One-Machine Scheduling with Time-Dependent Capacity via Efficient Memetic Algorithms
Abstract
:1. Introduction
- •
- First, new efficient local search procedures for the problem are proposed and their relevant properties, as correctness and worst-case complexity, are studied. As the previous local search approach, the new methods rely on the notion of C-paths in a feasible schedule. However, they incorporate mechanisms to make them more effective. These include a new condition for swapping pairs of consecutive jobs, the integration of a hill climbing approach, a procedure that operates on several C-paths at the same time and a new way of improving the quality of schedules by interchanging jobs between different C-paths.
- •
- Then, the local search procedures are exploited in combination with a genetic algorithm, giving rise to new memetic algorithms. These algorithms have been designed with the aim of achieving a proper balance between the exploration of the search space and the intensification in its most promising areas.
- •
- An extensive experimental study demonstrates that the memetic algorithms proposed in this work achieve conclusive improvements in practice. The results reveal that the new local search procedures enable the memetic algorithms to reach far better solutions than other methods, including the memetic algorithm proposed in [32] and a constraint programming approach.
2. Definition of the Problem
- The capacity of the machine cannot be exceeded at any time, i.e., for all , where denotes the total consumption of the machine in the interval due to the jobs scheduled. This corresponds to the number of jobs that are processed in parallel in that interval.
- The processing of a job cannot be preempted, i.e., for all , where denotes the completion time of job i.
i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
4 | 4 | 2 | 3 | 4 | 3 | 2 | 3 | 2 | 3 | 3 | 5 | |
4 | 9 | 13 | 4 | 7 | 8 | 10 | 3 | 13 | 5 | 9 | 7 |
3. Preliminaries
3.1. Schedule Builder
Algorithm 1 Schedule Builder ([31,32]) |
|
3.2. Genetic Algorithm
Algorithm 2 Genetic Algorithm ([31,32]) |
|
3.3. Local Search Procedure
Algorithm 3 Single C-path local search (SCP) |
Data: A feasible schedule S. Result: A feasible schedule with . ; ; return ; |
Algorithm 4 |
|
3.4. Memetic Algorithm
4. New Local Search Procedures
4.1. Enhancements to Single C-Path Local Search
4.1.1. Slack-Aware Improvement Condition
4.1.2. Hill Climbing on a Single C-Path
Algorithm 5 |
|
Algorithm 6 SCP with Hill Climbing (SCP+) |
Data: A feasible schedule S. Result: A feasible schedule with . ; ; return ; |
4.2. Cover-Based Local Search
Algorithm 7 |
|
Algorithm 8 Cover-based Local Search Procedure (CB) |
|
4.3. Interchanging Jobs between C-Paths
Algorithm 9 Local search by interchanging jobs between C-paths (ICP) |
|
4.4. Hybrid Approach
5. New Memetic Algorithms
6. Results
6.1. Analyzing the Slack-Aware Improvement Condition
6.2. Analyzing MA, MA, MA and MA
6.3. Final Comparison
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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n | MA | MA | |||||
---|---|---|---|---|---|---|---|
Best | Avg. | Best | Avg. | ||||
120 | 3 | 0.01 | 0.49 | 0.01 | 0.18 | ||
5 | 0.08 | 0.98 | 0.07 | 0.29 | |||
7 | 0.35 | 1.50 | 0.08 | 0.48 | |||
10 | 0.65 | 1.97 | 0.30 | 1.13 | |||
Avg. | 0.27 | 1.23 | 0.11 | 0.52 | |||
250 | 10 | 1.46 | 4.09 | 0.27 | 0.99 | ||
20 | 1.64 | 3.81 | 0.59 | 1.95 | |||
30 | 1.98 | 4.71 | 1.51 | 3.39 | |||
Avg. | 1.70 | 4.20 | 0.79 | 2.11 | |||
500 | 10 | 8.54 | 20.00 | 0.53 | 1.40 | ||
20 | 4.35 | 11.55 | 2.01 | 3.46 | |||
30 | 6.92 | 13.41 | 4.55 | 7.11 | |||
Avg. | 6.60 | 14.99 | 2.36 | 3.99 | |||
750 | 10 | 7.88 | 16.21 | 0.36 | 1.79 | ||
20 | 6.08 | 14.81 | 0.97 | 2.17 | |||
30 | 12.97 | 21.98 | 6.59 | 9.70 | |||
50 | 18.58 | 25.21 | 11.62 | 14.65 | |||
Avg. | 11.38 | 19.55 | 4.88 | 7.08 | |||
1000 | 10 | 15.72 | 41.02 | 0.80 | 2.88 | ||
20 | 5.86 | 11.63 | 1.61 | 3.03 | |||
30 | 21.08 | 31.39 | 9.42 | 13.50 | |||
50 | 27.97 | 41.59 | 12.55 | 17.17 | |||
100 | 43.66 | 50.27 | 36.71 | 42.45 | |||
Avg. | 22.86 | 35.18 | 12.22 | 15.81 | |||
All | 9.78 | 16.66 | 4.76 | 6.72 |
n | MA | MA | MA | MA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Best | Avg. | Best | Avg. | Best | Avg. | Best | Avg. | ||||||
120 | 3 | 0.00 | 0.16 | 0.00 | 0.11 | 0.12 | 2.20 | 0.00 | 0.01 | ||||
5 | 0.07 | 0.27 | 0.05 | 0.20 | 0.00 | 1.10 | 0.00 | 0.03 | |||||
7 | 0.09 | 0.42 | 0.07 | 0.32 | 0.06 | 0.75 | 0.00 | 0.07 | |||||
10 | 0.33 | 1.19 | 0.15 | 0.68 | 0.16 | 1.06 | 0.00 | 0.17 | |||||
Avg. | 0.12 | 0.51 | 0.07 | 0.33 | 0.08 | 1.28 | 0.00 | 0.07 | |||||
250 | 10 | 0.39 | 1.18 | 0.15 | 0.56 | 1.05 | 2.12 | 0.00 | 0.12 | ||||
20 | 0.81 | 2.10 | 0.25 | 0.80 | 0.51 | 1.17 | 0.00 | 0.14 | |||||
30 | 1.61 | 3.53 | 0.34 | 1.45 | 1.10 | 1.85 | 0.07 | 0.37 | |||||
Avg. | 0.94 | 2.27 | 0.25 | 0.94 | 0.89 | 1.71 | 0.02 | 0.21 | |||||
500 | 10 | 0.49 | 1.26 | 0.21 | 0.58 | 3.28 | 5.31 | 0.00 | 0.12 | ||||
20 | 2.08 | 3.63 | 0.65 | 1.21 | 2.71 | 3.92 | 0.12 | 0.37 | |||||
30 | 4.83 | 7.48 | 0.98 | 2.38 | 3.28 | 5.08 | 0.16 | 0.38 | |||||
Avg. | 2.47 | 4.12 | 0.61 | 1.39 | 3.09 | 4.77 | 0.09 | 0.29 | |||||
750 | 10 | 0.39 | 1.48 | 0.06 | 0.67 | 7.82 | 10.16 | 0.14 | 0.62 | ||||
20 | 1.13 | 2.27 | 0.16 | 0.70 | 2.71 | 3.84 | 0.17 | 0.28 | |||||
30 | 6.64 | 10.28 | 1.53 | 2.87 | 4.09 | 5.93 | 0.00 | 0.27 | |||||
50 | 11.23 | 14.00 | 1.79 | 3.23 | 7.13 | 8.78 | 0.00 | 0.29 | |||||
Avg. | 4.85 | 7.01 | 0.89 | 1.87 | 5.44 | 7.18 | 0.08 | 0.36 | |||||
1000 | 10 | 0.42 | 2.15 | 0.00 | 0.78 | 21.25 | 27.28 | 0.64 | 1.07 | ||||
20 | 1.48 | 2.40 | 0.12 | 0.90 | 6.21 | 7.56 | 0.23 | 0.39 | |||||
30 | 7.45 | 10.05 | 0.20 | 1.40 | 19.00 | 22.91 | 0.34 | 0.83 | |||||
50 | 11.45 | 14.98 | 2.72 | 4.21 | 4.07 | 5.89 | 0.08 | 0.14 | |||||
100 | 21.18 | 25.20 | 2.86 | 4.03 | 5.53 | 6.44 | 0.00 | 0.20 | |||||
Avg. | 8.40 | 10.95 | 1.18 | 2.26 | 11.21 | 14.02 | 0.26 | 0.53 | |||||
All | 3.79 | 5.47 | 0.65 | 1.42 | 4.74 | 6.49 | 0.10 | 0.31 |
n | MA | CPO | MA | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Best | Avg. | Best | Avg. | Best | Avg. | |||||
120 | 3 | 0.01 | 0.49 | 3.82 | 3.85 | 0.00 | 0.01 | |||
5 | 0.08 | 0.98 | 4.54 | 4.54 | 0.00 | 0.03 | ||||
7 | 0.35 | 1.50 | 5.47 | 5.47 | 0.00 | 0.07 | ||||
10 | 0.65 | 1.97 | 6.43 | 6.63 | 0.00 | 0.17 | ||||
Avg. | 0.27 | 1.23 | 5.06 | 5.12 | 0.00 | 0.07 | ||||
250 | 10 | 1.46 | 4.09 | 6.29 | 6.36 | 0.00 | 0.12 | |||
20 | 1.64 | 3.81 | 9.45 | 9.46 | 0.00 | 0.14 | ||||
30 | 1.98 | 4.71 | 12.74 | 12.74 | 0.07 | 0.37 | ||||
Avg. | 1.70 | 4.20 | 9.49 | 9.52 | 0.02 | 0.21 | ||||
500 | 10 | 8.54 | 20.00 | 8.97 | 8.97 | 0.00 | 0.12 | |||
20 | 4.35 | 11.55 | 9.89 | 9.94 | 0.12 | 0.37 | ||||
30 | 6.92 | 13.41 | 16.90 | 17.24 | 0.16 | 0.38 | ||||
Avg. | 6.60 | 14.99 | 11.92 | 12.05 | 0.09 | 0.29 | ||||
750 | 10 | 7.88 | 16.21 | 6.93 | 6.95 | 0.14 | 0.62 | |||
20 | 6.08 | 14.81 | 10.62 | 10.63 | 0.17 | 0.28 | ||||
30 | 12.97 | 21.98 | 15.70 | 16.14 | 0.00 | 0.27 | ||||
50 | 18.58 | 25.21 | 13.46 | 13.46 | 0.00 | 0.29 | ||||
Avg. | 11.38 | 19.55 | 11.68 | 11.79 | 0.08 | 0.36 | ||||
1000 | 10 | 15.72 | 41.02 | 9.23 | 9.39 | 0.64 | 1.07 | |||
20 | 5.86 | 11.63 | 5.64 | 5.67 | 0.23 | 0.39 | ||||
30 | 21.08 | 31.39 | 14.79 | 14.85 | 0.34 | 0.83 | ||||
50 | 27.97 | 41.59 | 14.46 | 14.46 | 0.08 | 0.14 | ||||
100 | 43.66 | 50.27 | 8.77 | 8.77 | 0.00 | 0.20 | ||||
Avg. | 22.86 | 35.18 | 10.58 | 10.63 | 0.26 | 0.53 | ||||
All | 9.78 | 16.66 | 9.69 | 9.76 | 0.10 | 0.31 |
Position | Algorithm | Ranking |
---|---|---|
1 | MA | 121.12 |
2 | CPO | 335.75 |
3 | MA | 399.62 |
Algorithm | ||||||||
---|---|---|---|---|---|---|---|---|
MA | 0.0 | 0.0 | ||||||
CPO | 0.0 | 0.0 |
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Mencía, R.; Mencía, C. One-Machine Scheduling with Time-Dependent Capacity via Efficient Memetic Algorithms. Mathematics 2021, 9, 3030. https://doi.org/10.3390/math9233030
Mencía R, Mencía C. One-Machine Scheduling with Time-Dependent Capacity via Efficient Memetic Algorithms. Mathematics. 2021; 9(23):3030. https://doi.org/10.3390/math9233030
Chicago/Turabian StyleMencía, Raúl, and Carlos Mencía. 2021. "One-Machine Scheduling with Time-Dependent Capacity via Efficient Memetic Algorithms" Mathematics 9, no. 23: 3030. https://doi.org/10.3390/math9233030