On index priority sequencing rules for scheduling with time-dependent job processing times

We consider single-machine scheduling problems with time-dependent processing times given by a general multiplicative functional form. We conclude that the linear function is the only function leading to an index priority rule to minimize the makespan. We also consider the case in which the base job processing times are neglected and conclude that the power functions and a logarithmic function are the only completion time functions that can be minimized by an index priority rule.