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Air Miles

Air Miles. Find the best route. Maths and transport. Planning the best routes for your company to fly or drive Detecting when people pose a threat to security on the Tube - CCTV analysis Designing new technologies to make transport faster and more energy efficient.

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Air Miles

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  1. Air Miles Find the best route

  2. Maths and transport • Planning the best routes for your company to fly or drive • Detecting when people pose a threat to security on the Tube - CCTV analysis • Designing new technologies to make transport faster and more energy efficient

  3. The travelling salesman problem Given a number of cities and the costs of travelling from any city to any other city, what is the least-cost round-trip route that visits each of the cities?

  4. Air miles Moscow New York London Dubai Ibiza Tokyo Rio de Janeiro Cape Town Sydney

  5. Scaling up • With these nine cities it’s not too hard to work out some possible cheapest routes. • With 90 cities you’d use a computer, but how long would it take? • For 9 cities, 362 880 possible routes. • For 90 cities, 90! = 90 x 89 x 88 x … x 1 = 1.49 x 10138 routes.

  6. A “good enough” answer • When you were choosing your route you didn’t have time to check every route • Instead, you may have tried a route which looked sensible and made small changes to see if they made a cheaper route

  7. Computer methods • Modern methods can find solutions for extremely large problems – millions of cities! – within a few minutes. • Such solutions have a high probability of being just two or three percent away from the best solution.

  8. Biology, physics and all that jazz • What makes a good method for solving problems like finding the cheapest route? • Mathematicians have taken inspiration from biology, physics and even jazz music to find good methods.

  9. Method 1: Survival of the Fittest • Pick some routes at random • Keep the best of those • Create a new generation by breeding routes together • Throw away the bad routes • Have some random mutations each generation • Thousands of generations later, you get good routes!

  10. Method 2: Simulated annealing • Annealing: heat a material like steel or glass and then cool it, to make it softer. • Simulated annealing exposes a "solution" to "heat" and cools producing a better solution.

  11. Method 3: Harmony search • In jazz music each musician tunes their notes to find a best harmony all together. • You can imagine each city having preferred previous and next destinations that “sound better”. • It’s possible to make this work mathematically!

  12. Where’s this maths used? • Water distribution • Computer network design • Environmental projects • Design of traffic networks • Music composition • Sudoku puzzle solving • Timetabling software

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