Capacitated vehicle routing problem (CVRP) plays a particular role in both optimization algorithmic research and real-life applications. Various extensions and methods have been proposed for the problem in the last few decades due to its main challenges as distribution, logistics, and transportation. This paper introduces a new variant of the CVRP problem that considers a new lower-bound capacity constraint. In our context, the load assigned to each vehicle is not greater than its capacity and not less than a predetermined minimum value. This new constraint creates a challenge for construction algorithms in which violations exist from assignment operators start until the required amount of goods is assigned to vehicles. We formulate our problem as an integer linear programming model and propose a local search algorithm to solve it. The experimental results on a large number of benchmark CVRPs revealed that the proposed algorithm got promising results compared to other well-known methods.