Development of the mathematical model and the method to solve a problem on the optimization of packing the ellipsoids into a convex container

Abstract

This paper addresses the problem on the optimal packing of the predefined set of ellipsoids into a convex container of minimum volume. The ellipsoids are assigned by the dimensions of semi-axes and arrangement parameters in the local coordinate system and may permit continuous rotation and translation. The container could be a cuboid (rectangular parallelepiped), a cylinder, a sphere, an ellipsoid, or a convex polyhedron. To analytically describe the non-overlapping relations between ellipsoids, we use the quasi-phi-functions. To model the inclusion relations, we apply the quasi-phi-functions or phi-functions depending on the shape of a container. By employing the appropriate modeling tools, we construct a mathematical model in the form of a non-linear programming task. The solution strategy is devised based on the method of a multistart. We propose a fast algorithm for generating the starting points from the region of feasible solutions, as well as the specialized optimization procedure that reduces the problem of large dimensionality O(n 2 ) with a large number of nonlinear inequalities to a sequence of sub-tasks in nonlinear programming with a smaller dimensionality O(n) with fewer non-linear inequalities. The optimization procedure makes it possible to significantly reduce (by 10 % to 90 %, depending on the dimensionality of a problem) computing resources, such as time and memory. Depending on the shape of a container, constraints for the orientation of ellipsoids (continuous turns, fixed orientation) and features in metric characteristics of ellipsoids, the result of solving the problem is the derived locally optimal or good feasible solutions. In the work we report numerical experiments for different containers (including a cylinder, a cuboid, a sphere, an ellipsoid).