Job Shop Reformulation of Vehicle Routing

Evgeny Selensky

University of Glasgow

evgeny@dcs.gla.ac.ukhttp://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

Why bother?

• Try to understand problem structure

• Improve performance of solution techniques

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)

This work is the first step towards understanding why this happens

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*

* Due to J.C.Beck

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It preserves cost! Proof. 3n1. Assume

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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

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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

Acknowledgements

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

Thanks a lot! Any questions?