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  • 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

  • Christofides and Hadjiconstantinou (1995)

    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

  • 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+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