Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search
Abstract
1. Introduction
- Adjusting machine speeds to achieve energy savings and emission reductions without compromising economic performance [6].
2. Problem Statement
2.1. Problem Description of FSSP_LB
2.2. Mathematical Model of GFSSP_LBAGV
- The time for jobs to arrive at the first machine (transportation time) is not considered;
- An AGV can only transport one job at a time;
- The speed of the AGV is unaffected by whether it is carrying a load or running empty;
- Collisions and failures during AGV transportation are not considered;
- The transportation time of the AGV is solely related to the current transportation distance;
- The number of AGVs is sufficient to ensure that a job is immediately transported to the designated location after being offloaded.
2.3. Energy Consumption of GFSSP_LBAGV
- Minimize the maximum completion time :
- Minimize the total energy consumption :
3. IDPFA for Solving GFSSP_LBAGV
3.1. Population Initialization
3.2. The Encoding and Decoding
3.3. Pathfinder Location Update
3.4. Improved Discrete Pathfinder Location Update
3.5. Improved Discrete Followers Location Update
3.6. Multi-Neighborhood Local Search
3.7. Algorithm Scheme
4. Experimental Simulation and Comparison
4.1. Experimental Setup
4.2. Comparison Between DPFA and Standard PFA
- (1)
- Minimum makespan (Min): The smallest value obtained from multiple independent algorithm runs. This metric reflects the algorithm’s optimal performance, representing the most compact scheduling arrangement achievable.
- (2)
- Maximum makespan (Max): The largest value observed across multiple runs. This metric indicates the algorithm’s stability in worst-case scenarios, demonstrating the upper bound of solution quality even under suboptimal conditions.
- (3)
- Average makespan (Avg): The arithmetic mean of values over multiple runs. As a core indicator of algorithm reliability, it measures overall performance. When the average value approximates the minimum while remaining substantially lower than the maximum, the algorithm demonstrates strong robustness with minimal result fluctuations.
- (4)
- Computational time (Time): The elapsed time from algorithm initiation to termination. This metric evaluates computational efficiency, where high-performance algorithms should consistently obtain high-quality solutions within reasonable time frames.
4.3. Comparison Between DPFA and Other Algorithms
- (1)
- For buffer size , 78% of non-dominated solutions in IDPFA outperforms SFLA;
- (2)
- At and , this advantage increases to 82%;
- (3)
- Notably, 67% of IDPFA’s solutions show significantly better quality than SFLA’s solutions.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
GFSSP_LBAGV | Green flow shop scheduling problem with limited buffers and automated guided vehicle |
FSSP_LB | Flow shop scheduling problem with limited buffers |
IDPFA | Improved discrete pathfinder algorithm |
AGV | Automated guided vehicle |
PFA | Pathfinder algorithm |
NEH | Nawaz–Enscore–Ham |
LOV | Largest-order-value |
OBX | Order-based crossover |
SEC | Subtour exchange crossover |
RNDS | Quality of non-dominated solutions in the set of non-dominated solutions |
ONSN | Number of non-dominated solutions in the set of non-dominated solutions |
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Mathematical Notations | Definitions |
---|---|
the number of jobs | |
the number of machines | |
the number of AGVs | |
the processing sequence of jobs | |
the buffer size between machines | |
the completion time of job on machine with AGVs | |
the processing time of job on machine | |
the transportation time of the job between machines | |
the transportation time of the job from machine to buffer | |
the transportation time of the job from buffer to buffer machine | |
the processing speed of job on machine | |
the maximum completion time of processing sequence | |
the energy consumption per unit time for machine when running at gear level | |
the machine operates at gear at time | |
the machine is in standby mode at time | |
the total energy consumption | |
the energy consumption of machines during the production process | |
the energy consumption of AGVs when transporting jobs between different machines or buffer zones | |
the average power consumption during the transportation process of the AGV | |
the energy consumption per unit time when the machine is in standby mode | |
the current iteration number of the algorithm | |
the maximum number of iterations | |
the position vector of the contemporary pathfinder | |
the position of the pathfinder from the previous generation | |
the pathfinder’s updated position vector | |
the step size factor | |
the multidirectional and random movement of the pathfinder | |
the current position of the follower | |
the follower’s updated position | |
the interaction coefficient among followers | |
the attraction coefficient from the pathfinder to followers | |
the randomness in follower movement |
Dimension | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
1.28 | 2.09 | 3.17 | 0.78 | 2.66 | |
4 | 3 | 1 | 5 | 2 | |
3 | 5 | 2 | 1 | 4 |
Problem , | |||||||||
---|---|---|---|---|---|---|---|---|---|
PFA | DPFA | ||||||||
Min | Max | Avg | Time | Min | Max | Avg | Time | ||
rec01 | 20, 5 | 1284 | 1370 | 1336.80 | 50.80 | 1249 | 1600 | 1351.75 | 42.95 |
rec03 | 20, 5 | 1119 | 1224 | 1161.90 | 46.15 | 1111 | 1424 | 1142.30 | 42.15 |
rec05 | 20, 5 | 1260 | 1332 | 1294.70 | 46.85 | 1245 | 1621 | 1280.15 | 40.60 |
rec07 | 20, 10 | 1586 | 1703 | 1642.05 | 55.50 | 1584 | 2000 | 1640.10 | 54.70 |
rec09 | 20, 10 | 1583 | 1735 | 1649.20 | 54.65 | 1560 | 2129 | 1664.70 | 54.75 |
rec11 | 20, 10 | 1487 | 1662 | 1561.10 | 56.15 | 1444 | 2033 | 1553.45 | 53.20 |
rec13 | 20, 15 | 2027 | 2167 | 2082.50 | 59.85 | 1957 | 2492 | 2038.85 | 68.25 |
rec15 | 20, 15 | 2033 | 2149 | 2077.95 | 60.90 | 1965 | 2431 | 2021.75 | 70.40 |
rec17 | 20, 15 | 1972 | 2107 | 2058.00 | 68.85 | 1943 | 2382 | 2001.20 | 71.00 |
rec19 | 30, 10 | 2219 | 2373 | 2297.40 | 91.50 | 2153 | 2603 | 2229.65 | 68.70 |
rec21 | 30, 10 | 2111 | 2282 | 2203.25 | 89.05 | 2050 | 2670 | 2143.60 | 73.45 |
rec23 | 30, 10 | 2152 | 2311 | 2225.85 | 89.95 | 2069 | 2824 | 2241.65 | 71.00 |
rec25 | 30, 15 | 2686 | 2848 | 2771.55 | 100.20 | 2595 | 3168 | 2701.45 | 88.85 |
rec27 | 30, 15 | 2539 | 2714 | 2653.85 | 100.70 | 2443 | 3211 | 2536.45 | 90.70 |
rec29 | 30, 15 | 2458 | 2667 | 2574.10 | 98.55 | 2374 | 3156 | 2492.75 | 91.25 |
rec31 | 50, 10 | 3333 | 3506 | 3436.25 | 189.15 | 3212 | 3920 | 3366.70 | 110.85 |
rec33 | 50, 10 | 3296 | 3540 | 3439.55 | 186.00 | 3171 | 3998 | 3303.10 | 107.00 |
rec35 | 50, 10 | 3404 | 3578 | 3491.50 | 188.95 | 3300 | 3919 | 3407.30 | 112.6 |
Problem | , | ||||||
---|---|---|---|---|---|---|---|
IDPFA | SFLA | INSGA-II | |||||
rec01 | 20, 5 | 0.58 | 5.60 | 0.68 | 3.20 | 0.08 | 0.25 |
rec03 | 20, 5 | 0.81 | 9.00 | 0.59 | 2.30 | 0.00 | 0.00 |
rec05 | 20, 5 | 0.69 | 5.85 | 0.65 | 2.90 | 0.00 | 0.00 |
rec07 | 20, 10 | 0.89 | 8.45 | 0.34 | 0.95 | 0.00 | 0.00 |
rec09 | 20, 10 | 0.73 | 8.15 | 0.48 | 1.70 | 0.00 | 0.00 |
rec11 | 20, 10 | 0.84 | 7.70 | 0.38 | 1.05 | 0.00 | 0.00 |
rec13 | 20, 15 | 0.64 | 5.20 | 0.06 | 0.20 | 0.25 | 0.80 |
rec15 | 20, 15 | 0.75 | 6.30 | 0.15 | 0.30 | 0.38 | 1.00 |
rec17 | 20, 15 | 0.53 | 4.05 | 0.33 | 0.80 | 0.46 | 1.10 |
rec19 | 30, 10 | 0.70 | 5.80 | 0.72 | 2.05 | 0.00 | 0.00 |
rec21 | 30, 10 | 0.71 | 6.40 | 0.67 | 1.40 | 0.00 | 0.00 |
rec23 | 30, 10 | 0.79 | 6.40 | 0.61 | 2.10 | 0.00 | 0.00 |
rec25 | 30, 15 | 0.83 | 7.25 | 0.32 | 0.75 | 0.00 | 0.00 |
rec27 | 30, 15 | 0.92 | 7.85 | 0.26 | 0.65 | 0.05 | 0.30 |
rec29 | 30, 15 | 0.82 | 8.00 | 0.57 | 1.35 | 0.00 | 0.00 |
rec31 | 50, 10 | 0.42 | 3.05 | 0.98 | 2.95 | 0.00 | 0.00 |
rec33 | 50, 10 | 0.52 | 4.25 | 0.92 | 2.50 | 0.00 | 0.00 |
rec35 | 50, 10 | 0.47 | 2.90 | 0.86 | 1.80 | 0.00 | 0.00 |
Problem | , | ||||||
---|---|---|---|---|---|---|---|
IDPFA | SFLA | INSGA-II | |||||
rec01 | 20, 5 | 0.69 | 7.65 | 0.59 | 2.20 | 0.03 | 0.10 |
rec03 | 20, 5 | 0.87 | 9.30 | 0.41 | 2.15 | 0.00 | 0.00 |
rec05 | 20, 5 | 0.76 | 6.60 | 0.72 | 2.30 | 0.00 | 0.00 |
rec07 | 20, 10 | 0.92 | 7.55 | 0.19 | 0.45 | 0.03 | 0.05 |
rec09 | 20, 10 | 0.94 | 8.85 | 0.32 | 1.10 | 0.00 | 0.00 |
rec11 | 20, 10 | 0.94 | 10.00 | 0.24 | 0.70 | 0.03 | 0.05 |
rec13 | 20, 15 | 0.73 | 5.65 | 0.20 | 0.65 | 0.28 | 0.40 |
rec15 | 20, 15 | 0.58 | 5.40 | 0.17 | 0.30 | 0.34 | 1.75 |
rec17 | 20, 15 | 0.64 | 4.90 | 0.27 | 0.80 | 0.51 | 0.90 |
rec19 | 30, 10 | 0.73 | 6.45 | 0.54 | 1.35 | 0.00 | 0.00 |
rec21 | 30, 10 | 0.75 | 5.80 | 0.48 | 1.20 | 0.05 | 0.10 |
rec23 | 30, 10 | 0.76 | 6.70 | 0.48 | 1.20 | 0.00 | 0.00 |
rec25 | 30, 15 | 0.89 | 8.15 | 0.30 | 0.85 | 0.00 | 0.00 |
rec27 | 30, 15 | 0.85 | 7.70 | 0.46 | 1.05 | 0.00 | 0.00 |
rec29 | 30, 15 | 0.81 | 8.20 | 0.26 | 0.60 | 0.07 | 0.45 |
rec31 | 50, 10 | 0.49 | 3.85 | 0.90 | 3.05 | 0.00 | 0.00 |
rec33 | 50, 10 | 0.44 | 3.35 | 0.84 | 2.25 | 0.00 | 0.00 |
rec35 | 50, 10 | 0.53 | 3.15 | 0.93 | 1.60 | 0.00 | 0.00 |
Problem | , | ||||||
---|---|---|---|---|---|---|---|
IDPFA | SFLA | INSGA-II | |||||
rec01 | 20, 5 | 0.77 | 7.10 | 0.46 | 1.90 | 0.02 | 0.05 |
rec03 | 20, 5 | 0.90 | 11.40 | 0.24 | 1.25 | 0.00 | 0.00 |
rec05 | 20, 5 | 0.67 | 5.70 | 0.69 | 2.55 | 0.00 | 0.00 |
rec07 | 20, 10 | 0.98 | 8.40 | 0.20 | 0.45 | 0.01 | 0.05 |
rec09 | 20, 10 | 0.94 | 9.00 | 0.37 | 1.10 | 0.00 | 0.00 |
rec11 | 20, 10 | 0.91 | 9.40 | 0.13 | 0.30 | 0.00 | 0.00 |
rec13 | 20, 15 | 0.51 | 4.15 | 0.05 | 0.05 | 0.75 | 1.35 |
rec15 | 20, 15 | 0.84 | 7.15 | 0.25 | 0.80 | 0.05 | 0.05 |
rec17 | 20, 15 | 0.83 | 7.70 | 0.24 | 0.60 | 0.15 | 0.45 |
rec19 | 30, 10 | 0.65 | 5.85 | 0.76 | 2.00 | 0.00 | 0.00 |
rec21 | 30, 10 | 0.70 | 6.60 | 0.52 | 1.50 | 0.00 | 0.00 |
rec23 | 30, 10 | 0.77 | 6.35 | 0.48 | 1.40 | 0.00 | 0.00 |
rec25 | 30, 15 | 0.84 | 7.15 | 0.32 | 0.90 | 0.00 | 0.00 |
rec27 | 30, 15 | 0.91 | 9.05 | 0.48 | 1.05 | 0.10 | 0.10 |
rec29 | 30, 15 | 0.82 | 8.70 | 0.25 | 0.55 | 0.15 | 0.20 |
rec31 | 50, 10 | 0.48 | 3.25 | 0.91 | 2.65 | 0.00 | 0.00 |
rec33 | 50, 10 | 0.41 | 3.15 | 0.93 | 2.80 | 0.00 | 0.00 |
rec35 | 50, 10 | 0.52 | 3.05 | 0.75 | 1.95 | 0.00 | 0.00 |
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Dong, Y.; Wang, S.; Liu, X. Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search. Processes 2025, 13, 2325. https://doi.org/10.3390/pr13082325
Dong Y, Wang S, Liu X. Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search. Processes. 2025; 13(8):2325. https://doi.org/10.3390/pr13082325
Chicago/Turabian StyleDong, Yuming, Shunzeng Wang, and Xiaoming Liu. 2025. "Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search" Processes 13, no. 8: 2325. https://doi.org/10.3390/pr13082325
APA StyleDong, Y., Wang, S., & Liu, X. (2025). Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search. Processes, 13(8), 2325. https://doi.org/10.3390/pr13082325