Attribute Charts for Monitoring a Dependent Process

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AbstractFor some repetitive production processes, the quality measure taken on the output is an attribute variable. An attribute variable classifies each output item into one of a countable set of categories. One of the simplest and most commonly used attribute variables is the one which classifies an item as either ‘conforming’ or ‘non‐conforming’. A tool used with a considerable amount of success in industry for monitoring the quality of a production process is the quality control chart. Generally a control charting procedure uses a sequence, $X_1 ,X_{2,} \ldots ,X_t, \ldots$ of the quality measures to make a decision about the quality of the process. How this sequence is used to make a decision defines the control chart. In order to design a control chart one must consider how the underlying sequence, $X_1 ,X_{2,} \ldots,X_t, \ldots,$ is modeled. The sequence is often modeled as a sequence of independent and identically distributed random variables. For many industrial processes, this model is appropriate, but in others it may not be. In this paper, a sequence of random variables, $X_i ,\ i = 1,2,\ldots,$ is used to classify an item as conforming or non‐conforming under a stationary Markov chain model and under 100% sequential sampling. Two different control charting schemes are investigated. Both schemes plot a sequence of measures on the control chart, $Y_i ,\ i = 1,2,\ldots$ that count the number of conforming items before a non‐conforming item. The first scheme signals as out‐of‐control if a value of $Y_i ,\ i = 1,2,\ldots$ falls below a certain lower limit. The second scheme signals as out‐of‐control if two out of two values of $Y_i ,\ i = 1,2,\ldots$ fall below a certain lower limit. The efficiency of both of the control charts is evaluated by the average run length (ARL) of the chart and the power of the chart to detect a shift in the process. The two out of two scheme is shown to have high power and a large ARL given certain parameter values of the process. An example of the two out of two scheme is provided for the interested reader. Copyright © 2006 John Wiley & Sons, Ltd.