T

1
Técnicas Distribuidas para Problemas de
Scheduling a Gran Escala

• Miguel A. Salido, M. Abril, F. Barber,
• P. Tormos, A. Llova, L. Ingolotti

GTI-IA, DSIC
Univerisdad Politécnica de Valencia España
IV Workshop Planificación, Scheduling y
Razonamiento Temporal Santiago de Compostela, 15
de Noviembre de 2005
2
Índice

• 1. Introducción
• 2. Modelo Distribuido
• 3. Ejemplo
• 4. Planificación de Rutas ferroviarias
• 5. Evaluación
• 6. Conclusiones y Trabajo Futuro

3
Definitions

• CSP
• Variables x1, x2, ....., xn
• Domains xi?Diai, bi i1..n
• Constraints c1,c2,..,ck

Distributed CSP
4
Distributed CSP

• A distributed CSP is a CSP in which the variables
  and constraints are distributed among block
  agents and there are interagent constraints.

• A block agent is a virtual entity that
  essentially has the following properties
  autonomy, sociability, reactivity and
  pro-activity.
• Each block agent has a semi-independent
  subproblem (variables and constraints) and
  attempts to determine the variable values.
• these values must also satisfy the interagent
  constraints.

5
Distributed Model
Preprocessing Agent
6
Preprocessing Agent

• Preprocessing Agent carries out a partition of
  the original problem in semi-independent
  subproblems.

• It uses a graph partitioning software called
  METIS.

• METIS solves the partition problem efficiently.

14.000 nodes
2 seconds
410.000 edges
www-users.cs.umn.edu/karypis/metis/index.html
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Block Agents
Subproblem
c12x1x32x5 c2x1-x3 c3x3-2x5x7 c42x3x5-2x7
Varx1,x3,x5,x7
c5x14x22x5 c6x1x2x3 c73x1-2x7x8 c82x3x5-
2x7 c92x2x3-2x8
Used Variablesx1,x3,x5,x7
x1 x3 x5 x7
x2 x4 x8
New Variablesx2,x4,x8
(3,-,1,-,2,-,3,-,...)
(3,4,1,1,2,-,3,-2,...)
Domains x2,x4,x8
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Example
Three variables x1, x2, x3 Domains d11,2,3
d21,2 d31,2,3 Constraints c1 x2 x3 c2
x1 ? x2
(-,1,1)
(-,1,2)
(-,2,1)
(1,1,1)
(2,1,1)
(2,2,2)
(3,2,2)
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Railway Scheduling Problem
TIME
Over a set of previously scheduled trains
The problem To assist to railway managers in
order to obtain a correct railway schedule.
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Our Approach CSP

• The railway scheduling problem as a Constraint
  Satisfaction Problem
• Variables (departure arrival time to stations)
  Traini_Arrivalj, Traini_Departurej
• Interpretation domain Integers
• Constraints
• Infrastructure Constraints
• Travel time between stations.
• Maintenance times
• Number of tracks in stations and sections,
• Traffic constraints
• Crossing and Overtaking.
• Reception-time, Expedition-time.
• Succession time between two consecutive trains,
• User Requirements
• Initial departure of first train, frequency,
  maximum travel time.
• Type and Number of trains.
• Commercial Stops,

Exponential Search Space
In typical and simple cases thousands and
thousands of alternatives
Distributed CSP
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How distribute the problem?

1. The preprocessing agent carries out the partition
   by means of METIS

1. The preprocessing agent carries out the partition
   by means of METIS
2. It divides the running map into clusters composed
   by contiguos stations

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How distribute the problem?

1. The preprocessing agent carries out the partition
   by type of train

13
Random Problems
Problems Distributed CSP Centralized CSP
lt50,25,10gt 12 3
lt100,25,10gt 12 14
lt150,25,10gt 15 37
lt200,25,10gt 16 75
lt250,25,10gt 17 98
lt300,25,10gt 19 140
lt350,25,10gt 23 217
lt400,25,10gt 30 327
lt450,25,10gt 32 440
lt500,25,10gt 42 532
gt
lt n,a,p gt n variables a arity p partition
size
lt
lt
lt
lt
lt
lt
lt
lt
lt
14
Distributed Railway
15
Distributed Railway
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Conclusions

• A distributed model for solving large-scale CSPs
  was present
• A preprocessing agent partitions the problem in
  semi-independent sub-CSPs
• A set of block agents incrementally and
  concurrently built partial solutions until a
  global solution is found.
• Each block agent may solve its subproblem with
  different algorithms
• Large CSPs (Railway Scheduling Problems) could
  be solved more efficiently.

17
Future Works

• The Railway Scheduling Problem can be
  partitioned by different ways.
• We are working on techniques to partition the
  problem, depending on the problems objective.
• Evaluate the ways in which Railway Scheduling
  Problem can be distributed.

18
Técnicas Distribuidas para Problemas de
Scheduling a Gran Escala

• Miguel A. Salido, M. Abril, F. Barber,
• P. Tormos, A. Llova, L. Ingolotti

GTI-IA, DSIC
Univerisdad Politécnica de Valencia España
IV Workshop Planificación, Scheduling y
Razonamiento Temporal Santiago de Compostela, 15
de Noviembre de 2005