A Stochastic Traveling Salesman Problem

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A Stochastic Traveling Salesman Problem

• ESI 6912
• Adam Czernikowski
• Chase Rainwater
• 11/23/2005

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Outline

• Problem Overview
• Formulation and Results
• Solving with Heuristics

ESI 6912 Czernikowski Rainwater 11/23/2005
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Objective

• To formulate a working model of a stochastic
  variant of the Traveling Salesman Problem that
  produces a solution in practical running time

ESI 6912 Czernikowski Rainwater 11/23/2005
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Conventional TSP

• Visit n cities once and only once
• Minimize total cost (e.g. distance traveled)

ESI 6912 Czernikowski Rainwater 11/23/2005
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Conventional TSP

• Solved using integer programming
• xij 1 if we travel from city i to city j
• 0 if not

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Problem Motivation

• TSP normally studied in deterministic sense
• Interest in exploring characteristics in
  stochastic setting
• Consider random factors impacting salesmans
  decision once at destination

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Stochastic Problem Description

• Two Decisions
• What is the tour?
• How much does salesman work when he visits a
  city?
• Random Influence
• Weather directly affects demand for salesmans
  products
• Salesmans items
• Selling two products (hot dogs and umbrellas)

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Two City Types

• Cities of each type have different demands
  depending on the weather
• Want to be in blue city when blue city demand is
  high and red low

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Weather Forecasts

• Three possible daily scenarios sunny, rainy, and
  stormy
• For n days, we have 3n possible scenarios
• In real world, all scenarios not equally likely
• Randomly generate a number of scenarios based on
  the size of n

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Scenario Generation

• Markov Chain used to account for behavior of
  weather

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Decision Variables

• xijk 1 if arc from city i to city j on day k
• 0 otherwise
• yiks amount of time worked in city i on day k
  in scenario s

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Objective Function

• Maximize profit

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Small Example

• Five cities
• Demand values

Blue City
Red City
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Example Data

• Arc Costs from city i to city j
• Revenue Hot Dog 2, Umbrella 5
• Fixed cost 22/hr

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Results

• Expected Profit from Stochastic Recourse Problem
• 77.08
• Expected Result of Expected Value Solutions, EV
• 73.24
• Value of Stochastic Solution, VSS
• 3.84

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CVaR Results/Interpretation

• CVaR and VaR for 75 and 90
• 47
• Average Profit
• -39.68
• For small problem, results are very similar for
  majority of solutions
• Result of weather?
• Not enough scenarios?
• Need to be able to consider larger problem

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Genetic Algorithm

• Belongs to class of stochastic search methods
• Work on a group of solutions instead of just one
• Based on theory of evolution
• A group of organisms will adapt to their
  environment over many generations
• Mutations in offspring fuel the adaptations

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Genetic Algorithm

• Organisms are actually data structures
• Tour through all n cities
• Hours worked on each day in each scenario
• Start out with initial population of organisms
• Sort based on organisms respective objective
  values
• Initiate mutations

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Mutating Tours and Hours

• Mating
• (1 2 3 4 5 6) (1 4 3 6 5 2) ? (1 2 5 6 3 4)
• City switch
• (1 2 3 4 5 6) ? (1 2 5 4 3 6)
• Inversion
• (1 2 3 4 5 6) ? (1 5 4 3 2 6)
• Mutate hours by increasing/decreasing

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Running the GA

• Take best organisms on to next generation
• Repeat mutation steps
• After a number of generations, will converge
  toward near-optimal solution
• User specifies number of generations

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Run of 20 Cities, 50 Generations
Best Organisms Tour After First Generation
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Run of 20 Cities, 50 Generations
Best Organisms Tour After Last Generation
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Best Expected Profit by Generation

• After 1 -63.25
• After 13 -18.89
• After 25 -15.13
• After 37 -13.26
• After 50 -12.60
• Seemingly converging toward optimal value
• Runtime 2h 10m

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Conclusions

• Stochastic formulation shown to successfully
  offer better solution than its deterministic
  counterpart
• CVaR may not be best deviation model due to
  skewed behavior of random components
• GA solves practical problems in manageable
  running time
• Applications to real-world problems
• Airline scheduling
• Inventory/transportation management

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Questions?
ESI 6912 Czernikowski Rainwater 11/23/2005