1 / 22

Graph Transformations for Vehicle Routing and Job Shop Scheduling Problems

This paper explores the use of graph transformations to solve vehicle routing and job shop scheduling problems. The authors propose various transformations and assess their performance through experiments on different graph sizes.

dustym
Download Presentation

Graph Transformations for Vehicle Routing and Job Shop Scheduling Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 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

  2. w1 w1,n w12 w2,n wn w2 w2,n-1 w1,n-1 wn-1,n wn-1 wi Basic Problem Find a cycle of min cost ICGT 2002, E. Selensky

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

  4. 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. Selensky

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

  6. 3 machines: M1, M2, M3 3 jobs: J1, J2, J3 M1 M2 J1: (M1,Dt11)  (M3,Dt13)  (M2,Dt12) J2: (M3,Dt23)  (M1,Dt21)  (M2,Dt22) J3: (M2,Dt32)  (M3,Dt33)  (M1,Dt31) M3 0 Makespan Time Job Shop Scheduling Go find a schedule with min Makespan NP-complete ICGT 2002, E. Selensky

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

  8. 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. Selensky

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

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

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

  12. Order Independent Transformation Example ICGT 2002, E. Selensky

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

  14. Performance measures • 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 ICGT 2002, E. Selensky

  15. Performance measures • Relative edge/node weights ordering: • Sort edge/node weights in ascending order: • e.g. {w11, w12, w13} 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 ICGT 2002, E. Selensky

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

  17. Experiments ICGT 2002, E. Selensky

  18. Experiments ICGT 2002, E. Selensky

  19. Experiments ICGT 2002, E. Selensky

  20. Experiments ICGT 2002, E. Selensky

  21. 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. Selensky

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

More Related