Traveling Salesman Problem

IEOR 4405 Production SchedulingProfessor Stein

Sally Kim James Tsai

April 30, 2009

TSP Defined

Given a list of cities and their pairwise distances, find the shortest tour that visits each city exactly once

Well-known NP-hard combinatorial optimization problem

Used to model planning, logistics, and even genome sequencing

Project Objectives

Perform a literature search of the TSP

Find interesting, real-life applications

Discover algorithms uncovering optimal solutions

Fuzzy Multi-objective LP Approach

“Fuzzy Multi-objective Linear Programming Approach for Traveling Salesman Problem” (Rehmat, Amna; 2007)

Ideal solution would solve every TSP to optimality

Proven not only to be difficult, but also unrealistic

Impossible to have all constraints and resources in exact form – always vagueness

“Fuzzy Logic”: vague or imprecise data off which decisions are made

Multi-objective LP

Takes a general linear multiple criteria decision making model and represents it as follows:

Find a vector xT = [x1, x2, … ,xn] which maximizes k objective functions, with n variables and m constraints

Opt Z = CX

s.t. AX <= b

Z = (z1, z2,…,zn) is the vector of objectives, C is a K x N matrix of constants and X is an Nx1 vector of decision variables, A is an M x N matrix of constants and b is a Mx1 vector of constants

Fuzzy Multi-objective LP Approach

Modify the multi-objective LP formulation to:

Max Cx >=~Z0

s.t. AX<=~b

Where Z0=(z10,z2

0,…zn0) are aspiration levels and

>=~ are fuzzy inequalities

Consider a case of TSP with 3 objectives: minimize cost, time, and overall distance

Ant Colony Optimization

“An interactive simulation and analysis software for solving TSP using Ant Colony Optimization algorithms” (Ugur, Aybars; 2008)

ACO is a population based probabilistic technique for solving NP-hard combinatorial problems

Ant Colony Optimization

Simulation and analysis software are developed for solving TSP using ACO algorithm

Web-based tool employing virtual ants and interactive graphics to produce near-optimal solutions to the TSP

Artificial ants build solutions and exchange them with others via a communication scheme

Ant Colony Optimization

ConstructSolutions: each ant starts at a particular state, then traverses the states one by one

ApplyLocalSearch: before updating the ant’s trail, a local search can be applied on each solution constructed

UpdateTrails: after the solutions are constructed and calculated, pheromone levels increase and decrease on paths according to favorability

Ant Colony Optimization

Simulator TSPAntSim provides analysis of algorithms textually and graphically

Best tour-so-far represents the best found thus far

Tour best represents the best any tour length after

Standard deviation illustrates the evolution of the standard deviation of populations’ tour length

Conclusions

While finding the exact solution is often desired in problems of optimality, this is sometimes not realistic

Relaxation and modification are some ways to approach a NP-hard problem that is otherwise difficult to solve