A note on the optimal makespan of a parallel machine scheduling problem

For the parallel machine scheduling problem under consideration, the authors in two literatures of 1961 and 2002 respectively gave the proofs for the optimal makespan under Level Algorithm. But, some errors in their proofs are found by us with three counterexamples, and no one has given the correct proof until now. In this paper, a new algorithm is proposed. And the new algorithm is more convenient and easier for theoretical analysis than Level Algorithm does. Then, it is showed that the result schedule obtained by using the new algorithm is consistent with that by Level Algorithm in the sense that they can give the same result schedule. Finally, by using the proposed new algorithm, the proof for the optimal makespan is accomplished.