Monitoring The Nonconforming Fraction With A Dynamic Scheme When Sample Sizes Are Time Varying

In many practical applications, it is more convenient to characterize the quality of 1 production processes or service operations throughout the count of nonconformities. In the context 2 of SPC, noncorfomities are usually assumed to appear according to the binomial probability model. 3 The conventional way for monitoring nonconformities invloves Shewhart-type control procedures 4 based on both constant and time varying sample sizes. In this article, an EWMA scheme is 5 proposed for monitoring the fraction of nonconforming items with time-varying sample sizes. The 6 proposed control chart is referred to as the EWMAG-B and can be easily adapted to work with 7 a constant sample size by fixing it at a needed value. By means of simulation, it was found out 8 that the EWMAG-B chart outperforms the conventional p control chart in Phase II while detecting 9 changes in the process level is wanted. 10


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SPC has been implemented for some time in many areas, including applications 14 in industry, service operations, health surveillance and other activities where the moni-15 toring of processes is required. Control charts have become the most widely used tool 16 for process monitoring. The main goal of using control chart is to check processes for 17 stability over time. In many practical applications, the quality of processes is often 18 characterized by a continuous random variable, whose distributional parameters (or 19 some function of them) need to be monitored in order to detect unwanted drops in them  However, it is not always easy or convenient in economic terms to record the 26 measure of a quality characteristic. It is better sometimes to classify the producto of a 27 process into one of two exhaustive and mutually exclusive categories: in compliant and 28 non-compliant, for example. In these situations, process quality is formally expressed by 29 a discrete counting random variable. Vargas [8], among many other authors, present the The p chart is a Shewhart-type control methodology and its design, as well as those 36 of the aforementioned charts, strongly depends on assumptions that must necessarily 37 be met. Although chart designing requires the size of available samples to be known 38 and fixed, it is also based on asymptotic normal distributional properties. In most cases, 39 as in healthcare surveillance applications, a fixed large enough sample size is almost 40 impossible to guarantee. As pointed out in Alvarado & Retamal [1], the normal approxi-41 mation works fairly acceptable just for n > 30, np ≥ 5 and np(1 − p) ≥ 5. Quesenberry 42 [6] states that when n ≤ 30 and p ≥ 0.20, the estimates of the normal parameters would 43 inevitably be unreliable and could lead to an unacceptable performance of the traditional 44 p control chart. Recall that the problem related to the independence of the available 45 samples, which is not minor, has not even touched.

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Recently, some methodologies have been proposed that work fairly well when the 48 assumptions for the p chart design are not fully met. Aytaçoglu & Woodall [2] propose a 49 CUSUM methodology that overcomes aforementioned shortcomings. Nevertheless, the 50 construction and interpretation of this proposal may represent some challenges for some 51 untrained practitioners, given the relative complexity of chart designing and implemen-52 tation. In SPC, a great effort has been focused on developing monitoring schemes whose 53 control limits can be drawn not as rigidly as is done in traditional schemes. In this sense, 54 Shen et al. [7] proposed the monitoring of Poisson-type counting data when sample sizes 55 vary over time using an EWMA chart with probability control limits.   Let X t , t = 1, 2, . . ., be the count of an adverse event during the fixed time period (t − 1, t]. Given the sample size n t at time t, suppose that X t independently follows the Poisson distribution with mean θ n t , where θ denotes the occurrence rate of the event of interest. To detect an abrupt change in the rate of occurrence from θ 0 to another unknown value θ 1 > θ 0 , it is proposed the EWMA statistic where Z 0 = θ 0 and λ ∈ (0; 1] is the smoothing parameter of the chart determining the weights of past observations. According to Shen et al. [7], the the upper control limit h t of the EWMA statistic (1) a each monitoring moment t has to satisfy where α is the previously established level of the false alarm rate.

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It is worth mentioning that the probability control limit is determined just after Carlo and Markov chains are available. For more details, the reader is addressed to 86 Shen et al. [7]. This methodology was also implemented by Morales & Vargas [5] for 87 monitoring aggregated Poisson data in processes with time-varying sample sizes. The upper control limit of the EWMAG-B chart must satisfy the expressions given 102 in (2) and is determined just after the value of n t is observed at time t. As mentioned 103 above, due to the complexity of the conditional probability given in (2) There are obtained M "pseudo EWMA" values at the first monitoring moment.   3.
Transform each of theX t,i , i = 1, . . . , M, into a "pseudo EWMA"Ẑ t,i as indicated belowẐ whereẐ t−1 is a value randomly picked from the set of M * "pseudo EWMA" 127 values obtained as said in Step 5-1. This is done to ensure that the process 128 works under stable conditions.     Table 1 Table 2 for α = 0.0027 and p 0 = 0.1. be 372 approximately, whatever the scheme with geometrically distributed run lengths.

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As can be seen in Table 2 Table 2 were obtained.