Shop Scheduling Reformulation of Vehicle Routing

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Shop Scheduling Reformulation of Vehicle Routing

• Evgeny Selensky
• Dept of Computing Science
• Glasgow University

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Overview

• Why is reformulation important?
• Examples of reformulation
• Vehicle Routing Problem and its instances
• Job shop scheduling problem and its instances
• Tools to study reformulation issues
• Models and search procedures used in study
• Some preliminary results
• Outline of research (big picture)
• Future work

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Why Reformulate?

• Better problem-solving (using representation
  benefits)
• Solution process improvement (speed of solvers,
  memory use)
• Human interface improvements (better
  understanding of problems)
• Software reuse (generic types of constraints,
  heuristics, etc)

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Examples of reformulation

• N-queens
• n2 0/1 variables
• n variables with domains of size n
• B.Nadel (reformulation of n-queens, case study,
  IEEE Expert, June 1990)
• Potential search space of each representation
• Benefits of each?
• One might be more pruningful than another
• Better heuristics might be available
• May reduce symmetries
• Graph colouring
• n-variables, k-colours, e edges, go colour them
• k set variables, partition the set of n nodes
  such that
• Crossword puzzles construction
• Stable marriage (research frontier)
• Scheduling and vehicle routing

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Vehicle routing problem (delivery)

• N identical vehicles of capacity C
• M customers with demands Digt0, i 1..M
• 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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VRP instances

• Repair/install equipment
• Pick up money from banks
• Deliver prisoners from jail to court etc
• Street cleaning, garbage collection
• Automated guided vehicles in a factory
• Ambulance routing
• Drilling circuit boards
• Robot arm movements
• Computer networks

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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)
• ? i ? j ? tr_costij ? 0
• ? 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)
• very hard, no polynomial time
  approximation
• within 5/4 from optimal solution
• Multipurpose machines
• ? j Oij requires alternative resource

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JSSP instances

• engineering job shop
• making engines
• forge and machine pistons
• cast block
• treat surfaces
• drill, machine, heat treat, etc.
• high speed communications networks
• navigation
• less like construction and assembly

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Tools

• ILOG Solver 5.0
• general constraint programming problems
• offers enhanced search facilities
• ILOG Scheduler 5.0
• scheduling and resource allocation
• ILOG Dispatcher 3.0
• advanced local search algorithms for routing

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ILOG Scheduler models

• Real-world processes represented by resources and
  activities
• Activity
• starting time, processing time,
  completion time, demand
• performed by resource
• Resource
• capacity or state
• Capacitated resources
• unary (capacity ? 1)
• - useful when dealing with
  transition costs
• discrete (capacity ? 1)
• - allows one to take into account
  capacity constraints

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An OSSP models of a TSP and VRP

• Vehicles/salesman are machines on the shop floor
• the visits are operations (aka activities)
• the visits (activities) pass through the vehicles
  (machines)!
• Relativity of representation

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A JSSP model of a TSP

• N1 activities as visits to N cities 1
  additional visit
• earliest and latest time on each activity
• 1 unary resource as salesman
• Transition costs as distances between
  cities
• First city picked out arbitrarily, ? i ?
  2 .. N1 end(act1) ? start(acti)
• Last visit in tour to first city, ? i ? 1
  .. N start(actN1) ? end(acti)
• Search goal minimise(?tr_cost)

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An OSSP model of a VRP

• N2M activities as visits to N customers
  and base for M vehicles
• Pair of resources ltunary, discretegt as
  vehicle
• Transition costs as distances
• M additional activities (starts of tours)
• ? i ? 1 .. M acti is setup
• ? i ? 1 .. M acti requires
  vehiclei
• N actual visits
• ? i ? M1 .. NM acti
  requires vehicle0 ? vehicle1 ? ? vehicleM-1
• M additional activities (ends of tours)
• ? i ? NM1 .. N2M acti is
  teardown
• ? i ? NM1 .. N2M acti
  requires vehiclei
• Search goal minimise(? ? tr_cost)

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An OSSP model of a VRP

• Why bother?
• Maybe that scheduling heuristics work for vrp
• maybe scheduling propagation works for vrp
• edge finding, energetic reasoning,
• maybe as vrp becomes urban it looks like jssp
• urban, low transition costs?
• Can you see the symmetric argument?
• Ossp modeled as a vrp

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Search

• Search facilities
• constraint propagation, heuristics
• goal should be specified
• performed by Solver engine
• Limited Discrepancy Search
• William D Harvey, Matthew L Ginsberg,
  
• August 1995, IJCAI
• Depth Bounded Discrepancy Search
• Toby Walsh, August 1997, IJCAI

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LDS trace
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Results. JSSP representation of TSP instances

• nearest neighbour heuristic (schedule activity
  with min earliest start time)
• Intel Pentium III 933 MHz, 1Gb RAM

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Results. An OSSP representation of VRP instances

• Capacitated VRP instances (CVRP)
• 9 - 12 customers served by 2 vehicles
• 13-14 customers served by 3 vehicles
• Intel Pentium III 933 MHz, 1Gb RAM
• Resource selection heuristics
• minimal capacity
• maximal capacity

• VRP instances with time windows (CVRPTW)
• harder to solve
• approx. 2 times worse in distance than
• with local search
• The best known worst-case performance ratio for
  3-machine dense OSSP schedules 3/2

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Interesting, but so what?
An early step in our research
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A sketch of our planned research in the field

• VRP model, ILOG Dispatcher
• model ossp as a vrp
• use vrp heuristics
• see what happens as we vary the vrp properties of
  the problem
• setups
• few operations per job
• increase alternative resources

• OSSP model, ILOG Scheduler
• model vrp as ossp
• use ossp heuristics
• see what happens as we vary ossp properties
• setups vs durations
• many operations per job
• few alternative resources

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Conclusion. Future work

• TSP and VRP empirically represented as JSSP
• Search for solutions
• Complete and quasi-complete embedded
  in binary chop
• Different resource and activity
  selection criteria
• So no conclusion yet
• In the future
• A VRP representation of OSSP
• Other problems (SM, SMTI, Car Sequencing, )