business & market intelligence / or 1 project saru july 2006 ivana ljubic university vienna...

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Business & Market Intelligence / OR

Project SARU

July 2006

Ivana Ljubic University ViennaBertram Wassermann Telekom Austria

How to situate Access Remote Units and construct a minimal cost fibre optic cable network

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Problem Definition Overview1

Introduction

Broadband demand increases.• New products and Services (ADSL TV) • Number of customers still increasing

Existing local area access networks are based on copper cables• Limited with respect to bandwidth and distance • Will not cover upcoming demand

Fibre optic technology is the alternative• nearly unlimited bandwidth• used for core – net• rarely for LAN

Consequence • Creating a new network -> network design problem

Terms known in the industry• FTTH, Fibre To The Home• FTTC, Fibre To The Curb

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Introduction

Fibre To The Home (FTTH)• No copper between customer and switching centre

anymore• Customers are directly connected via fibre optic cable• Probably no multiplexing (or only small scale)• Passive, no need electricity

Fibre To The Curb (FTTC)• A Access Remote Unit (ARU) is placed close (“at the curb”)

to several customers• “Last few meters” still copper• The ARU functions as a translator between copper

(electricity) and optical medium (light)• Serves also as a multiplexer (Customers share Fibre)

Solution to these network design problems:• Steiner Trees and its capacitated variants• Well studied• Although NP-hard, fast algorithms do exist

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Introduction

Search for an alternative• SARU, Situating Access Remote Unit• Fibre as close to the customer as necessary and as far as

possible

Problems with FTTH and FTTC• Expensive as a country-wide approach• Inefficient: Telekom Austria wants to be prepared for any

customer, but knows not all customers will come.• FTTH or FTTC probably suitable for certain LANs or specific

parts of LANs

Key idea• Within a certain distance (L) of the customer an ARU

(Access Remote Unit) has to be placed / situated which houses this customer.

• Copper network still supplies last mile• At the moment L = 600m

Distance Metric• Length of cable is used

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Switching nodes

Problem Definition Graph2

Typical LAN structure

Switching centre Root of the

Copper Tree Source node

Customer nodes

Copper cablesCopper tree

Leaves are customersBut customers need not be leaves

Graph structure should be tree-like.Big pre-processing problem!

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Potential Fibre Optics Network

Potential Fibre Optics Lineswith intersection nodes

All nodes should be connected to the switching centre

The Fibre Optics net should form a connected graph!

*) FON is not shaped like a rectangular grid!Shape indicates, that FON may be of different form then copper net. However, nets are superimposed

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Additional Nodes:Intersection points of FOL and Copper Net


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Potential ARU positions

Additional Nodes: Intersection points of FOL and Copper Net

Potential ARU Positions are chosen in the vicinity of intersections of copper net and fibre optic net


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Distance Condition L and Edge Directions

Assignement of Customer to potential ARUs under Distance Condition

Additional Condition:

Never go up the tree, always go down towards root.

But Fibre Optic edges may be used in one of the two directions.

Consequently:

Copper edges are directed.

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Distance Condition L and Edge Directions

Assignement of Customer to potential ARUs under Distance Condition

Alternative Representation of the Copper Net obeying Distance condition

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Optimization Problem

Find Positions for ARUs and create Fibre Optic Network such that

• all customers are served

• all ARUs are connected to the root by fibre optic lines

• all this is done at minimal cost

• all other constraints are met (length L)

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Comparison with FTTH and FTTC

In FTTH(C) the end-nodes (ARU positions) are given and therefore fixed.

No optimisation of their position is necessary.

This optimisation formulation corresponds to the Steiner Tree Problem.

In our problem the graph consist of two strictly separated layers (copper network, potential FON) and a set of nodes potentially connecting them.

In FTTH and FTTC there is “just” one layer and no set of designated nodes besides customer nodes.

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Related Problem3

Connected Facility Location Problem (ConFL)

Given a graph G=(V,E), with lengths on the edges, with a subset of facilities, their opening costs and client demands.

Our goal is to:• Pick a set of facilities to open• Assign each demand to an open facility• Connect all open facilities by a Steiner tree • Minimize the costs of opening and assigning facilities, plus the cost of the Steiner tree

Our problem reduces to ConFL if edge installation costs are M*length.

Approximation algorithms:

• Gupta et al. (2003): randomized 3.55-factor algorithm (no opening costs)

• Swamy & Kumar (2002): 9-approx. algorithm for general case

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Related Problem3

Capacitated Local Access Network Design (CapLAN)

Simplifying our problem:

• For already placed access nodes, find minimum-cost capacity installation of the fiber optic network.

•Also known as Network Loading Problem. Edge-cost function depends on capacity and may be piecewise-linear or step function.

Uniform capacities:

• Edge-cost function the same for all edges Single-sink buy-at-bulk• Approx. algorithms: Gupta et al. (2003)• Polyhedral approaches: Magnanti (1995), Günlück (1999)

Non-uniform capacities:

• Dahl & Stoer (1998): cutting plane approach

We propose our problem-specific non-uniform ILP formulation

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Problem Definition Cost4

Customer Demand

Demand in terms of copper lines (twisted pairs of copper lines)

With every customer a certain demand di is associated

Rule:Demand has to be completely satisfied

d1

d2

d3

d4

d5

d6

di

di+1

dn

Not in the sense of bandwidth

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Cost related to the Copper Net

The copper network has to be incorporated as it is.No alteration allowed!

No cost due to copper network.

d1

d2

d3

d4

d5

d6

di

di+1

dn

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Cost related to the potential ARU locations

costARU( location, demand)

Location cost factors are:• Outdoor or indoor• Electricity• Rental • Development

Demand cost factors are:• Type of ARU (mainly size = number of copper lines to be served)

Cost function is a step function (also in terms of demand)

Buy at Bulk principle:Price per unit (=served copper

line) decreases with increasing size of ARU

ARUs produce demand. #Fibre Optic Lines depends on type of ARU

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Cost related to the potential Fibre Optic Network

In general: costFON( length, demand) per edge

But: The potential FON is a union of 3 layers.

Layer 1: Dark FibreExisting Fibre Optic Lines which are not in use

Layer 2: Empty PipesEmpty pipes where fibre optic cables may be inserted

Layer 3: ExcavationExcavating trenches and laying new pipes

None of the layers need to form a connected graph.

New trenches usually follow roadmaps

Two adjacent nodes of the potential FON may be connected by any combination of the 3 edge types!

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Graph Structure of FON

Any combination of the 3 edge types may connect two nodes.

Into both directions

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Dark Fibre Edge: costDF ( length, demand) = const

The cost for dark fibre may by viewed as being constant.

It is independent of the length of the line.

The work cost resulting from lighting the lines is a constant compared to costs resulting from other layers.

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Insertion cost for fibre optic cables is linear in terms of edge length

Need to know cost of cables per unit length

Like for ARUsCost function is a step function with respect to demand.

Empty Pipes: costEP ( length, demand) = length*costEP/UL (demand)

Again Buy at Bulk principle:Price per unit (=optic fibre) decreases with increasing size of fibre optic cables

linear cost

0

50

100

150

200

250

300

350

400

450

0 20 40 60 80 100 120 140

#Optic Fibres

cost

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Excavation costs depend linearly on the edge length

Excavation costs depend on location in two ways:• regionally cost may differ (big city, small city, country-side)• surface conditions (concrete, soil, …)

Cost function obeys economies of scale (compare Buy at Bulk principle)


Excavation: costExT ( length, location, demand) = length*costExT/UL (location, demand)

Simplification:Costs are based on the assumption,

trenches are filled completely with pipes which are completely filled with fibre cable.

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Excavation: costExT ( length, location, demand) = length*costExT/UL (location, demand)

costFON( length, location, demand) per edge =

const + length * [costEP/UL (demand) +costExT/UL (location, demand)]

Cost function is dominated by excavation costs.

Cheapest contribution from Dark Fibre.

With respect to free capacities it will be the other way round.

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Solution Strategy5

Overview

Phase 1:Solving the simplified problem:• To improve the solution• To study the cost function

1

Phase 2:Solving the problem• To find an exact algorithm• Study the approximation qualities of heuristic solutions

2

Pre-Phase, Heuristic solution• A (really) fast algorithm for a first solution• Finding a feasible solution for a given instance• Initial upper bound for exact (branch-and-bound based)

algorithm

0

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Phase 0 & 16

Simplified Optimization Problem

Start with copper net0

Find “optimized” Positions for ARUs heuristically

1

Switch to Fibre Optic Net2

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Phase 0 & 16

Simplified Optimization Problem

Create fibre optic network with:

Phase 0Heuristic Algorithm

Phase 1Integer Linear Program (exact)

3

Start with copper net0

Find “optimized” Positions for ARUs heuristically

1

Switch to Fibre Optic Net2

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Pre-Phase, Heuristic Solution6

Minimal number of ARUs

Idea:Optimal solution will “minimize” the number of ARUs necessary to satisfy all demand.Hence, a set of ARUs satisfying all demand and minimal in number will approximate the optimal (=cost minimal) solution.

Algorithm

Pick customer furthest away from source.

1

Choose potential ARU node furthest away from this customer still valid under distance condition L

2

Install ARU at this position and serve all customers of sub tree rooted at this node

3

Ignore sub-tree and proceed form step 1

4

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Minimal number of ARUs

Solution is uniqueOf all solutions with minimal number of nodes it’s the one where no ARU can be moved closer to the source node without violating the distance condition L for at least one customer.

Dropping this condition gives rise to different solutions

For example:

Nice to have: We know minimal number of ARU nodes needed to provide complete service.

!

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Cost minimized Fibre Optic Net

Simple but fast approachto connect so found ARUs with source node via FON

Imitation of the Minimal Cost Flow algorithm for linear cost functions

Pick any unconnected ARU and determine shortest path through actual network.

1

Update network along shortest path:• cost-functions on used edges• free capacities• used capacities

2

Repeat from step 1 until all ARUs are connected.3

Works for network with “unlimited” capacities on edges.

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Connectors Type Set … N=N1 U N2 U N3

For every edge type several connectors are possibleEdge Type 3: excavation trenches of different size

and filling … N3

Edge Type 2: different (combination of) cables to fill empty pipes … N2

Edge Type 1: different (combination of) dark fibres … N1

Phase 17

CapLAN: Notation for ILP formulation

Different Edge Types:Edge Type 1: Dark fibre edgesEdge Type 2: Empty pipes edgesEdge Type 3: Excavation edges

Directed graph representing FON

with customer set (ARUs)

and sink (switching centre) s

VK

AVG

),(

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Phase 17

CapLAN: Notation for ILP formulation

length of edge (i,j):

building cost of connector type n:

indicator variable for connector type n beinginstalled on edge (i,j):

flow on edge (i,j) using connector type n

flow on edge (i,j) using connector type nfor customer k

customers demand (careful! customer=ARU)

capacity limit for edge (i,j) and connector type nnij

k

knij

nij

nij

n

ij

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ob

jectiv

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Flow preservation constraints

Capacity constraints

con

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ints

Phase 1

Vi

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7

CapLAN: ILP Single-commodity formulation

Aijxn

nij

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1N

N nAijx nij ,}1,0{,

Aijxn

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3N

33


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Capacity constraints

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Phase 1

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7

CapLAN: ILP Multi-commodity formulation with 0/1 variables

Aijxn

nij

,10 ,

1N

N nAijx nij ,}1,0{,

Aijxn

nij

,10 ,

3N

N

nAijudf nijKk

kknij ,,0 ,,

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Additional Connector Type Set For every edge type several connectors are possible

Edge Type 0: Copper Connectors … N0

(only one element) Edge Type A: potential ARUs … NA

Phase 28

Notation for ILP formulation

Additional Edge Types:Edge Type 0: Copper Connection of Customer and potential

ARU nodeEdge Type A: potential ARUs represented as edges

(Now real) Customer nodes

ARU nodes in (customer side)

ARU nodes out (sink side) VpARU

VpARU

VC

2

1

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Phase 28

Notation for ILP formulation

length of edge (i,j):

building cost of connector type n:

indicator variable for connector type n beinginstalled on edge (i,j):

flow on edge (i,j) using connector type n

flow on edge (i,j) using connector type nfor customer k

customers demand

ARU demand

capacity limit for edge (i,j) and connector type n nij

An

k

knij

nij

nij

n

ij

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nda

Ckd

f

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36


ob

jectiv

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Aij n nnijnnijn

nijnijn

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xcxclxc3211

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Copper Net

Input ARU

con

stra

ints

1

Phase 2

1pARUiCkx nki ,},1,0{0,

Aniij npARUix N1 ,},1,0{),(

8

ILP Single-commodity formulation

CkxpARUi

nki

,10,

1

21

N

pARUijpARUi

xuxdAn

nijnijCk

nkik

)(,

,,, 0

Aniijnniij npARUixdaf N1 ,),(),(

AijxAn

nij

,10 ,N

37


ob

jectiv

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Aij n nnijnnijn

nijnijn

A

xcxclxc3211

11 ,,,minNN NN

Fibre Optic Net


con

stra

ints

2

Phase 2

)(\

,0

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1

),(

,, 1

pARUCVi

else

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Aij

pARUv nnvvjn

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Aji nnji A

N

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Aijxn

nij

,10 ,

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N nAijxuf nijnijnij ,,0 ,,,

8

ILP Single-commodity formulation

Aijxn

nij

,10 ,

1N

N nAijx nij ,}1,0{,

Aijxn

nij

,10 ,

3N

38


Phase 28

Connected Facility Location in Multi-commodity Networks

How to find an exact solution for the stated optimisation problem?

39


Phase 28



40


Phase 28



41


Phase 28



42


Phase 28



43


Phase 28



44


Phase 28



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Generalisation9

Preparation of Land for Building

Difference 1:• Layer connecting node do not multiplex

The representation of this problem as a graph is very similar to the presented one:

• Customer demand has to be met through a potential network starting from a source node

• Graph of network consists out of two strictly separated layers (above ground, below ground) and a set of nodes potentially connecting the two layers)

Difference 2:• Design of network has to be optimised in both layers not

only in one.

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Contact:

Thank you!

Bertram WassermannMarketing Retail – Business & Market IntelligenceOperations Research

Telekom Austria AGLassallestrasse 9, A-1020 Wien Tel: +43 (0)59 059 1 31089

E-Mail: [email protected]: +43 (0)664 629 5527

Ivana LjubicUniversity of Vienna

E-Mail: [email protected]

business & market intelligence / or 1 project saru july 2006 ivana ljubic university vienna...

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