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**Problem Reformulation and Search**

Patrick Prosser & Evgeny Selensky

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**funded by EPSRC supported by ILOG A 3 year project 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 Formul’01 initial study of reformulation in the small 0/1 encodings using Solver and Choco Constrainedness of car sequencing meetings with IPG and BMS And now for vrp&ssp and 0/1 encoding

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**VRP and JSSP: Extremes on a Spectrum**

CVRPTW Dispatcher JSSP Scheduler 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 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

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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 <v,b,r,k,l> 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 1 2 3 4 5 7 9 8 6

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An Independent Set 1 2 3 4 5 7 9 8 6

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**The Largest Independent Set**

1 2 3 4 5 7 9 8 6

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**A Maximal Independent Set**

1 2 3 4 5 7 9 8 6

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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 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

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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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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?

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**Thanks for supporting us**

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Questions?

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Maximal Independent Sets of a Hypergraph IJCAI01.

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