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Download This PDF File 1170 Int. J Sup. Chain. Mgt Vol. 8, No. 2, April 2019 Monitoring Variability in Complex Manufacturing Process: Data Analysis Viewpoint with Application Maman Abdurachman Djauhari1, Norhaidah Mohd Asrah2, Irianto3, Noor Akma Ibrahim4 1Graduate School, Indonesian Institute of Education (IPI) Jl. Terusan Pahlawan 32, Garut 44151, Indonesia [email protected] 2Faculty of Applied Science and Technology, Universiti Tun Hussein Onn Malaysia Pagoh Campus, 84600 Pagoh, Johor, Malaysia [email protected] 3&4Institute for Mathematical Research, Universiti Putra Malaysia 43400 UPM Serdang, Selangor, Malaysia [email protected] [email protected] Abstract— To relate the control limits of Shewhart- 1. Introduction type chart to the p-value, the control charting techniques were constructed based on statistical The word “complex” in CPV refers to several inference scheme. However, in daily practice of interrelated critical to qualities (CTQs). It is complex process variability (CPV) monitoring customarily replaced by the word “multivariate” operation, these limits have nothing to do with the p- when the interrelationship among CTQs is value. We cannot put any number to p. Instead, p is quantified in terms of Pearson correlation. It just read as “most probably”. These words mean that in practice we are finally working under data analysis represents the mathematical and statistical scheme instead. For this reason, in this paper we complexity faced in monitoring CPV and generally introduce the application of STATIS in CPV in complex statistical process control (CSPC). As monitoring operation. It is a data analysis method to mentioned in [19] and repeated in [20], [21], label the sample(s) where anomalous covariance monitoring CPV is as important as monitoring structure occurs. This method is algebraic in nature process target. and dominated by principal component analysis (PCA) principles. The relative position of a covariance matrix Although it is an important part in CSPC, we learn among others is visually presented along the first two that the technique for monitoring CPV is still in eigenvalues of the so-called “scalar product matrix development. This is perhaps due to the fact that (1) among covariance matrices”. Its strength will be it requires large sample size, and (2) CPV is difficult illustrated by using a real industrial example and the results, compared with those given by the current to measure. And among the very limited number of methods, are very promising. Additionally, root causes CPV measures available, according to [25], analysis is also provided. However, since STATIS is a generalized variance (say GV for brevity) is the most PCA-like, it does not provide any control chart, i.e., the adopted measure. We can find its importance as a history of process performance. It is to label the CPV measure in, among others, [1], [12], [18], [23], anomalous sample(s). To the knowledge of the authors, [27], [28], [29], and [30]. the application of STATIS in complex statistical process control is an unprecedented. Thus, it will However, as mentioned in [5], it is unfortunate that enrich the literature of this field. GV converges in distribution very slow to normality. Keywords— conjoint analysis, Escoufier’s operator, It is then not favorable to be used in manufacturing generalized variance, Hilbert-Schmidt space, vector industrial practice where only small sample is variance. available. And when dealing with small sample, in the current practice, the control limits do not reflect ______________________________________________________________ the p-value. In other words, they have nothing to do International Journal of Supply Chain Management IJSCM, ISSN: 2050-7399 (Online), 2051-3771 (Print) with probability of false alarm (PFA). As a Copyright © ExcelingTech Pub, UK (http://excelingtech.co.uk/) consequence, p cannot be represented by any number or percentage and in practice we are happy 1171 Int. J Sup. Chain. Mgt Vol. 8, No. 2, April 2019 to read p as “most probably”. See [20]-[21] for the organized as follows. We start in the next section by details. recalling GV-based control chart (GV-chart for brevity); its theoretical background and practical To have a better method for CPV monitoring, [7] implementation will be highlighted. Later on, in introduced a technique constructed based on vector Section 3 we show the performance of convergence variance (VV). The notion of VV is originally in distribution of GV and that of VV to normality. appeared in [8] and introduced as CPV measure by Due to slow convergence, in Section 4 we adopt the [4]. The purpose of VV-based method is to reduce method of STATIS in CPV monitoring. Here, our the weakness of GV-based method. See also [18] focus is on its implementation without going into the and [26] for a discussion of these methods. Since VV details of theoretical background. In Section 5 a real converges in distribution faster than GV to example coming from the Centre for Indonesian normality, see [6], this method needs less sample Army Industry, Ltd., will be presented and size. In addition, VV-based method is more sensitive discussed. Finally, closing remarks in the last to the small shift in covariance structure. However, section will end this work. see [5], it is still not favorable in manufacturing industrial practice because, for small sample, the 2. The Most Adopted Shewhart- control limits have still nothing to do with the p- Type Chart value; they do not reflect PFA. Monitoring operation of CPV is basically conducted These two methods are constructed based on the based on m independent samples of the same size n. theory of statistical inference in order to have direct Statistically speaking, it is equivalent to testing relationship between the control limits and the p- repeatedly the following H : for all value. But finally, p must be interpreted as “most 0 i 0 probably”. This means that, in practice, these i 1,2,,m , versus H1 :i 0 for an i in the set methods are not an inferential method but a data 1,2,,m. Here, i is the covariance matrix of the analysis or also called data exploration method. This population where the i -th sample is drawn and 0 is the point that motivates us to write this paper. is the hypothetical covariance matrix. The p -value, This point of view leads us to introduce here the also called PFA, is set to be 0.0027 or 0.27% while application of what French statisticians called the two sided critical values represent the lower STATIS in CSPC to enrich data exploration results. control limit (LCL) and upper control limits (UCL). STATIS stands for “Structuration des Tableaux à Trois Indices de la Statistique”. It was first If Si is the i -th sample covariance matrix, under introduced in a thesis by [16] in order to do conjoint for all , [2], see also [22], analysis of three-way data matrix. Since then, its shows that, development and application in many areas can be found in, for example, [10], [15] and [17]. This d 2 nSi 0 N0,2p 0 (2.1) method is algebraic in nature and constructed based on PCA principles. For those who are interested, the 2 p 2 suggested reading in French is [9], and [13], and in We show that the parameters and in 0 n 0 English is [11] and [15]. (2.1) are not the true mean and true variance of S . We show that, through graphical representation, They are the limit of the true ones when n tends to STATIS could label the sample(s) where anomalous infinity. The true mean and variance are, covariance structure is present. In addition, it could lead to identify the root causes why at a given 2 ES b and VarS b , sample the covariance structure has been shifted. i 1 0 i 2 0 This facility cannot be provided by the two where, Shewhart-type methods mentioned above. To illustrate its strength, a real industrial example will p 1 be presented, and the results will be compared with b n k 1 p those given by GV-based and VV-based methods. n 1 k1 In the rest of the paper our discussion will be and 1172 Int. J Sup. Chain. Mgt Vol. 8, No. 2, April 2019 p p 1 1 b2 b1 n k 2 b1 . b4 b3 mn 1 j 3 b3 . n 1 p p k1 mn 1 j1 Accordingly, (2.3) provides these unbiased control That is the reason why (2.1) is not used in the b b b literature to construct the GV-chart. Instead, as can limits UCL S 1 3 2 , CL S 1 , b 2 b be seen in the next sub-section, (2.2) is used 3 b3 b4 3 b1 b2 1 d 2 and LCL max0, S 3 . This is S b N 0, b 2 i 1 0 0 (2.2) 3 b3 b4 b2 the latest form of GV-chart evolution. In addition, the convergence in (2.1) is slower than That is the scenario at a glance of GV-chart that in (2.2) evolution. It is worth noting that whichever the control limits we use, if n is sufficiently large, the p- 2.1 GV-chart value is 0.27%. Meanwhile, for small n, p must be read as “most probably” and does not represent the Suppose m independent samples each of size n PFA. drawn from a p -variate normal distribution with positive definite covariance matrix are available We conclude that no matter whether the sample size for CPV monitoring operation. We consider again is large or small, the control chart is always the same. The only difference lies in the way we S the covariance matrix of the i -th sample; i interpret the chart.
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