Published online by Cambridge University Press: 21 October 2008
In this paper we present a dual approximation scheme for the classconstrained shelf bin packing problem.In this problem, we are given bins of capacity 1, and n items ofQ different classes, each item e with class c e and sizes e . The problem is to pack the items into bins, such thattwo items of different classes packed in a same bin must be indifferent shelves. Items in a same shelf are packed consecutively.Moreover, items in consecutive shelves must be separated by shelfdivisors of size d. In a shelf bin packing problem, we have toobtain a shelf packing such that the total size of items and shelfdivisors in any bin is at most 1. A dual approximation scheme must obtain a shelf packing of all items into N bins, such that, thetotal size of all items and shelf divisors packed in any bin is atmost 1 + ε for a given ε > 0 and N is the number of bins usedin an optimum shelf bin packing problem.Shelf divisors are used to avoid contact between items of differentclasses and can hold a set of items until a maximum given weight.We also present a dual approximation scheme for the class constrainedbin packing problem. In this problem, there is no use of shelfdivisors, but each bin uses at most C different classes.