A bicriterion single-machine scheduling problem with step-improving processing times

Time-dependent scheduling problems, where the real processing time of jobs is dependent on the starting time, have received growing attention in recent decades. In particular, scheduling problems on a single machine have been widely studied in many facets that address the learning effect and diverse processing environments or time-dependent processing scheduling. Motivated by this observation, we introduce a variant based on the industrial procedure consideration; that is, owing to due date pressure, the processing time of the remaining jobs should be shortened after a period of manufacturing process. We consider a new single-machine scheduling problem with step-improving processing times where the objective function is to find a schedule to minimize a linear combination of the total weighted completion time and total tardiness of all jobs. The proposed problem without a critical date is an NP-hard problem. Therefore, a mixed integer programming model as well as a branch-and-bound (B&B) along with several dominance properties and a lower bound on the completion of an active partial schedule is utilized for solving the problem under study. Subsequently, four variants of the water wave optimization algorithm and four variants of the simulated annealing algorithms were proposed to solve this problem. The simulation results showed that the branch-and-bound method can solve instance problems for up to twelve jobs. The results also showed that all four variants of the wave optimization algorithm did not perform uniformly better than all three variants of the simulated annealing algorithms.