Techniques are provided for mapping storage volumes to storage processing nodes in a storage system. One method comprises determining a number of input/output operations associated with each of multiple storage volumes, wherein the input/output operations associated with a given storage volume are processed by a corresponding storage processing node based on a mapping of the storage volumes to the storage processing nodes; obtaining constraints that limit the input/output operations processed by at least a subset of the storage processing nodes; identifying at least one possible mapping of the storage volumes to the storage processing nodes that satisfy the constraints; and selecting an alternative mapping of the storage volumes to the storage processing nodes by applying a designated function to the at least one possible mapping. The input/output operations may comprise read operations and write operations, and the read and write operations may be balanced separately.