Hub and Spoke Network Design
1
Outline
Motivation
 Problem Description
 Mathematical Model
 Solution Method
 Computational Analysis
 Extension
 Conclusion

2
Motivation
1
7
5
2
4
3
8
9
3
Motivation
Spoke
s
σ = 0.25
Hubs
Spoke and Hub Network
4
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
5
Problem Description

Given a network of nodes with given flows
between each pair, determine:
Which nodes are set as hubs
 Which hub is a node assigned to


So that:

Every flow is first routed through one or two
hubs before being sent to its destination
6
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)
7
Mathematical Model
1 if spoke i assigned to hub
Z ik  
0, otherwise
X ijkm
k


1 if flow from i to j via hubs k and m in that order 


0, otherwise

m
i
k
j
8
Mathematical Model
Min
    Fijkm X ijkm
i
Subject to:
Fijkm  Wij (Cik   Ckm  Cmj )
(1)
j k m
 Z ik  1
for all i
(2)
for all i, k
(3)
k
Z ik  Z kk
 Z kk  p
(4)
k
 X ijkm  Z ik
m
 X ijkm  Z jm
for all i, j > i, k
(5)
for all i, j > i, m
(6)
k
X ijkm , Z ik  {0,1}
(7)
9
Mathematical Model

Problem size:

For number of nodes = n:
n3 (n  1)
No. of binary var iables 
 n2
2
That’s too
large!
No. of constraint s  n3  1
For
n = 15:
No. of binary var iables  23850
No. of constraint s  3376
10
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
11
Solution Method
Min
    Fijkm X ijkm
i
Subject to:
(1)
j k m
 Z ik  1
for all i
(2)
for all i, k
(3)
αi
k
Z ik  Z kk
 Z kk  p
(4)
k
 X ijkm  Z ik
m
 X ijkm  Z jm
for all i, j > i, k
(5)
βijk
for all i, j > i, m
(6)
Gijm
k
X ijkm , Z ik  {0,1}
(7)
12
Solution Method
Min
 C ik Zik    F ijkm X ijkm  i
i
Subject to:
i j i k m
k
Z ik  Z kk
i
for all i, k
 Z kk  p
(7)
(3)
(4)
k
Where,
C ik   i    ijk    jik
j i
j i
F ijkm  Fijkm   ijk   ijm
Sub
problem
2
Sub
problem
1
13
Solution Method
Constrained
added to
improve bound
[SUB2]:
  F ijkm X ijkm
Min
Subject to:
i j i k m
  X ijkm  1
for all i, j > i
k m
X ijkm {0,1}
[SUB1]:
Min
  C ik Z ik
Subject to:
Z ik  Z kk
i
k
for all i, k
 Z kk  p
k
Zik {0,1}
14
Solution Method
[MASTER]:
Max
   i  1   2
i
h
Subject to: 1    F ijkm X ijkm
for h = 1,2,….
i j i k m
 2    Cik Z h ik
i
for h = 1,2,….
k
 i ,1 , 2 free
15
Solution Algorithm

[SUB1]:

For each i, j:

Find F hn  Min ( F ijkm )
km

Set
X ijhn  1
16
Solution Algorithm

[SUB2]:
Let Z kk  1 for all k
for all i, k, if C ik  0, set Zik  1
Set S k   C ik Z ik
i
Sort Sk ' s in ascending order
Place in a set the index k of the first smallest p Sk ' s
For this set of indices,
set each associated Zkk  1
if C ik  0, set each Z ik  1, for all i, and set all other Z ik variables to zero.
17
Solution Algorithm

[Feasible Solution]:
If  Z ik  1, then
k
Set Z ik  0, for all k
Find C in  Min k (C ik | Z kk  1)
Set Z in  1
If  Zik  0, then
k
Find C in  Min k (C ik | Z kk  1)
Set Z in  1
For all i, j  i, m, k, set X ijkm  Z ik * Z jm
18
Solution Algorithm

Issues:


Slow convergence as master problem grows too
large
Could not converge in 30 minutes for 10 nodes
How
to
resolve??
?
19
Solution Algorithm

Subgradient Optimization to find lagrang
multipliers
Initialize α, β, γ;
Initialize step size
Solve SUB1; SUB2 and obtain LB
Construct a feasible solution and obtain
UB
If no improvement in LB since
long, decrease step size
No
stop
Is (UB-LB)/LB>ε?
α, β, γ
Yes
Adjust α, β, γ by the amount of
infeasibility
20
Computational Analysis
Original Model (Cplex)
# Nodes
# Hubs
5
2
0.012
8
2
10
Lagrangean Relaxation
Time (sec) Time (sec)
% Gap
Optimal ?
0.924
0.0
Y
0.112
5.048
0.0
Y
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
---
21
Analysis
Congested
22
Extended Model
b
Min
    Fijkm X ijkm 
i
Subject to:
j k m

 
 a  
WijWZijikXijkm 
k  i jjii m
 
 Z ik  1
b
(1)
for all i
(2)
for all i, k
(3)
k
Z ik  Z kk
 Z kk  p
(4)
k
 X ijkm  Z ik
m
 X ijkm  Z im
for all i, j > i, k
(5)
for all i, j > i, m
(6)
k
X ijkm , Z ik  {0,1}
Fijkm  Wij (Cik   Ckm  Cmj )
Congestion
Cost function
23
Extended Model cont..
Min
    Fijkm X ijkm 
i
j k m

h 

 ab  Wij Z ik 
 i j i

Subject to:
b


 
 max
a1  b    Wij Z ik  
h

H
k
 i j i
 

b 1
(2) – (7)
 Wij Z ik
i j i
Linear
Approximation
using tanget
planes for
congestion cost
function
24
Extended Model cont..
Min
    Fijkm X ijkm 
i
j k m
a  wk
k
Subject to:


wk  b   Wij Z ikh 
 i j i

(2) – (7)
b 1
b




   Wij Z ik   1  b     Wij Z ikh   h  H (8)
k
  i j i

 i j i




MIP with an
infinite number
of constraints
25
Solution Method (Langrangean Relaxation)
Min
 C ik Zik  aWk    F ijkm X ijkm  i
i
Subject to:
k
k
i j i k m
Z ik  Z kk
i
for all i, k
(3)
 Z kk  p
(4)
k


wk  b   Wij Z ikh 
 i j i

b 1
b




   Wij Z ik   1  b     Wij Z ikh   h  H (8)
k
  i j i

 i j i




X ijkm , Z ik  {0,1}
Where,
(7)
C ik   i    ijk    jik
j i
j i
F ijkm  Fijkm   ijk   ijm
Sub
Sub
problem
problem
12
26
Solution Method contd..
[SUB1]:
Min
  C ik Z ik  aWk
i
k
k
In absence of this
constraint, problem
separates into k
smaller problems;
each can be
solved using
cutting plane
method
Subject to:
Z ik  Z kk
for all i, k
(3)
 Z kk  p
(4)
k

h 

wk  b   Wij Z ik 
 i j i

b 1
b




   Wij Z ik   1  b     Wij Z ikh   h  H (8)
k
  i j i

 i j i




X ijkm , Z ik  {0,1}
(7)
27
Solution Method contd..

Solution implemented in MATLAB 7.0

[SUB1-k] solved using CPLEX 10

CPLEX called from MATLAB
28
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
29
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.
30
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
31
32