Sports leagues consist of conferences subdivided into divisions. Teams play a number of games within their divisions and fewer games against teams in different divisions and conferences. Usually, a league structure remains stable from one season to the next. However, structures change when growth or contraction occurs, and realignment of the four major professional sports leagues in North America has occurred more than 25 times since 1967. In this paper, we describe a method for realigning sports leagues that is flexible, adaptive, and that enables construction of schedules that minimize travel while satisfying other criteria. We do not build schedules; we develop league structures which support the subsequent construction of efficient schedules. Our initial focus is the NHL, which had a need for realignment following the recent move of the Atlanta Thrashers to Winnipeg, but our methods can be adapted to virtually any situation. We examine a variety of scenarios for the NHL, and apply our methods to the NBA, MLB, and NFL. We find the biggest improvements for MLB and the NFL, where adopting the best solutions would reduce league travel by about 20%.
The authors wish to thank Dirk Hoag for the NHL data, Michael Wilczynski for the NBA data, and Michael Peterson for feedback about the paper. We also acknowledge an improvement suggested by an anonymous referee to the MIP formulation described in the Appendix.
Table of results
|League||# of Conf||# of Divs||Tms per Div||Div Gms||Conf Gms||Non-Conf Gms||Solution||Travel (miles)||Miles over minimum|
|2||6||5||24||40||18||3 CAN tms||1,157,640||2249|
|4||0||7, 8||0||36, 38||46, 44||Best 4-conf||1,203,918||48,527|
|4||0||7, 8||0||36, 38||46, 44||Rivalries||1,204,272||48,881|
|4||0||7, 8||0||36, 38||46, 44||Proposed||1,212,740||57,349|
|2||6||5||24||20||6||Fix AL, NL||990,034||157,774|
|NFL||2||8||4||2||0, 1||0, 1||Best||398,617||0|
|2||8||4||2||0, 1||0, 1||Current||483,782||85,165|
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=(Duv=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, Gij 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 Gij 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 Gij equal to the average number of intra-divisional away games played by teams in the division.
Let b be an s element vector with bi equal to the number of teams required in division i. Note that
In the case of the 2008–2012 6-division NHL, n=30, s=6, The away game matrix is
In the case of the current 4-conference NHL, for example, we have n=30, s=4, The away game matrix is
For each team v and each division i we have a variable xvi=1 if team i is in division i and xvi=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:
We could express the objective function as a quadratic function of x, D and G as follows:
However, we formulate this as a MIP problem by defining a set of nonnegative variables yuvij as follows: For each pair (u, v) of teams and for each pair (i, j) of divisions, we have yuvij=1 if team u is assigned to division i and team v is assigned to conference j and yuvij=0 if not. The cost cuvij of yuvij is defined to be cuvij=Duv·Gij.
We require each pair of teams to play in exactly one pair of divisions.
The third set of constraints forces the x and y variables to behave consistently. We want to have yuvij=1 if xui and xvj both equal 1 and have yuvij=0 otherwise. We create the inequalities
We constrain xui to be a 0–1 variable for all u, i. Constraints (2) and (3) then force yuvij to be 0 unless both xui=1 and xvj=1. In this case, constraint (4) forces yuvij=1.
So, finally, our MIP formulation for obtaining an optimal league structure is
In the case of the s-division NHL, we have 30×6=180 zero-one variables xvi, nonnegative variables yuvij, and constraints. In the case of the 4-conference NHL, we have 30×4=120 zero-one variables xvi, nonnegative variables yuvij, and 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.
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