Trade-offs between optimal and robust scheduling in one-stage production

Given stochastic disturbances, such as variation in processing times, robust scheduling is recommended over optimal scheduling for production. Different from optimal scheduling that seeks an optimal solution to a key performance indicator (KPI), which relates to the average of a KPI, robust scheduling is to minimize the largest deviation from the optimum for the worst-case scenarios, which relates to the variance of a KPI. However, minimizing the variance does not necessarily optimize the average of a KPI. As one of the fundamental KPIs in production scheduling, total completion time (TCT) drives many other KPIs, such as average flow time, waiting time, due dates, and length of stay. Stochastic processing times and NP-hardness to minimize the variance of TCT, i.e., $\min \left(\mathrm{VTCT}\right)$, are two challenges in production scheduling. To investigate the trade-offs between optimal and robust scheduling, we apply the differentiation method to analyze the first and second moments of TCT. In our approach, we use three statistical measures for processing times, which are the lower bound, the expected value, and the upper bound. We also use three terms for sequencing, which are $x\left(1\right)$ the processing time of the initial job, $x\left(i\right)$ the processing time of a job in the current position i of a sequence, and $x\prime \left(i\right)$ the difference of processing times between two adjacent jobs. Applying the three measures for processing times to each of the three independent terms for sequencing, we generate $27=3·3·3$ sequences to analyze the dynamics of VTCT. Through numerical analysis in our case studies, we show that our sequencing scheme can generate optimal solutions to $\min \left(\mathrm{TCT}\right)$, and solid variation ranges of VTCT. Consequently, we can not only balance the trade-offs between $\min \left(\mathrm{TCT}\right)$ and $\min \left(\mathrm{VTCT}\right)$, but also analyze the trade-offs between optimal scheduling focusing on the first-moment of a KPI and robust scheduling focusing on the second moment. Moreover, our analysis approach using the differentiation method is unique for production scheduling, which enables us to develop analytical methods and heuristics for balancing trade-offs between optimal and robust scheduling.