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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
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 (u sually with larger vehicles). -Cost: i k m j : C ijkm = c ik + c km + c mj -Distance: i k m j = d ik + d km + d mj Hub nodes provide: -Sorting, switching and connection. -Consolidation/break-bulk to access reduced cost hub arcs. k j m i collection transfer distribution
11 Hub Location Applications Passenger and Freight Airlines: -Hubs are consolidation airports and/or sorting centers. -Non-hubs are feeder airports. Trucking: -LTL hubs are consolidation/break-bulk terminals. -Truckload hubs are relay points to change drivers/tractors. -Non-hubs are end-of-line terminals. Postal operations: -Hubs are sorting centers; non-hubs are regional post offices. Public transit: -Hubs are subway/light-rail stations. -Non-hubs are bus stations or patron o/d’s. Computer & telecom networks.
12 Hub Location Motivation Deregulation of transportation in USA: -Airlines (1978). -Trucking (1980). Express delivery industry (Federal Express began in 1973). -Federal Express experiences: Developed ILP models in ~1978 to evaluate 1 super-hub vs. 4 hubs. Used OR models in mid-1970s to evaluate adding “bypass hubs” to handle increasing demand. Large telecommunications networks.
13 Hub Location Research Strategic location of hubs and design of hub networks. -Not service network design, telecom, or continuous location research. Began in 1980’s in diverse fields: -Geography, Transportation, OR/MS, Location theory, Telecommunications, Network design, Regional science, Spatial interaction theory, etc. Builds on developments in “regular” facility location modeling.
14 Hub Location Foundations First hub publications: Morton O’Kelly (1985-1987): -Transportation Science, Geographical Analysis, EJOR: First math formulation (quadratic IP). 2 simple heuristics for locating 2-4 hubs with CAB data set. -Focus on single allocation and schedule delay. Continuous approximation models for many-to- many transportation. -Built on work with GM by Daganzo, Newell, Hall, Burns, etc. in 1980s. -Daganzo, 1987, “The break-bulk role terminals in many- to-many logistics networks”, Operations Research. Considered origin-hub-hub-destination, but without discounted inter-hub transportation.
15 Hub Location & Network Design Given: -Network G=(V,E) -Set of origin-destination flows, W ij -Discount factor for hub arcs, 0< <1 Design a minimum cost network with hub nodes and hub arcs to satisfy demand W ij. Select hub nodes and hub arcs. Assign each non-hub node to hubs.
16 Traditional Discrete Location Models Demand occurs at discrete points. Objective is related to the distance or cost between the facilities and demand points. “Classic” problems: -p-median (pMP): Minimize the total transportation cost (demand weighted total distance). -Uncapacitated facility location problem (UFLP): Minimize the sum of fixed facility and transportation costs. -p-center: Minimize the maximum distance to a customer. -Set Covering: Minimize the # of facilities to cover all customers. -Maximum covering: Maximize the covered demand for a given number of facilities (or given budget). Demand points are assigned to the closest (least cost) facility.
17 Discrete Hub Location Models Demand is flows between origins and destinations. Objective is usually related to the distance or cost for flows (origin-hub-hub-destination). -Usually, all flows are routed via at least one hub. Analogous “classic” hub problems: -p-hub median (pMP): Minimize the total transportation cost (demand weighted total distance). -Uncapacitated hub location problem (UHLP): Minimize the sum of fixed hub and transportation costs. -p-hub center: Minimize the maximum distance to a customer. -Hub Covering: Minimize the # of hubs to cover all customers. -Maximum covering: Maximize the covered demand for a given number of hubs (or given budget). Non-hubs can be allocated to multiple hubs.
18 Hub Location Research Very rich source of problems - theoretical and practical. Problems are hard!! A wide range of exact and heuristic solution approaches are in use. Many extensions: Capacities, fixed costs for hubs and arcs, congestion, hierarchies, inter-hub and access network topologies, competition, etc. Many areas still awaiting good research.
19 Hub Location Literature Early hub location surveys/reviews: -Campbell, 1994, Studies in Locational Analysis. 23 transportation and 9 telecom references. -O’Kelly and Miller, 1994, Journal of Transport Geography. -Campbell, 1994, “Integer programming formulations of discrete hub location problems”, EJOR. -Klincewicz, 1998, Location Science. Recent surveys: -Campbell, Ernst and Krishnamoorthy, 2002, in Facility Location: Applications and Theory. -Alumur and Kara, 2008, EJOR (106 references). -Computers & Operations Research, 2009, vol. 36. Much recent and current research…
20 Hub Median Model p-Hub Median: Locate p fully interconnected hubs to minimize the total transportation cost. Assume: (1) Every o-d path visits at least 1 hub. (2) Inter-hub cost per unit flow is discounted using . Boston Dallas Chicago Cleveland 3 Hub Median Optimal Solution
21 Hub Median Formulations Cost: i k m j : χc ik + c km + δc mj k j m i collection transfer distribution Single allocation: Z ik = 1 if node i is allocated to a hub at k ; 0 otherwise Z kk = 1 if node k is a hub; 0 otherwise Min Use p hubs Serve all o-d flows Link flows and hubs Subject to
22 Hub Median Formulations Min Subject to Multiple allocation: 4 subscripted “path” variables X ijkm = fraction of flow that travels i-k-m-j H k = 1 if node k is a hub; 0 otherwise Cost: i k m j : C ijkm = χc ik + c km + δc mj Use p hubs Link flows & hubs Serve all o-d flows
23 Hub Median Formulations Multiple allocation: 3 subscripted “flow” variables Z ik = flow from origin i to hub k Y i km = flow originating at i from hub k to hub m X i mj = flow originating at i from hub m to destination j k j m i collection transfer distribution Z ik Y i km X i mj Min
24 Hub Median Formulations Min Subject to Multiple allocation – 3 subscripted “flow” variables Use p hubs Link flows & hubs Serve all o-d flows Flow balance
25 Hub Center and Hub Covering Introduced as analogues of “regular” facility center and covering problems…but notion of covering is different. Campbell (EJOR 1994) provided 3 types of centers/covering: -Maximum cost/distance for any o-d pair -Maximum cost /distance for any single link in an o-d path. -Maximum cost/distance between an o/d and a hub. k j m i collection transfer distribution 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 Use p hubs Link flows & hubs Serve all o-d flows Hub radius Objective X ik = 1 if node i is allocated to hub k, and 0 otherwise X kk = 1 node k is a hub z is the maximum transportation cost between all o–d pairs. r k = “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)! Boston 85 Dallas Chicago 76 217 166 305 235 120 94 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 Median p=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 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 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.
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 hubs to 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 =
46 Geographic Expansion q=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 T ij 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 q A hub arcs, knowing that the follower will later locate its own hub arcs. -The follower optimally locates its own q B 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 D ij A : The distance for the trip from i to j that achieves the minimum cost for Firm A. D ij B : The distance for the trip from i to j that achieves the minimum cost for Firm B. Distance ratio (passengers): DR ij =(D ij A –D ij B ) /(D ij A +D ij B ) C ij A : The minimum cost for the trip from i to j for Firm A. Cost ratio (freight): C ij B : The minimum cost for the trip from i to j for Firm B. CR ij =(C ij A –C ij B ) /(C ij A +C ij B ) i k j l As D ij A (or C ij A ) 0, DR ij (or CR ij ) -1, and Firm A captures all revenue.
54 5-level Step Function for Customer Allocation CR ij or Dr ij –r 1 –r 1 to –r 2 –r 2 to r 2 r 2 to r 1 > r 1 Φ ij A (x A,x B ) 100% 75% 50% 25% 0% Φ ij A (x A,x B ) = fraction of demand captured by Firm A r 1 and r 2 determine selectivity level of customers. r 1 = r 2 = 0 is an “all-or-nothing” allocation. r 1 = 0.75, r 2 = 0.50 is insensitive to differences. Fraction of demand captured by Firm A
55 Notation Given: -V = set of demand nodes, V (|V |=n) -W ij = set of origin-destination flows -F ij = set of origin-destination revenues (e.g. airfares) -d ij = distance between i and j -C ijkl = unit cost for the path i k l j = d ik + d kl +d lj s - = cost discount factor for hub arcs, 0< ≤1. Decision variables: -x ijkl A ( x ijkl B ) = flow for i k l j for Firm A (B) - y kl A ( y kl B ) = 1 if there is a hub arc k–l for Firm A (B) -z k A ( z k B ) = 1 if there is a hub at city k for Firm A (B) i j k l
57 HALCE-A (Firm A’s Problem) Network Flow 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 q A hub arcs. For each set of Firm A’s hub arcs, use bounding tests to enumerate only some of Firm B’s q B 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: (r 1, r 2 )=(0.75,0.25) -medium: (r 1, r 2 )=(0.083,0.015) -high: (r 1, r 2 )=(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.
60 Results: High Customer Selectivity Red lines: Firm A’s optimal solution Blue lines: Firm B’s optimal solution Revenue = airfareRevenue = distance Distance ratio allocation q A =q B =2, =0.6
61 Hub Use with Distance Ratio Allocation 92.2% 86.3% 57.4% 47.8% 47.0% Top hub arcs for Firm A Top hub arcs for Firm B
62 Cost Ratio vs. Distance Ratio Revenue=distance, q A =q B =3, =0.6 Cost Ratio allocation (freight) Firm A’s hubs=4,6,8,12,17,22 Distance ratio allocation (passengers) Firm A’s hubs=1,4,12,14,17,22 Red lines: Firm A’s optimal solution Blue lines: Firm B’s optimal solution Only 15% of revenues are from paths with a hub arc. Over 67% of revenues are from paths with a hub arc.
63 Findings The leader (Firm A) usually has an advantage, but not always (“first entry paradox”). Distance ratio allocation encourages one-stop routes (as preferred by passengers). Cost ratio allocation encourages more circuitous two- stop routes (as in freight transportation). Large origins/destinations have a large advantage for hub location. -Peripheral cities have a geographic disadvantage for hub location. Though the optimal hub arcs vary considerably, the competitors generally use the same optimal hub nodes.
64 Competitive Model Conclusions There are some interesting differences between the leader’s and follower’s strategies: -The leader tends to use fewer hubs more intensively, but the follower performs about as well in many cases! -The leader tends to capture the higher revenue customers, while the follower captures more, but less valuable, customers. Optimal network design can be very sensitive to the customer allocation mechanisms.
65 VI. Hub Location Data Sets Much work has been done with only a few data sets: -CAB25: 25 cities in US. -AP: up to 200 postal locations in Sydney, Australia. -“Turkish data”: 81 nodes in Turkey What should alpha be?
66 CAB25 Data Set 25 US cities with symmetric flows based on air passenger traffic in 1970. No flow from a node to itself (W ii =0). Subsets are alphabetical.
67 AP Data Sets Up to 200 postal codes in Sydney with asymmetric flows of mail from 1993(?) and given collection, transfer and distribution costs. 42.4% of flows (including all flows W ii ) 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.
68 Turkish network: TR81 81 nodes for provinces in Turkey with asymmetric flows generated based on populations. Often used with =0.9 (from interhub travel time discount). Smaller versions selected in various ways.
71 Distribution of Demand Optimal hub locations and hub networks reflect the underlying distributions of flows (and aggregated flows). All data sets have flows heavily concentrated in a few large nodes. CAB is least centrally concentrated with large peripheral demand centers. AP has concentrated demand and is least evenly distributed over the region. -Subsets of AP may not be as similar to each other as “designed”. TR81 is most evenly distributed in space.
72 Alpha What is the “right” value of ? ValueModeLocationReference 0.25-0.375Truck-postalAustralia Ernst and Krishnamoorthy, Location Science 1996 0.7TruckEU Limbourg and Jourquin, Transportation Research E 2009 0.365Truck-railEU Limbourg and Jourquin, Transportation Research E 2009 0.7 – 1.0LTLBrazil Cunha and Silva, EJOR 2007 0.4946 Truck (time definite) Taiwan Chen, Networks and Spatial Economics 2010
73 New Directions for Hub Location Research Better, more realistic models: -Incorporate cost, service and competition. -Model relevant costs (especially economies of scale) more accurately. -More complex networks with longer paths and direct routes. Solve larger problems.(?) Link to service network design. Link to telecom hub location. Link to practice.