Packing Circles and Irregular Polygons using Separation Lines

Jeinny Peralta

, Marina Andretta

and Jos

e Fernando Oliveira

Instituto de Ci

encias Matem

aticas e de Computac¸

ao, Universidade de S

ao Paulo, Av. Trabalhador S

ao Carlense, 400,

13566-590, S

ao Carlos, Brasil

Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-590, Porto, Portugal

Keywords:

Irregular Packing Problems, Circle Packing, Separation Lines.

Abstract:

In this paper we propose a nonlinear mathematical model for the problem of packing circles, convex and non-

convex irregular polygons, within a rectangular envelope to be produced, obeying containment constraints and

non-overlapping constraints; the objective of the problem is to minimize the area of the rectangular envelope.

We consider free rotations of the polygons and use separation lines to ensure non-overlapping. Computational

tests were run using instances presented in the literature that deal with circles and polygons simultaneously;

different solutions, in which the area of the rectangular envelope is smaller than or equal to the ones found in

the literature were found in most cases, and the execution time is very low. This indicates that our model is

computationally efﬁcient.

1 INTRODUCTION

In production processes, such as garment manufac-

turing, shoe manufacturing, furniture making, among

others, the optimization of layouts of items of differ-

ent shapes is necessary. These items can be repre-

sented by circles or polygons. In the literature, there

are many works dealing with circle packing problem

(e.g., (Szab

o et al., 2007) and (Castillo et al., 2008))

and polygon packing problem (see (Oliveira and Fer-

reira, 1993), (Toledo et al., 2013), and (Jones, 2013)),

but few papers deal with the problem of packing cir-

cles and polygons simultaneously ((Kallrath, 2009;

Stoyan et al., 2012; Mundim et al., 2017)).

In packing problems, items must be positioned

into a certain shape, aiming to optimize an objective

function. In this work, we consider the positioning of

circles and polygons into a rectangle is such a man-

ner that the produced area is minimized. That is, the

objective is to minimize the multiplication of the rect-

angle dimensions which must be determined.

The main constraint in packing problems is the

non-overlapping between items, but it is generally

very complex for a computational program to deter-

mine if two items are overlapping, touching or sep-

arated. When dealing with polygons, there are tools

for solving this issue ((Bennell and Oliveira, 2008)),

such as the raster methods, direct trigonometry, no-ﬁt

polygon, and phi-function. When dealing with cir-

cles, the problem is commonly formulated as a prob-

lem of Euclidean geometry, either in cartesian coordi-

nates (George et al., 1995; Addis et al., 2008) or po-

lar and cartesian coordinates combined (Mladenovi

et al., 2005; L

opez and Beasley, 2013). In this paper,

we use direct trigonometry in cartesian coordinates to

ensure non-overlapping between items.

Both for circle packing problems and polygon

packing problems, different techniques of solution

have been developed: there are heuristics (Liu et al.,

2016; Oliveira and Ferreira, 1993), linear program-

ming models (Galiev and Lisaﬁna, 2013), linear

mixed integer programming models (Toledo et al.,

2013), nonlinear mixed integer programming mod-

els (George et al., 1995), and nonlinear programming

models (Birgin et al., 2013; Stoyan et al., 2016). For

the problem of packing circles and polygons simul-

taneously, there are fewer works, such as (Kallrath,

2009; Stoyan et al., 2012; Mundim et al., 2017). In

(Kallrath, 2009) and (Stoyan et al., 2012), nonlinear

programming models where the items can be both

translated and rotated were proposed. In (Mundim

et al., 2017), two heuristics based on bottom-left

moves and the no-ﬁt raster concept were proposed;

in these, the items can not be freely rotated.

In (Kallrath, 2009) a model for two cases of pack-

ing of circles and convex polygons was developed. In

the ﬁrst case, the objective is to pack all items pro-

vided minimizing the area of the rectangular enve-

Peralta, J., Andretta, M. and Oliveira, J.

Packing Circles and Irregular Polygons using Separation Lines.

DOI: 10.5220/0006602700710077

In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 71-77

ISBN: 978-989-758-285-1

lope, as it will be done here. In the second case, the

objective is to pack the items on stocked rectangles

of known geometric dimensions. Separation lines are

used to ensure non-overlapping. In (Kallrath, 2009),

the problem is addressed using Branch&Reduce Op-

timization Navigator (BARON) (Tawarmalani and

Sahinidis, 2005; Sahinidis, 2014) and LindoGlobal

from Lindo Systems, Inc., which is part of the GAMS

22.5 distributions.

In (Stoyan et al., 2012) a model for packing prob-

lem of circles and non-convex polygons into a strip

with prohibited regions was proposed. The objective

of the problem is to maximize space utilization. Fur-

thermore, the authors ensures non-overlapping by us-

ing the phi-function.

In (Mundim et al., 2017) two heuristics for

packing circles and non-convex polygons into two-

dimensional bin in order to minimize one or both di-

mensions of the bin are proposed. These heuristics

are based on bottom-left moves and the no-ﬁt raster

concept. For the development of the heuristics, the se-

quence in which items are packed must be determined

and this is done by a Biased Random Key Genetic Al-

gorithm.

The model proposed here is an extension of the

one presented in (Peralta et al., 2017), which is a non-

linear mathematical model for an irregular strip pack-

ing problem of polygons, which may be convex or

non-convex, that can rotate freely. This model yielded

good solutions for large instances in rather reasonable

execution time, and it should be noted that this model

has a signiﬁcantly smaller number of variables, when

compared to the model proposed in (Kallrath, 2009).

Here, we propose a nonlinear mathematical model for

a packing problem in which the items are not only

polygons, but also circles. The container is no longer

a strip, but a rectangular envelope; the objective is to

minimize the area of the rectangular envelope. In this

model, we use direct trigonometry, in particular sep-

aration lines, to ensure the non-overlapping between

items. A separation line is a straight line such that,

given two items, one of them is on one side of the line

and the other on the opposite side. A polygon is on

one side of the line if all its vertices are on that side

of the line or on the line. A circle is on one side of the

line if its center is on that side of the line and if the dis-

tance from center to the line is greater than or equal

to its radius. The use of separation lines allows us

to ﬁnd good solutions, even when dealing with non-

convex polygons, since in this case the non-convex

polygons are partitioned into convex polygons and we

must have lines separating each convex sub-polygon

belonging to the partition of the remaining items.

Both the polygons and the separation lines can ro-

tate freely. We use a code for nonlinear programming

to solve the problem, IPOPT (W

achter and Biegler,

2006).

This paper is organized as follows. In the next sec-

tion we develop a model for packing circles and poly-

gons from one rectangular envelope, which considers

free rotations of the polygons. In Section 3, the nu-

merical results obtained are presented and discussed.

We end up presenting some conclusions in Section 4.

2 MODEL FOR A CIRCLES AND

POLYGONS PACKING

PROBLEM

We assume that n items, which can rotate freely,

should be positioned into one rectangular envelope,

obeying containment constraints and non-overlapping

constraints, in order to minimize the area of the rect-

angular envelope. Items can be circles or irregular

(convex or non-convex) polygons. The ideas used to

model this problem are based in the ones presented in

(Peralta et al., 2017).

If a polygon is non-convex, it is partitioned into

convex polygons. The coordinates of a vertex of a

non-convex polygon P

are given by:

with k = 1,..., p

and l = 1, . . . , v

, being p

the num-

ber of convex polygons belonging to the partition of

the non-convex polygon P

and v

the number of ver-

tices of the convex polygon P

The problem is modeled using the following vari-

ables: for each circle i, the coordinates of its center

); for each polygon i, the coordinates of its ref-

erence point (x

) and its angle of rotation θ

; for

each separation line, which separates item i from item

j, the coordinates of its reference point (x

i, j

) and

its rotation angle α

i, j

; and, the width W and the length

L of the rectangular envelope.

The reference point of a polygon is used to repre-

sent all the vertices of the polygon, even after transla-

tions and/or rotations are undergone; a separation line

is given by two points, one is its reference point and

the other is rewritten in terms of its reference point

(see more details in (Peralta et al., 2017)).

The nonlinear mathematical model which is pro-

posed here, as said before, consists of minimizing the

area (W × L) of the rectangular envelope, subject to

constraints ensuring that all items do not exceed the

bounds of the rectangular envelope (that are presented

in Section 2.1), and others ensuring that the items do

not overlap (that are presented in Section 2.2).

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

2.1 Containment Constraints

We have to ensure that all items are fully inside the

rectangular envelope.

To garantee that the circles are inside the rectan-

gular envelope, we require that:

− x

≤ 0, (1)

+ x

≤ W, (2)

− y

≤ 0, (3)

and,

+ y

≤ L, (4)

where (x

) are the coordinates of the center of cir-

cle i and r

is its radius, for i = 1, ..., n

, with n

the

number of circles to be packed.

To garantee that the polygons are inside the rect-

angular envelope, we require that:

0 ≤ v

cosθ

− v

sinθ

+ x

≤ W, (5)

and,

0 ≤ v

sinθ

+ v

cosθ

+ y

≤ L, (6)

for i = 1,...,n

, where n

is the number of polygons

to be packed; k = 1,..., p

and l = 1, . . . , v

2.2 Non-overlapping Constraints

In the model, separation lines are used to ensure non-

overlapping. The equation that ensures that a line sep-

arates item i from item j is given by:

y = c

i, j

x + d

i, j

, (7)

with

i, j

k+1

− x

)sinα

i, j

+ (y

k+1

− y

)cosα

i, j

k+1

− x

)cosα

i, j

+ (y

− y

k+1

)sinα

i, j

and

i, j

= y

i, j

− c

i, j

being (x

) and (x

k+1

), the two points that de-

termine the initial separation line; (x

i, j

) and α

i, j

are the reference point and the rotation angle of the

separation line, respectively. When we are dealing

with non-convex polygons, we must have lines sep-

arating each pair of convex polygons, belonging to

the partition of two different non-convex polygons.

To see more details of how separation lines are used

when it comes to non-convex polygons, see (Peralta

et al., 2017). The non-overlapping constraints are

given below.

2.2.1 Avoiding Overlap Between Circles

In the constraints of non-overlap between circles, the

use of separation lines is not necessary. We ensure

that two circles do not overlap by enforcing that the

distance between their centers are greater than or

equal to the sum of their radii. Therefore, for two cir-

cles C

and C

, i = 1, . . . , n

− 1 and j = i + 1, . . . , n

the constraint is given by

− x

)

+ (y

− y

)

≥ (r

+ r

)

. (8)

2.2.2 Avoiding Overlap Between Polygons

Non-overlap between two polygons P

and P

is en-

forced by the condition that all vertices of P

and P

are on different sides of, or on a separation line.

Let

= v

cosθ

− v

sinθ

+ x

and

= v

sinθ

+ v

cosθ

+ y

be the coordinates of vertice l of a polygon P

. Then,

the constraints that ensure non-overlap between two

polygons are given by

− c

i, j

− d

i, j

≤ 0, for l = 1, . . . , v

(9)

and

− c

i, j

− d

i, j

≥ 0, for l = 1, . . . , v

. (10)

2.2.3 Avoiding Overlap Between Circles and

Polygons

We ensure that a circle and a polygon are not over-

lapping by using a separation line. A separation line

separates a circle C

and a polygon P

if (i) all vertices

of P

are on one side of (or on) the separation line,

and (ii) the center of C

is on the other side of the sep-

aration line and distant from at least the radius of C

The constraints that ensure non-overlapping between

a circle and a polygon are given by

− c

i, j

− d

i, j

≥ 0, for l = 1, . . . , v

, (11)

− c

i, j

− d

i, j

≤ 0, (12)

and

∆

i, j

≥ r

, (13)

with ∆

i, j

− y

+ d

i, j

+ 1

the distance from the

center of C

to the line that separates C

from P

Packing Circles and Irregular Polygons using Separation Lines

3 NUMERICAL RESULTS

In this section we present and compare the results ob-

tained when applying our model to solve the prob-

lems presented in (Kallrath, 2009) and (Stoyan et al.,

2012), which pack circles and polygons simultane-

ously. All the numerical experiments have been

performed using a code for nonlinear programming

to solve the problem, IPOPT (W

achter and Biegler,

2006) (an algorithm of interior points type designed

to ﬁnd local solutions of mathematical optimization

problems), which is part of the COIN-OR (W

achter

and Biegler, 2015), on an Ubuntu 12.04 laptop with an

Intel Core I7-4510U CPU @ 2.1GHz processor and 8

GB of memory.

In (Kallrath, 2009), as here, the items should be

positioned into a rectangular envelope. The rectangu-

lar envelope is subject to lower and upper bounds of

its width and length. We tested all those instances in

(Kallrath, 2009) with circles and polygons which has

data available and used the same bounds presented

there, which are

W = 4 for the upper bound of the

width and

L = 8 for the upper bound of the length for

all instances, except for instance c1p1-3, for which

we have

W = 2.5 and

L = 3, and for instance c6p3,

for which we have

W = 6 and

L = 10.

The results obtained by solving these instances us-

ing the model presented in Section 2 and those pre-

sented in (Kallrath, 2009) are summarized in Table 1.

The names of the instances are presented in the ﬁrst

column. The number of circles and polygons are

presented in second and third columns, respectively.

From the fourth to the seventh column, the data and

results in (Kallrath, 2009) are presented. From the

eighth to the eleventh column, our data and results

are presented. In these, n is the number of variables

of the model; m is the number of constraints of the

model; CPU is the execution time in seconds; and a is

the area of the rectangular envelope. In all instances,

IPOPT stopped in optimal local solutions. The solu-

tion obtained by the proposed model and the starting

point used to ﬁnd it, for each instance, are displayed

from Figures 1 to 8.

We can observe that the number of variables and

constraints in our model is signiﬁcantly smaller than

the number of variables and constraints of the model

presented in (Kallrath, 2009). The execution time is

considerably reduced, which can be attributed, not

only to the reduction of variables and constraints in

the model, but also to the differences in operating sys-

tem, hardware architecture, among others. We can

also observe in our results that the area is smaller

than or equal to the ones obtained in (Kallrath, 2009)

in most instances. This reduction in the number of

4.00

8.00

(a) Starting point

1.00

(b) Solution

Figure 1: Instance: c1p1-1.

4.00

8.00

(a) Starting point

1.00

1.50

(b) Solution

Figure 2: Instance: c1p1-2.

2.50

3.00

(a) Starting point

1.00

1.50

(b) Solution

Figure 3: Instance: c1p1-3.

variables, constraints and execution time allows us

to solve larger instances, as the ones presented in

(Stoyan et al., 2012).

In (Stoyan et al., 2012), the items should be posi-

tioned into a strip which can have prohibited regions;

only one instance without prohibited regions was pre-

sented in this paper. We use this instance to test our

model, ﬁxing the width W of the strip in the value

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

Table 1: Comparison of our results to those in (Kallrath, 2009).

(Kallrath, 2009) Our approach

Instance C P n m CPU a n m CPU a

c1p1-1 1 1 47 54 0.10 1.00 10 26 0.01 1.00

c1p1-2 1 1 47 54 42.00 1.50 10 26 0.02 1.50

c1p1-3 1 1 47 54 368.00 2.50 10 26 0.01 1.50

c1p5a 1 2 767 799 2000.00 15.90 19 80 0.89 11.70

c1p5b 1 5 791 838 1800.00 18.55 64 284 0.06 19.73

c1p6a 1 6 1110 1161 1800.00 20.37 85 376 0.05 20.06

c3p3 3 3 489 516 2000.00 16.75 53 177 0.03 16.86

c6p3 6 3 628 661 33000.00 35.10 86 219 0.04 33.19

4.00

8.00

(a) Starting point

3.90

3.00

(b) Solution

Figure 4: Instance: c1p5a.

4.00

8.00

(a) Starting Point

3.00

6.57

(b) Solution

Figure 5: Instance: c1p5b.

4.00

8.00

(a) Starting point

3.00

6.68

(b) Solution

Figure 6: Instance: c1p6a.

4.00

8.00

(a) Starting point

3.40

4.96

(b) Solution

Figure 7: Instance: c3p3.

Packing Circles and Irregular Polygons using Separation Lines

6.00

10.00

(a) Starting point

5.97

5.56

(b) Solution

Figure 8: Instance: c6p3.

presented in the paper, W = 25; for this, we setup the

option of IPOPT “ﬁxed variable treatment” which in-

cludes equality constraints to ﬁx the variable. This

instance consists of 44 polygons, being some non-

convex, and 5 circles of different size. To model this

instance, 19335 variables and 54860 restrictions were

used. In (Stoyan et al., 2012), a length of 33.9 in two

hours of runtime was achieved. We achieved a length

of 36.00 in 521.64 seconds. The layout obtained is

displayed in Figure 10.

4 CONCLUSIONS

This work deals with a mathematical model for a

packing problem of circles and non-convex polygons

with continuous rotations in a rectangular envelope;

the objective of the problem is to minimize the area of

the rectangular envelope. In the model, we use sepa-

ration lines to avoid overlap of the items. The model-

ing of separation lines uses the general equation of the

line. We compare our model with the one presented

in (Kallrath, 2009) for packing problem of circles and

convex polygons, in which separation lines are also

used, but these are modeled using the vector equation

of the line. The vector equation of the line implies the

employment of many variables and constraints. With

the use of the general equation of the line the num-

ber of variables and constraints in the model is sig-

niﬁcantly reduced, and therefore, the execution time

is also reduced; with these reductions, we were able

to extend the model to non-convex polygons and to

solve larger problems.

We tested seven instances of those presented in

(Kallrath, 2009), and to verify the effectiveness of our

model for larger problems with non-convex polygons,

we tested the only instance for packing problem of

circles and polygons without prohibited regions pre-

sented in (Stoyan et al., 2012). We achieved a very

good solution in a short runtime for this instance.

To solve this problem, we use a code for nonlinear

programming, IPOPT (W

achter and Biegler, 2006),

which depends substantially on a starting point. In all

the instances we have tested, we use a random starting

point. We will work on a method that constructs start-

ing points, with which we believe the results could be

improved.

ACKNOWLEDGEMENTS

This research was partially supported by CNPq

(grants 141072/2014-8 and 409043/2016-8) and

FAPESP (grants 2013/07375-0 and 2016/01860-1),

from Brazil.

REFERENCES

Addis, B., Locatelli, M., and Schoen, F. (2008). Efﬁciently

packing unequal disks in a circle. Operations Re-

search Letters, 36:37–42.

Bennell, J. A. and Oliveira, J. F. (2008). The geometry of

nesting problems: A tutorial. European Journal of

Operational Research, 184:397–415.

Birgin, E. G., Bustamante, L. H., Callisaya, H. F., and

Mart

ınez, J. M. (2013). Packing circles within el-

lipses. International Transactions in Operational Re-

search, 20:365–389.

Castillo, I., Kampas, F. J., and Pintr, J. D. (2008). Solving

circle packing problems by global optimization: Nu-

merical results and industrial applications. European

Journal of Operational Research, 191(3):786–802.

Galiev, S. I. and Lisaﬁna, M. S. (2013). Linear models for

the approximate solution of the problem of packing

equal circles into a given domain. European Journal

of Operational Research, 230:505–514.

George, J. A., George, J. M., and Lamar, B. W. (1995).

Packing different-sized circles into a rectangular con-

tainer. European Journal of Operation Research,

84:693–712.

Jones, D. R. (2013). A fully general, exact algorithm for

nesting irregular shapes. Journal of Global Optimiza-

tion, 59:367–404.

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

25.00

100.00

Figure 9: Starting point used to solve the instance from (Stoyan et al., 2012).

25.00

35,97

Figure 10: Solution found for the instance from (Stoyan et al., 2012).

Kallrath, J. (2009). Cutting circles and polygons from area-

minimizing rectangles. Journal of Global Optimiza-

tion, 43:299–328.

Liu, J., Zhang, K., Yao, Y., Xue, Y., and Guan, T. (2016).

A heuristic quasi-physical algorithm with coarse and

ﬁne adjustment for multi-objective weighted circles

packing problem. Computers and Industrial Engi-

neering, 101:416–426.

opez, C. O. and Beasley, J. E. (2013). Packing unequal cir-

cles using formulation space search. Computers and

Operations Research, 40(5):1276–1288.

Mladenovi

c, N., Plastria, F., and Urosevi

c, D. (2005).

Reformulation descent applied to circle packing

problems. Computers and Operations Research,

32(9):2419–2434.

Mundim, L. R., Andretta, M., and Queiroz, T. (2017). A

biased random key genetic algorithm for open dimen-

sion nesting problems using no-ﬁt raster. Expert Sys-

tems with Applications, 81:358–371.

Oliveira, J. F. and Ferreira, J. S. (1993). Algorithms for nest-

ing problems, applied simulated annealing. In: Vidal,

R.V.V. (Ed.), Lecture notes in econ. and Maths Sys-

tems. Springer Verlag, 396:255–274.

Peralta, J., Andretta, M., and Oliveira, J. F. (2017). Solving

irregular strip packing problems with free rotations us-

ing separation lines. https://arxiv.org/abs/1707.07177.

Sahinidis, N. V. (2014). BARON 14.3.1: Global Optimiza-

tion of Mixed-Integer Nonlinear Programs, User’s

Manual.

Stoyan, Y., Pankratov, A., and Romanova, T. (2016). Cut-

ting and packing problems for irregular objects with

continuous rotations: mathematical modelling and

non-linear optimization. Journal of the Operational

Research Society, 67(5):786–800.

Stoyan, Y. G., Zlotnik, M. V., and Chugay, A. M. (2012).

Solving an optimization packing problem of circles

and non-convex polygons with rotations into a mul-

tiply connected region. Journal of the Operational

Research Society, 63:379–391.

Szab

o, P. G., Mark

ot, M. C., Csendes, T., Specht, E.,

Casado, L. G., and Garcia, I. (2007). New approaches

to circle packing in a square: with program codes.

Optimization and its Applications.

Tawarmalani, M. and Sahinidis, N. V. (2005). A polyhe-

dral branch-and-cut approach to global optimization.

Mathematical Programming, 103:225–249.

Toledo, F. M. B., Carravilla, M. A., Ribeiro, C., Oliveira,

J. F., and Gomes, A. M. (2013). The dotted-

board model: A new mip model for nesting irregu-

lar shapes. International Journal of Production Eco-

nomics, 145(2):478–487.

achter, A. and Biegler, L. T. (2006). On the imple-

mentation of a primal-dual interior point ﬁlter line

search algorithm for large-scale nonlinear program-

ming. Mathematical Programming, 106(1):25–57.

achter, A. and Biegler, L. T. (2015). COIN OR project.

http://projects.coin-or.org/Ipopt.

Packing Circles and Irregular Polygons using Separation Lines