The paper proposes a methodology of accounting for illegal polygons in the process of computer-aided design of optimal layouts of production and other facilities on the territory designated for an industrial enterprise. A linear piecewise continuous approximation of an arbitrary contour of illegal polygons is used. The location of the contour of the base of objects in an illegal polygon is established based on the analysis of predicates of four types, three of which demonstrate the location of the vertices of objects relative to straight lines containing approximating segments, and one – the location of the object relative to the unacceptable area of the illegal polygon. The last of the mentioned predicates is implemented by the "ray" method. Using the mathematical apparatus of R-functions, 16 typical mutual positions of the contours of facilities foundations and approximating segments of polygons were discovered including facilities both inside and outside the closed polygon. Logical functions to identify the established typical mutual positions of facilities and illegal polygons are obtained, as well as analytical expressions for calculating partial and resultant penalty functions reflecting the degree of violation by facilities of the boundaries of illegal polygons. To correctly select the analytical expression of the penalty function from the proposed set, parameters are introduced that characterize the location of the approximating segments relative to the invalid area of the illegal polygon, relative to objects and coordinate axes. The numerical experiment performed using the gradient algorithm showed the adequacy of the proposed approach and the steady return of the facilities to the permissible area, which confirmed the possibility of applying the developed methodology in the computer-aided design.
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