A one-dimensional bin packing problem with shelf divisions

Given bins of size B, non-negative values d   and ΔΔ, and a list L   of items, each item e∈Le∈L with size sese and class cece, we define a shelf as a subset of items packed inside a bin with total item sizes at most ΔΔ such that all items in this shelf have the same class. Two subsequent shelves must be separated by a shelf division of size d. The size of a shelf is the total size of its items plus the size of the shelf division. The class constrained shelf bin packing problem (CCSBP) is to pack the items of L into the minimum number of bins, such that the items are divided into shelves and the total size of the shelves in a bin is at most B. We present hybrid algorithms based on the First Fit (Decreasing) and Best Fit (Decreasing) algorithms, and an APTAS for the problem CCSBP when the number of different classes is bounded by a constant C.