In a public health emergency, resupplying points of dispensing (PODs) with the smallest number of vehicles is an important problem in mass dispensing operations. To solve this problem, this paper describes the Continuous Replenishment Inventory Routing Problem (CRIRP) and presents heuristics for finding feasible solutions when the duration of vehicle routes cannot exceed a given bound. This paper describes a special case of the CRIRP that is equivalent to the bin-packing problem. For the general problem, the paper presents an aggregation approach that combines low-demand sites that are close to one another. We discuss the results of computational tests used to assess the quality and computational effort of the heuristics and the aggregation approach.