Job Shop Reformulation of Vehicle Routing Evgeny Selensky University of Glasgow [email protected]...
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Transcript of Job Shop Reformulation of Vehicle Routing Evgeny Selensky University of Glasgow [email protected]...
Job Shop Reformulation of Vehicle Routing
Evgeny Selensky
University of Glasgow
[email protected]://www.dcs.gla.ac.uk/~evgeny
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
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
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
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
Reformulation
• Machine Vehicle
• Operation Visit
• Operation duration Service time
• Transition time Distance
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)
Previous Studies. Outcome
• Local Search in general is better for routing problems
• Systematic search in general is better for scheduling problems
Why so huge a difference?Routing Scheduling
• many alternative resources
• few (or no) alternative resources
• large setups, small durations
• small (or no) setups, large durations
• few (or no) temporal relationships among activities
• many temporal relationships among activities (precedence constraints)
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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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
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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 P. 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 start
2/'iii eses
2/'iii lsls
Experiments. Test bed• Based on M.Solomon’s 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
Experiments. Tools (i)
• Scheduler 5.1• Scheduling Technology, core - global
constraint propagation: – slack-based heuristics– edge finder – timetable constraints
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
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
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
Scheduler Results
Scheduler solutions to Solomon’s benchmarks. Average differences, % of the non-transformed costs.Black - lex, grey - maxmin, white - minmin ordering
Dispatcher Results
Dispatcher solutions to Solomon’s benchmarks. Average differences, % of the non-transformed cost.Black - lex, grey - maxmin, white - minmin ordering
Analysis of Results
• For clustered problems Dispatcher’s performance degrades, Scheduler’s improves as expected
• Hard to draw any conclusions on the rest problems. We don’t have full control over instance structure. The test bed proved a bit peculiar for these experiments
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