Bicriteria scheduling in a two-machine permutation flowshop

In this paper we consider a production scheduling problem in a two-machine flowshop. The bicriteria objective is a linear combination or weighted sum of the makespan and total completion time. This problem is computationally hard because the special case concerning the minimization of the total completion time is already known to be strongly NP-hard. To find an optimal schedule, we deploy the Johnson algorithm and a lower bound scheme that was previously developed for total completion time scheduling. Computational experiments are presented to study the relative performance of different lower bounds. While the best known bound for the bicriteria problem can successfully solve test cases of 10 jobs within a time limit of 30 min, under the same setting our branch-and-bound algorithm solely equipped with the new scheme can produce optimal schedules for most instances with 30 or less jobs. The results demonstrate the convincing capability of the lower bound scheme in curtailing unnecessary branching during problem-solving sessions. The computational experience also suggests the practical significance and potential implications of this scheme.