  This paper proposes important and useful results relating to the minimization
of the sum of the flow-time and the tardiness of tasks or jobs with unequal
release dates (occurrence date), with application to maintenance planning and
scheduling. Firstly, the policy of real-time maintenance is defined for
minimizing the cost of tardiness and critical states. The required local
optimality rule (FTR) is proved, in order to minimize the sum or the linear
combination of the tasks' flow-time and tardiness costs. This rule has served
to design a scheduling algorithm, with O(n 3) complexity when it is applied to
schedule a set of n tasks on one processor. To evaluate its performance, the
results are compared to a lower bound that is provided, in a numerical case
study. Using this algorithm in combination with the tasks urgency criterion, a
real-time algorithm is developed to schedule the tasks on q parallel
processors. This latter algorithm is finally applied to schedule and assign
preventive maintenance tasks to processors in the case of a distributed system.
Its efficiency enables, as shown in the numerical example, to minimize the cost
of preventive maintenance tasks, expressed as the sum of the tasks tardiness
and flow-time. This corresponds to costs of critical states and of tardiness of
preventive maintenance.
