Strategies to Reduce the Probability of a Misleading Signal

Abstract

Standard practice in statistical process control (SPC) 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 probabilities of misleading signals (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.

Appendix

In this section we present the necessary results to compute the probabilities in the denominator and numerator of the PMS of types III and IV:

• $P_{\delta,\theta }\left (T^{(i)} \in [\mathrm{LCL}_{\mu }^{(i)},\mathrm{UCL}_{\mu }^{(i)}],\,U^{(i)} \in [0,\mathrm{UCL}_{\sigma }^{(i)}]\right )$ equals

  $$\displaystyle{\begin{array}{ll} \left [\varPhi \left (\frac{\gamma _{\mu }^{(1)}} {\theta } -\frac{\delta }{\theta }\right ) -\varPhi \left (-\frac{\gamma _{\mu }^{(1)}} {\theta } -\frac{\delta }{\theta }\right )\right ] \times F_{\chi _{(n-1)}^{2}}\left (\frac{\gamma _{\sigma }^{(1)}} {\theta ^{2}} \right ), &i = 1 \\ \int _{0}^{\frac{\gamma _{\sigma }^{(2)}} {\theta ^{2}} }\left [\varPhi \left ( \frac{\gamma _{\mu }^{(2)}} {\sqrt{n-1}}\sqrt{x} -\frac{\delta }{\theta }\right ) -\varPhi \left (- \frac{\gamma _{\mu }^{(2)}} {\sqrt{n-1}}\sqrt{x} -\frac{\delta }{\theta }\right )\right ] \times f_{\chi _{(n-1)}^{2}}(x)\,\mathit{dx},&i = 2 \\ \int _{ -\frac{\gamma _{\mu }^{(3)}} {\theta } -\frac{\delta }{\theta }}^{\frac{\gamma _{\mu }^{(3)}} {\theta } -\frac{\delta }{\theta } }\phi (z) \times F_{\chi _{(n)}^{2}}\left (\max \left \{0,\gamma _{\sigma }^{(3)}/\theta ^{2} - (z +\delta /\theta )^{2}\right \}\right )\,dz, &i = 3; \end{array} }$$

• $P_{\delta,\theta }\left (T^{(i)}\not\in [\mathrm{LCL}_{\mu }^{(i)},\mathrm{UCL}_{\mu }^{(i)}],\,U^{(i)} \in [0,\mathrm{UCL}_{\sigma }^{(i)}]\right )$ is given by

  $$\displaystyle{\begin{array}{ll} \left \{1 -\left [\varPhi \left (\frac{\gamma _{\mu }^{(1)}} {\theta } -\frac{\delta }{\theta }\right ) -\varPhi \left (-\frac{\gamma _{\mu }^{(1)}} {\theta } -\frac{\delta }{\theta }\right )\right ]\right \} \times F_{\chi _{(n-1)}^{2}}\left (\frac{\gamma _{\sigma }^{(1)}} {\theta ^{2}} \right ), &i = 1 \\ \int _{0}^{\frac{\gamma _{\sigma }^{(2)}} {\theta ^{2}} }\left \{1 -\left [\varPhi \left ( \frac{\gamma _{\mu }^{(2)}} {\sqrt{n-1}}\sqrt{x} -\frac{\delta }{\theta }\right ) -\varPhi \left (- \frac{\gamma _{\mu }^{(2)}} {\sqrt{n-1}}\sqrt{x} -\frac{\delta }{\theta }\right )\right ]\right \} \times f_{\chi _{(n-1)}^{2}}(x)\,\mathit{dx},&i = 2 \\ \int _{-\infty }^{-\frac{\gamma _{\mu }^{(3)}} {\theta } -\frac{\delta }{\theta } }\phi (z) \times F_{\chi _{(n-1)}^{2}}\left (\max \{0,\gamma _{\sigma }^{(3)}/\theta ^{2} - (z -\delta /\theta )^{2}\}\right )\,dz \\ +\int _{ \frac{\gamma _{\mu }^{(3)}} {\theta } -\frac{\delta }{\theta }}^{\infty }\phi (z) \times F_{\chi _{(n)}^{2}}\left (\max \left \{0,\gamma _{\sigma }^{(3)}/\theta ^{2} - (z +\delta /\theta )^{2}\right \}\right )\,dz, &i = 3; \end{array} }$$

• $P_{\delta,\theta }\left (T^{(i)} \in [\mathrm{LCL}_{\mu }^{(i)},\mathrm{UCL}_{\mu }^{(i)}],\,U^{(i)}\not\in [0,\mathrm{UCL}_{\sigma }^{(i)}]\right )$ is equal to

  $$\displaystyle{\begin{array}{ll} \left [\varPhi \left (\frac{\gamma _{\mu }^{(1)}} {\theta } -\frac{\delta }{\theta }\right ) -\varPhi \left (-\frac{\gamma _{\mu }^{(1)}} {\theta } -\frac{\delta }{\theta }\right )\right ] \times \left [1 - F_{\chi _{(n-1)}^{2}}\left (\frac{\gamma _{\sigma }^{(1)}} {\theta ^{2}} \right )\right ], &i = 1 \\ \int _{\frac{\gamma _{\sigma }^{(2)}} {\theta ^{2}} }^{\infty }\left [\varPhi \left ( \frac{\gamma _{\mu }^{(2)}} {\sqrt{n-1}}\sqrt{x} -\frac{\delta }{\theta }\right ) -\varPhi \left (- \frac{\gamma _{\mu }^{(2)}} {\sqrt{n-1}}\sqrt{x} -\frac{\delta }{\theta }\right )\right ] \times f_{\chi _{(n-1)}^{2}}(x)\,\mathit{dx},&i = 2 \\ \int _{ -\frac{\gamma _{\mu }^{(3)}} {\theta } -\frac{\delta }{\theta }}^{\frac{\gamma _{\mu }^{(3)}} {\theta } -\frac{\delta }{\theta } }\phi (z) \times \left [1 - F_{\chi _{(n)}^{2}}\left (\max \left \{0,\gamma _{\sigma }^{(3)}/\theta ^{2} - (z +\delta /\theta )^{2}\right \}\right )\right ]\,dz, &i = 3. \end{array} }$$

The derivation of $P_{\delta,\theta }(T^{(i)} \in [\mathrm{LCL}_{\mu }^{(i)},\mathrm{UCL}_{\mu }^{(i)}],\,U^{(i)} \in [0,\mathrm{UCL}_{\sigma }^{(i)}])$ follows closely Walsh (1952). The apparent differences are essentially due to the fact that Walsh (1952) represented the shift in the process mean (resp. standard deviation) by $a = \sqrt{n}(\mu _{0}-\mu )/\sigma$ (resp. $b =\sigma _{0}/\sigma$), i.e., $\delta = -a/b$ (resp. $\theta = 1/b$), and considered a chart for σ with a lower control limit.

The derivation of the remaining probabilities follows in a straightforward manner.