Weighted-loss-function control charts

Abstract

This article proposes a single weighted-loss-function chart (WL chart) for monitoring the process mean and variance simultaneously in statistical process control (SPC). The weights of the losses due to mean shift and variance shift in the loss function are balanced through a weighting factor λ, so that the WL chart is considerably more effective than the unadjusted loss function chart, the joint $\overline{X}$&S charts and many other charts as well. The improvement in detection effectiveness will directly lead to the reduction of a number of defects in the manufacturing process when out-of-control cases occur. Moreover, the WL chart provides a platform for the further development of the Cusum chart, runs rule, or adaptive chart for monitoring both process mean and variance.

Calculation of ARL of a WL chart

The statistic WL (Eq. 6) is greater than Γ when

$$\overline{x} <\mu _{0} - a,\,{\text{or}}\,\overline{x} >\mu _{0} + a,$$

(14)

or

$$\mu _{0} - a < \overline{x} < \mu _{0} + a\,\,{\text{and}}\,s^{2} > \frac{{\Gamma - {\left( {1 - \lambda } \right)}{\left( {\overline{x} - \mu _{0} } \right)}^{2} }}{\lambda }$$

(15)

where,

$$a = {\sqrt {\frac{\Gamma } {{1 - \lambda }}} }{\text{.}}$$

(16)

Thus, the probability that WL is greater than Γ can be calculated by conditioning on the value of the sample mean $\overline{x} .$.

$$\Pr {\left( {WL >\Gamma } \right)} = P_{1} + P_{2} + P_{3} {\text{.}}$$

(17)

$$P_{1} = \Pr {\left( {\overline{x} <\mu _{0} - a} \right)} = \Phi {\left[ {\frac{{ - {\sqrt n }{\left( {a + \delta _{\mu } \sigma _{0} } \right)}}} {{\delta _{\sigma } \sigma _{0} }}} \right]}{\text{.}}$$

(18)

$$P_{2} = \Pr {\left( {\overline{x} >\mu _{0} + a} \right)} = 1 - \Phi {\left[ {\frac{{{\sqrt n }{\left( {a - \delta _{\mu } \sigma _{0} } \right)}}} {{\delta _{\sigma } \sigma _{0} }}} \right]}{\text{.}}$$

(19)

$$P_{3} = {\int\limits_{\mu _{0} - a}^{\mu _{0} + a} {p{\left( {\overline{x} } \right)}} }\,f{\left( {\overline{x} } \right)}d\overline{x} {\text{.}}$$

(20)

where, $f{\left( {\overline{x} } \right)}$ is the density probability function of $\overline{x}$ that follows a normal distribution with the mean equal to ${\left( {\mu _{0} + \delta _{\mu } \sigma _{0} } \right)}$ and the variance equal to ${\left( {\delta _{\sigma } \sigma _{0} } \right)}^{2} /n;$ and $p{\left( {\overline{x} } \right)}$ is the probability that WL is greater than Γ when the value of $\overline{x}$ is given within the range between (μ₀–a) and (μ₀+a). From Eq. (15)

$$p{\left( {\overline{x} } \right)} = \Pr {\left( {s^{2} >\frac{{\Gamma - {\left( {1 - \lambda } \right)}{\left( {\overline{x} - \mu _{0} } \right)}^{2} }} {\lambda }} \right)} = \Pr {\left( {Q > b{\left( {\overline{x} } \right)}} \right)} = 1 - \chi ^{2}_{{n - 1}} {\left( {b{\left( {\overline{x} } \right)}} \right)}{\text{.}}$$

(21)

where, the random variable Q follows a Chi-square distribution with degrees of freedom of (n–1) and

$$b{\left( {\overline{x} } \right)} = \frac{{{\left( {n - 1} \right)}{\left[ {\Gamma - {\left( {1 - \lambda } \right)}{\left( {\overline{x} - \mu _{0} } \right)}^{2} } \right]}}}{{\lambda {\left( {\delta _{\sigma } \sigma _{0} } \right)}^{2} }}{\text{.}}$$

(22)

Finally, the ARL of the WL chart is calculated by

$$ARL = 1 \mathord{\left/ {\vphantom {1 {\Pr {\left( {WL > UCL_{{WL}} } \right)}}}} \right. \kern-\nulldelimiterspace} {\Pr {\left( {WL > UCL_{{WL}} } \right)}}$$

(23)

The weighted loss function WL can also be expressed as

$$\begin{array}{*{20}c} {WL = \sigma ^{2}_{0} w{\ell }} \\ {w{\ell } = \lambda {\left( {\frac{s}{{\sigma _{0} }}} \right)}^{2} + {\left( {1 - \lambda } \right)}{\left( {\frac{{\overline{x} - \mu _{0} }}{{\sigma _{0} }}} \right)}^{2} } \\ \end{array} $$

(24)

where, $w{\ell }$ is a special case of WL under the standard condition (μ ₀ =0, σ ₀ =1). Recall Eqs. 13 and 23,

$$\begin{array}{*{20}c} {ARL = 1 \mathord{\left/ {\vphantom {1 {\Pr {\left( {WL > UCL_{{WL}} } \right)}}}} \right. \kern-\nulldelimiterspace} {\Pr {\left( {WL > UCL_{{WL}} } \right)}}} \\{ = 1 \mathord{\left/ {\vphantom {1 {\Pr {\left( {\sigma ^{2}_{0} w{\ell } >\sigma ^{2}_{0} {\left[ {UCL_{{WL}} } \right]}_{{{\text{standard}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {\Pr {\left( {\sigma ^{2}_{0} w{\ell } >\sigma ^{2}_{0} {\left[ {UCL_{{WL}} } \right]}_{{{\text{standard}}}} } \right)}}} \\{ = 1 \mathord{\left/ {\vphantom {1 {\Pr {\left( {w{\ell } > {\left[ {UCL_{{WL}} } \right]}_{{{\text{standard}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {\Pr {\left( {w{\ell } > {\left[ {UCL_{{WL}} } \right]}_{{{\text{standard}}}} } \right)}}} \\\end{array} $$

(25)

It indicates that, for a given set of values of $\delta _{\mu }$ and $\delta _{\sigma }$, a general-condition WL chart with a control limit UCL_WL given by Eq. 13 produces the same ARL as a standard-condition WL chart with (UCL_WL)_standard.