Doshisha University, Kyoto, JAPAN Mitsunori MIKI, Tomoyuki HIROYASU, Jun’ya WAKO , Takeshi YOSHIDA

1. Adaptive Temperature ScheduleDetermined by Genetic Algorithmfor Parallel Simulated Annealing Adaptive Temperature ScheduleDetermined by Genetic Algorithmfor Parallel Simulated Annealing Doshisha University, Kyoto, JAPAN Mitsunori MIKI, Tomoyuki HIROYASU, Jun’ya WAKO, Takeshi YOSHIDA

2. Background • Optimization :Find values of the variables that minimize or maximize the objective function while satisfying the constraints. • Optimization Problem • Combinatorial Optimization Problem • Traveling Salesman Problems(TSPs) • Job shop Scheduling Problems(JSPs) • Continuous Opitimization Problem

3. Traveling Salesman Problems (TSPs) • Typical combinatorial optimization problem. • Finding a minimum tour to round all cities. Σd(vπ(i),vπ(i+1)+d(vπ(N), vπ(1)) v(i) : i-th city π : tour N : city size d(v(i),v(j)) : distance between two point from TSPLIB

4. Job shop Scheduling Problems(JSPs) • Job shop Scheduleing Problems(JSPs) • Given： • n jobs × m machines (resources) • each job consists of a sequence of operationsprocessed in a given order • Objective: • schedule that minimizes the makespan

5. Background • Optimization :Find values of the variables that minimize or maximize the objective function while satisfying the constraints. • Optimization Problem • Combinatorial Optimization Problem • Traveling Salesman Problems(TSPs) • Job shop Scheduling Problems(JSPs) • Continuous Opitimization Problem • Heuristic search methods • Simulated Annealing(SA) • Genetic Algorithm(GA)

6. Simulated Annealing (SA) Decreased Always accept. Increased Accepted in a certain probability, “P”. Low temperature High temperature Local minimum Global minimum • An effective algorithm to solve combinational optimization problems. • Algorithm • Cooling Schedule : The procedures for updating temperature. Move of a solution Judging acceptance by the “Energy”, Ecurrent. and “next Energy”, Enext “Temperature” is decreased. Disadvantages : (1) High computational costs.(2) The difficult determination of a proper cooling schedule.

7. Temperature in SA Max Temperature Temperature Min Temperature Time Conventional cooling schedule Max Temperature • A specific constant temperature in SAyields good solutions for TSPs.[Mark00] • A proportional cooling schedule Temperature Important Temperature Region Min Temperature Time Experiments to determine the region • many SAs with various constanttemperatures are performed. • comparing the qualities of the solutions obtained. • Objective problem is Traveling Salesman Problems (TSPs).

8. Important Temperature Region in TSPs - eil101 - • There is the important temperature region for each problem. • The values and ranges of important temperature region are problem-dipendent. Constant temperature

9. Purpose of this study • Adaptive cooling schedule • The mechanisum which specifies the important temperature region automatically by Genetic Algorithm • uses parallel SA + GA to Temperature Parallel SA Parallel SA with Adaptive Temperature determined by Genetic Algorithm (PSA/AT(GA)) Feature of PSA/AT(GA) • The cooling schedule of PSA/AT(GA) is automatically determined by GA, and the temperature on each SA converges on the important temperature region.

10. PSA / AT(GA) SA + evaluate Fitness 1 GA to temperature 2 SA Set new temperature 3 SA SA GA operation SA Solution GA operation Temperature (individual) E N D GA operation L L L : temperature change interval • PSA/AT(GA) is based on Parallel SA. • The different solutions and temperatures are assigned to different processors. • Sequential SA and evaluating “fitness” are performed on each processor. • The “fitness” isn’t Energy, but the value which evaluates the move of solution.

11. Characteristics of the transition of the solution - eil101 - • The characteristics of the important temperature • good solution • relatively medium fluctuations • It is able to judge goodness of the solution by temperature. • “Fitness” in PSA/AT(GA) is designed by this idea.

12. Evaluate “Fitness” • Fitness value is defined by the summation of the difference between a baseline and the energy value. L: temperature change interval • calculated only when the solution is accepted, and the energy dips from the baseline. • repeats till the syncronous interval, L. • E is an average of the energies of all SA processes performed in parallel. Solution Average Energy Annealing Steps: k

13. Effectiveness of “fitness” Energy Energy Baseline Energy Baseline Steps Steps Steps • The temperature is concerned with the fluctuations of the solution. • The fitness value is used to evaluate the fluctuations of the solution. • Searching at a important temperature, the fitness value is high. • SA processes with important temperatures are probably selected by GA. High temperature Important temperature Low temperature Baseline Fitness : low Fitness : high Fitness : low

14. PSA / AT(GA) SA + evaluate Fitness 1 GA to temperature 2 SA Set new temperature 3 SA SA GA operation SA Solution GA operation Temperature (individual) E N D GA operation L L L : temperature change interval • Sequential SA and evaluating fitness are performed on each processor for synchronous interval, L . • All temperatures are adjusted by GA synchronously.

15. PSA / AT(GA) SA + evaluate Fitness 1 GA to temperature 2 SA Set new temperature 3 SA SA GA operation Solution SA Temperature (individual) GA operation E N D GA operation Selection • In selection, individuals with ahigh fitness value would be selected. • In crossover & mutation, various individuals of temperature would be generated. Crossover Mutation

16. PSA / AT(GA) SA + evaluate Fitness 1 GA to temperature 2 Set new temperature 3 GA operation Solution GA operation Temperature (individual) E N D GA operation n n n : synchronous interval • PSA/AT(GA) repeats this cycle to end. • All temperatures of processors would be automatically adjusted to important temperature region.

17. Experiments • Objective Problem • TSPs (Traveling Salesman Problems) • JSPs (Job shop Scheduling Problems) • Compared method • TPSA (Temperature Parallel SA) [Konishi, 95]

18. Temperature Parallel SA(TPSA) SA TPSA • Algorithm • The features of TPSA High T Low T • The different temperatures are assigned to different processors. • Each processor performs on sequential SA with a constant temperature. • Two solutions with adjacent temperatures are exchanged. (1) Automatic determination of a cooling schedules (2) A good fit for parallel processing

19. Parameters used in the experiments for TSPs

20. Experimental results (Error Rate) in TSPs Error Rate (%) TSP Problems PSA/AT(GA) provides better results than TPSA.

21. Cooling schedule (eil101) - PSA/AT(GA) - - TPSA - Temperature Temperature Num. of annealing steps Num. of annealing steps A line : a cooling schedule on one SA. PSA/AT(GA) : Convergence on the important temperature region TPSA : All processes can’t always have good search. The cooling schedule of PSA/AT(GA) is more proper than TPSA’s.

22. Parameters used in the experiments for JSPs

23. Experimental results (Error Rate) in JSPs Error Rate (%) JSP Problems PSA/AT(GA) provides better results than TPSA.

24. Cooling schedule (ft10) - PSA/AT(GA) - - TPSA - Temperature Temperature Num. of annealing steps Num. of annealing steps A line : a cooling schedule on one SA. PSA/AT(GA) : Convergence on the important temperature region TPSA : All processes can’t always have good search. The cooling schedule of PSA/AT(GA) is more proper than TPSA’s.

25. Conclusions This study proposes a new hybrid method, Parallel Simulated Annealingwith Adaptive Temperature determined by Genetic Algorithm(PSA/AT(GA)). PSA/AT(GA) • is based on Parallel SA. • uses GA to determine temperature on each SA. • automatically converges important temperature region on SA Applying PSA/AT(GA) to TSPs and JSPs • PSA/AT(GA) has better searching ability than TPSA.

26. Fin

27. Important temperature region in JSPs - ft10 - • There is the important temperature region for each problem in JSP,too. • The values and ranges of important temperature region are problem-dipendent.

28. Coding of temperature • PSA/AT(GA) uses GA to optimize the cooling schedule. • Individual : Temperature on each processor • Design variable : The exponent of temperature function, X • The expression of temperature in PSA/AT(GA) is suitable for the exponential cooling schedule in SA. Real Value Bit Array Encoding X X Temperature = 10 X Decoding SA cooling method

29. Genetic Algorithm (GA) Individuals with high fitness survive. Selection Perform direct information exchange between Individuals. Crossover Change information of individuals. Mutation • Optimization method based on the mechanism of natural selection and natural genetics. • Searching points : Individuals • The new searching points are generated by GA operator. GA operator Individual Population

30. Temperature Parallel SA(TPSA) • The different temperatures are • assigned to different processors. • Each SA searches solution with a constant temperature. • Exchanging solution. • Algorithm High T Low T SA TPSA Feature of TPSA • Automatic determination of a cooling schedules • A good fit for parallel processing

31. The execution time The execution time [sec] (speedup) • The execution time of PSA/AT(GA) is a little longer than TPSA. • The speedup for PSA/AT(GA) and TPSA increases as the problem size become longer. • PSA/AT(GA) shows high parallel efficiency.

32. Energy histories • TPSA shows a good convergence at the beginning. • PSA/AT(GA) shows a better performance at the later stage. ※The values are the average of 20 trials. PSA/AT(GA) has a better performance in searching global optimum than TPSA.

33. Experimental results2 (Error Rate) Error Rate (%) TSP Problem PSA/AT provides the best result in the whole problems.

34. GA operator Selection of individuals 2 Individuals with higher fitness survive. Send Information 1 Individuals of a proper temperature for searching probably survive. Individuals of temperature and fitness are gathered to one processor. Crossover & Mutation 3 Various individuals of temperature are generated. Update temperature 4