Journal of Probability and Statistics

Volume 2019, Article ID 6187060, 20 pages

https://doi.org/10.1155/2019/6187060

## One-Sided and Two-Sided* w*-*of*-*w* Runs-Rules Schemes: An Overall Performance Perspective and the Unified Run-Length Derivations

Department of Statistics, College of Science, Engineering and Technology, University of South Africa, South Africa

Correspondence should be addressed to S. C. Shongwe; az.oc.skut@elidnas

Received 3 December 2018; Accepted 27 December 2018; Published 19 February 2019

Academic Editor: Aera Thavaneswaran

Copyright © 2019 S. C. Shongwe 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

The one-sided and two-sided Shewhart* w-of-w* standard and improved runs-rules monitoring schemes to monitor the mean of normally distributed observations from independent and identically distributed (iid) samples are investigated from an overall performance perspective, i.e., the expected weighted run-length (*EWRL*), for every possible positive integer value of* w*. The main objective of this work is to use the Markov chain methodology to formulate a theoretical* unified approach* of designing and evaluating Shewhart* w-of-w* standard and improved runs-rules for one-sided and two-sided schemes in both the zero-state and steady-state modes. Consequently, the main findings of this paper are as follows: (i) the zero-state and steady-state* ARL* and initial probability vectors of some of the one-sided and two-sided Shewhart* w-of-w* standard and improved runs-rules schemes are theoretically similar in design; however, their empirical performances are different and (ii) unlike previous studies that use* ARL* only, we base our recommendations using the zero-state and steady-state* EWRL* metrics and we observe that the steady-state improved runs-rules schemes tend to yield better performance than the other considered competing schemes, separately, for one-sided and two-sided schemes. Finally, the zero-state and steady-state unified approach run-length equations derived here can easily be used to evaluate other monitoring schemes based on a variety of parametric and nonparametric distributions.

#### 1. Introduction

Balakrishnan and Koutras [1] define a* run* as an uninterrupted sequence of the same elements bordered at each end by other types of elements. Supplementary runs-rules have been used since the 1950s to improve the performance of the basic Shewhart control charts; see some detailed discussions of some of these earlier works in [2–6]. Some of the commonly cited and recent research works on runs-rules are done in [7–21]. For a literature review on the parametric runs-rules charts that cover articles up to 2006, a reader is referred to Koutras et al. [22], whereas, for the full discussion of nonparametric runs-rules charts until 2017, see the book by Chakraborti and Graham [23]. While runs-rules have been mostly applied in statistical process control and monitoring to improve the detection rate of the basic Shewhart charts, more recently, these have also been used to further increase the detection rate of the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) schemes; see [24–28].

To differentiate between common and special causes of variation, control charts are the most used tools of statistical process control and monitoring to achieve this goal. That is, when a process has only common causes of variation present, a control chart will indicate that the process is in statistical control, or in short, in-control (IC); however, when a process has special causes of variation present, it is said to be in a state of out-of-control (OOC). Assume that is a sequence of samples from iid distribution. Let denote the plotting statistic calculated from at sampling point . In a production process, say, samples are usually taken at each sampling point to be inspected and then each of these samples is classified as either conforming or nonconforming depending on where the sample plots on the control charting regions are shown in Figure 1.