Standard practice in statistical process control is to run two individual charts, one for the process mean and another one for the process variance. The resulting scheme is known as a simultaneous scheme and it provides a way to satisfy Shewhart's dictum that proper process control implies monitoring both location and dispersion.

When we use a simultaneous scheme, the quality characteristic is deemed to be out-of-control whenever a signal is triggered by either individual chart. As a consequence, the misidentification of the parameter that has changed can occur, meaning that a shift in the process mean can be misinterpreted as a shift in the process variance and vice-versa. These two events are known as misleading signals (MS) and can occur quite frequently.

We discuss (necessary and) sufficient conditions to achieve values of PMS smaller than or equal to \(0.5\), explore, for instance, alternative simultaneous Shewhart-type schemes and check if they lead to PMS which are smaller than the ones of the popular \((\bar{X}, S^2)\) simultaneous scheme.