[PPT] - Hop Constrained Connected Facility Location Stefan Gollowitzer and PowerPoint Presentation

SLIDE 1

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Hop Constrained Connected Facility Location

Stefan Gollowitzer and Ivana Ljubi´ c

Department of Statistics and Decision Support Systems, Faculty of Business, Economics, and Statistics, University of Vienna, Austria

Aussois, January 2010

SLIDE 2

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Outline

Introduction ConFL Connectivity Concepts Quality of LP bounds Hop Constrained ConFL Cut Based Formulations Layered Graphs Polyhedral Results Computational Results Conclusion

SLIDE 3

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

What is Connected Facility Location?

SLIDE 4

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

What is Connected Facility Location?

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

SLIDE 5

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

What is Connected Facility Location?

⋆

⋆

⋆ ⋆

⋆

⋆

⋆

⋆

⋆

⋆ ⋆

SLIDE 6

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

What is Connected Facility Location?

⋆

⋆

⋆ ⋆

⋆

⋆

⋆

⋆

⋆

⋆ ⋆

SLIDE 7

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

What is Connected Facility Location?

⋆

⋆

⋆ ⋆

⋆

⋆

⋆

⋆

⋆

⋆ ⋆

SLIDE 8

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

What is Connected Facility Location?

⋆

⋆

⋆ ⋆

⋆

⋆

⋆

⋆

⋆

⋆ ⋆

SLIDE 9

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

What is Connected Facility Location?

⋆

⋆

⋆ ⋆

⋆

⋆

⋆

⋆

⋆

⋆ ⋆

SLIDE 10

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

ConFL

Given

a graph G = (V , E), a set of customers (R ⊆ V ), a set of facilities

(F ⊆ V ) and a set of Steiner nodes (˜ S ⊆ V ) such that S := ˜ S ∪ F and S ∩ R = ∅,

edge cost ce ≥ 0 for all e ∈ E and
facility opening costs fi for all i ∈ F

we try to find

a subset of open facilities such that
each customer is assigned to exactly one facility,
a Steiner tree connects the open facilities and
facility opening and edge costs are minimized.

SLIDE 11

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Applications and Previous Work

Applications

Telecommunications
Fiber-to-the-Curb strategy for fiber-optic broadband networks
Multicasting

Previous Work

Approximation Algorithms: 4.23 (4.00) (Eisenbrand et al., 2008), ...,

(Karger & Minkoff, 2000)

Heuristics: Dual-Ascent (Bardossy & Raghavan, 2010), GRASP

(Tomazic & Ljubi´ c, 2008), VNS (Ljubi´ c, 2007)

Exact: Branch-and-Cut (Gollowitzer & Ljubi´

c, 2010)

SLIDE 12

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Notation

Sets

AS: core graph (bidirected), connecting Steiner nodes and facilities
AR: assignment graph (directed), connecting facilities to customers
A := AR ∪ AS

Variables and Parameters

xij indicates whether arc ij is in the solution (1) or not (0)
zj indicates whether facility j is in the solution (1) or not (0)
cij costs for opening arc ij
fj costs for opening facility j

SLIDE 13

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Cuts based on customers

⋆ 3

4
≥ 1

r

1
2

SLIDE 14

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Model CUT R

min

ij∈A

xijcij +

i∈F

zifi s.t.

uv∈δ−(W )

xuv ≥ 1 ∀W ∩ R = ∅, r ∈ W (1) xjk ≤ zj ∀jk ∈ AR zr = 1 xij ∈ {0, 1} ∀ij ∈ A zi ∈ {0, 1} ∀i ∈ F

SLIDE 15

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Cuts based on facilities

⋆

= 1

3

4
r
1
2

SLIDE 16

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Cuts based on facilities

⋆ 3

≥ z3

4

r

1

2

SLIDE 17

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Model CUT F

min

ij∈A

xijcij +

i∈F

zifi s.t.

uv∈δ−(W )

xuv ≥ zi ∀W ⊆ S \ {r}, ∀i ∈ W ∩ F = ∅ (2)

jk∈AR

xjk = 1 ∀k ∈ R xjk ≤ zj ∀jk ∈ AR zr = 1 xij ∈ {0, 1} ∀ij ∈ A zi ∈ {0, 1} ∀i ∈ F

SLIDE 18

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Quality of LP bounds?

SLIDE 19

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Quality of LP bounds?

Example

Facility opening and assignment costs are 1. crs = L and csi = 1, for all i ∈ {1, . . . , n}. 1

2
r

L

s

. . . ⋆k n−1

n
Solutions

υLP = L/n + 4 υIP = L + 4

Lemma

There are instances for which υLP(CUT F) ≈ 1 |F|OPT.

SLIDE 20

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Quality of LP bounds?

Example

Facility opening and assignment costs are 1. crs = L and csi = 1, for all i ∈ {1, . . . , n}. 1

1/n

2
r

L 1/n

s

1/n

. . . ⋆k n−1

n
Solutions

υLP = L/n + 4 υIP = L + 4

Lemma

There are instances for which υLP(CUT F) ≈ 1 |F|OPT.

SLIDE 21

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Quality of LP bounds?

Example

Facility opening and assignment costs are 1. crs = L and csi = 1, for all i ∈ {1, . . . , n}. 1

1

2

r

L 1

s

1

.

. . ⋆k n−1 n

Solutions

υLP = L/n + 4 υIP = L + 4

Lemma

There are instances for which υLP(CUT F) ≈ 1 |F|OPT.

SLIDE 22

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Quality of LP bounds?

Example

Facility opening and assignment costs are 1. crs = L and csi = 1, for all i ∈ {1, . . . , n}. 1

1

2

r

L 1

s

1

.

. . ⋆k n−1 n

Solutions

υLP = L/n + 4 υIP = L + 4

Lemma

There are instances for which υLP(CUT F) ≈ 1 |F|OPT.

SLIDE 23

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Example: Neusiedl am See – Last Mile Network

customers Steiner nodes

pen facilities

SLIDE 24

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Example: Neusiedl am See – Last Mile Network

customers Steiner nodes

pen facilities

SLIDE 25

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Hop Constrained ConFL

HC ConFL:

Connected Facility Location problem with an additional constraint:

At most H nodes on each path between the root r and an open

facility.

Application:

When survivability of a network with respect to node failures is an issue.

SLIDE 26

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Path Constraints

Introduced for Hop Constrained Steiner Trees by Costa et al., 2009
Limit the number of open edges in each path P by H:
uv∈P

(xuv + xvu) ≤ H ∀P

Sufficient to consider paths originating in r with H + 1 edges.

SLIDE 27

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Path Constraints

Introduced for Hop Constrained Steiner Trees by Costa et al., 2009
Limit the number of open edges in each path P by H:
uv∈P

(xuv + xvu) ≤ H ∀P

Sufficient to consider paths originating in r with H + 1 edges.

SLIDE 28

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Path Constraints

Introduced for Hop Constrained Steiner Trees by Costa et al., 2009
Limit the number of open edges in each path P by H:
uv∈P

(xuv + xvu) ≤ H ∀P

Sufficient to consider paths originating in r with H + 1 edges.

SLIDE 29

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Path Constraints

Introduced for Hop Constrained Steiner Trees by Costa et al., 2009
Limit the number of open edges in each path P by H:
uv∈P

(xuv + xvu) ≤ H ∀P

Sufficient to consider paths originating in r with H + 1 edges.

SLIDE 30

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Path Constraints

Introduced for Hop Constrained Steiner Trees by Costa et al., 2009
Limit the number of open edges in each path P by H:
uv∈P

(xuv + xvu) ≤ H ∀P

Sufficient to consider paths originating in r with H + 1 edges.

SLIDE 31

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Jump Inequalities

Introduced for Hop Constrained MST by Dahl et al., 2006

Jump (S0, . . . , SH+1) for H = 3

S0

S4

S1

S3

S2

Definition
Partition of the core nodes with
r ∈ S0
k ∈ SH+1 ∩ F \ {r}

ij∈J xij ≥ zk

∀J, ∀k

SLIDE 32

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Jump Inequalities

Introduced for Hop Constrained MST by Dahl et al., 2006

Jump (S0, . . . , SH+1) for H = 3

S0

S4

S1

S3

S2

Definition
Partition of the core nodes with
r ∈ S0
k ∈ SH+1 ∩ F \ {r}

ij∈J xij ≥ zk

∀J, ∀k

SLIDE 33

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Jump Inequalities

Introduced for Hop Constrained MST by Dahl et al., 2006

Jump (S0, . . . , SH+1) for H = 3

S0

S4

S1

S3

S2

Definition
Partition of the core nodes with
r ∈ S0
k ∈ SH+1 ∩ F \ {r}

ij∈J xij ≥ zk

∀J, ∀k

SLIDE 34

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Jump Inequalities

Introduced for Hop Constrained MST by Dahl et al., 2006

Jump (S0, . . . , SH+1) for H = 3

S0

S4

S1

S3

S2

Definition
Partition of the core nodes with
r ∈ S0
k ∈ SH+1 ∩ F \ {r}

ij∈J xij ≥ zk

∀J, ∀k

SLIDE 35

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Jump Inequalities

Introduced for Hop Constrained MST by Dahl et al., 2006

Jump (S0, . . . , SH+1) for H = 3

S0

S4

S1

S3

S2

Definition
Partition of the core nodes with
r ∈ S0
k ∈ SH+1 ∩ F \ {r}

ij∈J xij ≥ zk

∀J, ∀k

SLIDE 36

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Jump Inequalities

Introduced for Hop Constrained MST by Dahl et al., 2006

Jump (S0, . . . , SH+1) for H = 3

S0

S4

S1

S3

S2

Definition
Partition of the core nodes with
r ∈ S0
k ∈ SH+1 ∩ F \ {r}

ij∈J xij ≥ zk

∀J, ∀k

SLIDE 37

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Cut based formulations

Path Jump customer based cuts CUT P

R

CUT J

R

facility based cuts CUT P

F

CUT J

F

Alternatives to separating hop constraints?

For Hop Constrained MST, Gouveia et al., 2010, considered a

transformation into STP on a Layered Graph.

For HC ConFL we can show three ways of such a disaggregation.

SLIDE 38

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Cut based formulations

Path Jump customer based cuts CUT P

R

CUT J

R

facility based cuts CUT P

F

CUT J

F

Alternatives to separating hop constraints?

For Hop Constrained MST, Gouveia et al., 2010, considered a

transformation into STP on a Layered Graph.

For HC ConFL we can show three ways of such a disaggregation.

SLIDE 39

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Layered Graphs: Disaggregated Core Graph (H = 3)

r

(5,1)
(4,1)
(3,1)
(2,1)
(1,1)
(5,2)
(4,2)
(3,2)
(2,2)
(1,2)

(5,3)

(4,3)
(3,3)
⋆

⋆ ⋆ ⋆ ⋆ 3

⋆

r

10

1

10 ◦2

4
⋆

5

⋆

SLIDE 40

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Layered Graphs: Disaggregated Core Graph (H = 3)

r

(5,1)
(4,1)
(3,1)
(2,1)
(1,1)
(5,2)
(4,2)
(3,2)
(2,2)
(1,2)

(5,3) (4,3)

(3,3)
⋆

⋆ ⋆ ⋆ ⋆ 3

⋆

r

10

1

10 ◦2

4
⋆

5 ⋆

SLIDE 41

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Layered Graphs: Disaggregated Core and Assignment Graph (H = 3)

r

(5,1)
(4,1)
(3,1)
(2,1)
(1,1)
(5,2)
(4,2)
(3,2)
(2,2)
(1,2)

(5,3)

(4,3)
(3,3)
⋆

⋆ ⋆ ⋆ ⋆ 3

⋆

r

10

1

10 ◦2

4
⋆

5

⋆

SLIDE 42

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Layered Graphs: Disaggregated Core and Assignment Graph (H = 3)

r

(5,1)

(4,1) (3,1)

(2,1)
(1,1)
(5,2)

(4,2) (3,2)

(2,2)
(1,2)

(5,3) (4,3)

(3,3)
⋆

⋆ ⋆ ⋆ ⋆ 3

⋆

r

10

1

10 ◦2

4
⋆

5 ⋆

SLIDE 43

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Hierarchy

LG xCUT F ↔ LG x,zCUT F

LG xCUT R ↔ LG x,zCUT R

HDF

HDR
HOPF ↔ HOPR
MCF F
CUT J

F

MCF R
CUT J

R

MTZ
CUT P

F

CUT P

R

Figure: Relations between LP-relaxations of MIP models for ConFL

SLIDE 44

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Computational study

Instances

Combining UFL instances from UflLib and STP instances from

OR-library

UFL instances: {mp,mq}-{1,2} with

|F| × |R| ∈ {200 × 200, 300 × 300}

STP instances: {c,d}{5,10,15,20} with 500 to 1000 nodes and

up to 25000 edges

Soft- & Hardware

Intel Core2 Quad 2.33 GHz processor, with 3.25 GB RAM
Branch-and-Cut implementation uses ILOG Concert Technology v2.7
Solver: IBM CPLEX v11.2
Setup: no CPLEX cuts; time limit 3600 secs; higher branching

priority to variables z

SLIDE 45

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Results

LG xCUT F HOP LG xCUT R H gapLP #OPT t gapLP #OPT t gapLP #OPT t 3 c mp 1.43 8 26.3 1.45 8 25.8 1.28 8 107.0 3 c mq 2.11 8 116.5 2.12 8 87.0 2.01 8 955.3 3 d mp 1.04 8 21.0 1.04 8 31.3 0.97 8 139.9 3 d mq 1.36 8 68.8 1.36 8 75.5 1.30 8 620.8 4 c mp 1.35 8 42.9 1.37 8 39.2 1.12 8 297.9 4 c mq 2.06 8 211.7 2.07 8 167.0 1.91 5 2058.6 4 d mp 1.71 8 41.1 1.72 8 52.2 1.53 8 222.6 4 d mq 2.21 8 167.2

6
2.10

6 1561.8 5 c mp 1.67 8 66.6 1.71 8 64.5 1.41 7 1207.9 5 c mq 2.38 8 235.9

6
2.24

3 2702.8 5 d mp 1.55 8 87.6

7
1.33

8 305.4 5 d mq 2.26 8 229.9

6
2.10

5 2279.8 96 89 82

SLIDE 46

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Conclusion

Our Branch-and-Cut approach: valuable tool for solving large-scale

real world instances

Layered Graphs are an alternative to the separation of Path or Jump

inequalities

Future Research

Stochastic ConFL: set of customers is not fixed
Multi-Objective optimization: hop limit versus coverage constraints

SLIDE 47

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion

Conclusion

Our Branch-and-Cut approach: valuable tool for solving large-scale

real world instances

Layered Graphs are an alternative to the separation of Path or Jump

inequalities

Future Research

Stochastic ConFL: set of customers is not fixed
Multi-Objective optimization: hop limit versus coverage constraints

SLIDE 48

Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion