In this paper we present a dual approximation scheme for the class
constrained shelf bin packing problem.
In this problem, we are given bins of capacity 1, and n items of
Q different classes, each item e with class ce and size
se. The problem is to pack the items into bins, such that
two items of different classes packed in a same bin must be in
Items in a same shelf are packed consecutively.
Moreover, items in consecutive shelves must be separated by shelf
divisors of size d. In a shelf bin packing problem, we have to
obtain a shelf packing such that the total size of items and shelf
divisors in any bin is at most 1. A dual approximation scheme
must obtain a shelf packing of all items into N bins, such that, the
total size of all items and shelf divisors packed in any bin is at
most 1 + ε for a given ε > 0 and N is the number of bins used
in an optimum shelf bin packing problem.
Shelf divisors are used to avoid contact between items of different
classes and can hold a set of items until a maximum given weight.
We also present a dual approximation scheme for the class constrained
bin packing problem. In this problem, there is no use of shelf
divisors, but each bin uses at most C different classes.