Mathematical model and solution method of optimization problem of placement of rectangles and circles taking into account special constraints

A mathematical model of the problem is constructed. A feasible region and peculiarities of the problem are investigated. Extremum of a linear objective function is searched at extreme points. To solve the problem a solution method based on a combination of the branch-and-bound algorithm and the reduced gradient method is utilized. A search tree for sorting all the extreme points of the feasible region is constructed. A stepwise passage from one extreme point to another towards decreasing the objective function is given. Appropriate software has been developed. To illustrate results of calculations the examples are given.