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• Web site: www.dcs.gla.ac.uk/pras• Industrial collaborator: , France

Why bother?

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

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

Job Shop Scheduling Problem

time

Earliest start time

Latest end time

• 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 : Oij requires 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 : Oij requires alternative resource

Reformulation

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

Tool

• Scheduler 5.1• Scheduling Technology:

– slack-based heuristics– edge finder – timetable constraints

TSP graph transformations

• Purpose: build part of transition times into operation durations to improve performance of temporal reasoning

• Based on preservation of cost

Example. Order independent transformation

22' ji

ijij ww

}{min2'kmEkkkk wwww

k

3'

23'

2'

12'

1'' wwwwwC

222222

311333

322322

211211

wwwwww

Cwwwwww 133232121

It preserves cost! Proof. 3n1. Assume

4n

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:

3131 2)1431()1321()14321( wwwCCC

1'1

0'0

3'

13'

1'

1'

0''

313110

2

2

CC

CC

wwwCCC

wwwCCC

CC '

2. Now let

4n

We can always split any cycle into a set of pairs of 3-node cycles with a common edge and starting node as before

CC '

Therefore for any n

3. Finally,

Example. Order dependent transformation*

Lexicographic ordering of nodes: A,B,C,D

* Due to Patrick Prosser

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

2/'iii eses

2/'iii lele

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

CapCap 95.0' CapCap 2'

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

Results I

Characteristic C1 C2 R1 R2 RC1 RC2Range, Lex -13..187 -110..39 -313..246 -114..148 -354..235 -194..163Range, MaxMin-46..184 -74..38 -361..337 -258..112 -135..177 -233..184Range, MinMin -13..124 -227..37 -323..166 -137..274 -239..247 -144..205Mean, Lex 25.8 -7.9 -19.5 13 -7 -9.5Mean, MaxMin 19.6 3.4 -5.7 -36.7 61.25 34.9Mean, MinMin 21 -23.9 -13.75 61 2.375 3.8Median, Lex 0 2 -2 14 13 -18Median, MaxMin 0 6.5 20.5 45 88 61.5Median, MinMin 0 1 -22 62 11.5 -19.5

Table 1. Pure VRPTWs

Ranges, means and medians of .' CC

Results II

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

Table 2. Influence of capacity

Results III

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

Table 3. Influence of time windows

Results IV

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

Table 4. Influence of capacity and time windows

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)

Acknowledgements

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