Introduction
The basis of variable sampling schemes is that if the drawn sample is
close enough to the centerline of the control chart, there is no reason
to think that there has been a shift in the process. In this state, the
next sample with smaller sample size and/or longer sampling interval
drawn on a control chart with more comprehensive control limits. On the
other hand, if the current drawn sample is close to the control limits,
it is likely that a shift has occurred in the process. Therefore, the
next sample with larger sample size and/or shorter sampling interval is
drawn on a control chart with tighter control limits to identify the
potential shift in the process.
The most common variable sampling schemes, which are also called
adaptive schemes, are VSS11Variable sample size, VSI22Variable
sampling intervals, and VSSI33Variable sample size and
sampling intervals, in which, sample size, sampling interval and
sample size with sampling interval are variable, respectively. The VSI
sampling scheme presented by Reynolds et al. [1] for the X control
chart, they evaluated the performance of this type of sampling scheme
from the statistical aspect. In their study, they conducted a
comprehensive review between the VSI sampling scheme and the traditional
sampling scheme or fix ratio sampling (FSSI) for different shifts in the
process mean. Comparison between standard deviation, coefficient of
variation, the average number of samples to signal, the average time to
signal indicated the acceptable performance of the VSI sampling scheme.
Saccucci et al. [2] also investigated the effect of using this
scheme compared with FSSI scheme on the performance of the exponentially
weighted moving average (EWMA) and the cumulative sum (CUSUM). Prabhu et
al. [3] and Costa [4] by presenting a VSS scheme and designing X
control chart with two sample sizes, conducted their studies on the
impact of this scheme on the measures and performance of the control
chart. Similarly, Castagliola et al. [5] by using two statistical
measures average sample sizes and truncated average run length,
evaluated the X control chart with the VSS sampling scheme. In more
practical applications, Nikolaidis et al. [6] concluded that the use
of variable control charts did not have much complexity compared than
traditional control charts, but better performance achieved from
statistically and economically aspects. Also, Lin and Chou [7]
evaluated the VSSI sampling scheme that presented by Prabhu et al.
[8] and analyzed the effect of using this sampling method on the X
control chart under normality and non-normality conditions. Zhou [9]
by considering estimated parameters, studied VSSI scheme for the X
control chart and evaluated average time to signal (ATS) in different
states. In general, the VSSI sampling scheme was more complex than the
VSS and VSI schemes, but it proved to be more efficient in many studies.
The researches expressed were just a few cases that examined the
statistical performance of sampling schemes for control charts. The
researches expressed were just a few cases that examined the statistical
performance of sampling schemes for control charts that more information
can be found on Khoo et al. [10], Costa and Machado [11], Cheng
et al. [12], Lim et al. [13] and Chong et al. [14].
In studies for the U control chart can be noted to Shojaie-Navokh et al.
[15] that their research to evaluate the economic-statistical U
control chart with variable sampling schemes. In their review by
considering the ANF and AATS as constraints and a cost function as the
objective function, developed the economic-statistical model and
evaluated and compared different sampling methods. Also, in their
research, the VSSI scheme was identified as the best scheme with the
lowest cost. However, since the main goal of providing variable sampling
methods was improvement the statistical measures, in this research, the
ANF, ANS, ANI, and AATS measures are reviewing to evaluate the
performance of VSSI scheme and other sampling schemes. In the next
section, the concept and approach of the Markov chain in designing
sampling schemes discussed. In the third section, calculating and
introducing statistical measures is explained. The fourth section
discussed the evaluation and comparison of statistical sampling schemes
under numerical examples and the results presented in the fifth section.