Simulated binary crossover

Simulated Binary Crossover (SBX) [Deb and Agrawal (1995)] operator for real valued variables is used to create two children from a pair of parents. 

Deb, K. and Agrawal, S. (1995). Simulated binary crossover for continuous search space. Complex Systems 9, 2, pp. 115-148. 

There are many ways to perform the crossover operation including linear crossover (Wright, 1991), BLX and its variants (Michalewicz and Janikow, 1991 Goldberg, 1991 Eshelman and Schaffer, 1993), a naive crossover (Deb, 2009) and simulated binary crossover (Deb and Agarwal, 1995 Deb and Kumar, 1995). 

Deb, K. and Kumar, A. 1995. Real-coded genetic algorithms with simulated binary crossover studies on multi-modal and multi-objective problems. 

Figure 47. Diffusion coefficient D as obtained from a molecular dynamics simulation study of a binary Lennard-Jones system reaching temperatures below the crossover temperature of mode coupling theory (MCT). Solid line represents interpolation by MCT power law note the large temperature range covered by the power law. (From Ref. 371.)...

In another popular definition of mutate and crossover, a binary representation of the unknown variables is required whereby the simulated DNA is converted into a concatenated sequence of binary numbers (0 s and l s). To obtain a binary representation of the ionic coordinates, the ions are constrained in that they can only sit on one of 2m discrete grid points across the unit cell (Fig. 1). For each grid point there is a unique binary number of length m. Note that the grid points can either be numbered 0 to 2m-l (000 to 111 for m=3) or, as shown in Fig. 1, have... 

Two implementations of genomes, a binary and a decimal (figure 2), were compared to study the dependence on a DNA-like and chemist-like crossover and mutation. Further variables were population size, mutation rate and the crossover strategy. In order to eliminate the stochastic differences between different GA runs, each parameter set was used in 100 parallel runs and the averages and standard deviations for the various results were calculated and displayed on the following figures. All results were compared with a "random" screening method that was simulated by a repeated random selection of new products from the library. 

Using the published Pitzer coefficients and applying them to the ternary NH,-COj-HjO yielded similar results. This is not surprising since NHi,OH(aq) was not considered. In addition to the scatter, several published points could not even be simulated. It appears that from a numerical point of view, the activity coefficient formulations, based upon the published interaction coefficients, were ill-behaved. This could be due to a number of factors such as a typographical error in one of the published interaction values. Since the numerical problems did not occur for either of the constituent binary systems, it would have to be an error in one of the crossover terms such as any NH2COI interaction. 

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