By Zenon Waszczyszyn

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The convex hull of a ﬁnite set of points S, is considered to be the convex polyhedron with the smallest volume that encloses S. Convex hulls can be used to form polyhedra with a polyhedron being deﬁned as a three dimensional object composed of a ﬁnite number of ﬂat faces, edges and vertices (Figure 32). It can also be described as the 3D generalisation of a 2D polygon. Within this work, every polyhedron will be convex with triangular faces and referred to as convex polyhedra. Polyhedral Figure 32.

As described above, the ﬁtness function contains 3 components, the aim of these being: Minimising cost Minimising clear span Maximising use of natural resources BGRID allows the user to weight the importance of each component using weight factors, which range from 0 (irrelevant) to 4 (extremely important). The next step is to activate the genetic algorithm, which generates an initial population of 50 solutions randomly within a conﬁned search space. As explained above, each solution contains information about the column grid, the structural system, the environmental strategy and details of the vertical dimensions of the building.

Szczepanik 58 T. Burczyński selected for the offspring population, which becomes a parent population and the algorithm works iteratively till the end of the computation. g. as the maximum number of iterations. In evolutionary algorithms the floating-point representation is applied, which means that genes included in chromosomes are real numbers. Usually, the variation of the gene value is limited. Figure 1. A flowchart of an evolutionary algorithm A single-chromosome individual (called a chromosome) chi, i=1,2,…,N, where N is the population size, may be presented by means of a column or a row matrix, whose elements are represented by genes gij, j=1,2,…,n, n-the number of genes in a chromosome.