A Study on EWMA charts with runs rules—the Markov chain approach

Quality & Reliability Eng

Malela‐Majika

Molahloe³

2019

In some industrial or health-related processes, it makes more practical sense to monitor either an increase only or a decrease only in the quality characteristic of interest. Consequently, in this paper, we propose four one-sided Shewhart X charts supplemented with runs-rules to monitor the mean of autocorrelated normally distributed samples using a stationary first-order autoregressive model. To counteract the negative effect of autocorrelation, we implement a sampling strategy which involves sampling of non-neighboring observations to form rational subgroups. The Markov chain technique is used to derive zero-state and steady-state closed-form expressions of the specific shift performance metric, ie, average run-length (ARL). Moreover, we compute the expected ARL metric which evaluates each monitoring scheme based on all specified range of possible values of the shift parameter, or more specifically, from a global point of view and thus gives a different perspective from the specific shift ARL metric. We observed that the steady-state improved w-of-w and the improved 2-of-(H + 1) schemes yield a better overall performance than their corresponding basic counterparts for all different levels of autocorrelation. A real-life example is provided to illustrate the implementation of the monitoring schemes proposed here.

Section: Introductionmentioning

confidence: 99%

One‐sided runs‐rules schemes to monitor autocorrelated time series data using a first‐order autoregressive model with skip sampling strategies

Quality & Reliability Eng

Malela‐Majika

Molahloe³

2019

“…whereas in steady-state H * ≥ 6, see Table 8. Furthermore, in Table 9 it can be seen that the values of the relative effectiveness of the MSS scheme computed using Equation (14) and the EQL values in Table 7 yield PCI values that are greater than 1 for all the competing schemes listed in Table 1 (except when H=1, see Remark 1).…”

Section: Remarkmentioning

confidence: 99%

A modified side-sensitive synthetic chart to monitor the process mean

Quality Technology & Quantitative Management

Graham

2016

In this paper, we propose a modified side-sensitive (MSS) synthetic ̅ chart which signals only if all the consecutive plotting statistics that lead to an out-of-control event fall on one side of the center line; unlike the non-side-sensitive, standard and revised side-sensitive synthetic ̅ charts that also signal even when some of the plotting statistics fall on opposite sides of the center line. Moreover, we use the Markov chain imbedding technique to study and compare the zero-state and steady-state average run-length (ARL), extra quadratic loss, average ratio of the ARLs and performance comparison index of the proposed MSS chart with other Shewhart-type synthetic and runs-rules charts. The synthetic ̅ chart with this MSS feature has a better overall zero-state and steady-state performance than the existing synthetic ̅ charts and hence makes it a strong contender in many applications where existing synthetic ̅ charts are currently used.

“…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][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

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

Journal of Probability and Statistics

Malela‐Majika

Rapoo

2019

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 X- 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.