Traveling Salesman Problem IEOR 4405 Production Scheduling Professor

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, … , xn] which maximizes k objective functions, with n variables and m constraints Opt Z = CX s. t. AX <= b Z = (z 1, z 2, …, zn) is the vector of objectives, C is a K x N matrix of constants and X is an Nx 1 vector of decision variables, A is an M x N matrix of constants and b is a Mx 1 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 10, z 20, …zn 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 developed for solving TSP using ACO algorithm Web-based tool employing virtual ants and interactive graphics to produce nearoptimal solutions to the TSP Artificial ants build solutions and exchange them with others via a communication scheme

Ant Colony Optimization Construct. Solutions: each ant starts at a particular state, then traverses the states one by one Apply. Local. Search: before updating the ant’s trail, a local search can be applied on each solution constructed Update. Trails: after the solutions are constructed and calculated, pheromone levels increase and decrease on paths according to favorability

Ant Colony Optimization Simulator TSPAnt. Sim 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