Discrete Optimisers

Many roads lead to the destination. How math simplifies our daily life 

Do you think mathematics is dusty and boring and something that nobody can do anything with in any case? That’s far from the truth! The researchers at the DFG Research Center MATHEON in Berlin have set themselves the task of solving real-life problems for a wide variety of problems in everyday life.

 The mathematicians Wiebke Höhn and Marco Lübbecke embark on a journey through an ocean of problems, models and algorithms, and on their travels they discover that maths is actually to be found in far more everyday things than one might expect. Their area of specialisation is discrete optimisation, which deals with the topic of choosing the single best option from a finite number of choices. This may, for example, be selecting either the shortest or the fastest route from Berlin to Munich out of all of the possibilities (satellite navigation devices use maths too) or finding the route with the shortest transfer times for every passenger travelling on the underground by scouring all of the network’s timetables.

And here’s another example: The German Federal Waterways and Shipping Administration is currently working on a large-scale project to extend the Kiel Canal, and here, again, the mathematicians from Berlin have been called upon to give advice. Ships of different sizes navigate this important canal in both directions, but large ships are only allowed to encounter each other at a few points along the canal. If in doubt, they have to wait for each other. Time spent waiting costs money though, and if they have to wait too long, using the canal becomes less appealing. So the mathematicians are developing models and algorithms to enable them to draw up plans for the ships that will minimise the total time spent waiting. It turns out that this task is closely related to entirely different problems, which at first sight appear completely unlike it. This is where mathematics shows its true power. Its abstract way of thinking allows an algorithm that has been discovered one to be used in any number of situations.

More and more companies are asking mathematicians if they can help find solutions to real-life complex planning and decision-making problems. To do so, the mathematicians take a careful look at the situation for themselves, ask probing questions and concentrate on what really matters, the abstract core of the task at hand. Then they create a mathematical model of the real situation. We’ll see that many of the problems faced in telecommunications, traffic management and manufacturing can be modelled using so-called networks. Once a suitable model has been found, it is necessary to find a method  an algorithm  that can be used to find the optimum solution, and when optimisers say “optimum”, they mean it! They will show us how they are able to prove that it is impossible for there to be a better solution than the one they have found, no matter how hard you try.

Once we have familiarised ourselves with the fundamental concepts, we will then be whisked away into the abstract world of mathematics, although we won’t ever lose touch with reality, as we will keep looking at real-life examples.
But why do mathematicians ask such unusual questions, why don’t they let up until they have found the last piece of the puzzle? What does their everyday work consist of when they are finding solution to real-life problems, and what do they do when they are concentrating on the theory? Why bother with maths at all, and who is going to give a mathematician a job once they have left university and why? Why does a mathematician have to be able to withstand a fair amount of frustration, and why are mathematicians artists too? The ten episodes of this series will answer these questions and will probably turn our image of mathematicians upside down.





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