1. Hub Location Problems Chapter 12
July 20, 2009
Fuad Ashwash

2. Outline Introduction Applications Fundamental Hub Location Models
Fixed Charge Location Problem
Basic p-Hub Location Model
p-hub/D/MA/•/Σflow
p-hub/D/SA/•/Σflow
Hub Location Example

3. Introduction Involve locating facilities and designing hub networks.
Goal to minimize total cost (as a function of distance) of transportation between hubs, facilities and demands.
Rather than serving every origin-destination demand with a direct link, a hub network provides service via smaller set of links between origin/destinations and hubs, and between pairs of hubs.
Hub network allows large set of origins to be connected with fewer links, via central hub facilities.
Use of fewer links in the network concentrates flows and allows economies of scale to be exploited.

4. Applications Transportation Telecommunication
Air passenger travel, air freight travel, express shipments, large trucking systems, postal operations and rapid transit systems.
Demand usually specified as flows of passengers or goods between city pairs; transported in vehicles of some type.
Telecommunication
Distributed data networks in computer communication, telephone networks, video teleconferences, distributed computer communication, telephone networks, etc.
Demand is transmission of information and occurs over telephone lines, fiber optic cables, co-axial cables, or satellite channels and microwave links.

5. Fundamental Hub Location Models
Multiple and single allocation versions exist for hub location problems
In single, each demand point must be allocated to communicate with one hub. All flows to and from each demand point travel via the same hub node.
In multiple, each demand point may be allocated to communicate with more than one hub. Greater flexibility allows lower cost solutions, and simplifies solution, since for a given set of hub nodes, each origin-destination flow can be routed separately from all others via the least cost path.
Assume every origin-destination path includes at least on hub node, and cost per unit flow is discounted between all hub pairs using
Problems that receive most attention are p-hub median problem, and the capacitated and uncapacitated hub location problem.

6. Fixed Charge Location Problem
Minimizes total facility and transportation costs.
Does not assume each site has the same fixed costs
Does not assume that sites are capacitated
Does not assume that there is a set number of facilities p that should be opened
Determines optimal number and locations of facilities, as well as assignments of demand to a facility.

7. Fixed Charge Location Problem
Notations
fj = fixed cost of locating a facility at candidate site j
Cj = capacity of a facility at candidate site j
α = cost per unit demand per unit distance
dij = distance between demand node i and candidate site j
hi = demand at node i
yij = 1 if demand node i is assigned to facility at node j, 0 otherwise

8. Fixed Charge Location Problem
(27)
(28)
(29)
(30)
(31)
(32)
(27) minimizes sum of fixed facility location costs and total travel costs for demand to be served.
(28) requires each demand node be assigned to exactly one facility
(29) restricts demand node assignments only to open facilities
(30) prohibits total demand assigned to a facility from exceeding capacity of the facility Cj
(31) establishes the siting decision variable as binary
(32) a binary constraint requiring all demand points be single sourced

9. Basic p-Hub Location Model
Minimize total cost (as a function of distance) of transportation between hubs, facilities and demands
Notations
hij = number of units of flow between nodes i and j
cij = unit cost of transportation between nodes i and j
α = discount factor for transport between hubs
Set N = all nodes
xj = 1 if a hub is located at node j, 0 otherwise
yij = 1 if demands from node i are assigned to a hub located at node j, 0 otherwise

10. Basic p-Hub Location Model
(33)
(34)
(35)
(36)
(37)
(38)
(33) Minimizes sum of cost of moving items between a non-hub node and the hub to which the node is assigned, the cost of moving from the final hub to the destination of the flow, and the interhub movement cost which is discounted by a factor of α.
(34) through (38) identical to constraints for p-median model
(35) each node should be assigned to exactly one hub

11. p-hub/D/MA/•/Σflow
Uncapacitated multiple allocation p-hub median model
Locate p hub facilities to minimize total cost
Notations
Wij = demand for flow between node i and node j
dij = distance from node i to node j
p = number of hubs to locate
χ = unit costs for collection (origin-hub)
α = unit costs for transfer (hub-hub)
δ = unit costs for distribution (hub-destination)
Zik = flow from origin i to hub k
Yikl = flow from hub k to hub l that originates at origin I
Xilj = flow from hub l to destination j that originates at origin I
Hk = 1 if node k is a hub, and 0 otherwise

12. p-hub/D/MA/•/Σflow

13. p-hub/D/MA/•/Σflow
(12.1) sums cost for collection, transfer and distribution
(12.2) ensures appropriate number of hubs are selected
(12.3) ensures all flow from each origin leaves the origin
(12.4) ensures all flow from each origin-destination pair arrives at the proper destination
(12.5) is the flow conversation equation at the hubs
(12.6) and (12.7) ensure that hub nodes are established for every distribution and collection movement

14. p-hub/D/SA/•/Σflow
Uncapacitated single allocation p hub median problem
Each non-hub node is restricted to be allocated to a single node
Notations
Formulated similarly to multiple allocation, but Zik can be restricted to be binary and eliminate Xilj variables
Zik = 1 if node i is allocated to a hub at node j, and 0 otherwise
Yikl = flow from hub k to hub l that originates at origin i

15. p-hub/D/SA/•/Σflow

16. p-hub/D/SA/•/Σflow
(12.11) sums the cost for collection, distribution and transfer
(12.12) ensures appropriate number of hubs are selected
(12.13) ensures that each non-hub is allocated to a single hub
(12.14) is the flow conservation equation at the hubs
(12.15) ensures that hub nodes are established for every distribution and collection movement

17. Hub Location Example
AC Cable is a company set up to handle calls at residential locations. They are locating service vehicle bases from which service trucks will be dispatched to handle calls. These bases will be located at up to p = 3 of the candidate sites as in the figure below and have a capacity Cj. Demands represented by each demand node are Di . Distances between nodes i and k are dik. Each facility has a fixed cost of fj. The per unit distance per item cost to respond to a demand from a facility is α. Formulate a linear programming model that will find locations for these facilities.