Realignment in the NHL, MLB, NFL, and NBA

Table of results

Minimizing total league travel with mixed integer programming

In Section 3 we described a heuristic method for finding a league structure that minimizes, or comes close to minimizing, total league travel subject to a variety of conditions. In this Appendix we outline how to find a provably optimal solution for most of these situations by formulating the problem as a mixed integer linear programming (MIP) problem which we solved using the CPLEX mixed integer programming solver. The computational time is much greater, but it enabled us to obtain a guaranteed optimal solution and so to show that the heuristic of Section 3 found the optimal solution for these cases.

The problem can also be formulated as a quadratic assignment problem (QAP), see Pardalos, Rendl and Wolkowicz (1994). Such an approach was proposed in Mitchell (2003) in the context of NFL realignment. His goal was to minimize intra-divisional travel distance, whereas ours is to minimize total league travel. These two problems are similar, but our objective facilitates the considering of a multi-level league structure. In Paul (2009), the author describes a heuristic for these problems by combining tabu search with an adaptation of the Lin-Kernighan heuristic for the traveling salesman problem.

We outline our approach to formulating the problem of minimizing the surrogate objective as a MIP problem. We have a set T of n teams/cities and a set S of s divisions. For any two teams u,v∈T recall that d(u, v)=d(v, u) is the travel distance between the home cities of u and v.

The input data consists of the following:

• D=(D_uv=d(u, v) : u, v∈T) is the n×n inter-city distance matrix.

• G is the s×s away game matrix, where s is the number of divisions in the league. For each pair (i, j) of divisions, G_ij specifies the number of away games to be played by teams in division i against teams in division j. In the case that this number is not the same for all pairs of teams in these divisions, we set G_ij equal to the average number of games over pairs of teams in the two divisions. For intra-divisional games, that is, when i=j, we set G_ij equal to the average number of intra-divisional away games played by teams in the division.

• Let b be an s element vector with b_i equal to the number of teams required in division i. Note that ∑ibi=n.

In the case of the 2008–2012 6-division NHL, n=30, s=6, b=[555555]. The away game matrix is

G=[3220.60.60.62320.60.60.62230.60.60.60.60.60.63220.60.60.62320.60.60.6223].

In the case of the current 4-conference NHL, for example, we have n=30, s=4, b=[7788]. The away game matrix is

G=[15/73/2113/215/7111114.5/63/2113/214.5/6].

For each team v and each division i we have a variable x_vi=1 if team i is in division i and x_vi=0 if not. We have the following sets of constraints on our x variables. The first set ensures that we have the correct number of teams in each division and that each team belongs to a division:

∑v∈Txvi=bi  for each division  i;∑i∈Sxvi=1  for each team  v.

We could express the objective function as a quadratic function of x, D and G as follows:

 Minimize  z(x)=∑i, j∈S∑u, v∈Txui⋅xvj⋅Gij⋅Duv.

However, we formulate this as a MIP problem by defining a set of nonnegative variables y_uvij as follows: For each pair (u, v) of teams and for each pair (i, j) of divisions, we have y_uvij=1 if team u is assigned to division i and team v is assigned to conference j and y_uvij=0 if not. The cost c_uvij of y_uvij is defined to be c_uvij=D_uv·G_ij.

We require each pair of teams to play in exactly one pair of divisions.

 For each pair u,v of cities,∑i, j∈Syuvij=1; For each pair i,j of divisions,∑u, v∈Tyuvij=bi⋅bj.

The third set of constraints forces the x and y variables to behave consistently. We want to have y_uvij=1 if x_ui and x_vj both equal 1 and have y_uvij=0 otherwise. We create the inequalities

(2)yuvij≤xui, (2)

(3)yuvij≤xvj, (3)

(4)yuvij≥xui+xvj−1  for all u, v∈T, i, j∈S. (4)

We constrain x_ui to be a 0–1 variable for all u, i. Constraints (2) and (3) then force y_uvij to be 0 unless both x_ui=1 and x_vj=1. In this case, constraint (4) forces y_uvij=1.

So, finally, our MIP formulation for obtaining an optimal league structure is

minimize∑u, v∈T, i, j∈Syuvij⋅cuvijsubject  to∑v∈Txvi=bi for each division i;∑i∈Sxvi=1 for each team v;∑i, j∈Syuvij=1 for each distinct pair u, v of cities;∑u, v∈Tyuvij=bi⋅bj  for each pair i, j of divisions;yuvij≤xui for all u, v∈T  and i, j∈S;yuvij≤xvj for all u, v∈T and i, j∈S;yuvij≥xui+xvj−1  for all  u,v∈T  and  i, j∈S.xuv∈{0, 1} for all u,v∈T  and yij≥0  for all u, v∈T and  i, j∈S.

In the case of the s-division NHL, we have 30×6=180 zero-one variables x_vi, 62⋅(302)=15,660 nonnegative variables y_uvij, and (6+30+435+36+3)⋅62⋅(302)=47,487 constraints. In the case of the 4-conference NHL, we have 30×4=120 zero-one variables x_vi, 42⋅(302)=6960 nonnegative variables y_uvij, and (4+30+435+16+3)⋅42⋅ constraints.

For the 4-division league we also “anchored” the two seven-team conferences to western cities and the eight-team divisions to eastern cities by requiring certain x variables to be 1. It is straightforward to add extra constraints to this model to require certain teams, or combinations of teams to be in specified divisions. We were able to reduce solution time by providing CPLEX with a starting solution equal to the best solution found by our heuristic for the problem. In all cases, this turned out to be an optimal solution. Also, constraining pairs of cities on opposite sides of the continent to be in different divisions significantly improved solution time.

The solution time required to solve these league structure problems ranged from several hours for the 4-division NHL to several days for the 6-division NHL on a moderately powerful four core workstation. However, as noted earlier, this only produced a single, provably optimal, structure. The set of optimal and near optimal solutions provided by the heuristic provide more options to league planners.

More details on our computational experience solving this problem as a MIP will be provided in a subsequent paper.

Additional figures

Figure 18

(Left) Optimal partition into two divisions for these six cities. This partition cannot be achieved by straight line cuts. (Right) Partition into two non-overlapping divisions. The solution on the left has a slightly lower weighted distance.

Figure 19

The difference in team travel between the 6-division configuration of 2011–2012 and our best 6-division solution.

Figure 20

The “best” configuration for a 4-conference structure, that satisfies the same additional constraints as before.

Figure 21

The difference in eastern conference team travel between the NHL’s new structure and our 4-conference solution given in Figure 11.

Figure 22

The best solutions if PHO moves to LV and if ONT and QUE are awarded expansion teams. These represent the best solutions assuming 4-team divisions (left) and 8-team divisions (right).