Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Location Problems on Networks with Routing Elena Fernández Universitat Politècnica

Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014

Location Problems on Networks with Routing

Elena Fernández

Universitat Politècnica de Catalunya-Barcelona TECH (UPC)

[email protected]


Ivan Contreras Elena Fernández

I. Contreras, E. Fernández, General network design: A unified view of combined location and network design problems. European Journal of Operational Research 219 (2012)

http://google.com/search?q=General+network+design%3A+A+unified+view+of+combined+location+and+network+design+problems.


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 related to Location

NETWORK OPTIMIZATION PROBLEMS


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


A classication of Network Optimization problems

based on the type of demand

User-facility demand (UF): Service relates users and facilities

User-User demand (UU): Service relates pairs of users;

facilities are used as intermediate locations in the routes that

connect pairs of users


Shortest spanning tree problem

(Borůvka, 1926)

UFRoutes between users and facilities

UURoutes between users through facilities

p-median problem

(Hakimi, 1964)

Facility Location –Network design problem

(Melkote and Daskin, 2001a)

Optimum communication spanning tree problem

(Hu, 1974)

Hub location problem

(O’Kelly, 1986)

Tree of hubs problem

(Contreras, Fernández, Marín, 2009)

Underlying network

Facility location

Network design

Facility location-Network design


Outline

Optimization criteria: design costs / service costs

UF: Facilities used to give service: Location + Network design

UU: Facilities used to re-route flows between customers: Hub location

(more general) HUB LOCATION


What optimization criteria are relevant?

Design costs • Facilities

Sum of set-up costs• Connections

Service (routing) costs

Sum of service costs of customers Maximum service cost


1

2

4

6

3

5

2

2

3

3

4

2

3 3 3

3

1

2

4

6

3

5

2

2

3

3

3

1

2

4

6

3

5

10

8

4

4

3

3

2 2 5

7

2

Per unit routing costs (dij ) Communication Requirements (Wij )

Minimum Spanning Tree Optimum Communication Spanning Tree

1

2

4

6

3

5

2

4

3

3

2

The Optimum communication Spanning Tree

Hu (1974); Ahuja and Murty (1987); Rothlauf (2009); C, F, Marín, (2010), F, Luna, Hildenbrandt, Reinelt, Wiesberg (2013)

To find a tree that minimizes the sum of all routing costs

3(3+3+3) 34


Minimize service cost

1

1

UU: Routing costs

51

1+1


Total set-up cost: 8+ 4 5 + 82Total service cost: 10 + 4 5 + 82Max service cost: 22

Total set-up cost: 16

Total service cost: 38

Max service cost: 4

UF: Routing (service) costs / Set-up costs

Routing costs Set-up costs

● Euclidean distances ● One facility is located


How to evaluate service costs?

Each arc is accounted for as many times as it is used

Trace paths


G = (V, E);

Vertices V={1, 2, …, n };

Edges e = (k,m), k, m V, k < m,

ckm 0: set-up cost for edge (k,m)

dkm 0: per unit routing cost from k to m

Commodities: C ={ l=(i,j): i,jV }

Wij: Flow that must be sent from i to j, (i,j)C

Modeling flows in Network Design

j

jiij

iji WDWO ;


i jkm m

⋮

⋮

⋮

⋮

xijkmxij

mk

Modeling flows in network optimization: path formulation (4-index)

xijkm: Fraction of flow from i to j routed via nodes k and m,

ykm=1 iff edge (k, m) activated

(i, j) C

Decision variables: For (k, m)E

For each commodity (i,j) define a path from i to j


i jkm m

⋮

⋮

⋮

⋮

xijkmxij

mk

Modeling flows in network optimization: path formulation (4-index)

otherwise0

1

1

jk

ik

xxm

ijkm

m

ijmk

For (i, j)C, for all k

Cji Emk

ijmkmk

ijkmkmij xdxdW

, ,

Routing costs

Ekmyxx kmijmk

ijkm ,

Emk

kmkm yc,

Set-up costs for links

Ekmyxx kmihmk

ijkm ,

For (i, j), (i, h) C


otherwise0

),(

1

Cki

ik

hhi

ikO

W

m

ikm

m

imk

For all i

i Emk

imkmk

ikmkmi hdhdO

,

Routing costs

Ekmyhh kmimk

ikm ,

Emk

kmkm yc,

Set-up costs for links

i jkm m⋮

⋮

⋮

⋮

hikm

himk

j

j

hikm: Fraction of flow emanating from i routed via arc (k, m)

Modeling flows in network optimization: flow formulation (3-index)

For each vertex i, send a flow of value Oi, with demand Wij at each j with (i, j)C


C),(

1

ji

ijkmij

i

ikm xW

Oh

Path formulation vs flow formulation

otherwise0

1

1

jk

ik

xxm

ijkm

m

ijmk

For (i, j)C, for all k

otherwise0

),(

1

Cki

ik

hhi

ikO

W

m

ikm

m

imk

For i fixed, adding on j with (i, j)C

otherwise0

jkW

ikW

xWxW ij

ij

m

ijkmij

m

ijmkij

otherwise0

),(,;

,;,;

CC

CCkiW

ikOW

xWxW ik

jijiij

jij m

ijkmij

jij m

ijmkij

otherwise0

),(,;,;

CCC

kiW

ikO

xWxW ik

i

m

ijkm

jijij

m jij

ijmkij


How to evaluate routing costs

if location decisions are involved ?

connecting customers and facilities

Trace paths

that use facilities as intermediate vertices

But facilities are not known before hand !!!


Outline






LOCATION

+

ROUTING CONNECTING CUSTOMERS & FACILITIES (UF)

Facility Location /Network Design Problem


fi 0: set up cost for facility i.

cij 0: set-up cost for edge (i,j)

dij 0: per unit routing cost k to m

Data

To find

A subset of vertices to locate facilities An allocation of non-facilites to facilities A subset of arcs to connect each vertex to its allocated facility That minimizes some objective function


• Total set-up cost• Total routing cost (total service cost)• Maximum routing cost from any vertex to its allocated facility• Any combination of the above


Set-up cost + routing cost Melkote and Daskin: (01, 01a)

Routing cost subject to maximum budget constraint

Cocking, Fleßa Reinelt (05), Cocking (08).

Maximum routing cost subject to budget constraint C, F & Reinelt (2010)

Maximum coverage within a given distanceMurawski & Church (09):





fi=1, i; cij=0 i, j; i, j; B=p

p-median or p-center (depending on the objective)

NP-hard


Set-up cost + routing cost

cij=0 i, j;

Network design trivial. Alternative formulation of UFLP

Routing cost subject to budget constraint

NP-hard

Some particular cases


Facility Location /Network Design ProblemThere is an optimal solution to FLNNDP which is a rooted forest

(If there are no capacity constraints on the facilities)

Decisions

Vertices that locate facilities

Paths connecting non facilities and facilities

Arcs of the rooted forests k

i

m

j


Vertices that locate facilities: zjj

vertex j is the root of a tree

Non-facility vertices: zij vertex i allocated to facility j

Paths connecting non xkmij

facilities and facilities: arc (k, m) in the path from i to

its allocated facility j

Arcs of the rooted forests ykm arc (k, m) is in the rooted forest

Path formulation

Facility Location /Network Design Problem:


i is a facility or assigned to one

Equilibrium on path from i to j

Path formulation for FL/ND

mkjixyz

Byczf

AmkjiVjiyxx

ijVjizx

kijVkjixx

ijVjizx

jiVjizz

Viz

xxd

ijkmkmij

Emkkmkm

Viiii

kmijmk

ijkm

kij

ijkj

m

ijmk

m

ijkm

ijEmi

ijim

jjij

Vjij

jiVji Emk

ijmk

ijkmkm

,,,,1,0,,

,,,,

,,

,,,,0

,,

,,

1

min

),(

),(

, ),(

Budget

Amk

kmkmVi

iii yczf),(

AB C jiVjiTxxd

T

Emk

ijkm

ijkmkm

,,

min

),(


Outline






LOCATION

+

ROUTING FOR CONNECTING FACILITIES

HUB LOCATION

Location problems on networks with routing E Fernández EUROXXIV

MINIMUM TOTAL COST

Set-up costs + Flow Routing costs

HUB LOCATION

Network design

A set of facilities (hubs) to open

Subset of edges to connect hubs among them

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 LOCATION• Alumur and Kara. 2008.• Campbell (1994), (1996)• Campbell, Ernst, Krishnamoorthy (2005a, 2005b)• Ernst and M Krishnamoorthy (1998), (1996), (1999)• O’Kelly (1986), (1987), (1986), (1992), (1992), (1994).• Skorin-Kapov, Skorin-Kapov, O’Kelly. (1996)• Cánovas, García, Marín (2007)• C PhD. Thesis (2009)• C, Cordeau, Laporte (2011), (2012) • C, Díaz, F. (2010), (2008)• C, F (2012), (2014)• C, F, Marín, (2008), (2009) • Ernst, Hamacher, Jiang, Krishnamoorthy, Woeginger. (2009)• García, Landete, Marín (2012)• Hamacher, Labbé, Nickel, Sonneborn. (2004).• Labbé, Yaman. (2004).• Labbé, Yaman, Gourdin. (2005).• Marín. C&OR, 2005, Top, (2007).• Marín, Cánovas, Landete. EJOR, (2006).

Campbell, O’Kelly (2012) Twenty-five years of hub location research. Tr. Sci.


Minimize set-up cost costSingle allocation / multiple allocationDoes it really matter ?

All flow leaving vertex i

goes through the same hub

Flow may leave vertex i

trough different hubs


Vi,kz

Vmki,jx

Vimjzx

Vjkizx

zbzO

Viz

kiVi,kzz

zdzzdzdWzfMin

ik

ijkm

jmVk

ijkm

ikVm

ijkm

Vikkkiki

Vkik

kkik

Vi Vj Vmjmjm

Vk Vmjmikkm

Vkikikij

Viiii

allfor 1,0

,, allfor 0

,, allfor

,, allfor

allfor 1

, allfor

Vi Vj Vk Vm

ijkm

ijkm

Viiii xFzfMin

SINGLE ALLOCATION

xijkm: Fraction of flow from vertex i to vertex j

routed via hubs k and m

O’Kelly 87, Campbell 94, Skorin-Karpov 96

Tight LP but many variables

jmkmikijkm

ij dddWF

Routing cost of the path i-k-m-j

Cijkm

zik=1 iff vertex i is assigned to hub k

xijkm {0,1}

Capacity constraints

HUB LOCATION: path formulation

i jk mdik

dkm

dkm

Uncapacitated: n=500 C, Cordeau Laporte, 2010

Capacitated: n= 200 vertices C, Díaz, F, 2010

xijkm=zikzjm

Even if the set of hubs is known

the allocation is still NP-HARD


Vi Vj Vk Vm

ijkm

ijkm

Viiii xFzfMin

Multiple Allocationzik0 Fraction of flow with origin at i routed via hub k

HUB LOCATION: path formulation

xijkm=zikzjm

The logic no longer holds

mmm

ijkm

kkm

ijkm

zx

zx

kkm

ijmk

m

ijkm zxx

Viz

kiVi,kzz

Vkik

kkik

allfor 1

, allfor

Can be substituted by the stronger

If the set of hubs is known the allocation is easy

(find best path for each commodity)

xijkm: Fraction of flow from vertex i to vertex j

routed via hubs k and m


hikm: Fraction of flow emanating from vertex i routed via hubs k and m

Vi,khOzOzWhOVl

ilkiiki

Vjjkij

Vm

ikmi

,

i

m

k

Oi

j

i

l

HUB LOCATION: flow formulationSingle allocation (Ernst and Krishnamoorthy 96)

Vi Vk Vi Vk Vl

ikliklikiiik hOdzDOd

Vli ,khVi,kz iklik , ,10; 1,0

Routing costs

Multiple allocation

(Ernst and Krishnamoorthy 98, Marin 05) rikj: Fraction of flow emanating from

vertex i routed via hub k and vertex j

i mkOi

j

l

zik

rikj

hikm

jk

hilk

rikj

Vi,khzrhVl

ilkik

Vj

ikj

Vm

ikm

,

jkjVi

ikji

ijVk

ikji

zDrO

Vi,jWrO






Outline


Hubs are not necessarily connected by

means of a complete graph


Tree of Hubs

Hub arc location


p Number of hubs to open

The Tree of Hubs Location ProblemC, F, Marín, 08, 09.

i

j

m

k a

MINIMIZE: Flow Routing costs

TO FIND

A set of p hubs to open

Subset of edges that define a tree to connect hubs among them

(to route flow between customers)

Subset of edges to connect customers to their allocated hubs

Assignment


m1i m2 mr jzjmrzim1 xij

m1m2xij

mr-1mr

i m1 jm2 mr

xijmrj

zim1 xijm1m2

xijmr-1mr

i m1 jm2 mr zjmrxij

im1 xijm1m2

xijmr-1mr

i m1 jm2 mr xijmrjxij

im1xij

m1m2xij

mr-1mr

Possible paths for sending the flow from i to j (ji)

i non – hubj non- hub

i non – hubj hub

i hubj non - hub

i hubj hub

jkjikjizxzx ikkm

ijmkjk

km

ijk m

,,,,0

The Tree of Hubs Location Problem


kmmky

kiz

kmmkjix

py

kmmkjiyxx

jkjikjizxzx

pz

iz

xdWzDdOdMin

km

ik

ijkm

k kmkm

mkijmk

ijkm

ikkm

ijmkjk

km

ijkm

kkk

kik

i j k km

ijkmkmijik

i kikiiik

,,1,0

,1,0

,,,,0

1

,,,,

,,,,0

1

1. There is an optimal solution such that xijkm{0,1}

2. The y variables define a spanning tree on subgraph induced the vertices s.t. zii=1

3. It holds that zikzkk, ik

Every vertex is allocated

Flow can only circulate in y edges

Connectivity and flow equilibrium

p hubs

p-1 y edges

The Tree of Hubs Location Problem


Hub arc location problems

i

ij A hub node is set-up at each

endnode of a hub arc

Commodities are routed via hub arcs


ge: set-up cost for hub arc e

cu : set-up cost for hub node u

Fek: cost for sending commodity k=(i,j) via hub arc e

Hub arc location problems: data

i

ij

ge

uv

i

j

dij

Fek = Wij ( diu + duv + dvj)cu

To find: the hub arcs to set-up The assignment of commodities to hub arcs

Such that the total costhubs set-up costs (arcs and nodes)+commodities distribution costs

ARE MINIMIZED


Further data

q: Maximum number of hub arcs

p: Maximum number of hub nodes implied by hub arcs

Some particular cases

q= p(p-1)/2 y ge=0, e (node) p-hub location problem

cu=0 u, only hub arc location

problem


Formulation I: decision variables

otherwise0

arc hub a is if1 eze

otherwise0

arc hub a of node-endan is if1 uyu

otherwise0

arc hub viarouted is commodity if1 ekxek

Hub arcs

Hub nodes

Allocation


Formulation

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)

(|K|+2)(1+|E|) constraints

Kk

ekzEe FMine 1:

Hub arc location NP-hard

A better formulation is based on

properties of supermodular functions

C, F, (2014)


Network optimization problems involve location and routing decisions which, in turn, imply network design decisions.

Objective function: Trade off between set-up costs and service (routing) costs Other objective functions (min-max, weighted average …)

Tight (effective) formulations require (too) many variables. Strengthen formulations with fewer variables (valid inequalities) Column generation, Benders decomposition, … Best formulations exploit structure of the problem

Research trend Forward: More general (complex) models Backward: Design and implement efficient algorithms for particular cases

of network design problems.

To conclude


THANK YOU

FOR YOUR ATTENTION

Documents

Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Location Problems on Networks with Routing Elena Fernández Universitat Politècnica