1 Hub Location & Hub Network Design James F. Campbell College of Business Administration & Center for Transportation Studies University of Missouri-St. Louis, USA Spring School on Supply Chain and Transportation Network Design HEC Montreal May 14, 2010
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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.
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.
• 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??
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.