Figure 2 A quality cycle
The average time from the start of the production until the signal by
control chart after the process shift is equal to the ATC. This measure
can be calculated based on the properties of exponential distribution
and Markov chain approach. The expected number of experiments at each
stage obtained from the following equation:
\(b\left(I-Q\right)^{-1}\) (3)
Where Q is a square matrix by deleting the elements corresponding to the
absorbing state of the transition probability matrix P, I is the
identity matrix, and b is the initial probabilities vector. Therefore,
ATC can be calculated as follows:
\(ATC=b\left(I-Q\right)^{-1}h\) (4)
Where h is the vector of interval sampling vector of different process
states. Also, assumed that the time before assignable cause occurs is
the exponential distribution with parameter λ. Therefore, the average
time that the process remains in-control state is λ-1.
The mean time from the occurrence of an assignable cause to the time
when the control chart detects an out-of-control signal, evaluated by
the AATS measure, which equals:
\(AATS=ATC-\lambda^{-1}\) (5)
AATS is the newest measure used to compare the effectiveness of
different sampling schemes. AATS shows the control chart sensitivity in
detecting shifts in the process. So, with the smaller AATS, the
performance of the control chart is better.
The use of variable control charts, moreover reduced AATS, is able to
reduce ANS and ANI. The ANS and ANI of different variable control charts
are less or equal in comparison with the FSSI sampling scheme. The
reduction in measures is also indicated the efficiency of control chart
and will have an impact on costs. The values of ANI and ANS for sampling
schemes are obtained as follows:
\(ANI=b\left(I-Q\right)^{-1}n\) (6)
\(ANS=b\left(I-Q\right)^{-1}s\) (7)
Where, n and s are vectors of the number of inspected items and the
number of samples, respectively. In designing control charts with
variable sampling methods because more focused on AATS, ANS, and ANI,
increasing the rate of false alarms is a possibility. Therefore,
researchers always have been suspicious to use of these types of control
charts. For this purpose, in this paper, we have also compared sampling
schemes based on the ANF. Reduction in ANF, as expressed in Fallahnezhad
et al. [20], Amiri et al. [21], Faraz and Saniga [22] and
Katebi et al. [23] improves the performance of the control chart.
The formula for calculating this measure with the vector of false alarms
f is similar to other measures based on the Markov chain concepts and is
as follows:
\(ANF=b\left(I-Q\right)^{-1}f\) (8)