Optimization Methods vs
Spatial Stochastic Models in
a Hierarchical Network
problem
Christian Destré, France Telecom R&D
[email protected]
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Research & Development
March 2006
Introduction
 Hierarchical network problem:
Location of network equipments
 Compare 2 different ways to solve this problem:
Spatial Stochastic Models
Operational Research
 Aim of this talk:
Present and discuss the approaches
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Outline
 3-level hierarchical network
Model and cost function
 Use of Spatial Stochastic models
Poisson-Voronoi Spanning Trees
 Another approach: Operational Research
Location Problem / Weber Problem
 Discussion
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3-level hierarchical network model
 3 types of equipment
2
 3-level hierarchical network
 Connections are done
according to shortest paths
1
 Subscriber line model
(PSTN)
0
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Costs
 For each link:
Capacity cost: between a point of level-i and the closest point of leveli+1 (medium used):
Ai ,i 1r
 i , i 1
Infrastructure cost (civil engineering):
Bi ,i 1r
 i ,i1
(r: distance, A,B,, non negative parameters)
 Constant equipment installation cost
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Cost function


1, 2
1 , 2
C1  B1, 2 y i

 A1, 2 yi
N yi



Link costs between level-1 and level- 2





yi N1 V0 ( N 2 ) 
 0 ,1
 0 ,1 
B0,1 x j  y i
 A0,1 x j  y i



 x j N 0 V yi ( N1 )



 
costs between level-1 and level-0
Link





Cost computed for level-1 equipments
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Cost function
2


B1, 2 yi 1, 2  A1, 2 yi 1, 2 N yi

1
Link costs between level-1 and level-2
yi
C1
 B
0 ,1
x j N 0 V y i ( N1 )
x j  yi
 0 ,1
 A0,1 x j  y i
 0 ,1

0 


Link costs between level-1 and level-0
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Level-1
 Level-0 and level-2 are given
 Number and location of equipments on level-1 in
order to minimize the total cost ?
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Use of spatial geometry
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Poisson-Voronoi Spanning Trees
 [Bacelli, Zuyev 96]
π2  3 independent
homogeneous
Poisson point
π1
processes π0 π1 π2
 Level-i is obtained
from realizations of πi
π0  λi is the intensity of πi
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λ1 intensity
 Given the intensities λ0 and λ2 and the installation cost
 Find λ1 that minimizes the average total cost
 Bacelli&Zuyev have proved that
  f (0 , 2 , cost parameters )
*
1
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Discussion
 Average method: no exact result for specific
instance
 Need realizations
 Analytical formula: direct result
 No computation time (without considering the
realizations of the point processes)
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Operational Research
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OR: Location problem
 Given a set of facility locations and a set of
customers:
Which facilities should be opened ?
Which customers served by which open facility ?
 Minimize the total cost of serving all the customers
 Facility locations are known
 Simple Plant Location Problem
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OR: Weber problem
 Location in the plane (continuous)
 Locate a given number m of new facilities to serve a
set of n customers
 To minimize the total service (transportation +
installation) cost
 NP-hard problem (depending on the number of
customers and equipments)
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OR: Formulation
m
n
Minimize Z   wij d ( X i , A j )  mC
i 1 j 1
m
subject to
w
i 1
ij
 w j , j  1,.., n
X i   , wij  0, i, j , m integer  1
2
 Aj = coordinates of customer j
 Xi = coordinates of new facility i
 wij = decision variable : connecting customer i to
facilty j
 d(X,Y) = distance function
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Brimberg et al. approach 04
 Fix m (unknown)
SPLP with customer = potential facility
 Improve the solution in continuous space
 Multi-phase heuristic approach
 Work to do:
Adapt this approach to our problem
- Distance function
Use the best heuristics
- Time consumption / instance size (number of points)
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OR: Discussion
 Exact or approximated solution for concrete
instances
 Time consuming
 Good for small instances
 What about big instances ? (to define)
 For example: Brimberg et al. solve instances with
1060 customers, 121 facilities (CPU time = 1370s
on a Sun Spare Station 10)
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Comparison
 Which optimization methods is interesting ?
According to the instance size & time consumption
 What about the quality of the solutions ?
Differences between the two approaches
 Can we validate the stochastic approach by OR ?
Are OR solutions Poisson ?
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Thank you
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