This article studies a problem of optimal scheduling and lot sizing a number of products on m unrelated parallel machines to satisfy given demands. A sequence-dependent setup time is required between lots of different products. The products are assumed to be all continuously divisible or all discrete. The criterion is to minimize the time at which all the demands are satisfied, C max, or the maximum lateness of the product completion times from the given due dates, L max. The problem is motivated by the real-life scheduling applications in multi-product plants. The properties of optimal solutions, NP-hardness proofs, enumeration, and dynamic programming algorithms for various special cases of the problem are presented. A greedy-type heuristic is proposed and experimentally tested. The major contributions are an NP-hardness proof, pseudo-polynomial algorithms linear in m for the case, in which the number of products is a given constant and the heuristic. The results can be adapted for solving a production line design problem.