Economic Design of Joint X and R Control Charts: A Markov Chain Method

Abstract

本研究以系統化的方式提供一個X和R管制圖的經濟設計方法，而此兩個管制圖在生產過程中同時被用來管制某一被選定的品質特性。研究中用X及R圖追蹤的一般化製程模式(a generalized process model)首度被建立，且被表示為再生過程(a renewalprocess)，在再生過程中每一個循環(cycle)又被表示為馬可夫過程(Markov process)。運用我們所提供的馬可夫鏈方法推導平均循環時間(expected cycle time)和平均循環成本(expected cycle cost)，繼而導出成本函數(Asymptotic cost function)會比擴展鄧肯(Duncan)或其他作者的方法簡單容易，尤其是當非隨機因素(assignable causes)有多個時。由於導出的單位時間成本是管制圖設計參數之函數，故用最佳化技巧即可決定設計參數的最佳值。唯非隨機因素越多時，此類問題的計算越複雜，但我們已設計出一般化的福傳(Fortran)程式來簡化此類複雜的計算。文中將給予一個只考慮二個非隨機因素的簡單例子及其資料分析結果。

Economic models for the design of control charts based on Duncan`s approach have been studied in the recent past. However, the economic design of control charts has not been developed in a systematic manner so far. Consequently, various assumptions and approaches have been made, and most researchers only consider process models involving a single assignable cause, for which a single control chart (X, P, or S) is used. In practice, these assumptions and approaches are not realistic and flexible, and the application of a single control chart is not sufficient. In this study, the design of control charts for one process variable is treated in a systematic manner. The structure of this study is: (1) To develop a generalized process model (a Markov process) with multiple assignable causes. (2) To apply the joint X and R control charts to the generalized process model. (3) To derive a cost model depending on the design parameters, (sample size, sampling interval and control limits of X and R charts), of joint X and R charts using the Markov properties. (4) To obtain optimal design parameters by optimizing the derived cost function. It is believed that the expected cycle time and expected cycle cost are more easily obtained by the proposed Markov chain method than by extending the Duncan`s approach and others approaches. The generalized process model, in which we use joint X and R charts, shows that it is more reasonable and flexible than a basic process model, in which a single control chart is used. An application of the method is presented using a simple example. A general Fortran program has been written to solved this type of problem. The results of data analyses tell us the critical parameters and show that this design method gives lower quality cost compared to Shewhart`s design. The design method can be applied to multiple process variables and to a variety of control charts.