Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Arc hub location problems as network design problems with routing
Elena Fernández- Dpt EIO-UPCIvan Contreras- CIRRELT- Montréal
Seminario de Geometría TóricaJarandilla 12-15 de noviembre 2010
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Which set of facilities to open ? Location
How to satisfy the customers demands from open facilities ?
From which facility does the customer receive service ?
Allocation
How is service provided ? Routing
Are facilities somehow connected ? Routing
Which are the possible (or preferable) connections between Network
design
customers or between customers and facilities ?
Decisions in discrete location problems on networks
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
where customers obtain service from
where flows between pairs of customers are consolidated and rerouted
connect customers and facilities
Connect customers and facilities Connect facilities between them
What are facilities used for?
Routing
Which are the possible connections ?
Network design
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
If customers move to
facilities to recieve service
⋮
the routing of each
customer is trivial
Customers receive service from/at facilities
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
If several customers are visited in the same route
⋮
the design of the routes may become difficult
Customers receive service from facilities
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
There exists communication between each pair of customers.
Flows are consolidated and re-routed at facilities
(which must be connected)
Facilities used to reroute flows between pairs of customers
HUB LOCATION
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Connection
between facilities by
means of a tree
HUB LOCATION
Facilities used to reroute flows between pairs of customers
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
G=(V, E)
fi: set-up cost for facilities iV
dij: per unit routing cost from i to j i, jV
Wij: flow between i and j i, jV
MINIMUM TOTAL COST
Set-up costs + Flow Routing costs
HUB LOCATION
TO FIND
Network design
Hubs are used to consolidate and reroute flow between customers
A set of facilities (hubs) to open
Subset of edges to connect hubs among them
Subset of edges to connect customers to their allocated hubs
Location
Assignment
i
j
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
i
j
m
k
HUB LOCATION: Typical asumptions
Transfer between hubs
Collection
Distribution
Discount factors to routing costs
Full interconnetion of hubs
Paths: i-k-m-jTriangle inequality
da
a
Hub location problems are NP-hard
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Hub arcs
i
i
j
j
j
Campbell JF, Ernst AT, Krishnamoorthy M (2005a) Hub arc location problems: Part i-introduction and results. Manag Sci 51(10):1540–1555
Campbell JF, Ernst AT, Krishnamoorthy M (2005b) Hub arc location problems: Part ii-formulations and optimal algorithms. Manag Sci 51(10):1556–1571
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Index
Hub arc location problems
Formulation based on properties of supermodular functions
Comparison of formulations
Some computational results
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Hub arc location problems
G=(V, E) complete undirected graph
n=|V|, m=|E|=n(n-1)/2
K={(i, j)VV: there is demand between i y j}; k K commodity
i
ij
q: Maximum (exact) number of hub arcs
p: Maximum (exact) number of hub nodes
Commodities demand is routed via hub arcs
If an arc hub is set-up then hub nodes are also
established at both endnodes
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
dij: unit routing cost from i to j i, jV
ge: set-up cost for hub arc e eE
cu : set-up cost for hub vertex u uV
Fek: routing cost for commodity k K via hub arc e=(u,v) kK, eE
i
ij
ge
uv
i
j
dij
To find:
Hub arcs to set-up Assignment of commodities to hub arcs
Such that the overall cost is minimizedHubs set-up cost (both arcs and nodes)
+Commodities routing costs
Fek =Wij( diu + duv + dvj)
cu
Hub arc location problems
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Hub arc location problems
i
ij
General model:
If we allow G to have loops, then we can locate both hub arcs and
independent hub nodes.
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
q < p(p-1)/2 unfeasible (if exactly p hub nodes must be open)
q= p(p-1)/2 y ge=0, e p-hub (nodes) location problem
q≥ min{m, p(p-1)/2} the constraint on the number of hub arcs is redundant
If cu=0 u, and p ≥ 2q, problem of locating only hub arcs.
Hub arc location problems
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Formulation I
otherwise0
edgeat destablishe is hub a If1 eze
Otherwise0
arc hub a of node-endan is If1 uyu
Otherwise0
arc hub viarouted is commodity If1 ekxek
Variables: |E| + |V| + |E||K|
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Formulation I
VuEeKkxyz
py
qz
Evueyz
Evueyz
EeKkzx
Kkx
xFyczgMin
ekue
Vuu
Eee
ve
ue
eek
Eeek
Ee Kk Eeekek
Vuuuee
,,,1,0,,
),(
),(
,
1
Extension of UFLP
Many variables (xek 4-index variables)
(|K|+2)(1+|E|) constraints
Kk
ekzEe FMine 1:
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Formulation II
(based on propertirs of supermodular functions)
Hub arc minimization problem
Minimization of supermodular function
Maximization of submodular functions
Nemhauser, Wolsey, Maximizing submodular set functions: formulations and analysis of algorithms, in P. Hansen, ed., Studies on Graphs and Discrete Programming, N-H (1981)
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Supermodular functions
Let E be a finite set, and f : P(E) ℝ
Definition: f is supermodular if f(S T) + f(S T) ≥ f(S) + f(T) S, T E
Characterization: f supermodular f(S {e}) - f(S) f(S {e’, e}) - f(S {e’})
Characterization: f supermodular and non-increasing
f(T) ≥ f(S) + eT\S [ f(S {e}) - f(S)] S, T E
The maximization of supermodular functions is “easy”
The minimization of supermodular functions is “difficult”
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
For S ⊆ E hk (S) = Min e’S Fe’k assignment cost associated with kK
Proposition: hk is supermodular and non-increasing , for all kK
Supermodular functions
Optimization problem: Find T* such that hk (T*) =Min S ⊆ E hk (S)
Find Min k = hk (T*)
k ≥ hk (S) + eT*\S ek(S) for all S E
Find (ze)eE, ze{0,1} s.t. Min k
k ≥ hk (S) + eS ek(S) ze for all S E
Find (ze)eE, ze{0,1} s.t. Min k
k ≥(Mine’S Fe’k)+eS(Fek-Mine’SFe’k)-ze for all S E
Corollary: hk (T) ≥ hk (S) + eT\S ek(S) for all kK , S, T E
where hk (S {e})- hk (S )= (Fek - Min e’S Fe’k )-
(a)- =min {a, 0}
Ske
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Formulation II
Min k
k ≥ (Mine’S Fe’k) + eS (Fek - Min e’S Fe’k )- ze for all S E
Remark:Even if there is an exponential number of constraints (subsets S) there is a small number of possible values of Min e’S Fe’k . The candidate values are Fehk for eh E.
Find (ze)eE, ze{0,1} s.t. Min k
k ≥ Fehk + eS (Fek - Fehk)- ze for all eh E
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Formulation II (based on propertirs of supermodular functions)
f(S) = g(S)+c(V(S))+kK hk(S) supermodular
g(S) = eE ge supermodular
ĉ(S) = c(V(S))=uV(S) cu supermodular
hk (S) =Min eS Fek supermodular and non-increasing
S ⊆ E,
Hub arc minimization problem
Minimization of supermodular function
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
VuEeKkxyz
py
qz
Kvueyz
Kvueyz
KkESzSSh
zczgMin
ekue
Vuu
Eee
ve
ue
eSe
ke
kk
Kk
k
Vuuu
Eeee
,,,1,0,,
),(
),(
,*
Formulation II
keke khhkFF
,1
KkkeeEe
Kkke
ekEeke
kFfFii
FFi
max)
max)
*
*
SheShS kkke
KkmhzFFF eEe
keekkek
hkhk
,11
|K|+|E|+|V| variables (variables with1-2 indices)
(|K|+2)(1+|E|) constraints
“Saving” in allocation cost for using additional hub arc e
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Formulation I vs Formulation II
|K|+|E|+|V|
(|K|+2)(1+|E|)
|E||K|+|E|+|V|
(|K|+2)(1+|E|)
Variables
Constraints
Theorem: (LP bounds ) vLPF1=vLPF2
FI FII
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Formulation I vs Formulation II: APset
Instance Natural Formulation Supermodular Formulation
|V| LP boundLP % gap
Final % gap
Time(sec) nodes
LP bound
LP % gap
Final % gap
Time(sec) nodes
10 0.2 167708 5.21 0.00 4.80 119 167708 5.21 0.00 20.00 101
10 0.5 173104 3.84 0.00 2.00 57 173104 3.84 0.00 9.50 53
10 0.8 176158 2.15 0.00 1.00 21 176158 2.15 0.00 5.80 29
20 0.2 188501 0.80 0.00 40.00 13 188501 0.80 0.00 1136.00 35
20 0.5 194095 0.33 0.00 9.90 5 194095 0.33 0.00 90.60 9
20 0.8 194737 0.00 0.00 3.10 0 194737 0.00 0.00 17.90 0
25 0.2 191717 0.52 0.00 303.40 21 191717 0.52 0.35 10800.00 32
25 0.5 196165 0.66 0.00 88.30 11 196165 0.66 0.00 811.00 13
25 0.8 197387 0.31 0.00 37.10 7 197387 0.31 0.00 258.60 5
40 0.2 196449 2.68 2.30 10800.00 28 memory
40 0.5 200711.45 0.00 0.00 828.70 0 memory
40 0.8 200711.45 0.00 0.00 407.60 0 memory
50 0.2 - - - 10800.00 - memory
50 0.5 200436.8 0.38 0.27 10800.00 53 memory
50 0.8 201074.02 0.06 0.00 3156.30 5 memory
p=3, q=9, Xpress, CPU limit:3 hours ge = (cu+ cv)coeff , e=(u,v); coeff = 0.15
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Formulation I
VuEeKkxyz
py
qz
Evueyz
Evueyz
EeKkzx
Kkx
xFyczgMin
ekue
Vuu
Eee
ve
ue
eek
Eeek
Ee Kk Eeekek
Vuuuee
,,,1,0,,
),(
),(
,
1
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
VuEeKkxyz
py
qz
Kvueyz
Kjieyz
KkESzSSh
zczgMin
ekue
Vii
Eee
je
ie
eSe
ke
kk
Kk
k
Vuuu
Eeee
,,,1,0,,
),(
),(
,*
Formulation II
KkmhzFFF eEe
keekkek
hkhk
,11 Separate
eEe
keekkek
zFFFhkhk
zGiven
¿kK, h t.q. ?
Brute force: |K||E| (O(|V4|)
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
px p0
Separation problem
Given a polyhedron P and a point x*, to identify if x* P. If it does not, to find a valid inequality for P, px p0 such that px*>p0.
x*
x*
px p0
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Separation of constraints
1
1
1
11
h
ieke
h
iekee
Eekeekkehk iiihkhkhk
zFzFzFFFS
eEe
keekkek
zFFFhkhk
zGiven to know if there exists kK, h s.t.
Proposition:
For k given, the maximum of Shk, is attained for h=rk
h
i
ek izhr1
1:min
eEe
ek zGFGGS
)( Concave, piecewise linear;
with break values Fek
(k fixed)
First index such that the slope is no longer positive
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Preliminary Results
nodes alpha LP%gap
Final % gap
Times(sec)
Nodes
LP % gap
Final % gap
Times (sec)
nodes
10 0.2 5.21 0 4.8 119 5.21 0 1.03 43
10 0.5 3.84 0 2 57 3.84 0 0.88 33
10 0.8 2.15 0 1 21 2.15 0 0.55 21
20 0.2 0.80 0 40 13 0.80 0 6.65 33
20 0.5 0.33 0 9.9 5 0.33 0 2.46 9
20 0.8 0.00 0 3.1 0 0.00 0 1.08 0
25 0.2 0.52 0 303.4 21 0.52 0 11.05 7
25 0.5 0.66 0 88.3 11 0.66 0 9.117 11
25 0.8 0.31 0 37.1 7 0.31 0 4.37 5
40 0.2 2.68 2.3 10800 28 2.68 0 923.4 124
40 0.5 0.00 0 828.7 0 0.00 0 62.36 0
40 0.8 0.00 0 407.6 0 0.00 0 29.37 0
50 0.2 - - 10800 - 1.24 0 9438.8 265
50 0.5 0.38 0.27 10800 53 0.38 0 810.59 59
50 0.8 0.06 0 3156.3 5 0.06 0 145.23 5
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Preliminary Results
nodes alpha LP % gap LP % gap Final % gap Times(sec) nodes
60 0.2 memory 2.10 0 6887.43 183
60 0.5 memory 1.56 0 1830 59
60 0.8 memory 0.96 0 931.22 35
75 0.2 memory 1.72 1.26 13483 74
75 0.5 memory 1.17 0 12216 105
75 0.8 memory 0.98 0 5922.39 55
90 0.2 memory 0.80 0 30479.6 95
90 0.5 memory 8.41 8.27 21293 58
90 0.8 memory 0.69 0 5822.45 33
100 0.2 memory - - 10800 -
100 0.5 memory 0.63 0 27033.7 55
100 0.8 memory 9.17 8.81 38200 28
Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12-15 Noviembre
Arc hub location problems (involve routing decisions)
General Problem
Two alternative formulacions
Minimization of supermodular function
Efficient solution of separation problem
Promising preliminary results
Summary