1

Hub Location & Hub Network Design

James F. CampbellCollege of Business Administration &

Center for Transportation StudiesUniversity of Missouri-St. Louis, USA

Spring School on Supply Chain and Transportation Network Design

HEC MontrealMay 14, 2010

2

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.

3

Design a Network to Serve 32 Cities

32 demand points (origins and destinations)

32*31/2 = 496 direct connections

4

One Hub

Single hub: Provides a switching, sorting and connecting (SSC) function.

Access arc connect non-hubs to hubs

Hub networks concentrate flows to exploit economies of scale in

transportation.

5

Two Hubs and One Hub Arc

1 hub arc & 2 connected hubs: Hubs also provide a consolidation and break-bulk (CB) function.

Multiple Allocation

Flows are further concentrated on hub arcs.

6

Multiple Allocation Four Hub Median

4 fully connected hubs38 access arcs

7

Single Allocation Four Hub Median

4 fully connected hubs28 access arcs

8

Multiple Hubs and Hub Arcs

9

Final Network

6 connected hubs, 1 isolated hub and 8 hub arcs

10

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

- Distance: i k m j = dik + dkm + dmj

• Hub nodes provide: - Sorting, switching and connection.- Consolidation/break-bulk to access reduced cost hub

arcs.

k

j

micollection transfer

dist

ribut

io

n

11

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.

12

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.

13

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.

14

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-to-many transportation.- Built on work with GM by Daganzo, Newell, Hall,

Burns, etc. in 1980s.

- Daganzo, 1987, “The break-bulk role terminals in many-to-many logistics networks”, Operations Research.

• Considered origin-hub-hub-destination, but without discounted inter-hub transportation.

15

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.

16

Traditional Discrete Location Models

• Demand occurs at discrete points.

•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).

•Demand points are assigned to the closest (least cost) facility.

17

Discrete Hub Location Models

• Demand is flows between origins and destinations.

•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).

•Non-hubs can be allocated to multiple hubs.

18

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.

19

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…

20

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 .

Boston

Dallas

Chicago

Cleveland

3 Hub Median Optimal Solution

21

Hub Median Formulations

• Cost: i k m j : χcik + ckm + δcmj

k

j

mi

collection transfer distributio

n

• Single allocation:

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

ji m

jmjm

k m

jmikkm

k

ikikij ZcZZcZcW,

iZk

ik 1

pZk

kk kiZik ,}1,0{

Use p hubs

Serve all o-d flows

kZpnZ kk

i

ik )1( Link flows and hubsSubject to

22

Hub Median Formulations

Min

Subject to

ji k m

ijkmijkm

j

jiij XCWW )(

• 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

jijiXk m

ijkm ,,1

pHk

k

jimkjiX ijkm ,,,,0

jikjiHXXXkm

kijmkijkmijkk

,,,)(

Use p hubs

Link flows & hubs

Serve all o-d flows

kHk 1,0

23

Hub Median Formulations

• Multiple allocation: 3 subscripted “flow” variables

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

k

j

mi

collection transfer distributio

nZik Y ikm

X imj

Min

i m

imj

j

mj

k m

ikmkm

k

ikik XcYcZc

24

Hub Median Formulations

kHk 1,0

Min

Subject to

• Multiple allocation – 3 subscripted “flow” variables

iWZj

ij

k

ik

pHk

k

mkjiXYZ imj

ikmik ,,,0,,

kiZYXYj

ik

m

imk

ikj

m

ikm ,0

Use p hubs

Link flows & hubs

Serve all o-d flows

i m

imj

j

mj

k m

ikmkm

k

ikik XcYcZc

jiWX ij

m

imj ,

Flow balance

kiWHZj

ijkik , jmWijHX

i

m

i

imj ,

25

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.

k

j

mi

collection transfer distributio

n

• Much recent attention: - Ernst, Hamacher, Jiang, Krishnamoorthy, and

Woeginger, 2009, “Uncapacitated single and multiple allocation p-hub center problems”, Computers & OR

26

Hub Center Formulation

Min

Subject to

z

iXk

ik 1

pXk

kk

kiX ik ,}1,0{

kiXcr ikikk ,

Use p hubs

Link flows & hubs

Serve all o-d flows

kiXX kkik ,

mkcrrz kmmk

Hub radius

Objective

• 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).

k

27

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.

28

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.

29

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.

30

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.

31

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.

32

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)!

Boston85

Dallas

Chicago76

217

166

305

235

12094

Cleveland

low flows on hub arcs

3 Hub Median Optimal Solution

33

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.

34

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-hub-hub-destination): model HAL1.

35

Hub Median and Hub Arc Location

Hub Medianp=3

Hub Arc Location q=3

3 hubs & 3 hub arcs

5 hubs & 3 hub arcs

36

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! Transit Drop-off Pickup

Dest Distance Days at STL at Dest

ATL 575 2 22:00 7:00

JFK 982 2 22:00 9:00

MIA 1230 3 22:00 8:00

ORD 308 1 22:00 9:00

SEA 2087 4 22:00 8:30

• Campbell, 2009, “Hub location for time definite transportation”, Computers & OR.

37

Service Levels

• Limit the travel distance via the hub network to ensure the schedule (high service level) can be met with ground transport.

• 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.

High Service LevelDirect o-d Distance Max Travel Distance 0 - 400 miles 600 miles 400 - 1000 miles 1200 miles 1000 - 1800 miles 2000 miles

38

Time Definite Hub Arc Solutions for CAB

Medium SL solution - 9 hubs!

High SL solution - 10 hubs

Low SL solution - 9 hubs!

=0.2, p=10, and q=5

39

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.

40

Models with Congestion

Elhedhli and Wu, 2010, “A Lagrangean heuristic for hub-and-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:

- Uses multiple capacity levels.- Solves small problems up to 4 hubs and 25

nodes to within 1% of optimality.

i j

ikijk

i j

ikij

kZWCapacity

ZW

Congestion

41

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:

- Solves with “favorable weight based evolutionary algorithm”.

k

i j m

ijkmij

k Capacity

XW

Congestion

42

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.

43

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.

44

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 p2q

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.

mjkmik ddd Cost: i-k-m-j =

45

Hub Network Expansion No SL, =0.6

# of hubs , # of hub arcs, # isolated hubs

Transportation Cost

Add a hub arc between existing

hubs

Add a new isolated hub

3, 2, 0

965.2

3, 3, 0

949.2

4, 2, 1

906.6

4, 3, 1

890.6

5, 2, 2875.7

5, 3, 1

859.1

6, 2, 3

862.7

5, 4, 1843.2

6, 3, 2

841.6

6, 4, 2

825.7

7, 3, 3

831.2

6, 5, 1

812.0

7, 4, 3

815.3

6, 6, 0803.5

7, 5, 2

801.7Start with a 3-

hub optimal solution

46

Geographic Expansionq=3 hub arcs

Allow 1 Isolated Hub 1 isolated hub,

Cost=914

Allow hub arcs to be moved 1 isolated hub,

Cost=864

Optimal with no west-coast cities, p=4

Add 5 West- Coast cities

No isolated hubs, Cost=1085

47

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.

48

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 .

49

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.

50

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 leader, knowing the leader’s hub arc locations.

• 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.

51

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.

52

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).

53

Distance Ratio and Cost Ratio

DijA: The distance for the trip from i to j that achieves the

minimum cost for Firm A.Dij

B : The distance for the trip from i to j that achieves the minimum cost for Firm B.

Distance ratio (passengers):

DRij =(Dij

A–DijB) /(Dij

A +DijB)

CijA : The minimum cost for the trip from i to j for Firm A.

Cost ratio (freight):

CijB : The minimum cost for the trip from i to j for Firm B.

CRij =(Cij

A–CijB) /(Cij

A +CijB)

i k

j

l

As DijA (or Cij

A) 0, DRij (or CRij) -1, and Firm A captures all

revenue.

54

5-level Step Function for Customer Allocation

CRij or Drij

–r1

–r1 to –r2

–r2 to r2

r2 to r1

> r1

ΦijA(xA,xB)

100%

75%

50%

25%

0%

ΦijA(xA,xB) = fraction of

demand captured by Firm A

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.

Fraction of demand captured by Firm A

55

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.

• Decision variables:- xijkl

A (xijklB) = flow for i k l j for Firm A (B)

- yklA (ykl

B) = 1 if there is a hub arc k–l for Firm A (B)

- zkA (zk

B) = 1 if there is a hub at city k for Firm A (B)

i jk l

iklY

56

HALCE-B (Firm B’s problem)

}1,0{,,

,,1

,,,,

,,,,,,

,

,1

,..

)),(1(

,

,

zyx

Vjix

ijVkjizx

klijVlkjiyx

Vkyyz

Vkzz

qyts

xxWFMaximize

Bk

Bkl

Bijkl

lk

Bijkl

Bk

Bijkk

Bkl

Bijkl

kl

Blk

kl

Bkl

Bk

Ak

Bk

B

lk

Bkl

Vi ij

BAAijijij

NetworkFlow

Hub arcs & hubs

Maximize B’s total revenue

57

HALCE-A (Firm A’s Problem)

).,,(],,[

},1,0{,,

,,1

,,,,

,,,,,,

,

,..

),(

,

AAABBB

Ak

Akl

Aijkl

lk

Aijkl

Ak

Aijkk

Akl

Aijkl

kl kl

Alk

Akl

Ak

A

Vk kl

Akl

Vi ij

BAAijijij

zyxzyx

zyx

Vjix

ijVkjizx

klijVlkjiyx

Vkyyz

qyts

xxWFMaximize

NetworkFlow

Hub arcs & hubs

Maximize A’s total revenue

Firm B finds an optimal solution

58

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…

59

540 Problem Scenarios with CAB data

• 2 OD revenue sets:- airfare : IATA Y class airfares - distance : direct OD distance

• 3 levels of customer selectivity:- low: (r1, r2)=(0.75,0.25)

- medium: (r1, r2)=(0.083,0.015)

- high: (r1, r2)=(0,0) (“all-or-nothing”)

• 2 Customer allocation schemes:- 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.

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

Direct Distance (miles)

Air

fare

(U

SD

)

60

Results: High Customer Selectivity

Red lines: Firm A’s optimal solutionBlue lines: Firm B’s optimal solution

Revenue = airfare Revenue = distance

Distance ratio allocationqA=qB=2, a=0.6

61

Hub Use with Distance Ratio Allocation

92.2%

86.3%

57.4% 47.8%

47.0%

Top hub arcs for Firm ATop hub arcs for Firm B

62

Cost Ratio vs. Distance Ratio

Revenue=distance, qA=qB=3, a=0.6

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 solutionBlue lines: Firm B’s optimal solution

Only 15% of revenues are from paths with a hub arc.

Over 67% of revenues are from paths with a hub arc.

63

Findings

• The leader (Firm A) usually has an advantage, but not always (“first entry paradox”).

• Distance ratio allocation encourages one-stop routes (as preferred by passengers).

• Cost ratio allocation encourages more circuitous two-stop 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.

64

Competitive Model Conclusions

• There are some interesting differences between the leader’s and follower’s strategies:- 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.

65

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?

66

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.

0 400 800 1200 1600 2000 2400 2800-100

300

700

1100

1500

COG

1-median

67

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.

0 10000 20000 30000 40000 50000 600000

10000

20000

30000

40000

50000

60000 AP200

COG

Median

10000 20000 30000 40000 50000 600000

10000

20000

30000

40000

50000

60000 AP20

COG

Median

68

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.

26 28 30 32 34 36 38 40 42 4436

37

38

39

40

41

42

43TR81

COG

1-median

69

Concentration of Demand

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cumulative Demand Curves

TR81 o-d flows

CAB25 o-d flows

AP200 nodes

TR81 nodes

CAB25 nodes

AP20 nodes

Cumulative % of o-d pairs or nodes

cum

ula

tive %

of

dem

and

70

Spatial Distribution of Demand

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Cumulative Distribution of Demand

CAB25AP20AP200TR81

% of max distance from median

% o

f fl

ow

71

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.

72

Alpha

• What is the “right” value of?

Value Mode Location Reference0.25-0.375 Truck-postal Australia

Ernst and Krishnamoorthy, Location Science 1996

0.7 Truck EULimbourg and Jourquin, Transportation Research E 2009

0.365 Truck-rail EULimbourg and Jourquin, Transportation Research E 2009

0.7 – 1.0 LTL Brazil Cunha and Silva, EJOR 2007

0.4946 Truck (time definite) Taiwan

Chen, Networks and Spatial Economics 2010

73

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.

• Link to practice.

74

Questions?