\textit{Weighted shortest processing time first} (WSPT) is one of the best known algorithms for total weighted completion time scheduling problems. For each job $J_j$, it first combines the two independent job parameters weight $w_j$ and processing time $p_j$ by simply forming the so called Smith ratio $w_j/p_j$. Then it schedules the jobs in order of decreasing Smith ratio values. The algorithm guarantees an optimal schedule for a single machine and the approximation factor $1.2071$ for parallel identical machines. For the corresponding online problem in a single machine environment with preemption, the \textit{weighted shortest remaining processing time first} (WSRPT) algorithm replaces the processing time $p_j$ with the remaining processing time $p_j(t)$ for every job that is only partially executed at time $t$ when determining the Smith ratio. Since more than 10 years, we only know that the competitive ratio of this algorithm is in the interval $[1.2157,2]$. In this paper, we present the tight competitive ratio $1.2259$ for WSRPT. To this end, we iteratively reduce the instance space of the problem without affecting the worst case performance until we are able to analyze the remaining instances. This result makes WSRPT the best known algorithm for deterministic online total weighted completion time scheduling in a preemptive single machine environment improving the previous competitive ratio of $1.5651$. Additionally, we increase the lower bound of this competitive ratio from $1.0730$ to $1.1038$.

翻译：\textit{加权最短剩余处理时间优先}（WSRPT）是解决总加权完成时间调度问题最著名的算法之一。对于每个作业$J_j$，该算法首先通过简单地构建所谓的史密斯比$w_j/p_j$，将两个独立的作业参数——权重$w_j$和处理时间$p_j$相结合。然后，它按照史密斯比值递减的顺序调度作业。该算法保证了单机环境下的最优调度，以及同构并行机环境下的近似比$1.2071$。对于相应的可抢占单机环境下的在线问题，\textit{加权最短剩余处理时间优先}（WSRPT）算法在计算史密斯比时，将每个在时间$t$仅部分执行的作业的处理时间$p_j$替换为剩余处理时间$p_j(t)$。过去十多年里，我们仅知道该算法的竞争比区间为$[1.2157,2]$。在本文中，我们给出了WSRPT的紧竞争比$1.2259$。为此，在不影响最坏情况性能的前提下，我们逐步缩减问题的实例空间，直至能够分析剩余实例。这一结果使得WSRPT成为可抢占单机环境中确定性在线总加权完成时间调度问题的最优已知算法，将先前$1.5651$的竞争比进行了改进。此外，我们将该竞争比的下界从$1.0730$提升至$1.1038$。