Shop Scheduling Reformulation of Vehicle Routing

Evgeny SelenskyDept of Computing ScienceGlasgow University

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

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)

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

• N identical vehicles of capacity C• M customers with demands Di>0, 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

Vehicle routing problem (delivery)

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

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 < Oij+1 (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) < start(Oij+1) start(Oij) > start(Oij+1) very hard, no polynomial time approximation within 5/4 from optimal solution

Multipurpose machines

j : Oij requires alternative resource

time

Earliest start time

Latest end time

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

Tools

• ILOG Solver 5.0 general constraint programming problems offers enhanced search facilities

• ILOG Scheduler 5.0 scheduling and resource allocation

• ILOG Dispatcher 3.0advanced local search algorithms for routing

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

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

A JSSP model of a TSP

• N+1 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 .. N+1] end(act1) start(acti)• Last visit in tour to first city, i [1 .. N] start(actN+1) end(acti)• Search goal minimise(tr_cost)

An OSSP model of a VRP

• N+2*M activities as visits to N customers and base for M vehicles• Pair of resources <unary, discrete> 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 [M+1 .. N+M] acti requires vehicle0 vehicle1 … vehicleM-1

• M additional activities (ends of tours) i [N+M+1 .. N+2*M] acti is teardown

i [N+M+1 .. N+2*M] acti requires vehiclei

• Search goal minimise( tr_cost)

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

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

LDS trace

Discrepancy = 0

Discrepancy = 2

Discrepancy = 1

Discrepancy = 3

Results. JSSP representation of TSP instances

nearest neighbour heuristic (schedule activity with min earliest start time)

Intel Pentium III 933 MHz, 1Gb RAM

Number of cities

CPU Time, s Distance OptimalDistance

Number ofBranching Points

Number ofBacktracks

14 1.046 3323 3323 1592 93516 28.703 6859 6859 127150 10047017 2.969 2085 2085 10009 492321 2.766 2707 2707 5683 323922 6195.86 7013 7013 19657050 1529959424 1031.734 1272 1272 2311096 164031426 78.438 937 937 167029 71832

Results. An OSSP representation of VRP instances

• 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

1

10

100

1000

10000

9 10 11 12 13 14

min capacity

max capacityTime, s

Number of Customers, CVRP

• 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

Interesting, but so what?

… An early step in our research

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

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, …)