Quasi-phi-functions and optimal packing of ellipses

Abstract

We further develop our phi-function technique for solving Cutting and Packing problems. Here we introduce quasi-phi-functions for an analytical description of non-overlapping and containment constraints for 2D- and 3D-objects which can be continuously rotated and translated. These new functions can work well for various types of objects, such as ellipses, for which ordinary phi-functions are too complicated or have not been constructed yet. We also define normalized quasi-phi-functions and pseudonormalized quasi-phi-functions for modeling distance constraints. To show the advantages of our new quasi-phi-functions we apply them to the problem of placing a given collection of ellipses into a rectangular container of minimal area. We use radical free quasi-phi-functions to reduce it to a nonlinear programming problem and develop an efficient solution algorithm. We present computational results that compare favourably with those published elsewhere recently.

Appendices

Appendix 1: Examples of phi-functions

Example 7

We consider the simplest example of phi-functions for two circles $C_i$ of radii $r_i$ and center points $(x_{C_i } ,y_{C_i } )$, $i=1,2$. An ordinary phi-function, a normalized phi-function and a pseudonormalized phi-function for circles $C_1$ and $C_2$ may be defined in the following forms respectively:

Example 8

Let $p_i^1 =(x_i^{\prime },y_i^{\prime })$, $i=1,\ldots ,m_1$, be the vertices of $K_1 (u_1 )$, and $p_j^2 =(x_j{}^{\prime \prime },y_j{}^{\prime \prime }), j=1,\ldots ,m_2$, those of $K_2 (u_2 )$, and $K_1 (u_1 )=\{(x,y){:}{\upvarphi } _i \le 0,\;i=1,\ldots ,m_1 \}$, ${\upvarphi } _i ={\upalpha } _i^{\prime }x+{\upbeta } _i^{\prime }y+{\upgamma } _i ^{\prime }$, and $K_2 (u_2 )=\{(x,y){:}{\uppsi } _j \le 0,\;j=1,\ldots ,m_2 \}, \,{\uppsi } _j ={\upalpha } _j^{{\prime }'}x+{\upbeta } _j^{{\prime }'}y+{\upgamma } _j ^{{\prime }'}$, where $u_1 =(x_1 ,y_1 ,{\uptheta } _1 )$ and $u_2 =(x_2 ,y_2 ,{\uptheta } _2 )$ are placement parameters of polygons $K_1$ and $K_2$. It should be noted that each point $(\tilde{x},\tilde{y})\in K(0,0,0)$ in the local coordinate system of a convex polygon K is transformed into point (x, y):

$$\begin{aligned} x=\tilde{x}\cdot \cos {\uptheta } _K +\tilde{y}\cdot \sin {\uptheta } _K +x_K ,y=-\tilde{x}\cdot \sin {\uptheta } _K +\tilde{y}\cdot \cos {\uptheta } _K +y_K . \end{aligned}$$

A phi-function for two convex polygons $K_1$ and $K_2$ can be defined in the form

$$\begin{aligned} {\Phi }^{K_1 K_2 }=\max \left\{ \mathop {\max }\limits _{1\le i\le m_1 } \mathop {\min }\limits _{1\le j\le m_2 } {\upvarphi } _{ij} ,\mathop {\max }\limits _{1\le j\le m_2 } \mathop {\min }\limits _{1\le i\le m_1 } {\uppsi } _{ji} \right\} , \end{aligned}$$

(18)

where ${\upvarphi } _{ij} ={\upvarphi } _i (p_j^2 )={\upalpha } _i^{\prime }x_j^{\prime \prime }+{\upbeta } _i^{\prime }y_j^{\prime \prime }+{\upgamma } _i ^{\prime }$, ${\uppsi } _{ji} ={\uppsi } _j (p_i^1 )={\upalpha } _j^{\prime \prime }x_i ^{\prime }+{\upbeta } _j^{\prime \prime }y_i^ {\prime }+{\upgamma } _j^{\prime \prime }$.

In general, each of our phi-functions (ordinary, normalized, pseudonormalized) is formed by operations of minimum and maximum of continuous and everywhere defined functions. The more operations of maximum take part in forming of a phi-function the more nonlinear programming subproblems we need to solve.

For example, in order to reach the global minimum for the problem of packing of two convex polygons $K_1$ and $K_2$ in a rectangle of minimum area, using phi-function (18), we need to solve $m_1 +m_2$ nonlinear programming subproblems optimally. See details in [10].

Alternatively, in order to reach the global minimum of the latter problem, using quasi-phi-function (5), we need to solve only one nonlinear programming problem optimally. However, in the case the problem dimension is increased by two.

We may reasonably combine phi-functions and quasi-phi-functions in our models depending on types of our objects.

Appendix 2: The motivation of the epsilon parameter of the LOFTR procedure

We study the effect of the value of the parameter ${\varepsilon }$ on the computational time in our computational experiments. We take the value of ${\varepsilon }$ from the collection $\{0.1,0.2,0.4,0.6,\ldots ,3.8,4.0,\ldots ,{\varepsilon } ^{*}\}$, where ${\varepsilon } ^{*}=\max \{l^{0},w^{0}\}$, and apply our algorithm to our instances, starting from the same feasible point $u^{0}$ (point $u^{0}$ is obtaned by SPA algorithm).

Fig. 13

Dependence of the computational time on ${\upvarepsilon }$ for the instance “TC50”

From our computational experiments follows that there always exists an interval $[{\varepsilon } ^{-}, {\varepsilon } ^{+}]$, where the computational time reaches its “minimal” value and ${\varepsilon }$ weakly effects to the computational time. We take ${\varepsilon } =\frac{1}{n}\cdot \mathop {\sum }\nolimits _{i=1}^n {b_i }$ in our LOFRT algorithm since $\frac{1}{n}\cdot \mathop {\sum }\nolimits _{i=1}^n {b_i } \in [{\varepsilon } ^{-},{\varepsilon } ^{+}]$ for our computational experiments. It shold be noted that the value of ${\varepsilon }$ is taken significantly greater than the computational accuracy of IPOPT.

As an example we provide a diagram for the instance “TC50”. The diagram given in Figure 13 shows the dependence of the computational time on the value of ${\varepsilon }$, where $[{\varepsilon } ^{-}=0.6,{\varepsilon } ^{+}=1.6]$.

Fig. 14

A diagram of the LOFRT procedure

Appendix 3: A diagram of the LOFRT procedure

Figure 14 illustrates our LOFRT procedure. In fact on k-th step of the iterative procedure we solve the k-th subproblem on a subset $W_k =W\cap {\Pi }_k^{\varepsilon }$, provided that we ignore redundant inequalities , and fix variables of $\tau _k$, that have no effect on the value of our objective function in the “${\varepsilon }$-neighborhood” of point $u^{(k)*}$ by variables $x_i^k, y_i^k ,i=1,2,\ldots ,n$. If $u_{w_k }^*$ belongs to the frontier of the “artificial” subset ${\Pi }_k^{\varepsilon }$, then we take the point as a center point for a subset $\Pi ^{\varepsilon }_{k+1} \subset R^{\sigma }$ and continue our optimization procedure, otherwise we stop our LOFRT procedure. Figure 14 shows that each point of local minima $u_{w_k }^*$ is the frontier point of the appropriate “artificial” subset ${\Pi }_k^{\varepsilon }$ for $k=1,2,3,4,$ and the point $u_{w_5 }^*$ is the interior point of subset ${\Pi }_5^{\varepsilon }$. We note that $\hbox {dist}(u_{w_k }^*,u_{w_{k+1} }^*)\ge {\varepsilon }$ for $k=1,2,3$.