Heuristic Ensemble Construction Methods of Automatically Designed Dispatching Rules for the Unrelated Machines Environment
Abstract
:1. Introduction
2. Literature Review
3. Background
3.1. The Unrelated Machines Scheduling Problem
Algorithm 1 SGS used by generated DRs |
|
3.2. Automated Design of DRs with GP
Algorithm 2 Standard steady state GP algorithm |
|
3.3. Ensemble Learning for Automatically Designed DRs
4. Deterministic Ensemble Construction Methods
4.1. Fitness-Based Ensemble Construction
Algorithm 3 Outline of the fitness-based heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
4.2. Optimal Count Heuristic
Algorithm 4 Outline of the optimal count heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
4.3. Optimal Unique Count Heuristic
4.4. Remaining Optimal Count Heuristic
Algorithm 5 Outline of the optimal unique count heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
Algorithm 6 Outline of the remaining optimal count heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
4.5. Fitness Coverage Heuristic
Algorithm 7 Outline of the fitness coverage heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
4.6. Remaining Fitness Heuristic
4.7. Weighted Fitness Heuristic
Algorithm 8 Outline of the remaining fitness heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
Algorithm 9 Outline of the weighted fitness heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
4.8. Adjusted Fitness Heuristic
Algorithm 10 Outline of the adjusted fitness heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
4.9. Standard Deviation Heuristic
4.10. Optimal Match Based Heuristic
Algorithm 11 Outline of the standard deviation heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
Algorithm 12 Outline of the optimal match-based heuristic |
Input: k—ensemble size, R—set of DRs, —fitness of rule r on instance i in dataset D Output: E—the constructed ensemble
|
5. Experimental Analysis
5.1. Experimental Setup
5.2. Experimental Results
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Terminal | Description |
---|---|
processing time of job j on machine i | |
minimal processing time (MPT) of job j | |
average processing time of job j across all machines | |
time until machine with the MPT for job j becomes available | |
time until machine i becomes available | |
time which job j spent in the system | |
time until which job j has to finish with its execution | |
w | weight of job j () |
slack of job j, |
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|
ATC | 16.63 | ||||||||
DR | 16.09 | ||||||||
BagGP | 16.28 | 16.11 | 16.05 | 16.04 | 16.05 | 16.00 | 16.00 | 16.02 | 16.07 |
BoostGP | 15.82 | 15.76 | 15.87 | 15.89 | 15.89 | 15.85 | 15.86 | 15.86 | 15.89 |
MGA | 15.66 | 15.46 | 15.49 | 15.48 | 15.47 | 15.46 | 15.45 | 15.45 | 15.77 |
SEC | 15.68 | 15.68 | 15.65 | 15.52 | 15.43 | 15.29 | 15.38 | 15.32 | 15.34 |
FBH (3) | 16.09 | 15.53 | 15.05 | 15.07 | 15.13 | 14.95 | 14.92 | 14.91 | 14.99 |
OCH (4) | 15.55 | 16.07 | 15.99 | 16.01 | 15.95 | 15.86 | 15.91 | 15.93 | 16.21 |
OUCH (5) | 15.70 | 15.65 | 15.38 | 16.48 | 16.82 | 16.17 | 16.86 | 16.89 | 15.79 |
ROCH (6) | 15.27 | 15.15 | 15.12 | 15.10 | 15.28 | 15.31 | 14.90 | 15.38 | 15.90 |
FCH (7) | 15.55 | 15.16 | 15.27 | 15.11 | 15.40 | 15.48 | 15.45 | 15.60 | 15.59 |
RFH (8) | 15.55 | 15.52 | 15.91 | 15.31 | 15.29 | 15.32 | 15.29 | 15.45 | 15.88 |
WFH (9) | 16.09 | 14.94 | 15.12 | 14.94 | 15.01 | 15.01 | 15.02 | 15.02 | 15.02 |
AFH (10) | 15.27 | 15.32 | 15.48 | 15.45 | 15.31 | 15.44 | 15.43 | 15.47 | 15.38 |
SDH (11) | 15.46 | 15.73 | 15.48 | 15.84 | 15.57 | 15.66 | 15.66 | 15.66 | 15.67 |
OMBH (12) | 15.95 | 15.38 | 16.00 | 15.60 | 15.67 | 15.68 | 15.68 | 15.68 | 15.70 |
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|
ATC | 16.63 | ||||||||
DRs | 16.09 | ||||||||
BagGP | 16.38 | 15.81 | 15.82 | 15.70 | 15.68 | 15.58 | 15.65 | 15.58 | 15.57 |
BoostGP | 16.02 | 15.72 | 15.67 | 15.50 | 15.64 | 15.56 | 15.68 | 15.53 | 15.55 |
MGA | 15.59 | 15.54 | 15.79 | 15.51 | 15.53 | 15.51 | 15.59 | 15.50 | 15.50 |
SEC | 15.76 | 15.60 | 15.67 | 15.67 | 15.64 | 15.55 | 15.57 | 15.52 | 15.55 |
FBH (3) | 15.73 | 15.80 | 15.80 | 15.87 | 15.80 | 15.66 | 15.51 | 15.74 | 15.52 |
OCH (4) | 15.43 | 15.71 | 15.52 | 15.74 | 15.71 | 15.60 | 15.65 | 15.46 | 15.50 |
OUCH (5) | 15.73 | 15.61 | 15.64 | 15.20 | 15.21 | 15.11 | 15.27 | 15.67 | 15.69 |
ROCH (6) | 16.17 | 15.46 | 15.49 | 15.29 | 15.34 | 15.32 | 15.50 | 15.49 | 15.58 |
FCH (7) | 15.62 | 15.73 | 15.56 | 15.24 | 15.08 | 15.89 | 15.42 | 15.77 | 15.45 |
RFH (8) | 15.62 | 15.53 | 15.24 | 15.67 | 15.71 | 15.59 | 15.56 | 15.31 | 15.58 |
WFH (9) | 15.73 | 15.52 | 15.29 | 15.33 | 15.57 | 16.03 | 15.26 | 15.94 | 16.23 |
AFH (10) | 16.17 | 15.64 | 15.54 | 15.39 | 15.53 | 15.59 | 15.40 | 15.75 | 15.61 |
SDH (11) | 16.65 | 16.14 | 16.04 | 15.75 | 15.81 | 15.96 | 15.98 | 15.87 | 15.90 |
OMBH (12) | 16.01 | 15.88 | 16.10 | 15.77 | 16.44 | 15.62 | 16.02 | 15.52 | 15.95 |
Sum | Vote | Both | ||||||
---|---|---|---|---|---|---|---|---|
# | Avg. Rank | Tot. Rank | # | Avg. Rank | Tot. Rank | Avg. Rank | Tot. Rank | |
FBH (3) | 7 | 2.89 | 2 | 1 | 6.67 | 8 | 4.78 | 6 |
OCH (4) | 1 | 8.67 | 10 | 4 | 4.56 | 5 | 6.61 | 8 |
OUCH (5) | 1 | 8.56 | 9 | 5 | 3.67 | 2 | 6.11 | 7 |
ROCH (6) | 6 | 3 | 3 | 7 | 3.55 | 1 | 3.28 | 1 |
FCH (7) | 5 | 4.78 | 5 | 5 | 4.33 | 4 | 4.56 | 4 |
RFH (8) | 4 | 5.22 | 6 | 4 | 3.67 | 2 | 4.44 | 3 |
WFH (9) | 8 | 2.56 | 1 | 4 | 5.33 | 7 | 3.94 | 2 |
AFH (10) | 5 | 4.44 | 4 | 4 | 5.11 | 6 | 4.78 | 5 |
SDH ¨(11) | 2 | 6.56 | 7 | 0 | 9 | 10 | 7.78 | 9 |
OMBH (12) | 1 | 7.56 | 8 | 0 | 8.22 | 9 | 7.89 | 10 |
DR Index | Fitness | # Optimal | # Unique Optimal |
---|---|---|---|
1 | 14.494 | 37 | 1 |
3 | 14.3753 | 36 | 1 |
13 | 14.6812 | 37 | 0 |
20 | 14.297 | 37 | 3 |
24 | 14.3575 | 35 | 1 |
27 | 14.5239 | 38 | 0 |
29 | 14.7101 | 37 | 1 |
32 | 14.6285 | 37 | 1 |
43 | 14.43 7 | 35 | 1 |
Average | 14.5969 | 35.14 | 0.32 |
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Đurasević, M.; Jakobović, D. Heuristic Ensemble Construction Methods of Automatically Designed Dispatching Rules for the Unrelated Machines Environment. Axioms 2024, 13, 37. https://doi.org/10.3390/axioms13010037
Đurasević M, Jakobović D. Heuristic Ensemble Construction Methods of Automatically Designed Dispatching Rules for the Unrelated Machines Environment. Axioms. 2024; 13(1):37. https://doi.org/10.3390/axioms13010037
Chicago/Turabian StyleĐurasević, Marko, and Domagoj Jakobović. 2024. "Heuristic Ensemble Construction Methods of Automatically Designed Dispatching Rules for the Unrelated Machines Environment" Axioms 13, no. 1: 37. https://doi.org/10.3390/axioms13010037