<p>This paper addresses a rescheduling problem where a set of original jobs has already been scheduled to optimize a specific objective on a single machine. However, before processing begins, a set of new jobs arrives and should be inserted into the original jobs with minimal disruption to the planned schedule, which creates a necessity for rescheduling. The disruption is quantified by the maximum deviation in completion time for any original job between the planned and the rescheduled sequences. To ensure an acceptable service level for the original jobs, the rescheduling process allows for the rejection of some new jobs, with each rejected job paying a corresponding rejection cost. We explore three general models in which the maximum completion time disruption is considered both as a constraint and as a component of the cost objective. The primary scheduling objective is to minimize the maximum delivery completion time. The overall cost objective is to minimize the sum of the scheduling objective for the original and newly accepted jobs, the maximum disruption penalty, and the total rejection cost. We present a corresponding dynamic programming exact algorithm that runs in pseudo-polynomial time. Additionally, by employing a trimming technique, we propose a fully polynomial-time approximation scheme.</p>