Economic design of two-stage X̄ charts: The Markov chain approach

When the X̄ chart is in use, samples are regularly taken from the process, and their means are plotted on the chart. In some cases, it is too expensive to obtain the X values, but not the values of a correlated variable Y. This paper presents a model for the economic design of a two-stage control chart, that is, a control chart based on both performance (X) and surrogate (Y) variables. The process is monitored by the surrogate variable until it signals an out-of-control behavior, and then a switch is made to the X̄ chart. The X̄ chart is built with central, warning, and action regions. If an X sample mean falls in the central region, the process surveillance returns to the Ȳ chart. Otherwise, the process remains under the X̄ chart's surveillance until an X sample mean falls outside the control limits. The search for an assignable cause is undertaken when the performance variable signals an out-of-control behavior. In this way, the two variables are used in an alternating fashion. The assumption of an exponential distribution to describe the length of time the process remains in control allows the application of the Markov chain approach for developing the cost function. A study is performed to examine the economic advantages of using performance and surrogate variables.