R，S和S2管制圖在製程變異數需估計時之修正管制界限

Abstract

管制圖在統計製程管制上是非常基本而且被廣泛運用的工具。在實務上，因為鮮少知道製程品質特性分配的真實參數，所以會將統計製程管制的執行分為階段一和階段二。很明顯地，階段二的監控能力會深受階段一時參數估計得是否準確而影響，因而造成假警報率和管制下的平均連串長度與設計管制圖時原先所期望的值，這在許多文獻中都有被提到。然而，在參數為某參數估計式之條件下，將其平均連串長度視為一隨機變數，我們可以將其期望值定義為整體平均連串長度。本研究針對R、S以及S2管制圖提供兩種修正管制界限使得管制下的整體平均連串長度能夠符合我們預期的值；一種修正管制界限調整了個別的假警報率，另一種修正管制界限則使得整體平均連串長度不偏。我們計算出各種階段一管制下資料量和各種子群樣本大小之下的修正管制界限，將之列成表格供業界查詢使用，並提供一個範例說明如何使用這些管制界限。

Control charts is a very basic and widely used tool for statistical process control. In practice, the true parameters of process quality characteristics are rarely known, so that the implementation of control charts is divided into phase I and phase II. Obviously, the estimation accuracy of parameters in phase I has a great impact on the monitoring capability in phase II, resulting in that the false-alarm rate and in-control average run length (ARL) are not the values the chart is designed for, as mentioned in the literature. By treating an ARL conditioned on the estimated parameter as a random variable, the overall ARL can be defined as its expectation. In this study, we propose adjusting the control limits for R, S and charts in order to control the overall in-control ARL at a prescribed level. Two approaches are considered, one adjusts the individual false-alarm rate, the other adjusts the control limits to make the control chart overall-ARL-unbiased. Adjusted control limits of R, S and S2 charts for various sizes of phase I in-control data and subgroup sizes are tabulated for practitioners. An illustrative example is provided to demonstrate the usage of the proposed method in practice.