Hub Location & Hub Network Design

1. 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

2. 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.

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 hubs 38 access arcs

7. Single Allocation Four Hub Median 4 fully connected hubs 28 access arcs

8. Multiple Hubs and Hub Arcs

9. Final Network 6 connected hubs, 1 isolated hub and 8 hub arcs

10. 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

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 • 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.

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. • 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).

17. 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).

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: • 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

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. • Dynamic hub location. • Models with stochasticity. • Competition. • 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. 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

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 Median p=3 Hub Arc Location q=3 5 hubs & 3 hub arcs 3 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! 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.

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 Level Direct o-d DistanceMax Travel Distance 0 - 400 miles 600 miles 400 - 1000 miles 1200 miles 1000 - 1800 miles 2000miles

38. 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

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.

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”.

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 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 =

45. 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

46. 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

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→jis 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.