This paper proposes three important and suitable results relating to the minimization of the sum of the flow-time and the tardiness of tasks or jobs with unequal release dates. The first result is a locally optimal priority rule (FTR), proposed for minimizing the sum or the linear combination of the task’s flow-time and tardiness. The second result is a scheduling algorithm based on these rules. The algorithm has a complexity of O(n3) when applied to schedule a set of n tasks on one processor. To evaluate its performance, we compare its results to a lower bound that we provide as the third result in a numerical case study. Using this algorithm combined with the tasks urgency criterion enabled us to develop an online algorithm for scheduling the tasks on q parallel processors. This latter algorithm corresponds to the fourth result that we applied for scheduling and assigning preventive maintenance tasks to processors in a distributed system. As shown in the appropriate numerical example, its efficiency minimizes the cost of preventive maintenance tasks, expressed as the sum of the tasks' tardiness and flow-time.