Traveling Salesman Problem IEOR 4405 Production Scheduling Professor 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 x T = [x 1, x 2, …,x n ] which maximizes k objective functions, with n variables and m constraints Opt Z = CX s.t. AX <= b Z = (z 1, z 2,…,z n ) 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 >=~Z 0 s.t. AX<=~b Where Z 0 =(z 1 0,z 2 0,…z n 0 ) 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