  Considering the successful theory of multiple summing multilinear operators
as a prototype, we introduce the classes of multiple Cohen strongly p-summing
multilinear operators and polynomials. The adequacy of these classes under the
viewpoint of the theory of multilinear and polynomial ideals and holomorphy
types is discussed in detail. Some abstract results are also proved in the
abstract setting of the full general Pietsch Domination Theorem due to
Pellegrino, Santos and Seoane-Sep\'{u}lveda.
