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Graph Transformations for Vehicle Routing and Job Shop Scheduling Problems J.C.Beck, P.Prosser, E.Selensky c.beck@4c.ucc.ie, {pat,evgeny}@dcs.gla.ac.uk

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ICGT 2002, E. Selensky2 w1w1 w2w2 w 12 wnwn wiwi w n-1 w 1,n w n-1,n w 1,n-1 w 2,n w 2,n-1 Find a cycle of min cost Basic Problem

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ICGT 2002, E. Selensky3 Lexicographic ordering of nodes: A,B,C,D Example

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ICGT 2002, E. Selensky4 Motivation Core problem in vehicle routing and shop scheduling Edge weights to node weights: –Large for VRP, small for JSP Can we use graph transformations to make VRP look like JSP and vice versa?

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ICGT 2002, E. Selensky5 Vehicle Routing [2:25pm 2:40am] [9:00am 9:15am] [3:00pm 5:00am] [3:00pm 5:00am] [9:00am 5:00am] [4:00pm 5:00am] NP-hard! Go find vehicle tours with min travel

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ICGT 2002, E. Selensky6 Job Shop Scheduling J 1 : (M 1, t 11 ) (M 3, t 13 ) (M 2, t 12 ) J 2 : (M 3, t 23 ) (M 1, t 21 ) (M 2, t 22 ) J 3 : (M 2, t 32 ) (M 3, t 33 ) (M 1, t 31 ) 3 machines: M 1, M 2, M 3 3 jobs: J 1, J 2, J 3 Go find a schedule with min MakespanNP-complete TimeMakespan0 M1M1 M2M2 M3M3

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ICGT 2002, E. Selensky7 Hypothesis Graph Transformation VRP Solver JSP Solver Graph Transformation VRP JSP Is it important?

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ICGT 2002, E. Selensky8 Cost-Preserving Transformations Assumptions: –Graphs: complete (true for VRP, JSP subsumed), undirected (directed case subsumed); –A solution is a cycle on the graph (for Hamiltonian paths everything is similar); –Transformations should preserve cost and order of nodes in a cycle.

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ICGT 2002, E. Selensky9 Caveat This is not a comprehensive study of all possible transformations Rather, we propose some transformations and study them

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ICGT 2002, E. Selensky10 Types of Transformations Direct : Reduce Edge Weights, Increase Node Weights Inverse : Increase Edge Weights, Reduce Node Weights

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ICGT 2002, E. Selensky11 lexicographic order of nodes choose a node whose cheapest incident edge is a maximum choose a node whose cheapest incident edge is a minimum Order Dependent Transformations MaxMin: MinMin: Lex:

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ICGT 2002, E. Selensky12 Example Order Independent Transformation

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ICGT 2002, E. Selensky13 Inverse Transformation Reminder: Increase Edge Weights, Reduce Node Weights Order-independent G G inv ; G G dod G inv ; G G doi G inv ; Express as if odd and if even

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ICGT 2002, E. Selensky14 Weight transfer from nodes to edges: –change in proportion of weight of cycle C: –a similar measure for the whole graph: where W and W are graph weights before and after transformation Performance measures

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ICGT 2002, E. Selensky15 Relative edge/node weights ordering: –Sort edge/node weights in ascending order: e.g. {w 11, w 12, w 13 } for edges (1,1), (1,2) and (1,3); –Apply transformations and count how many pair-wise changes there are: e.g. {w 13, w 11, w 12 }, so we have 2 changes; Two measures: and Performance measures

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ICGT 2002, E. Selensky16 Experiments Purpose: –Assess performance of the transformations on complete undirected graphs Layout: –Randomly generate 100-instance sets of graphs of different sizes; –Apply and MaxMin,MinMin,Lex,DirOrderInd Inverse.

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ICGT 2002, E. Selensky17 Experiments

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ICGT 2002, E. Selensky18 Experiments

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ICGT 2002, E. Selensky19 Experiments

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ICGT 2002, E. Selensky20 Experiments

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ICGT 2002, E. Selensky21 Analysis of Results Weight Transfer: Inverse >> Order Independent >> Order Dependent Changes in Edge/Node Ordering: Inverse: constant w.r.t. graph size; Inverse>>MaxMin >> Order Independent, Lex >> MinMin

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ICGT 2002, E. Selensky22 Future Work Systematically apply the transformations to VRP/JSP instances and study their performance in practice.

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