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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
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