Packing ellipses in an optimized rectangular container

Abstract

The paper studies packing ellipses in a rectangular container of minimum area. The problem has various applications in production, logistics, industrial design. New phi-functions are proposed to state containment constraints and quasi-phi-functions are used for analytical description of non-overlapping constraints. A mathematical model for the packing problem is stated as a nonlinear programming problem. Two algorithms to find feasible starting points for identical and non-identical ellipses are proposed. The optimization procedure is used as a compaction algorithm to search for local optimal solutions. Computational results are provided to show the efficiency of the proposed approach.

Appendices

Appendix 1: Definitions of a phi-function and quasi phi-function

Let there be two objects $A \subset R^{2}$ and $B \subset R^{2}$. The position of object $A$ is defined by a vector of placement parameters $u_{A} = (v_{A} ,\theta_{A} )$, where $v_{A} = (x_{A} ,y_{A} )$ is a translation vector and $\theta_{A}$ is a rotation angle. The object A, rotated by angle $\theta_{A}$ and translated by vector $v_{A}$ will be denoted by $A(u_{A} )$.

Phi-functions allow us to distinguish the following three cases: $A(u_{A} )$ and $B(u_{B} )$ are intersecting so that $A(u_{A} )$ and $B(u_{B} )$ have common interior points (Fig. 5a); $A(u_{A} )$ and $B(u_{B} )$ are in contact, i.e. $A(u_{A} )$ and $B(u_{B} )$ have only common frontier points (Fig. 5b); $A(u_{A} )$ and $B(u_{B} )$ do not intersect, i.e. $A(u_{A} )$ and $B(u_{B} )$ do not have common points (Fig. 5c).

Fig. 5

Relations between two objects: a interior overlapping, b touching, c non-overlapping

Definition 1.1

A continuous and everywhere defined function $\varPhi^{AB} (u_{A} ,u_{B} )$ is called a phi-function for objects $A(u_{A} )$ and $B(u_{B} )$ if

$$\begin{aligned} & \Phi ^{{AB}} (u_{A} ,u_{B} ) < 0,\quad {\text{if}}\quad \text{int} A(u_{A} ) \cap \text{int} B(u_{B} ) \ne \emptyset ; \\ & \Phi ^{{AB}} (u_{A} ,u_{B} ) = 0,\quad {\text{if}}\quad \text{int} A(u_{A} ) \cap \text{int} B(u_{B} ) = \emptyset \quad {\text{and}}\quad frA(u_{A} ) \cap frB(u_{B} ) \ne \emptyset ; \\ & \Phi ^{{AB}} (u_{A} ,u_{B} ) > 0,\quad {\text{if}}\quad A(u_{A} ) \cap B(u_{B} ) = \emptyset . \\ \end{aligned}$$

Figure 5 illustrates three types of relations between two ellipses: (a) interior overlapping, $\varPhi^{AB} (u_{A} ,u_{B} ) < 0;$ (b) touching, $\varPhi^{AB} (u_{A} ,u_{B} ) = 0;$ c) non-overlapping $\varPhi^{AB} (u_{A} ,u_{B} ) > 0.$

The inequality $\varPhi^{AB} (u_{A} ,u_{B} ) \ge 0$ provides the non-overlapping condition, i.e., $\text{int}\,A(u_{A} ) \cap \text{int}\,B(u_{B} ) = \emptyset$, and inequality $\varPhi^{{AB^{*} }} (u_{A} ,u_{B} ) \ge 0$ provides the containment condition $A(u_{A} ) \subset$$B(u_{B} )$, i.e. $\text{int}\,A(u_{A} ) \cap \text{int}\,B^{*} (u_{B} ) = \emptyset$, where $B^{ * } = R^{2} \backslash \text{int}\,B.$ (Fig. 6)

Fig. 6

Containment of A into B

We refer the reader for details to, e.g., Stoyan and Romanova [10].

Definition 1.2

A continuous and everywhere defined function $\varPhi^{{{\prime }AB}} (u_{A} ,u_{B} ,u^{{\prime }} )$ is called a quasi-phi-function for two objects $A(u_{A} )$ and $B(u_{B} )$ if $\mathop {\hbox{max} }\limits_{{u^{\prime}}} \varPhi^{{{\prime }AB}} (u_{A} ,u_{B} ,u^{{\prime }} )$ is a phi-function for the objects.

Here $u^{\prime}$ is a vector of auxiliary continuous variables that belongs to Euclidean space.

Based on features of a quasi-phi-function the non-overlapping constraint can be described in the form:

if $\varPhi^{{{\prime }AB}} (u_{A} ,u_{B} ,u^{{\prime }} ) \ge 0$ for some $u^{\prime}$, then $\text{int}\,A(u_{A} ) \cap \text{int}\,B(u_{B} ) = \emptyset$.

Example 1.1

A function defined by

$$\varPhi^{{{\prime }AB}} (u_{A} ,u_{B} ,u^{{\prime }} ) = \hbox{min} \{ \varPhi^{AP} (u_{A} ,u^{{\prime }} ),\varPhi^{{BP^{ * } }} (u_{B} ,u^{{\prime }} )\}$$

is a quasi-phi-function for a pair of convex objects $A$ and $B$, where $\varPhi_{{}}^{AP} (u_{A} ,u^{\prime})$ is a phi-function for $A(u_{A} )$ and a half-plane $P(u^{\prime})$ and $\varPhi_{{}}^{{BP^{ * } }} (u_{B} ,u_{P} )$ is a phi-function for $B(u_{B} )$ and a half-plane $P^{*} (u^{\prime}) = R^{2} \backslash \text{int}\,P(u^{\prime})$.

Figure 7 illustrates position of two convex objects and a separation line that provide $\varPhi^{{{\prime }AB}} (u_{A} ,u_{B} ,u^{{\prime }} ) > 0$.

Fig. 7

Illustration to a quasi-phi-function $\varPhi^{{{\prime }AB}}$ for objects A and B

We refer the reader for details to, e.g., Stoyan et al. [7].

Appendix 2: Illustrations to the SPA_1 algorithm

See Figures 8, 9, 10 and 11.

Fig. 8

Illustration to Steps 1, 2 of SPA_1

Fig. 9

Illustration to Steps 4 of SPA_1

Fig. 10

Illustration to Steps 5 of SPA_1

Fig. 11

Feasible placement of ellipses

Appendix 3: Illustrations to the LOFRT procedure

Starting from point $u^{0} = (l^{0} ,w^{0} ,u_{1}^{0} ,u_{2}^{0} , \ldots ,u_{n}^{0} ,\tau^{0} )$ the point $u^{*} = (l^{*} ,w^{*} ,u_{1}^{*} ,u_{2}^{*} , \ldots ,u_{n}^{*} ,\tau^{*} )$ of local minimum is found (Fig. 12).

Fig. 12

Arrangement of ellipses that corresponds to: the starting point u⁰ and point of local minimum u^*

For details of the LOFRT procedure we refer the reader to Stoyan et al. [7].