[Paper Review] Packing a fixed number of identical circles in a circular container with circular prohibited areas

C.O. López, J. E. Beasley|arXiv (Cornell University)|Nov 1, 2018

Optimization and Packing Problems31 references4 citations

TL;DR

This paper proposes a formulation space search (FSS) heuristic to maximize the radius of identical circles packed within a unit circular container containing fixed circular prohibited areas. The method iteratively reformulates the nonlinear optimization problem to escape local optima, demonstrating effective solutions for six test cases with up to 100 circles, including annular, union-based, and disconnected prohibited regions, achieving high-quality radius optimizations.

ABSTRACT

In this paper we consider the problem of packing a fixed number of identical circles inside the unit circle container, where the packing is complicated by the presence of fixed size circular prohibited areas. Here the objective is to maximise the radius of the identical circles. We present a heuristic for the problem based upon formulation space search. Computational results are given for six test problems involving the packing of up to 100 circles. One test problem has a single prohibited area made up from the union of circles of different sizes. Four test problems are annular containers, which have a single inner circular prohibited area. One test problem has circular prohibited areas that are disconnected.

Motivation & Objective

• To address the challenge of packing a fixed number of identical circles in a circular container with fixed circular prohibited areas, where standard packing methods are complicated by spatial obstructions.
• To maximize the radius of the identical circles under geometric constraints, including annular, union, and disconnected prohibited regions.
• To develop and evaluate a novel metaheuristic approach—formulation space search (FSS)—for nonlinear, non-convex circle packing problems with prohibited zones.
• To provide computational results for diverse prohibited area configurations, including real-world sequential packing scenarios where prior items become forbidden regions.
• To demonstrate the robustness and scalability of the FSS method across varying numbers of circles (up to 100) and prohibited area types.

Proposed method

• The method uses formulation space search (FSS), an iterative metaheuristic that dynamically alters the problem formulation at each iteration to explore different solution landscapes.
• At each FSS iteration, a new formulation is generated by perturbing the problem's constraints and objective function, enabling escape from local optima.
• The core optimization problem is a nonlinear program maximizing circle radius subject to non-overlapping and container/prohibited area boundary constraints.
• The formulation is solved using nonlinear programming solvers, with solutions evaluated for feasibility and radius quality.
• The algorithm maintains an external archive of elite solutions and uses adaptive perturbation strategies to guide the search toward high-quality packings.
• The method is applied to six test problems: one with a union of multiple prohibited circles, four annular (inner circular forbidden zones), and one with disconnected prohibited regions.

Experimental results

Research questions

• RQ1 How effective is formulation space search (FSS) in solving circle packing problems with fixed circular prohibited areas?
• RQ2 Can FSS consistently find high-quality solutions across diverse prohibited area configurations, including annular, union, and disconnected regions?
• RQ3 How does the performance of FSS scale with increasing numbers of identical circles (up to 100) and varying prohibited area complexity?
• RQ4 What is the impact of different formulation parameters (e.g., |F|, the number of formulations per iteration) on solution quality and computational time?
• RQ5 How do the resulting maximum radii compare across different prohibited area types, and what insights do they offer for sequential packing applications?

Key findings

• For test problem 1 (annular container with single inner prohibited circle), the FSS heuristic achieved a best radius of 0.25725385 for n=10, with total computation time of 262.81 seconds for |F|=4.
• With |F|=11, the best radius for n=100 was 0.08286614, achieved in 1184.00 seconds, indicating scalability to larger instances.
• In test problem 6 with disconnected prohibited areas, the method achieved a radius of 0.08427842 for n=100, computed in 962.60 seconds.
• For n=50 in test problem 1, the best radius was 0.12068548 with |F|=7, computed in 357.89 seconds, showing consistent improvement with higher |F|.
• The FSS method outperformed baseline approaches in terms of solution quality, particularly in complex configurations like disconnected prohibited regions.
• The results demonstrate that FSS is effective in navigating complex, non-convex search spaces, yielding high-quality solutions even with significant geometric constraints.