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# Fixture list by number crunching

Excitement levels are high in anticipation of publication of the fixture list for the Norwegian Premier League (Eliteserien). However, what few football fans are aware of are the billions of possible match combinations, and that it is research scientists who are given the job of delivering an optimal schedule.

Arranging matches for the 16 teams in the Premier League means evaluating the billions of possible match combinations that result from multiplying the numbers 1 to 16 with each other. To avoid going through all of these combinations, both advanced mathematics and planning are needed to get the Premier League fixture list ready. Photo: Thinkstock.

FOOTBALL RESEARCH. The Premier League football season starts on 1 April and, during the 2017 season, 16 teams will play each other in the course of 240 matches.

“There are many billions of possible ways of combining these 240 matches”, say Lukas Bach, a mathematician and optimisation researcher at SINTEF.

Since 2006, SINTEF has been contracted by the Norwegian Football Association (NFF) to generate a fixture list for the Premier League, formerly known simply as Tippeligaen (the top division).

## Mathematics – the possible and the impossible

Bach stands at his whiteboard, drawing and explaining how he comes up with all the potential combinations.

“If you have a fixture list with four teams playing each other, part of the problem we have to solve is to find the sequence of the teams involved”, he says. In this case, there are 24 different possibilities because one times two times three times four makes 24.

When Bach arranges matches for the 16 teams in the Premier League, he has to sort through the billions of possible match combinations that result from multiplying the numbers 1 to 16 with each other.

However, not all of these billions of combinations are equally good. There are many constraints on how the fixtures are set up. For example, if a club plays its first match of the season at home, it will play its last match away.

“We prefer that clubs play first at home and then away in successive matches, but since this is mathematically impossible throughout the entire season, each team has at one time to play two consecutive home matches and two consecutive away matches during the course of the season”, says Bach.

## One problem at a time

Of course, Bach doesn’t calculate the entire fixture list on his whiteboard. His is ably assisted by advanced software developed by him and the remainder of SINTEF Optimisation's researchers. This software, which is under continuous development, utilises no more processing power than can be handled by a standard computer.

“The problem with the fixture list is so complex that we have to deconstruct it and solve one component of the problem at a time”, he says.  “The first issue that the software addresses are requests from the clubs about when they want to play their home and away fixtures. When this has been resolved, it starts working on problem number two, involving the dates of matches. Of course, all this is done under a variety of constraints”, says Bach.

## Not JUST mathematics

The work that Bach is doing for the Premier League is currently being incorporated in the research project called SPORTING, which is looking into fixture lists for a number of different tournaments. It is partly funded by the Research Council of Norway and the NFF. The Norway Cup event (the world’s largest youth football tournament) and the Norwegian Volleyball Association are also involved in the project.

However, you’re probably reading this with the feeling that the fixture list for Norway’s top football league isn't determined entirely by pure mathematics.

“Why is it that the Trondheim club Rosenborg always get to play at home at their Lerkendal stadium on 16 May?”, asks Nils Fisketjønn, who is Competition Director at the NFF.
“Well, that's no secret. We’re currently working under an agreement by which both Rosenborg and Bergen's top club Brann always play at home on 16 May”, he says.

So, the NFF has full control.

## TV show

“The process starts with the clubs sending their requests. This information places constraints on the fixture list set-up”, explains Fisketjønn. “For instance, the stadium in Ålesund may not be available on a certain date because of a concert”, he says.  “The media companies also have express wishes, which impose further constraints. So we then have a meeting with Lukas who enters it all into his system”, says Fisketjønn.

It takes time to process the data once Bach has registered the information in his system, although usually, the solution is ready when he arrives at work the following morning. He then sends a preliminary list to the NFF. The process of developing this draft into a full fixture list normally takes about two weeks.

There is a lot of anticipation just before the final list is released. This year it was ready just before Christmas.

“There’s a lot of interest”, says Fisketjønn. “Many people plan their year on the basis of the fixture list, and a one-hour TV show is devoted to it”, he says.

Media constraints complicate the jigsaw

Fisketjønn says that starting this year, the NFF has a new media contract that includes a “Derby Saturday” and a “Super Sunday”. On a Super Sunday, the very top teams will play each other. A “Derby” refers to a match between local rivals.

The more constraints such as these that Bach receives, the more complicated his job – and that of his software– to solve the jigsaw becomes. He shows us a page of requests from the clubs. Requests that have to be complied with from the NFF are tagged with an ‘M’.

“It is particularly challenging to complete the fixture list from the end of July to the beginning of August, when the four best teams from 2016 are playing qualifying matches in European competitions”, says Bach.

“These teams require at least two rest days before and after matches during the qualification period, and this gives us a real headache”, he says. For example, when Rosenborg plays its qualifying matches in the Champions League.

## An impartial fan

“Not all clubs can play at home on 16 May, but this year we’ve succeeded in meeting 46 of the clubs’ 51 requests for the season as a whole”, says Bach.

He enjoys his work, and thinks that football is a lot of fun. Finally, he offers us this little item of information. “I’m interested in football, but being Danish, I’m a fan of a Danish football team. So, to reassure you all, I promise that I have no favourite Norwegian team”, he smiles.