Approximate Circle Packing in a Rectangular Container: Integer

Approximate Circle Packing in a Rectangular Container: Integer Programming Formulations and Valid Inequalities Igor Litvinchev, Luis Infante, and Edith Lucero Ozuna Espinosa Department of Mechanical and Electrical Engineering Nuevo Leon State University Monterrey, Nuevo Leon, Mexico {igor,infante,lucero}@yalma.fime.uanl.mx

Abstract. A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. Frequently, the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. A new formulation is proposed using a regular grid approximating the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. The resulting binary problem is then solved by commercial software. Two families of valid inequalities are proposed to strengthen the formulation. Numerical results are presented to demonstrate the efficiency of the proposed approach. Keywords: Circle packing, integer programming, large scale optimization.

1

Introduction

Packing problems constitute a family of natural combinatorial optimization problems, which occur in many fields of study such as computer science, industrial engineering, logistics and manufacturing and production processes. For instance, several real life industrial applications require the allocation of a set of pieces to a larger standardized rectangular stock unit. They generally consist of packing a set of items of known dimensions into one or more large objects in order to minimize a certain objective (e.g. the unused part of the objects or waste). The circle packing problem is a well studied problem [8, 13] whose aim is the packing of a certain number of circles, each one with a fixed known radius (not necessarily the same for each circle) inside a container. The shape of the container may vary from a circle, a square, a rectangular, etc. This problem has been applied in different areas, such as the coverage of a geographical area with cell transmitters, storage of a cylindrical drums into containers or R.G. Gonzáez-Ramírez et al. (Eds.) : ICCL 2014, LNCS 8760, pp. 47–60, 2014. © Springer International Publishing Switzerland 2014

48

I. Litvinchev, L. Infante, and E.L.O. Espinosa

stocking them into an open area, packaging bottles or cans into the smallest box, planting trees in a given region as to maximize the forest density and the distance between the trees, and so forth [2, 6, 9]. Other applications one can find in the motor cycle industry, circular cutting, communication networks, facility location and dashboard layout [6, 11, 12]. In this paper we address the problem of packing a set of circular items in a rectangular container. There are two principal types of objectives that have been used in the literature: a) regarding the circles (not necessary equal) as being of fixed size and the container as being of variable size and b) regarding the circles and the container as being of fixed size and minimize “waste”. Examples of the first approach include [18]: • For the square container minimize the length of the side and hence minimize the perimeter and area of the square; • Minimize the perimeter of the rectangle; • Minimize the area of the rectangle; • Considering one dimension of the rectangle as fixed, minimize the other dimension. Problems of this type are often referred to as strip packing problems (or as circular open dimension problems). For the second approach various definitions of the waste can be used. The waste can be defined in relation to circles not packed (e.g., the number of unpacked circles or the perimeter/area of unpacked circles), or introducing a value associated with each circle that is packed (e.g., area of the circles packed), etc. Many variants of packing circular objects in the plane have been formulated as nonconvex (continuous) optimization problems with decision variables being coordinates of the centres. The nonconvexity is mainly provided by no overlapping conditions between circles. These conditions typically state that the Euclidean distance separating the centres of the circles is greater than a sum of their radii. The nonconvex problems can be tackled by available nonlinear programming (NLP) solvers; however, most NLP solvers fail to identify global optima. Thus, the nonconvex formulation of circular packing problems requires algorithms which mix local searches with heuristic procedures in order to widely explore the search space. We refer the reader to review papers presenting the scope of techniques and applications for the circle packing problem (see, e.g., [1, 4, 5, 8, 17-19] and the references therein). In this paper we propose a new formulation for approximate solution of circular packing problems using a regular grid to approximate the container. The nodes of the grid are considered as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. Two classes of valid inequalities are proposed to strengthening the formulation. Numerical results are presented to demonstrate efficiency of the proposed approach. To the best of our knowledge, the idea to use a grid was first implemented by Beasley [3] in the context of cutting problems. This approach was recently applied in [10, 14, 15] for packing problems. This work is a continuation of [15].

Approximate Circle Packing in a Rectangular Container

2

49

The Model

Suppose we have non-identical circles C k of known radius Rk , k ∈ K = {1, 2,...K } . Let at most M k circles C k be available for packing and at least mk of them have to be packed. Denote by i ∈ I = {1,2..., n} the node points of a regular grid covering the rectangular container. Let F ⊆ I be the grid points lying on the boundary of the container. Denote by d ij the Euclidean distance between points i and j of the grid. Define binary variables xi = 1 if the centre of a circle C k is assigned to the point i ; k

xik = 0 otherwise. In order to the circle C k assigned to the point i be non-overlapping with other circles being packed, it is necessary that x lj = 0 for

j ∈ I , l ∈ K , such that

d ij < Rk + Rl . For fixed i, k let N ik = { j , l : i ≠ j , dij < Rk + Rl } . Let nik be the cardinality of N ik : nik = N ik . Then the problem of maximizing the area covered by the circles can be stated as follows:

max

R

2 k k i

x

(1)

i∈I k ∈K

subject to

mk ≤  xik ≤ M k , k ∈ K ,

(2)

x

(3)

k i

i∈I

≤ 1, i ∈ I \ F ,

k ∈K

Rk xik ≤ min d ij , i ∈ I , k ∈ K , j∈F

(4)

xik + x lj ≤ 1, for i ∈ I , k ∈ K , ( j , l ) ∈ N ik

(5)

xik ∈{0,1}, i ∈ I , k ∈ K

(6)

Constraints (2) ensure that the number of circles packed is between mk and M k ; constraints (3) guarantee that at most one centre is assigned to any grid point; constraints (4) ensure that the point i cannot be a centre of the circle C k if the distance from i to the boundary is less than Rk ; pair-wise constraints (5) guarantee that there is no overlapping between the circles; constraints (6) represent the binary nature of the variables. By definition, N ik = { j , l : i ≠ j , dij < Rk + Rl } and hence if ( j , l ) ∈ N ik , then

(i, k ) ∈ N jl . Thus, half of the constraints in (5) are redundant since we have: xik + x lj ≤ 1, for i ∈ I , k ∈ K , ( j , l ) ∈ N ik ,

50


x lj + xik ≤ 1, for j ∈ I , l ∈ K , (i, k ) ∈ N jl We may eliminate any (none) of these two constraints to get the reduced equivalent formulation. This can be represented by multiplying constraints (5) by fixed λ lj ∈ {0,1} :

xik λ lj + x lj λ lj ≤ λ lj , for i ∈ I , k ∈ K , ( j , l ) ∈ N ik ,

(7)

subject to λ lj + λik ≥ 1 . This way either one of the redundant constraints is eliminated ( λ lj + λik = 1 ) or none ( λ lj + λik = 2 ). Since eliminating redundant constraints does not affect the feasible set, the problem (1)-(6) is equivalent to (1)-(4), (6), (7) for any λ fulfilling the normalized condition λ ∈ Λ = {λ lj ∈ {0,1}: λ lj + λik ≥ 1, ( j , l ) ∈ N ik } . Similar to plant location problems [20] we can state non-overlapping conditions in a more compact form. Summing up constraints (7) over ( j , l ) ∈ N ik we get

xik



λ lj +

( j ,l )∈Nik



λ lj xlj ≤

( j , l )∈Nik



λ lj , for i ∈ I , k ∈ K .

(8)

( j ,l )∈Nik

Proposition 1. For any λ ∈ Λ constraints (5), (6) are equivalent to constraints (6), (8). Proof. If constraints (5) are fulfilled, than obviously constraints (8) hold by construction. Now let constraints (8) be fulfilled. Define N ik1 = {( j , l ) ∈ N ik : λ lj = 1} , N ik0 = {( j , l ) ∈ N ik : λ lj = 0} , N ik1 ∪ Nik0 = Nik ,

N ik1 = nik1 , N ik0 = nik0 . By (8) we have

xik nik1 +



xlj ≤ nik1

( j , l )∈Nik1

and hence, if xi = 1 , then x lj = 0 for ( j , l ) ∈ Nik . k

1

(9)

By definition, if ( j , l ) ∈ N ik , then (i, k ) ∈ N jl . Thus by (8) we have

xlj



λik +

( i , k )∈N jl



( i , k )∈N jl

λik xik ≤



λik for j ∈ I , l ∈ K .

(10)

( i , k )∈N jl

In particular, (10) is fulfilled for ( j , l ) ∈ Nik . Since λ lj + λik ≥ 1 , then for ( j , l ) ∈ Nik 0

0

all λi in (10) are positive ( λi = 1 ). Then by (10) we have: k

k

if x lj = 1 for at least one ( j , l ) ∈ Nik , then xi = 0 . 0

k

(11)


51

Note that constraints (5) can be interpreted in two ways. First, if xi = 1 , k

then x lj = 0 for all ( j , l ) ∈ Nik . Second, if x lj = 1 for at least one ( j , l ) ∈ Nik , then

xik = 0 . Combining (9) and (11) we may conclude that if constraints (8) are fulfilled, then constraints (5) hold.  Remark 1. In [10] the compact formulation

xi ni +

x

j

≤ ni for i ∈ I

(12)

j ∈N i

was used to represent non-overlapping conditions for the case of packing identical circles of radius R . Here ni is the cardinality of the set N i = { j : i ≠ j , d ij < 2 R} . This case corresponds to a singleton set K and all multipliers λ equal to 1 in (8). Remark 2. Proposition 1 remains true for nonnegative (not necessarily binary) multipliers λ subject to λ lj + λik ≠ 0 . The proof is similar. Example. Consider the same example as in [10]. Suppose we want to pack the maximum possible number of circles of radius 1 into a rectangle of width 4 and height 3. Take a rectangular uniform grid of size Δ = 1 along both sides of the container. The boundary grid points can be eliminated from the consideration (see [10] for details) leaving only 6 interior grid points to state the non-overlapping conditions as shown in Fig.1.

1

2

3

4

5

6

Fig. 1. Example.

For i = 1, 2...,6 the corresponding sets N i are as follows:

i

1

2

3

4

5

6

Ni

2,4,5

1,3,4,5,6

2,5,6

1,2,5

1,2,3,4,6

2,3,5

The corresponding pair-wise non-overlapping constraints (5) have the following form:

52


 x1 + x2 ≤ 1    x1 + x4 ≤ 1 ,  x + x ≤ 1  1 5 

 x2 x  2  x2 x  2  x2

+ x1 ≤ 1  + x3 ≤ 1  + x4 ≤ 1 , + x5 ≤ 1  + x6 ≤ 1

 x3 + x2 ≤ 1    x3 + x5 ≤ 1 ,    x3 + x6 ≤ 1

 x4 + x1 ≤ 1     x4 + x2 ≤ 1 ,    x4 + x5 ≤ 1

 x5   x5   x5 x  5  x5

+ x1 ≤ 1   + x2 ≤ 1  + x3 ≤ 1 , + x4 ≤ 1  + x6 ≤ 1

 x6 + x2 ≤ 1    x6 + x3 ≤ 1    x6 + x5 ≤ 1

(13)

Summing up constraints in each group (all multipliers λ equal to 1 in (8)) we get the same compact formulation as in [10], which is equivalent to (13) for binary x :

3 x1 + x2 + x4 + x5 ≤ 3 5 x2 + x1 + x3 + x4 + x5 + x6 ≤ 5 3 x3 + x2 + x5 + x6 ≤ 3

(14)

3 x4 + x1 + x2 + x5 ≤ 3 5 x5 + x1 + x2 + x3 + x4 + x6 ≤ 5 3 x6 + x2 + x3 + x5 ≤ 3 . We see that in (13) half of the constraints are redundant. Constraint x1 + x2 ≤ 1 appears in the first and in the second group of (13), constraint x1 + x4 ≤ 1 is presented in the first and in the fourth, etc. Obviously, eliminating redundant constraints does not change the set defined by (13). However, the way of eliminating redundant constraints affects the equivalent system obtained by summing up the remaining constraints. Suppose that we will eliminate redundant constraints beginning from the first group of (13) where redundancy appears. That is, x1 + x2 ≤ 1 will be eliminated from the first group, x2 + x3 ≤ 1 from the second, etc. Doing it this way we get a reduced set of constraints

 x5 + x1 ≤ 1     x4 + x1 ≤ 1   x5 + x2 ≤ 1 { x2 + x1 ≤ 1} , { x3 + x2 ≤ 1} ,  ,  ,  x4 + x2 ≤ 1  x5 + x3 ≤ 1  x5 + x4 ≤ 1  

 x6 + x2 ≤ 1    x6 + x3 ≤ 1  x + x ≤ 1 5  6 

(15)

The compact equivalent system obtained by summing up constraints in each group in (15) is as follows: x2 + x1 ≤ 1 ,


53

x3 + x2 ≤ 1 , 2 x4 + x1 + x2 ≤ 2 ,

(16)

4 x5 + x1 + x2 + x3 + x4 ≤ 4 , 3 x6 + x2 + x3 + x5 ≤ 3 . If we now will eliminate redundant constraints beginning from the last group, the reduced set of constraints becomes

 x1 + x2 ≤ 1    x1 + x4 ≤ 1 ,  x + x ≤ 1  1 5 

 x2 + x3 ≤ 1    x2 + x4 ≤ 1  x3 + x5 ≤ 1  ,   , { x4 + x5 ≤ 1} , { x5 + x6 ≤ 1} (17)  x2 + x5 ≤ 1  x3 + x6 ≤ 1  x2 + x6 ≤ 1

The equivalent compact system obtained by summing up constraints in each group in (17) is as follows: 3 x1 + x2 + x4 + x5 ≤ 3 ,

4 x2 + x3 + x4 + x5 + x6 ≤ 4 , 2 x3 + x5 + x6 ≤ 2 ,

(18)

x4 + x5 ≤ 1 , x5 + x6 ≤ 1 . Eliminating redundant constraints with the objective to balance the number of constraints in the groups of (13) we get the reduced system

 x2 + x1 ≤ 1   ,  x2 + x5 ≤ 1

 x3 + x2 ≤ 1  ,  x3 + x5 ≤ 1

 x4 + x2 ≤ 1   , { x5 + x6 ≤ 1} ,  x4 + x5 ≤ 1

 x6 + x2 ≤ 1  .  x6 + x3 ≤ 1

 x1 + x4 ≤ 1  ,  x1 + x5 ≤ 1

(19)

The corresponding compact system equivalent to (19) is as follows

2 x1 + x4 + x5 ≤ 2 , 2 x2 + x1 + x5 ≤ 2 , 2 x3 + x2 + x5 ≤ 2 , 2 x4 + x2 + x5 ≤ 2 ,

(20)

54


x5 + x6 ≤ 1 , 2 x6 + x2 + x3 ≤ 2 . As follows from Proposition 1, the non-overlapping conditions can be stated in different forms. We have a family of formulations equivalent to (5) and obtained for different multipliers λ in (8). To compare equivalent formulations, let

P1 = { x ≥ 0 : xik + x lj ≤ 1, for i ∈ I , k ∈ K , ( j , l ) ∈ N ik } ,

  P2 =  x ≥ 0 : xik  λ lj +  λ lj xlj ≤  λ lj , i ∈ I , k ∈ K  , ( j ,l )∈N ik ( j ,l )∈N ik ( j , l )∈N ik   where multipliers λ in P2 fulfil normalizing condition stated in Proposition 1. Proposition 2. P1 ⊂ P2 . Proof. Since constraints of P2 are a linear combination of those in P1 with nonnegative multipliers λ , then P1 ⊆ P2 . To show that P1 ⊂ P2 we need to find a point in

P2 that is not in P1 . This point can be constructed as follows. Choose (i, k ) ∈ N jl (and hence

( j , l ) ∈ N ik ) such that





λ lj ,

( j ,l )∈N ik k i

λik ≥ 2 . Set to zero all the variables ex-

( i , k )∈N jl

l j

cept x , x . Obviously all constraints in P2 corresponding to zero variables are fulfilled. Define xik , x lj to fulfil the two remaining constraints as equalities:

xik



λ lj + x lj =

( j , l )∈N ik

Denote nik =





λ lj , x lj



λik thus nik , n jl ≥ 2 . The corresponding so-

( j ,l )∈N ik

λ lj , n jl =

( j , l )∈N ik



λik + xik =

( i , k )∈N jl



λik

( i , k )∈N jl

( i , k )∈N jl

lution of the two equations above is

xik =

n jl ( nik − 1) n jl nik − 1

< 1 , x lj =

nik ( n jl − 1) n jl nik − 1

1.

This point violates corresponding constraint in P1 and hence P1 ⊂ P2 as desired.  As follows from Proposition 2, the pairwise formulation (1)-(6) is stronger than the compact one (1)-(4), (6), (8) in the sense of [20].


3

55

Valid Inequalities

We may expect that the linear programming relaxation of the problem (1)-(6) provides a poor upper bound for the optimal objective value. For example, for K = 1 and k suitable M k , mk the point xi = 0.5 for all i ∈ I may be feasible to the relaxed problem with the corresponding objective growing linearly with respect to the number of grid points. To tightening the LP-relaxation we consider two families of valid inequalities. The first ensures that no grid point is covered by two circles, while the second guarantees that there is at most one centre assigned to the area covered by a circle. To present the first family, define matrix α ijk  as follows. Let α ijk = 1 for d ij < Rk ,

α ijk = 0 , otherwise. By this definition, α ijk = 1 if the circle Ck centered at i covers point j . The following constraints ensure that no points of the grid can be covered by two circles:

α

k ij

x kj ≤ 1, i ∈ I

(21)

k∈K j∈I

Note that (21) is not equivalent to non-overlapping constraints (5). Constraints (21) ensure that there is no overlapping in grid points, while (5) guarantee that there is no overlapping at all. We will refer to (21) as point-covering valid inequalities. The second family of inequalities is stated as follows: (22) xik + x kj ≤ 1, for i ∈ I , k ∈ K .



j:d ij < Rk

To demonstrate that (22) is valid for the problem (1)-(6) assume that xi = 1 in (22). k

That is, the centre of the circle Ck



is assigned at i . By (22) we have

x ≤ 0, for i ∈ I , k ∈ K and then it follows that x kj = 0 for j : d ij < Rk . That is, k j

j:d ij < Rk

there are no other centres assigned to points inside the circle centred at i . For

xik = 0 we have



x kj ≤ 1 . This means that among all grid points covered by the

j:d ij < Rk

(imaginary) circle centred at i , at most one point can be assigned as a centre. This is true since the distance between any pair of these points is less than 2 Rk and assigning the centres of Ck violates non overlapping constraints.

4

Numerical Experiments

In this section we numerically compare different problem formulations and study the impact of introducing valid inequalities for the case of packing equal circles. A rectangular uniform grid of size Δ along both sides of the container was used. The test

56


bed set of nine instances from [10, Table 3] was used for packing a maximal number of circles into a rectangular container of width 3 and height 6. All optimization problems were solved by the system CPLEX 12.6. The runs were executed on a desktop computer with CPU AMD FX 8350 8-core processor 4 Ghz and 32Gb RAM. The following four formulations were compared: pairwise formulation (1)-(6) (complete), reduced formulation (1)-(6) without redundant constraints (half), compact formulation (12) as in [10] (compact), and the compact formulation obtained by summing up constraints in the reduced formulation (1)-(6) (compact half). The results of the numerical experiment are given in Table 1. Here the first five columns present instance number, circle radius, size of the grid Δ , number of binary variables and the number of circles packed. The last four columns give CPU time (in seconds) for different formulations. For all problem instances mipgap = 0 was set for running CPLEX. In this table and all tables below, an asterisk indicates that the computation was interrupted after the computation time exceeded 12-hour CPU time. For problem instances 7 and 9, where optimality was not achieved without valid inequalities neither for complete nor for compact formulations, the number of circles in column 5 corresponds to optimal solution obtained by using valid inequalities (see the tables below). As we can see from Table 1, CPU time for complete formulations is much lower than for the compact, especially for large problems. Eliminating redundant constraints typically (but not always) reduces CPU time. Although eliminating redundancy does not change corresponding LP-relaxation, it may affect the path selected by branch and bound technique and thus results in increase/decrease of CPU time. Table 1. Integer solution (gap 0%) without valid inequalities Problem

Instance

Circle radius

Δ

1

0.5

0.125

697

2

0.625

0.15625

3

Circle

compact

complete

half

compact

18

2

1

276

5

1403

10

71

41

1040

50

0.5625 0.0703125

2449

13

337

186

11666

831

4

0.375

0.046875

1425

32

6

4

2698

169

5

0.3125

0.078125

2139

45

96

114

*

*

6

0.4375 0.0546875

3666

21

17473

17654

*

*

dimension number

half

7

0.25

0.0625

3649

74

*

*

*

*

8

0.275

0.06875

2880

61

132

177

*

*

9

0.1875

0.046875

6897

140

*

*

*

*

In the second part of the experiment we study the effect of introducing valid inequalities in the problem formulations. Table 2 presents values of upper bounds obtained by the LP relaxation for complete formulation (1)-(6) (column 4) and for the complete formulation with valid inequalities (21) (column 5). The results for compact


57

formulations and valid inequalities (22) are quite similar and are omitted here. We see that a) LP-bounds without valid inequalities are very poor and b) valid inequalities improve significantly the quality of the bounds for all problem instances. Table 2. LP-relaxation Complete

Problem

Circle

dimension

number

1

697

18

348.5

19

2

1403

10

701.5

10

3

2449

13

1224.5

14.07

4

1425

32

712.5

36.33

5

2139

45

1069.5

53.4

6

3666

21

1833.5

23.86

7

3649

74

1824.5

90.98

8

2880

61

1440

72

9

6897

140

3448.5

162

Instance

complete

+C

Table 3 presents the impact of valid inequalities (21) and (22) (C1 and C2, respectively) on CPU time required to get integer solution with mipgap = 0 for pairwise formulation with (complete) and without (half) redundant constraints. Table 4 presents corresponding results for compact formulations. We see that introducing valid inequalities decreases CPU time for all problem instances and all problem formulations. Although introducing valid inequalities slightly increases time to solve the LP-relaxation, the effect of improving the quality of the LP-bound becomes more important for the convergence of the overall branch and bound scheme. That is why CPU times decrease significantly for hard instances 6, 7, 9, while for “easy” instances the decrease may be relatively modest. Moreover, with valid inequalities CPU times to get provably optimal solutions ( mipgap = 0 ) is comparable with those reported in [10] for their heuristic approach. Tables 5, 6 are similar to Tables 3, 4, but present CPU time for mipgap = 5% . Here numbers in parenthesis in the first column indicate the best integer solution obtained (circles packed). In the next column this number is indicated only if it is different from the previous column. We see that the impact of valid inequalities is less visible and even may be negative (see, e.g., instances 7-9 in Table 5). For larger values of the gap the quality of bounds is less important comparing with increasing CPU time to solve the LP-relaxation. Comparing valid inequalities C1 and C2, we may conclude that there is not much difference; they both produce similar impact on CPU time.

58

I. Litvinchev, L. Infante, and E.L.O. Espinosa Table 3. Integer solution (gap 0%) with valid inequalities, complete formulation Half

Half

Half

C1

C2

C1+C2

1

1

1

Complete

Complete

Complete

C1

C2

C1 + C2

2

2

1

2

2

71

15

12

18

41

11

9

10

3

337

82

81

86

186

75

72

73

Instance

half

complete

1

1

4

6

9

6

8

4

4

3

4

5

96

163

157

150

114

189

180

191

6

17473

1392

1355

1122

17654

1379

1364

776

7

*

3531

3540

5542

*

3178

3186

2978

8

132

87

103

88

177

87

103

104

9

*

17437

17193

24602

*

*

*

11892

Table 4. Integer solution (gap 0%) with valid inequalities, compact formulation Half

Half

Half

C1

C2

C1+C2

5

4

4

5

27

50

12

13

16

41

831

32

32

34

31

169

92

94

87

Compact

Compact

Compact

C1

C2

C1+C2

276

4

4

4

2

1040

35

33

3

11666

37

36

4

2698

29

28

Instance

compact

1

half

5

*

819

818

1019

*

1027

1029

1849

6

*

39347

39260

41154

*

*

*

*

7

*

*

*

*

*

*

*

*

8

*

2523

2525

2146

*

2860

2850

2188

9

*

*

*

*

*

*

*

*

Table 5. Integer solution (gap 5%) with valid inequalities, complete formulation Complete Instance 1

Complete

Complete

4 (18)

Half

Half

Half

C1

C2

C1+ C2

1

1

1

Half

Complete C1

C2

C1+C2

1

1

1

1

2

62 (10)

13

13

13

42

11

11

10

3

285 (13)

83

82

83

213

75

75

74

4

4 (32)

6

6

7

4

3

3

4

5

28 (45)

28

27

34

26

42

41

33

6

17447 (21)

1586

1579

1538

17654

1379

1364

776

7

1373 (73)

2161 (74)

2130

1782

1978

1716

1756

1862

8

39 (60)

77 (59)

71

58

19 (61)

81 (60)

65 (59)

69 (59)

9

2121 (135)

2601

2590

2316

3768

3458

3515

3194


59

Table 6. Integer solution (gap 5%) with valid inequalities, compact formulation Instance

Compact

Compact

Compact Half

compact C1

C2

C1+C2

Half C1

Half

Half

C2

C1 + C2

1

6 (18)

3

2

3

2

2

2

2

2

372 (10)

21

31

28

48

9

9

12

3

11163 (13)

30

30

32

1063

27

26

26

4

234 (32)

13

12

14

115

54

53

75

5

* (44)

417 (45)

424

1043

*

634

630

568

6

* (20)

39347 (21)

39260

33321

*

*

*

*

7

* (63)

* (74)

*

* (73)

*

7314

7341

7340

1999

1972

1701 (59)

*

*

*

8

* (50)

1819 (59)

1809

1325 (61)

31859 (60)

9

* (116)

* (135)

*

* (136)

* (135)

Summarizing the results of our numerical experiment, we may conclude that pairwise formulation (1)-(6), with or without redundant constraints, is computationally much more attractive than compact formulations. This is in good correspondence with Proposition 2. Valid inequalities proposed in the paper are very efficient if a provably optimal solution with zero gap is required.

5

Conclusions

We studied integer programming formulations obtained by using a grid to approximate a container in a packing problem. The equivalence of these formulations is demonstrated and a stronger formulation in the sense of [20] is indicated. Two classes of valid inequalities are used to strengthen the formulations. Numerical results are presented to demonstrate the usefulness of these inequalities. An interesting area for the future research is to use the LP-relaxation of integer formulations to get a feasible solution. The LP-relaxation without valid inequalities is trivial – all variables are equal to 0.5. However, aggregating valid inequalities changes significantly the structure of the relaxed solution and this impact becomes more visible for fine grids. Some results in this direction are in course. The other direction for the future research is to study the use of Lagrangian relaxation/decomposition [16] to cope with large dimension of the problem formulation. Other valid inequalities [7] can also be used.

References 1. Akeb, H., Hifi, M.: Solving the circular open dimension problem using separate beams and look-ahead strategies. Computers & Operations Research 40, 1243–1255 (2013) 2. Baltacioglu, E., Moore, J.T., Hill, R.R.: The distributor´s three-dimensional pallet-packing problem: a human intelligence-based heuristic approach. International Journal of Operational Research 1, 249–266 (2006)

60


3. Beasley, J.E.: An exact two-dimensional non-guillotine cutting tree search procedure. Operations Research 33, 49–64 (1985) 4. Birgin, E.G., Martínez, J.M., Nishihara, F.H., Ronconi, D.P.: Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization. Computers & Operations Research 33, 3535–3548 (2006) 5. Birgin, E.G., Gentil, J.M.: New and improved results for packing identical unitary radius circles within triangles, rectangles and strips. Computers & Operations Research 37, 1318–1327 (2010) 6. Castillo, I., Kampas, F.J., Pinter, J.D.: Solving circle packing problems by global optimization: Numerical results and industrial applications. European Journal of Operational Research 191, 786–802 (2008) 7. Costa, A.: Valid constraints for the point packing in a square problem. Discrete Applied Mathematics 161, 2901–2909 (2013) 8. Fasano, G.: Solving Non-standard Packing Problems by Global Optimization and Heuristics. Springer, Berlin (2014) 9. Frazer, H.J., George, J.A.: Integrated container loading software for pulp and paper industry. European Journal of Operational Research 77, 466–474 (1994) 10. Galiev, S.I., Lisafina, M.S.: 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 (2013) 11. George, J.A., George, J.M., Lamar, B.W.: Packing different–sized circles into a rectangular container. European Journal of Operational Research 84, 693–712 (1995) 12. George, J.A.: Multiple container packing: a case study of pipe packing. Journal of the Operational Research Society 47, 1098–1109 (1996) 13. Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: Models and methodologies. Advances in Operations Research Article ID 150624, 22 pages (2009), doi:10.1155/2009/150624 14. Litvinchev, I., Ozuna, E.L.: Packing circles in a rectangular container. In: Proc. Intl. Congr. on Logistics and Supply Chain, Queretaro, Mexico, pp. 24–25 (October 2013) 15. Litvinchev, I., Ozuna, E.L.: Integer programming formulations for approximate packing circles in a rectangular container. Mathematical Problems in Engineering (2014), doi:10.1155/2014/317697 16. Litvinchev, I., Rangel, S., Saucedo, J.: A Lagrangian bound for many-to-many assignment problem. Journal of Combinatorial Optimization 19, 241–257 (2010) 17. Lopez, C.O., Beasley, J.E.: A heuristic for the circle packing problem with a variety of containers. European Journal of Operational Research 214, 512–525 (2011) 18. Lopez, C.O., Beasley, J.E.: Packing unequal circles using formulation space search. Computers & Operations Research 40, 1276–1288 (2013) 19. Stoyan, Y.G., Yaskov, G.N.: Packing congruent spheres into a multi-connected polyhedral domain. International Transactions in Operational Research 20, 79–99 (2013) 20. Wolsey, L.A.: Integer Programming. Wiley, New York (1999)