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The floodlights are on, the crowd is breathless with anticipation, and months and years of preparation have come to fruition. Journalists have their keyboards ready: one of football’s major events, the Euros, is about to kick off. While it may not be as grand as the World Cup, it still commands a massive audience, featuring some of the world’s top players. Millions will be watching, and without a doubt, refereeing decisions will be a hot topic, sparking controversy, travesty, and outrage in the eyes of fans.

At Motulus, we may not be able to change every fan’s opinion, but we could certainly contribute to a smoother tournament for referees. We must recognize that referees are dedicated professionals who maintain the same levels of fitness and focus as the players. They too will be in Germany for the duration of the tournament, and their allocation to different games, along with rest periods and travel considerations, can affect their performance.

With the tournament kicking off in just a few hours, we decided to apply our mathematical expertise to assist UEFA with this challenge. Here’s the problem statement as we understand it:

• 36 group games (fixtures known) over 12 days
• 15 knockout fixtures (matchups determined throughout the tournament)
• 18 referee teams (including referees, assistants, and 4th officials)
• 10 venues across Germany, with referees based in Frankfurt

Given our knowledge of the costs of flights, trains, taxis, and hotels in the various cities, we can initially develop an objective function (which represents an optimization model aims) to minimize costs. An interesting variable to consider is that the cost in Frankfurt (the referees base)  might be zero since the accommodation is already committed. This raises the question: is it more cost-effective for referees to travel directly from venue to venue, or to return to Frankfurt after each game? Assumptions would include that a referee must arrive at a venue a certain amount of time before a game and leave a certain amount of time after the game.

If we only consider cost efficiency: creating the best plan with lowest cost, and probably the fewest number of referees, then a few problems would arise, the plan would be very impractical. For example:

• No buffers for travel delays would be applied, risking late arrivals and inadequate preparation time for referees.  Therefore, travel time buffers and restrictions on post-evening game travel must be included.
• Additionally, the referees would be extremely fatigued if they officiated games daily, sufficient rest must be scheduled between fixtures. This could be a fixed constraint, such as no refereeing duty for three days after a match, or it could be built into the optimization aims to achieve the ideal distribution (e.g., a four-day gap allows for adequate rest while keeping the referee ‘sharp’). This can be modelled as ‘soft’ costs that can be compared with the actual costs of the overall plan (e.g., hotel and travel expenses).
• There may also be hard rules, such as a referee not officiating a match involving their own nation, and possibly other nations where there might be a perceived bias. Referees are undoubtedly professional and unbiased, nevertheless organizers may want to avoid any speculation.
• Additionally, for high-profile fixtures, experienced referees might be preferred due to their familiarity with major tournaments. This can be modelled as a constraint or as a soft cost.
• Organizers might also consider past data on referees’ performance, aiming to balance the assignment of high-profile or challenging fixtures while ensuring fair distribution of workload among all referees.  All of this is possible to include in an optimization model.

When the knockout stage is reached, another optimization job will need to be run using a similar model, potentially with more emphasis on referee ratings and adding constraints to prevent referees from officiating the same team too frequently. So, UEFA, it’s not too late to get in touch with us!

This scheduling problem, with Europe, if not the world watching, is similar to challenges faced across various industries, particularly in aviation. Our pioneering crew scheduling solution can handle complexities far greater than those described here. Please reach out to learn more.

Photo by Diego Parodi on Unsplash