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)