchers continue to explore new approaches to tackle the MO-JSSP. The objective of this paper is to present an enhanced NSGA-III algorithm that incorporates efficient crossover and mutation mechanisms, along with a more effective individual selection method. These enhancements will enable the algorithm to meet diverse production requirements and effectively solve the MO-JSSP problem.

MO-JSSP can be described as _{1}, _{2}, _{3}, …, _{m}} for _{i} operations. The _{ij}. The processing time _{ij} for process _{ij} is determined by the specific performance of the machine _{ij}) being processed, and the delivery time of job _{i}. The scheduling task is to determine the optimal machining sequence of

This model evaluates the production efficiency of the entire scheduling model based on objectives such as makespan, flow time, tradiness time, average machine idle time and the Just-in-Time (JIT) production mode. The optimization objective set can be represented as _{1}, _{2}, _{3}, _{4}, _{5}), where minimizing the makespan and minimizing the total flow time aim to reduce job processing time and improve the overall efficiency of the production workshop. Minimizing total tardiness time and adhering to the JIT production mode are intended to maximize customer satisfaction by ensuring on-time delivery, thereby enhancing the company’s reputation. Minimizing average machine idle time aims to maximize machine utilization and shorten the overall project duration. The formulas for the objective functions are as follows:

Makespan:

Total flow time:

Total tradiness time:

Average machine idle time:

Just in Time (JIT) production mode time:

S.T.

_{i} in _{i} in _{m} represents the time required for the

In _{ij} on the optional processing machine. _{ij} process is _{ij}).

In scheduling, an effective encoding method is crucial for the overall optimization of scheduling solutions [

The key issue of JSSP in _{1} and _{2}. For example, the processing sequence of _{12} must be after _{11}. The processing sequence of _{1} is {_{11}, _{32}, _{22}, _{42}, _{51}}, and the processing sequence of _{2} is {_{52}, _{41}, _{31}, _{12}, _{21}}.

Since the NSGA-III algorithm is primarily designed for solving continuous optimization problems and cannot be directly applied to typical discrete combinatorial JSSP, it is necessary to decode the solutions into operation code formats at the end of each iteration. Furthermore, it is essential to handle any illegal solutions generated during the decoding process. In this regard, the paper has devised an effective decoding method to facilitate the transition between the population positions in the continuous solution space and the discrete JSSP encoding of operations, as depicted in

(1) Obtain the individual position

(2) The individual positions of the continuous solution are converted to job individual position codes by rounding down;

(3) Handle the generated illegal solutions effectively, following these steps:

i) Determine whether the number of operations for each job equals the maximum allowed. If it does, no illegal solutions are generated.

ii) If the number exceeds the maximum, identify the positions of all jobs with an excessive number of operations, and randomly replace the operations exceeding the maximum with operations from adjacent jobs, adjusting the counts accordingly.

iii) If the number does not exceed the maximum, randomly select positions outside the index of the illegal job and fill the missing operations with operations from that job.

(4) Establish a valid correspondence between individual positions and operation encoding.

After adopting this encoding and decoding scheme, each set of encoding schemes can be regarded as a chromosome, which facilitates the algorithm to perform corresponding crossover and mutation, thereby gradually optimizing the scheduling results. Moreover, the proposed approach in this paper allows for the rapid transformation of individual positions into valid discrete encodings, without introducing complex calculations that would burden the algorithm. It is a simple and efficient method.

The Simulated Binary Crossover (SBX) [_{1} and _{2}, and two offspring are generated through SBX, resulting in chromosomes _{1} and _{2}.

In

In

The probability of polynomial mutation is 1/D [

SBX and Polynomial mutation are the original evolutionary operators of NSGA-III. However, its overall search performance is poor and prone to premature convergence. Moreover, its dominance relation performs poorly in balancing convergence and diversity, often resulting in a small portion of the solution set being concentrated on the Pareto front. To enhance the algorithm’s performance, new evolutionary operators and dominance relations have been introduced based on experimental research, as described in the following text.

Introducing Differential Evolution (DE) into the algorithm’s evolution process can effectively enhance the diversity of solutions. However, it is important to note that traditional DE operators scale the difference vector based on the differences, which can result in mutation having a high degree of randomness and a lack of direction [_{m}, and the mutation probability is 0.15. Then restrict the difference vector

The specific constraint differential evolution strategy is shown in Algorithm 1.

The algorithm enhances the diversity of solutions by introducing a constrained differential evolution mechanism, which allows for better exploration of the search space. Additionally, by setting upper and lower parameter limits and using polynomial mutation, NSGA-III-SD can control the mutation intensity of individuals, thereby balancing the trade-off between exploration and exploitation during the search process and avoiding getting stuck in local optima. This evolutionary approach enables NSGA-III-SD to excel in global search and prevent premature convergence.

Elimination and selection of individuals will inevitably occur during the evolution of algorithms. In multi-objective optimization, the pareto dominance relation is widely used to distinguish the superiority and inferiority of candidate solutions. Reviewing the existing dominance relations, there are four representative classes:

1. The first class of dominance relations aims to improve selection pressure by expanding the dominance region. Examples include methods like S-CDAS [

2. The second class of dominance relations is based on the grid-based approach in the objective space, such as

3. The third class of dominance relations introduces a new dominance relation defined using fuzzy logic, as seen in

4. The fourth class of dominance relations is defined by a set of weight vectors, as seen in

However, most existing dominance relations increase the algorithm’s selection pressure and struggle to balance the trade-off between convergence and diversity [

The SDR is defined as follows:

According to the first equation in

To better illustrate this process, _{1} is located within the niche range of _{1}. On the other hand, _{2} is located outside the niche range of _{2}. The dominance region of

The non-dominated region identified by SDR covers the entire Pareto front, while other dominance relations often shrink to a small region or fail to comprehensively cover the Pareto front. In comparison, SDR maintains a better balance between convergence and diversity. Additionally, SDR does not rely on aggregation functions or weight vectors but can adaptively select candidate solutions with better convergence and diversity. This adaptability allows SDR to handle Pareto front of various shapes.

This algorithm operates within the framework of the NSGA-III algorithm and is executed as follows:

Step 1: Initialize the population and algorithm parameters.

Step 2: Generate the mating pool using tournament selection. Check if the evaluation count is less than or equal to one-third of the maximum evaluation count. If it is, use constrained differential evolution as the genetic operator to generate offspring population. Otherwise, use simulated binary crossover and polynomial mutation to generate offspring population.

Step 3: Merge the parent and offspring populations.

Step 4: Apply the SDR fast non-dominated sorting strategy to divide the merged population into non-dominated layers. Use the NSGA-III algorithm based on reference points to select suitable individuals from the last front to determine the next population.

Step 5: According to the encoding and decoding scheme, the individual position is converted into the operation encoding.

Step 6: Check if the termination condition is met. If yes, end the iteration. Otherwise, return to Step 2.

The specific NSGA-III-SD algorithm is detailed in Algorithm 2.

This study employs a set of 20 widely recognized job shop scheduling test datasets to validate the algorithm’s effectiveness. These datasets are sourced from Fisher & Thompson (FT) [

To validate the effectiveness of the NSGA-III-SD algorithm, it is compared with five advanced MOEAs: VaEA [

To ensure experimental fairness, consistent parameter settings are applied to all algorithms. According to the parameters mentioned by He et al. [

To evaluate the quality, diversity, and distribution of the algorithm's solutions, two commonly used multi-objective evaluation metrics, Coverage of two sets (C) [

_{1} and _{2} represent two different Pareto solution sets, _{2} that are either dominated by or equal to solutions in _{1}, and its value ranges between 0 and 1. Specifically, when _{2} can be found in _{1}, and they are either dominated by or equal to solutions in _{1}. Conversely, when _{2} is dominated by any solution in _{1}.

In this equation,

a: NSGA-III-SD b: VaEA c: SRA d: MaOEACSS e: NSGA-III f: NSGA-II | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Problem | C (a,b) | C (b,a) | C (a,c) | C (c,a) | C (a,d) | C (d,a) | C (a,e) | C (e,a) | C (a,f) | C (f,a) | |

FT06 | 6 × 6 | 0.741 | 0.701 | 0.770 | 0.789 | 0.346 | |||||

FT10 | 10 × 10 | 0.785 | 0.412 | 0.759 | 0.731 | 0.244 | |||||

FT20 | 20 × 5 | 0.891 | 0.776 | 0.902 | 0.860 | 0.868 | |||||

ABZ7 | 15 × 20 | 0.972 | 0.931 | 0.965 | 0.964 | 0.967 | |||||

ABZ8 | 15 × 20 | 0.946 | 0.854 | 0.960 | 0.960 | 0.964 | |||||

ABZ9 | 15 × 20 | 0.944 | 0.816 | 0.948 | 0.952 | 0.960 | |||||

SWV01 | 20 × 10 | 0.927 | 0.854 | 0.933 | 0.906 | 0.926 | |||||

SWV04 | 20 × 10 | 0.939 | 0.840 | 0.939 | 0.917 | 0.925 | |||||

SWV06 | 20 × 15 | 0.923 | 0.810 | 0.941 | 0.943 | 0.950 | |||||

SWV10 | 20 × 10 | 0.950 | 0.910 | 0.966 | 0.944 | 0.973 | |||||

SWV12 | 50 × 10 | 0.926 | 0.877 | 0.941 | 0.933 | 0.950 | |||||

SWV15 | 50 × 10 | 0.940 | 0.911 | 0.955 | 0.940 | 0.953 | |||||

LA01 | 10 × 5 | 0.787 | 0.656 | 0.815 | 0.768 | 0.785 | |||||

LA06 | 15 × 5 | 0.902 | 0.864 | 0.914 | 0.872 | 0.922 | |||||

LA11 | 20 × 5 | 0.918 | 0.886 | 0.900 | 0.902 | 0.918 | |||||

LA21 | 15 × 10 | 0.935 | 0.880 | 0.921 | 0.936 | 0.950 | |||||

LA26 | 20 × 10 | 0.966 | 0.925 | 0.956 | 0.956 | 0.955 | |||||

LA31 | 30 × 10 | 0.966 | 0.912 | 0.950 | 0.951 | 0.962 | |||||

LA33 | 30 × 10 | 0.962 | 0.921 | 0.952 | 0.960 | 0.962 | |||||

LA35 | 30 × 10 | 0.950 | 0.896 | 0.935 | 0.957 | 0.949 | |||||

NSGA-III-SD | VaEA | SRA | MaOEACSS | NSGA-III | NSGA-II | ||||||

>/< | 19/1 | 20/0 | 20/0 | 20/0 | 18/2 |

These comparative findings underscore the outstanding performance of the NSGA-III-SD algorithm in addressing MO-JSSP, attributed to its exceptional evolutionary strategy. Unlike the singular search approach of NSGA-III, NSGA-III-SD incorporates advanced evolutionary techniques like constrained differential evolution and simulated binary crossover. These techniques enable a diverse evolution mode and robust global exploration capability, mitigating the risk of premature convergence. The algorithm’s effective exploration of the solution space contributes to the production of higher-quality solutions and overall performance in addressing the MO-JSSP.

In addition to conducting a comparative analysis, we performed further analysis to provide a deeper understanding of the results. This section explores the algorithm’s performance across various datasets and problem complexities. Notably, the NSGA-III-SD algorithm consistently exhibited exceptional performance even when faced with an increasing number of jobs and machines. This indicates its robustness and scalability in handling most instances of the MO-JSSP. Furthermore, analysis reveals that the competitive edge of the NSGA-III-SD algorithm becomes increasingly prominent as problem complexity escalates. This suggests that the algorithm’s powerful capacity for exploratory endeavors remains potent when confronted with intricate and demanding problems. The effective exploration of the solution space and the ability to adapt to complex problems contribute to the NSGA-III-SD algorithm’s superior performance and the production of higher-quality solutions.

Following 20 independent runs of each algorithm, where the average HV values are computed, a Friedman rank-sum test is performed.

MOEAs | HV | |
---|---|---|

Rank | ||

NSGA-III-SD | 4.75 | 3.2915E-16 |

VaEA | 2.30 | |

SRA | 5.55 | |

MaOEACSS | 4.70 | |

NSGA-III | 1.70 | |

NSGA-II | 2.00 |

The superiority of NSGA-III-SD can be attributed to its strategic utilization of the SDR approach during individual selection. This approach not only selects individuals with superior fitness but also takes advantage of individuals with potential benefits. In comparison to methods that solely rely on dominance relations for individual selection and elimination, SDR incorporates a niche technique based on candidate solutions. By autonomously determining the niche size according to the distribution of candidate solutions, NSGA-III-SD maintains a more favorable balance between convergence and diversity within each niche. This ability allows the algorithm to avoid solution overcrowding and enhance the diversity of the solution set, especially in complex MO-JSSP scenarios. Furthermore, NSGA-III-SD’s rank significantly surpasses that of NSGA-III, demonstrating a substantial improvement in the performance of the enhanced algorithm and effectively illustrating the effectiveness of various strategies.

In conclusion, the hybrid evolutionary approach of NSGA-III-SD exhibits enhanced search capability in solving MO-JSSP. Moreover, employing the SDR non-dominance ranking strategy, instead of the original sorting method in the NSGA-III algorithm, effectively balances the trade-off between population convergence and diversity.

Of course, this experimental study has certain limitations that should be acknowledged. One of the limitations is that the scalability of NSGA-III-SD has not been extensively tested, especially when dealing with hyperscale scheduling problems. Hyperscale problems may involve more complex constraints and decision variables, potentially impacting the performance of NSGA-III-SD. Therefore, further research can explore the performance of NSGA-III-SD on hyper-scale scheduling problems and evaluate its adaptability in handling highly discontinuous or non-convex Pareto front.

For the MO-JSSP with five objectives: makespan, total flow time, total tardiness time, average machine idle time, and just-in-time production mode, a strengthened dominance relation NSGA-III algorithm based on differential evolution, NSGA-III-SD, is proposed. NSGA-III-SD employs a constrained differential evolution strategy during the initial evolution stage to prevent premature convergence and enhance the trait of convergence. Furthermore, it replaces the original dominance relation of NSGA-III with the SDR non-dominance sorting strategy, effectively managing the trade-off between population convergence and diversity. The experimental results on MO-JSSP tests undeniably demonstrate the superior performance of NSGA-III-SD. It outperforms other algorithms in both the C metric and the HV metric, indicating higher-quality solutions as well as improved diversity and distribution of the solution set. These results highlight the strong overall performance and practical advantage of NSGA-III-SD in solving MO-JSSP.

The proposed algorithm has significant theoretical implications as it contributes to the understanding of multi-objective optimization in job shop scheduling. It provides insights into the effective utilization of a strengthened dominance relation and the integration of a constrained differential evolution strategy, contributing to the advancement of optimization algorithms in complex manufacturing systems.

Due to the need to maintain the Pareto front in NSGA-III-SD, the size of the set increases with the problem scale. Therefore, for hyperscale problems, NSGA-III-SD may have low convergence accuracy and require more computational resources. Future research can focus on designing more efficient evolutionary approaches to address this limitation.

In future work, we plan to further enhance the algorithm and expand its applications. We consider incorporating a dynamic resource allocation mechanism into the algorithm to solve the flexible job shop scheduling problem [

Not applicable.

This work was in part supported by the Key Research and Development Project of Hubei Province (Nos. 2020BAB1141, 2023BAB094), the Key Project of Science and Technology Research Program of Hubei Educational Committee (No. D20211402), the Teaching Research Project of Hubei University of Technology (No. XIAO2018001), and the Project of Xiangyang Industrial Research Institute of Hubei University of Technology (No. XYYJ2022C04).

The authors confirm contribution to the paper as follows: study conception and design: Liang Zeng; data collection: Junyang Shi; analysis and interpretation of results: Shanshan Wang; draft manuscript preparation: Yanyan Li; manuscript proofreading: Weigang Li. All authors reviewed the results and approved the final version of the manuscript.

Data is available on request from the authors. The data that support the findings of this study are available from the corresponding author, S. Wang, upon reasonable request.

The authors declare that they have no conflicts of interest to report regarding the present study.