The other side of ranking schemes: generating weights for specified outcomes

This paper treats the problem of how to determine weights in a ranking, which will cause a selected entity to attain the highest possible position. We establish that there are two types of entities in a ranking scheme: those which can be ranked as number one and those which cannot. These two types of entities can be identified using the ‘ranking hull’ of the data; a polyhedral set that envelops the data. Only entities with data points on the boundary of this hull can attain the number one position. There are no weights that will make an entity whose data point is in the interior of the hull to ever attain the number one position. We deal with these two types of entities separately. In the first case, we propose an approach for finding a set of weights that, under special conditions, will result in a selected entity achieving the top of the ranking without ties and without ignoring any of the attributes. For the second category of entities, we devise a procedure to guarantee that these entities will attain their highest possible position in the ranking. The first case will require using interior point methods to solve a linear program (LP). The second case involves a binary mixed integer formulation. These two mathematical programs were tested on data from a well‐known university ranking.