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Learning the heuristic distribution by an evolutionary hyper-heuristic

Learning the heuristic distribution by an evolutionary hyper-heuristic. Edmund Burke, Nam Pham, Rong Qu. Outline. Motivation Constructive hyper-heuristics Heuristic representation Our evolutionary hyper-heuristic approach An application to graph colouring

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Learning the heuristic distribution by an evolutionary hyper-heuristic

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  1. Learning the heuristic distribution by an evolutionary hyper-heuristic Edmund Burke, Nam Pham, Rong Qu

  2. Outline • Motivation • Constructive hyper-heuristics • Heuristic representation • Our evolutionary hyper-heuristic approach • An application to graph colouring • An application to examination timetabling • Conclusions and Future Work Nam Pham

  3. Motivation • Hyper-heuristics are at higher generality • Applicable for different problems without many modifications • Hyper-heuristics may learn effective and ineffective heuristics • It is observed by human that a heuristic may be better at giving decisions at different stages during the search • Example: ‘largest degree’ and ‘saturation degree’ for graph colouring Nam Pham

  4. Constructive Hyper-heuristics • Constructive heuristics • Consist of step-by-step decisions • Build a complete solution from scratch • Different problems have different set of decisions at each step • Constructive hyper-heuristics • Are heuristics • Intelligently choose different constructive heuristics for different situations Nam Pham

  5. A Heuristic Representation • Sequences of low-level heuristics • Each element represents a set of heuristics • That set of heuristics provides a number of decisions to construct one step towards a complete solution • Elements in a sequence are applied consecutively until a complete solution is obtained • We use this representation for problems in this talk. Nam Pham

  6. Our Evolutionary Hyper-heuristic • An evolutionary algorithm on the high level search • Chromosomes: sequence representation • Divide a sequence into a number of intervals • Learn the appearance frequency of simple low-level heuristics in different intervals of sequences • Maintain a list of heuristic probability distribution • Update the list by using fittest sequences • With a good probability distribution, we are likely to generate fitter sequences Nam Pham

  7. Our Evolutionary Hyper-heuristic • An evolutionary algorithm on the high level search • Chromosomes: sequence representation • Divide a sequence into a number of intervals • Learn the appearance frequency of simple low-level heuristics in different intervals of sequences • Maintain a list of heuristic probability distribution • Update the list by using fittest sequences • With a good probability distribution, we are likely to generate fitter sequences Nam Pham

  8. A Sequence of Heuristics H1 H1 H1 H1 H2 H3 H1 H1 H2 H3 H2 Nam Pham

  9. Intervals of a sequence H1 H1 H1 H1 H2 H3 H1 H1 H2 H3 H2 Nam Pham

  10. Our Evolutionary Hyper-heuristic • An evolutionary algorithm on the high level search • Chromosomes: sequence representation • Divide a sequence into a number of intervals • Learn the appearance frequency of simple low-level heuristics in different intervals of sequences • Maintain a list of heuristic probability distribution • Update the list by using fittest sequences • With a good probability distribution, we are likely to generate fitter sequences Nam Pham

  11. List of heuristic probability distribution H1 H1 H1 H1 H2 H3 H1 H1 H2 H3 H2 H1 100% H1 33.3% H1 0% H2 0% H2 33.3% H2 66.6% H3 0% H3 33.3% H3 33.3% Nam Pham

  12. Update the list H1 H1 H1 H1 H2 H3 H1 H1 H2 H3 H2 H1 H1 H1 H1 H2 H3 H1 H1 H2 H3 H2 H1 H1 H1 H1 H2 H3 H1 H1 H2 H3 H2 Average Probabilities H1 100% H1 33.3% H1 0% H2 0% H2 33.3% H2 66.6% H3 0% H3 33.3% H3 33.3% Nam Pham

  13. Our Evolutionary Hyper-heuristic • An evolutionary algorithm on the high level search • Chromosomes: sequence representation • Divide a sequence into a number of intervals • Learn the appearance frequency of simple low-level heuristics in different intervals of sequences • Maintain a list of heuristic probability distribution • Update the list by using fittest sequences • With a good probability distribution, we are likely to generate fitter sequences Nam Pham

  14. Our Hyper-heuristic Outline • Initialise heuristic probability distribution such that all low-level heuristics have the same chance to appear in every interval • Do • Sort the evaluation function of all chromosomes • Update the list of heuristic distribution using p = 10% of the best fit chromosomes • Generate chromosomes for the next generation. • Until stopping condition is met Nam Pham

  15. The Creation of New Chromosomes • Keep chromosomes that always select a particular low-level heuristic • Keep the best 10% from the previous generation • 10% of the population will be generated randomly • 10% of the population will be generated based on the probabilities in the list of heuristic distribution • The remaining chromosomes are generated by using a crossover operator • Fitter chromosomes are more likely to be selected Nam Pham

  16. Crossover Operator Chromosome 1 H1 H1 H1 H1 H2 H3 H1 H1 H2 H3 H2 H1 100% H1 33.3% H1 0% H2 0% H2 33.3% H2 66.6% H3 0% H3 33.3% H3 33.3% Nam Pham

  17. Crossover Operator Chromosome 2 H1 H2 H1 H1 H2 H1 H1 H1 H2 H3 H2 H1 66.6% H1 66.6% H1 0% H2 33.3% H2 33.3% H2 66.6% H3 0% H3 0% H3 33.3% Nam Pham

  18. Crossover Operator H1 100% H1 33.3% H1 0% H2 0% H2 33.3% H2 66.6% H3 0% H3 33.3% H3 33.3% H1 H1 66.6% 83.3% H1 H1 66.6% 50% H1 H1 0% 0% H2 H2 33.3% 16.6% H2 H2 33.3% 33.3% H2 H2 66.6% 66.6% H3 H3 0% 0% H3 H3 0% 0% H3 H3 33.3% 33.3% Average Probabilities Nam Pham

  19. Generality of our Hyper-heuristic • Not designed with only a problem in mind • No problem dependent information • Learn from the frequency of low-level heuristics in sequences • Applicable to different problems • We test on the graph colouring problem and the examination timetabling problem • The only modifications are the pool of low-level heuristics and the fitness evaluation for sequences Nam Pham

  20. An Application to Graph Colouring • Problem description • Pool of low-level heuristics • Fitness evaluation for sequences • Computational results Nam Pham

  21. Problem description • NP-hard combinatorial optimisation problem (Papadimitriou and Steiglitz, 1982) • Assigning colours to vertices such that adjacent vertices receive different colours • Find a colouring that requires as few colours as possible Nam Pham

  22. Low-level heuristics • Sequence representation • Each element in a sequence consists of a vertex-selection(VS) heuristic and a colour-selection(CS) heuristic • We currently use the same colour-selection heuristic and concern only vertex-selection heuristics • Strategy: a vertex that is likely to cause trouble if deferred until later should be coloured first VS1 CS VS1 CS VS1 CS VS1 CS VS2 CS VS3 CS VS1 CS VS1 CS VS2 CS VS3 CS VS2 CS Nam Pham

  23. Pool of low-level heuristics • SD: Least Saturation Degree: Vertices are ordered increasingly by the number of available colours during the colouring process • LD: Largest Degree: Vertices are ordered decreasingly by the number of neighbours in the graph • LCD: Largest Coloured Degree: Vertices are ordered decreasingly by the number of coloured neighbours • SD2, SD3, LD2, LD3, LCD2, LCD3 Nam Pham

  24. Fitness Evaluation for Sequences • We currently use the evaluation of a colouring as the sequence evaluation • the number of colours required to obtain a non-conflict colouring • breaking ties by the distance to a colouring using one colour less than the current colouring Nam Pham

  25. Computational Results • Each interval consists of 10 decisions to select heuristics • For each instance, we evolved 5000 generations • Compare our hyper-heuristic with results obtained from other approaches for the Toronto benchmark (Qu et al., 2008) • The probability distribution of heuristics are reported Nam Pham

  26. Computational Results Nam Pham

  27. Heuristic Probability Distribution Nam Pham

  28. Effective and Ineffective Heuristics Nam Pham

  29. Effective and Ineffective Heuristics Nam Pham

  30. An Application to Exam Timetabling • Problem description • Pool of low-level heuristics • Fitness evaluation for sequences • Computational results Nam Pham

  31. Problem description • A generalisation of the graph colouring problem • Assigning exam to timeslots such that • students do not have to sit two exams at the same time • the timetable with the best spread of exams for students is preferred • Using graph colouring to model exam timetabling • vertices represent exams • colours represent timeslots • Find a non-conflict timetable that has the smallest penalty for the spread of exams for students Nam Pham

  32. Low-level heuristics • Sequence representation • Each element in a sequence consists of a exam-selection(ES) heuristic and a timeslot-selection(TS) heuristic • We currently use the same timeslot-selection heuristic and concern only exam-selection heuristics • Use the same strategy to select exams as for the graph colouring problem ES1 TS ES1 TS ES1 TS ES1 TS ES2 TS ES3 TS ES1 TS ES1 TS ES2 TS ES3 TS ES2 TS Nam Pham

  33. Pool of low-level heuristics • SD: Least Saturation Degree: Exams are ordered increasingly by the number of available timeslots • LD: Largest Degree: Exams are ordered decreasingly by the number of neighbours in the graph • LCD: Largest Coloured Degree: Exams are ordered decreasingly by the number of already assigned neighbours • LE: Largest Enrolment: Exams are ordered decreasingly by the enrolment • LWD: Largest Weighted Degree: Exams are ordered decreasingly by the total number of students in conflict • SD2, SD3, LD2, LD3, LCD2, LCD3, LE2, LE3, LWD2, LWD3 Nam Pham

  34. Fitness Evaluation for Sequences • If a feasible timetable is found • The evaluation is the average penalty of students sitting in exams close together • If a feasible timetable cannot be found • Record the first position, i, that causes infeasibility • We use the evaluation as follows where l is the length of the sequence and controls the degree of involvement for infeasible solutions Nam Pham

  35. Computational Results • Each interval consists of 10 decisions to select heuristics • For each instance, we evolved 5000 generations • Compare our hyper-heuristic with results obtained from other constructive approaches for the Toronto examination timetabling benchmark (Qu et al., 2008) • The probability distribution of heuristics are reported Nam Pham

  36. Computational Results Nam Pham

  37. Heuristic Probability Distribution Nam Pham

  38. Conclusions and Future Work • This evolutionary hyper-heuristic is re-usable for different problems and requires little problem specific information • We will experiment on other problems in future work • Produce solutions of acceptable quality. Some of them are competitive. • The probability distribution of heuristics shows how such heuristics work at different stages of the search • We can learn whether a heuristic is effective or ineffective in different situations Nam Pham

  39. Reference • PAPADIMITRIOU, C. H. & STEIGLITZ, K. (1982) Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall. • QU, R., BURKE, E. K., MCCOLLUM, B., MERLOT, L. T. G. & LEE, S. Y. (2008) A Survey of Search Methodologies and Automated Approaches for Examination Timetabling. to appear in Journal of Scheduling. Nam Pham

  40. ? Thank you! ? ? Questions/Comments Nam Pham

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