Graph Transformations for Vehicle Routing and Job Shop Scheduling Problems

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