Simple matching vs linear assignment in scheduling models with positional effects: A critical review

► A critical review of scheduling models with positional job-independent weights. ► Identified drawbacks of known approaches to these models. ► The use of models with general positional non-monotone factors. ► A unified approach to the models based on minimizing a linear form over permutations. ► Improved performance of the resulting algorithms. This paper addresses scheduling models in which a contribution of an individual job to the objective function is represented by the product of its processing time and a certain positional weight. We review most of the known results in the area and demonstrate that a linear assignment algorithm as part of previously known solution procedures can be replaced by a faster matching algorithm that minimizes a linear form over permutations. Our approach reduces the running time of the resulting algorithms by up to two orders, and carries over to a wider range of models, with more general positional effects. Besides, the same approach works for the models with no prior history of study, e.g., parallel machine scheduling with deterioration and maintenance to minimize total flow time.