Self-Learning Genetic Algorithm for a Timetabling Problem with Fuzzy Constraints

Transcription

1 Self-Learning Genetic Algorithm for a Timetabling Problem with Fuzzy Constraints Radomír Perzina, Jaroslav Ramík perzina(ramik)@opf.slu.cz Centre of excellence IT4Innovations Division of the University of Ostrava Institute for Research and Applications of Fuzzy Modeling Ostrava, Czech Republic

2 Content Genetic Algorithms Self-Learning Genetic Algorithm Course Timetabling Problem Fuzzy Preferences Conclusions

3 Genetic Algorithms A class of probabilistic optimization algorithms Inspired by the biological evolution process Uses concepts of Natural Selection and Genetic Inheritance (Darwin 1859) Originally developed by John Holland (1975) Particularly well suited for hard problems where little is known about the underlying search space

4 Create an initial generation (usually randomly) Evaluate the fitness of each individual in the population Create a new generation by using genetic operators: selection, crossover, mutation Is satisfied the end condition? No Yes The solution is the fittest individual in the population

5 Properties of Genetic Algorithms Efficiency depends on their parameters Setting up of GA parameters Recommendations of experts Two-level genetic algorithm Self-adaptation

6 Requirements for Self-Learning GA Self-adaptation of all possible parameters Separate parameters for each part of a chromosome Robustness

7 Encoding Structure Population Individual 1 Individual 2 Individual N p Gene 1 Gene 2 Gene N g Gene elem 1 Gene elem 2 Gene elem N e

8 The Structure of a Gene Element 1 x - Optimized variable q m - Parameter of mutation r m - Radius of mutation p c - Probability of crossover r c - Ratio of crossover q d - Parameter of deletion q u - Parameter of duplication s m Identifier of myself for mating s w - Wanted partner for mating

9 The Structure of a Gene Element 2 r r - Ratio of replacement r t - Ratio of population for selection r p - Ratio of population for 2 nd partner selection c d - Coefficient of death N p - Wanted size of population

10 Selection Simple GA based on fitness Nature based on individual s preferences s w - individual s preferences for mating s m - individual s phenotype for mating 1 st parent tournament selection with variable ratio of population r t 2 nd parent individual s preferences s w -s m

11 Crossover Probability of crossover p c E E E E X X X X r 3 E where X stands for all parameters of a gene element, r c is a ratio of crossover of the first parent defined in this gene element, the lower index 1 denotes the gene element of the first parent, the index 2 the second parent and the index 3 denotes the child of both parents. c

12 Mutation Probability of mutation p m E new E old E E E E X X U - r r X X, max min where X stands for all parameters of the gene element, U(a,b) is a random variable with uniform probability distribution in the interval <a,b>, X new is the value of the parameter after mutation, X old is the original value of the parameter, X max (X min ) is the maximal (minimal) allowed gene element value of the parameter. m m

13 Course Timetabling Problem Finding the exact time allocation within a limited time period for a number of events (lectures, seminars) and assigning them to a number of resources (teachers, students and classrooms) so that the constraints are satisfied. Hard constraints (no resource may be assigned to different events at the same time, suitable rooms for events, ) Soft constraints (teacher preferences)

14 Course Timetabling Problem Notation 1 n R - number of available rooms (classrooms, offices), {R 1, R 2,, } set of available rooms, n E - number of events (actions, lectures, seminars), {E 1, E 2,, } set of events (actions), n T - number of teachers, {T 1, T 2,, } set of teachers, n S - number of students (group of students), {S 1, S 2,, } set of students (group of students), n P - number of time slots (time periods), {P 1, P 2,, } set of time slots (time periods),

15 Course Timetabling Problem Notation 2 n G - number of time-room slots, {G 1, G 2,, } set of time-room slots, C = {c ij } - clash matrix with elements c ij ; i = 1, 2,, n E ; j = 1, 2,, n E, A = {a ij } - room acceptance matrix with elements a ij ; i = 1, 2,, n R ; j = 1, 2,, n E.

16 Fuzzy Preferences 1 Teacher preference membership function p(t) Scheduled timetable membership function r(t)

17 Fuzzy Preferences 2 Membership function of satisfaction with teacher preferences s(t)

18 The Timetable Optimization Model

19 Solving the Timetabling Problem Each gene of a chromosome represents one real variable within the interval <0;1> The chromosome is divided into two parts The first part A: n E genes - parameters for events The second part B: n G genes - parameters for timeroom slots

20 Timetable Builder Sort events in ascending order according to values of parameters in the part A of the chromosome Sort time-room slots in ascending order according to values of parameters in the part B of the chromosome Assign events to the first suitable unused time-room slot

21 Timetable Builder - Example Part A B Description E 1 E 2 E 3 G 1 G 2 G 3 G 4 Value E 2 E 1 E 3 G 2 G 4 G 1 G 3 X X X E 1 G 1, E 2 G 2, E 3 G 4

22 Timetable at Silesian University Number of rooms n R = 43 Number of events n E = 705 Number of students n S = 1807 Number of teachers n T = 112 Number of time slots n M = 60 Number of time-room slots n G = 2580 All hard constraints satisfied

23 Conclusions Genetic algorithms are able to effectively solve the timetabling problem No need for finding values of the parameters, as there are no parameters set in advance Soft constraints formulation using fuzzy sets is more natural for real life timetabling problems Future work - parallel representation of the genetic algorithm

24 Thank you for your attention