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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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2

3

4

5

7

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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

1

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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