INTEGRATING RANKED SET SAMPLING IN MEMORY CONTROL CHARTS: A STUDY ON PEPSI COLA Mu’azu Ramat Abujiya Presented at the CMS 2019 and MMEI 2019 University of Technology, Chemnitz, Germany March 27 - 30, 2019 1
Outline Introduction Methodology Performance Evaluation Application Conclusions 2
Introduction The industrial revolution that brought along highly advanced machineries has made quality of manufactured products a thing of concern. To ensure conformity to certain quality criteria, a statistical quality control chart is often used to establish a criterion for determining when the process is in-control or out-of-control. Upper Control Limit (UCL) Center Line (CL) Lower Control Limit (LCL) 3
Introduction The primary objective of the control chart is to quickly detect the formation of assignable causes so that investigation of the process and corrective measure may be taken before many nonconforming units are manufactured. The Shewhart control charts is the most widely used procedure for monitoring a process but they are only sensitive to large shifts. 4
Introduction To monitor small and moderate changes in a process, memory control charts such as the exponentially weighted moving average (EWMA) and the cumulative sum (CUSUM) were introduced. Today, both the schemes are gaining their share of practical uses as a result of recent technological advancements in computers and technical expertise. Combined applications of the Shewhart, EWMA and the CUSUM charts have also been suggested to detect changes that may escapes the individual charts. 5
Introduction Most of these charts, however, are based on the assumption that samples are drawn from a process using the traditional simple random sampling (SRS). But since sampling techniques are very essential in all statistical applications, developing control charts using well structured sampling technique such as ranked set sampling (RSS) can improve estimation process and reduce manufacturing cost. 6
Introduction RSS is a data collection method that is more effective than random sampling in practical problems where the actual measurements of quality characteristics may be costly, destructive or time-consuming but could be ranked by visual inspection or some inexpensive method without actual measurements. Why RSS …… . ? RSS utilizes extra information from specific units in the population to guide its search for a truly representative sample data. 7
RSS Methodology RSS procedure: Involves drawing n random samples, each of subgroup size n from a population and rank the units within each set with respect to a variable of interest. Then n measurements are obtained by take the smallest observation from the 1st set….... Define 𝑌 𝑗:𝑜 𝑘 , 𝑗 = 1, 2, … , 𝑜 and 𝑘 = 1, 2, … , 𝑛 to be the 𝑗 th order statistic for RSS of size 𝑜 in the 𝑘 th cycle from a process. The RSS estimator for the mean and variance are respectively, given by ത 𝑛 𝑜 1 𝑜𝑛 σ 𝑘=1 Τ σ 𝑗=1 𝑌 𝑆𝑇𝑇 = 𝑌 𝑗:𝑜 𝑘 8
RSS Methodology 2 . 1 𝑛 𝑜 2 𝑌 𝑗:𝑜 𝑘 − ത • 𝑜𝑛−1+𝑤 𝑜 σ 𝑘=1 σ 𝑗=1 𝑇 𝑆𝑇𝑇 = 𝑌 𝑆𝑇𝑇 In this study, several individual and combined control charts for location and dispersion were developed for effective monitoring of wide range of changes in a process to ensure high performance accuracy, and adequate handling of uncertainties associated with manufacturing process. Most often, these uncertainties results from shifts in mean, standard deviation or both. 9
RSS based Control Charts Combined Shewhart-EWMA ത 𝑌 charts: CS-EWMA control limit for the Shewhart component 𝑉𝐷𝑀 𝑆𝑇𝑇−𝑇 = 𝜈 0 + 𝑙 Var ത 𝑌 𝑆𝑇𝑇 𝑀𝐷𝑀 𝑆𝑇𝑇−𝑇 = 𝜈 0 − 𝑙 Var ത 𝑌 𝑆𝑇𝑇 CS-EWMA control limit for the EWMA component 𝜇Var ത 𝑌 𝑆𝑇𝑇 1 − 1 − 𝜇 2𝑗 𝑉𝐷𝑀 𝑆𝑇𝑇−𝐹 = 𝜈 0 + 𝑀 2−𝜇 𝜇Var ത 𝑌 𝑆𝑇𝑇 1 − 1 − 𝜇 2𝑗 𝑀𝐷𝑀 𝑆𝑇𝑇−𝐹 = 𝜈 0 − 𝑀 2−𝜇 10
RSS based Control Charts The design parameters k and L are chosen based on the choice of the smoothing constant λ to satisfy the required process in-control needs. The RSS based CS-EWMA control chart gives an out- of-control signal when either 𝑌 𝑆𝑇𝑇 𝑘 < 𝑀𝐷𝑀 𝑇ℎ𝑓𝑥 or ത ത 𝑌 𝑆𝑇𝑇𝑘 > 𝑉𝐷𝑀 𝑇ℎ𝑓𝑥 and/or EWMA 𝑆𝑇𝑇𝑘 < 𝑀𝐷𝑀 𝐹𝑋𝑁𝐵 or EWMA 𝑆𝑇𝑇𝑘 > 𝑉𝐷𝑀 𝐹𝑋𝑁𝐵 where EWMA 𝑆𝑇𝑇 𝑘 = 𝜇 ത 𝑌 𝑆𝑇𝑇𝑘 + 1 − 𝜇 EWMA 𝑆𝑇𝑇 𝑘−1 11
RSS based Control Charts Combined Shewhart-CUSUM 𝑇 charts: The CUSUM component for the CS-CUSUM chart using RSS is based on the statistic + + 𝑇 𝑆𝑇𝑇 𝑘 𝜏 0 − 𝑙 + Τ 𝐷𝑉𝑇𝑉𝑁 𝑆𝑇𝑇 𝑘 = max 0, 𝐷𝑉𝑇𝑉𝑁 𝑆𝑇𝑇 𝑘−1 + − − 𝑇 𝑆𝑇𝑇 𝑘 𝜏 0 + 𝑙 − Τ 𝐷𝑉𝑇𝑉𝑁 𝑆𝑇𝑇 𝑘 = max 0, 𝐷𝑉𝑇𝑉𝑁 𝑆𝑇𝑇 𝑘−1 − where 𝑙 + = ∗ 1 + 𝜐 2; 𝜐 ≥ 1 and 𝑙 − = ∗ 1 + 𝜐 Τ Τ 𝑑 4 𝑑 4 2 ; 𝜐 ≤ 1 thus, the CS-CUSUM 𝑇 chart gives an out-of-control signal when + Τ 𝐷𝑉𝑇𝑉𝑁 𝑆𝑇𝑇 𝑘 > 𝐼 𝑇 𝑆𝑇𝑇 𝑘 𝜏 0 > 𝑉𝐷𝑀 S or − Τ 𝐷𝑉𝑇𝑉𝑁 𝑆𝑇𝑇 𝑘 > 𝐼 𝑇 𝑆𝑇𝑇 𝑘 𝜏 0 < 𝑀𝐷𝑀 𝑇 or 12
Performance Evaluation Average Run Length (ARL) Average Ratio of ARLs (ARARL) 𝜀 max 1 ARL 𝜀 ARARL = න d𝜀 𝜀 max − 𝜀 min ARL best 𝜀 𝜀 min Average Extra Quadratic Loss (AEQL) 𝜀 max 1 𝜀 2 ARL 𝜀 AEQL = න d𝜀 𝜀 max − 𝜀 min 𝜀 min Performance Comparison Index (PCI) Τ PCI = AEQL AEQL best 13
Performance Evaluation Τ where δ = 𝑜 𝜈 out − 𝜈 0 𝜏 0 , is the process mean shift σ 0 µ out UCL µ 0 LCL ARL δ is the ARL value of a control chart at 𝜀 ; ARL best δ and AEQL best are generated by the best performing charts. 14
Performance Comparison 15
Applications • Here, we present practical examples to illustrate the application of the RSS based memory control charts using real dataset on fill volume of a soft drink bottle. • The original dataset was obtained from a production line of the Pepsi-Cola production company, Al- Khobar, Saudi Arabia. We used re-sampling approach to obtain new dataset. 16
Applications 9 MRSS based CS-CUSUM Z_srs 8 C+_srs ത 𝑌 control chart using real Shewhart control limit 7 CUSUM control limit dataset. 6 Cumulative sum 5 10 4 Z_mrss 9 C+_mrss 3 Shewhart control limit 8 2 CUSUM control limit 7 1 Cumulative sum 6 0 5 1 5 9 13 17 21 25 29 33 37 41 45 49 53 4 Observation number 3 ത 𝑌 Classical CS-CUSUM 2 1 control chart using real data. 0 1 5 9 13 17 21 25 29 33 37 41 45 49 53 Observation number 17
Applications X bar Z+ srs SCL ECL 6.6 MRSS based CS-EWMA 6.5 ത 6.4 𝑌 control chart using real 6.3 6.2 dataset. Ploting statistics 6.1 6.0 X bar Z+dmrss SCL ECL 5.9 6.5 5.8 Indroduction of shift 6.4 5.7 6.3 5.6 6.2 5.5 5.4 6.1 Ploting statistics 5.3 6.0 1 4 7 10 13 16 19 22 25 28 31 34 5.9 Observation number 5.8 Indroduction of shift ത 5.7 𝑌 Classical CS-EWMA 5.6 5.5 control chart using real 5.4 dataset. 1 4 7 10 13 16 19 22 25 28 31 34 Observation number 18
Applications 14 ERSS based CS-CUSUM Classical Shewhart-R Classical CUSUM-R 12 𝑆 control chart using real Shewhart limit Indroduction of shift CUSUM limit 10 dataset. Ploting statistics 8 14 Shewhart-R DERSS 6 CUSUM-R DERSS Indroduction of shift 12 Shewhart limit 4 CUSUM limit 10 2 Ploting statistics Mean line for Shewhart R plot 8 0 6 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Observation number 4 𝑆 Classical CS-CUSUM 2 Mean line for Shewhart R plot control chart using real 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 dataset. Observation number 19
Conclusion In this work, the RL properties of the charts were evaluated and found to be substantially more effective in the detection of different sizes of shifts than most of the existing schemes without increasing the false alarm rate. We demonstrated the practicability for the application of the proposed charts using Pepsi-Cola data sets. It is found that apart from the detection effectiveness of the charts, the design procedures are easy to follow and implement. 20
Setup of RSS Scheme Site Randomly allocate x to sets x x x x x x x x x x x x x x x x x x x x x x x x ` Sets Ranked Sets 1 2 3 4 5 1 2 3 4 5 x 11 x 12 x 13 x 14 x 15 x (11) x (12) x (13) x (14) x (15) Rank within x 21 x 22 x 23 x 24 x 25 x (21) x (22) x (23) x (24) x (25) sets x 31 x 32 x 33 x 34 x 35 x (31) x (32) x (33) x (34) x (35) x 41 x 42 x 43 x 44 x 45 x (41) x (42) x (43) x (44) x (45) x 51 x 52 x 53 x 54 x 55 x (51) x (52) x (53) x (54) x (55)
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