CUSUM中位数控制图设计

Design of the CUSUM Median Chart

针对生产过程中质量特性数据存在个别异常以及离群值的情形，采用中位数统计量（X）代替传统均值（X）统计量，提出一种累积和（Cumulative Sum，CUSUM）X控制图来监控过程均值的偏移。采用马尔科夫链方法，首先构建CUSuMx控制图的状态转移矩阵，推导各状态的转移概率，进一步可推导出CUSUMX控制图的链长分布特性。通过最优化过程处于失控状态下的CUSUMX控制图的平均运行链长（Average Run Length，ARL）指标，获得控制图的最优决策变量和性能指标。仿真结果表明，CUSUMX控制图优于传统的休哈特（Shewhart）X以及指数加权滑动平均（Exponentially Weighted Moving Average，EWMA）X控制图，尤其针对较小的均值偏移，其性能优势更加明显。

Abstract： Considering the outliers in the dataset of the quality character in production processes, the statistic median （X） is used instead of the mean （X） and a CUSUM （Cumulative Sum）X chart is proposed to monitor the process mean shift. The transition probability matrix of the CUSUM X chart are first obtained using the Markov chain method, and the run length （RL） properties of the CUSUM X chart can be further obtained using this method. The optimal parameters and properties of the CUSUM X chart are given through the minimization of the out-of-control ARL （Average Run Length） of the CUSUM X chart. The simulation results show that the CUSUM X chart outperforms the Shewhart and EWMA （Exponentially Weighted Moving Average） X charts, especially for smaller mean shifts.