Min-max relative regret for scheduling to minimize maximum lateness

We study the single machine scheduling problem under uncertain parameters, with the aim of minimizing the maximum lateness. More precisely, the processing times, the release dates, and the delivery times of the jobs are uncertain, but an upper and a lower bound of these parameters are known in advance. Our objective is to find a robust solution, which minimizes the maximum relative regret. In other words, we search for a solution which, among all possible realizations of the parameters, minimizes the worst-case ratio of the deviation between its objective and the objective of an optimal solution over the latter one. Two variants of this problem are considered. In the first variant, the release date of each job is equal to 0. In the second one, all jobs are of unit processing time. Moreover, we also consider the min-max regret version of the second variant. In all cases, we are interested in the sub-problem of maximizing the (relative) regret of a given scheduling sequence. The studied problems are shown to be polynomially solvable.