Hi I'm looking for a variation of the Orthogonal 3D-BinPacking algorithm with only one bin of unknown size.

I have a set $S$ of $n$ cuboids items $i_j$ with $j=1...n$. The dimensions of the items are $w_k, l_k, h_k$ with $k=1...n$.

I need to find the smallest cuboid $B$ that can contain all the items with no overlap and the positions $(x_k, y_k, z_k)$ with $k=1...n$ of the items in container for the optimal solution found.

I'm looking at the Martello et al. approach for the single bin filling. What should I use in the branch backtrack condition since I don't know the volume $B$ of the bin?

The original condition is: $\sum_{i \in I}{v_i + (B-V(I)) \le F}$ where $I$ is the subset of items currently placed in the bin, $v_i$ is the volume of the item $i$, $V(I)$ is the volume occupied by the current set of items placed in the bin and $F$ is the current minimum found solution.

Any help is appreciated



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