In Part 1 of our look at optimizing realignment, we explained how we estimate team travel, and total league travel, without knowing the schedule ahead of time. This was needed in order to compare different realignment proposals. In Part 2, we will:
talk about how we generated lots of realignment solutions;
give our top 100 solutions;
show that we can easily add other conditions (like "PIT and PHI must stay in the same division"), and give the best solutions that satisfy those conditions.
Generating lots of solutions
We developed a method for generating thousands of good solutions to realignment. We were able to prove that the best solution that we found using this method was in fact the optimal solution. We feel that a strength of our approach is that instead of finding just one "best" solution, we generate lots of very good solutions. Many of the top solutions are similar in terms of required travel distance, and the NHL could choose from among these and base their final decision on other criteria time zones, TV ratings, and rivalrieswithout significantly increasing the league's travel.
We generate our realignment solutions in the following way. First, we take every pair of cities in the league and draw a line through them:
Not all of these lines split the league into two 15-team conferences. For example, the line thru LA and VAN does not split the league in half. So we remove all lines that don't split league in half:
Some of these remaining lines would still not be good for dividing the league into two conferences. Lines that are too horizontal would lead to some undesirable results. For example, a horizontal line through SJ and an east coast team would split the league into a northern conference and a southern conference. Both conferences would span four different time zones, which is bad for TV. Also, the travel requirements would be huge, and these solutions would never appear near the top of our list of best solutions anyway. So we remove these horizontal lines:
Finally, we use these remaining lines to divide the league into two 15-team conferences.
We then repeat a similar process to divide each conference into three five-team divisions. See the paper for details.
Once we have generated the solutions, we can sort them by the estimated league travel as we described in Part 1.
Best six-division solutions
The top 100 solutions are shown in the animated .gif below. A .pdf slideshow, which allows you to scroll through more slowly, is here.
Let's freeze this animation to show the top solution:
Notice that FLA and TB are not in the same division for our top solution and for many of the other solutions shown in the animation. This is a great example of why we wanted generate lots of good solutions instead of one best solution: now we can easily add other requirements, such as "we want FLA and TB to stay together", and remove the solutions from our list that do not meet these requirements. We add these additional requirements:
1. These teams must be in the same division: TB and FLA; ANA and LA; NYR, NYI, and NJ; PHI and PIT; CGY and EDM.
2. A division can span at most two different time zones (example: the division with VAN and MIN in the upper left spans three time zones, so we remove it).
3. At most three Canadian teams can be in one division (example: the division with VAN and MIN has only one American team).
Here is animation of the top 10 solutions that satisfy these requirements, and here is a .pdf slideshow.
We will again freeze this animation and show the top solution:
This configuration requires a little bit more travel for the league, but it is only an increase of a couple of thousand miles, and this configuration may be more desirable because of other reasons.
In Part 3, we will discuss the four-conference realignment proposed by the NHL, and compare it to our best six-division solution. We will also give our best four-conference solution, and some solutions for some hypothetical franchise moves (like PHO to QUE) and expansion to 32 teams.
Brian Macdonald is an author of Hockey Prospectus.
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