Mathematical programming formulations forthe orthogonal 2d knapsack problem:

a survey

Cédric Joncour, Arnaud Pêcher, Pierre Pesneau, François Vanderbeck

Université Bordeaux 1, Institut de Math (IMB)

&

INRIA Bordeaux Sud Ouest

2d knapsack problem formulations – p. 1/28

Outline

1. Problem

2. FormulationsExact vs relaxedcompact vs pseudo-polynomial vs exponential

3. Bounds comparison

2d knapsack problem formulations – p. 2/28

Outline

1. Problem2. Formulations

Exact vs relaxedcompact vs pseudo-polynomial vs exponential

3. Bounds comparison

2d knapsack problem formulations – p. 3/28

Definition for packing problem

Consider

Items i ∈ I = {1, . . . , n},

∀i : wi (width), hi (height), pi (profit),

a bin of size W × H.

Problem:Find a selection of items that:

maximize profit fit

into the bin (feasible placement).

2d knapsack problem formulations – p. 4/28

Example

5

3

4

0

2

1

0

2

0

2

3

0

x

with W=H=5

y

1

5

1

3

4

2

2

1

X

Y

2d knapsack problem formulations – p. 5/28

Outline

1. Problem

2. FormulationsExact vs relaxedcompact vs pseudo-polynomial vs exponential

3. Bounds comparison

2d knapsack problem formulations – p. 6/28

Beasley (1985)

max∑

i∈I

piδi

s. t.∑

i∈I

wihiδi ≤ WH

δi ∈ {0, 1} ∀ i ∈ I

Relaxed and compact

2d knapsack problem formulations – p. 7/28

Fekete and Schepers (1997)

Definition: Dual Feasible FunctionA function u : [0,W ] → [0, 1] is called dual feasible, if

∀S ⊆ I :∑

i∈S

wi ≤ W ⇒∑

i∈S

uw(wi) ≤ 1

max∑

i∈I

piδi

s. t.∑

i∈I

uw(wi′)uh(hi′)δi ≤ 1

δi ∈ {0, 1} ∀ i ∈ I

Relaxed and compact

2d knapsack problem formulations – p. 8/28

Beasley (1985)

Let

Xi = {0, . . . ,W − wi} and Yi = {0, . . . ,H − hi} ∀i ∈ I,

δi x y =

1 if item i has bottom-leftcorner in position (x, y)

0 otherwise

∀i ∈ I, (x, y) ∈ Xi × Yi.

max∑

i∈I

∑

(x,y)∈(Xi,Yi)

piδi x y

s. t.∑

i∈I

x∑

ν=x−wi+1

y∑

τ=y−hi+1

δi ν τ ≤ 1 ∀ (x, y) ∈ (X,Y )

∑

(x,y)∈(Xi,Yi)

δi x y ≤ 1 ∀ i ∈ I

δi x y ∈ {0, 1} ∀ i ∈ I, (x, y) ∈ (Xi, Yi)

Exact and pseudo-polynomial2d knapsack problem formulations – p. 9/28

Christofides and Hadjiconstantinou (1995)

Let

δi x =

1 if item i has lower cornerat x-coordinate x

0 otherwise

∀i ∈ I, x ∈ Xi,

δi y =

1 if item i has lower cornerat y-coordinate y

0 otherwise

∀i ∈ I, y ∈ Yi,

δx y =

{

1 if position (x, y) is free0 otherwise

∀(x, y) ∈ X × Y.

2d knapsack problem formulations – p. 10/28

Christofides and Hadjiconstantinou (1995)

max∑

i∈I

∑

x∈Xi

piδi x

s. t.∑

x∈Xi

δi x =∑

y∈Yi

δi y ≤ 1 ∀ i ∈ I

∑

i∈I

x∑

ν=x−wi+1ν∈Xi

hiδi ν +∑

y∈Y

δx y = H ∀x ∈ X

∑

i∈I

y∑

τ=y−hi+1τ∈Yi

wiδi τ +∑

x∈X

δx y = W ∀ y ∈ Y

x+wi−1∑

ν=x

y+hi−1∑

τ=y

δν τ ≤ wihi(2 − δi x − δi y) ∀ i ∈ I, (x, y) ∈ (Xi, Yi)

δi x, δi y ∈ {0, 1} ∀ i ∈ I, (x, y) ∈ (Xi, Yi)

δx y ∈ {0, 1} ∀ (x, y) ∈ (X,Y )

2d knapsack problem formulations – p. 11/28

Only a relaxation: Amaral et Letchford(2002)

��������������������������������������������������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������������������������������������������������

Relaxed and pseudo-polynomial

2d knapsack problem formulations – p. 12/28

Pisinger and Sigurd (2001)

Let

δi =

{

1 if item i is selected0 otherwise

∀i ∈ I,

lij =

{

1 if item i is on the left of item j

0 otherwise∀i, j (i 6= j) ∈ I,

bij =

{

1 if item i is below item j

0 otherwise∀i, j (i 6= j) ∈ I,

xi ≥ 0 indicating the x-coordinate of item i ∀i ∈ I,

yi ≥ 0 indicating the y-coordinate of item i ∀i ∈ I.

2d knapsack problem formulations – p. 13/28

Pisinger and Sigurd (2001)

max∑

i∈I

piδi

s. t. li j + lj i + bi j + bj i ≥ 1 − (1 − δi) − (1 − δj) ∀ i, j ∈ I

xi + wi ≤ xj + W (1 − li j) ∀ i, j ∈ I

yi + hi ≤ yj + H(1 − bi j) ∀ i, j ∈ I

0 ≤ xi ≤ W − wi ∀ i ∈ I

0 ≤ yi ≤ H − hi ∀ i ∈ I

δi, li j , bi j ∈ {0, 1} ∀ i, j ∈ I

Exact and compact2d knapsack problem formulations – p. 14/28

Tsai, Malstorm and Meeks (1988)

Let

λv ∈ N number of vertical pattern v ∀v ∈ V .

δi =

{

1 if item i is selected0 otherwise

∀i ∈ I.

ii

i

i

Y

X

horizontal preemption allowed2d knapsack problem formulations – p. 15/28

Tsai, Malstorm and Meeks (1988)

Master:

max∑

i∈I

piδi

s. t.∑

v∈V

yvi λv = wiδi ∀ i ∈ I

∑

v∈V

λv ≤ W

λv ∈ N ∀ v ∈ V

δi ∈ {0, 1} ∀ i ∈ I

Subproblem:

V = {y ∈ {0, 1}n :∑

i∈I

hiyi ≤ H}

Relaxed and exponential

2d knapsack problem formulations – p. 16/28

Only a relaxation

33

1

1

1

2

32

X

Y

2d knapsack problem formulations – p. 17/28

Scheithauer (1999)

Let

λv ∈ N number of vertical pattern v ∀v ∈ V ,

µh ∈ N number of horizontal pattern h ∀h ∈ H,

δi =

{

1 if item i is selected0 otherwise

∀i ∈ I.

Subproblem:

V = {y ∈ {0, 1}n :∑

i∈I

hiyi ≤ H}

H = {x ∈ {0, 1}n :∑

i∈I

wixi ≤ W}

2d knapsack problem formulations – p. 18/28

Scheithauer (1999)

Master:

max∑

i∈I

piδi

s. t.∑

v∈V

yvi λv = wiδi ∀ i ∈ I

∑

h∈H

xhi µh = hiδi ∀ i ∈ I

∑

v∈V

λv ≤ W

∑

h∈H

µh ≤ H

λv ∈ N ∀ v ∈ V

µh ∈ N ∀h ∈ H

δi ∈ {0, 1} ∀ i ∈ I

Relaxed and exponential

2d knapsack problem formulations – p. 19/28

Only a relaxation

2

2 2

2

5

5

5

1

2

3 4

1 1

3

3

3

4

4

4

X

Y

X

Y

2d knapsack problem formulations – p. 20/28

Boschetti, Hadjiconstantinou andMingozzi (2002)

Let

λv x = 1 iff vertical pattern v is selected in position x,

µh y = 1 iff horizontal pattern h is selected in position y,

δi = 1 iff item i is selected,

δi x = 1 iff item i has lower corner at x-coordinate x,

δi y = 1 iff item i has lower corner at y-coordinate y,

xij = 1 iff item i is on the side of j,

yij = 1 iff item i is above or under j.

Subproblem:

sets V and H do not change

2d knapsack problem formulations – p. 21/28

Boschetti, Hadjiconstantinou andMingozzi (2002)Master:

max∑

i∈I

piδi

s. t. δi =∑

x∈X

δi x =∑

y∈Y

δi y ∀ i ∈ I

∑

v∈V

∑

x∈X

yvi λv x = wiδi ∀ i ∈ I

∑

h∈H

∑

y∈Y

xhi µh y = hiδi ∀ i ∈ I

∑

v∈V

λv x ≤ 1 ∀x ∈ X

∑

h∈H

µh y ≤ 1 ∀ y ∈ Y

2d knapsack problem formulations – p. 22/28

Boschetti, Hadjiconstantinou andMingozzi (2002)

∑

v∈V

yvi λv x =

x∑

ν=x−wi+1

δi ν ∀ i ∈ I, x ∈ X

∑

h∈H

xhi µh y =

y∑

τ=y−hi+1

δi τ ∀ i ∈ I, y ∈ Y

∑

h∈H

xhi xh

j µh y ≤ xi j ∀ i, j(i 6= j) ∈ I, y ∈ Y

∑

v∈V

yvi yv

j λv x ≤ yi j ∀ i, j(i 6= j) ∈ I, x ∈ X

xi j + yi j ≤ 1 ∀ i, j ∈ I

λv x ∈ {0, 1} ∀ v ∈ V , x ∈ X

µh y ∈ {0, 1} ∀h ∈ H, y ∈ Y

xi j , yi j , δi, δi x, δi y ∈ {0, 1} ∀ i, j ∈ I, x ∈ X, y ∈ Y

Exact and exponential2d knapsack problem formulations – p. 23/28

Outline

1. Problem

2. FormulationsExact vs relaxedcompact vs pseudo-polynomial vs exponential

3. Bounds comparison

2d knapsack problem formulations – p. 24/28

Dominance study

KNPsurf : Beasley bound. (knapsack on item surfaces),

λv: optimal solution of Tsai, Malstorm and Meeks (columngeneration of a relaxed problem),

λv, µh: optimal solution of Scheithauer (column generation of arelaxed problem),

opt: optimal solution.

KNPsurf ≥ λv ≥ λv, µh ≥ opt

2d knapsack problem formulations – p. 25/28

Numerical test: optimality gap (%)

approach KNP 1d pseudo-poly compact Column gen

δi ∈ {0, 1} δi . ∈ [0, 1] δi ∈ {0, 1} δi ∈ {0, 1}

formulation KNPsurf KNPDFF δi x y δi x, δi y li j , bi j λv λv , µh

ngcut1 22,56 22,56 25,51 26,12 102 0,00 0,00

ngcut2 10,00 10,00 11,96 11,96 153 0,45 0,00

ngcut3 7,69 2,83 3,44 5,41 128 2,43 0,81

ngcut4 2,61 0,00 1,12 1,12 13 0,00 0,00

ngcut5 4,19 0,00 0,38 0,38 72 3,35 0,00

ngcut6 9,69 9,69 9,69 9,69 77 0,00 0,00

ngcut8 12,47 0,00 8,63 8,63 69 10,21 3,36

ngcut9 4,11 0,00 1,19 2,35 108 3,03 0,00

ngcut10 4,48 0,00 3,42 4,61 81 0,00 0,00

ngcut11 10,43 2,25 5,27 5,42 72 8,00 0,00

ngcut12 7,88 2,57 4,32 4,82 134 0,54 0,00

moyen 8,74 4,54 6,81 7,32 92 2,55 0,38

2d knapsack problem formulations – p. 26/28

Conclusion

Summary:

Various formulations: in terms ofposition indicator, relative position, feasible sets.

advantages and drawbacks.

we should emphasize symmetry drawback.

Future research:

complet picture of theoritical dominance relationship.

test Boschetti et al. formulation with quadratic subproblems.

new column generation formulation .

2d knapsack problem formulations – p. 27/28

Bibliography

A. Amaral and A. N. Letchford (2002). Comment on "An exact algorithm for general,orthogonal, two-dimensional knpasack problems". Working paper.

J.E. Beasley (1985). An exact two-dimensional non-guillotine cutting tree searchprocedure. Operations Research, 33: 49-64.

M.A. Boschetti, E. Hadjiconstantinou and A. Mingozzi (2002). New upper bounds forthe two-dimensional non-guillotine cutting stock problem. IMA J Man. Math., 13: 95-119.

A. Caprara and M. Monaci (2005). Bidimensional packing by bilinear programming.Lect. Notes in Comp. Sciences, 377-391.

N. Christofides and E. Hadjiconstantinou (1995). An exact algorithm for orthogonal 2dcutting problems using guillotine cuts. European Journal of Operational Research, 83: 21-38.

S.P. Fekete and J. Schepers (2004). A general framework for bounds forhigher-dimensional orthogonal packing problems. Math. Methods of Oper. Res., 60: 81-94.

D. Pisinger and M. Sigurd (2007). Using decomposition techniques and constraintprogramming for solving the two-dimensional bin packing problem. Informs J. on Comp.,19(1): 36-51.

G. Scheithauer (1999). LP-based bounds for the container and multi-container loadingproblem. International Transactions in Operational Research, 6: 199-213.

R.D. Tsai, E.M. Malstorm and H.D. Meeks (1988). A two-dimensional palletizingprocedure for loading operations. IIE Transactions, 20:418-425.

2d knapsack problem formulations – p. 28/28