Conditional median: A parametric solution concept for location problems

Classical approaches to location problems are based on the minimization of the average distance (the median concept) or the minimization of the maximum distance (the center concept) to the service facilities. The median solution concept is primarily concerned with the spatial efficiency while the center concept is focused on the spatial equity. The k-centrum model unifies both the concepts by minimization of the sum of the k largest distances. In this paper we investigate a solution concept of the conditional median which is a generalization of the k-centrum concept taking into account the portion of demand related to the largest distances. Namely, for a specified portion (quantile) of demand we take into account the entire group of the corresponding largest distances and we minimize their average. It is shown that such an objective, similar to the standard minimax, may be modeled with a number of simple linear inequalities. Equitable properties of the solution concept are examined.