Discretization-Based Solution Approaches for the Circle Packing Problem

Abstract

The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as automobile, textile, food, and chemical industries. The so-called circle packing problem can be cast as a nonconvex quadratically constrained program, and is difficult to solve in general. An iterative solution approach based on a bisection-type algorithm on the radius of the larger circle is provided. The present algorithm discretizes the container into small cells and solves two different integer linear programming formulations proposed for a restricted and a relaxed version of the original problem. The present algorithm is enhanced with solution space reduction, bound tightening and variable elimination techniques. Then, a computational study is performed to evaluate the performance of the algorithm. The present algorithm is compared with BARON and Gurobi that solve the original nonlinear formulation and heuristic methods from literature, and obtain promising results.

Figures (4)

• Figure 1: Example feasible and infeasible placements of circles for CPP.
• Figure 2: Example discretizations of the circle and example placements of circles.
• Figure 3: Example placements of circles for the relaxed version.
• Figure 4: An example of solution space reductions.