Doshisha Univ., Kyoto, Japan CEC2003 Adaptive Temperature Schedule Determined by Genetic Algorithm for Parallel Simulated Annealing Doshisha University, Kyoto, JAPAN Mitsunori MIKI, Tomoyuki HIROYASU, Jun’ya WAKO, Takeshi YOSHIDA Adaptive Temperature Schedule Determined by Genetic Algorithm for Parallel Simulated Annealing

Doshisha Univ., Kyoto, Japan CEC2003 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

Doshisha Univ., Kyoto, Japan CEC2003 Traveling Salesman Problems (TSPs) Typical combinatorial optimization problem. Finding a minimum tour to round all cities. Σd(v π (i),v π (i+1)+d(v π (N), v π (1)) v(i) : i-th city d(v(i),v(j)) : distance between two point N : city size π : tour TSP Num. of City Optimum eil kroA lin pr rat d from TSPLIB

Doshisha Univ., Kyoto, Japan CEC2003 Job shop Scheduling Problems(JSPs) Job shop Scheduleing Problems(JSPs) Given ： –n jobs × m machines (resources) –each job consists of a sequence of operations processed in a given order Objective: –schedule that minimizes the makespan

Doshisha Univ., Kyoto, Japan CEC2003 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)

Doshisha Univ., Kyoto, Japan CEC2003 An effective algorithm to solve combinational optimization problems. Algorithm Cooling Schedule : The procedures for updating temperature. 1.Move of a solution 2.Judging acceptance by the “Energy”, E current. and “next Energy”, E next 3.“Temperature” is decreased. High temperature Local minimum Global minimum Low temperature Always accept. Decreased Increased Accepted in a certain probability, “P”. Simulated Annealing (SA) (1) High computational costs. (2) The difficult determination of a proper cooling schedule. Disadvantages :

Doshisha Univ., Kyoto, Japan CEC2003 Temperature in SA Temperature Time Max Temperature Min Temperature Important Temperature Region A specific constant temperature in SA yields good solutions for TSPs.[Mark00] Experiments to determine the region Temperature Max Temperature Min Temperature Time many SAs with various constant temperatures are performed. comparing the qualities of the solutions obtained. Objective problem is Traveling Salesman Problems (TSPs). Conventional cooling schedule A proportional cooling schedule

Doshisha Univ., Kyoto, Japan CEC2003 Important Temperature Region in TSPs There is the important temperature region for each problem. The values and ranges of important temperature region are problem-dipendent. - eil101 - TSPOptimumT opt region eil ~ 2.5 kroA ~ 52.7 lin ~ 39.0 pr ~ 72.3 rat ~ 3.9 d ~ 26.8 Constant temperature

Doshisha Univ., Kyoto, Japan CEC2003 Adaptive cooling schedule –The mechanisum which specifies the important temperature region automatically by Genetic Algorithm –uses parallel SA Purpose of this study -The cooling schedule of PSA/AT(GA) is automatically determined by GA, and the temperature on each SA converges on the important temperature region. Parallel SA with Adaptive Temperature determined by Genetic Algorithm (PSA/AT(GA)) GA to Temperature + Parallel SA Feature of PSA/AT(GA)

Doshisha Univ., Kyoto, Japan CEC2003 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. PSA / AT(GA) L : temperature change interval Temperature (individual) Solution GA operation E N D SA + evaluate Fitness GA to temperature 3 Set new temperature 2 1 GA operation L L SA

Doshisha Univ., Kyoto, Japan CEC2003 Characteristics of the transition of the solution 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. - eil101 -

Doshisha Univ., Kyoto, Japan CEC2003 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. Fitness value is defined by the summation of the difference between a baseline and the energy value. Evaluate “Fitness” Energy Annealing Steps: k Average Solution L: temperature change interval

Doshisha Univ., Kyoto, Japan CEC2003 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. Effectiveness of “fitness” Important temperature Fitness : high Energy Steps Baseline Low temperature Fitness : low Energy Steps Baseline High temperature Fitness : low Energy Steps Baseline

Doshisha Univ., Kyoto, Japan CEC2003 PSA / AT(GA) Sequential SA and evaluating fitness are performed on each processor for synchronous interval, L. All temperatures are adjusted by GA synchronously. L : temperature change interval Temperature (individual) Solution GA operation E N D SA + evaluate Fitness GA to temperature 3 Set new temperature 2 1 GA operation L L SA

Doshisha Univ., Kyoto, Japan CEC2003 PSA / AT(GA) GA operation E N D SA + evaluate Fitness GA to temperature 3 Set new temperature 2 1 In selection, individuals with a high fitness value would be selected. In crossover & mutation, various individuals of temperature would be generated. GA operation Selection Crossover Mutation Temperature (individual) Solution SA

Doshisha Univ., Kyoto, Japan CEC2003 PSA / AT(GA) n : synchronous interval Temperature (individual) Solution GA operation E N D SA + evaluate Fitness GA to temperature 3 Set new temperature 2 1 GA operation n n PSA/AT(GA) repeats this cycle to end. All temperatures of processors would be automatically adjusted to important temperature region.

Doshisha Univ., Kyoto, Japan CEC2003 Experiments Objective Problem –TSPs (Traveling Salesman Problems) –JSPs (Job shop Scheduling Problems) Compared method –TPSA (Temperature Parallel SA) [Konishi, 95]

Doshisha Univ., Kyoto, Japan CEC2003 Temperature Parallel SA(TPSA) Algorithm The features of TPSA High T Low T TPSA SA 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

Doshisha Univ., Kyoto, Japan CEC2003 Parameters used in the experiments for TSPs AlgorithmPSA/AT(GA)TPSA Number of Processes 32 Temperature change intervalCity size × 5 Annealing steps on each process(City size × 5) × 160 Maximum temperature The maximum energy of uphill move is accepted at a rate of 50%. Minimum temperature One uphill move of solution is accepted for the temperature change interval. Neighborhood Structure 2-opt Chromosome length10- Crossover method Roulette - Crossover ratio0.3- Mutation ratio0.01-

Doshisha Univ., Kyoto, Japan CEC2003 Experimental results (Error Rate) in TSPs PSA/AT(GA) provides better results than TPSA. TSP Problems Error Rate (%)

Doshisha Univ., Kyoto, Japan CEC2003 Cooling schedule (eil101) - TPSA - Num. of annealing steps Temperature - PSA/AT(GA) - Num. of annealing steps Temperature 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.

Doshisha Univ., Kyoto, Japan CEC2003 Parameters used in the experiments for JSPs AlgorithmPSA/AT(GA)TPSA Number of Processes 32 Temperature change interval200 Annealing steps on each process32000 Maximum temperature The maximum energy of uphill move is accepted at a rate of 50%. Minimum temperature One uphill move of solution is accepted for the temperature change interval. Neighborhood Structure Critical Block Neighborhood [Yamada, ‘94] Modification of solution GT Method [Giffler and Tompson, ‘60 ] Chromosome length10- Crossover method Tournament (size=2) - Crossover ratio0.3- Mutation ratio0.01-

Doshisha Univ., Kyoto, Japan CEC2003 Experimental results (Error Rate) in JSPs PSA/AT(GA) provides better results than TPSA. JSP Problems Error Rate (%)

Doshisha Univ., Kyoto, Japan CEC2003 Cooling schedule (ft10) - TPSA - Temperature - PSA/AT(GA) - Temperature 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. Num. of annealing steps The cooling schedule of PSA/AT(GA) is more proper than TPSA’s.

Doshisha Univ., Kyoto, Japan CEC2003 Conclusions This study proposes a new hybrid method, Parallel Simulated Annealing with 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.

Doshisha Univ., Kyoto, Japan CEC2003 Fin

Doshisha Univ., Kyoto, Japan CEC2003 Important temperature region in JSPs TSPOptimumT opt region ft ~ 14.2 ft ~ 9.7 orb ~ 14.2 orb ~ 16.0 la ~ 12.5 la ~ 12.5 There is the important temperature region for each problem in JSP,too. The values and ranges of important temperature region are problem-dipendent. - ft10 -

Doshisha Univ., Kyoto, Japan CEC2003 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. Coding of temperature Real ValueBit Array Decoding Temperature = 10 X Encoding SA cooling method XX

Doshisha Univ., Kyoto, Japan CEC2003 Genetic Algorithm (GA) Optimization method based on the mechanism of natural selection and natural genetics. Searching points : Individuals The new searching points are generated by GA operator. Individual GA operator Selection Individuals with high fitness survive. Crossover Perform direct information exchange between Individuals. Mutation Change information of individuals. Population

Doshisha Univ., Kyoto, Japan CEC2003 Temperature Parallel SA(TPSA) Algorithm High T Low T - The different temperatures are assigned to different processors. - Each SA searches solution with a constant temperature. - Exchanging solution. TPSA SA - Automatic determination of a cooling schedules - A good fit for parallel processing Feature of TPSA

Doshisha Univ., Kyoto, Japan CEC2003 The execution time 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. ProblemsPSA/AT(GA)TPSASequential SA eil (9.3)10.1(20.0)202(1.0) kroA (19.0)41.5(25.1)1043(1.0) lin (22.5)76.0(26.6)2025(1.0) The execution time [sec] (speedup)

Doshisha Univ., Kyoto, Japan CEC2003 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.

Doshisha Univ., Kyoto, Japan CEC2003 Experimental results2 (Error Rate) PSA/AT provides the best result in the whole problems. TSP Problem Error Rate (%)

Doshisha Univ., Kyoto, Japan CEC2003 GA operator Individuals of temperature and fitness are gathered to one processor. 1 2 Selection of individuals Individuals with higher fitness survive. Send Information Individuals of a proper temperature for searching probably survive. 3 Crossover & Mutation Various individuals of temperature are generated. 4 Update temperature