Job Shop Reformulation of Vehicle Routing

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Job Shop Reformulation of Vehicle Routing

• Evgeny Selensky
• University of Glasgow
• evgeny_at_dcs.gla.ac.uk
• http//www.dcs.gla.ac.uk/evgeny

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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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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

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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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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

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Reformulation

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

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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)

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Previous Studies. Outcome

• Local Search in general is better for routing
  problems
• Systematic search in general is better for
  scheduling problems

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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

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This work is the first step towards understanding
why this happens
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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
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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 start

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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

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Experiments. Tools (i)

• Scheduler 5.1
• Scheduling Technology, core - global
• constraint propagation
• slack-based heuristics
• edge finder
• timetable constraints

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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

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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

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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

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Scheduler Results
Scheduler solutions to Solomons benchmarks.
Average differences, of the non-transformed
costs. Black - lex, grey - maxmin, white - minmin
ordering
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Dispatcher Results
Dispatcher solutions to Solomons benchmarks.
Average differences, of the non-transformed
cost. Black - lex, grey - maxmin, white - minmin
ordering
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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

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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

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Acknowledgements

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

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Thanks a lot! Any questions?