Multi4Objective Genetic Algorithms Engineering University Osaka 593 F - PDF document

1 Multi4-Objective Genetic Algorithms Engi
Multi4-Objective Genetic Algorithms Engineering, University Osaka 593, Fax: +81-722-59-3340 propose a .framework of genetic algorithms to show optimal solutions to the maker. Then one the Pareto solutions can be chosen depending on the the Pareto optimal solutions algorithms, the variety individuals should kept in each generation. Recently Recently proposed the Niched Pareto domination in procedure and applying to spread along the Pareto front. this paper, we propose Genetic Algorithm with various shown in the selection procedure, our MOGA uses multiple objective functions to combine them into fitness function. The our MOGA is that weights attached to the multiple functions are not constant but randomly selection. Therefore the direction the search the MOGA tentative set preserved in the execution the MOGA. individuals in this set are inherited the next generation as elite be minimized Multi-Objective Genetic GENETIC OPERATIONS Selection Procedure One simple combine multiple objective functions into a scalar fitness function the following weighted sum (we assume functions should be maximized): a string fitness function, i-th objective a constant genetic algorithms we propose selection procedure with random weights to search utilizing various search directions shown in pair of strings are selected for a we assign random real to each we can real number in the closed interval interval 11. Next pair strings are selected with different weight values newly given the execution the MOGA, optimal solutions is and updated at generation. A certain number (say, individuals are randomly selected from the set at each generation. Those solutions are used individuals in our MOGA. elite preserve strategy has an in keeping the va

2 riety each population our MOGA. shown in
riety each population our MOGA. shown in is described below. an initial population functions for the generated strings. optimal solutions. Calculate the using the random pair of strings from current population according to the following selection probability. selection probability a string block diagram the proposed MOGA This step repeated for strings from the current selected pair, apply operation to generate two new strings. Step 0: Initialization 1 Step 2: Selection li Step 4: Mutation the tentative set of Fig.4 The block diagram Multi-Objective Genetic value of the strings the crossover operation, apply mutation operation a prespecified mutation the set of the strings generated the previous operations, and replace them strings randomly selected a tentative Pareto optimal solutions. a prespecified stopping condition is not satisfied, return to Step MOGA shows the final Pareto optimal solutions to the decision best solution is then to the decision Randomly remove The genetic operations such crossover and mutation are decided depending the feature at hand. this section, first compare the proposed the VEGA VEGA and the Niched Pareto GA GA om the test problem used in [5]. Then we demorutrate the computer simulations a flowshop scheduling problem and a selection problem. show the simulation results the three genetic algorithms Niched Pareto GA and for a simple test problem problem called “unitation versus pairs”. This problem has objectives to be maximized: unitation and complementary adjacent is simply the number the fixed len@h bit string is the number complementary adjacent bits 01 or applied to this problem, the fitness function as as . I/nit(x) + Wprs . Prs(x) , where w~J~~~ and wprs are randomly sp

3 ecified negative weights for Pareto opti
ecified negative weights for Pareto optimal solutions problem with bit strings are are where “P“ indicates a Pareto optimal solution denotes a feasible solution dominated Pareto optimal solution(s). solution(s). In all the three algorithms, the one-point crossover was employed the crossover The number population size) specified as Fig.8 show the the three algorithms afier each numeral plotted space shows the number of obtained solutions the corresponding coordinate. Simulation results and Fig.8 are obtained our simulation, and Fig.7 quoted from from Pareto optimal solutions these figures are encircling the corresponding numerals these figures, can observe the each approach. were driven to the extreme solutions, there were 38 and 21 individuals at (Pairs, Unitation) (see Fig.6). Pareto optimal solutions (Pairs, Unitation) 10) and generation obtained also Fig.6). The Niched Pareto maintaining equal size subpopulations, but optimal solution (Pairs, Unitation) the final generation (see Fig.7). could find all the Pareto optimal solutions (see llnitation I 11) 07 06 05 04 03 2 h3’ 51 124 9 88 I 65 / 0 1 2 3 4 5 6 7 8 9 10 11 12 Pairs Fig.8 Simulation result by the MOGA B. A Flowshop Scheduling Problem Flowshop scheduling problems are quite well- known in the Recently, several approaches based on iterative improvement procedures were applied to the flowshop scheduling. applied genetic algorithms to scheduling problems (for example, see see Ishibuchi et al. [6], Manderick er al. [9] and Syswerda [ 1 11). General assumptions of the flowshop scheduling problems can described as follows follows )Jobs (work orders) are to available continuously. is processed on one machme time without preemption, and a machin

4 e processes more than at a time. paper,
e processes more than at a time. paper, we are processed in the same order on machines. This means that our flowshop scheduling is purpose is to determine This sequence is denoted a string in genetic completion time the duedate is known correlation between these two MOGA, the fitness written as are processed in the is the tardiness, and randomly specified non-negative weights for should be negative sign is attached weight in test problem, we generated scheduling problem with randomly specifying the processing time of machine as the closed interval interval 991. We employed the two-point crossover and the shift change mutation as in following parameter In this problem, we assign the mutation probability means that each individual is mutated once the VEGA the MOGA to flowshop scheduling stopping condition, total number solutions). When solutions were evaluated in each algorithm, the algorithm was these algorithms where the horizontal makespan and show the obtained and the MOGA respectively. From Fig.10, we can see that better solutions were the MOGA because the MOGA Simulation results Rule Selection Problem Next, we show the simulation VEGA and MOGA for fuzzy rule selection selection 81. The fuzzy rule selection problem is small number rules from the possible rules to construct a compact classification an individual) is denoted the following patterns by the selected if-then rules. then rules. fitness function in the correctly classified the number the selected fuzzy if-then rules randomly specified non-negative respectively. Because should be minimized, negative sign to the We applied and the rule selection problem for the iris data data The classification problem problem with four attributes. each class, patterns

5 are employed six sets: “small”
are employed six sets: “small”, “medium large”, “large” and “don’t Therefore we as candidate rules because the sets are for each the four attributes. Each individual candidate rules, bit assumes means that the corresponding rule could not be means that the rule is selected, and indicates that the rule is not uniform crossover was employed with crossover probability the mutation was biased were used and the MOGA in the same manner. Simulation results rule selection problem the obtained solutions MOGA respectively. that better solutions were obtained the MOGA because by the VEGA the MOGA Simulation results the rule selection multi-objective optimization problems have concave Pareto fronts, weighted sum approaches tend entire Pareto fronts the Pareto optimal solutions). approach, however, can handle multi-objective optimization problems with concave Pareto fronts. This is shown the following example. We applied to a test problem with following two objectives to be subject to we obtained relation between problem. This Pareto front the concave space as shown in Fig.12. proposed MOGA, the was specified minimized, the negative sign attached to each In the MOGA, point crossover was employed with crossover probability the number individuals was Fig. 12, final solutions Fig. 12, we can that the MOGA can find many solutions the concave Pareto front. the objective also applied single objective genetic algorithm (SOGA) where weights l:15, 1:20, 1:50, single objective algorithm with different weight values these simulation results, we can single objective algorithm with constant cannot find the Pareto front various weight values were we proposed a framework of genetic algorithms for

6 multi-objective optimization features. F
multi-objective optimization features. Firstly, weights used multiple objectives into a scalar fitness function randomly specified for each selection. is, the not constant but at the selection of each pair Secondly, multiple elite individuals selected from a tentative set next generation. computer simulations, demonstrated that the (Multi-Objective Genetic Algorithm) could find better solutions than (Vector Evaluated Evaluated REFERENCES [I] R.A.Dudek, S.S.Panwalkar and M.L.Smith, “The lessons of flowshop Operations Research, pp.65-74, 1992. 1992. R.A.Fisher, “The use of multiple measurements in in B.R.Fox and M.B.McMahon, “Genetic operations for sequencing problems”, in G.J.E.Rawlins (ed.) Foundations of Genetic Algorithms Publishers, San Mateo), pp.284-300, pp.284-300, C.A.Glass, C.N.Potts and PShade, “Genetic algorithms and neighborhood scheduling unrelated parallel parallel J.Hom, N.Nafpliotis and Genetic Algorithm Algorithm H.Ishibuchi, N.Yamamoto, T.Murata H.Tanaka, “Genetic Algorithms and Neighborhood Search Algorithms Scheduling Problems”, Problems”, H.Ishibuctu, K.Nozaki, N.Yamamoto H.Tanaka, “Construction fuzzy classification with rectangular rules using genetic genetic H.Ishibuchi, T.Murata and linguistic classification rules objective genetic genetic B.Manderick and P.Spiessens, “How to operators for combinatorial optimization analyzing their fitness landscape”, landscape”, J.D.Schaffer, “Multiple objective optimization with vector evaluated genetic algorithms”, algorithms”, 111 G.Syswerda, “Scheduling optimization using ge- netic algorithms”, Nostrand Reinhold, New York), 82-87, 1994. pp.237-253, 1994. 170-181, 1994.