### New Models for Hub Location and Network Design

```Hub Location & Hub
Network Design
Spring School on Supply Chain and
Transportation Network Design
HEC Montreal
May 14, 2010
James F. Campbell
Center for Transportation Studies
University of Missouri-St. Louis, USA
1
Outline
• Introduction, examples and background.
• “Classic” hub location models.
• Interesting “recent” research.
I.
Better solutions for classic models.
II. More realistic and/or complex problems
III. Dynamic hub location.
IV. Models with stochasticity.
V.
Competition.
VI. Data sets.
• Conclusions.
2
Design a Network to Serve 32 Cities
32 demand points (origins and destinations)
32*31/2 = 496 direct connections
3
One Hub
Access arc connect non-hubs to hubs
Single hub: Provides a
switching, sorting and
connecting (SSC) function.
Hub networks concentrate flows to exploit
economies of scale in transportation.
4
Two Hubs and One Hub Arc
Multiple Allocation
1 hub arc & 2 connected hubs:
Hubs also provide a consolidation
and break-bulk (CB) function.
Flows are further concentrated on hub arcs.
5
Multiple Allocation Four Hub Median
4 fully connected hubs
38 access arcs
6
Single Allocation Four Hub Median
4 fully connected hubs
28 access arcs
7
Multiple Hubs and Hub Arcs
8
Final Network
6 connected hubs,
1 isolated hub and
8 hub arcs
9
Hub Networks
• Allow efficient “many-to-many” transportation:
- Require fewer arcs and concentrate flows to exploit
transportation economies of scale.
• Hub arcs provide reduced cost transportation between
two hubs (usually with larger vehicles).
- Cost: i  k  m j : Cijkm = cik + ckm + cmj
j
- Distance: i  k  m j = dik + dkm + dmj
i
• Hub nodes provide:
m
k
- Sorting, switching and connection.
- Consolidation/break-bulk to access reduced cost hub arcs.
10
Hub Location Applications
• Passenger and Freight Airlines:
- Hubs are consolidation airports and/or
sorting centers.
- Non-hubs are feeder airports.
• Trucking:
- LTL hubs are consolidation/break-bulk terminals.
- Truckload hubs are relay points to change drivers/tractors.
- Non-hubs are end-of-line terminals.
• Postal operations:
- Hubs are sorting centers; non-hubs are regional post offices.
• Public transit:
- Hubs are subway/light-rail stations.
- Non-hubs are bus stations or patron o/d’s.
• Computer & telecom networks.
11
Hub Location Motivation
• Deregulation of transportation in USA:
- Airlines (1978).
- Trucking (1980).
• Express delivery industry (Federal Express began
in 1973).
- Federal Express experiences:
• Developed ILP models in ~1978 to evaluate 1 super-hub
vs. 4 hubs.
• Used OR models in mid-1970s to evaluate adding
“bypass hubs” to handle increasing demand.
• Large telecommunications networks.
12
Hub Location Research
• Strategic location of hubs and design of hub
networks.
- Not service network design, telecom, or continuous
location research.
• Began in 1980’s in diverse fields:
- Geography, Transportation, OR/MS, Location theory,
Telecommunications, Network design, Regional science,
Spatial interaction theory, etc.
• Builds on developments in “regular” facility location
modeling.
13
Hub Location Foundations
• First hub publications: Morton O’Kelly (1985-1987):
- Transportation Science, Geographical Analysis, EJOR:
• First math formulation (quadratic IP).
• 2 simple heuristics for locating 2-4 hubs with CAB data set.
- Focus on single allocation and schedule delay.
• Continuous approximation models for many-tomany transportation.
- Built on work with GM by Daganzo, Newell, Hall, Burns,
etc. in 1980s.
- Daganzo, 1987, “The break-bulk role terminals in manyto-many logistics networks”, Operations Research.
• Considered origin-hub-hub-destination, but without
discounted inter-hub transportation.
14
Hub Location & Network Design
Given:
- Network G=(V,E)
- Set of origin-destination flows, Wij
- Discount factor  for hub arcs, 0<<1
Design a minimum cost network with hub nodes
and hub arcs to satisfy demand Wij.
Select hub nodes and hub arcs.
Assign each non-hub node to hubs.
15
• Demand occurs at discrete points.
• Demand points are assigned to the closest (least
cost) facility.
• Objective is related to the distance or cost between
the facilities and demand points.
• “Classic” problems:
- p-median (pMP): Minimize the total transportation cost (demand
-
weighted total distance).
Uncapacitated facility location problem (UFLP): Minimize the sum of
fixed facility and transportation costs.
p-center: Minimize the maximum distance to a customer.
Set Covering: Minimize the # of facilities to cover all customers.
Maximum covering: Maximize the covered demand for a given
number of facilities (or given budget).
16
Discrete Hub Location Models
• Demand is flows between origins and destinations.
• Non-hubs can be allocated to multiple hubs.
• Objective is usually related to the distance or cost for
flows (origin-hub-hub-destination).
- Usually, all flows are routed via at least one hub.
• Analogous “classic” hub problems:
- p-hub median (pMP): Minimize the total transportation cost
-
(demand weighted total distance).
Uncapacitated hub location problem (UHLP): Minimize the sum of
fixed hub and transportation costs.
p-hub center: Minimize the maximum distance to a customer.
Hub Covering: Minimize the # of hubs to cover all customers.
Maximum covering: Maximize the covered demand for a given
number of hubs (or given budget).
17
Hub Location Research
• Very rich source of problems - theoretical and
practical.
• Problems are hard!!
• A wide range of exact and heuristic solution
approaches are in use.
• Many extensions: Capacities, fixed
costs for hubs and arcs, congestion,
hierarchies, inter-hub and access
network topologies, competition,
etc.
• Many areas still awaiting good
research.
18
Hub Location Literature
• Early hub location surveys/reviews:
- Campbell, 1994, Studies in Locational Analysis.
 23 transportation and 9 telecom references.
- O’Kelly and Miller, 1994, Journal of Transport Geography.
- Campbell, 1994, “Integer programming formulations of
discrete hub location problems”, EJOR.
- Klincewicz, 1998, Location Science.
• Recent surveys:
- Campbell, Ernst and Krishnamoorthy, 2002, in Facility
Location: Applications and Theory.
- Alumur and Kara, 2008, EJOR (106 references).
- Computers & Operations Research , 2009, vol. 36.
• Much recent and current research…
19
Hub Median Model
• p-Hub Median: Locate p fully interconnected hubs
to minimize the total transportation cost.
• Assume:
(1) Every o-d path visits at least 1 hub.
(2) Inter-hub cost per unit flow is discounted using .
3 Hub Median
Optimal Solution
Boston
Chicago
Cleveland
Dallas
20
Hub Median Formulations
• Cost: i  k  m j : χcik + ckm + δcmj
i
• Single allocation:
transfer
k
j
m
Zik= 1 if node i is allocated to a hub at k ; 0 otherwise
Zkk= 1 if node k is a hub; 0 otherwise
Min
Subject to

Wij  
cik Z ik  

i, j
k
 k



ckm Z ik Z jm  
c jm Z jm 

m
m



Z  (n  p 1)Z  k Link flows and hubs
Z ik  1  i
Serve all o-d flows

k
ik
kk
i
Z
kk
 p
Use p hubs
k
Z ik  {0,1}  i, k
21
Hub Median Formulations
• Multiple allocation: 4 subscripted “path” variables
Xijkm= fraction of flow that travels i-k-m-j
Hk = 1 if node k is a hub; 0 otherwise
Cost: i  k  m j : Cijkm = χcik + ckm + δcmj
Min
Subject to
  (W
ij
i j
 W ji )
j
 C
k
 X  1
Hk  p

k
X
 ( X
ijkm
k
ijkm X ijkm
m
 i, j , i  j
Serve all o-d flows
m
ijkk
Use p hubs
ijkm
 X ijmk )  H k
 i, j, k , i  j
mk
X ijkm  0  i, j, k , m, i  j
H k  0,1
k
22
Hub Median Formulations
• Multiple allocation: 3 subscripted “flow” variables
i
transfer
Zik
k
Y ikm
m
X imj
j
Zik= flow from origin i to hub k
Y ikm= flow originating at i from hub k to hub m
X imj= flow originating at i from hub m to destination j
Min


cik Z ik  


i 
k
k
 

m
i
ckmYkm


m
j
i
cmj X mj





23
Hub Median Formulations
• Multiple allocation – 3 subscripted “flow” variables
Min
Subject to


cik Z ik  

 k
i 
k
 

m
 Z  W
ik


m
j
j

i
cmj X mj





i
ij
k
i
ckmYkm
Serve all o-d flows
i
X mj
 Wij
 i, j
m
Hk  p

k
Y
i
km

m
Use p hubs
 X  Y
i
kj
j
i
mk
Z ik  0
 i, k
m
W  i, k
 X mji  H m W ij  m, j
Z ik  H k
ij
j
i
Flow balance
i
H k  0,1  k
i
i
Zik , Ykm
, X mj
0
 i, j, k , m
24
Hub Center and Hub Covering
• Introduced as analogues of “regular” facility center and
covering problems…but notion of covering is different.
• Campbell (EJOR 1994) provided 3 types of
centers/covering:
- Maximum cost/distance for any o-d pair
- Maximum cost /distance for any single link in an o-d path.
- Maximum cost/distance between an o/d and a hub.
i
transfer
k
j
m
• Much recent attention:
- Ernst, Hamacher, Jiang, Krishnamoorthy, and Woeginger,
2009, “Uncapacitated single and multiple allocation p-hub
center problems”, Computers & OR
25
Hub Center Formulation
• Xik = 1 if node i is allocated to hub k, and 0 otherwise
• Xkk = 1 node k is a hub
z is the maximum transportation cost between all o–d pairs.
rk = “radius” of hub k (maximum distance/cost between hub k and
the nodes allocated to it).
Min
Subject to
z
X
ik
1  i
k
X ik  X kk
X
kk
 i, k
 p
k
rk  cik X ik  i, k
z  rk  rm  ckm  k  m
X ik  {0,1}  i, k
Serve all o-d flows
Use p hubs
Objective
k
26
Hub Location Themes
I. Better solution algorithms for “classic” problems.
II. More realistic and/or complex problems.
- More general topologies for inter-hub network and access
network.
- Objectives with cost + service.
- Other: multiple capacities, bicriteria models, etc.
III. Dynamic hub location.
IV. Models with stochasticity.
V. Competition.
VI. Data sets.
27
I. Better solutions for “classic” problems
• Improved formulations lead to better solutions and
solving larger problems…
 Hamacher, Labbé, Nickel, and Sonneborn, 2004 “Adapting
polyhedral properties from facility to hub location
problems”, Discrete Applied Mathematics.
 Marín, Cánovas, and Landete, 2006, “New formulations for
the uncapacitated multiple allocation hub location
problem”, EJOR.
- Uses preprocessing and polyhedral results to develop tighter
formulations.
- Compares several formulations.
28
Better solutions for “classic” problems
• Contreras, Cordeau, and Laporte, 2010, “Benders
decomposition for large-scale uncapacitated hub location”.
- Exact, sophisticated solution algorithm for UMAHLP.
- Solves very large problems with up to 500 nodes (250,000
commodities).
- ~2/3 solved to optimality in average ~8.6 hours.
• Contreras, Díaz, and Fernández, 2010, “Branch and price for
large scale capacitated hub location problems with single
assignment”, INFORMS Journal on Computing.
- Single allocation capacitated hub location problem.
- Solves largest problems to date to optimality (200 nodes) up to
12.5 hrs.
- Lagrangean relaxation and column generation and branch and
price.
29
II. More Realistic and/or Complex Problems
• More general topologies for inter-hub network and
access network.
- Inter-hub network: Trees, incomplete hub networks, isolated
hubs, etc.
- Access network: “Stopovers”, “feeders”, routes, etc.
• Better handling of economies of scale.
- Flow dependent discounts, flow thresholds, etc.
- Restricted inter-hub networks.
• Objectives with cost + service.
• Others: multiple capacities, bicriteria models, etc.
30
Weaknesses of “Classic” Hub Models
• Hub center and hub covering models:
- Not well motivated by real-world systems.
- Ignore costs: Discounting travel distance or
time while ignoring costs seems “odd”.
• Hub median (and UHLP) models:
- Assume fully interconnected hubs.
- Assume a flow-independent cost discount on
all hub arcs.
- Ignore travel times and distances.
31
Hub Median Model
• p-Hub Median: Locate p fully interconnected hubs
to minimize the total transportation cost.
- Hub median and related models do not accurately model
economies of scale.
- All hub-hub flows are discounted (even if small) and no access
arc flows are discounted (even if large)!
3 Hub Median
Optimal Solution
Boston
Chicago
Cleveland
Dallas
low flows on hub arcs
32
Better Handling of Economies of Scale
• Flow dependent discounts: Approximate a non-linear
discounts by a piece-wise linear concave function.
- O’Kelly and Bryan, 1998, Trans. Res. B.
- Bryan, 1998, Geographical Analysis.
- Kimms, 2006, Perspectives on Operations
Research.
• More general topologies for inter-hub network and
access network
- “Tree of hubs”: Contreras, Fernández and Marín, 2010, EJOR.
- “Incomplete” hub networks: Alumur and Kara, 2009,
Transportation Research B
- Hub arc models: Campbell, Ernst, and Krishnamoorthy, 2005,
Management Science.
33
Hub Arc Model
• Hub arc perspective: Locate q hub arcs rather than p
fully connected hub nodes.
- Endpoints of hub arcs are hub nodes.
• Hub Arc Location Problem: Locate q hub arcs to
minimize the total transportation cost.
 q hub arcs and ≤2q hubs.
 Assume as in the hub median model that:
• Every o-d path visits at least 1 hub.
• Cost per unit flow is discounted on q hub arcs using .
• Each path has at most 3 arcs and one hub arc (origin-hubhub-destination): model HAL1.
34
Hub Median and Hub Arc Location
Hub Median
p=3
3 hubs &
3 hub arcs
Hub Arc Location
q=3
5 hubs &
3 hub arcs
35
Time Definite Hub Arc Location
• Combine service level (travel time) constraints with cost
minimization to model time definite transportation.
• Motivation: Time definite trucking:
- 1 to 4 day very reliable scheduled service between terminals.
- Air freight service by truck!
Dest
ATL
JFK
MIA
ORD
SEA
•
Transit
Distance Days
575
2
982
2
1230
3
308
1
2087
4
Drop-off
at STL
22:00
22:00
22:00
22:00
22:00
Pickup
at Dest
7:00
9:00
8:00
9:00
8:30
Campbell, 2009, “Hub location for time definite transportation”,
Computers & OR.
36
Service Levels
• Limit the travel distance via the hub network to ensure
the schedule (high service level) can be met with
ground transport.
Direct o-d Distance
0 - 400 miles
400 - 1000 miles
1000 - 1800 miles
High Service Level
Max Travel Distance
600 miles
1200 miles
2000 miles
• Problems with High service levels (High SL) have
reduced sizes, since long paths are not feasible.
• Formulate as MIP and solve via CPLEX 10.1.1.
37
Time Definite Hub Arc Solutions for CAB
=0.2, p=10, and q=5
Low SL solution - 9 hubs!
Medium SL solution - 9 hubs!
High SL solution - 10 hubs
38
Time Definite Hub Locations
• High service levels make problems “easier”.
• High service levels “force” some hub locations.
• Good hub cities:
- Large origins and destinations.
• Chicago, New York, Los Angeles.
- Large isolated cities near the perimeter.
• Miami, Seattle.
- Some centrally located cities.
• Kansas City, Cleveland.
• Poor hub cities:
- Medium or small cities near large origins & destinations.
• Tampa.
39
Models with Congestion
 Elhedhli and Wu, 2010, “A Lagrangean heuristic for huband-spoke system design with capacity selection and
congestion”, INFORMS Journal on Computing.
- Single allocation.
- Minimize sum of transportation cost, fixed cost and congestion
“cost”.
- Congestion at hub k:
W Z
Congestion 
Capacity  W Z
ij ik
i
j
k
k
ij ik
i
j
- Uses multiple capacity levels.
- Solves small problems up to 4 hubs and 25 nodes to
within 1% of optimality.
40
Another Model with Congestion
 Koksalan and Soylu, 2010, “Bicriteria p-hub location
problems and evolutionary algorithms”, INFORMS
Journal on Computing.
- Two multiple allocation bicriteria uncapacitated p-HMP models.
• Model 1: Minimize total transportation cost and minimize total
collection and distribution cost.
• Model 2: Minimize total transportation cost and minimize maximum
delay at a hub.
- Delay (congestion) at hub k:
Congestionk 
Wij  X ijkm

i
j
m
Capacityk
- Solves with “favorable weight based evolutionary
algorithm”.
41
III. Dynamic Hub Location
How should a hub network respond to changing demand??
 Contreras, Cordeau, Laporte, 2010, “The dynamic hub
location problem”, Transportation Science.
- Multiple allocation, fully interconnected hubs.
- Dynamic (multi-period) uncapacitated hub location with up to
10 time periods.
- In each period, adds new o-d pairs (commodities) and increase
or decrease the flow for existing o-d pairs.
- Hubs can be added, relocated or removed.
- Solves up to 100 nodes and 10 time periods with branch and
bound with Langrangean relaxation.
42
Isolated Hubs
• Isolated hubs are not endpoints of hub arcs.
- Provide only a switching, sorting, connecting function; not a
consolidation/break-bulk function.
- Give flexibility to respond to expanding demand with
incremental steps.
• How can isolated hubs be used, especially in response
to increasing demand in a fixed region and demand in
an expanding region.
 Campbell, 2010, “Designing Hub Networks with
Connected and Isolated Hubs”, HICSS 43
presentation.
43
Hub Arc Location with Isolated Hubs
• Locate q hub arcs with p hubs to minimize the
total transportation cost.
 If p>2q there will be isolated hubs; When p2q isolated
hubs may provide lower costs.
 Each non-hub is connected to one or more hubs.
Key assumptions:
1. Every o-d path visits at least 1 hub.
2. Hub arc cost per unit flow is discounted
using .
3. Each path has at most 3 arcs and one hub
arc: origin-hub-hub-destination.
Cost: i-k-m-j = d ik  d km  d mj
44
Hub Network Expansion
No SL, =0.6
# of hubs , # of hub arcs, # isolated hubs
Transportation Cost
existing hubs
3, 3, 0
949.2
3, 2, 0
965.2
6, 6, 0
803.5
4, 3, 1
890.6
4, 2, 1
906.6
a 3-hub
optimal
solution
5, 4, 1
843.2
5, 3, 1
859.1
5, 2, 2
875.7
6, 5, 1
812.0
6, 4, 2
825.7
6, 3, 2
841.6
6, 2, 3
862.7
7, 5, 2
801.7
7, 4, 3
815.3
7, 3, 3
831.2
45
Geographic Expansion
q=3 hub arcs
Optimal with no west-coast
cities, p=4
WestCoast
cities
Allow 1 Isolated Hub
1 isolated hub, Cost=914
No isolated hubs, Cost=1085
Allow hub arcs to be moved
1 isolated hub, Cost=864
46
Findings for Isolated Hubs
• Isolated hubs are useful to respond efficiently to:
- an expanding service region and
- an increasing intensity of demand.
• Adding isolated hubs may be a more cost effective than
adding connected hubs (and hub arcs).
• Isolated hubs seem most useful in networks having:
few hub arcs, small  values (more incentive for
consolidation), and/or high service levels.
• With expansion, the same hubs are often optimal – but
the roles change from isolated to connected.
47
IV. Models with Stochasticity
How should stochasticity be incorporated??
 Lium, Crainic and Wallace, 2009, “A study of demand
stochasticity in service network design, Transportation Science.
- Does not assume particular topology and shows hub-and-spoke
structures arise due to uncertainty.
“consolidation in hub-and-spoke networks takes place not necessarily
because of economy of scale or other similar volume-related reasons,
but as a result of the need to hedge against uncertainty”
 Sim, Lowe and Thomas, 2009, “The stochastic p-hub center
problem with service-level constraint”, Computers & OR.
- Single assignment hub covering where the travel time Tij is
normally distributed with a given mean and standard deviation.
- Locate p hubs to minimize  so that the probability is at least 
that the total travel time along the path i→k→l→j is at most .
48
V. Competitive Hub Location
• Suppose two firms develop hub networks to compete for
customers.
• Sequential location - Maximum capture problem:
- Marianov, Serra and ReVelle, 1999, “Location of hubs in a
competitive environment”, EJOR.
- Eiselt and Marianov, 2009, “A conditional p-hub location
problem with attraction functions”, Computers & OR.
• Stackelberg hub problems:
- Sasaki and Fukushima, 2001, “Stackelberg hub location
problem”, Journal of Operations Research Society of Japan.
- Sasaki, 2005, “Hub network design model in a competitive
environment with flow threshold”, Journal of Operations
Research Society of Japan.
49
Stackelberg Hub Arc Location
• Use revenue maximizing hub arc models with Stackelberg
competition.
• Two competitors (a leader and follower) in a market.
- The leader first optimally locates its own qA hub arcs, knowing
that the follower will later locate its own hub arcs.
- The follower optimally locates its own qB hub arcs after the
• Assume:
- Competitors cannot share hubs.
- Customers travel via the lowest cost path in each network.
• The objective is to find an optimal solution for the leader given the follower will subsequently design its optimal hub
arc network.
50
How to Allocate Customers among
Competitors?
• Customers are allocated between competitors based on
the service disutility, which may depend on many
factors:
- Fares/rates, travel times, departure and arrival times,
frequencies, customer loyalty programs, etc.
• For a strategic location model, we assume revenues
(fares/rates) are the same for each competitor.
• We focus on disutility measures in terms of travel
distance (time) and travel cost.
• Key factors may differ between passenger and freight
transportation.
51
Cost & Service
• For freight, a shipper does not care about the path
as long as the freight arrives “on time”.
- Often pick up at end of day and deliver at the
beginning of a future day.
- Allocate between competitors based on relative cost
of service.
• Passengers are more sensitive to the total travel
time (though longer trips allow more circuity).
- Allocate between competitors based on relative
service (travel time or distance).
52
Distance Ratio and Cost Ratio
Distance ratio (passengers):
DijA: The distance for the trip from i to j that achieves the minimum
cost for Firm A.
DijB : The distance for the trip from i to j that achieves the minimum
cost for Firm B.
j
DRij =(DijA–DijB) /(DijA +DijB)
i
l
k
Cost ratio (freight):
CijA : The minimum cost for the trip from i to j for Firm A.
CijB : The minimum cost for the trip from i to j for Firm B.
CRij =(CijA–CijB) /(CijA +CijB)
As DijA (or CijA)  0, DRij (or CRij) -1, and Firm A captures all revenue.
53
5-level Step Function for Customer Allocation
Fraction of demand
captured by Firm A
ΦijA(xA,xB) = fraction of
demand captured by Firm A
CRij or Drij
 –r1
–r1 to –r2
–r2 to r2
r2 to r1
> r1
ΦijA(xA,xB)
100%
75%
50%
25%
0%
r1 and r2 determine selectivity level of customers.
r1 = r2 = 0 is an “all-or-nothing” allocation.
r1 = 0.75, r2 = 0.50 is insensitive to differences.
54
Notation
• Given:
-
V = set of demand nodes, V (|V |=n)
Wij = set of origin-destination flows
Fij = set of origin-destination revenues (e.g. airfares)
dij = distance between i and j
Cijkl = unit cost for the path i  k  l  j = dik+dkl+dlj s
 = cost discount factor for hub arcs, 0<≤1.
Ykli
• Decision variables:
i
k
l
j
- xijklA (xijklB) = flow for i  k  l  j for Firm A (B)
- yklA (yklB) = 1 if there is a hub arc k–l for Firm A (B)
- zkA (zkB) = 1 if there is a hub at city k for Firm A (B)
55
HALCE-B (Firm B’s problem)
Maximize  FijWij (1  ijA ( x A , x B ))
iV j i
s.t.
Maximize B’s total
revenue
B

q
,
y
 kl
B
k ,l
z  1 z
B
k
A
k
k V ,
z   y  y
B
kl
B
k
l k
xijkl  y kl
B
x
B
ijkk
B
z
B
k
B
x
ijkl
 1
l k
B
lk
k V ,
Hub arcs
& hubs
i, j , k , l  V , j  i, l  k ,
i, j , k  V , j  i,
Network
Flow
i, j  V ,
k ,l
B
B
,
,
 {0,1}
y
x
z
k
kl
B
ijkl
56
HALCE-A (Firm A’s Problem)
Maximize  FijWijijA ( x A , x B )
Maximize A’s total
revenue
iV j i
s.t.
A
A
y

q
,
 kl
kV l  k
z kA   yklA   ylkA
l k
x
A
ijkl
l k
 y klA
i, j , k , l  V , j  i, l  k ,
xijkk  z kA
i, j , k  V , j  i,
A
x
A
ijkl
k V ,
Hub arcs
& hubs
1
i, j  V ,
Network
Flow
k ,l
A
xijkl
, yklA , z kA  {0,1},
[ x , y , z ]   ( x , y , z ).
B
B
B
A
A
A
Firm B finds an
optimal solution 57
Optimal Solution Algorithm
• “Smart” enumeration algorithm:
 Enumerate all of Firm A’s sets of qA hub arcs.
 For each set of Firm A’s hub arcs, use bounding tests to
enumerate only some of Firm B’s qB hub arcs and only
some OD pairs.
• Bounding tests are effective and allow problems with
up to 3 hub arcs for Firm A and Firm B to be solved to
optimality.
• But we would still like to solve larger problems…
58
540 Problem Scenarios with CAB data
• 2 OD revenue sets:
- airfare : IATA Y class airfares
2500
- distance : direct OD distance
- low: (r1, r2)=(0.75,0.25)
- medium: (r1, r2)=(0.083,0.015)
Airfare (USD)
• 3 levels of customer selectivity:
2000
- high: (r1, r2)=(0,0) (“all-or-nothing”)
• 2 Customer allocation schemes:
1500
1000
500
0
0
1000
2000
3000
Direct Distance (miles)
- Distance ratio allocation (passenger)
- Cost ratio allocation (freight)
• 5 values of : 0.2, 0.4, 0.6, 0.8, 1.0
• Up to 3 hub arcs for Firms A and B.
59
Results: High Customer Selectivity
Distance ratio allocation
qA=qB=2, =0.6
Revenue = airfare
Revenue = distance
Red lines: Firm A’s optimal solution
Blue lines: Firm B’s optimal solution
60
Hub Use with Distance Ratio Allocation
92.2%
86.3%
47.0%
57.4%
47.8%
Top hub arcs for Firm A
Top hub arcs for Firm B
61
Cost Ratio vs. Distance Ratio
Revenue=distance, qA=qB=3, =0.6
Over 67% of revenues are
from paths with a hub arc.
Only 15% of revenues are
from paths with a hub arc.
Cost Ratio allocation (freight)
Firm A’s hubs=4,6,8,12,17,22
Distance ratio allocation (passengers)
Firm A’s hubs=1,4,12,14,17,22
Red lines: Firm A’s optimal solution
Blue lines: Firm B’s optimal solution
62
Findings
• Distance ratio allocation encourages one-stop routes
(as preferred by passengers).
• Cost ratio allocation encourages more circuitous twostop routes (as in freight transportation).
• Large origins/destinations have a large advantage for
hub location.
- Peripheral cities have a geographic disadvantage for hub
location.
• Though the optimal hub arcs vary considerably, the
competitors generally use the same optimal hub nodes.
63
Competitive Model Conclusions
• There are some interesting differences between the
- The leader tends to use fewer hubs more intensively, but the
follower performs about as well in many cases!
- The leader tends to capture the higher revenue customers,
while the follower captures more, but less valuable, customers.
• Optimal network design can be very sensitive to the
customer allocation mechanisms.
64
VI. Hub Location Data Sets
• Much work has been done with only a few
data sets:
- CAB25: 25 cities in US.
- AP: up to 200 postal locations in Sydney, Australia.
- “Turkish data”: 81 nodes in Turkey
• What should alpha be?
65
CAB25 Data Set
• 25 US cities with symmetric flows based on air passenger traffic
in 1970.
• No flow from a node to itself (Wii=0).
• Subsets are alphabetical.
1500
1100
700
COG
1-median
300
-100 0
400
800
1200
1600
2000
2400
2800
66
AP Data Sets
• Up to 200 postal codes in Sydney with asymmetric flows of mail
from 1993(?) and given collection, transfer and distribution costs.
• 42.4% of flows (including all flows Wii) are at minimum level of
0.01 (mean flow=0.0995)
• Smaller data sets are created to be “ a reasonable approximation”
of the larger problem.
AP200
60000
AP20
60000
50000
50000
40000
40000
COG
30000
COG
30000
Median
Median
20000
20000
10000
10000
0
0
0
20000
40000
60000
0
20000
40000
60000
67
Turkish network: TR81
• 81 nodes for provinces in Turkey with asymmetric flows
generated based on populations.
• Often used with =0.9 (from interhub travel time discount).
• Smaller versions selected in various ways.
TR81
43
COG
42
1-median
41
40
39
38
37
36
26
28
30
32
34
36
38
40
42
44
68
Concentration of Demand
Cumulative Demand Curves
1
0.9
cumulative % of demand
0.8
0.7
TR81 o-d flows
0.6
CAB25 o-d flows
0.5
AP200 nodes
0.4
TR81 nodes
0.3
CAB25 nodes
0.2
AP20 nodes
0.1
0
0
0.2
0.4
0.6
0.8
1
Cumulative % of o-d pairs or nodes
69
Spatial Distribution of Demand
Cumulative Distribution of Demand
1
% of flow
0.8
0.6
CAB25
AP20
0.4
AP200
TR81
0.2
0
0
0.2
0.4
0.6
0.8
1
% of max distance from median
70
Distribution of Demand
• Optimal hub locations and hub networks reflect the
underlying distributions of flows (and aggregated
flows).
• All data sets have flows heavily concentrated in a few
large nodes.
• CAB is least centrally concentrated with large
peripheral demand centers.
• AP has concentrated demand and is least evenly
distributed over the region.
- Subsets of AP may not be as similar to each other as
“designed”.
• TR81 is most evenly distributed in space.
71
Alpha
• What is the “right” value of?
Value
Mode
Location
0.25-0.375
Truck-postal
Australia
0.7
Truck
EU
0.365
Truck-rail
EU
0.7 – 1.0
LTL
Brazil
0.4946
Truck (time
definite)
Taiwan
Reference
Ernst and Krishnamoorthy,
Location Science 1996
Limbourg and Jourquin,
Transportation Research E 2009
Limbourg and Jourquin,
Transportation Research E 2009
Cunha and Silva,
EJOR 2007
Chen, Networks and Spatial
Economics 2010
72
New Directions for Hub Location Research
• Better, more realistic models:
- Incorporate cost, service and competition.
- Model relevant costs (especially economies of scale)
more accurately.
- More complex networks with longer paths and direct
routes.
• Solve larger problems.(?)
• Link to service network design.
• Link to telecom hub location.