ABSTRACT

Recent studies have shown that an adaptive T² chart with double sampling and variable sampling interval $DSVSI{T}^2$ chart shows a good performance in detecting small to moderate shifts in the mean vector. This paper develops an economic statistical model for the $DSVSI{T}^2$ chart. The statistical performance of the proposed establish is evaluated using the average number of false alarms ($ANF$) and the adjusted average time to signal ($AATS$) criteria. The Wald's identity approach is used to derive an expression for the $AATS$. The unified Lorenzen and Vance's cost model is generalized to establish the economic model. Using numerical examples, we illustrate the application of the proposed model and compare its performance with the other $T^2$ and $MEWMA$ schemes. A sensitivity analysis is also carried out to investigate the effects of model parameters on the solution of the economic statistical design. The results indicate that the $DSVSI{T}^2$ chart economically outperforms the existing standard $T^2$ and other adaptive $T^2$ charts. The economic performance of the $DSVSI{T}^2$ chart is also favourable compared to the $MEWMA$ chart.

Abbreviations: SPC Statistical process control; FRS Fixed ratio sampling; DS Double sampling; VSS Variable sample size; VSI Variable sampling interval; VSSI Variable sample size and sampling interval; VP Variable parameters; FA Full adaptive; DSVSI Double sampling and variable sampling interval; MEWMA Multivariate exponentially weighted moving average; ANF Average number of false alarms; ATS Average time to signal; AATS Adjusted average time to signal; ANS Average number of samples; ARL Average run length; SD Statistical design; ED Economic design; ESD Economic statistical design; GA Genetic algorithm

摘要

最近的研究表明，具有双采样和可变采样间隔的自适应T ²图$D秒伏秒\mathrm{一世}{吨}^2$图表显示了在检测均值向量中小到中等变化方面的良好性能。本文建立了一个经济统计模型$D秒伏秒\mathrm{一世}{吨}^2$图表。建议建立的统计性能使用误报的平均数（$\mathrm{一种}NF$) 和调整后的平均信号时间 ($\mathrm{一种}\mathrm{一种}吨秒$） 标准。Wald 恒等式方法用于推导$\mathrm{一种}\mathrm{一种}吨秒$. 将统一的 Lorenzen 和 Vance 成本模型推广到建立经济模型。通过数值例子，我们说明了所提出模型的应用，并将其性能与其他模型进行了比较。${吨}^2$ 和 $米乙宽米\mathrm{一种}$计划。还进行了敏感性分析，以研究模型参数对经济统计设计解决方案的影响。结果表明，$D秒伏秒\mathrm{一世}{吨}^2$ 图表在经济上优于现有标准 ${吨}^2$ 和其他自适应 ${吨}^2$图表。经济表现$D秒伏秒\mathrm{一世}{吨}^2$ 图表也比 $米乙宽米\mathrm{一种}$ 图表。

缩写： SPC 统计过程控制；FRS 固定比率采样；DS 双采样；VSS 可变样本量；VSI 可变采样间隔；VSSI 可变样本量和采样间隔；VP 变量参数；FA 全自适应；DSVSI 双采样和可变采样间隔；MEWMA 多元指数加权移动平均线；ANF 平均误报数；ATS 平均发出信号时间；AATS 调整后的平均信号时间；ANS 平均样本数；ARL 平均运行长度；SD统计设计；ED 经济设计；ESD 经济统计设计；GA遗传算法