Gumen O. Geometric modeling of multicriteria technical problems

In the dissertation has found the further development the decision of problems of multicriteria compromise optimization by many criteria of optimization simultaneously for complex dependences between many variables when functions of optimization are influenced with arguments in various combinations. The essence of the propounded approach consists in consideration of a system of projections or the equations corresponding to them which in common display a variety as a geometrical model of the general dependence. The geometrical decision of the multicriteria problem is reduced to definition on the variety an extreme point as the point of contact a linear k-subspace, parallel to the hyperplane ?1 x1 + ... + ?n-k xn-k = 1, where ?i - weights of functions of optimization which are set depending on priority of the functions of optimization. Besides to weights of functions of optimization signs "+" or "-" are attributed depending on necessity to maximize or minimize the function. Therefore it is called multicriteria compromise optimization as to some functions the others are preferred. As special cases of varieties in the work the surfaces of the fourth order are considered: a hyperbolic paraboloid of two sheet and a parabolic paraboloid of two sheet with research of their characteristic sections and forms. The equation of flat curves of the fourth order as geometrical places of points in plane which are considered as generalization of Persey-Cassini curves, in particular Bernoulli lemniscate is examined also. The method developed in the dissertation is introduced on ZAT "Artservice" (Lviv) for calculation of optimum technological parameters by preparation of manufacture of photopolymeric printed forms. The generalizations received in the work - lemniscates compressed and extended - are introduced into manufacture on UMT "Lvivtransgas" for interface of pipelines on angular joints. Results of work are introduced also into educational process at physical and mathematical faculty of National technical university ofUkraine "Kyiv politechnical institute".