Minimizing maintenance cost involving flow-time and tardiness penalty with unequal release dates

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. First, the policy of real-time maintenance is defined for minimizing the cost of tardiness and critical states. The required local optimality rule (flow time and tardiness rule) 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(n³) complexity when it is applied to schedule a set of n tasks on one processor. To evaluate its performance, the results are compared with 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 a 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, the cost of preventive maintenance tasks expressed as the sum of the tasks' tardiness and flow time to be minimized. This corresponds to the costs of critical states and of tardiness of preventive maintenance.