Applications of the Quadratic AssignmentProblem

Axel Nyberg

CENTER OF EXCELLENCE INOPTIMIZATION AND SYSTEMS ENGINEERINGÅBO AKADEMI UNIVERSITY

OSE SEMINAR, NOVEMBER 15, 2013

Introduction 2 | 20

I Introduced by Koopmans and Beckmann in 1957

I Cited by ≈ 1500

I Among the hardest combinatorial problems

I Real and test instances easily accessible (QAPLib - A quadraticassignment problem library)

I Instances of size N = 30 are still unsolved

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Introduction 3 | 20

A

B

C

D

E

5

4

2

1

3

I Optimal assignment of factories to the cities marked in greenI Distances between the cities and flows between the factories

shown below

A=

0 3 6 4 23 0 2 3 36 2 0 3 44 3 3 0 12 3 4 1 0

B=

0 10 15 0 7

10 0 5 6 015 5 0 4 20 6 4 0 57 0 2 5 0

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Introduction 3 | 20

I Optimal solution = 258I Optimal permutation=[2 4 5 3 1]

A=

0 3 6 4 23 0 2 3 36 2 0 3 44 3 3 0 12 3 4 1 0

B24531 =

0 6 0 5 106 0 5 4 00 5 0 2 75 4 2 0 15

10 0 7 15 0

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Facility Layout Examples 4 | 20

I Hospital Layout - Germanuniversity hospital, KlinikumRegensburg, built 1972

I Optimality proven in the year2000

I [Krarup and Pruzan(1978)]

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Facility Layout Examples 5 | 20

I Airport gate assignment

I Minimize total passengermovement

I Minimize total baggagemovement

I [Haghani and Chen(1998)]

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Applications–Three Examples in Electronics 6 | 20

I Steinberg wiring problemI [Steinberg(1961)]I component placing on circuit boardsI [Rabak and Sichman(2003)]I Minimizing the number of transistors needed on integrated

circuitsI Burkard et al.(1993)

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Applications–Keyboard Layout 7 | 20

I Optimal placing of letters onkeyboards

I Language specific

I Burkard andOffermann(1977)

I Optimal placing of letters ontouchscreen devices

I Only one or two fingers used

I Dell’Amico et al.(2009)

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Applications–Electricity 8 | 20

I Turbine runner in electricitygeneration

I The weight of the blades candiffer up to ±5%

I Objective: To balance theturbine runner

I [Laporte and Mercure(1988)]

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Applications–A not so Serious Example 9 | 20

I Seating order at tonight’sdinner

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

DNA MicroArray Layout 10 | 20

I Microarrays can have up to 1.3 million probes

I Small subregions can be solved as QAPs

I Objective: To reduce the risk of unintended illumination ofprobes

AGTCGCACGCGTAGA

ATTTGGAGCCGTCGA

TGTTGATGCGGAGGC

CGTCGCCTCCGAGGT

ACTCGAGGAAGATGT

Sub-region (size N = 25)

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Even More Applications 11 | 20

I Bandwith minimization of a graph

I Image processing

I Economics

I Molecular conformations in chemistry

I Scheduling

I Supply Chains

I Manufacturing lines

In addition to all the above, many well known problems incombinatorial optimization can be written as QAPs, e.g.

I Traveling salesman problem

I Maximum cut problem

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 12 | 20

Three Objective Functions

I Koopmanns-Beckmann

minA ·XBXT (1)

I SDPmintr(AXBXT ) (2)

I DLRminXA ·BX (3)

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 13 | 20

Koopmans Beckmann form

N¼i=1

N¼j=1

N¼k=1

N¼l=1

aijbkl · xikxjl

N¼i=1

xij = 1, j = 1, ...,N ;

N¼j=1

xij = 1, i = 1, ...,N ;

xij ∈ {0,1}, i , j = 1, ...,N ;

I This formulation has N2(N −1)2 bilinear terms.

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 13 | 20

Koopmans Beckmann form

N¼i=1

N¼j=1

N¼k=1

N¼l=1

aijbkl · xikxjl

N¼i=1

xij = 1, j = 1, ...,N ;

N¼j=1

xij = 1, i = 1, ...,N ;

xij ∈ {0,1}, i , j = 1, ...,N ;

I This formulation has N2(N −1)2 bilinear terms.

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 14 | 20

minXA ·BX

minN¼i=1

N¼j=1

a ′ijb′ij

a ′ij =n¼

k=1

akjxik ∀i , j

b ′ij =n¼

k=1

bik xkj ∀i , j

A=

0 3 5 9 63 0 2 6 95 2 0 8 109 6 8 0 26 9 10 2 0

B=

0 4 3 7 74 0 4 10 4

3 4 0 2 37 10 2 0 47 4 3 4 0

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 14 | 20

minXA ·BX

minN¼i=1

N¼j=1

a ′ijb′ij

a ′ij =n¼

k=1

akjxik ∀i , j

b ′ij =n¼

k=1

bik xkj ∀i , j

A=

0 3 5 9 63 0 2 6 95 2 0 8 109 6 8 0 26 9 10 2 0

B=

0 4 3 7 74 0 4 10 4

3 4 0 2 37 10 2 0 47 4 3 4 0

a ′23 = 5x21 +2x22 +0x23 +8x24 +10x25

b ′23 = 4x13 +0x23 +4x33 +10x43 +4x53Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 15 | 20

Discrete Linear Reformulation (DLR)

minn¼

i=1

n¼j=1

Mi¼m=1

Bmi zmij

zmij ≤ Aj¼

k∈Kmi

xkj m = 1, ...,Mi

Mi¼m=1

zmij = a′ij

∀i , j

Example for one bilinear term a′23b′23

a′23 = 5x21 +2x22 +0x23 +8x24 +10x25

b ′23 = 4x13 +0x23 +4x33 +10x43 +4x53

x13 + x23 + x33 + x43 + x53 = 1

x21 + x22 + x23 + x24 + x25 = 1

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 15 | 20

Discrete Linear Reformulation (DLR)

minn¼

i=1

n¼j=1

Mi¼m=1

Bmi zmij

zmij ≤ Aj¼

k∈Kmi

xkj m = 1, ...,Mi

Mi¼m=1

zmij = a′ij

∀i , j

Example for one bilinear term a′23b′23

a′23 = 5x21 +2x22 +0x23 +8x24 +10x25

b ′23 = 4x13 +0x23 +4x33 +10x43 +4x53

x13 + x23 + x33 + x43 + x53 = 1

x21 + x22 + x23 + x24 + x25 = 1

4z123 +10z2

23

z123 ≤ 10(x13 + x33 + x53)

z123 + z2

23 = a ′23

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 16 | 20

05

10

05

100

50

100

05

10

05

100

50

100

Figure 1: Bilinear term a ′23b′23 discretized in b ′23 (to the left) and in a ′23 (to the right)

I The size of the MILP problem is dependent on the number ofunique elements per row.

I Tightness of the MILP problem is dependent on the differencesbetween the elements in each row.

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 17 | 20

I The size of the model is dependent on the number of uniqueelements per row.

I The tightness of the model is dependent on the differencesbetween the elements in each row.

I A can be modified to any matrix A, where aij + aji = aij +aji .

I By solving an LP a priori, we can decrease the model size,tighten the formulation and improve the lower bound.

A=

0 1 2 2 3 4 4 51 0 1 1 2 3 3 42 1 0 2 1 2 2 32 1 2 0 1 2 2 33 2 1 1 0 1 1 24 3 2 2 1 0 2 34 3 2 2 1 2 0 15 4 3 3 2 3 1 0

A=

0 2 2 2 6 6 6 60 0 0 0 4 4 4 42 2 0 2 2 2 2 22 2 2 0 2 2 2 20 0 0 0 0 0 0 02 2 2 2 2 0 2 22 2 2 2 2 2 0 24 4 4 4 4 4 0 0

I Does not break the symmetries in the problem.

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 17 | 20

I The size of the model is dependent on the number of uniqueelements per row.

I The tightness of the model is dependent on the differencesbetween the elements in each row.

I A can be modified to any matrix A, where aij + aji = aij +aji .

I By solving an LP a priori, we can decrease the model size,tighten the formulation and improve the lower bound.

A=

0 1 2 2 3 4 4 51 0 1 1 2 3 3 42 1 0 2 1 2 2 32 1 2 0 1 2 2 33 2 1 1 0 1 1 24 3 2 2 1 0 2 34 3 2 2 1 2 0 15 4 3 3 2 3 1 0

A=

0 2 2 2 6 6 6 60 0 0 0 4 4 4 42 2 0 2 2 2 2 22 2 2 0 2 2 2 20 0 0 0 0 0 0 02 2 2 2 2 0 2 22 2 2 2 2 2 0 24 4 4 4 4 4 0 0

I Does not break the symmetries in the problem.

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

Formulations 18 | 20

Instance Size opt old LB DLR Time(minutes)esc32a 32 130 103 130 1964esc32b 32 168 132 168 3500esc32c 32 642 616 642 254esc32d 32 200 191 200 10esc64a 64 116 98 116 48

Table 1: Solution times for the instances esc32a, esc32b, esc32c, esc32d and esc64a

from the QAPLIB to optimality using Gurobi 4.1 with default settings.

I Instances presented in 1990 and solved 2010 with our models

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

19 | 20

A few references

Ali Haghani and Min-Ching Chen.

Optimizing gate assignments at airport terminals.Transportation Research Part A: Policy and Practice, 32(6):437 – 454, 1998.

Jakob Krarup and Peter Mark Pruzan.

Computer-aided layout design.In M.L. Balinski and C. Lemarechal, editors, Mathematical Programming in Use, volume 9 of Mathematical Programming Studies,pages 75–94. Springer Berlin Heidelberg, 1978.

Gilbert Laporte and Helene Mercure.

Balancing hydraulic turbine runners: A quadratic assignment problem.European Journal of Operational Research, 35(3):378 – 381, 1988.

Axel Nyberg and Tapio Westerlund.

A new exact discrete linear reformulation of the quadratic assignment problem.European Journal of Operational Research, 220(2):314 – 319, 2012.

C.S Rabak and J.S Sichman.

Using a-teams to optimize automatic insertion of electronic components.Advanced Engineering Informatics, 17(2):95 – 106, 2003.

Leon Steinberg.

The backboard wiring problem: A placement algorithm.SIAM Review, 3(1):37–50, January 1961.

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University

20 | 20

Thank you for listening!

Questions?

Axel Nyberg

Optimization and Systems Engineering at Åbo Akademi University