School Timetabling Optimisation Using Artificial Bee Colony Algorithm Based on a Virtual Searching Space Method
Abstract
:1. Introduction
2. Problem Formulation
2.1. Terminology
2.2. School Timetabling Concept Model
2.3. Symbols and Notations
L | Total number of educators |
O | Total number of units |
K | Total number of classes in a school week |
P | Maximum preference value |
E | Maximum expertise value |
V | Number of classes that an educator is allowed to deliver in a school week |
H | Number of hours in a school day (e.g., from 8:00 a.m. to 6:00 p.m., there are 10 h) |
D | Number of days in a school week |
M | Maximum number of hours a class lasts |
d | |
t | |
r | |
c | |
p | * |
e | * |
A | |
u | Number of units that cannot be allocated with educators |
* When p = 0, it indicates that an educator does not prefer a unit. When e = 0, it means that the educator is not capable to deliver the unit. By default, all the p and e are 0. They need to be set by educators and the head of school. |
Set of timeslots that educator is available in the school week. If educator is available during a whole school week, then . | |
Set of timeslots that class is scheduled to in the school week. If class is scheduled in the first three hours of the school week, then . |
2.4. Hard Constraints
2.5. Soft Constraints
- Educators will be assigned to the most preferred units.
- Units will be allocated to the educators who are more capable to teach.
- All units must be allocated to educators.
2.6. Objective Function
3. Related Bio-Inspired Optimisation Methods
4. The Proposed Approach
4.1. The Basic Concept of Educator Allocation
4.2. ABC Algorithm
4.3. A Modified ABC Algorithm for Proposed STP
Algorithm 1. Pseudocode of the modified ABC algorithm. |
/*Initialisation stage*/ 01: Retrieve the scheduled course timetable and educators’ profiles 02: Define the neighbour search range, number of Traits and read parameters 03: Initialise the food source and construct school timetable, satisfying all hard constraints as in Equations (1)–(4) 04: Evaluate school timetable’s objective values with Equation (6) 05: Send the employed bees to the current food sources 06: Iteration = N 07: FOR (each iteration) /∗Employed Bee Phase∗/ 08: FOR (each employed bee) 09: Seek new food source from neighbourhood in VSS, satisfying Equations (1)–(4) 10: Construct school timetables and evaluate their objective value with Equation (6); apply greedy selection 11: END FOR 12: Calculate the probability p for each food source with Equation (9) /∗Onlooker Bee Phase∗/ 13: FOR (each onlooker bee) 14: Send onlooker bees to food sources based on p 15: Find a new food source in its neighbourhood in VSS, satisfying Equations (1)–(4) 16: Construct school timetables and evaluate their objective value with Equation (6); apply greedy selection and set Trait + 1 if applicable 17: END FOR /∗Scout Bee Phase∗/ 18: IF (any onlooker bee becomes scout bee) 19: Send scout bee to a randomly produced food source, satisfying Equations (1)–(4) 20: END IF 21: Memorise the best solution achieved so far 22: END FOR 23: Output the best solution achieved |
- Integrate VSS construction approach in neighbour population process (Steps 9 and 15 in Algorithm 1) instead of forming a whole searching space beforehand.
- Integrate school timetable constructor approach to generate solutions in every objective value evaluation process (Steps 3, 10 and 16 in Algorithm 1) instead of picking up solutions directly when populate.
4.4. Virtual Searching Space Construction
4.4.1. How VSS Works
4.4.2. The Implementation of VSS in Modified ABC Algorithm
Algorithm 2. Pseudocode of VSS. |
01: Randomly select first educator (index is ) from the given educator list. 02: Randomly generate neighbour search range . 03: Randomly generate number . 04: Set the second educator’s index 05: If , repeat Step 2 ( is the number of educators). 06: Swap and . |
4.5. School Timetable Construction
Algorithm 3. Pseudocode of school timetable solution construction. |
1: Receive an educator list and the number of units an educator can take 2: FOR (V) 3: FOR (K) 4: FOR (each unallocated educator in the educator list) 5: IF (the educator is available and prefer to teach the class) 6: Allocate the educator to the class 7: Modify the educator’s availability 8: IF (all classes are allocated with an educator) 9: Return constructed school timetable 10: FOR (each unallocated class in the course timetable) 10: FOR (L) 12: IF (the educator can and is available to teach the class) 13: Allocate the educator to the class 14: Modify the educator’s availability 15: Return constructed school timetable |
5. Experimental Section
- The course timetable is prescheduled and fixed based on the course enrolment information. The class activities have been scheduled in the weekly timeslots.
- Before planning a school timetable, educators will submit their expression of interest to the administration, including the units they want to teach and the preferences towards each unit.
- Educators need to provide their unavailability form in the school week. The number of unavailable hours cannot be more than 3 h.
- The head of school will confirm and adjust each educator’s expertise against a unit.
- The goal of the school timetabling is to satisfy educators’ preferences and classes’ qualities and attempt to ensure every unit is allocated to an educator.
5.1. Experimental Settings
- K = 300 (Total number of classes in a school week)
- L = 150 (Total number of educators)
- V = 5 (Number of classes that an educator is allowed to deliver in a school week)
- H = 8 (Number of hours in a school day)
- D = 5 (Number of days in a school week)
- P = 3 (Maximum preference value)
- E = 3 (Maximum expertise value)
- d = 1 or 2 (Duration of each class)
- Maximum unavailable timeslots an educator can have in a week: 3
- Operating System: Windows 10 Pro Edition.
- Integrated development environment: IntelliJ IDEA 2020.3.2.
- Programming language: Java.
- Computer hardware system: Intel® Core™ i7-1065G7 1.30GHz Processor with 16.0 GB of memory.
5.2. Experimental Results
- The proposed approach can obtain a feasible solution in a gigantic searching space in a short time. In Sample A, the time spent for the best solution is 4.742 s.
- Deploying more bees can help improve objective value. Sample B has a better result than Sample A in objective value (43.4970 to 42.4299 on average), as Sample B populates twenty bees and Sample A deploys five bees.
- Increasing the number of program iterations can obtain a better result. Sample D has a better objective function value over Sample A (43.6596 over 42.4299 on average), as Sample D has 10,000 iterations and Sample A has 1000 iterations.
- Increasing both the number of bees and the number of iterations can improve the result. Comparing Sample A and Sample F, the objective value improves from 42.4299 to 44.7148 on average, but the time spent also increased significantly (from 4.983 s to 197.513 s on average).
- Expanding the neighbourhood can slightly improve the result. Sample C expands the neighbourhood three times compared to Sample A, but the result only improves 0.85% (from 42.4299 to 42.7910 on average).
- Increasing the exploitation will not benefit the solution and instead provides worse results. Compared to Sample A, although Sample E triples the number of traits, the objective value decreases and the number of unallocated units increases. This is because exploitation reduces the opportunity of exploration, which is also proven by comparing the results between Sample D and Sample H as well as between Sample F and Sample G.
- Sample J enlarges the variables. Although the improvement of the result is obvious, the time spent is significant. Compared to Sample A, the objective value is increased by 10.3% (from 42.4299 to 46.8359), but the time Sample J spends is 165 times than that of Sample A.
- All the CVs are lower than 8%, indicating that the proposed ABC can provide solutions stably.
6. Comparison Study
Items | Proposed ABC | CP | ||
---|---|---|---|---|
Best | Average | Best | Average | |
Time spent (seconds) | 4.742 | 4.983 | 3.5 | 3.6108 |
Unallocated classes 1 | 0 | 0.5 | 12 | 12 |
Objective function values | 44.0740 | 42.4299 | 30.5128 | 30.1158 |
- CP is slightly faster than the modified ABC algorithm in solution-seeking (3.6108 s over 4.983 s on average).
- The modified ABC algorithm can allocate all the non-conflict classes. However, CP has 12 allocatable classes left.
- The modified ABC algorithm can seek a solution that is 40.88% better than CP in terms of objective function values gaining (42.4299 against 30.1158 on average).
7. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Class ID | Preference | Expertise |
---|---|---|
2 | 3 | |
1 | 1 | |
… | … | … |
3 | 2 | |
… | … | … |
0 | 0 |
Mon | Tue | Wed | Thu | Fri | |
---|---|---|---|---|---|
8:00 a.m. | Y | N | Y | N | N |
9:00 a.m. | Y | N | N | N | N |
10:00 a.m. | Y | Y | N | N | N |
11:00 a.m. | Y | Y | Y | N | N |
12:00 p.m. | N | Y | Y | N | N |
1:00 p.m. | N | N | Y | N | Y |
2:00 p.m. | N | N | Y | N | Y |
3:00 p.m. | N | Y | Y | N | Y |
4:00 p.m. | N | N | Y | N | Y |
5:00 p.m. | Y | N | Y | N | Y |
6:00 p.m. | Y | N | Y | N | Y |
A region | B region | ||||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
A | A | A | A | A | A | B | B | B | B | B | B |
B | B | C | C | D | D | A | A | C | C | D | D |
C | D | B | D | B | C | C | D | A | D | A | C |
D | C | D | B | C | B | D | C | D | A | C | A |
C region | D region | ||||||||||
13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
C | C | C | C | C | C | D | D | D | D | D | D |
A | A | B | B | D | D | A | A | B | B | C | C |
B | D | A | D | A | B | B | C | A | C | A | B |
D | B | D | A | B | A | C | B | C | A | B | A |
Sample | Num of Bees | Neighbour Range | Num of Iterations | Num of Traits |
---|---|---|---|---|
A | 5 | 5 | 1000 | 10 |
B | 20 | 5 | 1000 | 10 |
C | 5 | 30 | 1000 | 10 |
D | 5 | 5 | 10,000 | 10 |
E | 5 | 5 | 1000 | 30 |
F | 20 | 30 | 10,000 | 10 |
G | 20 | 30 | 10,000 | 30 |
H | 5 | 5 | 10,000 | 30 |
I | 20 | 30 | 10,000 | 30 |
J | 40 | 60 | 20,000 | 60 |
Sample | Objective Values | Time Spent (Seconds) | Unallocated Classes | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Avg. | Best | Std. Dev. | CV | Avg. | Best | Std. Dev. | CV | Avg 1 | Best | |
A | 42.4299 | 44.0740 | 0.9697 | 2.285% | 4.983 | 4.742 | 0.2712 | 5.443% | 0.5 | 0 |
B | 43.4970 | 44.7037 | 0.6937 | 1.594% | 18.8521 | 18.453 | 0.4208 | 2.232% | 0.1 | 0 |
C | 42.7910 | 43.7407 | 0.6779 | 1.584% | 5.0719 | 4.834 | 0.1950 | 3.846% | 0.5 | 0 |
D | 43.6596 | 44.8518 | 0.6690 | 1.532% | 48.843 | 47.238 | 2.4431 | 5.002% | 0 | 0 |
E | 41.9709 | 43.9256 | 1.0925 | 2.603% | 4.9919 | 4.818 | 0.1732 | 3.470% | 1 | 0 |
F | 44.7148 | 45.3703 | 0.4201 | 0.939% | 197.513 | 193.957 | 2.2012 | 1.114% | 0 | 0 |
G | 44.5255 | 45.1481 | 0.2849 | 0.639% | 201.0127 | 194.826 | 6.5969 | 3.281% | 0 | 0 |
H | 42.5502 | 43.2222 | 0.4757 | 1.118% | 50.3867 | 46.775 | 3.9439 | 7.827% | 0.3 | 0 |
I | 44.4592 | 45.8148 | 0.5484 | 1.228% | 206.4878 | 197.141 | 7.1228 | 3.449% | 0 | 0 |
J | 46.8359 | 47.1481 | 0.2448 | 0.522% | 831.6302 | 813.494 | 441.1553 | 4.467% | 0 | 0 |
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Zhu, K.; Li, L.D.; Li, M. School Timetabling Optimisation Using Artificial Bee Colony Algorithm Based on a Virtual Searching Space Method. Mathematics 2022, 10, 73. https://doi.org/10.3390/math10010073
Zhu K, Li LD, Li M. School Timetabling Optimisation Using Artificial Bee Colony Algorithm Based on a Virtual Searching Space Method. Mathematics. 2022; 10(1):73. https://doi.org/10.3390/math10010073
Chicago/Turabian StyleZhu, Kaixiang, Lily D. Li, and Michael Li. 2022. "School Timetabling Optimisation Using Artificial Bee Colony Algorithm Based on a Virtual Searching Space Method" Mathematics 10, no. 1: 73. https://doi.org/10.3390/math10010073
APA StyleZhu, K., Li, L. D., & Li, M. (2022). School Timetabling Optimisation Using Artificial Bee Colony Algorithm Based on a Virtual Searching Space Method. Mathematics, 10(1), 73. https://doi.org/10.3390/math10010073