Exact and Heuristic MIP models for Nesting Problems

Exact and Heuristic MIP Models for Nesting Problems Matteo Fischetti, Ivan Luzzi

DEI, University of Padova

presented at the EURO meeting, Istanbul, July 2003

Slide 1

Exact and Heuristic MIP models for Nesting Problems

The Nesting Problem

Given a set of 2-dimensional pieces of generic (irregular) form and a 2-dimensional container, find the best non-overlapping position of the pieces within the container. small pieces big pieces

Pieces: 45/76

Length: 1652.52

Eff.: 85.86%

Complexity: NP-hard (and very hard in practice)

Slide 2

Exact and Heuristic MIP models for Nesting Problems

Literature Heuristics • J. Blazewicz, P. Hawryluk, R. Walkowiak, Using a tabu search approach for solving the two-dimensional irregular cutting problem, AOR 1993 • J.F.C. Oliveira, J.A.S. Ferreira, Algorithms for nesting problems, Springer-Verlag 1993 • K.A. Dowsland, W.B. Dowsland, J.A. Bennel, Jostling for position: local improvement for irregular cutting patterns, JORS 1998 • ... Containment & Compaction • K. Daniels, Z. Li, V. Milenkovic, Multiple Containment Methods, Technical Report TR-12-94, Harvard University, July 1994. • Z. Li, V. Milenkovic, Compaction and separation algorithms for non-convex polygons and their applications, EJOR 1995 • K. Daniels, Containment algorithms for non-convex polygons with applications to layout, PhD thesis 1995 • ... Branch & Bound • R. Heckmann, T. Lengauer, Computing closely matching upper and lower bounds on textile nesting problems, EJOR 1998 • ...

Slide 3

Exact and Heuristic MIP models for Nesting Problems

A MIP model for the nesting problem Input • We are given a set P of n := |P| pieces. The form of each piece is defined by a simple polygon described through the list of its vertices. In addition, each piece i is associated with an arbitrary reference point whose 2-dimensional coordinates vi = (xi , yi ) will be used to define the placement of the piece within the container. • The container is assumed to be of rectangular form, with fixed height maxY and infinity length.

top 

(x i, yi )

i

maxY



bottom

left

i

right

i

length

i

Variables • vi = (xi , yi ) : coordinates of the reference point of piece i • length : right margin of the used area within the container ("makespan") Objective Minimize length, i.e., maximize the percentage efficiency computed as: Pn

efficiency =

areai ∗ 100 length ∗ maxY i=1

Slide 4

Exact and Heuristic MIP models for Nesting Problems

How to check/model the overlap between two pieces? The Minkowski sum of two polygons A and B is defined as: A ⊕ B = {a + b : a ∈ A, b ∈ B} The no-fit polygon between two polygons A and B is defined as UAB := A ⊕ (−B) y−y B

A

B touches A vB

vB B does not intersect A

vA = (0,0)

x − xA B

vB UAB

B overlaps A

Interpretation: place the reference point of polygon A at the origin; then the no-fit polygon represents the trajectory of the reference point of polygon B when it is moved around A so as to be in touch (with no overlap) with it.

Slide 5

Exact and Heuristic MIP models for Nesting Problems

The Minkowski difference between polygons A and B is defines as: \ Ab AªB = b ∈B

The containment polygon corresponding to two polygons A and B is defined as: VAB := A ª (−B) and represents the region of containment (without overlap) of a piece B inside a hole A.

B A VAB

Slide 6

Exact and Heuristic MIP models for Nesting Problems

Using the no-fit polygon How to express the non-overlapping condition between two pieces i and j?  

 vj −vi = 

xj yj

−

 xi

 6∈ Uij ⇐⇒ vj −vi ∈ U ij ,

yi

∀ i, j ∈ P : i < j

Partition the non-convex region U ij into a collection of mij disjoint k

polyhedra U ij called slices. y −y j

i

__5 Uij

__6 Uij

__4 Uij

__7 Uij

__8 Uij

O

__3 Uij

Uij __9 Uij

xj−x i

__2 Uij

__1 Uij

Each slice can be represented through a set of linear constraints of the form: k

U ij = {u ∈ IR2 : Akij · u ≤ bkij }

Slide 7

Exact and Heuristic MIP models for Nesting Problems

The MIP model A variant of a model by Daniels, Li, and Milenkovic (1994)

Variables • vi = (xi , yi ) : coordinates of the reference point of piece i • length : rightmost used margin of the container   1 if v − v ∈ U k j i ij k • zij = ∀ i, j ∈ P : i < j, k = 1 . . . mij  0 otherwise

Model

min

length + ε

X

(xi + yi )

i∈P

s. t.

lef ti ≤ xi ≤ length − righti



bottomi ≤ yi ≤ maxY − topi

∀i∈P

k Akij (vj − vi ) ≤ bkij + M (1 − zij )·1

∀ i, j ∈ P : i < j, k = 1 . . . mij mij X

k zij =1

k=1 k zij ∈

∀ i, j ∈ P : i < j

{0, 1}

∀ i, j ∈ P : i < j, k = 1 . . . mij

Slide 8

Exact and Heuristic MIP models for Nesting Problems

Constraint coefficient lifting Issue: the use of big-M coefficients makes the LP relaxation of the model quite poor kf kf kf k αij (xj − xi ) + βij (yj − yi ) ≤ γij + M (1 − zij )

∀ f = 1 . . . tkij

Replace the big-M coefficient by: kf h δij :=

max

h

(vj −vi ) ∈ U ij ∩B

kf kf αij (xj − xi ) + βij (yj − yi )

so as to obtain (easily computable) lifted constraints of the form: kf αij (xj

− xi ) +

kf βij (yj

− yi ) ≤

mij X

kf h h δij zij

h=1 y −y j

i

h

2 * maxY

Uij

O

xj−x i

Uij

k

Uij

2 * maxX

Slide 9

Exact and Heuristic MIP models for Nesting Problems

Some computational results

4

4

5

4

6

5

3

6 3

5 7

8 7

3

9

1

1

2

INSTANCE Glass1 Glass2 Glass3

2

PIECES

INT

PRIOR

5

73

7 9

173 302

1

2

NODES

TIME

no

470

0.26”

0%

yes

111

0.11”

0%

no

100,000

97.40”

32.08%

yes

11,414

13.29”

0%

no

100,000

157.76”

59.82%

yes

100,000

203.48”

58.70%

PRIOR yes/no refers to the use of a specific branching strategy based on "clique priorities" Solved with ILOG-CPLEX 7.0 on a PC AMD Athlon/1.2 GHz "Not usable in practice for real-world problems"

Slide 10

GAP

Exact and Heuristic MIP models for Nesting Problems

Multiple Containment Problem An important subproblem: after having placed the "big pieces", find the best placement of the remaining "small pieces" by using the holes left by the big ones.

A greedy approach for placing the small pieces can produce poor results Aim: Define an approximate MIP for guiding the placement of the small pieces Idea: Small pieces can be approximated well by rectangles Input • Set P of n small pieces • Set H of m irregular polygons representing the available holes

Slide 11

Exact and Heuristic MIP models for Nesting Problems

Geometrical considerations • rectangular approximation of the small pieces • original holes and usable holes

• placement grid within each hole.





  



 

       Slide 12



 

Exact and Heuristic MIP models for Nesting Problems

An approximate multiple-containment MIP model

min

X

h

(holeAreah · U −

+ ε s. t.

hp pieceAreap · Zrc )

r=1 c=1 p∈P

h∈H

X

Rh X Ch X X

Rh X Ch XX

h h (Xrc + Yrc )

h∈H r=1 c=1 hp Zrc ≤ Uh

∀ h ∈ H, r = 1 . . . Rh , c = 1 . . . Ch

p∈P

X

pieceAreap

p∈P h Xrc +

X

Rh X Ch X

hp ≤ holeAreah Zrc

∀h∈H

r=1 c=1 hp lengthp Zrc ≤ Xr, c+1

p∈P

h + Xrc

X

∀ h ∈ H, r = 1 . . . Rh , c = 1 . . . Ch − 1 hp ≤ rowEndhr lengthp Zrc

p∈P

∀ h ∈ H, r = 1 . . . Rh , c = Ch X

lengthp

p∈P h Yrc +

X

Ch X

hp Zrc ≤ rowLengthhr

∀ h ∈ H, r = 1 . . . Rh

c=1 hp widthp Zrc ≤ Yr+1, c

p∈P

h + Yrc

X

∀ h ∈ H, r = 1 . . . Rh − 1, c = 1 . . . Ch hp ≤ colEndhc widthp Zrc

p∈P

∀ h ∈ H, r = Rh , c = 1 . . . Ch X p∈P

widthp

Rh X

hp Zrc ≤ colW idthhc

r=1

Slide 13

∀ h ∈ H, c = 1 . . . Ch

Exact and Heuristic MIP models for Nesting Problems

Bounds on the variables

h max(rowStarthr , origXh + (c − 1) · cellLengthh ) ≤ Xrc

≤ min(rowEndhr , origXh + (c) · cellLengthh ) ∀ h ∈ H, r = 1 . . . Rh , c = 1 . . . Ch h max(colStarthc , origYh + (r − 1) · cellW idthh ) ≤ Yrc

≤ min(colEndhc , origYh + r · cellW idthh ) ∀ h ∈ H, r = 1 . . . Rh , c = 1 . . . Ch U h ∈ {0, 1}

∀h∈H

hp Zrc ∈ {0, 1}

∀ h ∈ H, p ∈ P, r = 1 . . . Rh , c = 1 . . . Ch

Remark 1: Solvable in short computing time

Remark 2: To be followed by a greedy post-processing procedure for fixing possible overlaps Remark 3: Sequential approach: big pieces placed first, "special pieces" of intermediate size/difficulty second, and "trims" last.

Slide 14

Exact and Heuristic MIP models for Nesting Problems

Example: Smart vs. greedy placement of "special pieces"

Pieces: 34/76

Length: 1643.53

Pieces: 30/76

Length: 1634.55

Slide 15

Eff.: 83.97%

Eff.: 81.54%

Exact and Heuristic MIP models for Nesting Problems

Example (cont’d): Smart vs. greedy placement of "trims"

Pieces: 44/76

Pieces: 42/76

Length: 1665.50

Length: 1660.87

Slide 16

Eff.: 86.13%

Eff.: 85.67%

Exact and Heuristic MIP models for Nesting Problems

Pieces: 44/50

Pieces: 42/50

Length: 3840.28

Efficiency: 82.12 %

Length: 3838.27

Efficiency: 81.57 %

Pieces: 44/54

Length: 4697.05

Efficiency: 83.74 %

Pieces: 39/54

Length: 4671.81

Efficiency: 83.58 %

Slide 17

Exact and Heuristic MIP models for Nesting Problems

Preliminary Computational Results INSTANCE

PIECES

TRIMS

LENGTH

EFFIC.

smart

34/76

14

1643.53

83.97%

greedy

30/76

10

1634.55

81.54%

smart

44/76

10

1665.50

86.13%

greedy

42/76

8

1660.87

85.67%

smart

44/50

10

3840.28

82.12%

greedy

42/50

8

3838.27

81.57%

smart

44/54

22

4697.05

83.74%

greedy

39/54

17

4671.81

83.58%

82 - group 1

82 - group 2

101

385

Slide 18

Exact and Heuristic MIP Models for Nesting Problems

Exact and Heuristic MIP models for Nesting Problems. Exact and Heuristic ... The no-fit polygon between two polygons A and B is defined as. UAB := A ⊕ (−B).

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