A tardiness-augmented approximation scheme for rejection-allowed multiprocessor rescheduling

Abstract

In the multiprocessor scheduling problem to minimize the total job completion time, an optimal schedule can be obtained by the shortest processing time rule and the completion time of each job in the schedule can be used as a guarantee for scheduling revenue. However, in practice, some jobs will not arrive at the beginning of the schedule but are delayed and their delayed arrival times are given to the decision-maker for possible rescheduling. The decision-maker can choose to reject some jobs in order to minimize the total operational cost that includes three cost components: the total rejection cost of the rejected jobs, the total completion time of the accepted jobs, and the penalty on the maximum tardiness for the accepted jobs, for which their completion times in the planned schedule are their virtual due dates. This novel rescheduling problem generalizes several classic NP-hard scheduling problems. We first design a pseudo-polynomial time dynamic programming exact algorithm and then, when the tardiness can be unbounded, we develop it into a fully polynomial time approximation scheme. The dynamic programming exact algorithm has a space complexity too high for truthful implementation; we propose an alternative to integrate the enumeration and the dynamic programming recurrences, followed by a depth-first-search walk in the reschedule space. We implemented the alternative exact algorithm in C and conducted numerical experiments to demonstrate its promising performance.

A An illustration of the dynamic programming exact algorithm

Below is an illustration of executing the dynamic programming exact algorithm on the 10-job instance described in Table 1, for the combination $(s_1, s_2) = (r_\mathcal{D}+13, r_\mathcal{D}) = (63, 50)$. The following Table 4 lists all the transitions from a true state with j to a true state with $j+1$, for $j = 0, 1, 2, 3, 6, 7, 8, 9, 10$, in each of which we give every true state a number, and the third column records from which true state and in which case it is first time derived true, except the initial true state. The last column contains additional comments, if any, for possibly better implementations of the exact algorithms (see Sect. 5), and the objective values for those true states $(n; M_1, M_2, P_1, P_2, Z, T, R)$ with $s_i = \max \{M_i, r_\mathcal{D}\}$ for each machine i.

There are six true states of the form $(10; 63, \cdot , \cdot , \cdot , \cdot , \cdot , \cdot )$ corresponding to six feasible reschedules under the constraint $(s_1, s_2) = (63, 50)$, respectively, and the minimum objective value is 314. One can trace back the optimal reschedule in which the earlier schedule on machine 1 processes the jobs 7, 9, the later schedule on machine 1 is empty, the earlier schedule on machine 2 processes the jobs 1, 2, 3, and the later schedule on machine 2 processes the job 10.