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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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
Problem Definition Overview1
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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Business & Market Intelligence / OR
Problem Definition Overview1
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
Problem Definition Graph2
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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Business & Market Intelligence / OR
Problem Definition Graph2
Potential Fibre Optics Network
Additional Nodes:Intersection points of FOL and Copper Net
Potential Fibre Optics Lineswith intersection nodes
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Business & Market Intelligence / OR
Problem Definition Graph2
Potential Fibre Optics Network
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
Potential Fibre Optics Lineswith intersection nodes
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Business & Market Intelligence / OR
Problem Definition Graph2
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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Business & Market Intelligence / OR
Problem Definition Graph2
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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Business & Market Intelligence / OR
Problem Definition Graph2
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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Business & Market Intelligence / OR
Problem Definition Graph2
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
Problem Definition Cost4
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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Business & Market Intelligence / OR
Problem Definition Cost4
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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Business & Market Intelligence / OR
Problem Definition Cost4
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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Business & Market Intelligence / OR
Problem Definition Cost4
Graph Structure of FON
Any combination of the 3 edge types may connect two nodes.
Into both directions
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Business & Market Intelligence / OR
Problem Definition Cost4
Cost related to the potential Fibre Optic Network
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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Business & Market Intelligence / OR
Problem Definition Cost4
Cost related to the potential Fibre Optic Network
Dark Fibre Edge: costDF ( length, demand) = const
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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Business & Market Intelligence / OR
Problem Definition Cost4
Cost related to the potential Fibre Optic Network
Dark Fibre Edge: costDF ( length, demand) = const
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)
Empty Pipes: costEP ( length, demand) = length*costEP/UL (demand)
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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Business & Market Intelligence / OR
Problem Definition Cost4
Cost related to the potential Fibre Optic Network
Dark Fibre Edge: costDF ( length, demand) = const
Empty Pipes: costEP ( length, demand) = length*costEP/UL (demand)
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
Pre-Phase, Heuristic Solution6
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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Business & Market Intelligence / OR
Pre-Phase, Heuristic Solution6
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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
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
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u
Kkd
f
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x
nc
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,
...
}1,0{
...
...
N
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Business & Market Intelligence / OR
ob
jectiv
e
Aij nnijn
nijnijn xclxc
3211
11 ,,minNNN
Flow preservation constraints
Capacity constraints
con
stra
ints
Phase 1
Vi
else
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i
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,0
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7
CapLAN: ILP Single-commodity formulation
Aijxn
nij
,10 ,
1N
N nAijx nij ,}1,0{,
Aijxn
nij
,10 ,
3N
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Business & Market Intelligence / OR
ob
jectiv
e
Aij nnijn
nijnijn xclxc
3211
11 ,,minNNN
Flow preservation constraints
Capacity constraints
con
stra
ints
Phase 1
KkVi
else
si
ki
ffAij n
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nji
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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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Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
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
u
nda
Ckd
f
f
x
nc
Aijl
,
,
,
,
...
...
}1,0{
...
...
N
N
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Business & Market Intelligence / OR
ob
jectiv
e
Aij n nnijnnijn
nijnijn
A
xcxclxc3211
11 ,,,minNN NN
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
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Business & Market Intelligence / OR
ob
jectiv
e
Aij n nnijnnijn
nijnijn
A
xcxclxc3211
11 ,,,minNN NN
Fibre Optic Net
Flow preservation constraints
con
stra
ints
2
Phase 2
)(\
,0
,
1
),(
,, 1
pARUCVi
else
sixdaff
Aij
pARUv nnvvjn
nnij
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Aijxn
nij
,10 ,
2N
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8
ILP Single-commodity formulation
Aijxn
nij
,10 ,
1N
N nAijx nij ,}1,0{,
Aijxn
nij
,10 ,
3N
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Business & Market Intelligence / OR
Phase 28
Connected Facility Location in Multi-commodity Networks
How to find an exact solution for the stated optimisation problem?
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Business & Market Intelligence / OR
Phase 28
Connected Facility Location in Multi-commodity Networks
How to find an exact solution for the stated optimisation problem?
40
Business & Market Intelligence / OR
Phase 28
Connected Facility Location in Multi-commodity Networks
How to find an exact solution for the stated optimisation problem?
41
Business & Market Intelligence / OR
Phase 28
Connected Facility Location in Multi-commodity Networks
How to find an exact solution for the stated optimisation problem?
42
Business & Market Intelligence / OR
Phase 28
Connected Facility Location in Multi-commodity Networks
How to find an exact solution for the stated optimisation problem?
43
Business & Market Intelligence / OR
Phase 28
Connected Facility Location in Multi-commodity Networks
How to find an exact solution for the stated optimisation problem?
44
Business & Market Intelligence / OR
Phase 28
Connected Facility Location in Multi-commodity Networks
How to find an exact solution for the stated optimisation problem?
45
Business & Market Intelligence / OR
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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Business & Market Intelligence / OR
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]