Modeling and solving of a radio antennas deployment support application with discrete and interval...

Michael Heusch - IntCP 2006

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Modeling and solving of a radio antennasdeployment support application with discreteand interval constraints


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Outline of the talk

Presentation of the application Modeling with discrete and interval constraints Defining search heuristics Modeling the problem with the distn constraint Experimental results on solving the progressive

deployment problem


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Presentation of the LocRLFAP

Informal description of the de radio antennas deployment problem :

Constraints involved : Distance between

frequencies depends on distance between antennas


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Minimal and maximal distances between antennas

Presentation of the LocRLFAP

Informal description of the de radio antennas deployment problem :

Difficulties : Hybrid combinatorial

optimisation problem non-linear continuous

constraints

Constraints involved : Distance between

frequencies depends on distance between antennas


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Specification of the problem

Formulation as a constrained optimisation problem: Data

Fixed set of antennas (transmitter-receiver) Dispatched on n sites {P1, … , Pn} The links to establish is known in advance

Variables of the problem: A solution associates one frequency to each antenna and a position to

each site Pi = (Xi,Yi): Position of a site

fi,j : frequency allocated to the link from P i to Pj

Optimisation problem: Minimise the maximal frequency used


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Constraints of the problem

Constraints of the problem discrete constraints:

Compatibility between antennas Forbidden frequencies

continuous constraints Maximum distance between antennas (range) Minimum distance between the antennas (security, interference)

mixed constraints Compatibility between the allocation and the deployment


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Comparing the RLFAP/LocRLFAP with 5 sites

RLFAP LocRLFAP


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Comparing the RLFAP/LocRLFAP with 5 sites

RLFAP LocRLFAP

dist² (Si,Sj) = Σi (Xi - Xj)²


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Comparing the RLFAP/LocRLFAP


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Hybrid solving with collaborating solvers

Original approach Modeling with the finite domain constraint solver Eclair Full discretization of the problem

Modeling three types of constraints Discrete constraints Continuous constraints Mixed constraints


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Discrete constraints

Co-site transmitter-receiver interference constraints:

Duplex distance constraints for each bidirectional link

Forbidden portions in the frequency range


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Continuous and mixed constraints

Elementary continuous constraints:dist²(Pi,Pj) > mij² , for all i<j

dist²(Pi,Pj) < Mij² , if there exists a radio link between Pi and Pj

Mixed constraints: Compatibility constraints

If dist(Pi,Pj)< d1, great interference If d1 <= dist(Pi,Pj)< d2, limited interference

Expression with elementary constraints { dist(Pi,Pj)< d1 } v { |fik-fjl| > Δ1 }, (i,j,k), i≠j, i≠k, j≠k { dist(Pi,Pj)< d2 } v { |fik-fjl| > Δ2 }, (i,j,k), i≠j, i≠k, j≠k

d1

d2


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Test set

Full deployment of networks with 5 to 10 sites

RLFAP

LocRLFAP


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Progressive deployment of networks with 9 and 10 sites

P

P

P

P

P

P

P

P P

P


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Solving with elementary constraints

Full deployment in both models


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Improvements to the search algorithm

Usage of a naïve Branch & Bound with: Distinction of the type of variables

The problem is under-constrained on positions

Branch on disjunctions? Branch first on constraints entailing a strong interdistance?

Variable selection heuristics minDomain min(dom/deg) minDomain+maxConstraints


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Results with minDomain+maxConstraints

9 sites 10 sites

99% of the backtracks are performed on the continuous part of the search tree

A bit less backtracks on the hybrid model

Hybrid solving is 1 to 3 times slower

Progressive deployment in both models


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Introducing the distn global constraint

distn ([P1, … , Pn], V)

Pi = Xi x Yi : Cartesian coordinates of the point pi V i,j : distance to maintain between Pi and Pj

distn(p1, … , pn], v)

satisfied if and only if dist(pi,pj) = vi,j

Filtering algorithm uses geometric approximation techniques


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Applications of the constraint

Molecular conformation

Robotics

Antennas deployment


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Using distn in the model

Second formulation of the problem with the global constraint: Simple continuous constraints

Introduction of a matrix {Vi,j} of distance variables:Domain(Vi,j)=[mi,j , Mi,j]

Expression of the set of min and max distance constraints:distn([P1, … , Pn], V)

Expression of the mixed « distant compatibility » disjunctions distn([P1, … , Pn], V)

{ Vij<d 1 } v { |fik-fjl| > Δ 1 }, (i,j,k), i≠j, i≠k, j≠k

{ Vij<d 2 } v { |fik-fjl| > Δ 2 }, (i,j,k), i≠j, i≠k, j≠k


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Results using distn (9 sites)

Simple heuristics Advanced heuristics

hybrid model / discrete model comparison:

1.8 times slower

1.5 times more backtracks

Similar performance of both models

wrt. simple model, distn divides by 2 the nb. of backtracks


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Results using distn (10 sites)

Simple heuristics Advanced heuristics

hybrid model / discrete model comparison:

4 additional instances are solved

Performance on the solved instances:

• 63% less backtracks

All instances are solved


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Quality of solutions

9 sites 10 sites


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Conclusion and perspectives

We showed the relevance of coupling discrete and continuous constraints Obtain solution of greater quality Better performance when solving Independence w.r.t. the discretization step Validation on one industrial application

Key points Definition of appropriate search heuristics Usage of the distn global constraint


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Perspectives on the application

Validation on instances of greater size Take forbidden zone constraints into account Provide deployment zones using polygons


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Other approaches for solving the RLFAP

Other approaches for solving the classical RLFAP Graph coloring Branch & Cut CP

LDS [Walser – CP96] Russian Doll Search [Schiex et. al - CP97]

Heuristics Tabou [Vasquez – ROADEF 2001] Simulated annealing, evolutionary algorithms…

Motivations for an approach using CP Robustness wrt modification of the constraints of the problem


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Sketch of distn’s filtering algorithm


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Filtering algorithm on polygons

Method using polygons for representing domains Theorem by K. Nurmela et P. Östergård (1999)

M. Markót et T. Csendes: A New Verified Optimization Technique for the ``Packing Circles in a Unit Square'' Problems. SIAM Journal of Optimization, 2005

pi2

pik-1

pik

pi1

Pi

Pj


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P1

P2


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-

-

++

+

+

P1

P2


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P1

P2


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Interval extension of the algorithm

P1

P2


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Filtering algorithm of distn

P1

P2

Adjusting bounds of the distance variables

Modeling and solving of a radio antennas deployment support application with discrete and interval...

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