Exact Methods for Recursive Circle Packing

Stephen J. Maher

joint work with

Ambros M. Gleixner, Benjamin Müller, and João Pedro Pedroso

Lancaster University Management School,

[email protected]

20th July 2017

Exact Methods for Recursive Circle Packing (Stephen J. Maher) 1/19

Motivation

I originates from tube industry

I reduce large shipping costs

I recursive packing of tubes into containers

I can be modelled as a 2D problem

Recursive Circle Packing Problem (RCPP)

input . . .

I set of ring types T := {1, . . . ,T}I external radius Rt and internal radius rt for each t ∈ TI demand Dt ∈ N for each t ∈ TI in�nitely many rectangles with width W and height H

objective . . .

I minimize z∗ = number of rectangles

I s.t.⋃

t{Dt many rings of type t} are packable into z∗ rectangles

TheoremRCPP is NP-hard.

Proof.For r = 0 : RCPP → Circle Packing Problem (CPP)

input . . .

objective . . .

I s.t.⋃

input . . .

objective . . .

I s.t.⋃

Compact MINLP Formulation

I consider each ring individually

I (xi , yi ) center of ring i

I φr ∈ {0, 1} if rectangle r is used

I ui,j ∈ {0, 1} if ring i is directly placed inside ring j

I wi,r ∈ {0, 1} if ring i is directly placed inside rectangle r

min∑r

∥∥∥∥(xiyi)−(xjyj

)∥∥∥∥2 ≥ (Ri + Rj)2(wi,r + wj,r − 1) ∀i 6= j ∀r∥∥∥∥(xiyi

)−(xjyj

)∥∥∥∥2 ≥ (Ri + Rj)2(ui,k + uj,k − 1) ∀i 6= j 6= k

Hopeless to solve with a general MINLP solver!

min∑r

)−(xjyj

)∥∥∥∥2 ≥ (Ri + Rj)2(ui,k + uj,k − 1) ∀i 6= j 6= k

min∑r

)−(xjyj

)∥∥∥∥2 ≥ (Ri + Rj)2(ui,k + uj,k − 1) ∀i 6= j 6= k

Cutting stock formulation

min∑P∈P

zP∑P∈P

αt,P · zP ≥ Dt ∀t ∈ T (cons-demand)

zp ∈ Z+ ∀P ∈ P

Pricing Problem (PP)

I (λ1, . . . , λT ) dual multipliers of (cons-demand)

I λt gain for packing a ring of type t

I �nd P ∈ P minimizing 1−∑

t αt,Pλt

Drawbacks of PP

I recursive structure of RCPP is completely in PP

I PP is the maximization variant of RCPP

I can be modeled as nonconvex MINLP → hopeless to solve

min∑P∈P

zP∑P∈P

zp ∈ Z+ ∀P ∈ P

t αt,Pλt

Drawbacks of PP

I can be modeled as nonconvex MINLP → hopeless to solve

min∑P∈P

zP∑P∈P

zp ∈ Z+ ∀P ∈ P

t αt,Pλt

Drawbacks of PP

I can be modeled as nonconvex MINLP → hopeless to solveExact Methods for Recursive Circle Packing (Stephen J. Maher) 5/19

Pattern-based Dantzig-Wolfe Decomposition

I inspired by multi-stage cutting stock formulation [Muter, Birbil, Bülb�l

I zC ∈ Z+ for C ∈ CPI zR ∈ Z+ for R ∈ RPI αt,R / αt,C number of circles of type t in pattern R / C

min∑

R∈RP

zR∑C=(d,t)∈CP

zC ≥ Dt ∀t ∈ T

∑C=(d,t)∈CP

zC ≤∑

R∈RP

αt,RzR +∑

C∈CP

αt,C zC ∀t ∈ T (cons-pack)

zR ∈ Z+ ∀R ∈ RPzC ∈ Z+ ∀C ∈ CP

Pattern-based Dantzig-Wolfe Decomposition

I inspired by multi-stage cutting stock formulation [Muter, Birbil, Bülb�l

I zC ∈ Z+ for C ∈ CPI zR ∈ Z+ for R ∈ RPI αt,R / αt,C number of circles of type t in pattern R / C

min∑

R∈RP

zR∑C=(d,t)∈CP

zC ≥ Dt ∀t ∈ T

∑C=(d,t)∈CP

zC ≤∑

R∈RP

αt,RzR +∑

C∈CP

αt,C zC ∀t ∈ T (cons-pack)

zR ∈ Z+ ∀R ∈ RPzC ∈ Z+ ∀C ∈ CP

One-level Packings

De�nitionA tuple (d , t) ∈ ZT

+ × T is a circular pattern :⇔I⋃

i{di many circles with radius Ri} packable inside a ring of type t

Exampleinput

all circular patterns

One-level Packings

De�nitionA tuple (d , t) ∈ ZT

+ × T is a circular pattern :⇔I⋃

i{di many circles with radius Ri} packable inside a ring of type t

Exampleinput

all circular patterns

One-level Packings

RP the set of all rectangular patterns

RP = { , , , . . .}

I elements of RP are vectors d = (d1, . . . , dT ) ∈ ZT+

I |RP| depends on T ,R,W ,H

Main Idea

I using R ∈ RP → possibility to use �tting C ∈ CPI using C ∈ CP → possibility to use �tting C ′ ∈ CP

One-level Packings

RP = { , , , . . .}

Main Idea

One-level Packings

RP = { , , , . . .}

Main Idea

Recursive Packings

How to solve it?

1. enumeration of CP2. pattern-based branch-and-price

Enumeration of CPI The feasible placement of circles within rings is veri�ed by solving an

NLP. ∥∥∥∥(xiyi)−(xjyj

)∥∥∥∥2

≥ Ri + Rj ∀i , j ∈ C : i < j ,∥∥∥∥(xiyi)∥∥∥∥

≤ rt − Ri ∀i ∈ C ,

xi , yi ∈ R ∀i ∈ C .

I All dominated circle packings are elimated.

I Two sets of circle packings are formed, CP feas and CPunknown,

CP∗ ⊆ CP feas ∪ CPunknown.

Enumeration of CPI The feasible placement of circles within rings is veri�ed by solving an

NLP. ∥∥∥∥(xiyi)−(xjyj

)∥∥∥∥2

≥ Ri + Rj ∀i , j ∈ C : i < j ,∥∥∥∥(xiyi)∥∥∥∥

≤ rt − Ri ∀i ∈ C ,

xi , yi ∈ R ∀i ∈ C .

I All dominated circle packings are elimated.

I Two sets of circle packings are formed, CP feas and CPunknown,

Pattern-based Branch-and-Price

Restricted Master Problem (RMP ′)

I contains recursive structure ofRCPP

I decides how to use patterns(→ arrows)

I after solving RMP ′ →branching

Pricing Problem (PP ′)

I task: �nd improving zR forR ∈ RP

I (λ1, . . . , λT ) dual multipliersfor (cons-pack)

I λt gain for packing a circle oftype t

I classical Circle Packing Problem

Pattern-based Branch-and-Price

Restricted Master Problem (RMP ′)

I contains recursive structure ofRCPP

I decides how to use patterns(→ arrows)

I after solving RMP ′ →branching

Pricing Problem (PP ′)

I task: �nd improving zR forR ∈ RP

I (λ1, . . . , λT ) dual multipliersfor (cons-pack)

I λt gain for packing a circle oftype t

I classical Circle Packing Problem

The Pricing Challenge

I PP ′ still NP-hardI can be modelled as nonconvex MINLP

I but much easier to solve than PP

I useful: Farley Bound [Farley 1990]

Corollaryd ν(RMP′)1−ν(PP′)e ≤ OPT (RCPP) is a valid lower bound for RCPP.

Could not solve PP ′ . . .

I LB(PP ′) ≤ ν(PP ′) ⇒ d ν(RMP′)1−LB(PP′)e ≤ OPT (RCPP)

I improvement LB(PP ′) ⇒ improvement for LB(RCPP)

I strengthen relaxation of PP ′ viaI symmetry breakingI positive semi-de�nite relaxationI disjunctive cuts [Saxena, Bonami, Lee 2010]

Valid bounds

I The lower bound is provided by the Farley Bound [Farley 1990]

I Valid primal bounds can be found using the partitioning of thecircular patterns.

CP feas ⊆ CP and

I Thus, any primal feasible solution using only patterns from CP feas isa valid primal bound.

Valid bounds

I The lower bound is provided by the Farley Bound [Farley 1990]

I Valid primal bounds can be found using the partitioning of thecircular patterns.

CP feas ⊆ CP and

I Thus, any primal feasible solution using only patterns from CP feas isa valid primal bound.

Test Set

I contains 800 random generated instances

I |T | ∈ {3, 4, 5, 10}I maxi ri

mini Ri∈ {2.0, 2.3, . . . , 5.0}

I max{W ,H}maxi Ri

∈ {2.0, 2.3, . . . , 5.0}

I Di ∈[0.8·W ·Hπ(Ri )2

, 1.2·W ·Hπ(Ri )2

]· γ for γ ∈ {5, 10}

Experiments

I Time limit of 7200 seconds

I Computing primal and dual bounds,I Primal bound with tNLP = 30s and tCPP = 600,I Dual bound with tNLP = 100s and tCPP = ∞

I Comparison with GRASP heuristic

Test Set

I contains 800 random generated instances

I |T | ∈ {3, 4, 5, 10}I maxi ri

mini Ri∈ {2.0, 2.3, . . . , 5.0}

I max{W ,H}maxi Ri

∈ {2.0, 2.3, . . . , 5.0}

I Di ∈[0.8·W ·Hπ(Ri )2

, 1.2·W ·Hπ(Ri )2

]· γ for γ ∈ {5, 10}

Experiments

I Time limit of 7200 seconds

I Computing primal and dual bounds,I Primal bound with tNLP = 30s and tCPP = 600,I Dual bound with tNLP = 100s and tCPP = ∞

I Comparison with GRASP heuristic

Results: Reached Gaps

I gap = |primal−dual|min{primal,dual}

I dual = dual bound for CPLB

I primal = primal bound for CPUB

0% gap 0− 25% gap 25− 50% gap 50− 100% gap > 100% gap

0% gap

0− 25% gap

25− 50% gap

50− 100% gap

> 100% gap

# instances

|T |0%

3 4 5 10

Results: Reached Gaps

I gap = |primal−dual|min{primal,dual}

I dual = dual bound for CPLB

I primal = primal bound for CPUB

0% gap 0− 25% gap 25− 50% gap 50− 100% gap > 100% gap

0% gap

0− 25% gap

25− 50% gap

50− 100% gap

> 100% gap

# instances

|T |0%

3 4 5 10

Results: Primal Bounds

I compare to randomized greedy heuristic GRASP [Pedroso, Cunha, Tavares

I time limit 7200s

Instance branch+price GRASP

Name T maxt rtmint Rt

dual primal primal

s03i1 3 2.2 1 1 1s03i2 3 2.2 9 10 10s03i3 3 2.2 82 100 95s05i1 5 5.1 1 1 1s05i2 5 5.1 8 10 10s05i3 5 5.1 73 94 98s16i1 16 6.3 1 1 1s16i2 16 6.3 8 10 10s16i3 16 6.3 � 96 96

Instances from Tube Industry

I maxi rimini Ri

large → many and hard veri�cation NLPs to solveI rectangles too large compared to rings → PP ′ too hardI still: Comparable performance with GRASP (one better and one

worse).

Instances from Tube Industry

I maxi rimini Ri

large → many and hard veri�cation NLPs to solveI rectangles too large compared to rings → PP ′ too hardI still: Comparable performance with GRASP (one better and one

worse).Exact Methods for Recursive Circle Packing (Stephen J. Maher) 18/19

Recursive Circle Packing Problem

I pack set of rings; minimize the number of rectangles

I generalization of Circle Packing Problem

Pattern-based Dantzig-Wolfe decomposition

I recursive master problem

I nonconvex MINLP pricing

First exact algorithm: one-level enumeration +branch-and-price

I optimal solutions for many small and medium-sized instances

I improved primal solutions compared to existing heuristic

Future work

I tighter convex relaxations of PP ′?

I branching on original variables?

Exact Methods for Recursive Circle Packing

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