Problem Reformulation and Search

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Problem Reformulation and Search
Patrick Prosser Evgeny Selensky
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• 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
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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 Formul01
• initial study of reformulation in the small
• 0/1 encodings
• using Solver and Choco
• paper at Formul01
• Constrainedness of car sequencing
• meetings with IPG and BMS

And now for vrpssp and 0/1 encoding
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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

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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

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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

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The study continues VRP and OSSP problem
generation

• use benchmark vrps
• 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

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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 ltv,b,r,k,lgt
• Two common constraints
• summation of variables
• biconditional
• Variety of encodings
• for summation
• for biconditional
• Two toolkits
• ILOG Solver
• Choco

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• 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!

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A Hypergraph
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An Independent Set
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The Largest Independent Set
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A Maximal Independent Set
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More Generally
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Independent Set

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

ind1S and ind2S in Solver ind1C and ind2C in Choco
Hypergraphs are bibds where blocks are
hyperedges I.e. regular degree hypergraphs
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• 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

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Maximality and the biconditional
Three encodings of the biconditional
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• p and q or not p and not q is best for Solver
• worst for Choco
• can be 3 times

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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

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Thanks for supporting us
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Questions?