Let me set the scene. For the first time in a few years, your organization has decided to plan a major corporate event with attendees from various regions around the world. You want to spark networking between individuals and groups that are typically siloed and never, or rarely, interact. The challenge you face is not only arranging the seating for a large number of people, many of whom approach such interactions with trepidation, but also doing so in a way that will maximize the diversity of the guests at each table. Thankfully there is a solution.

After struggling to find adequate software to help with table arrangements, I aptly solved the problem for a recent Oliver Wyman event. Based on my operations research background, I built an open-source program to do the job of mixing up 1,500 people. The three-day event was a huge success. While some company veterans griped about not seeing many old friends at their table, most, including staff who joined during and pre-COVID, raved about how well it helped them meet and network with new people.

## What to consider for the perfect seating plan

Our criteria for having a mixture of people at each table was based on the information available to us in company profiles. In our system, we had access to details including a person’s job title, office location, level of seniority, and gender identity. Loaded with this data we were able to utilize the algorithm to ensure a perfect representation of personalities at each table, mixing new joiners and veterans as well as people across a range of roles (partners, specialist, associate consultants, core consultants, support), office locations, seniority level (pre-COVID and COVID joiner), and genders.

We quantified maximum diversity using a penalty score that had a low value when the entire table was highly diversified and a high value when it was not. For example, imagine there are 31 specialists at an event, which means we don’t want to seat more than four at one table. Consider a score that is the square of the number of people with the same role at a table. For two tables, one with two specialists and the other with five specialists, we would have a score of 2^{2} + 5^{2} = 29. If we rearranged the seating so that there were three and four specialists at the two tables, that would be 3^{2 }+ 4^{2} = 25, which is a lower score, and better distributes the specialists.

This equation could be extended to consider all the attributes and is a simple penalty score. Highly complex diversity penalty scores could be handled as well.

## How it works: table assignment and seating plans using an algorithm

The overall algorithm has two primary roles:

- The placement of the employees at tables that maximize diversity based on simple rules that yield a low overall penalty score, although not the lowest possible penalty score.
- The iterative improvement process that remixes a small set of employees to find rearrangements that further reduce the overall penalty score.

The algorithm will be explained using an example with 67 people, split by roles, offices, COVID or pre-COVID joiner, and gender. Below is the split among roles. Not shown is that the group has five offices ranging from Atlanta with 26 to London with five people. Twenty-four people joined during COVID and 43 joined pre-COVID. There are 20 females, 44 males, and three people who identify as non-binary.

## What’s new in this approach

Many papers have been written on using heuristics to solve this problem, but it is a very hard problem to solve to provable optimality. In these papers, the heuristics involve testing to see if swapping two or three attendees at a time could improve the solution. Our approach that uses a network flow for the improvement step simultaneously tries to swap many attendees at a time. In the small example above, each improvement step looked at swapping nine attendees at a time. For the offsite, it swaps 150 attendees at a time. When we tested this compared to one of the existing algorithms, we achieved a much higher diversity score.

While this algorithm does not guarantee a perfectly optimal amount of diversity at each table, it comes very close, and it definitely sparked a lot of valuable connections for our company. Try the algorithm yourself at your next event.