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While conceptually appealing, this real number model of encoding imposes some restrictions on the GA machinery. Fundamentally, bearing in mind the origin of the entries in the string, we should become aware that any improper recombination and mutation may result in the formation of strings that are totally infeasible (and thus meaningless) from the viewpoint of the original search space.

As an illustration, consider a simple codebook composed of several triangular membership functions with 1/2 overlap between successive terms. The legitimate strings are those with only two successive nonzero entries that sum up to 1, or a single nonzero entry equal to 1. For example,

are valid strings. The arithmetic recombination with α = 0.2 produces the strings

that are no longer valid. This emphasizes a need for extra provisions when it comes to the basic form of the GA operations. NB, note that the decoding of any valid string is lossless - this is due to the peculiar form of the fuzzy sets in the codebook.

While it is possible to properly redefine GA operations to generate only valid strings, we can pursue an alternative route and consider a notion of the weak encoding as offered by fuzzy sets.

8.2.2. Weak encoding with fuzzy sets

The underlying idea is to pursue a one - to - many type of relationship when moving from the genotype space to its phenotype counterpart. This allows us to take full advantage of the linguistic labels while restricting ourselves to pure binary coding. The mechanism is self-evident: GA search space is coded binary where each binary combination describes a certain linguistic term. For four linguistic terms we may have the assignment: NS: 00, Z: 01, PS: 10, PM: 11 (of course, there are other possible code arrangement as well). This encoding looks as if the fuzziness has been completely ignored. The encoding (NS, Z, PS, PM) leads to symbols and at best these could be regarded as sets (or numerical intervals). Hopefully, this is not correct. Fuzzy sets arise at the level of the computations of the fitness function. What is the fitness of PS, fit(PS)? Naturally, a determination of fit(PS) is not evident as the encoding is weak and the label is not represented by a single number. To come up with the fitness value for the linguistic terms one should consider some sampling techniques of these fuzzy sets and compute a certain representative such as a weighted average fitness assumed over the entire linguistic term of interest. In other words, the fitness of A, fit(A), reads as

where x(1), x(2), …, x(p) are the sampling points. Here A(x(k)) describes a contribution of x(k) to the overall fitness value due to its membership content while fit(x(k)) is the fitness over this specific sampling point, Fig. 8.4.


Figure 8.4  General concept of weak encoding realized in the setting of fuzzy sets

Another important feature of the fuzzy encoding needs to be underlined. The formation and granularity of the linguistic terms can help express the objectives of the entire search. A multimodal function shown in Fig. 8.5 exhibits two extrema (maxima): one is very narrow, the other one, somewhat lower, covers a broad range of the universe of discourse.


Figure 8.5  Multimodal function with two different extrema

Which maximum should we select? This depends upon the problem at hand, however the flat one can be preferred as not being highly sensitive. By defining fuzzy sets of a certain granularity we modulate our interest in some class of extrema. Very narrow fuzzy sets will promote search for needle-like extrema. Fuzzy sets of lower granularity would tend to ignore this form of the maximum; the fitness over the needle-like maximum would be far lower because of the reduced integration (aggregation) effect,

where Ω is a support of the corresponding element of the codebook. The weak encoding comes with the weak format of decoding: an element in the GA search space maps on a number of structures in the original search space. More interestingly, these structures can be uniformly embraced in the form of a fuzzy set. Any further numerical adjustments are usually done by far more refined optimization techniques including those gradient based instruments.

8.3. Fuzzy crossover operations

The ideas of fuzzy set operations are exploited in the development of a broad spectrum of crossover (or more precisely, recombination operators). Let us denote by x and y two n-bit strings in the n-dimensional unit hypercube; furthermore denote their i-th coordinates by xi and yi. The idea is to produce offsprings whose coordinates are situated outside the range (min(xi, yi), max(xi, yi)) of the original string. In Herrera et al. (1995) defined are four classes of functions, called F, S, M, and L, such that

The new chromosome coming as a result of the recombination of x and y is defined as

i=1, 2, …, n, where Q ∈ {F, S, M, L}. F are just t-norms, S forms a family of s-norms, M stands for a family of averaging operators and L are compensation operators. Let us recall that the averaging operator M: [0, 1]2 → [0, 1] satisfies the property of boundness (that is min(a, b) ≤ M(a, b) ≤ max(a, b)), obeys two boundary conditions M(0, 0) = 0, M(1, 1) = 1 and is commutative, M(a, b) = M(b, a). It is also increasing and continuous. In particular, a so-called Quasi Arithmetic Average is introduced as follows

where f: [0, 1] → R+ is strictly monotone and continuous with f(0) = 0; the parameter p is situated in the unit interval.

The compensation operator C: [0, 1]2 → [0, 1] is defined as

or alternatively

where the t-norm and s-norm are duals; as before p ∈ [0, 1].


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