Abstract
We examine the efficacy of two linear programming models for optimizing the linear ordering problem (LOP) of ranking teams in major American sports, whether measured by head-to-head win counts or by collective victory margins. Using standard solution software, both models are tested over 42 problems involving full seasons from the National Football League, the National Basketball Association, the National Hockey League, and Major League Baseball, as well as major college football, baseball, and men’s basketball. We find that optimal solutions can be achieved rapidly for all four professional sports from either model, regardless of whether victory margins or win counts are used. However, solution speeds were much slower for the collegiate sports, particularly for baseball and men’s basketball where the number of teams were the largest, and especially when win counts were the performance measure. Such problems have a large and sparse pairwise comparison matrix with low values and variation in its non-zero elements, characteristics that have been found to be challenging in the general LOP literature. However, a modified minimum violations model demonstrated dramatic efficiency advantages when solving the collegiate cases, and illustrated that optimal solutions are largely achievable in reasonable time frames even for those sports.