Air MilesAir Miles

Find the best routeFind the best route

Maths and transportMaths and transport

• Planning the best routes for Planning the best routes for your company to fly or driveyour company to fly or drive

• Detecting when people Detecting when people pose a threat to security on pose a threat to security on the Tube - CCTV analysisthe Tube - CCTV analysis

• Designing new technologies Designing new technologies to make transport faster to make transport faster and more energy efficientand more energy efficient

The travelling salesman The travelling salesman problemproblem

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

Air milesAir miles

Scaling upScaling up

• With these nine cities it’s not too hard to With these nine cities it’s not too hard to work out some possible cheapest routes.work out some possible cheapest routes.

• With 90 cities you’d use a computer, but With 90 cities you’d use a computer, but how long would it take?how long would it take?

• For 9 cities, 362 880 possible routes.For 9 cities, 362 880 possible routes.

• For 90 cities, 90! For 90 cities, 90! = 90 x 89 x 88 x … x = 90 x 89 x 88 x … x 11

= 1.49 x 10= 1.49 x 10138138 routes. routes.

A “good enough” answerA “good enough” answer

• When you were choosing When you were choosing your route you didn’t have your route you didn’t have time to check every routetime to check every route

• Instead, you may have tried Instead, you may have tried a route which looked a route which looked sensible and made small sensible and made small changes to see if they made changes to see if they made a cheaper routea cheaper route

Computer methodsComputer methods

• Modern methods can find Modern methods can find solutions for extremely large solutions for extremely large problems – millions of cities! problems – millions of cities! – within a few minutes.– within a few minutes.

• Such solutions have a high Such solutions have a high probability of being just two probability of being just two or three percent away from or three percent away from the best solution.the best solution.

Biology, physics and all that Biology, physics and all that jazzjazz

• What makes a good method What makes a good method for solving problems like for solving problems like finding the cheapest route?finding the cheapest route?

• Mathematicians have taken Mathematicians have taken inspiration from biology, inspiration from biology, physics and even jazz music physics and even jazz music to find good methods.to find good methods.

Method 1: Survival of the Method 1: Survival of the FittestFittest• Pick some routes at randomPick some routes at random• Keep the best of thoseKeep the best of those• Create a new generation by Create a new generation by

breeding routes togetherbreeding routes together• Throw away the bad routesThrow away the bad routes• Have some random Have some random

mutations each generationmutations each generation• Thousands of generations Thousands of generations

later, you get good routes!later, you get good routes!

Method 2: Simulated Method 2: Simulated annealingannealing

• Annealing: heat a material Annealing: heat a material like steel or glass and then like steel or glass and then cool it, to make it softer.cool it, to make it softer.

• Simulated annealing Simulated annealing exposes a "solution" to exposes a "solution" to "heat" and cools producing "heat" and cools producing a better solution. a better solution.

Method 3: Harmony searchMethod 3: Harmony search

• In jazz music each musician In jazz music each musician tunes their notes to find a tunes their notes to find a best harmony all together. 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?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 solving Sudoku puzzle solving • Timetabling softwareTimetabling software