An efficient method for calculating the minimum distance from an operating point to a specific (hyperbolic) efficient frontier

This paper is concerned with movement from a current operating point so as to reach a 2D efficient frontier. After a discussion of different criteria for deciding on which point on the frontier to target, we focus, as an illustration, on a particular inventory management context and the use of the criterion of minimum distance from the current point to the frontier. Specifically, the efficient frontier turns out to be a hyperbola in a 2D representation of total (across a population of items) average stock (in monetary units) versus total fixed costs of replenishments per year. Any current (or proposed) operating strategy, differing from the class along the frontier, is located above the frontier. Finding the minimum distance from the current point to the frontier requires determining the smallest root of a quartic equation within a restricted range.