Journal of Probability and Statistics

Volume 2019, Article ID 5269357, 11 pages

https://doi.org/10.1155/2019/5269357

## Generalized Performance Measures of Control Charts Based on Different Sampling Schemes

^{1}Department of Mathematical Sciences, Universiti Teknologi Malaysia, Malaysia^{2}Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin, Saudi Arabia^{3}Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Correspondence should be addressed to Muhammed Hisyam Lee; ym.mtu@lhm

Received 17 February 2019; Accepted 2 May 2019; Published 4 June 2019

Academic Editor: Aera Thavaneswaran

Copyright © 2019 Rashid Mehmood et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 10^{5} 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 .