Scheduling with a discounted profit criterion on identical machines

In this paper, we introduce a novel scheduling criterion named as a discounted profit (to be maximized), which could be considered as a generalization of early work (also to be maximized). The goals of such scheduling models are to maximize ${\sum }_{j=1}^n(X_j+\delta Y_j)$, where $X_j$ ($Y_j$) is the early (late) work of job $J_j$, and $0\le \delta <1$ is a discount factor. When $\delta =0$, these models are reduced to the ones with early work maximization. We focus on the models of scheduling on identical machines when jobs share a common due date. For the online case, we prove that the competitive ratio of the classical List Scheduling (LS) algorithm is exactly $\frac{4}{3+\delta }$, improving the seminal result ($\sqrt{2}$) and covering the very recent result ($\frac{4}{3}$) when $\delta =0$. Moreover, when the number of machines $m=2$, we propose a new optimal online algorithm with a competitive ratio $\frac{\sqrt{2\delta +5}+2\delta -1}{\delta \sqrt{2\delta +5}+1}$, matching the previous best known result ($\sqrt{5}-1$) when $\delta =0$. For the offline case, we prove that the Longest Processing Time first (LPT) algorithm has an approximation ratio $\frac{\sqrt{2}+1}{2+(\sqrt{2}-1)\delta }$, extending the existed results when $\delta =0$ and $m=2$.