適應性計數值損失函數管制圖之設計 - 政大學術集成

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(1)國立政治大學統計學系碩士學位論文. 指導教授：楊素芬博士政治蔡紋琦博士. 立. 大. ‧ 國. 學 ‧. 適應性計數值損失函數管制圖之設計 Nat. io. sit. y. Design of the Adaptive Loss Function er. n. Control aChart for Binomial Data iv l C n hengchi U. 研究生：李宜臻撰中華民國九十九年六月.

(2) 謝辭首先，誠摯感謝我的指導教授們，感謝楊素芬老師二年來的諄諄教誨與關心，老師就像良師益友般，總是不斷地砥礪我們、勉勵我們，要求將事情做到最好，在學習的過程中，獲益匪淺。回想起研一時，被老師頻繁「叮嚀」的日子裡，不禁嚇出一身冷汗，好在通過統計所上的種種認證與老師們的要求，本文才得以順利產出。感謝蔡紋琦老師細心閱讀論文並給予建議。老師們對學問的嚴謹與要求更是我們學習的典範。感謝我的口試委員蔡小蓁教授及黃榮臣教授，謝謝老師們耐心閱讀與指. 政治大究所時期，不時地關心與鼓勵，讓我有信心、有勇氣去完成一切，就像站在巨人立教，並給予中肯地建議，提醒論文裡該注意的細節。感謝爸爸、媽媽、弟弟在研. ‧ 國. 學. 的肩膀上，可以信心滿滿地勇往直前。感謝同師門好夥伴，伊萱、亮妤、政憲，一路上之陪伴與幫忙，只能說，有你們真好。感謝好朋友們，娜娜、丹丹、鹹魚. ‧. 等，溫暖的擁抱。男朋友安安在背後的默默支持更是我前進的動力，沒有安安的. sit. y. Nat. 體諒、包容，相信這兩年的生活將是很不一樣的光景。最後，謹以此文獻給我. al. er. io. 尊敬的教授、摯愛的家人以及所有陪伴或關心過我的朋友們。. v. n. 本研究承蒙行政院國家科學委員會補助，計畫編號. Ch. engchi. i n U. NSC-96-2118-M-004-001-MY2、NSC-98-2118-M-004-005-MY2，及政治大學商學院服務創新頂尖研究中心(CSI) 補助，特此感謝。. 李宜臻謹致中華民國九十九年六月.

(3) Abstract This article proposes the algorithm of a new control chart (loss function control chart) based on the Taguchi loss function with an adaptive scheme for binomial data. The loss function control chart is able to monitor cost variation from the process by applying loss function in the design. This new angle economically explores production cost. This research provides designs of the loss function control chart with specified VSI, optimal VSI, VSS and VP, respectively. Numerical analyses show that the specified VSI loss function chart, the optimal VSI loss function chart, the optimal VSS. 政治大 function chart significantly立 and show costs can be controlled systematically.. loss function chart and the optimal VP loss function chart outperform the Fp loss. ‧. ‧ 國. 學. KEYWORDS: Attribute Control Chart; Taguchi Loss Function; Variable. y. sit. n. al. er. io. Chain.. Nat. Sampling Intervals; Variable Sample Sizes; Variable Parameters; Markov. Ch. engchi. i n U. v.

(4) CONTENTS 1. Introduction…………………………………………………………………...….1 2. Distribution of the Loss Function for binomial data………………….…….…4 3. Design of the VSI Loss Function Chart for binomial data………….....……....6 3.1 Construction of the VSI Loss Function Chart for binomial data……........6 3.2 Performance Measurment...............................................................................9 3.3 Determination of the UCL, WCL of the Specified VSI Loss Function Chart……………………………………………………………………..…11. 政治大 Chart……………………………………………………………………..…12 立. 3.4 Determination of the UCL, WCL of the Optimal VSI Loss Function. 3.5 Performance Comparisons…………………………………………………15. ‧ 國. 學. 3.5.1 Specified VSI Loss Function Chart for binomial data…………..….15. ‧. 3.5.2 Optimal VSI Loss Function Chart for binomial data………………17. sit. y. Nat. 3.6 Example……………………………………………...…………………..….22. io. er. 4. Design of the VSS Loss Function Chart for binomial data……………………27 4.1 Construction of the VSS Loss Function Chart for binomial data…...….27. al. n. v i n Ch 4.2 Performance Measurement………………………………………………...30 engchi U 4.3 Determination of the UCLi, WCLi of the Optimal VSS Loss Function. Chart.............................................................................................................32 4.4 Performance Comparisons…………………………………………………35 4.5 Example……………………………………………………………………...39 5. Design of the VP Loss Function Chart for binomial data…...…………………43 5.1 Construction of the VP Loss Function Chart for binomial data…………43 5.2 Performance Measurement………………………………………………...46. I.

(5) 5.3 Determination of the UCLi, WCLi of the Optimal VP Loss Function Chart…………..……………………………………………………………48 5.4 Performance Comparisons………………………………………………....52 5.5 Example……………………………………………………………………...56 6. Conclusion and Future Study……………………………………………………60 References…………………………………………………………………………...61 Appendix…...………………………………………………………………………..63. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. II. i n U. v.

(6) LIST OF TABLES Table1. State Definition of the VSI Loss Function Chart for binomial data…………..9 Table2. ATS of the VSI and Fp under various p with specified h1=1.8, h2=0.1, R=500.............................................................................................................15 Table3. ATS of the VSI and Fp under various p with specified h1=1.8, h2=0.1, R=700…….....................................................................................................16 Table4. ATS of the Optimal VSI and Fp under various p with R=500……………….18 Table5. ATS of the Optimal VSI and Fp under various p with R=400……………….20. 政治大 Table7. Sampling data and VSI Loss Function Chart……………………………..…24 立 Table6. Optimal Design of the VSI Loss Function Chart……………………………23. Table8. Fp Loss Function Chart……………………………………………………...25. ‧ 國. 學. Table9. Comparison of the Fp and VSI Loss Function Chart………………………..26. ‧. Table10. State Definition of the VSS Loss Function Chart for binomial data……….30. sit. y. Nat. Table11. Optimal VSS and Fp with R=500, p=0.001, 0.01, 0.02……………………36. io. er. Table12. Optimal VSS and Fp with R=500, p=0.05, 0.1, 0.3, 0.5…………………...37 Table13. Optimal Design of the VSS Loss Function Chart………………………….39. al. n. v i n CLoss Table14. Sampling data and VSS Chart……………………………...41 h eFunction ngchi U. Table15. Comparison of the Fp and VSS Loss Function Chart……………………...42 Table16. State Definition of the VP Loss Function Chart for binomial data………...47 Table17. Optimal VP and Fp with R=500, p=0.001, 0.01, 0.02, 0.05………………..53 Table18. Optimal VP and Fp with R=500, p=0.1, 0.2, 0.3, 0.5………………………54 Table19. Optimal Design of the VP Loss Function Chart……………………………57 Table20. Sampling data and VP Loss Function Chart………………………………..58 Table21. Comparison of the Fp and VP Loss Function Chart………………………..59. III.

(7) LIST OF FIGURES Figure1. VSI Loss Function Chart…………………………………………………….6 Figure2. Flow Chart of the Design of the Optimal VSI Loss Function Chart……….14 Figure3. Main Effect Plot of the Optimal VSI Loss Function Chart (p, R=500)…….18 Figure4. Main Effect Plot of the Optimal VSI Loss Function Chart (δ, R=500)…...19 Figure5. Main Effect Plot of the Optimal VSI Loss Function Chart (p, R=400)…….21 Figure6. Main Effect Plot of the Optimal VSI Loss Function Chart (δ, R=400)…...21 Figure7. Optimal VSI Loss Function Chart………………………………………….23. 政治大 h )...……………………………………………….……………………….25 立. Figure8. Optimal VSI Loss Function Chart (red points adopt h2; blue points adopt 1. Figure9. Fp Loss Function Chart……………………………………………………..25. ‧ 國. 學. Figure10. VSS Loss Function Chart………………………………………………….27. ‧. Figure11. Flow Chart of the Design of the Optimal VSS Loss Function Chart….…..34. sit. y. Nat. Figure12. Main Effect Plot of the Optimal VSS Loss Function Chart (p, R=500)…..38. io. er. Figure13. Main Effect Plot of the Optimal VSS Loss Function Chart (δ, R=500)…38 Figure14. Optimal VSS Loss Function Chart………………………………………..39. al. n. v i n C h Chart (yellow points Figure15. Optimal VSS Loss Function adpot n engchi U. 1. and right scale;. green points adpot n2 and left scale)……...……….……………..……….41. Figure16. VP Loss Function Chart…………………………………………………...43 Figure17. Flow Chart of the Design of the Optimal VP Loss Chart…………………51 Figure18.Main Effect Plot of the Optimal VP Loss Function Chart (p, R=500)…….55 Figure19.Main Effect Plot of the Optimal VP Loss Function Chart (δ, R=500)…...56 Figure20. Optimal VP Loss Function Chart………………………………………….57 Figure21. Optimal VP Loss Function Chart (yellow points adpot n1, h1and right scale; green points adpot n2,h2 and left scale)…..................................................59 IV.

(8) 1. Introduction Control charts are powerful statistical tools for monitoring production quality. The technology industry has widely applied the Shewhart control charts for detecting process variations. A Shewhart control chart involves three design parameters, including sample size n, sampling interval h and control limit factor k. From a statistical viewpoint of a control chart design, a special cause (S.C.) is detected when the process operation is no longer stable. The determination of design parameters of a control chart depends on the required false alarm rate and power.. 政治大 about the speed of detecting 立an unstable process, but also focuses on the cost (loss). However, from a producer viewpoint, the company may not only be concerned. ‧ 國. 學. produced from processes. Duncan (1956) first developed the economic control chart several decades ago, based on the cost-control concept, which considered several cost. ‧. parameters in the model. The economic model and minimal cost occurring within the. sit. y. Nat. production cycle determine the n, h, and k of a control chart. Many researchers have. n. al. er. io. followed Duncan’s work, and been reviewed in the literature by Montgomery (1980). i n U. v. and Vance (1983). Nevertheless, the complicated nature of the economic model is a. Ch. engchi. fact that cannot be ignored. Woodall (1986) considered weaknesses of the economic control chart to include “awkward time interval, difficulties in calculation and imprecise process parameters.” Although computer technology has improved many of the calculation tasks, estimating the cost parameters still hinders using the model. Taguchi (1984) commented on quality loss as “the loss to society caused by the product after it is shipped out.” Researchers have applied the concept of using loss function to economic design to make more reasonable cost estimates. Koo and Lin (1992) adopted this approach into the economic X control chart, using the Taguchi loss function to replace traditional cost parameters. Nevertheless, why does not jump 1.

(9) out of economic model, and use Taguchi loss function to establish a new chart directly. In contrast with economic charts, the Taguchi loss function provides a different approach to control cost (loss). The Taguchi loss function presents loss through the difference between target value and measured value. The concept of obtaining quality and loss information at the same time is what this article wants to catch. Researchers have recently demonstrated that adaptive control charts can help companies detect a S.C. more rapid than the fixed parameters control charts (Fp. 政治大 interval (VSI), variable sample size (VSS), variable sample size and sampling 立. control charts). Studies have widely investigated adopting the variable sampling. interval (VSSI) and variable parameters (VP), especially for variable data. Reynolds. ‧ 國. 學. et al. (1988) proposed the first VSI X charts which outperformed the Fp X charts.. ‧. Many researchers extended the ideas to improve detective ability. Tagaras (1998). sit. y. Nat. published a reviewed paper of recent adaptive control charts. According to it, Rendtel. io. er. (1987, 1990) proposed VSS and VSI charts for attribute data with a CUSUM-scheme. Vaughan (1993) published a VSI np chart by the Bayesian model, Calabrese (1995),. al. n. v i n C h adaptive controlUcharts for attribute data by the Porteus and Angelus (1997) proposed engchi Bayesian model. However, compared to the prosperous researches of variable charts. with attribute charts, the latter still leave much to explore. In recent researches, from 1999 to 2009, Costa (2001) proposed the VSS np chart with the Markov chain method, Luo and Wu (2002) developed one optimal algorithm for the VSS and VSI np charts, Costa (2003) developed a general method for the VP with c, np, u, and p charts, Luo and Wu (2004) worked on the VSSI np chart by an optimal algorithm that used attribute data on an adaptive control chart. The attribute control chart includes several attractive characteristics, such as easy implementation and measurement time saving, not to mention that variable data is sometimes undesirable because of process 2.

(10) restrictions or technological limits. Therefore, this study is particularly interested in binomial data with an adaptive scheme. This study establishes a new type of control chart, bases on the Taguchi loss function, and abbreviated here as a loss function chart. The purposes of this new control chart are monitoring the cost variation (loss) and maintaining a certain level of process quality. The chart controls cost in real-time and ensures production quality at the same time. The current study also develops an adaptive scheme. Section 2 introduces the distribution of the loss function for binomial data. Section 3 displays. 政治大 design of the VSS loss function chart. Section 5 demonstrates the design of the VP 立 the design of the VSI loss function chart for binomial data. Section 4 shows the. loss function chart. Data analyses and example are given and Section 6 is the. ‧ 國. 學. conclusion and future study.. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. 3. i n U. v.

(11) 2. Distribution of the Loss Function for binomial data A loss function chart is a new control chart, bases on the statistics of a Taguchi loss function. This method is an easy and efficient way to monitor process costs. From an economic viewpoint, the complicated economic model can estimate and control process loss. However, practical applications need to eliminate calculation obstacles for easy-usage, making the loss function chart a good alternative. Loss functions are used to describe quality cost (loss) while the product quality characteristic is far from a specified target (T). An appropriate type of loss function is. 政治大 on the industry and process. 立Although there are various forms of loss functions, this. important when using the loss function chart. The choice of loss function may depend. ‧ 國. 學. study focuses on the quadratic loss function.. Let random variable X be the number of nonconforming units found within each. ‧. sampling when the process is in control and it follows a binomial distribution with. sit. y. Nat. sample size n, fraction nonconforming p. The chosen quadratic loss function L as a. n. al. specified target. The quadratic loss function L is. Ch. L = ( X − T )2. engchi. er. io. loss function describes the cost while the product quality characteristic is far from the. i n U. v. (1). If the process is out-of-control, the random variable X turns to follow a binomial distribution with sample size n, fraction nonconforming δp , where the δ is a proportion scale of the process, 1 < δ < 1 . The target value T is set as zero (T=0) in this p. article because the best target value of found nonconforming units ought to be zero. Thus, the loss function can express as L = ( X ) 2 . Equations (2) and (3) show the results of E (L) and Var (L) . Appendix refers to the derivation of E (L) and Var (L) . E ( L) = E[ X 2 ] = np − np 2 + n 2 p 2. 4. (2).

(12) Var ( L) = 6n 2 p 2 − 16n 2 p 3 + 10n 2 p 4 + 4n 3 p 3 − 4n 3 p 4 + np − 7 np 2 + 12np 3 − 6np 4. From equations (2) and (3), the mean and variance of L are the function of n and p. Thus, the mean and variance of L are close to zero when p is extremely small.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 5. i n U. v. (3).

(13) 3. Design of the VSI Loss Function Chart for binomial data 3.1 Construction of the VSI Loss Function Chart for binomial data The fixed parameters (Fp) loss function chart is built with the upper control limit (UCL), central line (CL) and lower control limit (LCL). If the statistic L falls within the UCL and the LCL, the process is still in control. If the L falls outside the UCL, the process is out of control and requires search action. Take L as the statistic of a loss function chart and its UCL, CL, and LCL can be expressed as follows. UCL = E ( L) + k ⋅ Var ( L). (4). 政治大 CL = E (L) 立. (5). ‧ 國. 學. LCL = 0. (6). The LCL is set as zero because cost cannot be negative. Nevertheless, UCL is the. ‧. function of n and p. When p is extremely small, UCL is close to zero. Thus, the loss. io. sit. y. Nat. function chart is not appropriate to be used for extremely small p.. n. al. er. The VSI loss function chart is a control chart with adaptive sampling intervals. A. i n U. v. VSI loss function chart is built with a UCL, a WCL (warning control limit), and an. Ch. engchi. LCL (see Fig. 1). The LCL is set as zero in the model because the nonconforming fraction is the degree of deterioration. Thus, LCL=0 is the best level of quality loss.. I3 (Action region). UCL I2 (Warning region). WCL CL. I1 (Central region). 0 Figure1. VSI Loss Function Chart 6.

(14) A VSI loss function chart can be expressed as follows. See equations (7)-(10). UCL = E ( L) + k ⋅ Var ( L). (7). WCL = E ( L) + w ⋅ Var ( L). (8). CL = E (L). (9). LCL = 0. (10). where k is control limit factor and w is warning limit factor, 0 < w < k < ∞ . Refer to equations (2)-(3), the UCL and WCL can be expressed mathematically as equations (11)-(12).. 政治大. UCL = np − np 2 + n 2 p 2 + k ⋅ 6n 2 p 2 − 16n 2 p 3 + 10n 2 p 4 + 4n3 p 3 − 4n3 p 4 + np − 7np 2 + 12np 3 − 6np 4. (11). WCL = np − np 2 + n 2 p 2 + w ⋅ 6n 2 p 2 − 16n 2 p 3 + 10n 2 p 4 + 4n3 p 3 − 4n3 p 4 + np − 7 np 2 + 12np 3 − 6np 4. (12). 立. ‧ 國. 學. In a VSI loss function chart, WCL is a key to shorten the time of detecting the process shift. A VSI loss function chart has a long sampling interval h1 and a short. ‧. sampling interval h2. WCL is the guard to decide the use of h1 or h2 between the. io. sit. y. Nat. samples.. n. al. er. When using the VSI loss function chart, two different sampling intervals, h1 and. i n U. v. h2, are adopted. Users have to decide on a long sampling interval and a short sampling. Ch. engchi. interval, where h1 >h2. If the data point is plotted on the central region (I1), use the long interval h1 as the next sampling interval. If the data point is plotted on the warning region (I2), use the short interval h2 as the next sampling interval. If the data point is plotted on the action region (I3), find the S.C. and repair the process. For comparing the VSI loss function chart with the Fp loss function chart under the same standard, the average sampling interval needs to be the same when the process is in control. By Reynolds et al. (1988), the equation (13) needs to be satisfied. h1 ⋅ p0 + h2 ⋅ (1 − p0 ) = h0. (13). The average sampling interval of the VSI loss function chart is the same as the 7.

(15) fixed sampling interval of the Fp loss function chart, where h0 is the fixed sampling interval of the Fp loss function chart, 0 < h2 < h0 < h1 < ∞ , and p0 is the probability of being in the central region (I1) when the process is in control. p0 is defined as equation (14). p0 = P(0 ≤ L < WCL | 0 ≤ L < UCL) =. P(0 ≤ L < WCL) P(0 ≤ L < UCL). (14). From equation (14), WCL can be expressed in terms of P(0 ≤ L < WCL) = p0 ⋅ P(0 ≤ L < UCL) . Since L = X 2 , replace L as X 2 , P(0 ≤ X 2 < WCL) = p 0 ⋅ P(0 ≤ X 2 < UCL) . It is equivalent. 政治大. to P(0 ≤ X < WCL ) = p0 ⋅ P(0 ≤ X < UCL ) .. 立. . . WCL − 1 ) = p 0 ⋅ F X (. . 學. FX (. ‧ 國. Using the CDF of X to express the equation. . UCL − 1 ).. . . y. [ p 0 ⋅ FX ( UCL − 1 )].. sit. −1 X. Nat.  WCL − 1 = F. ‧. Take the inverse function of both sides.. n. al. er. io. Then, the WCL can be determined approximately as equation (15). −1. WCL ≈ [ FX [ FX (. Ch. . . i n U. UCL − 1 ) ⋅ p 0 ] + 1 ] 2. engchi. v. (15). where x  is the smallest integer not less than the corresponding elements of x. The value of p0 can be calculated through equation (13) and expressed as p0 =. h0 − h2 h1 − h2. (16). The UCL can be determined when n, p, k are given by using equation (11). The value of n and k are decided by in-control average run length (use ARL0 ). The ARL0 can be expressed as equation (17). ARL0 is a function of n, p and k. ARL0 =. 1. α. 1. = 1−. a −1. ∑C i =0. 8. n i. (17). p (1 − p ) i. n −i.

(16) 2 2 2 2 2 2 3 2 4 3 3 3 4 2 3 4 where a = [ np − np + n p + k ⋅ 6 n p −16 n p +10 n p + 4 n p − 4 n p + np − 7 np +12 np − 6 np. 3.2 Performance Measurement The control chart performance can be measured by many indexes, such as average run length (ARL) and average time to signal (ATS), where ATS = h ⋅ ARL . When the process is out of control, the smaller ARL or ATS means better detective ability of the control chart. When the process is in control, the larger ARL (see equation (17)) or ATS can result in fewer false alarms and costs.. 政治大. This study applies the Markov chain method to calculate ATS (Prabhu et al.. 立. (1993, 1995)) of the VSI loss function chart. Calculating the ARL and ATS of the VSI. ‧ 國. 學. loss function chart includes two assumptions. First, the loss function chart assumes that only one S.C. may occur during the process. Second, the process is out-of-control. ‧. at the beginning of the process. The ATS is calculated under the zero-state mode. Due. y. Nat. io. sit. to the assumptions, the process has two transient states and one absorbing state of the. er. Markov chain approach (see Table1).. al. n. v i n Cthe Table1. State Definition of VSI Loss Function he i U Chart for binomial data h n c g State S.C. occur Location of the VSI Loss Function Chart 1. Yes. I1. 2. Yes. I2. 3. Yes. I3. Denote Pi , j as a transition probability from the previous state i to the current state j, i=1, 2, 3, j=1, 2, 3. The transition probabilities are as follows: P1,1 = p (0 ≤ L < WCL | X ~ B ( n, δp )) = p (0 ≤ X 2 < WCL | X ~ B ( n, δp )) = p (0 ≤ X < WCL | X ~ B ( n, δp )) =. . . WCL −1. ∑C i =0. n i. 9. (δp ) i (1 − δp ) n −i.

(17) P1, 2 = p (WCL ≤ L < UCL | X ~ B (n, δp )) = p (WCL ≤ X 2 < UCL | X ~ B (n, δp )) . = p ( WCL ≤ X < UCL | X ~ B (n, δp )) = i=. P1,3 = p ( L ≥ UCL | X ~ B( n, δp )) = p ( X = p ( X ≥ UCL | X ~ B (n, δp )) = (1 −. . 2. UCL −1. . ∑C. (δp ) i (1 − δp ) n −i. n i WCL −1 +1. . . ≥ UCL | X ~ B (n, δp )) . UCL −1. ∑C i =0. n i. (δp ) i (1 − δp ) n −i ). P2, j = P1, j , j = 1,2,3 P3,1 = P3, 2 = 0, P3,3 = 1. From the elementary properties of the Markov chain, the average time to signal is ATS = b ( I − Q) −1 h. (18). 政治大. where b is a (1 × 2) vector of the initial probability of state 1 and 2, I is the identity. 立. matrix of order 2, Q is a 2 by 2 transition probability matrix, h is the vector of the. ‧ 國. 學.  p11 , p12   and  p21 , p22   . next sampling interval for state 1 and state 2. b = [ p 0' , 1 - p 0' ] , Q = . ‧. h ' = [h1 , h2 ] respectively, where p0' is the probability of being at state 1 at the. sit. y. Nat. io. n. al. er. beginning of the process when the process is out-of-control. 1 − p0' is the probability. i n U. v. of being state 2 at the beginning of the process when the process is out-of-control.. p0' is expressed as. Ch. engchi. p0' = P(0 ≤ L* < WCL | 0 ≤ L* < UCL) =. P(0 ≤ L* < WCL) P(0 ≤ L* < UCL). (19). where L* belongs to an out-of-control process. The ATS of the fixed parameters loss function chart is Fp _ ATS = h0 ⋅ ARL. (20). where ARL is ARL =. 1 = 1− β. 1 1−. b −1. ∑C i =0. 10. n i. (21). (δp ) (1 − δp ) i. n −i.

(18) 2 2 2 2 2 2 3 2 4 3 3 3 4 2 3 4 where b = np −np + n p + k ⋅ 6 n p −16 n p +10 n p + 4 n p −4 n p + np −7 np +12 np −6 np .. 3.3 Determination of the UCL, WCL of the Specified VSI Loss Function Chart This research establishes a design of the VSI loss function chart with a specified sampling interval. The design parameters, ( n,WCL, k ), can be determined by minimizing ATS under specified hi and subjects to (1) a specified ARL0 , (2) the range of sample size, 2 ≤ nL ≤ n ≤ nU < ∞ , (3) the range of average inspection rate. n 治 and B and 政 Bh +B h 大. (AIR), 0 < AIR ≤ R , where AIR =. 1. 1 1. 立. B2 are the stabilized. 2 2. p. p . ‧ 國. 學. probabilities for the out-of-control process. Since [ B1 B2 ] = [ B1 B2 ] 11 12   p21 p22  and B1 + B2 = 1 , solving the two equations. B1 , B2 are expressed as 1 − p11 p21 , B2 = . 1 − p11 + p21 1 − p11 + p 21. ‧. B1 =. n. al. Ch. Minimum ATS = f (n,WCL, k ) Subject to. engchi. (1) ARL0 (2) 2 ≤ nL ≤ n ≤ nU < ∞ (3) 0 < AIR ≤ R. 11. er. io. sit. y. Nat The mathematical model can be expressed as. i n U. v.

(19) The procedures search the optimal design parameters (n, WCL, k) are described as follows. Step1: Specify n L , nU , ARL0 , p, δ , h0 , h1 , h2 , R. Step2: p0 can be determined by equation (16) when h1 and h2 are specified. Step3: Searching available combinations (n, k) under the ARL0 . Step4: Determine UCL by using equation (11) when n, k and p are known. Step5: Determine WCL by using by equation (15) when UCL and p0 are known. Step6: Check if h1 , h2 and n satisfy the constraint AIR ≤ R . Then, the design. 政治大. parameters n*, k *, WCL * can be determined under the minimum ATS.. 立. ‧ 國. 學. 3.4 Determination of the UCL, WCL of the Optimal VSI Loss Function Chart If the five design parameters ( n, h1 , h2 ,WCL, k ) of the VSI loss function chart are. ‧. not known. Then, this section provides the application technique to determine the. sit. y. Nat. optimal design parameters through the direct search approach. The objective function. al. er. io. of the optimization is ATS which is the function of the five design parameters and. n. subjects to (1) a specified ARL0 , (2) the range of sample size, 2 ≤ nL ≤ n ≤ nU < ∞ , (3). Ch. engchi. the range of sampling interval, 0 < h2 < h0 < h1 ≤ hU 0 < AIR ≤ R . The mathematical model can be expressed as Minimum ATS = f (n, h1 , h2WCL, k ) Subject to (1) ARL0 (2) 2 ≤ nL ≤ n ≤ nU < ∞ (3) 0 < h2 < h0 < h1 ≤ hU (4) 0 < AIR ≤ R 12. iv n U , (4) the range of AIR,.

(20) The procedures search the optimal design parameters ( n, h1 , h2WCL, k ) are described as follows. Step1: Specify n L , nU , ARL0 , p, δ , h0 , hU , R . Step2: Searching available combinations (n, k) under the ARL0 . Step3: Determine UCL by using equation (11) when n, k and p are known. Step4: Searching h1 within the feasible region of ( h0 , hU ] and searching h2 within the feasible region of (0, h0 ) for minimizing ATS. p0 can be determined by equation (16) when h1 and h2 are known.. 治政 R . Then, the design Step6: Check if h , h and n satisfy the constraint AIR ≤大立. Step5: Determine WCL by using equation (15) when UCL and p0 are known. 1. 2. parameters n*, k *, h1 *, h2 *, WCL * can be determined under the minimum. ‧. ‧ 國. 學. io. sit. y. Nat. n. al. er. ATS.. Ch. engchi. 13. i n U. v.

(21) Input: Specify n L , nU , p, δ , h0 , hU , R, ARL0. Using equation (17) to determine various combinations of (n, k) under the specified ARL0. Using equation (11) to determine UCL. Let h1 ∈ (h0 , hU ] Let h2 ∈ (0, h0 ). 治政 Using equation (15) to calculate WCL 大. 立. by knownUCL , p0. ‧ 國. 學 ‧. Using equation (16) to calculate p0 by known h1 , h2. Nat. sit. io. n. er. satisfy AIR ≤ R. al. YES. Ch. engchi. Not Available. y. Check n, h1 and h2 to. i n U. NO. v. Find the optimal design ( n*, k *, h1*, h2 *,WCL * ) from all feasible solutions with minimal ATS. Figure2. Flow Chart of the Design of the Optimal VSI Loss Function Chart. 14.

(22) 3.5 Performance Comparisons 3.5.1 Specified VSI Loss Function Chart for binomial data Find the design parameters of the specified VSI loss function chart by approach describe in Section 3.3. First, specify 5 ≤ n ≤ 200 , ARL0 =355, h1 =1.8, h2 =0.1, h0 =1 and R=500. Let the in-control p = 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.5 andδ=1.5,. 2, 2.5. Table2 shows the design parameters (n*, k*, WCL*) and percentage of saved ATS by comparing with Fp loss function chart. The percentage of saved ATS is Saved ATS % =. Fp ATS − Specified VSI ATS ⋅ 100% Fp ATS. (22). 政治大 Table2. ATS of the VSI and Fp under various p with specified h =1.8, h =0.1, R=500 立 δ p n* k* WCL* VSI_ATS* Fp_ATS ARL AIR Saved ATS% 1. 2. 0. 1.5. NA. ‧ 國. NA. NA. NA. NA. NA. NA. ‧. NA. 2. NA. NA. NA. NA. NA. NA. NA. NA. 2.5. NA. NA. NA. NA. NA. NA. 1.5. 49. 5.90. 9. 20.72. 35.27. 357.25 490.00. 41.27. 2. 49. n. 9. 2.95. 8.94. 357.25 490.00. 67.02. 2.5. 49. 5.90. v ni. 79.35. 1.5. NA. NA. 0.78 3.78 357.25 490.00 e nNAg c h iNAU NA NA. 2. NA. NA. NA. NA. NA. NA. NA. NA. 2.5. NA. NA. NA. NA. NA. NA. NA. NA. 1.5. NA. NA. NA. NA. NA. NA. NA. NA. 2. NA. NA. NA. NA. NA. NA. NA. NA. 2.5. NA. NA. NA. NA. NA. NA. NA. NA. 1.5. 33. 4.05. 121. 1.29. 5.66. 355.48 330.00. 77.13. 2. 19. 4.44. 49. 0.28. 2.05. 354.28 190.00. 86.16. 2.5. 19. 4.44. 49. 0.11. 1.08. 354.28 190.00. 89.82. 1.5. NA. NA. NA. NA. NA. 0.1. 0.2. 0.3 0.5. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. io. 0.05. NA. Nat. 0.02. NA. al. 5.90. C9 h NA. *NA is not available. 15. y. 2.5. NA. sit. 2. NA. NA. er. 0.01. 學. NA. 1.5. NA. NA. NA. NA. NA.

(23) From Table2, the specified VSI loss function chart saves much ATS than the Fp chart when p=0.05 and 0.3. When p=0.05, the VSI loss function chart saves at least 41.27% on ATS and saves at most 79.35%. When p=0.3, the VSI loss function chart saves at least 77.13% on ATS and saves at most 89.82%.. Table3. ATS of the VSI and Fp under various p with specified h1=1.8, h2=0.1, R=700 1.5. NA. NA. NA. NA. NA. 2. NA. NA. NA. NA. 2.5. NA. NA. NA. NA. 1.5. 70. 6.79. 4. 2. 70. 2.5 1.5. NA. NA. NA. NA. NA. NA. NA. NA. NA. NA. 70. 6.79. 4. 2.58. 7.29. 355.50 700.00. 64.54. 49. 5.90. 9. 20.72. 35.27 357.25 490.00. 41.27. 49. 5.90. 9. 2.95. 8.94. 357.25 490.00. 67.02. 49. 5.90. 9. 0.78. 3.78. 357.25 490.00. 79.35. 62. 4.72. 49. 5.37. 13.71 355.71. 620. 60.81. 2. 62. 4.72. 49. 0.46. 2.83. 355.71. 620. 83.75. 2.5. 62. 4.72. 49. 0.16. 1.40. 355.71. 620. 88.72. 1.5. 65. 4.01. 196. 0.99. 4.84. 358.16. 650. 79.64. 2. 65. 196. 0.13. 1.23. 358.16. 650. 89.55. 2.5. 54. al. 4.13. 540. 88.92. 1.5. 33. 4.05. 330.00. 77.13. 2. 19. 4.44. 49. 0.28. 2.05. 354.28 190.00. 86.16. 2.5. 19. 4.44. 49. 0.11. 1.08. 354.28 190.00. 89.82. 1.5. 70. 3.28. 1296. 0.11. 1.06. 358.25. 89.98. ‧. n. 4.01. y. 2. 立. io. 0.5. NA. 6.79. Nat. 0.3. Saved ATS%. 24.99. 1.5. 0.2. AIR. 40.25 治 53.66 355.50 700.00 政大355.50 700.00 4 8.29 16.34. 2.5 0.1. WCL* VSI_ATS* Fp_ATS ARL0. 學. 0.05. k*. sit. 0.02. n*. er. 0.01. δ. ‧ 國. P. v i 144 0.15 1.39 n358.27 Ch 121 e n1.29 g c h i5.66U 355.48. 700. 49.28. *NA is not available. However, the parameter design set cannot be found when p=0.01, 0.02, 0.1, 0.2 and 0.5 because the restriction of AIR is too small (R=500). In Table3, the range of R is enlarged to 700 and the design parameters results are existent, except=0.01. When R=700, the VSI loss function chart saves at least 24.99% ATS and at most 89.98% ATS compared with the Fp loss function chart. The performance of the specified VSI 16.

(24) loss function chart outperforms the Fp loss function chart with R=700. We summarize results from data analyses as follows. (1) The value of k becomes smaller when p is increasing, except p=0.2, 0.3. (2) The bigger theδis, the smaller the ATS is. (3) The larger the p is, the smaller the ATS is. (4) The specified VSI loss function chart has better performance than the Fp loss function chart. (5) It is better to set R=700 rather than 500 of the specified VSI loss function chart. 3.5.2 Optimal VSI Loss Function Chart for binomial data Find the design parameters of the optimal VSI loss function chart by approach. 政治大 and R=500. The in-control立 p = 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.5 andδ=1.5, 2, 2.5.. describe in Section 3.4. First, specify 5 ≤ n ≤ 200 , ARL0 =355, 0 < h2 < h0 = 1 < h1 ≤ 2 ,. ‧ 國. 學. Table4 shows the optimal design parameters (n*, k*, h1*, h2*, WCL*) and percentage of saved ATS compared to Fp loss function chart. The percentage of saved ATS is Fp ATS − Optimal VSI_ATS ⋅ 100% Fp ATS. ‧. Saved ATS % =. Nat. y. (23). io. sit. In Table4, the optimal VSI loss function chart saves ATS at least 37.09% and at. n. al. er. most 89.97% ATS compared to the Fp loss function chart under various p.. Ch. engchi. 17. i n U. v.

(25) Table4. ATS of the Optimal VSI and Fp under various p with R=500. 0.1. 4. 28.02. 44.54. 356.65 298.63. 37.09. 2. 185 6.39. 2. 0.2. 4. 5.30. 12.54. 356.65 457.13. 57.76. 2.5. 185 6.39. 1.7. 0.1. 4. 8.36. 16.73. 357.37 290.03. 50.03. 1.5. 119 6.00. 1.7. 0.1. 9. 21.89. 36.49. 354.76 203.04. 40.02. 2. 119 6.00. 1.7. 0.1. 9. 3.35. 9.50. 354.76 365.55. 64.79. 2.5. 119 6.00. 1.6. 0.2. 9. 1.27. 4.07. 354.76 419.68. 68.93. 1.5. 187 4.40. 2. 0.2. 100. 3.64. 9.09. 356.15 493.94. 59.93. AIR. ATS%. 2. 83. 5.22. 1.6. 0.1. 25. 1.16. 4.87. 354.45 396.18. 76.22. 2.5. 49. 5.90. 1.8. 0.1. 9. 0.78. 3.78. 357.25 274.75. 79.35. 1.5. 169 3.95. 2. 0.3. 354.84 497.36. 64.73. 355.71 449.19. 84.12. 355.71 305.42. 78.95. 356.76 439.16. 59.78. 356.76 440.00. 60.00. 治 289 1.44 4.09 政大 1.7 0.1 49 0.45 2.83 立 1.6 0.2 49 0.29 1.40. 2. 62. 4.72. 2.5. 62. 4.72. 1.5. 176 3.52. 2. 0.4 1156. 0.62. 1.55. 2. 176 3.52. 2. 0.4 1156. 0.40. 1.00. 2.5. 176 3.52. 2. 0.4 1156. 0.40. 1.00. 356.76 440.00. 60.00. 1.5. 93. 3.53. 1.9. 0.2. 784. 0.34. 1.64. 355.74 460.42. 79.49. 2. 19. 4.44. 2. 0.1. 36. 0.23. 2.05. 354.28 179.54. 88.86. 2.5. 19. 4.44. 2. 0.1. 36. 0.11. 1.08. 354.28 189.96. 89.97. 1.5. 70. 3.28. 2. 0.2 1225. 0.21. 1.06. 358.25 349.99. 79.99. n. al. Ch. engchi. ‧. io. 0.5. 0.2. Nat. 0.3. 2. ARL0. 學. 0.2. 185 6.39. h1* h2* WCL* VSI_ATS* Fp_ATS. y. 0.05. 1.5. k*. sit. 0.02. n*. er. 0.01. Saved. δ. ‧ 國. p. i n U. v. Figure3. Main Effect Plot of the Optimal VSI Loss Function Chart (p, R=500) 18.

(26) 立. 政治大. Figure4. Main Effect Plot of the Optimal VSI Loss Function Chart (δ, R=500). ‧ 國. 學. To investigate the effects of various p andδon optimal design parameters (n*,. ‧. k*, h1*, h2*) and ATS, the main effect plots are illustrated in Figures 3 and 4. We. y. Nat. io. sit. summarize the results from data analyses and plots as follows. (1)The average of. n. al. er. optimal sample size n becomes small when p increases, except p=0.2 and 0.5, the. i n U. v. average of optimal sample size n also becomes small whenδincreases. (2)The. Ch. engchi. average ATS decreases when p orδincreases. (3)The average optimal k decreases when p is getting large, except p=0.2, the average optimal k increases whenδis getting large. (4)The distance between h1 and h2 is a fixed constant under various p andδ. If user has known the length of h1 or h2, the other one sampling interval can be obtained by the fixed distance between long and short sampling intervals.. 19.

(27) Table5. ATS of the Optimal VSI and Fp under various p with R=400 δ n*. k*. h2*. 2. 0.2. 4. 28.02. 44.54. 356.65 298.63. 37.09. 2 185 6.39 1.9. 0.3. 4. 6.24. 12.54. 356.65 383.85. 50.23. 2.5 138 6.85 1.9. 0.1. 4. 2.90. 7.50. 357.37 396.31. 61.36. 1.5 119 6.00 1.7. 0.1. 9. 21.89. 36.49. 354.76 203.04. 40.02. 2 119 6.00 1.7. 0.1. 9. 3.35. 9.50. 354.76 365.55. 64.79. 2.5 119 6.00 1.9. 0.2. 9. 1.36. 4.07. 354.76 394.76. 66.56. 1.5 187 4.40. 0.3. 100. 4.45. 9.09. 356.15 399.01. 51.05. 1.5 185 6.39 0.01. 0.02. 0.05. 2. 2. 25. 1.16. 4.87. 354.45 396.18. 76.22. 2.5 49. 5.90 1.8. 0.1. 9. 0.78. 3.78. 357.25 274.75. 79.35. 0.2. 121. 355.84 383.88. 69.90. 357.10 381.03. 77.09. 2. 2.5 62. 4.72 1.6. 0.2 立. 1.5 176 3.52 1.9. 1.40. 355.71 305.42. 78.95. 0.5. 1156. 0.78. 1.55. 356.76 351.53. 49.81. 2 176 3.52 1.9. 0.5. 1156. 0.50. 1.00. 356.76 352.00. 50.00. 2.5 176 3.52 1.9. 0.5. 1156. 0.50. 1.00. 356.76 352.00. 50.00. 3.53 1.1. 0.2. 1156. 0.48. 1.64. 355.74 394.10. 70.79. 4.44. 0.1. 36. 0.23. 2.05. 354.28 179.54. 88.86 89.97 79.99. 19. ‧ 國. 0.29. 2. 學. 49. y. 0.2. sit. 4.47 1.7. 2.10 6.98 治政 81 0.48 2.09 大. ‧. 82. 2. Nat. 0.5. ATS%. 0.1. 1.5 93 0.3. AIR. 5.22 1.6. 2. 0.2. ARL0. 83. 1.5 110 4.24 0.1. WCL* VSI_ATS* Fp_ATS. Saved. h1*. 2.5 19. 4.44. 2. 0.1. 36. 0.11. 1.08. 354.28 189.96. 1.5 70. 3.28. 2. 0.2. 1225. 0.21. 1.06. 358.25 349.99. io. n. al. er. p. Ch. i n U. v. Although the optimal VSI loss function chart saves more detective time than the. engchi. Fp loss function chart, the AIR and n are quite big. The bigger AIR or n means cost consuming. Thus, R may reduce to 400 (see Table5) for saving loss. Compared the optimal design parameters among Table4 (R=500) and Table5 (R=400), (p=0.01,δ =2.5, R=400) and (p=0.1,δ=1.5, R=400) save more sample size than design parameters with R=500, n from 185 decreases to 138 and from 169 decreases to 110. Only (p=0.1, δ=2) with R=400 increases sample size from 62 to 82. The other sets stay the same parameters results. The AIR are smaller averagely when R=400. Use R=400 is a good alternative to save cost, although we may sustain slightly increase of ATS. 20.

(28) 政治大. Figure5. Main Effect Plot of the Optimal VSI Loss Function Chart (p, R=400). 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure6. Main Effect Plot of the Optimal VSI Loss Function Chart (δ, R=400). Figures 5 and 6 are the main effect plots of the optimal VSI loss function chart with R=400 under various p andδ. We summarize the results from main effect plots as follows. (1) The average of optimal sample size n becomes smaller when p increases, except p=0.2 and 0.5, the average of optimal sample size n becomes smaller whenδ increases. (2) The average of ATS decreases when p orδincreases. 21.

(29) (3) The average optimal k decreases when p becomes large, except p=0.3, the average optimal k increases whenδbecomes large. (4) The distance between h1 and h2 is a fixed constant under various p andδ. From data analyses in Section 3.5, we summarize that (1) the optimal VSI loss function and the specified VSI outperform the Fp loss function chart, (2) the ATS of the optimal VSI loss function chart is smaller than the specified VSI loss function chart (see Table2 and Table4), (3) it is important to give proper R when using specified VSI loss function chart, or the design parameters results may not be. 政治大 loss function chart, (5) n decreases when p andδincreases. Consequently, it is 立. available, (4) R=400 is an good alternative to reduce loss when using the optimal VSI. recommended to adopt an optimal VSI loss function chart for detecting a S.C. rapidly.. ‧. ‧ 國. 學 sit. y. Nat. 3.6 Example. io. er. A manager is concerned about the defect proportion of baked pies at store. From the analyses in Section 3.5, the optimal VSI loss function chart outperforms the. al. n. v i n C Hence, specified VSI loss function chart. managerU determines to construct an h e nthe i h gc. optimal VSI loss function chart for monitoring loss caused by the process proportion shift. The historical record shows the in-control proportion of defected pies, p, is 0.02. The store can accept the sample size within [5, 200], the maximum sampling interval is 2, and an AIR is less than 500 due to process capacity. Consider ARL0 =355 and proportion scaleδ=2.5. Bases on those information,. p = 0.02, n L = 5, nU = 200, hU = 2, ARL0 = 355, R = 500, δ = 2.5 . The manager uses the approach describes in Section 3.4 to determine the optimal design parameters of the VSI loss function chart as follows (see Table6). Use those 22.

(30) design parameters, the optimal VSI loss function chart for the pie store can be established as Figure7. Table6. Optimal Design of the VSI Loss Function Chart p. n. h1. h2. UCL. WCL. ARL0. 0.02. 119. 1.6. 0.2. 36. 9. 354.76. I3 (Action region). UCL=36 (h2=0.2). I2 (Warning region). WCL=9. 立. 治政 I (Central region) 大. (h1=1.6). 1. LCL=0. ‧ 國. 學. Figure7. Optimal VSI Loss Function Chart. ‧. Bases on this plan, the store collects 24 samples with n=119 and constructs an. y. Nat. io. sit. optimal VSI loss function chart to monitor the quality of pies. If the current sample. n. al. er. point is located within the central region I1 , the next sample should adopt h1 =1.6 as. Ch. i n U. v. sampling interval. If the current sample point is located within the warning region I 2 ,. engchi. the next sample should adopt h2 =0.2 as sampling interval. If the current sample point is located outside the UCL, the occurred S.C. should be searched and removed from the process. Table7 shows the sampling results using the optimal VSI loss function chart scheme. The first sampling interval was decided randomly by the probability p0 ' =0.13 of using h1 =1.6 and the probability (1- p0 ' )=0.87 of using h2 =0.2. In this example, first sample used h1 =1.6 as sampling interval. The 1st data point L=1 is located within WCL=9, thus, the 2nd sample should adopt 1.6 as sampling interval. After 1.6 hours, the 2nd sample was taken. The 2nd data point L=9 is located between 23.

(31) 9 and 36, thus, the 3rd sample should adopt 0.2 as its sampling interval. After 1.6 hours from 2nd sample, the 3rd was taken. The 3rd data point L=9 is located between 9 and 36, thus, the 4th sample should still adopt 0.2 as its sampling interval. After 0.2 hour from the 3rd sample, the 4th sample was taken. The other data points follow the same rule to determine the use of sampling intervals.. Table7. Sampling data and VSI Loss Function Chart Sample. hi random. 1. choose h1=1.6. X. L. 1. 1. 9 立 3 9 3. 3. h2=0.2. 4. h2=0.2. 3. 5. h2=0.2. 6. hi. 13 h =1.6 政I 治大h =0.2 I 14. X. L. Located Region. 1. 1. 3. 9. I2. 2. 2. 2. 4. I1. 2. 4. I1. h1=1.6. 9. I2. 16. h1=1.6. 2. 4. I1. 1. 1. I1. 17. h1=1.6. 5. 25. I2. h1=1.6. 1. 1. I1. 18. h2=0.2. 1. 1. I1. 7. h1=1.6. 2. 4. I1. 19. h1=1.6. 2. 4. I1. 8. h1=1.6. 4. 16. I2. 20. h1=1.6. 10. 100. I3. 9. h2=0.2. 3. 9. I2. 21. 3. 9. I2. 10. h2=0.2. I2. 22. 2. 4. I1. 11. h2=0.2. 2. a l9. 3. 9. I2. 12. h1=1.6. 1. 2. 4. I1. io. Ch I 23 e n g c24h i 1 I 4. 1 1. y. sit. Nat. 3. random choose h1=1.6. er. ‧ 國. 15. 學. I2. ‧. h1=1.6. Region. Sample. n. 2. Located. iv n U h =1.6 h2=0.2 1. Figures8 shows the constructed optimal VSI loss function chart. The point on the 20th sample falls on action region, thus, the occurred S.C. should be searched and removed from the process. ATS of the optimal VSI loss function chart is 1.27 after calculation.. 24.

(32) Figure8. Optimal VSI Loss Function Chart (red points adopt h2; blue points adopt h1). 政治大 save for the pie store by comparing with Fp loss function chart. Table8 shows 立. The manager wants to know how much the optimal VSI loss function chart can. parameters of the Fp loss function chart. Bases on the setting, Fp loss function chart. ‧ 國. 學. was built as Figure9. The point on the 20th sample was out of the UCL and the. al. n. 0.02. Table8. Fp Loss Function Chart. n. 119. er. io p. sit. y. Nat. calculation.. ‧. occurred S.C. should be searched. ATS of the Fp loss function chart is 4.07 after. i n Ch 1 e n g c h36i U h. UCL. Figure9. Fp Loss Function Chart. 25. v. WCL. ARL0. 0. 354.76.

(33) Compared performance between the Fp and optimal VSI loss function chart, the latter saves around 68.8% ATS (see Table9). The VSI loss function chart outperforms the Fp loss function chart significantly and it can help store to monitor defect proportion of pies more effective. Thus, it is better to apply an optimal VSI loss function chart to control the loss and quality of pies.. Table9. Comparison of the Fp and VSI Loss Function Chart Chart. ATS. VSI. 1.27. Fp. 立. Saved ATS% 68.8. 政治大 4.07. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 26. i n U. v.

(34) 4. Design of the VSS Loss Function Chart for binomial data 4.1 Construction of the VSS Loss Function Chart for binomial data As Section 3.1 describe, the Fp loss function chart is built with UCL, CL and LCL. The UCL, CL, and LCL of Fp loss function chart can be expressed as equations (4)-(6). The VSS loss function chart is a control chart with adaptive sample sizes. A VSS loss function chart is built with two control limit UCL1 ,UCL2 , two warning limit WCL1 ,WCL2 and a LCL s Fig.10). The LCL is set as zero because fraction. 政治大. nonconforming is the degree of deterioration. Thus, LCL=0 is the best level of quality. 立. loss.. ‧ 國. 學 I3 (Action region). Nat. WCL1. sit. WCL2. I1 (Central region). io. n. er. 0. al. y. UCL2. I2 (Warning region). ‧. UCL1. Ch. i n U. v. Figure10. VSS Loss Function Chart. engchi. A VSS loss function chart can be expressed as follows. See equations (24)-(28).. UCL1 = E ( L1 ) + k1 Var ( L1 ). (24). WCL1 = E ( L1 ) + w1 Var ( L1 ). (25). UCL2 = E ( L2 ) + k 2 Var ( L2 ). (26). WCL2 = E ( L2 ) + w2 Var ( L2 ). (27). LCL=0. (28). 0 < WCL1 < UCL1 , 0 < WCL2 < UCL2 , where k1, k2 are control limit factors and w1, w2 27.

(35) are warning limit factors, 0 < w1 < k1 < ∞ , 0 < w 2 < k 2 < ∞ . Refer to equations (2)-(3), E ( L1 ) = E ( L | n = n1 ) , E ( L2 ) = E ( L | n = n2 ) , Var ( L1 ) = Var ( L | n = n1 ) and. Var ( L2 ) = Var ( L | n = n2 ) . The UCL1, WCL1, UCL2 and WCL2 can be expressed mathematically as equations (29)-(32).. 2 2 2 2 3 3 UCL1 = n1 p − n1 p 2 + n1 p 2 + k1 [6n1 p 2 − 16n1 p 3 + 10n1 p 4 + 4n1 p 3 − 4n1 p 4 + n1 p − 7n1 p 2 + 12n1 p 3 − 6n1 p 4 (29). 2 2 2 2 3 3 WCL1 = n1 p − n1 p 2 + n1 p 2 + w1 [6n1 p 2 − 16n1 p 3 + 10n1 p 4 + 4n1 p 3 − 4n1 p 4 + n1 p − 7n1 p 2 + 12n1 p 3 − 6n1 p 4 (30). UCL2 = n 2 p − n 2 p 2 + n 2 p 2 + k 2 [6n 2 p 2 − 16n 2 p 3 + 10n 2 p 4 + 4n 2 p 3 − 4n 2 p 4 + n 2 p − 7n 2 p 2 + 12n 2 p 3 − 6n 2 p 4 (31) 2. 2. WCL2 = n 2 p − n 2 p + n 2 p + w2 2. 2. 2. 2. 2. 3. 3. 治政 [6n p − 16n p + 10n p + 4n p − 4大 n p + n p − 7n p + 12n p − 6n p (32) 立 2. 2. 2. 2. 2. 2. 3. 2. 3. 4. 3. 3. 2. 2. 4. 2. 2. 3. 2. 2. 4. 2. where n1 is small sample size, n2 is large sample size.. ‧ 國. 學 ‧. In a VSS loss function chart, WCL1 and WCL2 are the keys to shorten the time. sit. y. Nat. of detecting process shift. A VSS loss function chart has a large sample size n2 and a. io. between samples.. er. small sample size n1. WCL1 and WCL2 are the guard to decide the use of n1 or n2. al. n. v i n C hchart, two differentUsample size, n When using VSS loss function engchi. 1. and n2, are. adopted. Users have to decide on a large sample size n2 and a small sample size n1, where n2 >n1. If the data point is plotted on the central region (I1), use the small sample size n1, UCL1 and WCL1 of the next sample. If the data point is plotted on the warning region (I2), use the large sample size n2, UCL2 and WCL2 of the next sample. If the data point is plotted on the action region (I3), find the S.C. and repair the process. For comparing the VSS loss function chart with the Fp loss function chart under the same standard, the average sample size needs to be the same when the process is in control. The equation (33) needs to be satisfied.. 28.

(36) n1 ⋅ p0 + n2 ⋅ (1 − p0 ) = n0. (33). The average sample size of the VSS loss function chart is the same as fixed sample size of the Fp loss function chart, where n0 is fixed sample size of the Fp loss function chart, 0 < n1 < n0 < n2 < ∞ , and p0 is the probability of being in central region (I1) when the process is in control. p0 is defined as equation (34). p0 = P(0 ≤ L < WCL1 | 0 ≤ L < UCL1 ) = P(0 ≤ L < WCL2 | 0 ≤ L < UCL2 ) =. P(0 ≤ L < WCL1 ) P(0 ≤ L < WCL2 ) = P(0 ≤ L < UCL1 ) P(0 ≤ L < UCL2 ). (34). 治政大 ) = 1−α P(0 ≤ L < WCL ) = p ⋅ P(0 ≤ L < UCL ) . Let P(0 ≤ L < UCL 立. From equation (34), WCL1 can be expressed in terms of 1. 0. 1. 1. 1. and. ‧ 國. 學. P(0 ≤ L < UCL2 ) = 1 − α 2 . Then, P(0 ≤ L < WCL1 ) = p0 ⋅ (1 - α 1 ) . Replace L as X 2 ,. P(0 ≤ X 2 < WCL1 ) = p0 ⋅ (1 - α 1 ) . It is equivalent to P(0 ≤ X < WCL1 ) = p 0 ⋅ (1 - α 1 ) .. ‧. Using the CDF of X to express the equation. y. . WCL1 − 1 ) = p 0 ⋅ (1 − α 1 ). io. sit. . Nat. FX (. n. al. er. Take the inverse function of both sides..  WCL. 1. . −1. − 1 = FX [ p0 ⋅ (1 − α 1 )]. Ch. engchi. i n U. v. Consequently, the WCL1 can be determined approximately as equation (35). −1. WCL1 ≈ [ FX ( p 0 (1 − α 1 )) + 1 ] 2. (35). The value of p 0 can be calculated through equation (33) and expressed as. p0 =. n0 − n2 n1 − n2. By using similar derivation, WCL2 ≈ [ FX −1 ( p0 (1 − α 2 )) + 1 ]2 . The UCL0 is the upper control limit of the Fp loss function chart and can be defined as equation (37). 29. (36).

(37) UCL0 = n0 p − n0 p 2 + n0 p 2 + k 0 [6n 0 p 2 − 16n 0 p 3 + 10n 0 p 4 + 4n 0 p 3 − 4n 0 p 4 + n 0 p − 7n 0 p 2 + 12n 0 p 3 − 6n 0 p 4 2. 2. 2. 2. 3. 3. (37). where k0 is the control limit factor of the Fp loss function chart. The UCL0 can be determined when n0, p, k0 are given by using equation (37). The UCL1 can be determined when n1, p, k1 are given by using equation (29). The UCL2 can be determined when n2, p, k2 are given by using equation (31).The value of (n0, k0), (n1, k1) and (n2, k2) are decided by ARL0 (see equation (17)).. 政治大 Performance of the VSS loss function chart can be measured by ARL, ATS and 立. 4.2 Performance Measurement. ANOS (average number of observations to signal), where ANOS = n ⋅ ARL . When. ‧ 國. 學. the process is out of control, the smaller ARL, ATS or ANOS means better detective. ‧. ability of the control chart. When the process is in control, the larger ARL can result. sit. y. Nat. in fewer false alarms and costs.. io. er. The Markov chain method is applied to calculate those performance measurements. There are two assumptions of calculating ARL, ATS and ANOS of the. al. n. v i n C hloss function chart U VSS loss function chart. First, the assumes only one S.C. may engchi. occur during the process. Second, process is out-of-control at the beginning of the process starts. ATS and ANOS are calculated under the zero-state mode. Due to the assumptions, the process has two transient states and one absorbing state of Markov chain approach (see Table10).. Table10. State Definition of the VSS Loss Function Chart for binomial data State. S.C. occur. Location of the VSS loss function Chart. 1. Yes. I11. 2. Yes. I12. 3. Yes. I13 30.

(38) Transition probabilities are as the following: P1,1 ( n1 ,UCL1 ,WCL1 ) = p (0 ≤ L < WCL1 | X ~ B( n1 , δp )) = p (0 ≤ X 2 < WCL1 | X ~ B( n1 , δp )) = p (0 ≤ X < WCL1 | X ~ B( n1 , δp )) =. . . WCL1 −1. ∑C i =0. n1 i. (δp ) i (1 − δp ) n1 −i. P1, 2 ( n1 , UCL1 , WCL1 ) = p(WCL1 ≤ L < UCL1 | X ~ B( n1 , δp)) = p(WCL1 ≤ X 2 < UCL1 | X ~ B( n1 , δp )) . = p ( WCL1 ≤ X < UCL1 | X ~ B( n1 , δp )) = i=. . UCL1 −1. . ∑C. n1 i WCL1 −1 +1. . (δp ) i (1 − δp ) n1 −i. P1,3 (n1 , UCL1 , WCL1 ) = p ( L ≥ UCL1 | X ~ B (n1 , δp )) = p ( X 2 ≥ UCL1 | X ~ B (n1 , δp )) = p( X ≥. UCL1 | X ~ B (n1 , δp )) = (1 −. . UCL1 −1. . ∑C i =0. n1 i. (δp ) i (1 − δp ) n1 −i ). P2, j (n2 , UCL2 , WCL2 ) = P1, j (n2 , UCL2 , WCL2 ) , j = 1,2,3. 政治大. P3,1 = P3, 2 = 0, P3,3 = 1. 立. 學. VSS _ ATS = b ( I − Q) −1 h. VSS _ ANOS = b ( I − Q) −1 n. ‧. ‧ 國. From the elementary properties of the Markov chain, the ATS and ANOS are (38) (39). sit. y. Nat. where b is a (1 × 2) vector of the initial probability of state 1 and 2, I is the identity. io. er. matrix of order 2, Q is a 2 by 2 transition probability matrix, n is the vector of the next sample size for state 1 and state 2, h is the vector of the next sampling interval. n. al. for state 1 and state . b = [ p0'. v i n , 1 -Cp h] , n = [n , n ] , h U e n g c h i = [1 , 1] and ' 0. '. '. 1. 2.  P1,1 (n1 , UCL1 , WCL1 ), P1, 2 (n1 , UCL1 , WCL1 )  ' Q=  respectively, where p0 is the  P2,1 (n 2 , UCL2 , WCL2 ), P2, 2 (n 2 , UCL2 , WCL2 ). probability of being at state 1 at the beginning of the process when the process is out-of-control. 1 − p0' is the probability of being state 2 at the beginning of the process when the process is out-of-control.. 31.

(39) p0' is expressed as p0' = P(0 ≤ L* < WCL1 | 0 ≤ L* < UCL1 ) = P(0 ≤ L* < WCL2 | 0 ≤ L* < UCL2 ) =. (40). P(0 ≤ L* < WCL1 ) P(0 ≤ L* < WCL2 ) = P(0 ≤ L* < UCL1 ) P(0 ≤ L* < UCL2 ). where L* belongs to an out-of-control process. The ANOS of the fixed parameters loss function chart is Fp _ ANOS = n0 ⋅ ARL. (41). ARL can refer to equation (21).. 治政 4.3 Determination of the UCL , WCL of the Optimal 大VSS Loss Function Chart 立 ( n , n ,UCL ,UCL ,WCL ,WCL ) of the VSS loss If the six design parameters i. i. 1. 2. 1. 2. 1. 2. ‧ 國. 學. function chart are not known. Then, this section provides the application technique to determine the optimal design parameters through the direct search approach. The. ‧. objective function of the optimization is ATS which is the function of the six design. Nat. sit. y. parameters and subjects to (1) α1 = α 2 = α 0 , where α 0 = P( L ≥ UCL0 ) , (2) the range. n. al. range of AIR, 0 < AIR ≤ R , where AIR =. Ch. i n U. B1n1 + B2 n2 . h. engchi. The mathematical model can be expressed as Minimum ATS = f (n1 , n2 ,UCL1 ,UCL2 ,WCL1 ,WCL2 ) Subject to (1) α1 = α 2 = α 0 (2) 2 ≤ n L ≤ n1 < n0 < n2 ≤ nU < ∞ (3) 0 < WCLi < UCLi ,i=1,2 (4) 0 < AIR ≤ R. 32. er. io. of sample size, 2 ≤ n L ≤ n1 < n0 < n2 ≤ nU < ∞ , (3) 0 < WCLi < UCLi , i=1,2, (4) the. v.

(40) The procedures search the optimal design parameters ( n1 , n2 , UCL1 , UCL2 , WCL1 , WCL2 ) are described as follows.. Step1: Specify n L , nU , α 0 , p, δ , h , R. Step2: Searching available combinations (n0, k0) under the α 0 . Step3: Searching n1 within feasible region of [ nL , n0 ) and searching n2 within the feasible region of ( n0 , nU ] for minimizing ATS. p0 can be determined by equation (36) when n0, n1 and n2 are known. Step 4: k1 can be obtained when α 0 and n1 are known, k2 can be obtained when α 0 and. 政治大 Step5: Determine UCL by立 using equation (29) when n , k and p are known. n2 are known.. 1. 1. 1. ‧ 國. 學. Determine UCL2 by using equation (31) when n2, k2 and p are known. Step6: Determine WCL1 by using equation (30) when UCL1 and p0 are known.. ‧. Determine WCL2 by using equation (32) when UCL2 and p0 are known.. sit. y. Nat. Step7: Check if n1 , n2 and h satisfy the constraint AIR ≤ R . Then, the design. n. al. er. io. parameters n1 *, n2 *,UCL1 *,UCL2 *,WCL1 *,WCL2 * can be determined under the minimum ATS.. Ch. engchi. 33. i n U. v.

(41) Input: Specify n L , nU , α 0 , p, δ , h, R. Using equation (17) to determine various combinations of (n0, k0) under the specified α 0. Determine k1 by known α 0 , p and n1 Determine k2 by known α 0 , p and n2. Searching n1 ∈ [n L , n0 ) , n2 ∈ (n0 , nU ]. Using equation (36) to calculate p0 by known n1 , n2. 立. 政治大. ‧ 國. 學. Using equation (29) to calculate UCL1 by known n1, k1 and p. Using equation (31) to calculate UCL2 by known n2, k2 and p.. ‧. n. al. er. io. sit. y. Nat. Using equation (30) to calculate WCL1 by known UCL1 and p0 . Using equation (32) to calculate WCL2 by known UCL2 and p0 .. Ch. engchi. Check n1 , n2 and h to. i n U. satisfy AIR ≤ R. v. Not Available NO. YES. Find the optimal design ( n1 *, n2 *,UCL1 *,UCL2 *,WCL1 *,WCL2 * )from all feasible solutions with minimal ATS Figure11. Flow Chart of the Design of the Optimal VSS Loss Function Chart. 34.

(42) 4.4 Performance Comparisons Find the design parameters of the optimal VSS loss function chart by approach describe in Section 4.3. First, specify 5 ≤ n ≤ 200 , α 0 =0.00283( ARL0 =355), h=1 and R=500. The in-control p =0.001, 0.01, 0.02, 0.05, 0.1, 0.3, 0.5 andδ=1.5, 2, 2.5. Table11 and Tables12 show the optimal design parameters n1 *, n2 *,UCL1 *,UCL2 *,WCL1 *,WCL2 * , percentage of saved ATS and percentage of saved ANOS compared to Fp loss function chart. The percentage of saved ATS is Fp ATS − Optimal VSS_ATS ⋅ 100% Fp ATS. (42). Fp ANOS − Optimal VSS_ANOS ⋅ 100% Fp ANOS. (43). 政治大 The percentage of saved ANOS 立 is Saved ATS % =. ‧ 國. 學. Saved ANOS % =. Table11 shows optimal design of the VSS loss function chart with p=0.001, 0.01. ‧. and 0.02. Table12 shows optimal design of the VSS loss function chart with p=0.05,. y. Nat. io. 1 ) ARL0 _ 1. al. er. 1 ) are not easy to be exactly the same. Thus, control ARL0 _ 2. n. and α 2 (. sit. 0.1, 0.3 and 0.5. Due to the trait of discrete distribution, binomial, α 1 (. Ch. engchi. i n U. v. ARL0 _ 1 − ARL0 < 5 and ARL0 _ 2 − ARL0 < 5 of calculation, except p=0.001. When p=0.001, let ARL0 _ 1 − ARL0 < 10 and ARL0 _ 2 − ARL0 < 10 for existent results.. 35.

(43) Table11. Optimal VSS and Fp with R=500, p=0.001, 0.01, 0.02 p. 0.001. 0.01. 0.02. δ. 1.5. 2. 2.5. 1.5. 2. 2.5. 1.5. 2. 2.5. n0. 77. 77. 77. 138. 138. 138. 119. 119. 119. k0. 11.61. 11.61. 11.61. 6.85. 6.85. 6.85. 6.00. 6.00. 6.00. n1*. 76. 76. 76. 58. 58. 58. 70. 70. 70. n2*. 78. 78. 78. 185. 185. 185. 174. 174. 174. WCL1*. 1. 1. 1. 1. 1. 1. 4. 4. 4. UCL1*. 1.08. 1.08. 1.08. 9.96. 9.96. 9.96. 27.37. WCL2*. 1. 1. 1. 4. 4. 4. 16. 16. 16. UCL2*. 4. 4. 4. 49. 49. 49. 100. 100. 100. VSS_ATS*. 166.99. 96.11. 62.95. 52.20. 14.06. 5.93. 30.73. 6.77. 2.84. 27.37 27.37. Fp_ANOS. 61.91 54.51 16.73 7.50 36.49 9.50 4.07 治政 12725.33 7330.20 4804.27 7699.83 2297.96 大 1011.32 4436.72 1082.63 469.05 12602.90 7266.82 立 4767.00 7521.74 2309.02 1035.30 4342.62 1131.08 484.90. ARL0. 359.24 359.24 359.24 357.37 357.37 357.37 354.76 354.76 354.76. 163.67. 76.28. 76.35 148.77 167.03 176.81 145.91 164.31 171.25. -2.02. -1.84. -1.68. 4.23. 15.97. 20.91. 15.80. 28.81 30.29. -0.87. -0.78. -2.37. 0.48. y. Saved ANOS%. 76.22. ‧. Saved ATS%. 350.26 359.24 359.24 356.65 356.65 356.65 351.65 351.65 351.65. Nat. AIR. 368.57 359.24 359.24 362.24 362.24 362.24 355.50 355.50 355.50. 2.32. -2.17. 4.28. -0.97. io. sit. ARL0_2. 學. ARL0_1. ‧ 國. VSS_ANOS. 94.37. er. Fp_ATS. al. In Table 11 and Table 12, the optimal VSS loss function chart can save more. n. v i n C h except p=0.001 and ATS than the Fp loss function chart, e n g c h i U 0.3. The optimal VSS loss function chart can save at least 3.86% and at most 30.47% without considering. p=0.001 and 0.3. When p is 0.001 or 0.3, the VSS loss function did not have better performance than the Fp loss function chart. Compared ANOS of the VSS and Fp loss function chart, VSS consumes more observations frequently. It is better to adopt the Fp loss function when p is too small or too large.. 36. 3.27.

(44) Table12. Optimal VSS and Fp with R=500, p=0.05, 0.1, 0.3, 0.5 P. 0.05. 0.1. 0.3. 0.5. δ. 1.5. 2. 2.5. 1.5. 2. 2.5. 1.5. 2. 2.5. 1.5. n0. 83. 83. 83. 110. 110. 82. 19. 19. 19. 70. k0. 5.22. 5.22. 5.22. 4.24. 4.24. 4.47. 4.44. 4.44. 4.44. 3.28. n1*. 49. 49. 49. 62. 89. 62. 19. 19. 19. 48. n2*. 95. 95. 95. 154. 154. 110. 33. 33. 33. 89. WCL1*. 4. 4. 4. 49. 121. 64. 144. 144. 144. 625. UCL1*. 55.48 55.48 55.48 172.61 302.55 172.61 133.29 133.29 133.29 1089.37. WCL2*. 16. 16. 16. 256. 324. 169. 324. 324. 324. 2025. UCL2*. 144. 144. 144. 729. 729. 441. 324. 324. 324. 3364. VSS_ATS*. 19.93. 4.23. 1.89. 4.85. 1.26. 1.07 11.48 2.05. 1.08. 1.01. 6.98 1.58 1.18 11.48 2.05 治 116.78 218.07 38.95 政 1823.08 397.44 178.99 710.04 190.68 大 1815.00 404.58 立 177.80 767.65 174.15 96.87 218.07 38.95. 1.08. 1.06. ARL0_2. 357.77 357.77 357.77 353.30 353.30 355.84 355.48 355.48 355.48 360.69 91.74 94.46 94.93 150.05 153.85 109.99 20.34 28.84 32.95 89.00 8.86. 13.25 11.62 30.47 20.51 9.68. 0.00. Nat. Saved ATS%. 357.25 357.25 357.25 355.71 351.26 355.71 354.28 354.28 354.28 362.63. ‧. AIR. 20.60 73.86. 354.45 354.45 354.45 355.84 355.84 357.10 354.28 354.28 354.28 358.25. 1.76. y. ARL0_1. 20.60 90.25. 學. ARL0. 2.14. Saved ANOS% -0.44. -0.67. 7.50. 0.00. 0.00. 3.86. -9.49 -20.56 -0.00 -0.00. 0.00. -22.19. io. sit. Fp_ANOS. 4.87. er. VSS_ANOS. 21.87. ‧ 國. Fp_ATS. al. Figures 12 and 13 are the main effect plots of the optimal VSS loss function. n. v i n chart under various p andδ. WeCsummarize resultsUfrom data analyses and plots h e n gthe h c i. as follows. (1) The average of ATS decreases when p increases, except p=0.3. (2) The average of WCL1 and WCL2 increase when p increases. The average of UCL1 and UCL2 increase when p increases, except p=0.3. (3) The average of ATS decreases whenδincreases. (4) The average of UCL1, UCL2, WCL1 and WCL2 decrease when δincreases. (5) The VSS loss function chart outperforms Fp loss function chart,. except p=0.001 and p=0.3.. 37.

(45) 立. 政治大. ‧ 國. 學. Figure12. Main Effect Plot of the Optimal VSS Loss Function Chart (p, R=500). ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure13. Main Effect Plot of the Optimal VSS Loss Function Chart (δ, R=500). 38.

(46) 4.5 Example A manager is concerned about the defect proportion of baked pies at store. From the analyses in Section 4.4, the optimal VSS loss function chart outperforms the Fp loss function chart when p is not extremely large of small. Hence, the manager determines to construct an optimal VSS loss function chart for monitoring loss caused by the process proportion shift. The historical record shows the in-control proportion of defected pies, p, is 0.02. The store can accept the sample size within [5, 200], the sampling interval is set as 1 hour, and an AIR is less than 500 due to the process capacity. Consider α 0 = 0.00282 ( ARL0 =355) and proportion scaleδ=2.5. Base on those information,. 政治大. 立. p = 0.02, n L = 5, nU = 200, h = 1, α 0 = 0.00282, R = 500, δ = 2.5 .. ‧ 國. 學. The manager uses the approach in describes Section 4.3 to determine the optimal. ‧. design parameters of the VSS loss function chart as follows (see Table13). Use those. sit. y. Nat. design parameters, the optimal VSS loss function chart for the pie store can be. io. er. established as Figure14.. al. ARL0. 119. 354.76. n. v i n Table13. OptimalC Design of the VSS Loss h e n g c h i U Function Chart n k n n WCL UCL WCL UCL. p 0.02. 1. 6. 70. 2. 174. 1. 4. 1. 27.37. 2. 16. 2. 100. I3 (Action region) UCL1=27.37. UCL2=100. I2 (Warning region). (n2=174). (n2=174) WCL2=16. WCL1=4. I1 (Central region). (n1=70) LCL=0. (n1=70) LCL=0. Figure14. Optimal VSS Loss Function Chart 39.

(47) Bases on this plan, the store collectes 24 samples with h=1 hour and constructs an optimal VSS loss function chart to monitor the quality of pies. If the current sample point is located within the central region I1 , the next sample should adopt n1 =70 as sample size. If the current sample point is located within the warning. region I 2 , the next sample should adopt n2 =174 as sample size. If the current sample point is located outside the UCL, the occurred S.C. should be searched and removed from the process. Table14 shows the sampling results using the optimal VSS loss function chart. 治政 using n =70 and the probability (1- p ' )=0.85 of using 大 n =174. In this example, first 立 scheme. The first sample size was decided randomly by the probability p0 ' =0.15 of 1. 0. 2. sample used n1 =70 as sample size with UCL1=27.37 and WCL1=4. The 1st data point. ‧ 國. 學. L=1 is located within WCL1=4, thus, the 2nd sample should adopt 70 as sample size. ‧. with UCL1=27.37 and WCL1=4. The 2nd data point L=9 is located between WCL1=4. y. Nat. and UCL1=27.37, thus, the 3rd sample should adopt 174 as its sample size with UCL2. er. io. sit. =100 and WCL2=16. The 3rd data point L=9 is located within WCL2=16, thus, the 4th sample should adopt 70 as its sample size. The other data points follow the same rule. n. al. i to determine the use of sample C sizes, UCL and WCL U . n hengchi i. 40. i. v.

(48) Table14. Sampling data and VSS Loss Function Chart Sample. ni random choose. 1. n1=70. X. L. 1. 1. Located. Located. Sample. ni. X. L. I1. 13. n1=70. 3. 9. I2. Region. Region. 2. n1=70. 3. 9. I2. 14. n2=174. 2. 4. I1. 3. n2=174. 3. 9. I1. 15. n1=70. 2. 4. I2. 4. n1=70. 3. 9. I2. 16. n2=174. 2. 4. I1. 5. n2=174. 1. 1. I1. 17. n1=70. 5. 25. I2. 6. n1=70. 1. 1. I1. 18. n2=174. 1. 1. I1. 7. n1=70. 2. 4. I2. 19. n1=70. 2. 4. I2. 8. n2=174. 4. 16. I2. 20. n2=174. 10. 100. I3. 9. n2=174. 3. 9. 3. 9. I2. 立9. 3. 11. n2=174. 2. 12. n1=70. 1. 1. I2. 22. n2=174. 2. 4. I1. 4. I1. 23. n1=70. 3. 9. I2. 1. I1. 24. n2=174. 2. 4. I1. ‧. ‧ 國. n1=70. 1. 學. 10. random choose 政I 治 21 大n =70. Figures15 shows the constructed optimal VSS loss function chart. The point on. y. Nat. er. io. sit. the 20th sample falls on action region, thus, the occurred S.C. should be searched and removed from the process. ATS of the optimal VSS loss function chart is 2.84 and. n. al. Ch. ANOS is 469.05 after calculation.. engchi. i n U. v. Figure15. Optimal VSS Loss Function Chart (yellow points adpot n1 and right scale; green points adpot n2 and left scale) 41.

(49) The manager wants to know how much the optimal VSS loss function chart can save for the pie store by comparing with Fp loss function chart. Figure9 shows. the. Fp loss function chart. The ATS of Fp loss function chart is 4.07 and the ANOS of Fp loss function chart is 484.9 after calculation. Compared performance between the Fp and optimal VSS loss function chart, the latter saves around 30.29% ATS and 3.27 %ANOS (see Table15). The VSS loss function chart outperforms Fp loss function chart significantly and it can help store to monitor defect proportion of pies more effective. Thus, it is better to apply an optimal. 政治大. VSS loss function chart to control the loss and quality of pies.. 立. ATS. VSS. 2.84. Fp. 4.07. Saved ATS% 30.29. ANOS. Saved ANOS%. 469.05. 3.27. 484.9. ‧. Chart. 學. ‧ 國. Table15. Comparison of the Fp and VSS Loss Function Chart. n. er. io. sit. y. Nat. al. Ch. engchi. 42. i n U. v.

(50) 5. Design of the VP Loss Function Chart for binomial data 5.1 Construction of the VP Loss Function Chart for binomial data The VP loss function chart is a control chart with variable sampling intervals ( h1 , h2 ), variable sample sizes ( n1 , n2 ) and variable control limit factors ( k1 , k 2 ) to detect the process. A VP loss function chart is built with two control limit UCL1 ,UCL2 , two warning limit WCL1 ,WCL2 and a LCL (see Fig.16). The LCL is set as zero because fraction nonconforming is the degree of deterioration. Thus, LCL=0 is the best level of quality loss.. 立 UCL. 政治大 I (Action region) 3. UCL2. I2 (Warning region). 學. WCL1. WCL2. I1 (Central region). ‧. 0. io. y. sit. Nat. Figure16. VP Loss Function Chart. n. al. er. ‧ 國. 1. i n U. v. A VP loss function chart can be expressed as follows. See equations (44)-(48).. Ch. engchi. UCL1 = E ( L1 ) + k1 Var ( L1 ). (44). WCL1 = E ( L1 ) + w1 Var ( L1 ). (45). UCL2 = E ( L2 ) + k2 Var ( L2 ). (46). WCL2 = E ( L2 ) + w2 Var ( L2 ). (47). LCL=0. (48). 0 < WCL1 < UCL1 , 0 < WCL2 < UCL2. where k1, k2 are control limit factors and w1, w2 are warning limit factors,. 0 < w1 < k1 < ∞ , 0 < w 2 < k 2 < ∞ . Refer to equations (2)-(3), E ( L1 ) = E ( L | n = n1 ) , E ( L2 ) = E ( L | n = n2 ) , Var ( L1 ) = Var ( L | n = n1 ) and Var ( L2 ) = Var ( L | n = n2 ) . 43.