Download - 1.3- Traveling Salesman Problems
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RAVELING ALESMAN ROBLEMS
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Traveling Salesman Problem
NP-complete Given a set of ncities and
All known techniques for obtaining an exact solution
require an exponentially increasing number of steps(computing resources) as the problems become larger
distances for each pair of cities,find a roundtrip of minimal totallen th visitin each cit exactlonce.
n!/2n possible tours (for n=60, 69x1078)
Symmetric TSP: Asymetric TSP: Euclidean TSP (triangle inequality)
,, =),(),( ijdjid
TSP is one of the most intensely studied problems in computational
mathematics, yet no effective solution method is known for thegeneral case.
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History of TSP
http://www.tsp.gatech.edu//history/index.html
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Milestones
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Germany Cities in TSP
Earliest known reference to TSP is published inGermany 1832
thun hat, um Auftraege zu erhalten und eines
gluecklichen Erfolgs in seinen Geschaeften gewiss zu-
In 1960, Schrijver solved a 45-city problem from the
ten omm s- oyageur In 1977, Groetschel solved a 120-city problem and
ublished in Mathematical Pro rammin Stud , 1980
The 120 cities include two cities in Switzerland and
one in Austria.
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15,112 Cities in Germany
Optimal result found in2001 has length 1,573,084units a rox. 66 000kilometer)
The computation wascarried out on a network
at Rice and Princeton. The total computer time
used was 22.6 CPU years,
scaled to a Compaq Alphaprocessor running at 500MHz.
13,509 city tour throughthe United States wassolved in 1998.
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24,978 Cities in Sweden
In May 2004, the travelingsalesman problem of visiting
,
was solved: a tour of length855,597 TSPLIB units
(approximately 72,500kilometers) was found and itwas proven that no shortertour exists. This is currently
the lar est solved TSP
instance,
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71,009 Cities in China
Best known result by Hung Dinh Nguyen, 4,566,563.
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1,904,711 Cities in the World
Best known result by Keld Helsgaun, 7,516,146,716
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Benchmark Repository
http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/
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Heuristic Approaches for TSP
Simulated annealing Exponential neighborhood
local search
Memetic algorithm Elastic net
Space filling heuristic LKH heuristic Tabu search
Self-organizing map Genetic algorithm Evolutionar strate
Branch and bound (40-60) Simulated electric field Petri net
Genetic programming Artificial immune
algorithm
ap ve r ng Cutting plane
n co ony sys em Particle swarm
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OB HOP CHEDULING ROBLEMS
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Problem Definition
n Jobs that must be processed on m Machines
with the following rules:
Each job must be processed in a certain order (precedent constraints)
Each machine can only process one job at a time ac o can on y e processe y one mac ne at a t me
Each job must be process by each machine exactly once
Objective: Determine schedule with minimum makespan, or theminimum time for all jobs to finish processing, for njobs on m
machines while adhering to the problems constraints
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Machine Sequence (Time)
Job 1: 2 (21) 1 (53) 5 (95) 4 (55) 3 (34) Job 2: 1 (21) 4 (52) 5 (16) 3 (26) 2 (71)Job 3: 4 (39) 5 (98) 2 (42) 3 (31) 1 (12) Job 4: 2 (77) 1 (55) 4 (79) 2 (66) 3 (77)Job 5: 1 (83) 4 (34) 3 (64) 2 (19) 5 (37) Job 6: 2 (54) 3 (43) 5 (79) 1 (92) 3 (62)Job 7: 4 (69) 5 (77) 2 (87) 3 (87) 1 (93) Job 8: 3 (38) 1 (60) 2 (41) 4 (24) 5 (83)Job 9: 4 (17) 2 (49) 5 (25) 1 (44) 3 (98) Job 10: 5 (77) 4 (79) 3 (43) 2 (75) 1 (96)
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Classifications of Schedules
Semi-Active Schedules
Semi-active schedules
operations are scheduled
at the earliest allowable.idle unnecessarily.
Job 1: 1(3); 2(5); 3(2)
Job 3: 2(4); 1(2); 3(1)
No operation can be started earlier without altering the operatingsequence of any machine
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Active Schedules
Active schedules are schedules whereno operation can be started earlier
Semi-activeSchedules
without delaying the total processing timeof any machine or breaking a precedentconstraint. An optimal schedule is anactive schedule.
Schedules
If possible to alter the operating sequence of the machine to produce aschedule with a smaller makespan and preserve the precedent constraints
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Non-Delay Schedules
Non-delay schedules are thesmallest class of schedules. These
Semi-activeSchedules
are active schedules in which nomachine is kept idle at any time, whenit could be processing an operation.
ActiveSchedules
Non-delaySchedules
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Shown together below, it is easy to see that non-delay schedules are not always optimal.
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Optimal Schedule
Schedules that are not semi-active schedule are not optimal. However, semi-active schedule is not necessary optimal. In
Optimal schedule lie in the space of active schedules.
Optimal schedule is not necessary a non-delay schedule.
However, it should be obvious that optimal schedule will most
likely be schedules where the amount of delay times for anygiven machine is kept to a minimum.
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Parameterized Active Schedules
Allows machines to be idle for aSemi-activeSchedules
.
Non-delay schedules are
parameterized active schedules with a
c ve
Schedules
Parameterized
Active Schedules
parameter o 0.
May or may not contain the optimalschedule.
Non-delay
Schedules
Many times optimal solutions will lie just out side the non-delay
set, so non-delay schedule building will not work, but searchingthe whole space of active schedules is very inefficient
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LA03 10 x 5 O timal Solution Solution that is just barely outside of non-delay space and intoparameterized active schedule space.
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More Combinatorial Optimization
0-1 Knapsack Problem
Prisoner Dilemma Problem