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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.

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hub location hub network design

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
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
Design a Network to Serve 32 Cities

32 demand points (origins and destinations)

32*31/2 = 496 direct connections

one hub
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
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.

multiple allocation four hub median
Multiple Allocation Four Hub Median

4 fully connected hubs

38 access arcs

single allocation four hub median
Single Allocation Four Hub Median

4 fully connected hubs

28 access arcs

final network
Final Network

6 connected hubs, 1 isolated hub and 8 hub arcs

hub networks

j

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
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
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
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
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
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
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
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 research18
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
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
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

m

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 formulations22
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 formulations23

m

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 formulations24
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

m

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

k

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
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
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
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
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
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.
hub median model32

217

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
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 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 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
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
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
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
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
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
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
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
  • 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
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 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 =

hub network expansion no sl 0 6
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 expansion q 3 hub arcs
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
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
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
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
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
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
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

l

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 s tep function for customer allocation
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
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 b firm b s problem
HALCE-B (Firm B’s problem)

Maximize B’s total

revenue

Hub arcs

& hubs

Network

Flow

halce a firm a s problem
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
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
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
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
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
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
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
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
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
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
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
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
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
Alpha
  • What is the “right” value of?
new directions for hub location research
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.