  We analyze a recently introduced concept, called the price of clustering, for
variants of bin packing called open-end bin packing problems (OEBP). Input
items have sizes, and they also belong to a certain number of types. The new
concept deals with the comparison of optimal solutions for the cases where
items of distinct types can and cannot be packed together, respectively. The
problem is related to greedy bin packing algorithms and to batched bin packing,
and we discuss some of those concepts as well. We analyze max-OEBP, where a
packed bin is valid if by excluding its largest item, the total size of items
is below 1. For this variant, we study the case of general item sizes, and the
parametric case with bounded item sizes, which shows the effect of small items.
Finally, we briefly discuss min-OEBP, where a bin is valid if the total size of
its items excluding the smallest item is below 1, which is known to be an
entirely different problem.
