The research object is the process of optimal packing and cutting of geometry objects having arbitrary shape. The purpose of the work is to improve the efficiency of solving optimisation problems of placing geometric objects by the means of modern information technology development on the basis of constructive tools of mathematical and computer modeling, new mathematical models and effective methods of optimisation. Research methods: general and homotopic topology, functional analysis, the method of phi-functions, analytical geometry, geometric design methods, the methods of non-linear and non-differentiable optimisation. Powerful tools for relationships mathematical modeling of arbitrary non-oriented geometric objects that are bounded by circular arcs and line segments with variable metric characteristics are developed. The generalized mathematical model in the form of non-differentiable optimisation problem is constructed and investigated. The variety of model realisations covers a wide spectrum of scientific and applied packing and cutting problems. Strategies, methods and algorithms for solving optimisation problems of arbitrary objects placement taking into consideration technological restrictions are suggested. The developed methods and algorithms are extended to a number of covering and 3D-packing problems. The application system “2D-Arrangment” for automatic solving of optimisation problems of arbitrary objects placing is developed. Scientific novelty: use of constructive tools for mathematical modeling of the geometric objects relationships permitted to describe in an analytical form the main restrictions for location problems of arbitrary nonoriented objects with variable metric characteristics, construct the generalized mathematical model and propose the generating method for the solution space of optimal placement based on the use of phi-trees. New strategies, fast methods for construction of starting points, effective techniques of global optimisation and searching for approach to the global extremum, the original method for decomposition of nonlinear optimisation problems are developed. The results of the work are used in the UK (University of Southampton, United Kingdom), Germany (Dresden University of Technology, Institute of Numerical Mathematics), in the Head Office of Civil Service of Ukraine of Emergency Situations in the Kharkiv region for modeling and solving the problem of locating of fire stations, fire hydrants, fire detectors and for the development of rational plans for the evacuation from high-rise buildings, in Spain (TDM Solution SL, Premia de Mar), introduced in the educational process at the Kharkov National University of Radio Electronics and at the National University of civil Defense in Ukraine. The main areas of use are machinery, light industry, chemical industry, building industry.
