Post on 24-Feb-2016
description
Traveling Salesman Problem (TSP)Chris SetoAndrea Smith
Problem description Given a list of a cities and distance
between each pair of cities, what is the shortest possible route that visits each?
NP-hard problem Problem usually modeled as an
undirected graph Produces a Hamiltonian Cycle
History Studies started in the 1800s by Sir William
Hamilton and Thomas Kirkman of related problems
Icosian game invented in 1857 TSP first studied in 1930s by Karl Menger,
Hassler Whitney, and Merrill Flood Solutions appeared in papers in mid-1950s Determined to be NP-hard in 1972 by
Richard M. Karp
Application: Pick And Place Printed circuit board (PCB) with locations
where chips must be placed by robot The faster the robot can place all
components, the faster the PCB will be assembled
The faster the PCB can be assembled, the more PCBs can be made in the same amount of time
Application: Logistics Warehouse with many parts in various
locations Order is received for several parts which
must be picked by robot or employee What path should the picker follow to
ensure that they fill the order in smallest amount of time?
Any number of TSP solutions can be applied and compared in parallel
Application: UPS ORION Route optimization used by UPS Implementation started in 2008 Saves ~35 million miles per year Increased projected annual savings
NP Hard Solution Methods Devise an algorithm for an exact
solution, even though it may only work efficiently for a small problem
Devise “Suboptimal” heuristic algorithms to yield good, but inexact solutions.
Find special cases for the problem for which better or exact heuristics are developed.
Possible ApproachesBrute-force Method
Best for small number of nodesGreedy Algorithm
Simplest algorithm for larger number of nodes
Genetic Algorithm Generates “close to optimal” solutions
Approach: Brute Force• Best for small number of nodes• Number of Hamiltonian Circuits = (n-1)!• Guaranteed an Optimal Solution• Complexity: ((n-1)!)• Tries all possible permutations and
compares costNode Count Approximate Completion Time
20 2 Minutes25 20 Years30 284 Million Years
Brute Force Demo
Approach: Greedy• Prim’s Algorithm, Kruskal’s Algorithm• Not guaranteed the optimal solution• “Close enough” solution• Prim’s Complexity: O()• Kruskal’s Complexity: O()
Approach: Genetic Algorithm New approach which uses natural
selection to create close to optimal solutions
Hamiltonian cycles are continually “bred” with mutations
Crossover occurs between solutions Relatively quickly produces a solution
which is probably close to optimal
Conclusion Studies originally began 1800s, again in
1950s Optimal solution found in ((n-1)!) time Close enough solution found in O() Used by…
UPS in ORION system Pick and place Many other graph representable systems
Questions?
References http://
en.wikipedia.org/wiki/Travelling_salesman_problem
https://xkcd.com/399/ (Comic) http://www.forbes.com/sites/alexkonrad/
2013/11/01/meet-orion-software-that-will-save-ups-millions-by-improving-drivers-routes/
http://www.theprojectspot.com/tutorial-post/applying-a-genetic-algorithm-to-the-travelling-salesman-problem/5