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. FitnessBased Ensemble Construction
Algorithm 3 Outline of the fitnessbased heuristic 
Input: k—ensemble size, R—set of DRs, ${f}_{r,i}$—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, ${f}_{r,i}$—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, ${f}_{r,i}$—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, ${f}_{r,i}$—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, ${f}_{r,i}$—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, ${f}_{r,i}$—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, ${f}_{r,i}$—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, ${f}_{r,i}$—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, ${f}_{r,i}$—fitness of rule r on instance i in dataset D Output: E—the constructed ensemble

Algorithm 12 Outline of the optimal matchbased heuristic 
Input: k—ensemble size, R—set of DRs, ${f}_{r,i}$—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 

$pt$  processing time of job j on machine i 
$pmin$  minimal processing time (MPT) of job j 
$pavg$  average processing time of job j across all machines 
$PAT$  time until machine with the MPT for job j becomes available 
$MR$  time until machine i becomes available 
$age$  time which job j spent in the system 
$dd$  time until which job j has to finish with its execution 
w  weight of job j (${w}_{j}$) 
$SL$  slack of job j, $max({d}_{j}{p}_{ij}t,0)$ 
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