Finding all subsets of a set of MultiSets made of elements from a single MultiSet (without replacement)
This problem (at least assuming you wish to find some sort of maximal subset) is simply the multidimensional knapsack problem, see e.g. &quot;The Multidimensional Knapsack Problem: Structure and Algorithms&quot; by Jakob Puchinger, Günther Raidl, Ulrich Pferschy.
To read that paper in your terminology, a quick translation.
Your 'components' are known as resources, their multiplicities as capacities. The $i$th resource $S_i$ has capacity $c_i$. Each of your multisets $M_j$ is an item, and it consumes $w_{ij}$ of the $j$th resource. Thus $w_{ij}$ is the multiplicity of $S_i$ in $M_j$.
Finally in the optimization version that's studied in the linked paper we wish to know the most profitable solution. In this context each of your multisets $M_j$ (AKA items) has an associated profit $p_j$. The paper selects $x_j \in \{0, 1\}$ to determine whether item $j$ is part of the optimal solution, and thus solves the following problem:
\begin{align}
\text{max }\;\; &amp; z = \sum_{j=1}^n p_jx_j\\
\text{s.t. }\;\; &amp;\sum_{j=1}^nw_{ij}x_j \leq c_i, \quad i =1,\dots,m,\\
           &amp;x_j\in\{0, 1\},\quad j = 1,\dots,n.
\end{align}
Note that the paper uses $m$ as the number of components/resources and $n$ as the number of multisets/items.