The Weber problem with demand uniformly generated in discs
提出圆盘韦伯问题，将需求表示为圆盘而非点，证明其严格凸性并给出精确与近似公式，发现点模型会低估目标函数且最优解差异显著。
We propose a Disc-Weber (DW) problem in which demand is represented as discs rather than points. We show exact formulations of the new model, study its properties, and propose a very close approximation. The new DW problem is strictly convex, and its analytical continuous derivative exists for the entire domain when all areas (radii) are positive. Therefore, unlike the original Weber problem, it lacks non-continuous derivatives at fixed points. We solve several illustrative examples to global optimality using the exact and approximate formulations. Our analysis and experiments show that the original point-based Weber formulation underestimates the objective function when area demand is present. Furthermore, the optimal solution for point and disc representations of demand can be substantially different. We also show that the DW problem approximation results are virtually identical to its exact formulation.