Here's a problem I need to solve that's clearly related to the standard bin-packing problem, but I'm not sure how to approach it.Suppose you have some finite set of bins $B$, and each $b_i \in B$ has some integer capacity $\mathit{cap}(b_i)$. You also have some set of items $L$ where each item $l_j \in L$ has some integer volume $vol(l_j)$. The question is, is there some way of assigning each item in $L$ to some bin in $B$ where no bin is overful, i.e. to find any total mapping $M : L \rightarrow B$ such that $\forall b_i \in B$, the sum of all $vol(l_j)$ with $M(l_j) = b_i$ is less than $cap(b_i)$.The notable differences from the classic bin-packing problem are: first, we have only a finite set of bins, not an infinite number of bins to draw from. Second, we have no notion of cost; we only want to find some total mapping or determine that no such mapping exists. Third, bins are not homogeneous; we might have one bin with capacity 5 and another with capacity 10.I'm not sure how to approach this problem or whether standard bin-packing approaches can be adapted to work with it. What is a good algorithm for answering the question?For extra credit, there are a few extra bells and whistles that I would like to add, though these are optional:Add a constraint that some bins can only contain a maximum number of items as well as having a volume limitSome bins may have a minimum per item volume as well as a maximum volume; i.e. you don't have to put an item in the bin, but if you that item must have at least a given volume
Bin-packing satisfiability rather than minimization
algorithms;packing