A Salient Feature for Enhancing the Detection Speed …v t w t t w t t (3) The integration of Equations (2) and (3) will be precisely the same as Equation (1), if w is equal to one

Proceedings of the International Conference on Industrial Engineering and Operations Management Dubai, UAE, March 10-12, 2020

© IEOM Society International

A Salient Feature for Enhancing the Detection Speed of EWMA Scheme

Salah Haridya Department of Industrial Engineering and Engineering Management,

University of Sharjah, Sharjah 27272, United Arab Emirates. Benha Faculty of Engineering, Benha University, Benha, Egypt.

[email protected]

Mohammad Shamsuzzamanb, Hamdi Bashirc, Imad Alsyoufd Department of Industrial Engineering and Engineering Management,

University of Sharjah, Sharjah 27272, United Arab Emirates. [email protected]; [email protected]; [email protected]

Ahmed Magede Department of Systems Engineering and Engineering Management, City University of Hong

Kong, Kowloon, Hong Kong. Benha Faculty of Engineering, Benha University, Benha, Egypt.

[email protected]

Nadia Bhuiyanf Department of Mechanical and Industrial Engineering,

Concordia University, Montreal, Quebec, Canada H3G 1M8 [email protected]

Abstract

This article proposes a salient feature to enhance the detection speed of the EWMA scheme based on the power of the difference between the actual number and the target in-control number of nonconforming items. We refer to it as wEWMA scheme and show that the traditional EWMA chart is a special case of the proposed scheme. We discuss the optimal design procedure of the scheme, including the determination of its charting parameters to ensure the best overall performance. The results reveal that the developed wEWMA scheme has better overall detection effectiveness than the traditional EWMA chart. Specifically, the former outperforms the latter by 35% for the tested cases under different specifications.

Keywords Control chart, manufacturing industry, quality control, statistical process monitoring.

1. IntroductionUnlike Shewhart charts that only use data concerning the number d of nonconforming items in the last sample, the binomial Exponentially Weighted Moving Average (EWMA) chart comprises all the data in the sequence of observed values of d. As a result, EWMA charts typically outperform Shewhart charts (Castagliola et al. 2016). Roberts (1959) first introduced the notion of the EWMA schemes in the context of monitoring the mean of a normally distributed quality characteristic. When the EWMA chart is used for detecting p shifts, the statistic Et is updated for each tth sample as follows:

0

0 1

0( ) (1 )t t t

EE d d Eλ λ −

== − + −

(1)

where dt is the number of nonconforming items existed in the tth sample, d0 (= n × p0 where n and p0 are the sample size and in-control fraction nonconforming, respectively) is the in-control value of dt, and λ (0 < λ < 1) is the weighting parameter.

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The binomial Exponentially Weighted Moving Average (EWMA) scheme emerged as a powerful and leading procedure for finding moderate and small shifts in fraction nonconforming p (Haridy et al. 2017). This has occurred mostly because of the rapid advances in automated inspection and online measurement systems in current Statistical Process Monitoring (SPM) software (Melo et al. 2017; Bezerra et al. 2018). Yeh et al. (2008) proposed EWMA charts based on non-transformed Bernoulli and geometric attribute characteristics. The suggested EWMA charts outperformed the Cumulative Sum (CUSUM) charts that Chang and Gan (2001) had developed for monitoring processes with a very low fraction nonconforming. Epprecht et al. (2010) optimized the charting parameters of the attribute EWMA chart with variable sampling intervals. Spliid (2010) put forward an EWMA chart for the data following Bernoulli distribution. The proposed chart is very applicable for examining many varied activities in the production and service industries. Chen (2012) developed an EWMA chart for monitoring the number of defects that follow a geometric-Poisson distribution. Graham et al. (2012) presented a two-sided nonparametric EWMA chart for detecting a shift in the location parameter of a continuous distribution. Shu et al. (2012) extended EWMA charts that could quickly detect increases in the Poisson rate. Saghir and Lin (2014) proposed a flexible and generalized attribute EWMA chart for monitoring count data. Khaliq et al. (2016) developed an effective EWMA chart based on a Tukey chart for skewed distributions. Some researchers such as Reynolds and Stoumbos (2004), and Jiao and Helo (2006) showed that the variable CUSUM scheme could be a powerful tool for detecting large process shifts if 2

0( )tx µ− replaces the sample mean shift

0( )tx µ− in formulating the cumulative sum. Wu et al. (2008) used this to successfully increase the detection effectiveness of the binomial CUSUM scheme for monitoring attributes. He found that replacing (dt – d0) by (dt – d0)2 could increase the detection speed of the binomial CUSUM scheme to detect large shifts in p. In light of this, this current research presents a new binomial EWMA scheme, named the wEWMA scheme, which uses a power w to the difference (dt – d0) for monitoring attributes. The statistic of this new scheme is calculated as follows:

0

1

0(1 )t t t

EE v Eλ λ −

== + −

(2)

where

<−−−≥−−

=.0)(

0)(

00

00

ddifddddifdd

vt

wt

tw

tt (3)

The integration of Equations (2) and (3) will be precisely the same as Equation (1), if w is equal to one. This indicates that the traditional EWMA scheme represents a special case of the wEWMA scheme. When an increasing p shift occurs, the statistic Et tends to surpass the control limit H of the wEWMA control scheme, which results in an out-of-control signal. Contrary to the previous researches such as Reynolds and Stoumbos (2004), Jiao and Helo (2006) and Wu et al. (2008) recommending w = 2 to increase the performance of the CUSUM scheme for detecting large shifts, w is optimized in this paper to improve the overall detection speed of the wEWMA scheme against a wide range of p shifts. In this study, sampling inspection is adopted to manage the constraints on the manpower and measurement instruments, and the sampling interval h is considered as the time unit (i.e., h = 1). The performance of the wEWMA scheme is then compared with the performance of the traditional EWMA scheme in terms of the Average Number of Defectives (AND) as an objective function (Haridy et al. 2013). The random number of defective items dt is assumed to follow a binomial distribution with a known p0, and the process shift δ follows a uniform distribution with a maximum shift of δmax. One rule of thumb in SPM is that attribute schemes are most frequently utilized to detect a deterioration in quality (i.e., increasing p shifts) because decreasing p shifts are usually not risky and do not jeopardize product quality (Reynolds and Stoumbos 1999). Hence, the emphasis of this study is in detecting increasing p shifts. The paper is arranged as follows. Section 2 explains the operation of the wEWMA scheme. Section 3 describes the design of the wEWMA scheme. Section 4 compares the EWMA and wEWMA schemes under different settings of the design specifications. Finally, the conclusion is presented in Section 5. 2. The Operation of the wEWMA Scheme The wEWMA scheme can be implemented similarly to the traditional EWMA scheme. The only difference is that (dt – d0)w replaces (dt – d0) in formulating the EWMA statistic. The procedure for implementing the wEWMA is: (1) Use E0 = 0 as an initial value for the statistic in Equation (2). (2) Sample n items at the end of each h and determine dt in the sample. Then, calculate the corresponding value of

vt (Equation (3)). (3) Update Et (referring to Equation (2)).

1(1 )t t tE v Eλ λ −= + − (4) (4) If Et ≤ H, the process is thought to be in control, then return to step (2) for the next sample.

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(5) Otherwise (i.e., if Et > H), the process is considered out of control and should be terminated as quickly as possible for identifying the assignable causes and making a corrective action.

3. Design of the wEWMA Scheme If a shift occurs in the process, the fraction nonconforming p will shift to: p pδ= × 0 (5) where δ (1 ≤ δ ≤ δmax) represents the shift with an in-control status when δ = 1 (i.e., p = p0) and an out-of-control status when (1 < δ ≤ δmax). The maximum possible shift is δmax. A sound measure of the overall performance of attribute schemes is the Average Number of Defectives (AND) (Haridy et al. 2013).

01

( ) ( )max

AND p ATS f dδ

δδ δ δ δ= × × ×∫ (6)

where ATS(δ) represents the out-of-control ATS value at a shift δ, and fδ(δ) is the probability density function of δ. The index AND is actually a weighted average of ATS that uses δ as the weight. The scheme producing a smaller AND is considered to be more effective over different shifts δ. The random shift δ in p is presumed to follow a uniform distribution (Sparks 2000) with a probability density function fδ(δ) of:

1( )

1max

fδ δδ

=−

(7)

Prior to the design of the wEWMA scheme, four parameters τ, n, p0 and δmax should be determined. τ is the minimum value of ATS0. In order to lower the false alarm rate when managing the Type I error is costly, a larger τ is used. However, a large τ may cause a reduction in the effectiveness of the control scheme (Wu et al. 2006). The parameter n is determined with regards to the availability of resources and the required sensitivity of the scheme in detecting a shift, while the value of p0 is specified based on the data obtained in Phase I. Additionally, the quality engineer can choose δmax according to the maximum possible p shift or the interested shift range. The wEWMA scheme can be designed using the following model: Objective: Minimize AND Constraint: ATS0 ≈ τ (8) Design variables: w, λ, H where w and λ are the independent charting parameters, while H depends on w, λ and the specified τ. The main goal of the model is to determine the best values of w and λ that produce the smallest AND across a shift range of (1 < δ ≤ δmax) and in the meantime, H is tuned so that ATS0 ≈ τ. 4. Comparative Studies 4.1. Comparison under One Case This section studies and compares the performance of the EWMA and wEWMA schemes. First, the two schemes are studied under one case. The design specifications are the following: τ = 740, p0 = 0.008, δmax = 6 and n = 125 The two schemes are designed and the results are as follows: EWMA control scheme: w = 1, λ = 0.485, H = 2.155, AND = 0.113. wEWMA control scheme: w = 0.90, λ = 0.245, H = 1.167, AND = 0.097. The values ATS0 (where δ = 1) and ATS (where 1 < δ ≤ 6) of both schemes are calculated within the range of the p shift, and the results are shown in Table 1. Figure 1 illustrates the normalized ATS (i.e., ATSEWMA/ATSwEWMA) of both schemes. The normalized ATS shows the percentage by which the wEWMA scheme outperforms the traditional EWMA in terms of ATS at a particular δ. For example, if ATS/ATSwEWMA = 1.66 at δ = 1.5, then the ATS value of the wEWMA scheme is less than that of the traditional EWMA scheme by 66% at δ = 1.5. The following can be observed from Table 1 and Figure 1: (1) The two schemes generate an ATS0 value very near to τ (constraint (8)) when there is no shift in p. This shows

that the two schemes have almost the same false alarm rate and ensures a fair comparison. (2) The wEWMA scheme is superior to the traditional EWMA scheme when δ ≤ 3.5, while the traditional EWMA

scheme outperforms the wEWMA scheme when δ > 3.5. (3) The superiority of the wEWMA scheme over the traditional EWMA scheme when δ ≤ 3.5 is more observable

than that of the latter over the former when δ > 3.5. This reflects that the wEWMA is more likely to exhibit better overall detection speed than the EWMA scheme over the entire range of p shifts (where 1 < δ ≤ 6).

Table 1. ATS values of the EWMA and wEWMA schemes

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Figure 1. Normalized ATS of the EWMA and wEWMA schemes

The values of AND (Equation (6)) of both schemes are also evaluated, and the ratio of ANDEWMA/ANDwEWMA is found to be 0.113/0.097 = 1.16. This value indicates that, for this case (where τ = 740, p0 = 0.008, δmax = 6 and n = 125), the mean number of defects resulted when using the wEWMA scheme is 16% less than that produced by the traditional EWMA scheme against the entire range of p shifts. 4.2. Comparison under Different Cases The EWMA and wEWMA schemes are further studied in additional different conditions for which the design specifications (τ, p0, δmax and n) are utilized as the input factors, and all of them has two levels as follows: τ: 500, 1500. p0: 0.005, 0.01. δmax: 3, 8. n: 50, 200. The levels are determined with reference to those that many authors commonly use (Bourke 2008; Haridy et al. 2019). The different combinations of τ, p0, δmax and n resulted in six cases as shown in Table 2. For all cases, the wEWMA and EWMA schemes are designed, and both produce an ATS0 close to τ. Reflecting AND, the overall performance and the charting parameters (w, λ and H) for the six cases are summarized in Table 2. The ratio of ANDEWMA/ANDwEWMA is always larger than 1. This demonstrates that the wEWMA scheme always performs better than or as equal as the traditional EWMA scheme. Of note is that the wEWMA scheme adopts a relatively large w when p0 and δmax are large. This finding is consistent with the fact concluded by previous studies (Reynolds and Stoumbos 2004; Wu et al. 2008) that a large w is preferred to detect large shifts. Finally, the grand mean 𝐴𝐴𝐴𝐴𝐴𝐴EWMA/𝐴𝐴𝐴𝐴𝐴𝐴𝑤𝑤EWMA�� representing the mean of the ANDEWMA/ANDwEWMA values over all the six cases in Table 2 is calculated. The result of 𝐴𝐴𝐴𝐴𝐴𝐴EWMA/𝐴𝐴𝐴𝐴𝐴𝐴𝑤𝑤EWMA�� is 1.35. This shows that, from an overall

δ ATS EWMA scheme (w = 1) wEWMA scheme (w = 0.9)

1 730.7376 727.6995 1.5 67.6361 40.8391 2 17.6465 11.6891 2.5 7.6551 6.0543 3 4.4467 3.9919 3.5 2.9686 2.9585 4 2.2489 2.3410 4.5 1.7548 1.9567 5 1.4242 1.6589 5.5 1.2186 1.4531 6 1.0536 1.2920

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viewpoint considering various values of τ, n, p0 and δmax, the wEWMA scheme is outperforming the EWMA scheme by 35%. In this context, it is worth mentioning that the traditional EWMA scheme is unlikely to possess a better overall detection speed than the wEWMA scheme as the EWMA scheme is merely a special case of the wEWMA scheme. If w of a wEWMA scheme is equal to 1 and its λ and H are equal to those of an EWMA scheme, then this wEWMA scheme will work exactly as the EWMA scheme. Consequently, a wEWMA scheme can be always designed so that it will act either better than or at least equal to the EWMA scheme.

Table 2: Comparison of the EWMA and wEWMA schemes under different conditions

Case τ p0 n δmax Scheme w λ H AND ANDEWMA / ANDwEWMA

1 500 0.005 50 3 EWMA 1 0.365 0.956 0.3230 1.470

wEWMA 1.20 0.065 0.306 0.2193 1.000

2 500 0.005 200 8 EWMA 1 0.485 2.043 0.0528 1.121

wEWMA 1.35 0.170 1.346 0.0471 1.000

3 500 0.01 50 3 EWMA 1 0.485 1.544 0.4113 1.578

wEWMA 0.90 0.125 0.516 0.2606 1.000

4 1500 0.005 200 8 EWMA 1 0.245 1.423 0.0612 1.034

wEWMA 1.35 0.155 1.516 0.0592 1.000

5 1500 0.01 50 3 EWMA 1 0.395 1.527 0.6246 1.813

wEWMA 1.05 0.080 0.501 0.3445 1.000

6 1500 0.01 200 8 EWMA 1 0.410 2.738 0.0680 1.056

wEWMA 1.50 0.170 2.959 0.0644 1.000 5. Conclusion This article proposes a wEWMA control scheme for monitoring increasing shifts in proportion p of nonconforming items in a binomial model. The traditional binomial EWMA chart is just a special case of the proposed scheme. Our monitoring statistic uses (dt – d0)w instead of difference (dt – d0). By applying this salient feature, the wEWMA is able to considerably outperform its traditional counterpart for detecting p shifts much quickly in terms of AND in different scenarios. The binomial EWMA scheme is, in general, not as efficient as its generalized version, namely, the wEWMA scheme. As a result, the wEWMA scheme should replace the EWMA scheme for all SPM applications. This article assumes that the number of nonconforming items dt follows a binomial distribution with a known p0, and the shift δ follows a uniform distribution. Thus, studying the performance of the wEWMA scheme when dt and δ follow other distribution rather than the binomial and uniform, respectively, is worthwhile. Another prospective future work would be to investigate how the detection speed of the wEWMA scheme is affected when p0 is estimated from a limited number of samples. Acknowledgments This research is supported by the University of Sharjah, UAE, under Competitive Research

Project No. 18020405112.

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Biographies Salah Haridy is an assistant professor in the Department of Industrial Engineering and Engineering Management at the University of Sharjah, UAE. He received his M.Sc. and Ph.D. degrees from Benha University, Egypt and Nanyang Technological University, Singapore in 2008 and 2014, respectively. He is the recipient of the 2013 Mary G. and Joseph Natrella Scholarship awarded by the American Statistical Association (ASA) and the 2014 Richard A. Freund International Scholarship awarded by the American Society for Quality (ASQ). His research interests cover quality engineering, statistical process control and design of experiments. Mohammad Shamsuzzaman is currently an associate professor in the Department of Industrial Engineering and Engineering Management at the University of Sharjah, UAE. He obtained his Ph.D. in Systems and Engineering Management in 2005 from Nanyang Technological University, Singapore. His current research focuses on quality control and improvement, reliability, simulation, and multi-criteria decision-making. He is a member of the American Society for Quality. Imad Alsyouf is an associate professor of Industrial Engineering, employed by the University of Sharjah, UAE. He is the founder and coordinator of the Sustainable Engineering Asset Management (SEAM) Research Group. He has produced more than 67 conference and journal papers. He has about 30 years of industrial and academic experience in various positions in Jordan, Sweden, and UAE. His research interests include reliability, quality, maintenance, and optimization. He has developed and taught more than 25 post and undergrad courses. He delivered training courses in Kaizen, TQM, and organizational excellence. Hamdi Bashir received his PhD degree in 2000 from McGill University, Montreal, Canada. Currently, he is an Associate Professor of Industrial Engineering and Engineering Management at the University of Sharjah. Prior to joining this university, he held faculty positions at Sultan Qaboos University, University of Alberta, and Concordia University. His research interests are in the areas of project management, manufacturing systems, quality management, and healthcare management. He is a senior member of the Institute of Industrial and Systems Engineers (IISE). Ahmed Maged is a Ph.D. student in the Department of Systems Engineering and Engineering Management at City University of Hong Kong. He received his M.Sc. degree in Industrial Engineering from Benha University. His research interests are focused on quality engineering, statistical process monitoring and machine learning. Nadia Bhuiyan obtained her bachelor’s degree in Industrial Engineering from Concordia University and her MSc and PhD in Mechanical Engineering from McGill University. She taught at McGill and Queen’s University. She is a Professor in the Department of Mechanical, Industrial and Aerospace Engineering at Concordia University, and is currently serving as the Vice-Provost of Partnerships and Experiential Learning. Her research focuses on product development processes, dealing with the design, development, production, and distribution of goods and services.

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A Salient Feature for Enhancing the Detection Speed …v t w t t w t t (3) The integration of Equations (2) and (3) will be precisely the same as Equation (1), if w is equal to one