assumption identically change method. 1. Introduction1 .iust.ac.ir/ ABSTRACT identifying KEYWORDS estimation, Correlation,

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1 International Journal Industrial Engineering & Production Research (207) December 207, Volume 28, Number 4 pp DOI: /ijiepr Change-Point Estimation High-Yiel ld Processes in the Presence Serial Correlation Rassoul Noorossana*& Mahnam Najafi Rassoul Noorossana, Industrial Engineering Department, Iran University Science and Technology Mahnam Najafi, Industrial Engineering Department, Institution Higher Education Kar-Qazvin Branch KEYWORDS Change point estimation, High yield process, Maximum likelihood estimation, Correlation, Statistical process control. ABSTRACT Change-point estimation is as an effective method for identifying the time a change in production and service processes. In most the statistical quality control literature, it is usually assumed that the quality characteristic interestt is independently and identically distributedd over time. It is obvious that this assumption could be easily violated in practice. In this paper, we use maximum likelihood estimation method to estimate when a step change occurs in a high-yield process by allowing a serial correlation between observations. Monte Carlo simulation is used as a vehicle to evaluate the performance the proposed method. Results indicate satisfactory performance for the proposed method. 207 IUST Publication, IJIEPR. Vol. 28, No. 4, All Rights Reserved Shewhart np control chart. Noorossana et al.. Introduction (202) compared the performance the The problem change-point estimation high- maximum likelihood estimator and those built- sum yield processes was studied first by Noorossana in change-point estimators cumulative et al. (2009) where a maximum likelihood (CUSUM) and exponential weighted moving estimator (MLE) for a step-change point in the average (EWMA) control charts. They proposed process fraction non-conforming was propose. a confidence set for the change point. The results Zandi et al. (20) introduced a model for a Monte-Carlo simulation show the superiority change-point estimation the process fraction the MLE in identifying the real time the non-conforming with a linear trend and applied a change. Niakii and Khedmati (203) proposed a maximum likelihood estimator when a linear maximum likelihood estimator for the change trend disturbance was present. Then, Monte Carlo point in a high-yield process when a linear trend simulation was applied in order to evaluate the disturbance occurs in the proportion process accuracy and precision the proposed change- was. The performance the proposed point estimator. Next, the proposed estimator change-point estimator in terms both accuracy compared to the maximum likelihood estimator and precision is compared to that the MLE the change-point process fraction non- and Khedmati (204) proposed a maximum the change-point designedd for step changes. Niaki conforming derived under simple-step and monotonic changes following signals from a likelihood estimator a change point in highthe change yield processes while assuming that * Corresponding author: Rassoul Noorossana belonged to a family monotonic changes. rassoul@iust.ac.ir Following a signal from the cumulative count Received 3 August 207; revised 4 October 207; accepted 4 conforming (CCC) control chart, the performance December 207 International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

2 368 Rassoul Noorossana*& Mahnam Najafi the proposed monotonic change-point estimator was evaluated by comparing its performance to the ones designed for step- extensive simulation experiments involving different single step-changes, linear-trend disturbances, and multiple-step changes. In Section 2, the effect correlation in high- yield process is discussed. Section 3 describes change-point estimation high-yield process with single-step change. Section 4 provides a changes and linear-trend disturbances through maximum likelihood estimation method when a step change occurs. Some numerical results are presented in Section 5. The proposed model s performance is evaluated in Section 6, and conclusions are provided in the final section. 2. Effect Correlation on The High- Yield Process Madsen (993) considered several generalized binomial models, one which is referred to as the correlation model. The key feature this model is the allowance a serial correlation for Bernoulli trials. For low values p, i.e., the fraction nonconforming the process, the generalized binomial distribution is approximated by a generalized Poisson distribution known as a modified Poisson distribution. Expression for P(Z n = k), where Z n is the sum n Bernoulli random variables defined as follows: 0 0 ( ) where 0 is the correlation coefficient between any pair observations,, (2) Madsen (993) also showed that the methods moment estimator for are as follows: ρ (3) where denotes the sample variance the observed data. Consider a manufacturing process that produces individual items, each which is inspected in the order production. Let Y denote the total number trials to obtain the first non- conforming item, and Y is conditioned on X 0 =. So, we have: Change-Point Estimation High-Yield Processes in the Presence Serial Correlation, 0, 0,, 0, (4) Now, the probability function Y is as follows: 0, 0,, 0,, 0, 0,, 0, / 2 (5) For i =, we have, (6) If, equation () tells us that sequence {X i } is completely dependent; therefore,. Hence, equation (5) does make sense. Combining equations (4) and (5), we have 2 (7) It is clear that, when 0, equation () reduces to the geometric distribution. 3. Change-Point Identification in Correlated High-Yield Processes with A Single-Step Change Following an unknown point in the time, the fraction non-conformity gets out control. It is assumed that level changes and the process control chart gives a signal at the time T, meaning that observation gets out control limits. Given that our goal is to identify changeobservations,,, belong to the in-control process with fraction non-conforming, while point with step change, observations,,, come from the out--control process with fraction non- used for monitoring the process is geometric chart or g conformity level. The control chart chart proposed by Benneyan ( 99) and Kaminsky et al. (992). Control limits this chart are defined as follows: (8) (9) (0) International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

3 Change-Point Estimation High-Yield Processes in the Presence Serial Correlation 4. Maximum Likelihood Estimation for Correlated Model with A Single-Step Change The proposed estimator for the period when a step change occurs in level is obtained using the maximumm likelihood estimation (MLE) method. When g control chart signals an out- control condition, the proposed method can be applied to determine the period when the step change occurs in the process parameter,. When value 0 maximizes the likelihood function, the step change occurs in this period. Using the following probability distribution function, we can derive the estimator for the change point: 2 (),, (2) The logarithm the likelihood function is as follows:, (3) The value that maximizes the likelihood function is, / The maximum likelihood estimate the change point τ is: max (4) Therefore,,,,,, (5) 5. Numerical Example Suppose that we have a process operating at non-conformity level; subsequently, the control limits for the chart can be calculated as follows: Given that the lower control limit is negative, it is rounded up to zero. Thus, if 0 or , the chart signals an out--control condition, indicating a change in the process non- estimator conformity level. The change-point proposed herein for the period, when the step change occurs in the process level, is based on the MLE method. When g control chart signals an out- control condition, the proposed method can be applied to determine the period when the step change occurs in the process parameter When the value 0 maximizes, the step change occurs in thatt period. We consider the derivation MLE for ; the process change-point period using the MLE technique was proposed by Casella and Berger(990) ). We consider MLE change point as in 0. The value that maximizes the likelihood function is as follows: Rassoul Noorossana*& Mahnam Najafi,. 3..,,,,, is , the estimate process fraction level, and, is the sum inspected units in the periods i+, i+2,...,t.the value i that maximizes is the estimate the last period from the in-control process, and, is its corresponding estimate the changed fraction level. Now, using a simple numerical example can point to the effectivenesss this method. Table () shows the number inspected items,, the values,, and, for 0,,2,,29., 369 (6) International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

4 370 Rassoul Noorossana*& Mahnam Najafi Change-Point Estimation High-Yield Processes in the Presence Serial Correlation According to this table, g control chart signalss a change in the processs non-conformity level at likely occurred at this step. It means that estimate τ, the period when the step change occurs in period T =30. To determine the period when the the process nonconformit ty level, would be τ step change occurs, we have to check the above 26. Therefore, we have to check the records table for the largest value. The largest value the process assigned to a cause that exists around is equal to period 26, showing that the period 26. change in the level has most Tab.. Change-point estimation Period i , Then, we want to generate this case for ρ0.04.,, International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

5 Change-Point Estimation High-Yield Processes in the Presence Serial Correlation Rassoul Noorossana*& Mahnam Najafi Tab. 2. Change-point estimation. Period i,,, In table(2), to determine the period when the step change has occurred, we have to check the above table for the largest value that is equal to period 8, showing that the change in the level has most likely occurred at this step. It means that estimate τ, the period when the step change occurs in the process level, would be τ 8. Therefore, we have to check records in the process for an assignable cause thatt exists around period 8. It is noticeable that a correlation is applied in this case. 6. Performance Evaluation Model Performance the proposed estimator for the period when the step change has occurred in the level is investigated using Monte Carlo simulation while a correlation exists throughoutt the process. Using a geometric distribution, 00 observations from an in-control process with are generated first. If any the observations exceeds the control limits, it is assumed to be a false alarm. We simply replace this observation with an in-control one. This procedure is repeated until all 000 observations are placed between the two control limits. At the 37 beginning 0 period, a shock is induced to the process and nonconformi ity level changes from to δ. Then, observations are generated until an out--control estimatee the period when a change has signal is detected. The point occurred, τ, which should be close to 00, is computed. To estimate the expected period, when the first alarm is given by g control chart,, the period number whose signal is determined has to be recorded. This process is repeated 0,000 times for and different values δ. The mean and standard error 0000 the estimates the periods that the step change has occurred, and, along with the estimate the expected period when the first alarm is given,, for different values increases (δ> ) and decreases (δ <) in the fraction nonconforming is shown in the following tables. The average run length (ARL) in the above expression for refers to the number points plotted on the control chart prior to observing any signal, and it can easily be obtained by τ, subtracted from. International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

6 372 Rassoul Noorossana*& Mahnam Najafi Tab. 3. Average change-point estimates and standard error.,, 0 Tab. 4. Average change-point estimates and standard error.,, Table (3) reveals that a 40% increase in the fraction level. and would be detected by g control chart on average 38.7 periods after the change has actually occurred in the process. However, the MLE provides an average estimatee 24.5 for the period when the step change occurred in fraction nonconforming level that is very close to real change point. The standard error the estimates is 4.555, which is small. The results in Table (3) indicate that the estimates the period when the step change has occurred get closer to the true value as the size the shift in the process nonconforming level increases, which is reasonable enough. According to Table (4), g control chart signals 57.2 periods on average after the process fraction nonconforming level drops by 40%, The change-point estimator performs relatively well by yielding 04.5 as the average estimate for the period when the step change has occurred. The results in Table (4) reveal that the performance the MLE improves as the value the fraction non-conforming decreases. In other words, as the deviation from the in-control value fraction level increases, the standard error estimates decreases. Tab. 5. Average change-point estimate and standard errorr for.,,, and Change-Point Estimation High-Yield Processes in the Presence Serial Correlation Tab. 6. Average change-point estimate and standard error for.,,, and Tab. 7. Average change-point estimates and standard error for.,,, and Tab. 8. Average change point estimate and standard error for.,,, and Tab. 9. Average change point estimate and standard error for.,,, and Tab. 0. Average change point estimate and standard error for.,,, and Table (5) shows that an increase in the fraction level. and. would be detected by g control chart 25.4 periods on average after the change has occurred in the process. The MLE provides an average estimate 07.5 for the period when the step change has occurred in fraction level that is very close to real change point. The standard errorr the estimates is..60 that is very small. Table (5) indicates that the estimate for the change-point period in the case a correlation is closer to the true value. Table (6) indicates that g control chart signals 03.4 periods on average after the process fraction level decreasess by 40% to..the change-point estimator performs relatively well by yielding 03 as the International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

7 Change-Point Estimation High-Yield Processes in the Presence Serial Correlation average estimate for the period when the step change has occurred. The resultss in Table (6) reveal that the performance the MLE improves as the value the fraction nonconforming decreases and correlation increases. Table (7) shows that the increase in the fraction level. and. would be detected by g control chart 9 periods on average after the change has occurred in the process. The MLE provides an average estimate 05 for the period when the step change has occurred in fraction level that is very close to real change point. The standard error the estimates is that is very small. Table (7) indicates that the estimate for period when the correlation exists in process is closer to the true value. Table (8) indicates that g control chart signals 08.9 periods on average after a decrease in the process fraction level to.. The change-point estimator performs relatively well by yielding 02 as the average estimate for the period when the step change has occurred. The results in Table (8) reveal that the Rassoul Noorossana*& Mahnam Najafi 373 performance the MLE improves as the value the fraction nonconforming decreases and correlation increases. Table (9) shows that an increase in the fraction level to. and. would be detected by g control chart 0 periods on average after the change has occurred in the process. The MLE provides an average estimate 00.5 for the period when the step change has occurred in fraction level thatt is very close to real change point. The standard error the estimates is which is very small. Table (9) indicates that the estimates for the period with the correlation in process when the step change occurred gets closer to the true value. We now consider the frequency with which the change point estimate is within m periods the true change point for m=, 2, 3, 4, 5, 0, 5, 20, 25, 30, and 40. The resultss derived from the same simulation study for the increase and decrease in the process fraction levels are given in Tables () and ( 2). Tab.. Precision the estimator:.,, τ τ Tab. 2. Precision the estimator:.,, τ τ / to Table (), the proposed MLE estimates the Table (3) shows that, for a 40% increase in the fraction level., the control chart yields an ARL According true change point 4.% the times correctly. The change point is estimated 6.4% the times within five periods the process change International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

8 374 Rassoul Noorossana*& Mahnam Najafi point. Similarly, for a 60% increase in the fraction level., the control chart yields an ARL For this step change, the proposed MLE estimates the process change point 8.2% the times correctly. The change point is estimated 22% the times within 5 periods the process change point. The results in Table () indicate that the performance the estimator improves as the magnitude the change increases. Table (4) indicates thatt for a 40% decrease in the process fraction level.,the control chart ARL drops to For a step change this magnitude, according to Table (2), the true process change point is estimated 8.2% the times correctly. For a 60% decrease. in the fraction Change-Point Estimation High-Yield Processes in the Presence Serial Correlation level, the control chart yields an ARL 3.9, and the true process change point is estimated 4.75% the times correctly. Simulation results indicate that the change point is estimated 49.86% the times within five periods the true process change point. According to Tables () and (2), it can be shown that with an increase or decrease in the change magnitude, a closer estimate the change point is expected. We now consider the number times a change point estimation is around m periods m=,,2,3,4,5. The results the simulations with different increasing and decreasing in the fraction nonconforming when correlation is present are shown in the following tables. Tab. 3. Precision the estimator:.,,, Tab. 4. Precision the estimator:.,,, Table (5) shows thatt for a 40% increase in., the control chart yields an ARL 3.. Table (3) shows that the MLE estimates the true change point correctly 33% the times. The change point is estimated 67.4% the times within five periods the process change point. Similarly, for a 60% increase., the control chart yields an ARL 2.2. The proposed MLE estimates the process change point correctly 49.7% the times. The change point is estimated 78.8% the times within 5 periods the process. The results in Table (3) indicate that the performance the estimator improves as the magnitude the change and correlation increase. Table (6) indicates that for.,the control chart ARL decreases to According to Table (4), the true process change point is estimated correctly 4.8% the times. For., ARL is 4.4 and the true process change point is estimated correctly 52.4% the times. Resultss indicate that the change point is estimated correctly 85.3% the times within five periods the true process change point. Tab. 5. Precision the estimator:.,,, International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

9 Change-Point Estimation High-Yield Processes in the Presence Serial Correlation Rassoul Noorossana*& Mahnam Najafi Tab. 6. Precision the Estimator:.,,, process. Table (5) indicates that the Table (7) shows thatt for., the control chart yields an ARL 29. Table (5) shows thatt the proposed MLE estimates the true change point correctly 56.7% the times. The change point is estimated 94.7% within five periods the processs change point. For., ARL is The proposed MLE estimates the processs change point correctly 59.9% the times. The change point is estimated performance the estimator improves as the magnitude the change and correlation increase. Table (8) indicates that for., the control chart ARL decreases to 2... According to Table (6), the true process change point is estimated correctly 60.6% the times. Results indicate that the change point is estimated95.6% the times within five periods the true process change point. 97.6% the times within 5 periods the Tab. 7. Precision the estimator:.,,, τ τ Tab. 8. Precision the estimator:.,,, τ τ estimator based on MLE method when Tables (7) and (8) show that by increasing or decreasing the amount change and correlation, the accurate probability estimatee the change correlation can be present to identify the period step change for fraction level in high-yield processes. The performance the point increases. For example, for a 40% increase proposed estimator with different values in., the proposed MLE estimates the true change point correctly 98.8% the times, and 00% times estimated change point is within 4 periods the true processs change point. fraction nonconforming level and correlation was investigated. Results show that the estimator has a reasonable performancee for different levels. Similarly, for a 40 % reduction in the fraction nonconforming , according to Table (8) in 98..8% the times, the change point is correctly estimated within one period and 00% the times within 4 periods the true change point. References [] Noorossana, R., Saghaei, A., Paynabar, K., & Abdi, S.Identifying the period a step change in high-yield processes. Quality and Reliability Engineering International; Vol. 25; 2009, pp Conclusion Knowing the real-time change in the process not only helps process engineers to discover and [2] Zandi, F.., Niaki, S. T. A., Nayeri, M. A., & Fathi, M. Change-point estimation the eliminate sources assignable causes process fraction non-conforming with a effectively, but also increases production linear trend in statistical process control. efficiency in industry. This paper proposed an 375 International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4

10 Noorossana R, Najafi M. Change Point Estimation in High Yield Processes in Presence Serial Correlation. IJIEPR. 207; 28 (4) : URL: Rassoul Noorossana*& Mahnam Najafi International Journal Computer Integrated Manufacturing; Vol. 24, 20, pp [3] Noorossana, R., Paynabar, K., Moradimanesh M. Estimating Time a Simple Step Change in Non-conforming Items in High-Yield Processes.International Journal Industrial Engineering & Production Management;Vol. 22, No. 4, 202, pp [4] Niaki, S. T. A., & Khedmati, M. Change point estimation high-yield processes with a linear trend disturbance. International Journal Advanced Manufacturing Technology; Vol. 69, 203, pp Change-Point Estimation High-Yield Processes in the Presence Serial Correlation [7] Benneyann JC (99): Statistical Control Charts Based on Geometric and Negative Binomial Populations, masters thesis, University Massachusetts, Amherst. [8] Kaminsky FC, Benneyan JC, Davis RB, and Burke RJ "Statistical Control Charts Based on a Geometric Distribution", Journal Quality Technology Vol. 24, No. 2, 992, pp [9] Casella G, Berger RL(990). Statistical Inference. Wadsworth and Brooks /Cole :Belmont, CA,. [5] Niaki, S. T. A., & Khedmati, M. Change point estimation high-yield processes experiencing monotonic disturbances. Computers & Industrial Engineering; Vol. 67, 204, pp [6] Madsen, R. W. Generalised binomial distributions, Communications in Statistics: Theory and Methods, Vol. 22, 993, pp Follow This Article at The Following Site the International Journal Industrial Engineering & Production Research, December 207, Vol. 28, No. 4