1 hub and spoke network design. 2 outline motivation problem description mathematical model solution...
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Hub and Spoke Network Hub and Spoke Network DesignDesign
Hub and Spoke Network Hub and Spoke Network DesignDesign
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Outline
MotivationProblem DescriptionMathematical ModelSolution MethodComputational AnalysisExtensionConclusion
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Motivation
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Motivation
Spoke and Hub Network
σ = 0.25
Spokes
Hubs
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Motivation
Hub and Spoke Network design:
Cited as “seventh in the American Marketing Association series of ‘Great Ideas in the Decade of Marketing’ (Marketing News, June 20, 1986)
Predominant architecture for airline route system since deregulation in 1978
Finds applications in telecommunication network, express cargo
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Problem Description
Given a network of nodes with given flows between each pair, determine:
Which nodes are set as hubsWhich hub is a node assigned to
So that:
Every flow is first routed through one or two hubs before being sent to its destination
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Methodologies
Enumeration heuristics - O’Kelly (1986)Meta-heuristics:
Tabu Search – Klincewicz (1991); Kapov & Kapov (1994)
Simulated Annealing – Ernst & Krishnamoorthy (1996)
Lagrangian relaxation – Pirkul & Schilling (1998); Aykin (1994); Elhedhli & Hu (2005)
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Mathematical Model
ijk
m
otherwise 0,
k hub toassigned i poke 1 sifZik
otherwise 0,
orderin that m andk hubs viaj toi from low 1 fifX ijkm
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Mathematical Model
i j k m
ijkmijkmXF
Subject to: 1k
ikZ
kkik ZZ
for all i (2)
(1)
for all i, k (3)
pZk
kk (4)
ikm
ijkm ZX for all i, j > i, k (5)
jmk
ijkm ZX for all i, j > i, m (6)
Min
}1,0{, ikijkm ZX (7)
) ( mjkmikijijkm CCCWF
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Mathematical Model
Problem size: For number of nodes = n:
23850 iablesbinary var of . No
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2
)1( iablesbinary var of . n
nnNo
1 sconstraint of . 3 nNo
For n = 15:
3376 sconstraint of . No
That’s too large!
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Solution Method
Lagrangian Relaxation
31 different lagrangian relaxations possible
Review on Lagrangian Relaxation: Fisher (1981, 2005); Geoffrion (1974)
In current study, constriant sets (2), (5), (6) relaxed
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Solution Method
i j k m
ijkmijkmXF
Subject to: 1k
ikZ
kkik ZZ
for all i (2) αi
(1)
for all i, k (3)
pZk
kk (4)
ikm
ijkm ZX for all i, j > i, k (5) βijk
jmk
ijkm ZX for all i, j > i, m (6) Gijm
}1,0{, ikijkm ZX
Min
(7)
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Solution Method
Subject to:
kkik ZZ
(7)
for all i, k (3)
pZk
kk (4)
Min i
ii ij k m
ijkmijkmi k
ikik XFZC
ijjik
ijijkiikC Where,
ijmijkijkmijkm FF Sub
problem 1
Sub problem
2
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Solution Method
[SUB2]:
}1,0{ijkmX
i ij k m
ijkmijkmXFMin
Subject to: 1k m
ijkmX for all i, j > i
[SUB1]:i k
ikikZCMin
Subject to:kkik ZZ
pZk
kk for all i, k
}1,0{ikZ
Constrained added to
improve bound
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Solution Method
[MASTER]:
Max
Subject to: for h = 1,2,….
21 i
i
i ij k m
ijkmh
ijkmXF1
ikh
i kikZC2 for h = 1,2,….
freei ,, 21
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Solution Algorithm
[SUB1]: For each i, j:
Find
Set 1ijhnX
)( ijkmkmhn FMinF
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Solution Algorithm
[SUB2]:k allfor 1 kkZLet
1 ,0 if k,i, allfor ikik ZsetC
iki
ikk ZCSSet
order ascendingin s'SSort k
s'S psmallest first theofk index set the ain Place k
indices, ofset For this
1 Zassociatedeach set kk
zero. to variablesother Z allset and i, allfor ,1each Zset ,0 ikik ikCif
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Solution Algorithm
[Feasible Solution]: then,1Z If
kik
k allfor ,0Set Z ik
1)Z|C(Min C Find kkikkin 1Set Zin
then ,0Z Ifk
ik
1)Z|C(Min C Find kkikkin
1Set Zin
jmik ZZ *Xset k, m, i,j i, allFor ijkm
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Solution Algorithm
Issues: Slow convergence as master problem grows too
large Could not converge in 30 minutes for 10 nodes
How to resolve???
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Solution Algorithm
Subgradient Optimization to find lagrang multipliers Initialize α, β, γ;
Initialize step size
Is (UB-LB)/LB>ε?
Solve SUB1; SUB2 and obtain LB
Construct a feasible solution and obtain UB
stop
Yes
No
α, β, γ
Adjust α, β, γ by the amount of infeasibility
If no improvement in LB since long, decrease step size
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Computational Analysis
Original Model (Cplex) Lagrangean Relaxation
# Nodes # Hubs Time (sec) Time (sec) % Gap Optimal ?
5 2 0.012 0.924 0.0 Y
8 2 0.112 5.048 0.0 Y
10 2 0.699 94.810 0.07 Y
12 3 353.76 202.928 0.84 Y
15 3 > 1 Hour 922.911 0.96 ---
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Analysis
Congested
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Extended Model
Subject to: 1k
ikZ
kkik ZZ
for all i (2)
(1)
for all i, k (3)
pZk
kk (4)
ikm
ijkm ZX for all i, j > i, k (5)
imk
ijkm ZX for all i, j > i, m (6)
) ( mjkmikijijkm CCCWF
}1,0{, ikijkm ZX
Min
Congestion Cost function
i j k m
ijkmijkmXFb
i ij mijkmij
kXWa
b
i ijikij
kZWa
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Extended Model cont..
i j k m
ijkmijkmXF
k
b
i ijikij
HhZWba 1maxMin
iki ij
ij
b
i ij
hikij ZWZWab
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Linear Approximation
using tanget planes for
congestion cost function
Subject to: (2) – (7)
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Extended Model cont..
i j k m
ijkmijkmXFMin k
kwa
Subject to:
(2) – (7) MIP with an infinite number of constraints
b
i ij
hikijik
i ijij
b
i ij
hikijk ZWbZWZWbw 1
1
kHh (8)
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Solution Method (Langrangean Relaxation)
Subject to:
kkik ZZ for all i, k (3)
pZk
kk (4)
Min i
ii ij k m
ijkmijkmk
ki k
ikik XFWaZC
ijjik
ijijkiikC Where,
ijmijkijkmijkm FF
b
i ij
hikijik
i ijij
b
i ij
hikijk ZWbZWZWbw 1
1
kHh (8)
}1,0{, ikijkm ZX (7)
Sub problem
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Sub problem
2
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Solution Method contd..
[SUB1]: k
ki k
ikik WaZCMin
Subject to:
kkik ZZ for all i, k (3)
pZk
kk
b
i ij
hikijik
i ijij
b
i ij
hikijk ZWbZWZWbw 1
1
kHh (8)
}1,0{, ikijkm ZX (7)
(4)
In absence of this constraint, problem
separates into k smaller problems;
each can be solved using cutting plane
method
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Solution Method contd..
Solution implemented in MATLAB 7.0
[SUB1-k] solved using CPLEX 10
CPLEX called from MATLAB
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Computational Analysis
# Nodes # Hubs Time (sec) Hubs % Gap
5 2 3.113 4,5 0.38
8 2 86.322 4,9 1.00
10 2 42.049 3,7 0.66
12 3 719.763 1,3,8 0.98
15 3 1800.00 2,14,15 2.81
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Discussion
Solution speed can be improved by using a compiled code (in C or Fortran). MATLAB is inefficient in executing loops as it is interpreted line by line.
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Conclusion
A model for Hub and Spoke Network Design solved using lagrangean relaxation
Model extended to address the issue of congestion
Good solutions obtained in reasonable time Solution speed can be further improved if
implemented in a language that uses a compiler
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