On characterizing supremum andl _p -efficient facility designs

Abstract

Define a design to be any planar setD of known areaa, but of unknown shape and location; more generally, a design can be any set inR ^d of measurea. For example, a design might be one floor of a warehouse, or a sports arena of known seating capacity. Suppose that the design has, say,m users, or evaluators, with user/evaluatori having a design disutility functionu _i , 1≤i≤m, which can be defined for all points in the plane independently of the designs of interest. Given any designD, denote byG _i (D) the disutility ofD to user/evaluatori where, by definition,G _i (D) is the supremum ofu _i over the setD, 1≤i≤m. LetG(D) be the vector with entriesG _i (D), 1≤i≤m, and define a design to be efficient if it solves the vector minimization problem obtained using the set of vectors {G(D):D a design}. Given mild assumptions about the disutility functions, and a slight refinement of the design definition to rule out certain pathologies, we present necessary and sufficient conditions for a design to be efficient, and study properties of efficient designs. In the final section, we extend the analysis to more generall _p -measures of design disutility.