Self-Adaptive Genetic Algorithms with Simulated Binary Crossover

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i?x1;t+1 i q=8
Step 1: Choose a random numberu2[0;1). Step 2: Calculatequsing equation 4. x(1;t) withinq2[(0:01)1 1+;1=(0:01)1 i=2:0andx(2;t) x(2;t+1) i?x(1;t+1) i =qx(2;t) i?x(1;t) i: Step 3: Compute children solutions using equations 5 and 6. Note that two children solutions are symmetric about the parent solutions. This is deliberately used to avoid any bias towards any particular parent solution in a single crossover operation. Another interesting aspect of this crossover operator is that for a fixed the children solutions has a spread which is proportional to that of the parent solutions. With simple algebraic manipulations, it can be shown that 99% of crossovers lie 1+]. For=2, this range isq2[0:215;4:64]. However, for a fixedq, the difference in children solutions is proportional to that of parent solutions: (7) This has an important implication. Let us consider two scenarios: (i) Two parents are far away from each other, and (ii) two parents are closer to each other. For illustration, both these cases (with parent solutions i=5:0in the first case and with parent solutionsx(1;t) i=2:0andx(2;t) i=2:5in the second case) and the corresponding probability distributions with=2are shown in Figures 2 and 3, respectively. For an identical random number,q, as calculated by using equation 4, are the same for both Probability density function per child 3 2.5 2.5 2 2 1.5 1.5 parentx(1;t) 1 1 solutionx(1;t+1) parentx(1;t) solutionx(1;t+1) i=1:464 0.5 0.5 0 o o 0 o o -1 0 1 2 3 4 5 6 7 8 -1 0 1 2 3 4 5 6 7 8 x_i x_i Figure 2: Probability distribution of children so- Figure 3: Probability distribution of children solutions with distant parents. lutions with closely spaced parents. cases. From equation 7, it is clear that in the first case the children are likely to be more widely spread than in the second case. Figures2 and 3 also show the corresponding children solutions (marked with a box) foru=0:8orq=2:51=3. Figures clearly show that if the parent values are far from each other (the first case), solutions away from parents are possible to be created. Compare the child with i=2:0. But if the parent values are close by (the second case), distant children solutions are not likely. Compare the child i=1:911with i=2:0creating using the same random number as in the first case. In initial populations, where the solutions are randomly placed (like the first case), this allows almost any values to be created as a child solution. But when the solutions tend to converge due to the action of genetic operators (like the second case), distant solutions are not allowed, thereby focusing the search to a narrow region. It is interesting to note that both equations 5 and 6 can be written in the form of equation 1 with following relationships:=0:5(1q):However, it is important that, unlike in BLX- operator, the equivalent term in SBX operator is not uniformly distributed (refer to equation 4). The SBX operator biases solutions near each parent more favorably than solutions away from parents. Essentially, SBX operator has two properties: 5 Probablity density function per child 3
Page 1 and 2: Self-Adaptive Genetic Algorithms wi
Page 3: (?0:5x(1;t) suggested. [x(1;t) i+1:
Page 7 and 8: x(t+1) i=(t) i=x(t) (t+1) for(0) ie
Page 9 and 10: F1-1:MinimizeNXi=1(xi?xi)2; 5.1.1 F
Page 11 and 12: Population-best distance from optim
Page 13 and 14: that when a small initial mutation
Page 15 and 16: Population standard deviation in x
Page 17 and 18: Population-best objective function
Page 19 and 20: tical~vis used for all algorithms.
Page 21 and 22: help us to create problem specific
Page 23 and 24: Deb, K. and Agrawal, R. B. (1995) S

Self-Adaptive Genetic Algorithms with Simulated Binary Crossover

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