You can solve the problem via integer linear programming as follows. Let $T=\{1,\dots,10\}$ be the set of polyomino types. Let $P$ be the set of polyominoes (one for each subset of cells in the $20 \times 36$ grid that matches one of the ten types in any of the four orientations). For each polyomino $p$, let $t_p$ be the type, let $C_p$ be the set of cells it contains, and let binary decision variable $x_p$ indicate whether that polyomino appears. The constraints are \begin{align} \sum_{p\in P: (i,j)\in C_p} x_p &= 1 && \text{for $(i,j)\in [20] \times [36]$} \tag1\label1 \\ \sum_{p\in P: t_p = t} x_p &= 9 && \text{for $t\in T$} \tag2\label2 \end{align} Constraint \eqref{1} covers each cell with exactly one polyomino. Constraint \eqref{2} selects exactly $9$ polyominoes of each type.

Here is one possible tiling, with one color per type and the cells numbered according to copy ($1$ through $9$ for each type):

You can solve the problem via integer linear programming as follows. Let $T=\{1,\dots,10\}$ be the set of polyomino types. Let $P$ be the set of polyominoes (one for each subset of cells in the $20 \times 36$ grid that matches one of the ten types in any of the four orientations). For each polyomino $p\in P$, let $t_p$ be the type, let $C_p$ be the set of cells it contains, and let binary decision variable $x_p$ indicate whether that polyomino appears. The constraints are \begin{align} \sum_{p\in P: (i,j)\in C_p} x_p &= 1 && \text{for $(i,j)\in [20] \times [36]$} \tag1\label1 \\ \sum_{p\in P: t_p = t} x_p &= 9 && \text{for $t\in T$} \tag2\label2 \end{align} Constraint \eqref{1} covers each cell with exactly one polyomino. Constraint \eqref{2} selects exactly $9$ polyominoes of each type.

Here is one possible tiling, with one color per type and the cells numbered according to copy ($1$ through $9$ for each type):