Abstract
Different versions of control chart structure are available under various ranked set strategies. In these control charts, computation of performance measures was carried out through Monte Carlo simulation method (MCSM). In this article, we have defined a generalized structure of control charts under variant sampling strategies followed by derivation of their different performance measures. For the derivation of different performance measures, we have proposed pivotal quantity. For comparative analysis, we have presented results of generalized performance measures by involving numerical method (NM) as computation. We found that values of generalized performance measures based on NM are almost similar to values of performance measures based on MCSM. Also, NM is time efficient and can be considered as an alternative of MCSM.
1. Introduction
Shewhart (1931) proposed control chart under simple random sampling for monitoring the variations (natural or unnatural) in a process’s parameters. Afterwards, control charts became popular among researchers and practitioners of various fields. Therefore, different researchers and practitioners constructed improved forms of control charts in order to cope with different situations, for instance, when we have extra information (also termed as auxiliary information) associated with the variable of interest. Then, for such situation, a number of authors proposed and recommended auxiliary information based control charts (see [1–10]).
Salazar & Sinha [1] attempted Shewhart control chart based on ranked set sampling (RSS) for monitoring of location parameter. They utilized the auxiliary information for ranking the units of interest. Muttlak & Al-Sabah [2] extended the idea of Salazar and Sinha [1] by offering control charts under median ranked set sampling (MRSS) and extreme ranked set sampling (ERSS). Abujiya & Muttlak [3] contributed control chart under double ranked set sampling. All of the illustrated studies were designed for Shewhart control chart based on one point decision rule, that is, if any plotting statistic goes outside the control limits. Later on, Mehmood et al. [4] enhanced the performance of ranked set sampling based control charts for the detection of small-to-moderate shifts by incorporating more runs rules with the existing structures. It is worth mentioning that the common methodology used in the prescribed studies was the computation of performance measures through MCSM. One may think of NM such as quadratic and cubic polynomial (see [11–13]) as an alternate methodology for computing the values of performance measures. The most commonly used performance measures dealing with quality control charts are false alarm rate, power, and average run length (see [14]).
This study is projected to define a generalized design structure of control chart under different ranked set strategies along with derivation of generalized performance measures. For the said purpose, we will propose constant and pivotal quantity. It is important to mention that we are intended to show the significance of NM in terms of producing almost similar results as one can expect through MCSM. This practice may highlight more scope of NM to researchers especially those who have been working in the direction of quality control charts under variant ranked set schemes. Another purpose of introducing the NM in the place of MCSM is to prove it as time efficient methodology.
The Rest of the article is organized as follows: In Section 2 we will define generalized design structure of control chart under different ranked set strategies. Based on Section 2, we will propose pivotal quantity in Section 3. In Section 4, we will define performance measures under proposed constant and pivotal quantity. Section 5 will contain computation of performance measures under NM as well as MCSM, and results and discussion. Lastly, the study will end up with summary and conclusion.
2. Design Structure
In this section we define a generalized design structure of control chart under different ranked set strategies. The sampling strategies include simple random sampling (SRS), ranked set sampling (RSS), median ranked set sampling (MRSS), extreme ranked set sampling (ERSS), double ranked set sampling (DRSS), double median ranked set sampling (DMRSS), median double ranked set sampling (MDRSS), double extreme ranked set sampling (DERSS), and extreme double ranked set sampling (EDRSS). Now we define the structure as follows.
Let , and , denote the observation of sample collected under a sampling strategy as mentioned above with in-control known mean and standard deviation . The details of these sampling strategies can be seen in the article of Mehmood et al. [4]. Thus, generalized two sided control chart structure for monitoring the statistic is given below: where represents the sampling strategy (as discussed earlier); indicates the sample size; is the amount of prefix false alarm rate of a process (see [14]); is the amount of correlation between study variable and concomitant variable which can attain different values, i.e., when case of perfect sampling strategy is considered and when case of imperfect sampling strategy is involved (see [4]); , are and quantiles of standard normal distribution, respectively. Furthermore, is a constant and given below: where is the variance of order statistic in a sample of size from standard normal distribution under a ranked set scheme .
2.1. Construction of Constant
A brief procedure is illustrated for the construction of constant by considering the study of Mehmood et al. [15] as guidance. For readers concern, a brief procedure is illustrated as follows: generate 105 random samples of size from standard normal distribution for the given choice of , , and followed by calculating the statistic () and standard deviation of statistic . The resulting values of are given in Table 1 at different choices of and for some choices of and . Likewise, one may approach other choices of and .
3. Proposed Pivotal Quantity
Before defining the performance measures (see [14]) for structure (1), we have defined a pivotal quantity as follows: where is standard normal variable. The theoretical justification for the relationship between and has been shown through Gaussian Kernel Density Plot (see Figures 1–6). The details of Gaussian Kernel Density Plot can be seen in the article of Sheether [16]. In short, Figures 1–6 showed that for a given choice of , , and the Gaussian Kernel Density Plots of standardized random variable (denoted as and random variable almost overlap with each other. This shows that probability distribution of random variable can be approximated by the random variable.
4. Derivation of Generalized Performance Measures
In this section we define performance measures (see [14]) for structure (1). The performance measures include false alarm rate, power, and average run length.
4.1. False Alarm Rate
False alarm rate (symbolically denoted as ) is a popular measure used for computing the probability that sample statistic goes outside the control limits (LCL, UCL) when actually the process is in-control. An expression for representing the definition of is given below: and (5) can be written based on (3):where is the area between two quantiles points to ) of standard normal curve (see [14]). Similarly is the area between two quantiles points ( to ).
4.2. Power
Power is defined as the probability that sample statistic goes outside the control limits (LCL, UCL) when actually a process is out-of-control. The out-of-control process implies that the in-control mean is shifted to new level , where denotes the amount of shift. An expression for representing the power is as follows:
4.3. Average Run Length
Average run length is defined as the average number of sample points that must be plotted on a control chart before an out-of-control signal is triggered. Moreover, average run length can be classified as in-control average run length and out-of-control average run length . is defined as the average number of sample points of an in-control process that must be plotted on the control chart before a plotting statistic breaches the control limits (LCL, UCL), whereas is defined as the average number of sample points of an out-of-control process that must be plotted on control chart before a plotting statistic breaches the control limits (LCL, UCL). For computing the and , one can use the results of (11).
5. Computation of Performance Measures and Discussion
In this section we have considered (8), (11), (12), and (13) to compute the false alarm rate, power, in-control average run length, and out-of-control average run length. For the said purpose, we have considered a prefix value of false alarm rate , various values of shifts (, and ), sampling strategies , sample size , and . Afterwards, for each choice of , , and we have solved (6), (9), (11), and (12) by using quadratic integration as numerical method (NM), and results are reported in Table 2. The details of quadratic and cubic polynomial integration can be seen in various articles (see [11–13]). Also, we have computed results of structure (1) through MCSM by taking into account all procedural details from the study of Mehmood et al. [4]. The number of simulations for implementing the MCSM is motivated from several existing studies such as Mehmood et al. [4] and Mehmood et al. [17]. Besides, we have recorded the time (see Table 3) utilized by both methods for obtaining the power . The time required for getting the is dependent on the choice of sampling strategy , sample size , and number of simulations . A method can prove to be time efficient if it consumes minimum time as compared to competitors given that all methods produced almost similar results.(i)Table 2 shows that results obtained through both methods (MCSM and NM) are almost close to each other and particularly at large shifts results are similar. The following findings remain true for any choice of , , , and .(ii)Behavior of different sampling strategies remains consistent under the both methods as reported and discussed in the study of Mehmood et al. [4].(iii)NM has proved to be time efficient which reduced computation burden as compared to MCSM (see Table 3). In more detail, NM has consumed minutes to gain the intended results. In contrast, MCSM has utilized more time, depending on the choice of sampling strategy , sample size , and number of simulations .(iv)Structure (1) and equations (8), (11), (12), and (13) represent a generalized form of several studies such as Shewhart (1931) and Montgomery [14] when ; Salazar & Sinha [1] when ; Muttlak & Alsabah [2] when ; Abujiya & Muttlak [3] when and for varying choices of sampling strategy . Also, structure (1) represents the study of Mehmood et al. [4] under the one point decision rule.
6. Summary and Conclusions
In this study we have defined a generalized design structure of control charts and derived their different performance measures. In this regard, we have proposed constant and pivotal quantity. Afterwards, we have computed values of different generalized performance measures based on numerical method (NM). In conclusion, we have observed that NM has significant scope in terms of producing desired result as one can expect based on MCSM. Also, NM method may be considered as a time efficient methodology for handling performance measures of control chart under ranked set strategies.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.