Economic statistical design of adaptive $\bar{X}$ control charts based on quality loss functions

Abstract

An economic-statistical model is developed for variable parameters $(VP){\bar{X}}$ control chart is developed in which the constraints are imposed on the expected times to signal when the process is out of control and the expected number of false alarms per cycle. To improve chart effectiveness, the cost function is extended based upon the Taguchi philosophy of social loss of quality using the common loss functions including the linear, quadratic, exponential, and Linex. All of the loss functions that have been used up so far are symmetric, but what actually happens is that the imposed costs are not the same for over estimation and under estimation of the target value. For this reason, in this paper for the first time in the literature we use the Linex asymmetric loss functions, that results in the least average cost in all adaptive designs, compared to other symmetric loss functions. Using numerical example, we compare the performances of the $VP{\bar{X}}$ control charts with the others adaptive charts based on the loss function, and investigate the sensitivity of the chart parameters to changes in process parameters and loss functions. Results indicate a satisfactory performance for the proposed models.

Appendices

Appendix 1: Calculate $J_{0}$ and $J_{1}$ under linear loss function

Lemma 1

Let $Z \sim N\left( {0,1} \right)$, then

$$E\left[ {\left| {Z - T} \right|} \right] = 2\left[ {\emptyset \left( T \right) + T\varPhi \left( T \right)} \right] - T,$$

where $\emptyset \left( . \right)$ is the standard normal pdf and $\varPhi \left( . \right)$ is the standard normal cdf and $T$ is a constant.

Proof

$$\begin{aligned} E\left[ {\left| {Z - T} \right|} \right] & = \mathop \int \limits_{ - \infty }^{T} \left( {T - z} \right)\emptyset \left( z \right)dz + \mathop \int \limits_{T}^{\infty } \left( {z - T} \right)\emptyset \left( z \right)dz, \\ & = 2T \varPhi \left( T \right) - T - \mathop \int \limits_{ - \infty }^{T} z \times \left( {\frac{1}{{\sqrt {2\pi } }}} \right){ \exp }\left(\frac{{ - z^{2} }}{2}\right) dz + \mathop \int \limits_{T}^{\infty } z \times \left( {\frac{1}{{\sqrt {2\pi } }}} \right){ \exp }\left(\frac{{ - z^{2} }}{2}\right) dz, \\ & = 2\left[ {\emptyset \left( T \right) + T\varPhi (T} \right)] - T. \\ \end{aligned}$$

For a normal random variable with mean $\mu_{0}$ and standard deviation $\sigma_{0}$, we have

$$\begin{aligned} J_{0} & = E\left[ {K\left| {X - T} \right|} \right] = K\sigma_{0} E\left[ {\left| {Z - \frac{{T - \mu_{0} }}{{\sigma_{0} }}} \right|} \right] \\ & = 2K\left[ {\left( {T - \mu_{0} } \right)\varPhi \left( {\frac{{T - \mu_{0} }}{{\sigma_{0} }}} \right) + \sigma_{0} \emptyset \left( {\frac{{T - \mu_{0} }}{{\sigma_{0} }}} \right)} \right] - K\left( {T - \mu_{0} } \right). \\ \end{aligned}$$

Similarly, for a normal random variable with mean $\mu_{1}$ and standard deviation $\sigma_{0}$, we have

$$J_{1} = K\sigma_{0} E\left[ {\left| {Z - \frac{{T - \mu_{1} }}{{\sigma_{0} }}} \right|} \right] = 2K\left[ {\left( {T - \mu_{0} } \right)\varPhi \left( {\frac{{T - \mu_{1} }}{{\sigma_{0} }}} \right) + \sigma_{0} \emptyset \left( {\frac{{T - \mu_{1} }}{{\sigma_{0} }}} \right)} \right] - K\left( {T - \mu_{1} } \right).$$

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Appendix 2: Calculate $J_{0}$ and $J_{1}$ under exponential loss function

Lemma 2

Let $Z \sim N\left( {0,1} \right),$ Then

$$E\left( {e^{{r\left| {Z - T} \right|}} } \right) = \exp \left( {\frac{{r^{2} }}{2}} \right)\left[ {e^{rT} \varPhi \left( {T + r} \right) + e^{ - rT} \left( {1 - \varPhi \left( {T - r} \right)} \right)} \right],$$

where $r > 0$ and $T$ are constants.

Proof

$$\begin{aligned} E\left( {e^{{r\left| {Z - T} \right|}} } \right) & = \mathop \int \limits_{ - \infty }^{T} e^{{r\left( {T - z} \right)}} \emptyset \left( z \right)dz + \mathop \int \limits_{T}^{\infty } e^{{r\left( {z - T} \right)}} \emptyset \left( z \right)dz, \\ & = e^{rT} \mathop \int \limits_{ - \infty }^{T} e^{ - rz} \times \frac{1}{{\sqrt {2\pi } }}\exp \left( {\frac{{ - z^{2} }}{2}} \right)dz + e^{ - rT} \mathop \int \limits_{T}^{\infty } e^{ + rz} \times \frac{1}{{\sqrt {2\pi } }}\exp \left( {\frac{{ - z^{2} }}{2}} \right)dz, \\ & = \exp \left( {\frac{{r^{2} }}{2}} \right)\left[ {e^{rT} \varPhi \left( {T + r} \right) + e^{ - rT} \left( {1 - \varPhi \left( {T - r} \right)} \right)} \right] \\ \end{aligned}$$

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For a normal random variable with mean $\mu_{0}$ and standard deviation $\sigma_{0}$, we have

$$\begin{aligned} J_{0} & = KE\left[ {K\left( {e^{{r\left| {X - T} \right|}} - 1} \right)} \right] = KE\left[ {e^{{r\sigma_{0} \left| {Z - \frac{{T - \mu_{0} }}{{\sigma_{0} }}} \right|}} - 1} \right] \\ & = K\exp \left( {\frac{{\left( {r\sigma_{0} } \right)^{2} }}{2}} \right)\left[ {e^{{r\sigma_{0} b_{0} }} \varPhi \left( {b_{0} + r\sigma_{0} } \right) + e^{{ - r\sigma_{0} b_{0} }} \left( {1 - \varPhi \left( {b_{0} - r\sigma_{0} } \right)} \right)} \right] - K, \\ & = K\exp \left( {\frac{{\left( {r\sigma_{0} } \right)^{2} }}{2}} \right)[\exp \left( {r\left( {T - \mu_{0} } \right)} \right){{\varPhi }}\left( {b_{0} + r\sigma_{0} } \right) - K \\ & \quad + \exp \left( {r\left( {\mu_{0} - T} \right)} \right)\left( {1 - {{\varPhi }}\left( {b_{0} - r\sigma_{0} } \right)} \right)] - K, \\ \end{aligned}$$

where $b_{0} = \frac{{T - \mu_{0} }}{{\sigma_{0} }}$.

Similarly, for a normal random variable with mean $\mu_{1}$ and standard deviation $\sigma_{0}$, we have

$$\begin{aligned} J_{1} & = K\exp \left( {\frac{{\left( {r\sigma_{0} } \right)^{2} }}{2}} \right) \times [{ \exp }\left( {r\left( {T - \mu_{1} } \right)} \right){{\varPhi }}\left( {b_{1} + r\sigma_{0} } \right) \\ & \quad + \exp \left( {r\left( {\mu_{1} - T} \right)} \right)\left( {1 - {{\varPhi }}\left( {b_{1} - r\sigma_{0} } \right)} \right)] - K, \\ \end{aligned}$$

where $b_{1} = \frac{{T - \mu_{1} }}{{\sigma_{0} }}$.

Appendix 3: Calculate $J_{0}$ and $J_{1}$ under Linex loss function

Lemma 3

Let $Z \sim N\left( {0,1} \right)$, Then

$$E\left[ {\exp \left( {v\left( {Z - T} \right)} \right) - v\left( {Z - T} \right) - 1} \right] = \exp \left( { - vT + \frac{{v^{2} }}{2}} \right) + vT - 1,$$

where $v$ and $T$ are constants.

Proof

$$\begin{aligned} E\left[ {\exp \left( {v\left( {Z - T} \right)} \right) - v\left( {Z - T} \right) - 1} \right] & = \exp \left( { - vT} \right)E\left( {e^{Zv} } \right) - vE\left( Z \right) + vT - 1, \\ & = \exp \left( { - vT} \right)M_{Z} \left( v \right) + vT - 1 \\ & = \exp \left( { - vT + \frac{{v^{2} }}{2}} \right) + vT - 1. \\ \end{aligned}$$

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For a normal random variable with mean $\mu_{0}$ and standard deviation $\sigma_{0}$, we have

$$\begin{aligned} J_{0} & = K\varphi E\left[ {\exp \left( {v\left( {X - T} \right)} \right) - v\left( {X - T} \right) - 1} \right] \\ & = K\varphi E\left[ {\exp \left( {v\left( {\sigma_{0} Z + \mu_{0} - T} \right)} \right) - v\left( {\sigma_{0} Z + \mu_{0} - T} \right) - 1} \right], \\ & = K\varphi E\left[ {\exp \left( {v\sigma_{0} \left( {Z - b_{0} } \right)} \right) - v\sigma_{0} \left( {Z - b_{0} } \right) - 1} \right], \\ & = K\varphi \left[ {\exp \left( { - v\sigma_{0} b_{0} + \frac{{\left( {v\sigma_{0} } \right)^{2} }}{2}} \right) + v\sigma_{0} b_{0} - 1} \right], \\ & = K\varphi \left[ {\exp \left( { - v\left( {T - \mu_{0} } \right) + \frac{{\left( {v\sigma_{0} } \right)^{2} }}{2}} \right) + v\left( {T - \mu_{0} } \right) - 1} \right], \\ \end{aligned}$$

where $b_{0} = \frac{{T - \mu_{0} }}{{\sigma_{0} }}$.

Similarly, for a normal random variable with mean $\mu_{1}$ and standard deviation $\sigma_{0}$, we have

$$J_{1} = K\varphi \left[ {\exp \left( { - v\left( {T - \mu_{1} } \right) + \frac{{\left( {v\sigma_{0} } \right)^{2} }}{2}} \right) + v\left( {T - \mu_{1} } \right) - 1} \right].$$

Appendix 4: Summary of the notations and abbreviations

Notation	Definition
$X$	Quality characteristic
$\mu_{0}$	In-control mean
$\sigma_{0}$	In-control standard deviation
$\delta$	Magnitude of the shift in the process mean due to assignable cause
$\mu_{1}$	Mean of the quality characteristic when the assignable cause is present ($\mu_{1} = \mu_{0} + \delta \sigma_{0}$)
λ	Occurrence rate of the assignable cause
$T_{0}$	Time needed to detect a false alarm
$T_{1}$	Time to search and remove assignable cause
$a$	The cost associated with locating and repairing the process
$\mathop a^{\prime }$	The cost of false alarms
$S$	The sampling cost
$C_{0}$	The expected cost associated with producing while the process is in control
$C_{1}$	The expected cost associated with producing while the process is out of control
$AATS$	Adjusted average time to signal
$ANF$	The expected number of false alarms per cycle
$ANI$	The average number of inspected items per cycle
$ATC$	The average time of the cycle
$E(C)$	The expected cost per cycle
$E(T)$	The expected length of a production cycle
$E(A)$	The expected cost per time unit
$FRS$	The fixe ratio sampling scheme
$VSI$	The variable sampling intervals scheme
$VSS$	The variable sample size scheme
$VSSI$	The variable sample size and sampling intervals scheme
$VP$	The variable parameters scheme