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Job Shop Reformulation of Vehicle Routing

Job Shop Reformulation of Vehicle Routing. Evgeny Selensky. Details of the Talk. PRAS project Problems addressed Two-level Reformulation TSP graph transformations Experiments and results. PRAS project. Problem Reformulation and Search Principal Investigator: Patrick Prosser

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Job Shop Reformulation of Vehicle Routing

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  1. Job Shop Reformulation of Vehicle Routing Evgeny Selensky

  2. Details of the Talk • PRAS project • Problems addressed • Two-level Reformulation • TSP graph transformations • Experiments and results

  3. PRAS project • Problem Reformulation and Search • Principal Investigator: Patrick Prosser • Web site: www.dcs.gla.ac.uk/pras • Industrial collaborator: , France

  4. Why bother? • Try to understand problem structure • Improve performance of solution techniques

  5. Vehicle Routing Problem • N identical vehicles of capacity C • M customers with demands Di>0 • Each vehicle serves subset of customers • Side constraints may be present (e.g., time windows, precedence constraints) • Find tours for subset of vehicles such that: • all customers served, each once • one tour per vehicle • total distance minimal

  6. time Latest end time Earliest start time Job Shop Scheduling Problem • M machines, i = 1..M, M  2 • N jobs each of S operations, j = 1..S, of duration dij •  j : Oij < Oij+1 (chain-type precedence constraints) •  j : Oijrequires specific resource • No preemption • Minimise makespan = LatestEnd - EasliestStart • Open shop relaxation •  j : start(Oij) < start(Oij+1)  start(Oij) > start(Oij+1) • Multipurpose machines  j : Oijrequires alternative resource

  7. Reformulation • Machine Vehicle • Operation Visit • Operation duration Service time • Transition time Distance

  8. Tool • Scheduler 5.1 • Scheduling Technology: • slack-based heuristics • edge finder • timetable constraints

  9. TSP graph transformations • Purpose: build part of transition times into operation durations to improve performance of temporal reasoning • Based on preservation of cost

  10. Example. Order independent transformation

  11. It preserves cost! Proof. 1. Assume

  12. 2. Now let Possible 4-node cycles: 1-2-3-4-1, 1-2-4-3-1, 1-3-2-4-1, 1-3-4-2-1, 1-4-2-3-1, 1-4-3-2-1. Consider 1-2-3-4-1:

  13. 3. Finally, We can always split any cycle into a set of pairs of 3-node cycles with a common edge and starting node as before Therefore for any n

  14. Example. Order dependent transformation* Lexicographic ordering of nodes: A,B,C,D * Due to Patrick Prosser

  15. A Few More Remarks • Both transformations change time bounds on operations • We don’t know yet how order independent transformation changes time bounds • Order dependent transformation makes a symmetric change: • earliest start • latest end

  16. Experiments. Data generation • Based on M.Solomon’s suite of 56 VRPTW benchmarks • pure problems: • classes C1, R1, RC1 – small capacities, short TWs • classes C2, R2, RC2 – large capacities, wide TWs • changed capacity: • classes C1’, R1’, RC1’ – reduced capacities • classes C2’, R2’, RC2’ – increased capacities • changed TWs: • classes C1’’, R1’’, RC1’’ – TW width reduced by 5% • classes C2’’, R2’’, RC2’’ – TW width increased by a factor of 2 • changed capacity and TWs: • classes C1’’’ – RC2’’’ analogously

  17. Experiments. Tools and Layout • Windows NT, Intel Pentium III 933 MHz, 1Gb RAM • Scheduler 5.1 • Search for first solutions: • LDS • slack-based heuristics • Time Limit 600s • Run each instance 4 times: • No transformation • Lex ordering • MaxMin ordering • MinMin ordering

  18. Results I Ranges, means and mediansof CharacteristicC1 C2 R1 R2 RC1 RC2 Range, Lex -13..187 -110..39 -313..246 -114..148 -354..235 -194..163 Range, MaxMin -46..184 -74..38 -361..337 -258..112 -135..177 -233..184 Range, MinMin -13..124 -227..37 -323..166 -137..274 -239..247 -144..205 Mean, Lex 25.8 -7.9 -19.5 13 -7 -9.5 Mean, MaxMin 19.6 3.4 -5.7 -36.7 61.25 34.9 Mean, MinMin 21 -23.9 -13.75 61 2.375 3.8 Median, Lex 0 2 -2 14 13 -18 Median, MaxMin 0 6.5 20.5 45 88 61.5 Median, MinMin 0 1 -22 62 11.5 -19.5 Table 1. Pure VRPTWs

  19. Results II Table 2. Influence of capacity Characteristic C1’ C2’ R1’ R2’ RC1’ RC2’ Range, Lex -1..187 -110..39 -313..246 -114..148 -354..235 -194..163 Range, MaxMin -66..184 -74..38 -361..337 -258..112 -135..177 -233..184 Range, MinMin -13..124 -227..37 -323..166 -137..274 -239..247 -144..205 Mean, Lex 35.3 -7.9 -19.5 13 -7 -9.5 Mean, MaxMin 23.8 3.4 -5.7 -36.7 61.25 34.9 Mean, MinMin 24.6 -23.9 -13.75 61 2.375 3.8 Median, Lex 1 2 -2 14 13 -18 Median, MaxMin 6 6.5 20.5 45 88 61.5 Median, MinMin 3 1 -22 62 11.5 -19.5

  20. Results III Table 3. Influence of time windows Characteristic C1’’ C2’’ R1’’ R2’’ RC1’’ RC2’’ Range, Lex -300..117 -184..110 -376..267 -139..265 -216..102 -370..474 Range, MaxMin -305..27 -8..418 -513..332 -237..98 -243..196 -461..263 Range, MinMin -284..124 -258..194 -341..67 -196..180 -347..136 -314..342 Mean, Lex -16.7 -7.9 -4.6 41.2 -53.9 70.1 Mean, MaxMin -23 82.8 -77 -21 -69.9 -41.8 Mean, MinMin -13.7 -16.5 -75.6 25.8 -90.1 63 Median, Lex 2 2 10.5 53 -56 87 Median, MaxMin 12 16 -129.5 42 -127 -48 Median, MinMin 3 1 -18.5 48 -24 118

  21. Results IV Table 4. Influence of capacity and time windows Characteristic C1’’’ C2’’’ R1’’’ R2’’’ RC1’’’ RC2’’’ Range, Lex -300..19 -164..118 -376..267 -139..265 -216..102 -370..474 Range, MaxMin -305..26 -8..463 -513..332 -237..98 -243..196 -461..263 Range, MinMin -284..44 -71..224 -341..67 -196..180 -347..136 -314..342 Mean, Lex -36 8.3 -4.6 41.2 -53.9 70.1 Mean, MaxMin -34.6 87.1 -77 -21 -69.9 -41.8 Mean, MinMin -35 19.4 -75.6 25.8 -90.1 63 Median, Lex -1 2 10.5 53 -56 87 Median, MaxMin -1 16 -129.5 42 -127 -48 Median, MinMin 0 1 -18.5 48 -24 118

  22. Analysis of Results • Influence of changing capacity alone dominated by influence of changing TW width • Transformation tends to improve solution quality with small TWs. • Lex: improves on C1, RC1, degrades on R1 • MaxMin: improves on C1, R1, RC1 • MinMin: improves on C1, R1, RC1 • Conversely, with large TWs solution quality degrades: • Lex: degrades on R2, RC2, the same on C2 • MaxMin: degrades on C2, R2 (still negative but worse), improves on RC2 (negative) • MinMin: degrades on C2 (still negative but worse), RC2, improves on R2 (positive but better)

  23. Acknowledgements • Thanks to Chris Beck ( ) for his suggestions on the order independent transformation

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