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Job Shop Reformulation of Vehicle Routing

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Side constraints may be present (e.g., time windows, precedence constraints) ... many temporal relationships among activities (precedence constraints) ... – PowerPoint PPT presentation

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Title: Job Shop Reformulation of Vehicle Routing


1
Job Shop Reformulation of Vehicle Routing
  • Evgeny Selensky
  • University of Glasgow
  • evgeny_at_dcs.gla.ac.uk
  • http//www.dcs.gla.ac.uk/evgeny

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
Joint work with
  • Patrick Prosser (University of Glasgow),
  • John Christopher Beck (still ILOG, France but
    soon Cork Constraint Computation Center, Ireland)
  • submitted a paper to SARA 2002

6
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

7
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

8
Similarities
  • Execution of tasks
  • Tasks use resource(s) and have durations
  • Resources constrained by capacity
  • Sets of alternative resources may exist
  • Setups, temporal constraints on tasks present
  • Solution is an assignment of tasks to resources,
    start times to tasks
  • Similar minimisation criteria may be specified

9
Reformulation
  • Machine Vehicle
  • Operation Visit
  • Operation duration Service time
  • Transition time Distance

10
Previous Studies
  • Scheduling and local search
  • Davenport Beck 1999 - alternative resources (up
    to 8 alternatives)
  • Focacci et al, 2000 - alternative resources (up
    to 3 alternatives) and setups
  • Selensky 2000 - extreme cases of performance of
    the routing and scheduling techniques (25
    alternatives, large setups)

11
Previous Studies. Outcome
  • Local Search in general is better for routing
    problems
  • Systematic search in general is better for
    scheduling problems

12
Why so huge a difference?
Scheduling
Routing
  • many alternative resources
  • few (or no) alternative resources
  • small (or no) setups, large durations
  • large setups, small durations
  • many temporal relationships among activities
    (precedence constraints)
  • few (or no) temporal relationships among
    activities

13
This work is the first step towards understanding
why this happens
14
TSP graph transformations
  • Purpose build part of transition times into
    operation durations to improve performance of
    temporal reasoning
  • Based on preservation of cost

15
Example. Order independent transformation
16
It preserves cost! Proof.
1. Assume
17
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
18
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
19
Example. Order dependent transformation
Lexicographic ordering of nodes A,B,C,D
20
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 start

21
Experiments. Test bed
  • Based on M.Solomons suite of 56 VRPTW
    benchmarks
  • classes C1, R1, RC1 small capacities, short TWs
  • classes C2, R2, RC2 large capacities, wide TWs
  • C1 (9 instances), C2 (8 instances) clustered
    distribution of customers
  • R1 (12 instances), R2 (11 instances) random
    distribution of customers
  • RC1 (8 instances), RC2 (8 instances)
    random-clustered distribution of customers
  • within a class, customer coordinates and demands
    are identical

22
Experiments. Tools (i)
  • Scheduler 5.1
  • Scheduling Technology, core - global
  • constraint propagation
  • slack-based heuristics
  • edge finder
  • timetable constraints

23
Experiments. Tools (ii)
  • Dispatcher 3.1
  • Routing Technology, core - local search
  • different first solution generation heuristics
  • plain local search, guided local search, tabu
    search
  • path constraints

24
Experiments. Layout (i)
  • Windows NT, Intel Pentium III 933 MHz, 1Gb RAM
  • Scheduler 5.1
  • Search for solutions
  • Discrepancy Bounded Depth First Search
  • slack-based heuristics
  • Max cpu time of 600s
  • Run each instance 4 times using
  • No transformation
  • Lex ordering
  • MaxMin ordering
  • MinMin ordering

25
Experiments. Layout (ii)
  • Windows NT, Intel Pentium III 933 MHz, 1Gb RAM
  • Dispatcher 3.1
  • Search for solutions
  • First solution generation using savings heuristic
  • Guided Local Search
  • Max cpu time of 600s
  • Run each instance 4 times using
  • No transformation
  • Lex ordering
  • MaxMin ordering
  • MinMin ordering

26
Scheduler Results
Scheduler solutions to Solomons benchmarks.
Average differences, of the non-transformed
costs. Black - lex, grey - maxmin, white - minmin
ordering
27
Dispatcher Results
Dispatcher solutions to Solomons benchmarks.
Average differences, of the non-transformed
cost. Black - lex, grey - maxmin, white - minmin
ordering
28
Analysis of Results
  • For clustered problems Dispatchers performance
    degrades, Schedulers improves as expected
  • Hard to draw any conclusions on the rest
    problems. We dont have full control over
    instance structure. The test bed proved a bit
    peculiar for these experiments

29
Future Work
  • Create our own problem generator with the purpose
    of being able to move from VRP to JSSP smoothly,
    varying
  • width of time windows
  • customer locations
  • capacities
  • vehicle specialisation
  • precedence constraints on visits
  • rejection/acceptance of visits on a vehicle

30
Acknowledgements
  • Thanks to Vincent Furnon ( ) and Barbara
    Smith (Huddersfield)

31
Thanks a lot! Any questions?
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