Air Miles Find the best route

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

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?

Air miles

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

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

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

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

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

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

Method 3: Harmony search In jazz music each musician tunes their notes to find a best harmony all together.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!

Where’s this maths used? Water distributionWater distribution Computer network designComputer network design Environmental projectsEnvironmental projects Design of traffic networksDesign of traffic networks Music compositionMusic composition Sudoku puzzle solvingSudoku puzzle solving Timetabling softwareTimetabling software