Inferred Weber weights: interpretation and an information minimizing solution

This article studies the inverse Weber problem, in which weights are inferred to rationalize a proposed optimal facility location. Building on classical formulations of the Weber location problem, we develop two complementary approaches: (i) minimum-information inference via Kullback–Leibler divergence minimization, yielding an exponential tilting of prior weights, and (ii) a quadratic optimization model that minimizes deviations from estimated weights. Both approaches satisfy equilibrium conditions corresponding to the Varignon frame interpretation of force balance. Extensive computational experiments on uniformly distributed and spatially clustered demand points demonstrate that the two methods produce comparable solutions under random configurations, but diverge when the proposed location is far from the true optimum. Results further highlight the dominant role of initialization and spatial structure in determining convergence behavior and information gain. The findings contribute to inverse optimization and facility location theory by linking entropy-based inference, convex optimization and geometric equilibrium, while providing practical insights for reconstructing spatial demand weights from the observed locational selections.