# Graph Transformations for Vehicle Routing and Job Shop Scheduling Problems J.C.Beck, P.Prosser, E.Selensky

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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

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

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

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?

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

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

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

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.

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

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

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:

ICGT 2002, E. Selensky12 Example Order Independent Transformation

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

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

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

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.

ICGT 2002, E. Selensky17 Experiments

ICGT 2002, E. Selensky18 Experiments

ICGT 2002, E. Selensky19 Experiments

ICGT 2002, E. Selensky20 Experiments

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

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

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