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.