WSRPT is 1.2259-competitive for Weighted Completion Time Scheduling

\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 $1.1038$.