Packing into designated and multipurpose bins: A theoretical study and application to the cold chain

We consider a multitype bin packing problem and focus on the particular case of an online setting with two types of items and three bin types: two designated bins and a multipurpose bin that can store both types of items. The flexibility of multipurpose bins comes at a greater cost per bin and the objective is to minimize the cost of bins used. First, we establish a competitive ratio lower bound for the unit size problem as a function of the bin cost parameters; over all bin costs the resulting worst-case competitive ratio is [formula presented]≈1.618. Next, we show that the first-fit algorithm׳s competitive ratio is tight (it equals the established lower bound) for the two-size standard bin packing problem (in the absence of item and bin types) with an absolute competitive ratio of [formula presented]. Then, we generalize our analysis for the problem with two item types, where each item type has a distinct size; the worst-case absolute competitive ratio is shown to be [formula presented] as in the unit size case. Finally, we apply our results to analyze mixed load packing of perishable items given current spot prices of dry and refrigerated shipping containers.