On bin packing with clustering and bin packing with delays

We continue the study of two recently introduced bin packing type problems, called with clustering, and online delays. A input consists items sizes not larger than 1, goal is to partition or pack them into bins, where total size every valid cannotexceed 1. In also have colors associated them. globally optimal solution can combine different while a clustered only monochromatic bins. The compare an solution, under certain constraints on coloring provided input. show close bounds worst-case ratio between these costs, price clustering , improving simplifying previous results. Specifically, we that does exceed 1.93667, over upper bound 1.951, it at least 1.93558, lower 1.93344. delays, are presented time. Items may wait be packed, algorithm create new any time, subset already existing unpacked it, condition valid. created cannot used again in future, all packed bins eventually. objective minimize number plus sum waiting costs items, build work modify simple phase-based algorithm. modification careful analysis improve previously known competitive from 3.951 below 3.1551.