No abstract

A class of basic 2D objects is introduced whose F-functions are known, and the decomposition theorem is proved for arbitrary j-objects whose boundaries are formed by circular arcs and line segments. A step-by-step decomposition algorithm is proposed for arbitrary two-dimensional j-objects. The algorithm efficiently constructs F-functions of arbitrary j-objects in mathematical and computer modeling of packing and cutting problems. Results of numerical experiments are presented.Packing and cutting problems (placement problems in what follows) arise in many industries including mechanical engineering, shipbuilding, and textile, paper, and light industries and also in designing nesting patterns of industrial materials [1][2][3][4]. These problems are connected with searching for a rational placement of objects on an industrial material with a view to maximizing material utilization or minimizing waste products. In the majority of problems, the objects being arranged have an arbitrary form. At the present time, individual (exclusive) production becomes most widespread and asked-for. In this connection, the need arises for searching for an optimal placement of objects of arbitrary form with minimal expenditure of time and maximal saving in material.Placement problems are optimization problems and, as a result, the problem of using well-known local and global optimization methods arises that is conditioned by the absence of constructive means of mathematical modeling allowing for the representation of a problem of placement of arbitrary geometric objects in the form of a mathematical programming problem.In [1-3], modern mathematical modeling means and methods for solving placement problems for arbitrary geometric objects are described. However, the approaches proposed are based on heuristics and modifications of the method of optimization by groups of variables.The development of efficient methods for solving optimization placement problems requires the construction of adequate mathematical models. The fundamental basis of mathematical modeling of problems of this class is an analytical description of placement restriction for arbitrary objects in a given domain.As is well known [2], in the class of 2D placement optimization problems, a constructive means for mathematical modeling of relations between geometric objects is the method of F-functions [5]. The form of a F-function is directly determined by the space form of geometric objects. In [6-8], F-functions are constructed for primary objects. Principles of construction of F-functions for composed objects represented in the form of unions and intersections of primary objects are considered in sufficient detail in [9]. In this case, strict requirements are imposed on composed objects, and these requirements considerably restrict the class of space forms of geometric objects used as models of real objects. In solving practical problems (for example, in creating modern information technologies of nesting patterns of industrial materials), information on objects is speci...