Online packing of arbitrary sized items into designated and multipurpose bins

We consider an online multitype bin-packing problem with two item types. This setting can be motivated by transport applications in which some items may be shipped either in dry shipping containers or more costly refrigerated ones, while other items can only be transported in refrigerated containers. The problem was introduced by Goldberg and Karhi [Omega 71:85-92, 2017] who focused on the case of up to two item types, three bin types and only two item sizes. For this special case a tight (also known as optimal) absolute competitive ratio was shown of approximately 1.618. Here we consider the general problem of arbitrary item sizes and show a lower bound on the absolute competitive ratio of any online algorithm that is a function of the bin costs. This bound in the worst-case is approximately 1.781. We then extend the first-fit method to our problem setting and prove an absolute competitive ratio bound that is a function of the bin costs. In the worst case this upper bound is approximately 1.930. In addition an upper bound of 1.750 is established on the worst-case asymptotic (as the number of items grows large) competitive ratio.