2D Bin Packing Problem with Side Constraints

The Two Dimensional Bin Packing Problem with additional Layer Constraints Markus Seizinger Andreas Fügener Jens O. Brunner 09.02.2017 DTU

Lehrstuhl für Health Care Operations / Health Information Management

Motivation: Paint shop as bottleneck resource Problem description • Items are set on racks • Each rack has two faces (front and rear) • Two different types of items • Different types may not face each other (“layer constraint”) Target • Minimize needed racks

Problem Type: Special 2D Bin Packing Problem (2DBPP) Markus Seizinger

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1. Problem Description

Example: Layer Constraint

Front

Rear

Rack Markus Seizinger

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Front

Rear


3



Front

Rear


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Front

Rear


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Agenda 1. Problem Description 2. Bounds 3. Column Generation Algorithm 1. 2. 3.

Master Problem Restricted Master Problem Subproblem Decomposition

4. Computational Study 5. Conclusion and Future Agenda

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Problem Description: 2DBPPLC Items and bins • Set of items 𝑖 ∈ 𝐼 and of item types 𝑡 ∈ 𝑇 • Each item has width 𝑤𝑖 , height ℎ𝑖 , and type 𝑡𝑖 • Every bin is equal with width 𝑊, height 𝐻, and 𝑆 number of layers • No rotation of items allowed Problem task • Assign every item to one bin and assign a concrete position (𝑥|𝑦|𝑧) • Items may not overlap each other or the edges of the bin • Items of different types may not be assigned face to face on the same bin • Objective: Minimize the number of required bins

2DBPP with Layer Constraints (2DBPPLC) Markus Seizinger

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Complexity: 2DBPPLC may be reduced to 2DBPP and 3DBPP

Reduction to 2DBPP

Reduction to 3DBPP

All bins have one layer only, i.e., 𝑆 = 1

Only one type of items exists, i.e., 𝑇 = 1

 Regular 2DBPP (strongly NP-hard)

 3DBPP, where all items have a size of one at the third dimension  Decomposes into 2DBPP (strongly NP-hard)

2DBPPLC is strongly NP-hard Markus Seizinger

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Approaches exist for 2DBPP and 3DBPP Martello & Vigo, 1998, Management Science • Lower Bounds (2DBPP) • Definition of instance classes Martello et al., 2000, Operations Research • Lower Bounds (3DBPP) Lodi et al., 2002a/b, EJOR/Discrete Applied Mathematics • Heuristics (2DBPP) Pisinger & Sigurd, 2007, INFORMS Journal on Computing • CG-based algorithm (2DBPP) • Pre-stage of subproblem with 1DBPP • Updating lower bound with RMP • Valid inequalities Markus Seizinger

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2. Bounds

Upper and lower bounds from literature may be adapted Upper Bounds/Heuristics

Lower Bounds/Relaxations

Adaptation of Extreme Point (EP) packing heuristic • Packs all items as greedy algorithm • EP-selection based on best fit • EP-generation had to be adapted to layer constraints

Adaptation of analytical bounds from literature • L0: Geometric bound • L1: Bound of large items • L2: Hybrid of L0 and L1

• Also used during column generation to pack a single bin

Development of new model-based lower bound 𝑳𝑳𝑪 𝟏𝑫𝑩𝑷 based on 1D-BPP • NP-hard, yet solvable for realistic instances • Dominates all other bounds

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2. Bounds

Greedy Extreme-Point based Packing Heuristic 𝜋𝑖 𝑖 ℎ𝑖

•

Order items by non-increasing relative value 𝑤

•

Place every item in the best feasible extreme point – –

Extreme points are generated by projecting the upper-right corner of item (w.r.t. type) Best fit rule maximizing type-overlay opposite side

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2. Bounds


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2. Bounds


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2. Bounds

Lower Bounds 𝑳𝟎 - Geometric bound

σ𝑖∈𝐼 𝑤𝑖 ℎ𝑖 𝐿0 = 𝑊⋅𝐻⋅𝑆

𝑳𝟏 - Bound of large items

𝐼

𝑙𝑎𝑟𝑔𝑒

𝑊 𝐻 (𝜏) = {𝑖 ∈ 𝐼: 𝑤𝑖 > ∧ ℎ𝑖 > ∧ 𝜏𝑖 = 𝜏} 2 2

𝐿𝐿𝐶 1

𝐼𝑙𝑎𝑟𝑔𝑒 𝜏 =෍ 𝑆 𝜏∈Τ

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2. Bounds

Lower Bounds 𝑳𝟐 - Bound of geometry and large items (dominates both 𝐿0 and 𝐿1 )

𝐼1 (𝑝, 𝑞) = {𝑖 ∈ 𝐼: 𝑤𝑖 ≥ 𝑊 − 𝑝 ∧ ℎ𝑖 ≥ 𝐻 − 𝑞} 𝑊 𝐻 𝐼2 𝑝, 𝑞 = {𝑖 ∈ 𝐼 \ 𝐼1 𝑝, 𝑞 : 𝑤𝑖 > ∧ ℎ𝑖 > } 2 2 𝐼3 𝑝, 𝑞 = {𝑖 ∈ 𝐼 \ 𝐼1 𝑝, 𝑞 ∪ 𝐼2 𝑝, 𝑞

𝐿𝐿𝐶 2

=

𝐿𝐿𝐶 1

+

max

𝑊 𝐻 1≤𝑝≤ 2 ; 1≤𝑞≤ 2

0,

: 𝑤𝑖 ≥ 𝑝 ∧ ℎ𝑖 ≥ 𝑞}

σ𝑖∈𝐼2(𝑝,𝑞)∪𝐼3(𝑝,𝑞) 𝑤𝑖 ℎ𝑖 − 𝑆 ⋅ 𝐿1 − 𝐼1 𝑝, 𝑞

⋅𝑊⋅𝐻

𝑊⋅𝐻⋅𝑆

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2. Bounds

Lower Bounds 𝑳𝟐 - Bound of geometry and large items (dominates both 𝐿0 and 𝐿1 )

𝐼1 (𝑝, 𝑞) = {𝑖 ∈ 𝐼: 𝑤𝑖 ≥ 𝑊 − 𝑝 ∧ ℎ𝑖 ≥ 𝐻 − 𝑞} 𝑊 𝐻 𝐼2 𝑝, 𝑞 = {𝑖 ∈ 𝐼 \ 𝐼1 𝑝, 𝑞 : 𝑤𝑖 > ∧ ℎ𝑖 > } 2 2 𝐼3 𝑝, 𝑞 = {𝑖 ∈ 𝐼 \ 𝐼1 𝑝, 𝑞 ∪ 𝐼2 𝑝, 𝑞

: 𝑤𝑖 ≥ 𝑝 ∧ ℎ𝑖 ≥ 𝑞}

Geometric 𝐿𝐿𝐶 2

=

𝐿𝐿𝐶 1

+

max

𝑊 𝐻 1≤𝑝≤ 2 ; 1≤𝑞≤ 2

0,

Large Items

σ𝑖∈𝐼2(𝑝,𝑞)∪𝐼3(𝑝,𝑞) 𝑤𝑖 ℎ𝑖 − 𝑆 ⋅ 𝐿1 − 𝐼1 𝑝, 𝑞

⋅𝑊⋅𝐻

𝑊⋅𝐻⋅𝑆

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4. 5. 6. 7.


Column Generation Computational Study Literature Conclusion and Future Agenda

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3. Column Generation Algorithm

Solution Approach Master Problem (MP): Set Covering • Requires all feasible packings of items in bins • Choose packings to cover all items • Minimize number of bins

Decomposition and Column Generation • Restricted Master Problem (RMP): Set Covering, as MP, but: – Linearized variables – Only subset of columns Subproblem (SP): 2D Packing Problem with layer constraints • Create attractive packings (see Pisinger & Sigurd, 2007) • Iteratively improves solution of RMP

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Linearized Restricted Master Problem (RMP) Number of bins with packing 𝑝

(I)

𝑧 𝑅𝑀𝑃 = min ෍ 𝜆𝑝 𝑝∈𝑃

1, if item 𝑖 is included in packing 𝑝

s.t.:

Set of Items

∀𝑖 ∈𝐼

෍ 𝑎𝑝𝑖 𝜆𝑝 ≥ 1 𝑝∈𝑃

(II) Set of Packings

∀𝑝 ∈𝑃

𝜆𝑝 ≥ 0

(III)

Note: 𝑃 is a subset of all possible packings Markus Seizinger

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Notation: 2D Packing Problem with layer CTs (2DPPLC) Sets, Indices, and Parameters 𝑖∈𝐼 Set of items

𝑡∈𝑇

Set of item types

𝑤𝑖 ℎ𝑖 𝑡𝑖 𝜋𝑖

Width of item 𝑖 Height of item 𝑖 Type of item 𝑖 Dual value of item 𝑖

Decision Variables 𝑥𝑖 x-coordinate of item 𝑖

𝑙𝑖𝑗

1, iff item 𝑖 is placed left of 𝑗

𝑦𝑖

y-coordinate of item 𝑖

𝑏𝑖𝑗

1, iff item 𝑖 is placed below 𝑗

𝑠𝑖

layer of item 𝑖

𝑓𝑖𝑗

1, iff item 𝑖 is placed in front of 𝑗

𝑎𝑖

1, iff item 𝑖 included in bin

𝑊 𝐻 𝑆

Width of bin Height of bin Number of layers

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Example: 2D Packing Problem with layer CTs (2DPPLC) 𝑥1 = 0; 𝑥2 = 4; 𝑦1 = 0; 𝑦2 = 5; 𝑠1 = 0; 𝑠2 = 0;

𝑙12 = 0; 𝑙21 = 0; 𝑏12 = 1; 𝑏21 = 0; 𝑓12 = 0; 𝑓21 = 0;

1

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Example: 2D Packing Problem with layer CTs (2DPPLC) 𝑥1 = 0; 𝑥2 = 4; 𝑦1 = 0; 𝑦2 = 5; 𝑠1 = 0; 𝑠2 = 0;

𝑙12 = 0; 𝑙21 = 0; 𝑏12 = 1; 𝑏21 = 0; 𝑓12 = 0; 𝑓21 = 0;

2

1

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Example: 2D Packing Problem with layer CTs (2DPPLC) 𝑥1 = 0; 𝑥32 = 6; 4; 𝑦1 = 0; 𝑦32 = 5; 𝑠1 = 0; 𝑠32 = 0; 1;

𝑙13 0; 𝑙23 12 = 1; 21 = 0; 𝑏13 12 = 1; 𝑏23 21 = 0; 𝑓13 0; 𝑓23 12 = 1; 21 = 0; 3

1

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Model: 2D Packing Problem with layer CTs (2DPPLC) 𝑧 2𝐷𝑃𝑃

=

min 1 − ෍ 𝜋𝑖 𝑎𝑖 ⇔ max ෍ 𝜋𝑖 𝑎𝑖 𝑖∈𝐼

s.t.: 𝑓𝑖𝑗 + 𝑓𝑗𝑖 + 𝑙𝑖𝑗Items + 𝑙𝑗𝑖 + 𝑏𝑖𝑗 +not 𝑏𝑗𝑖 overlap + 1 − 𝑎𝑖 + 1 − 𝑎𝑗 ≥ may

(I)

𝑖∈𝐼

1

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∀ 𝑖, 𝑗 ∈ 𝐼: Items may not 𝑙𝑖𝑗 + 𝑙𝑗𝑖 + 𝑏𝑖𝑗 + 𝑏𝑗𝑖 overlap + 1 − 𝑎𝑖(items + 1 −of𝑎𝑗different ≥ 1 type) 𝑖 < 𝑗 ∧ 𝑡𝑖 ≠ 𝑡𝑗

(II) (III)

𝑥-coordinate

𝑥𝑖 − 𝑥𝑗 + 𝑊𝑙𝑖𝑗

≤

𝑊 − 𝑤𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(IV)

𝑦-coordinate

𝑦𝑖 − 𝑦𝑗 + 𝐻𝑏𝑖𝑗

≤

𝐻 − ℎ𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(V)

𝑧-coordinate

𝑠𝑖 − 𝑠𝑗 + S𝑓𝑖𝑗

≤

S−1

∀ 𝑖, 𝑗 ∈ 𝐼

(VI)

Item may not exceed bin (width) 𝑥𝑖 ≤

𝑊 − 𝑤𝑖

∀𝑖 ∈𝐼

(VII)

𝑦𝑖 ≤ Item may not exceed bin (height)

𝐻 − ℎ𝑖

∀𝑖 ∈𝐼

(VIII)

Item may not exceed bin (layers) 𝑠𝑖 ≤

𝐷−1

∀𝑖 ∈𝐼

(IX)

0

∀𝑖 ∈𝐼

(X)

∀ 𝑖, 𝑗 ∈ 𝐼

(XI)

𝑥𝑖 , 𝑦𝑖 , 𝑠𝑖 𝑎𝑖 , 𝑙𝑖𝑗 , 𝑏𝑖𝑗 , 𝑓𝑖𝑗

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𝑏𝑖𝑛𝑎𝑟𝑦

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=


s.t.: 𝑓𝑖𝑗 + 𝑓𝑗𝑖 + 𝑙𝑖𝑗 + 𝑙𝑗𝑖 + 𝑏𝑖𝑗 + 𝑏𝑗𝑖 + 1 − 𝑎𝑖 + 1 − 𝑎𝑗 ≥

(I)

𝑖∈𝐼

1

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∀ 𝑖, 𝑗 ∈ 𝐼: Items may not 𝑙𝑖𝑗 + 𝑙𝑗𝑖 + 𝑏𝑖𝑗 + 𝑏𝑗𝑖 overlap + 1 − 𝑎𝑖(items + 1 −of𝑎𝑗different ≥ 1 type) 𝑖 < 𝑗 ∧ 𝑡𝑖 ≠ 𝑡𝑗

(II) (III)

𝑥-coordinate


≤

𝑊 − 𝑤𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(IV)

𝑦-coordinate


≤

𝐻 − ℎ𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(V)

𝑧-coordinate


≤

S−1

∀ 𝑖, 𝑗 ∈ 𝐼

(VI)


𝑊 − 𝑤𝑖

∀𝑖 ∈𝐼

(VII)


𝐻 − ℎ𝑖

∀𝑖 ∈𝐼

(VIII)


𝐷−1

∀𝑖 ∈𝐼

(IX)

0

∀𝑖 ∈𝐼

(X)

∀ 𝑖, 𝑗 ∈ 𝐼

(XI)


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=



1

𝑙𝑖𝑗 + 𝑙𝑗𝑖 + 𝑏𝑖𝑗 + 𝑏𝑗𝑖 + 1 − 𝑎𝑖 + 1 − 𝑎𝑗 ≥

1

(I)

𝑖∈𝐼

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∧ 𝑡𝑖 ≠ 𝑡𝑗

(II) (III)

𝑥-coordinate


≤

𝑊 − 𝑤𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(IV)

𝑦-coordinate


≤

𝐻 − ℎ𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(V)

𝑧-coordinate


≤

S−1

∀ 𝑖, 𝑗 ∈ 𝐼

(VI)


𝑊 − 𝑤𝑖

∀𝑖 ∈𝐼

(VII)


𝐻 − ℎ𝑖

∀𝑖 ∈𝐼

(VIII)


𝐷−1

∀𝑖 ∈𝐼

(IX)

0

∀𝑖 ∈𝐼

(X)

∀ 𝑖, 𝑗 ∈ 𝐼

(XI)


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=



1


1

(I)

𝑖∈𝐼

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∧ 𝑡𝑖 ≠ 𝑡𝑗

(II) (III)


≤

𝑊 − 𝑤𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(IV)


≤

𝐻 − ℎ𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(V)


≤

S−1

∀ 𝑖, 𝑗 ∈ 𝐼

(VI)


𝑊 − 𝑤𝑖

∀𝑖 ∈𝐼

(VII)


𝐻 − ℎ𝑖

∀𝑖 ∈𝐼

(VIII)


𝐷−1

∀𝑖 ∈𝐼

(IX)

0

∀𝑖 ∈𝐼

(X)

∀ 𝑖, 𝑗 ∈ 𝐼

(XI)


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≥


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=



1


1

(I)

𝑖∈𝐼

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∧ 𝑡𝑖 ≠ 𝑡𝑗

(II) (III)


≤

𝑊 − 𝑤𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(IV)


≤

𝐻 − ℎ𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(V)


≤

S−1

∀ 𝑖, 𝑗 ∈ 𝐼

(VI)

𝑥𝑖

≤

𝑊 − 𝑤𝑖

∀𝑖 ∈𝐼

(VII)

𝑦𝑖

≤

𝐻 − ℎ𝑖

∀𝑖 ∈𝐼

(VIII)

𝑠𝑖

≤

𝐷−1

∀𝑖 ∈𝐼

(IX)

𝑥𝑖 , 𝑦𝑖 , 𝑠𝑖

≥

0

∀𝑖 ∈𝐼

(X)

∀ 𝑖, 𝑗 ∈ 𝐼

(XI)

𝑎𝑖 , 𝑙𝑖𝑗 , 𝑏𝑖𝑗 , 𝑓𝑖𝑗

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3. Column Generation Algorithm RMP: Restricted Master Problem 2DPPLC: 2D Packing Problem w. layer CTs

Iteration scheme Dual values 𝜋

RMP

2DPPLC

New packing Markus Seizinger

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Problems: 2D Packing Problem with layer CTs (2DPPLC) Re-formulation

Selection Algorithm

Current problems • Number of decision variables rises exponentially to 𝐼 • Excessive runtime for realistic problem instances

1. Exclude items with non-positive duals

Alternate approach (Fekete et al. 2007) • Reduces symmetries • Finding special interval graphs • Tree search algorithm • But: no good results as well

3. Apply 1D Knapsack Problem (1DKPLC) as pre-stage • 1DKP with layer constraints • Solution time < 0.1s • Pre-selected items

2. Create start solution based on heuristics

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3. Column Generation Algorithm RMP: Restricted Master Problem 1DKPLC: 1D Knapsack Problem w. layer CTs 2DPPLC: 2D Packing Problem w. layer CTs


1DKPLC

RMP

Item subset

2DPPLC


21

3. Column Generation Algorithm RMP: Restricted Master Problem 1DKPLC: 1D Knapsack Problem w. layer CTs 2DPPLC: 2D Packing Problem w. layer CTs


1DKPLC

RMP

Forbid previous subset

Item subset

2DPPLC


21


Dominating items A dominated item 𝑗 can not be part of the optimal packing unless all of it’s dominating items 𝑖 are packed as well. Definition 𝑖≽𝑗⟺ 𝑤𝑖 ≤ 𝑤𝑗 ∧ ℎ𝑖 ≤ ℎ𝑗 ∧ 𝜋𝑖 ≥ 𝜋𝑗 ∧ 𝑡𝑖 = 𝑡𝑗 𝑖≻𝑗 ⟺ 𝑖 ≽ 𝑗 ∧ (𝑤𝑖 < 𝑤𝑗 ∨ ℎ𝑖 < ℎ𝑗 ∨ 𝜋𝑖 > 𝜋𝑗 ) Implied constraints 𝑥𝑖 ≥ 𝑥𝑗

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 ≻ 𝑗

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Conflicting items ሚ that are in strict (weak) conflict can not be packed on the same A set of items 𝐶 (𝐶) bin (layer) We process weak conflicts in preprocessing and check whether or not pairs of two and groups of three items can be possibly be packed. Strict conflicts are created by taking item types into account. Implied constraints ෍ 𝑎𝑖

≤

𝐶 −1

∀𝐶 ∈𝒞

𝑖∈𝐶

෍

𝑓𝑖𝑗

≥

1

∀ 𝐶ሚ ∈ 𝒞ሚ

ሚ 𝑖,𝑗∈𝐶:𝑖≠𝑗

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Strengthened model: 2DPPLC 𝑧 2𝐷𝑃𝑃

=


(I)

𝑖∈𝐼

s.t.: …

…

…

𝑥𝑖

≥

𝑥𝑗

෍ 𝑎𝑖

≤

Item dominances

Strict conflicts

𝐶 −1

(II-IX) ∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 ≻ 𝑗

(X)

∀𝐶 ∈𝒞

(XI)

𝑖∈𝐶

Weak conflicts

෍

𝑓𝑖𝑗

≥

1


(XII)


≥

0

∀𝑖 ∈𝐼

(XIII)

∀ 𝑖, 𝑗 ∈ 𝐼

(XIV)



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=


(I)

𝑖∈𝐼

s.t.:

Strict conflicts

…

…

…

𝑥𝑖

≥

𝑥𝑗

෍ 𝑎𝑖

≤

𝐶 −1

(II-IX) ∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 ≻ 𝑗

(X)

∀𝐶 ∈𝒞

(XI)

𝑖∈𝐶

Weak conflicts

෍

𝑓𝑖𝑗

≥

1


(XII)


≥

0

∀𝑖 ∈𝐼

(XIII)

∀ 𝑖, 𝑗 ∈ 𝐼

(XIV)



Markus Seizinger


24



=


(I)

𝑖∈𝐼

s.t.: …

…

…

𝑥𝑖

≥

𝑥𝑗

෍ 𝑎𝑖

≤

𝐶 −1

(II-IX) ∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 ≻ 𝑗

(X)

∀𝐶 ∈𝒞

(XI)

𝑖∈𝐶

Weak conflicts

෍

𝑓𝑖𝑗

≥

1


(XII)


≥

0

∀𝑖 ∈𝐼

(XIII)

∀ 𝑖, 𝑗 ∈ 𝐼

(XIV)



Markus Seizinger


24



=


(I)

𝑖∈𝐼

s.t.: …

…

…

𝑥𝑖

≥

𝑥𝑗

෍ 𝑎𝑖

≤

𝐶 −1

(II-IX) ∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 ≻ 𝑗

(X)

∀𝐶 ∈𝒞

(XI)

𝑖∈𝐶

෍

𝑓𝑖𝑗

≥

1


(XII)


≥

0

∀𝑖 ∈𝐼

(XIII)

∀ 𝑖, 𝑗 ∈ 𝐼

(XIV)



Markus Seizinger


24




Markus Seizinger

25

4. Computational Study

Instance Classes of Martello & Vigo (1998) 4 Classes of items –

Dimensions of items are uniformly distributed in an interval (depending on it‘s class) 2. High items

1. Wide items –

𝑤𝑖 ∈

2 3

𝑊; 𝑊 ; ℎ𝑖 ∈ 1;

𝐻 2

;

𝑤𝑖 ∈

𝑊 2

+ 1; 𝑊 ; ℎ𝑖 ∈

𝑤𝑖 ∈ 1;

𝑊 2

; ℎ𝑖 ∈

2 3

𝐻; 𝐻 ;

4. Small items

3. Large items –

–

𝐻 2

+ 1; 𝐻 ;

–

𝑤𝑖 ∈ 1;

𝑊 2

; ℎ𝑖 ∈ 1;

𝐻 2

;

4 Classes of instances • Instance (𝑥) contains 70% class 𝑥 items and 10% of every other item class 2 Subclasses • A: Types are equally distributed over all items • B: For instance (𝑥) all items of class 𝑥 are of the same type

Markus Seizinger

26


Instance Classes Instance class|subclass

1|A

“Wide items” 2 𝑊, 𝑊 3 1 ℎ𝑖 ~𝑈 1, 𝐻 2

𝑤𝑖 ~ 𝑈

35 %

1|B

70 %

2|A

5%

2|B 3|A

5%

3|B 4|A 4|B

35 %

“High items” 1 𝑤𝑖 ~ 𝑈 1, 𝑊 2

ℎ𝑖 ~𝑈

5%

3

𝐻, 𝐻

5%

1 𝑊 + 1, 𝑊 2 1 ℎ𝑖 ~𝑈 𝐻 + 1, 𝐻 2

𝑤𝑖 ~ 𝑈

5%

10 % 5%

35 %

10 %

70 %

5%

5%

10 % 5%

2

“Large items”

5% 10 %

5%

35 %

5%

“Small items” 1 𝑤𝑖 ~ 𝑈 1, 𝑊 2 1 ℎ𝑖 ~𝑈 1, 𝐻 2

5%

10 % 5%

5%

10 % 5%

10 % 5%

35 %

10 %

70 %

5%

5%

10 %

Markus Seizinger

35 %

5%

5% 10 %

5%

5% 10 %

5%

35 %

10 %

70 %

35 %

27


Setup Instances • 10 instances of every combination of class (4), subclass (2), and number of items (5) randomly generated • Total of 400 instances • 𝑊 = 𝐻 = 10; 𝑆 = 2; 𝛵 = 2

BFDH: Best-fit decreasing height algorithm L0: Geometric bound L1: Bound of large items L2: Hybrid of L0 and L1 𝐿𝐿𝐶 1𝐷𝐵𝑃 : Model-based lower bound

Initialization • Initial columns generated by BFDH-LC algorithm • Lower bounds 𝐿0 , 𝐿1 , and 𝐿2 • 𝐿𝐿𝐶 1𝐷𝐵𝑃 with runtime limit of ten minutes (if reached, best lower bound is used) Column Generation • Total runtime limited to five hours • Terminated when root node relaxation was solved Markus Seizinger

28

4. Computational Study BFDH: Best-fit decreasing height algorithm 𝐿𝐿𝐶 1𝐷𝐵𝑃 : Model-based lower bound

Bounds Analytical Bound

L0: Geometric bound L1: Bound of large items L2: Hybrid of L0 and L1

Model-Based

Class

Size

𝐿0

𝐿1

𝐿2

𝐿1𝐷𝐵𝑃

BFDH Algo.

abs. Gap

rel. Gap

#optimal

1|A 1|A 1|A 1|A 1|A 1|B 1|B 1|B 1|B 1|B 2|A 2|A 2|A 2|A 2|A 2|B 2|B 2|B 2|B 2|B

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

3,3 6,0 8,6 11,4 14,3 3,3 6,0 8,6 11,4 14,3 3,3 6,1 8,6 11,3 13,8 3,3 6,1 8,6 11,3 13,8

1,0 2,0 2,5 3,3 3,8 0,9 1,7 2,5 3,1 3,6 1,0 2,0 2,5 3,3 3,8 0,9 1,7 2,5 3,1 3,6

3,3 6,0 8,6 11,4 14,3 3,3 6,0 8,6 11,4 14,3 3,3 6,1 8,6 11,3 13,8 3,3 6,1 8,6 11,3 13,8

3,3 6,0 8,6 11,4 14,3 3,5 6,1 8,6 11,4 14,3 3,3 6,1 8,6 11,3 13,8 3,7 6,1 8,6 11,3 13,8

3,9 7,2 10,1 13,2 16,6 4,0 7,1 10,3 13,6 16,8 4,2 7,3 10,2 13,0 15,8 4,2 6,8 10,3 13,5 16,0

0,6 1,2 1,5 1,8 2,3 0,5 1,0 1,7 2,2 2,5 0,9 1,2 1,6 1,7 2,0 0,5 0,7 1,7 2,2 2,2

19% 20% 17% 16% 16% 16% 17% 20% 19% 17% 28% 20% 19% 15% 15% 14% 12% 20% 20% 16%

4 1 0 0 0 5 0 0 0 0 1 0 0 0 0 5 3 0 0 0

Markus Seizinger

29

4. Computational Study BFDH: Best-fit decreasing height algorithm 𝐿𝐿𝐶 1𝐷𝐵𝑃 : Model-based lower bound

Bounds Analytical Bound

L0: Geometric bound L1: Bound of large items L2: Hybrid of L0 and L1

Model-Based

Class

Size

𝐿0

𝐿1

𝐿2

𝐿1𝐷𝐵𝑃

BFDH Algo.

abs. Gap

rel. Gap

#optimal

3|A 3|A 3|A 3|A 3|A 3|B 3|B 3|B 3|B 3|B 4|A 4|A 4|A 4|A 4|A 4|B 4|B 4|B 4|B 4|B

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

5,2 9,9 14,7 19,0 24,0 5,2 9,9 14,7 19,0 24,0 2,3 4,0 5,8 7,3 8,9 2,3 4,0 5,8 7,3 8,9

6,1 11,2 16,0 21,1 26,9 6,0 11,1 15,5 20,8 26,7 1,2 2,1 2,5 3,4 4,2 1,1 1,9 2,3 3,1 3,9

6,8 13,2 18,4 24,8 31,1 6,8 13,2 18,4 24,8 31,1 2,3 4,0 5,8 7,3 8,9 2,3 4,0 5,8 7,3 8,9

7,2 13,3 18,4 24,8 31,1 7,2 13,3 18,4 24,8 31,1 2,4 4,0 5,8 7,3 8,9 2,4 4,0 5,8 7,3 8,9

7,6 13,8 19,4 25,8 32,5 7,8 14,3 19,9 26,6 33,7 2,8 4,7 6,4 8,1 9,8 2,9 4,8 6,6 8,1 10,1

0,4 0,5 1,0 1,0 1,4 0,6 1,0 1,5 1,8 2,6 0,4 0,7 0,6 0,8 0,9 0,5 0,8 0,8 0,8 1,2

6% 4% 5% 4% 5% 8% 8% 8% 7% 8% 20% 18% 11% 11% 10% 22% 21% 13% 11% 14%

6 6 2 0 0 4 2 0 0 0 6 3 4 2 1 5 3 2 3 0

Markus Seizinger

30


Key Results •

𝐿0 is equal for pairs of subclasses X|A and X|B –

•

Item dimensions are not influenced by subclass

BFDH: Best-fit decreasing height algorithm L0: Geometric bound L1: Bound of large items L2: Hybrid of L0 and L1 𝐿𝐿𝐶 1𝐷𝐵𝑃 : Model-based lower bound

𝐿1 does differ for pairs of subclasses X|A and X|B – –

Bound can take item types explicitly into account 𝐿𝐴1 ≥ 𝐿𝐵1

•

𝐿2 can improve 𝐿0 and 𝐿1 , if difference between these both bounds is small

•

Model-based bound 𝐿𝑆𝐶 1𝐷𝐵𝑃 dominates all other bounds

•

Relative gap is not significantly influenced by instance size

•

Mostly small instances could be solved to optimality by BFDH algorithm

Markus Seizinger

31


Column Generation All instances Class Size #instances

1|A 20 1|A 40 1|A 60 1|A 80 1|A 100 1|B 20 1|B 40 1|B 60 1|B 80 1|B 100 2|A 20 2|A 40 2|A 60 2|A 80 2|A 100 2|B 20 2|B 40 2|B 60 2|B 80 2|B 100

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

Succesfully terminated

time #columns #instances

0:18 4:07 5:00 4:46 5:00 0:10 3:44 4:22 4:25 5:00 0:34 3:23 4:20 4:56 5:00 0:00 1:58 4:56 4:40 5:00

40 268 591 904 1280 52 247 451 865 1411 86 366 667 1100 1096 26 138 517 820 816

10 4 0 2 0 10 4 3 2 0 10 5 3 1 0 10 7 1 2 0

#LB #UB time #columns improved improved

0:18 2:47

40 220

3:51

1121

0:10 1:50 2:54 2:06

52 206 470 904

0:34 1:46 2:49 4:20

86 245 841 319

0:00 0:40 4:24 3:20

26 102 1231 900

Markus Seizinger

3 2 0 2 0 2 3 4 3 0 4 4 2 1 0 0 2 0 2 0

2 2 0 0 0 2 0 1 2 0 5 4 1 0 0 4 2 4 3 1

CG LB

CG UB

3,6 6,2 8,6 11,7 14,3 3,7 6,4 9,1 11,9 14,3 3,7 6,5 8,8 11,5 13,8 3,7 6,4 8,6 11,6 13,9

3,7 7,0 10,1 13,2 16,6 3,8 7,1 10,2 13,4 16,8 3,7 6,9 10,1 13,0 15,7 3,8 6,6 9,9 13,2 16,0

initial LB initial UB

3,3 6,0 8,6 11,4 14,3 3,5 6,1 8,6 11,4 14,3 3,3 6,1 8,6 11,3 13,8 3,7 6,1 8,6 11,3 13,9

3,9 7,2 10,1 13,2 16,6 4,0 7,1 10,3 13,6 16,8 4,2 7,3 10,2 13,0 15,7 4,2 6,8 10,3 13,5 16,1 32


Column Generation All instances Class Size #instances

3|A 20 3|A 40 3|A 60 3|A 80 3|A 100 3|B 20 3|B 40 3|B 60 3|B 80 3|B 100 4|A 20 4|A 40 4|A 60 4|A 80 4|A 100 4|B 20 4|B 40 4|B 60 4|B 80 4|B 100

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

Succesfully terminated

time #columns #instances

0:00 0:00 1:57 1:36 3:31 0:00 0:01 3:09 1:01 2:47 0:01 2:27 3:54 3:55 4:26 0:02 3:10 2:44 3:25 5:00

6 7 27 36 38 4 21 54 66 67 30 1014 1415 2454 2001 45 857 587 1785 2554

10 10 7 8 5 10 10 4 9 6 10 7 4 3 1 10 6 6 4 0

#LB #UB time #columns improved improved

0:00 0:00 0:39 0:45 2:20 0:00 0:01 0:24 0:35 1:41 0:01 1:22 2:16 1:23 0:00 0:02 1:58 1:13 1:03

6 7 28 34 35 4 21 49 66 60 30 614 0 950 0 45 535 544 716

Markus Seizinger

1 2 6 5 6 0 2 6 8 6 2 2 0 1 0 2 1 4 1 0

1 1 3 0 0 5 7 5 5 8 1 2 0 0 0 3 2 1 0 0

CG LB

CG UB

7,3 13,5 19,0 25,3 32,1 7,2 13,5 19,0 25,6 32,0 2,6 4,2 5,8 7,4 8,9 2,6 4,1 6,2 7,4 8,9

7,5 13,7 19,1 25,8 32,8 7,3 13,6 19,3 26,0 33,1 2,7 4,5 6,4 8,1 9,8 2,6 4,6 6,5 8,1 10,1

initial LB initial UB

7,2 13,3 18,4 24,8 31,3 7,2 13,3 18,4 24,8 31,3 2,4 4,0 5,8 7,3 8,9 2,4 4,0 5,8 7,3 8,9

7,6 13,8 19,4 25,8 32,8 7,8 14,3 19,9 26,6 34,0 2,8 4,7 6,4 8,1 9,8 2,9 4,8 6,6 8,1 10,1 33


Key Results •

Results for classes 1|A, 1|B, 2|A, 2|B similar –

•

Best results could be achieved for 3|A and 3|B – –

•

Problem structure is closely related

Only few columns needed to terminate Termination criterion could be found fast

4|A and 4|B are hardest instances – –

Most columns generated, mainly by packing heuristic Hard to prove no new negative reduced-cost column exists

•

Last iteration not solvable

•

Using heuristic as warm-start speeds up subproblem dramatically

Markus Seizinger

34




Markus Seizinger

35

5. Conclusion and Future Agenda

Conclusion Problem class • 2DBP with layer constraints has never been published • No effective exact methods for 2DBP with layer constraints known Solution approach • Decomposition necessary • Column generation serves to provide an additional lower bound • Convergence of column generation quite bad Problems • Hard to prove that no new negative reduced cost column exists • Speed of subproblem is limiting factor

Markus Seizinger

36

5. Conclusion and Future Agenda

Future Agenda Improve subproblem formulation • Reduce symmetries • Provide better lower bounds • Restrict packing to certain Guillotine Cuts

Evaluate Constraint Satisfaction Approach • CP-solvers might find infeasible packings faster • Layer constraints restrict possible domains of pairs of items Graph representation of packing • Check results of Fekete et al. • Special interval graphs are equivalent to packings • Easy to adapt formulation to apply to layer constraints

Markus Seizinger

37

Questions and comments!

Markus Seizinger

38

(1/2)

Graphs representing feasible packings 4

“Translating” packings to a set of graphs • One graph 𝐺𝑖 for every dimension 𝑖 • An edge 𝑎𝑏 exists in 𝐺𝑖 exists, iff projections of items 𝑎 and 𝑏 overlap in dimension 𝑖

𝑖=1

3 2 1

6

5

𝑖=2 1

1

6

2

6

2

5

3

5

3

4

4 Markus Seizinger

39

(2/2)

Graphs representing feasible packings N

𝑖=1

𝑖=2 1

1

6

2

6

2

5

3

5

3

4

4 Markus Seizinger

40

(2/2)

Graphs representing feasible packings N : 𝐸 𝑖 =∅

𝑖=1 2 𝐸 𝑖 =∅

for Edges 𝐸 𝑖 connecting to vertices of different type

𝑃4:

‫ځ‬2𝑖=1 𝐸𝑖 = ∅

for Edges 𝐸𝑖 connecting to vertices of different type

𝑖=1

𝑖=2 1

1

6

2

6

2

5

3

5

3

4

4 Markus Seizinger

40

(1/2)

1DBPPLC as lower bound Sets, Indices, and Parameters 𝑖∈𝐼 Set of items 𝜏∈Τ Set of item types 𝑊 𝐻

Width of bin Height of bin

𝑠∈𝑆 𝑏∈𝐵

Set of bin layers Set of bins

𝑤𝑖 ℎ𝑖 𝜏𝑖

Width of item 𝑖 Height of item 𝑖 Type of item 𝑖

Additional Sets and Expressions 𝐼ሚ𝑖 = 𝑖 ′ |𝑖 ′ ∈ 𝐼: ℎ𝑖 + ℎ𝑖 ′ > 𝐻 ∧ 𝑤𝑖 + 𝑤𝑖 ′ > 𝑊 ∧ 𝑖 ≠ 𝑖 ′ 𝐼𝑖ҧ = 𝑖 ′ |𝑖 ′ ∈ 𝐼ሚ𝑖 : 𝜏𝑖 ≠ 𝜏𝑖 ′

Set of conflicting items for 𝑖

𝐴ҧ𝜏𝑏𝑠 = σ𝑖∈𝐼𝜏 𝑤𝑖 ℎ𝑖 𝑥𝑖𝑏𝑠

Area used by items of type 𝜏

Set of conf. items of diff. type

Decision Variables 𝑥𝑖𝑏𝑠 1, iff item 𝑖 is assigned to layer 𝑠 of bin 𝑏 𝑦𝑏 1, iff bin 𝑏 is used 41

(2/2)

1DBPPLC as lower bound 𝐿𝑆𝐶 1𝐷𝐵𝑃

=

min ෍ 𝑦𝑏

(I)

𝑏∈𝐵

𝑠. 𝑡. :

෍ ෍ 𝑥𝑖𝑏𝑠

=

1

∀𝑖 ∈𝐼

(II)

𝑥𝑖𝑏𝑠 − 𝑦𝑏

≤

0

∀ 𝑖 ∈ 𝐼, 𝑏 ∈ 𝐵, 𝑠 ∈ 𝑆

(III)

෍ 𝑤𝑖 ℎ𝑖 𝑥𝑖𝑏𝑠

≤

𝑊𝐻

∀ 𝑏 ∈ 𝐵, 𝑠 ∈ 𝑆

(IV)

≤

𝑊𝐻

∀𝑏 ∈ 𝐵, 𝑠, 𝑠′ ∈ 𝑆, 𝜏 ∈ T

(V)

𝑥𝑖𝑏𝑠 + 𝑥𝑖 ′ 𝑏𝑠

≤

1

(VI)

𝑥𝑖𝑏𝑠 + 𝑥𝑖 ′ 𝑏𝑠′

≤

1

∀ 𝑖 ∈ 𝐼, 𝑖 ′ ∈ 𝐼ሚ𝑖 , 𝑏 ∈ 𝐵, 𝑠 ∈ 𝑆 ∀ 𝑖 ∈ 𝐼, 𝑖 ′ ∈ 𝐼𝑖ҧ , 𝑏 ∈ 𝐵, 𝑠, 𝑠′ ∈ 𝑆

(VII)

𝑏∈𝐵 𝑠∈𝑆

𝑖∈𝐼

𝐴ҧ𝜏𝑏𝑠 +

෍ 𝐴ҧ𝜏′ 𝑏𝑠′ 𝜏′ ∈Τ\{𝜏}

𝑥𝑖𝑏𝑠

𝑏𝑜𝑜𝑙𝑒𝑎𝑛

∀𝑖 ∈ 𝐼, 𝑏 ∈ 𝐵, 𝑠 ∈ 𝑆

(VIII)

𝑦𝑏

𝑏𝑜𝑜𝑙𝑒𝑎𝑛

∀𝑏 ∈𝐵

(IX) 42

(1/2)

1DKPLC as pre-stage Sets, Indices, and Parameters 𝑖∈𝐼 Set of items 𝑊 𝐻 𝐷= 𝑆

Width of bin Height of bin Number of Layers

𝑡∈𝑇

Set of item types



Additional Sets and Expressions 𝐼ሚ𝑖 = 𝑖 ′ |𝑖 ′ ∈ 𝐼: ℎ𝑖 + ℎ𝑖 ′ > 𝐻 ∧ 𝑤𝑖 + 𝑤𝑖 ′ > 𝑊 ∧ 𝑖 ≠ 𝑖 ′ 𝐼𝑖ҧ = 𝑖 ′ |𝑖 ′ ∈ 𝐼ሚ𝑖 : 𝜏𝑖 ≠ 𝜏𝑖 ′ 𝐴ҧ𝜏𝑠 = σ𝑖∈𝐼𝜏 𝑤𝑖 ℎ𝑖 𝑥𝑖𝑠

Set of conflicting items for 𝑖 Set of conf. items of diff. type

Area used by items of type 𝜏 on layer 𝑠

Decision Variables 𝑥𝑖𝑠 1, iff item 𝑖 is placed on layer 𝑠 of bin 𝑎𝑖 1, iff item 𝑖 is placed in bin 43

(2/2)

1DKPLC as pre-stage 𝑧1𝐷𝐾𝑃

=

min 1 − ෍ 𝜋𝑖 𝑎𝑖

(𝐼)

𝑖∈𝐼

෍ 𝑥𝑖𝑠

𝑠. 𝑡. :

=

𝑎𝑖

∀𝑖 ∈𝐼

(𝐼𝐼)

≤

𝑊𝐻

∀𝑠 ∈𝑆

(𝐼𝐼𝐼)

≤

𝑊𝐻

∀ 𝑠, 𝑠′ ∈ 𝑆, 𝜏 ∈ T

(𝐼𝑉)

𝑥𝑖

≥

𝑥𝑗

(𝑉)

𝑥𝑖𝑠 + 𝑥𝑖 ′ 𝑠

≤

1

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 ≻ 𝑗 ∀ 𝑖 ∈ 𝐼, 𝑖 ′ ∈ 𝐼ሚ𝑖 , 𝑠 ∈ 𝑆

𝑥𝑖𝑠 + 𝑥𝑖 ′ 𝑠′

≤

1

∀ 𝑖 ∈ 𝐼, 𝑖 ′ ∈ 𝐼𝑖ҧ , 𝑠, 𝑠′ ∈ 𝑆

𝑠∈𝑆

෍ 𝑤𝑖 ℎ𝑖 𝑥𝑖𝑠 𝑖∈𝐼

𝐴ҧ𝜏𝑠 +

෍ 𝐴ҧ𝜏′𝑠′ 𝜏′ ∈Τ\{𝜏}

𝑥𝑖𝑠


∀ 𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆

𝑎𝑖


∀𝑖 ∈𝐼

(𝑉𝐼) (𝑉𝐼𝐼) (𝑉𝐼𝐼𝐼)

(IX) 44

(1/3)

2D Packing Problem with layer CTs (2DPPLC) Sets, Indices, and Parameters 𝑖∈𝐼 Set of items 𝑡∈𝑇 Set of item types

s∈𝑆

layers of bin



Decision Variables 𝑥𝑖 x-coordinate of item 𝑖 𝑠𝑖 layer of item 𝑖

𝑦𝑖 𝑎𝑖

y-coordinate of item 𝑖 1, iff item 𝑖 is placed in bin

𝑙𝑖𝑗

1, iff item 𝑖 is placed left of 𝑗

𝑏𝑖𝑗

1, iff item 𝑖 is placed below 𝑗

𝑓𝑖𝑗

1, iff item 𝑖 is placed in front of 𝑗

𝑊 𝐻 𝐷= 𝑆

Width of bin Height of bin Number of layers

Markus Seizinger

45

(2/3)


=



1


1

(I)

𝑖∈𝐼

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 < 𝑗 ∧ 𝑡𝑖 ≠ 𝑡𝑗

(II) (III)


≤

𝑊 − 𝑤𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(IV)


≤

𝐻 − ℎ𝑖

∀ 𝑖, 𝑗 ∈ 𝐼

(V)


≤

S−1

∀ 𝑖, 𝑗 ∈ 𝐼

(VI)

𝑥𝑖

≤

𝑊 − 𝑤𝑖

∀𝑖 ∈𝐼

(VII)

𝑦𝑖

≤

𝐻 − ℎ𝑖

∀𝑖 ∈𝐼

(VIII)

𝑠𝑖

≤

𝐷−1

∀𝑖 ∈𝐼

(IX)

𝑥𝑖

≥

𝑥𝑗

∀ 𝑖, 𝑗 ∈ 𝐼: 𝑖 ≻ 𝑗

(X)


≥

0

∀𝑖 ∈𝐼

(XI)

∀ 𝑖, 𝑗 ∈ 𝐼

(XII)

𝑎𝑖 , 𝑙𝑖𝑗 , 𝑏𝑖𝑗 , 𝑓𝑖𝑗 Markus Seizinger


46

(3/3)

2DPPLC (B) 𝑧 2𝐷𝑃𝑃

=

min 1 − ෍ 𝜋𝑖 𝑎𝑖

(I)

𝑖∈𝐼

෍ ෍ ෍ 𝑥𝑖𝛼𝛽𝑠

𝑠. 𝑡. :

=

𝑎𝑖

=

0

∀𝑖 ∈𝐼

(II)

𝛼∈Α 𝛽∈Β 𝑠∈𝑆

𝑥𝑖𝛼𝛽𝑠

𝑀 𝑥𝑖𝛼𝛽𝑏𝑠 − 1 + ෍

෍

෍

෍

𝑖 ′ ∈𝐼\𝐼 ′ (𝜏𝑖 ) 𝛼 ′ ∈Α′

𝑖,𝑖 ′ ,𝛼

෍ 𝛽 ′ ∈Β′

𝑖,𝑖 ′ ,𝛽

(III)

𝑥𝑖 ′ 𝛼 ′ 𝛽 ′ 𝑠

≤

1

∀ 𝑖 ∈ 𝐼, 𝛼 ∈ Α, 𝛽 ∈ Β, 𝑠 ∈ 𝑆

(𝐼𝑉)

෍ 𝑥𝑖 ′ 𝛼 ′ 𝛽 ′ 𝑠 ′

≤

0

∀ 𝑖 ∈ 𝐼, 𝛼 ∈ Α, 𝛽 ∈ Β, 𝑠 ∈ 𝑆

(𝑉)

∀ 𝑖 ∈ 𝐼, 𝛼 ∈ Α, 𝛽 ∈ Β, 𝑠 ∈ 𝑆

(𝑉𝐼)

෍

𝑖 ′ ∈𝐼 𝛼 ′ ∈Α′ 𝑖,𝑖 ′ ,𝛼 𝛽 ′ ∈Β′ 𝑖,𝑖 ′ ,𝛽

𝑀 𝑥𝑖𝛼𝛽𝑏𝑠 − 1 +

∀ 𝑖 ∈ 𝐼, 𝛼 ∈ Α, 𝛽 ∈ Β, 𝑠 ∈ 𝑆: 𝛼 + 𝑤𝑖 > 𝑊 ∨ 𝛽 + ℎ𝑖 > 𝐻

𝑠 ′ ∈𝑆\{𝑠}

𝑥𝑖𝛼𝛽𝑠


𝑎𝑖


Markus Seizinger

∀𝑖 ∈𝐼

(𝑉𝐼𝐼) 47

2D Bin Packing Problem with Side Constraints

2D Bin Packing Problem with Side Constraints

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