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### Hub Location & Hub Network Design

Spring School on Supply Chain and Transportation Network Design

HEC Montreal

May 14, 2010

James F. Campbell

College of Business Administration &

Center for Transportation Studies

University of Missouri-St. Louis, USA

Outline

- Introduction, examples and background.
- “Classic” hub location models.
- Interesting “recent” research.
- Better solutions for classic models.
- More realistic and/or complex problems
- Dynamic hub location.
- Models with stochasticity.
- Competition.
- Data sets.
- Conclusions.

Design a Network to Serve 32 Cities

32 demand points (origins and destinations)

32*31/2 = 496 direct connections

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.

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.

Final Network

6 connected hubs, 1 isolated hub and 8 hub arcs

m

k

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 nodesprovide:
- Sorting, switching and connection.
- Consolidation/break-bulk to access reduced cost hub arcs.

distribution

transfer

collection

i

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.

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.

Hub Location Research

- Strategiclocation 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.

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.

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.

Traditional Discrete Location Models

- 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).

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).

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.

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…

Hub Median Model

- p-Hub Median: Locate p fully interconnected hubs to minimize the total transportation cost.
- Assume:
- Every o-d path visits at least 1 hub.
- Inter-hub cost per unit flow is discounted using .

3 Hub Median Optimal Solution

Boston

Chicago

Cleveland

Dallas

Hub Median Formulations

- Cost: i k m j : χcik + ckm + δcmj

distribution

j

collection

transfer

i

- 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

k

Min

Subject to

Link flows and hubs

Serve all o-d flows

Use p hubs

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

Serve all o-d flows

Use p hubs

Link flows & hubs

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 ifrom hub m to destination j

distribution

j

collection

transfer

X imj

i

Zik

k

Y ikm

Min

Hub Median Formulations

- Multiple allocation – 3 subscripted “flow” variables

Min

Subject to

Serve all o-d flows

Use p hubs

Flow balance

Link flows & hubs

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.

distribution

j

collection

transfer

i

k

- Much recent attention:
- Ernst, Hamacher, Jiang, Krishnamoorthy, and Woeginger, 2009, “Uncapacitated single and multiple allocation p-hub center problems”, Computers & OR

Hub Center Formulation

- Xik= 1 if node i is allocated to hub k, and 0 otherwise
- Xkk= 1 node k is a hub
- zis 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

Serve all o-d flows

Link flows & hubs

Use p hubs

Hub radius

Objective

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.
- Dynamic hub location.
- Models with stochasticity.
- Competition.
- Data sets.

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.

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.

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.

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.

235

305

Chicago

120

94

85

Cleveland

76

Dallas

166

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

low flows on hub arcs

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.

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.

Hub Median and Hub Arc Location

Hub Median

p=3

Hub Arc Location

q=3

5 hubs &

3 hub arcs

3 hubs &

3 hub arcs

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!

TransitDrop-off Pickup

DestDistance 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

SEA2087 4 22:00 8:30

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

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 Level

Direct o-d DistanceMax Travel Distance

0 - 400 miles 600 miles

400 - 1000 miles 1200 miles

1000 - 1800 miles 2000miles

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

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.

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.

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”.

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.

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.

Hub Arc Location with Isolated Hubs

- Locate q hub arcs with p hubsto 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.

Cost: i-k-m-j =

Hub Network Expansion No SL, =0.6

Add a hub arc between existing hubs

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

Transportation Cost

Add a new isolated hub

3, 3, 0

949.2

6, 6, 0

803.5

3, 2, 0

965.2

4, 3, 1

890.6

5, 4, 1

843.2

6, 5, 1

812.0

4, 2, 1

906.6

5, 3, 1

859.1

6, 4, 2

825.7

7, 5, 2

801.7

Start with a 3-hub optimal solution

5, 2, 2

875.7

6, 3, 2

841.6

7, 4, 3

815.3

6, 2, 3

862.7

7, 3, 3

831.2

Geographic Expansionq=3 hub arcs

Add 5 West- Coast cities

Optimal with no west-coast cities, p=4

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

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.

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→jis at most .

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.

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.

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.

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).

Distance Ratio and Cost Ratio

Cost ratio (freight):

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

DRij=(DijA–DijB)/(DijA+DijB)

Distance ratio (passengers):

DijA: The distance for the trip from ito j that achieves the minimum cost for Firm A.

j

DijB: The distance for the trip from ito j that achieves the minimum cost for Firm B.

CijB: The minimum cost for the trip fromito j for Firm B.

i

k

CRij=(CijA–CijB)/(CijA+CijB)

As DijA (or CijA) 0, DRij(or CRij) -1, and Firm A captures all revenue.

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.

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+dljs
- = cost discount factor for hub arcs, 0<≤1.
- Decision variables:
- 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)

l

k

j

i

HALCE-A (Firm A’s Problem)

Maximize A’s total revenue

Hub arcs

& hubs

Network

Flow

Firm B finds an

optimal solution

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…

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.

Results: High Customer Selectivity

Distance ratio allocation

qA=qB=2, a=0.6

Revenue = airfare

Revenue = distance

Red lines: Firm A’s optimal solution

Blue lines: Firm B’s optimal solution

Hub Use with Distance Ratio Allocation

92.2%

86.3%

47.0%

47.8%

57.4%

Top hub arcs for Firm A

Top hub arcs for Firm B

Cost Ratio vs. Distance Ratio

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

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.

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.

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?

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.

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.

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.

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.

Alpha

- What is the “right” value of?

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.

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