location problems on networks with routing ▪ e fernández ▪ tgs 2010 ▪ jarandilla 12-15...

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


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


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


If customers move to

facilities to recieve service

⋮

the routing of each

customer is trivial

Customers receive service from/at facilities


If several customers are visited in the same route

⋮

the design of the routes may become difficult

Customers receive service from facilities


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


Connection

between facilities by

means of a tree

HUB LOCATION

Facilities used to reroute flows between pairs of customers


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


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


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


Index

Hub arc location problems

Formulation based on properties of supermodular functions

Comparison of formulations

Some computational results



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


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




i

ij

General model:

If we allow G to have loops, then we can locate both hub arcs and

independent hub nodes.


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.



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|


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:


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)


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”


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


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


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.


k ≥ Fehk + eS (Fek - Fehk)- ze for all eh E


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



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


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


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


Formulation I

VuEeKkxyz

py

qz

Evueyz

Evueyz

EeKkzx

Kkx

xFyczgMin

ekue

Vuu

Eee

ve

ue

eek

Eeek

Ee Kk Eeekek

Vuuuee

,,,1,0,,

),(

),(

,

1


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|)


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


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


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


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


Arc hub location problems (involve routing decisions)

General Problem

Two alternative formulacions


Efficient solution of separation problem

Promising preliminary results

Summary

location problems on networks with routing ▪ e fernández ▪ tgs 2010 ▪ jarandilla 12-15...

Documents

routing e fernndez tgs

networkslocation problems

discrete location problems

facilities hubs

arc hub location problems

network design problems

trivial customers

difficult customers