An asymptotically optimal online algorithm to minimize the total completion time on two multipurpose machines with unit processing times

AbstractIn the majority of works on online scheduling on multipurpose machines the objective is to minimize the makespan. We, in contrast, consider the objective of minimizing the total completion time. For this purpose, we analyze an online-list scheduling problem of n jobs with unit processing times on a set of two machines working in parallel. Each job belongs to one of two sets of job types. Jobs belonging to the first set can be processed on either of the two machines while jobs belonging to the second set can only be processed on the second machine. We present an online algorithm with a competitive ratio of ρLB+O(1n), where ρLB is a lower bound on the competitive ratio of any online algorithm and is equal to 1+(−α+4α3−α2+2α−12α2+1)2 where α=13+16(116−678)1/3+(58+378)1/33(2)2/3≈1.918. This result implies that our online algorithm is asymptotically optimal

Robust Two-Stage Packing into Designated and Multipurpose Bins

International audienceMultitype bin packing is a natural extension of the classical bin packing with applications to shipping using climate-controlled containers and plain dry containers. In transportation and other logistics applications there may be significant uncertainty with respect to the exact quantities of different variants of products (or item types) that may need to be shipped at the time when the containers and packaging are procured. In the current paper we model the problem as a robust two-stage two-item type bin packing problem. In the first stage bins of different types are acquired (e.g., reefer containers and dry containers). In the second stage the items are packed into bins. The bins that are secured in the first phase must allow for all of the items to be packed in the "worst-case" demand scenario. We first develop an algorithm for the robust two-stage two-item type bin packing problem with general item-number uncertainty sets and certain box uncertainty sets for item sizes (or equivalently two item sizes). We then consider the special case of identical (or unit) item sizes. In this special case we develop closed-form solutions for the optimal solution. Our closed-form solution reveals that it is optimal to use a number of multipurpose bins that is linear in the number of items. This is in contrast with solutions of the online and offline deterministic version of our problem that use at most one multipurpose bin. Finally, we consider computational methods that are efficient in practice for a generalization with unit item sizes but with an arbitrary number of item and bin types and arbitrary compatibility structures