On ARL-Unbiased Control Charts

Abstract

Manufacturing processes are usually monitored by making use of control charts for variables or attributes. Controlling both increases and decreases in a parameter, by using a control statistic with an asymmetrical distribution, frequently leads to an ARL-biased chart, in the sense that some out-of-control average run length (ARL) values are larger than the in-control ARL, i.e., it takes longer to detect some shifts in the parameter than to trigger a false alarm. In this paper, we are going to:

Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables

“…In this case, control limits of the two‐sided ARL‐unbiased EWMA‐RZ control chart L C L un and U C L un are chosen in such way that the ARL takes its maximum value at τ =1 with in‐control A R L = A R L 0 . The related algorithms are taken from Knoth and Morais and suitably modified.…”

“…The transition probabilities are calculated as follows: To obtain a certain target in‐control A R L , we determine the specific values for the first 2 designs, symmetric and equal‐tailed probability limits, by deploying the aforementioned numerical A R L routines in procedures such as bisection or secant method. To calculate the limits of the ARL‐unbiased EWMA‐RZ chart, we modify an algorithm given in the appendix of Knoth and Morais . This algorithm is detailed here in the Appendix.…”

“…When we use control charts for monitoring process characteristics with an asymmetric distribution to detect changes in both directions (two‐sided case), they might possess an biased A R L profile. In the sense, some out‐of‐control A R L values are larger than the in‐control ARL, ie, it takes longer to detect some shifts in the parameter than to trigger a false alarm, see Knoth and Morais . To overcome this problem, ARL‐unbiased control charts have been proposed in the literature, for further details, see, for instance, Zhang et al, Guo et al, Knoth and Morais, Yang and Arnold .…”

Design and implementation issues for a class of distribution-free Phase II EWMA exceedance control charts

“…The use of symmetrically placed control limits, for a two-sided chart for monitoring both decreasing and increasing shifts, with a plotting statistic having an asymmetrical distribution may lead to an ARL-biased chart, that is, some values are larger than the ARL 0 value. Recently, Knoth and Morais (2015) noted that: "problem of choosing the control limits of EWMA charts meant to monitor both increases and decreases in the process variance and based on asymmetrically distributed control statistics is not properly discussed in literature." They also pointed out that there are many instances in the literature where EWMA charts, for monitoring spread, have been developed that are ARL-biased (see, e.g.…”

Monitoring Process Variance Using an ARL‐unbiased EWMA‐p Control Chart

“… Step 1Let n , p v 0 , λ , and ARL v 0 be specified values. Step 2Let the out‐of‐control proportion, p v 1 , be a proportion of the in‐control proportion, p v 0 . That is, p v 1 = δp v 0 , δ ≠ 1, and 0 < p v 1 ≤ 1. Step 3Combine the Markov chain approach (e.g., Yang) and a numerical algorithm (e.g., Knoth and Morais) to determine the parameters of control limits, L 1 and L 2 , that satisfy the specified ARL v 0 when δ = 1, and let ARL v 1 be smaller than ARL v 0 for any δ ≠ 1. Here, the numerical algorithm used is routine nsga2 in R program. Step 4After the parameters of control limits L 1 and L 2 are determined, their control limits are calculated.…”