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### Transcript of Mathematical programming formulations for the orthogonal ... Mathematical programming formulations...

• Mathematical programming formulations for the 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. Formulations Exact vs relaxed compact vs pseudo-polynomial vs exponential

3. Bounds comparison

2d knapsack problem formulations – p. 2/28

• Outline

1. Problem 2. Formulations

Exact vs relaxed compact 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. Formulations Exact vs relaxed compact 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 Function A 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-left corner 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-polynomial 2d knapsack problem formulations – p. 9/28

Let

δi x =

1 if item i has lower corner at x-coordinate x

0 otherwise

∀i ∈ I, x ∈ Xi,

δi y =

1 if item i has lower corner at y-coordinate y

0 otherwise

∀i ∈ I, y ∈ Yi,

δx y =

{

1 if position (x, y) is free 0 otherwise

∀(x, y) ∈ X × Y.

2d knapsack problem formulations – p. 10/28

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+1 ν∈Xi

hiδi ν + ∑

y∈Y

δx y = H ∀x ∈ X

i∈I

y ∑

τ=y−hi+1 τ∈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)

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Relaxed and pseudo-polynomial

2d knapsack problem formulations – p. 12/28

• Pisinger and Sigurd (2001)

Let

δi =

{

1 if item i is selected 0 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 compact 2d 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 selected 0 otherwise

∀i ∈ I.

ii

i

i

Y

X

horizontal preemption allowed 2d 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

3 2

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 selected 0 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 and Mingozzi (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