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Problem Reformulation and Search. Patrick Prosser & Evgeny Selensky. A 3 year project funded by EPSRC supported by ILOG. Car Sequencing define constrainedness derive heuristics investigate reformulations. Routing & Scheduling investigate reformulations

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slide1

Problem Reformulation and Search

Patrick Prosser & Evgeny Selensky

slide5

A 3 year project

  • funded by EPSRC
  • supported by ILOG
  • Car Sequencing
    • define constrainedness
    • derive heuristics
    • investigate reformulations
  • Routing & Scheduling
    • investigate reformulations
    • use Scheduler and Dispatcher

… and other things

slide6

Current Status … 4 things

  • investigations of stable marriage problem
    • paper at CP01
  • initial study of reformulation in the large
    • VRP & OSSP & JSSP
    • using Scheduler and Dispatcher
    • paper at Formul’01
  • initial study of reformulation in the small
    • 0/1 encodings
    • using Solver and Choco
    • paper at Formul’01
  • Constrainedness of car sequencing
    • meetings with IPG and BMS

And now for vrp&ssp and 0/1 encoding

slide7

JSSP

Scheduler

CVRPTW

Dispatcher

VRP and JSSP: Extremes on a Spectrum

  • Extremes
  • will there be problems somewhere in between?
  • where you might use Dispatcher and/or Scheduler
  • a pragmatic study
slide8

Encoding VRP as an Open Shop Scheduling Problem

  • vehicles
    • with capacities
  • visits
    • with loads/demand
    • with time windows
  • distance between visits
  • minimise distance traveled
    • reduce vehicles used
  • machines
    • with consumable resource
  • operations/activities
    • with resource requirement
    • with time windows
  • transition times between operations
  • minimise make span
  • Translate CVRPTW into OSSP
  • solve OSSP with Scheduler
  • solve CVRPTW with Dispatcher
  • compare, using tools as intended
  • an extreme … expect to be bad
slide9

Encoding Job Shop Scheduling Problem as a VRP

  • vehicles
  • visits
    • specified vehicles
    • with time windows
    • with durations
  • sequence constraints between visits
  • zero travel times
  • respect time windows on vehicles
  • minimise make span
  • machines
  • operations/activities
    • specified resource
    • with time windows
    • with duration
  • jobs
    • sequence of operations
  • minimise make span
  • Translate JSSP to VRPTW with zero travel
  • solve VRPTW with Dispatcher
  • solve JSSP with Scheduler
  • again, compare, using tools as intended
  • an extreme … expect to be bad
slide10

The study continues: VRP and OSSP problem generation

  • use benchmark vrp’s
  • select R local (nearby) visits
  • R visits in same vehicle
  • produce an optimal tour for R
  • write out sequence constraints
  • iterate
  • the R sequences/tours maybe like a job
  • but on one resource
slide11

Problem Reformulation in the Small

  • Investigate problems with 0/1 variables
    • independent set of a hypergraph
    • maximal independent set of a hypergraph
    • construction of bibd <v,b,r,k,l>
  • Two common constraints
    • summation of variables
    • biconditional
  • Variety of encodings
    • for summation
    • for biconditional
  • Two toolkits
    • ILOG Solver
    • Choco
slide12

A hypergraph G = (V,E)

  • V is a set of vertices
  • E is a set of hyperedges
    • an edge with 2 or more vertices
  • An independent set S
  • assume vertices(e) is set of vertices in hyperedge e
  • Maximal independent set S
  • there is no independent set S’ that subsumes S
    • add anything to S and you lose independence!
slide13

A Hypergraph

1

9

2

3

4

5

7

8

6

slide18

Independent Set

  • ind1
  • the sum of the variables is less than or equal to k
  • ind2
  • the number occurrences of 1’s is less than or equal to k

ind1S and ind2S in Solver

ind1C and ind2C in Choco

Hypergraphs are bibd’s where blocks are hyperedges

I.e. regular degree hypergraphs

slide19

Nodes are same for all (same level of consistency?)

  • summation 3 times faster than occurrence in Solver
  • occurrence 20 times faster than sum in Choco
slide20

Maximality and the biconditional

Three encodings of the biconditional

slide22

p and q or not p and not q is best for Solver

    • worst for Choco
  • can be 3 times
slide26

Conclusion?

  • On 0/1 encodings
    • big variations within a toolkit
    • variation across toolkits
    • experiment!
  • On VRP/Dispatcher and OSSP/Scheduler
    • extremes explored
    • experiments being designed
  • Car Sequencing
    • on the stack (one pop away)
  • Stable Marriage
    • need a long term project
slide27

Other Things

  • 4th year Student Projects
    • student handbook, a design problem
    • anaesthetist’s rota
  • 3d year Student Projects
    • vrp system (3d year)
  • Teaching Goal
    • 4th year course in constraint programming
  • Research
    • SAC, a new algorithm (with Kostas)
    • stable marriage and consraint programming
    • bioinformatics?