6. The Total Tardiness of Tasks Minimization on Identical Parallel Machines with Arbitrary Fixed Times of Their Start and a Common Due Date

We solve a NP-hard problem of constructing a schedule for identical parallel machines that minimizes the total tardiness of tasks in relation to a common due date in case when the start times of machines are fixed at arbitrary time points less than the due date. We present an efficient PSC-algorithm of its solving which is a generalization of our previously developed results: for the problem with equal start times of machines we have derived two sufficient signs of optimality of a feasible solution and constructed two PSC-algorithms. Each of the algorithms checks one of these signs. In this chapter we propose a generalized PSC-algorithm for equal start times of machines that combines the best properties of both PSC-algorithms. We have obtained a modification of the generalized PSC-algorithm for the case of arbitrary start times of machines, its complexity is determined by O(n²m) function. The first polynomial component of the PSC-algorithm coincides with its second polynomial component. We obtain an efficient estimate of the deviation from an optimal solution for an approximation algorithm of the problem solving. We also present the statistical studies of the PSC-algorithm that showed its high efficiency (efficient exact solutions have been obtained for problems with tens of thousands of variables which is unique for NP-hard combinatorial optimization problems).