# Packing unequal rectangles and squares in a fixed size circular container using formulation space search

## TL;DR

This paper presents a mixed-integer nonlinear programming formulation and a heuristic for packing unequal rectangles and squares into a fixed circular container, optimizing for packing efficiency or quantity, with computational validation.

## Contribution

It introduces a novel formulation space search heuristic for the packing problem, including methods to handle nonlinear constraints and rotation options.

## Key findings

- Heuristic effectively solves test problems with up to 30 rectangles.
- Formulation can be adapted for maximizing number or area of packed rectangles.
- Nonlinear terms are successfully eliminated for improved computational performance.

## Abstract

In this paper we formulate the problem of packing unequal rectangles/squares into a fixed size circular container as a mixed-integer nonlinear program. Here we pack rectangles so as to maximise some objective (e.g. maximise the number of rectangles packed or maximise the total area of the rectangles packed). We show how we can eliminate a nonlinear maximisation term that arises in one of the constraints in our formulation. We indicate the amendments that can be made to the formulation for the special case where we are maximising the number of squares packed. A formulation space search heuristic is presented and computational results given for publicly available test problems involving up to 30 rectangles/squares. Our heuristic deals with the case where the rectangles are of fixed orientation (so cannot be rotated) and with the case where the rectangles can be rotated through ninety degrees.