1.3- Traveling Salesman Problems

8/2/2019 1.3- Traveling Salesman Problems

1/23

RAVELING ALESMAN ROBLEMS


2/23

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.


3/23

History of TSP

http://www.tsp.gatech.edu//history/index.html


4/23

Milestones


5/23

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.


6/23

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.


7/23

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,


8/23

71,009 Cities in China

Best known result by Hung Dinh Nguyen, 4,566,563.


9/23

1,904,711 Cities in the World

Best known result by Keld Helsgaun, 7,516,146,716


10/23


11/23

Benchmark Repository

http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/


12/23

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


13/23

OB HOP CHEDULING ROBLEMS


14/23

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


15/23

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)


16/23

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


17/23

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


18/23

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


19/23

Shown together below, it is easy to see that non-delay schedules are not always optimal.


20/23

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.


21/23

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


22/23

LA03 10 x 5 O timal Solution Solution that is just barely outside of non-delay space and intoparameterized active schedule space.


23/23

More Combinatorial Optimization

0-1 Knapsack Problem

Prisoner Dilemma Problem

1.3- Traveling Salesman Problems

Documents

Transcript of 1.3- Traveling Salesman Problems