Job Shop Reformulation of Vehicle Routing

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Job Shop Reformulation of Vehicle Routing

• Evgeny Selensky

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Details of the Talk

• PRAS project
• Problems addressed
• Two-level Reformulation
• TSP graph transformations
• Experiments and results

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PRAS project

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

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Why bother?

• Try to understand problem structure
• Improve performance of solution techniques

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Vehicle Routing Problem

• N identical vehicles of capacity C
• M customers with demands Digt0
• 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

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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 lt Oij1 (chain-type precedence
  constraints)
• ? j Oij requires specific resource
• No preemption
• Minimise makespan LatestEnd - EasliestStart
• Open shop relaxation
• ? j start(Oij) lt start(Oij1) ? start(Oij) gt
  start(Oij1)
• Multipurpose machines
• ? j Oij requires alternative resource

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Reformulation

• Machine Vehicle
• Operation Visit
• Operation duration Service time
• Transition time Distance

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Tool

• Scheduler 5.1
• Scheduling Technology
• slack-based heuristics
• edge finder
• timetable constraints

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TSP graph transformations

• Purpose build part of transition times into
  operation durations to improve performance of
  temporal reasoning
• Based on preservation of cost

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Example. Order independent transformation
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It preserves cost! Proof.
1. Assume
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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
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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
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Example. Order dependent transformation
Lexicographic ordering of nodes A,B,C,D
Due to Patrick Prosser
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A Few More Remarks

• Both transformations change time bounds on
  operations
• We dont know yet how order independent
  transformation changes time bounds
• Order dependent transformation makes a symmetric
  change
• earliest start
• latest end

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Experiments. Data generation

• Based on M.Solomons 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

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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

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Results I
Ranges, means and medians of
Table 1. Pure VRPTWs

• Characteristic C1 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..27
  4 -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

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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..27
  4 -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

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Results III
Table 3. Influence of time windows

• Characteristic C1 C2 R1 R2 RC1 RC2
• Range, Lex -300..117 -184..110 -376..267 -139..26
  5 -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..1
  80 -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

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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

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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)

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Acknowledgements

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