Separable convex resource allocation problem aims at finding an allocation of a discrete resource to several activities that minimizes a separable convex function representing the total cost or the total loss. In this paper, we consider the separable convex resource allocation problem with an additional constraint that the L₁-distance between a given vector and a feasible solution is bounded by a given positive constant. We prove that the simplest separable convex resource allocation problem with the L₁-distance constraint can be reformulated as a submodular resource allocation problem. This result implies that the problem can be solved in polynomial time by existing algorithms for the submodular resource allocation problem. We present specialized implementations of the existing algorithms and analyze their running time.

This paper proposes a continuous network model for determining the size of the toll area and toll level in cordon and area road pricing. Cordon pricing charges a toll to vehicles passing a cordon line surrounding a designated area, whereas area pricing charges a toll to all vehicles driving inside the area. Analytical expressions for the traffic volume and toll revenue are obtained for a circular city with a radial-arc network. The analytical expressions demonstrate how the size of the toll area and toll level affect the traffic volume and toll revenue. Comparing cordon and area pricing shows that area pricing is superior to cordon pricing in both reducing traffic volume in the toll area and generating revenue.