Suppose I have set $z = \{ 1, 5, 6, 10, 11, 35, 36 \}$
How can I find the smallest multiset $x$ that can add up to all the components of $z$?
For example, $x = \{1, 5, 5, 30\}$
1 => 1
5 => 5
1 + 5 => 6
5 + 5 => 10
1 + 10 => 11
5 + 30 => 35
1 + 5 + 30 => 36
This is sort of like a spanning set, except it's a multiset, and all scalars $\lambda_i$ in the span must $\in \{0,1\}$
You can solve the problem via integer linear programming as follows. Let $T$ (your $z$) be the given set of target values, and let $C=\{1,\dots,\max(T)\}$ be the values of the potential coins. For $c\in C$, let nonnegative integer decision variable $x_c$ be the number of copies of coin $c$. For $c\in C$ and $t \in T$, let nonnegative integer decision variable $y_{c,t}$ be the number of copies of coin $c$ used to represent $t$. The problem is to minimize $\sum_{c\in C} x_c$ subject to \begin{align} \sum_{c\in C} c y_{c,t} &= t &&\text{for all $t\in T$} \\ y_{c,t} &\le x_c &&\text{for all $c\in C$ and $t\in T$} \end{align}
For the example data, your solution $x_1=1, x_5=2, x_{30}=1$ is optimal, as is $x_1 = x_4 = x_6 = x_{30} = 1$. There are 12 optimal solutions.