Too Vague to Forecast, Too Smart to Pay for One: When Narrowing the Demand Box Buys Nothing

Abstract

Abstract - Demand that can only be bracketed is the normal condition of a facility siting study, and minimax regret is the standard way to turn a bracket into a site. What it will not tell you is whether the bracket did any work, since closed-form conditions for when an uncertainty set stops changing a decision exist only where the problem separates coordinate wise, so the reflex is to buy a sharper forecast and hope it earns its fee. We reparametrise the worst-case pairwise margin by box width; because freight geometry fixes the adversarial vertex against each rival, the margin is affine in width and changes sign at most once, so the binding frontier is closed form, a minimum of ratios over rival sites, which we evaluate on parametrised line and star networks. On both families that minimum collapses to the adjacent rival whenever the fixed-cost field is affine in position, leaving frontiers written in demand mass alone. Positions, gaps, spoke lengths and the freight rate all cancel, so distance settles which site wins and says nothing about when the box begins to bind, which is awkward, because distance is the one quantity a siting study measures carefully. Nonnegative floors also cap any derived box at total central mass, a ceiling a posited box does not carry. The frontier therefore prices the forecast: below it, narrowing the box buys nothing; above it, better data buys assurance but not a different warehouse; higher still, it changes the site.

Keywords - facility location, minimax regret, interval uncertainty, partial identification, value of information