Mathematical model and method of searching for a local extremum for the non-convex oriented polygons allocation problem

This work deals with the problem of optimal allocation of objects of a so-called irregular form. The objects are allocated on a strip of given width and with defects. This problem is insufficiently studied, but it is typical for many industries and is also interesting for developing the theory of solving cutting and packing problems. An analytical model of the problem using only continuous variables is written in terms of classical mathematical programming, and it is constructed on the basis of the original theory of Φ-functions and structures of linear inequalities. The presented theory allows one to easily describe the conditions of mutual non-overlapping of objects and their allocation in the stock region. The exact method for searching a local minimum of the problem from any feasible initial point is based on the application of the active set strategy ideas. A number of examples of solving practical problems are considered.