Traveling%20Salesman%20Problem

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Traveling Salesman Problem

• IEOR 4405 Production Scheduling
• Professor Stein
• Sally Kim
• James Tsai
• April 30, 2009

2
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

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Project Objectives

• Perform a literature search of the TSP
• Find interesting, real-life applications
• Discover algorithms uncovering optimal solutions

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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

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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 lt 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

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Fuzzy Multi-objective LP Approach

• Modify the multi-objective LP formulation to
• Max Cx gtZ0
• s.t. AXltb
• Where Z0(z10,z20,zn0) are aspiration levels
  and gt are fuzzy inequalities
• Consider a case of TSP with 3 objectives
  minimize cost, time, and overall distance

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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

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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

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Ant Colony Optimization

• ConstructSolutions each ant starts at a
  particular state, then traverses the states one
  by one
• ApplyLocalSearch before updating the ants
  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

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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

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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