Identification of the change point: an overview - Springer Link

Int J Adv Manuf Technol (2013) 64:1663–1683 DOI 10.1007/s00170-012-4131-2

ORIGINAL ARTICLE

Identification of the change point: an overview Karim Atashgar

Received: 5 November 2011 / Accepted: 2 April 2012 / Published online: 6 May 2012 # Springer-Verlag London Limited 2012

Abstract When a control chart addresses an out-of-control condition, a root-cause analysis should be started to identify and eliminate the special cause(s) of variation manifested in the process. The change point refers to the time when a special cause(s) takes place in the process and leads it to a departure from the in-control condition to an out-of-control condition. Identification of the change point is considered as an essential step for a root-cause analysis in both univariate and multivariate processes. If a change manifests in a normally distributed process mean, variance, or both, then the change point should be identified in the process mean, variance, or both, respectively. This paper attempts to comprehensively review the researches that considered the mean change point in different environment corresponding to univariate and multivariate normal processes. Keywords Change point . Monotonic . Isotonic . Antitonic . Statistical process control . Soft computing

1 Introduction Nowadays, statistical process control (SPC) is known as one of the quality professionals’ most powerful methods. Control charts are commonly used as an effective and efficient tool when SPC is approached by quality engineers. A control chart can help quality engineers to monitor the process parameter(s) and to recognize the process variability. In other words, control charts are used to detect whether the

K. Atashgar (*) Industrial Engineering Department, Iran University of Science and Technology, Tehran, Iran, 16846-13114 e-mail: [email protected]

identified problems are due to special causes or common causes of variation. Common causes correspond to systematic problems, while special causes are referred to as non random sources in the nature. Special causes can affect the process parameter(s) in two different ways. One way is to shift the parameter(s) then the change disappears and with a possibility appears again in the process. In this case, the identification of the out-of-control condition is referred to as the isolated special causes. Another way is the sustained special cause in which the change affecting the process remains until it is detected and eliminated. In this case, the new condition leads process to an out-of-control condition and it remains until some corrective actions are taken. There are different types of control charts for different types of data of the observations to statistically control a process. Accordingly, one can find two broad categories of control charts in literature: (a) variable control charts for continuous data, and (b) attribute control charts for attributes data (see Pyzdek [1]; Fig. 1). The application of control charts are first referred to Shewhart [2] when he launched a new approach to quality in 1931. Shewhart [2] showed that the variation of a process cannot be completely eliminated. The novel approach to the nature of the process variation led him to calculate the limits of the process variation when the process is operating in a normal state. The basic rule of SPC in Shewhart’s approach is defined as the variation should be released to chance where it belongs to the common cause and it should be identified and eliminated where it belongs to a special cause. Quality engineers to properly use control charts can conduct control charts under a standard like that introduced by the American Society for Testing and Materials (ASTM) [3] or the American National Standards Institute ANSI/ASQ [4]. In the quality specification control charting arena, Shewhart’s control charts were approached to monitor the

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Int J Adv Manuf Technol (2013) 64:1663–1683

When only COMMON CAUSES are present, the distribution is constant and predictable.

When SPECIAL CAUSE are present, the, distribution varies and is not predictable.

Prediction

Prediction

Time Time Size

Size

Fig. 1 Common cause versus the special cause affects (source: Pyzdek [1])

measurement of the value of the quality process affecting the common cause and special cause; however these charts are only concerned with the location of the recent observations. This fact addresses that Shewhart’s control charts ignore the behavior of the other observations representing the process trend. The major deficiency of these types of control charts is relatively insensitive to small and moderate magnitudes of mean shifts. In order to overcome this drawback of Shewhart control charts several researchers including Nelson [5,6] Roberts [7], Duncan [8], and Grant and Leavenworth [9] introduced the idea of using the supplementary run rules and developed schemes related to control charting. They discussed that the run rules approach increases the sensitivity of the charts when the control chart does not signal. Although run rules approach help quality engineers in identifying the out-of-control conditions where the charts are incapable of signaling them, they cannot radically overcome the Shewhart’s charts weakness point stated above. Accordingly, different researchers developed the Exponential Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) schemes to identify these out-of-control conditions along with increasing the chart’s sensitivity to small and moderate shifts. Page [10] proposed the CUSUM chart to identify small and moderate shifts affect the process. Then Lucas [11] showed that the CUSUM scheme can perform better than Shewhart chart when small changes take place in the process, however, CUSUM scheme is not as effective as Shewhart chart when large shifts manifest in the process. Later, Lucas and Crosier [12] proposed a combined Shewhart-CUSUM chart to detect both small and large shifts. Roberts [13] proposed the EWMA control chart to also enhance the performance of control charts when being interested in detecting small and moderate shifts. Lucas and Saccucci [14] evaluated the EWMA scheme and reported that the performance properties of the EWMA and the

CUSUM charts and concluded that the performance of the both charts are the same. For further details about the control charts and the details of the specifications, the reader is directed to one of the most popular textbook on SPC written by Montgomery [15]. To enhance the performance of control charts several researchers developed control charts using artificial neural network (ANN). Velasco and Rowe [16], Hwarng and Hubele [17,18], Cheng [19,20], Chang and AW [21], Guh and Tannock [22], and Noorossana et al. [23] approached monitoring the process using ANN. They reported the results of their proposed models in comparison with traditional control charts when different types of changes affect the univariate processes. Shewhart charts are used when one variable is involved. A univariate control chart is related to a single variable and it makes a simple condition to evaluate and analyze when the process parameter(s) shifts to an out-of-control condition. However, there are many cases in the real in which the simultaneous controlling of two or more related quality characteristics is required. The process involved several interrelated variables when being interested in monitoring is referred to as the multivariate process control. In the case of a process with several related characteristics is relatively more complex compared to a univariate case. In the case of controlling a process when more than one variable is involved Hotelling [24] proposed the T2 control chart. Hotelling [24] developed the T2 Hotelling chart in the case of bombsight. T2 control chart is similar to Shewhart chart in which it uses the most recent observation and thus it is not sensitive to the small and moderate shifts induced to the process. To overcome this weakness, the Multivariate Cumulative Sum (MCUSUM) and the Multivariate Exponentially Weighted Moving Average (MEWMA) control charts were developed. Properties and


performance of MCUSUM control chart have been investigated over time by many researchers including Woodall and Ncube [25], Healy [26], Crosier [27], Pignatiello and Runger [28], Ngai and Zhang [29], Chan and Zhang [30], Qiu and Hawkins [31,32], and Runger and Testik [33]. Several authors including Lowry et al. [34], Rigdon [35], Yumin [36], Runger and Prabhu [37], Kramer and Schmid [38], Prabhu and Runger [39], Fasso [40], Borror et al. [41], Runger et al. [42], Tseng et al. [43], Yeh et al. [44], Testik et al. [45], Testik and Borror [46], and Chen et al. [47] also contributed to the development of MEWMA control chart. The importance of the multivariate environment monitoring in the real cases and the restrictions in the level of the performance when being approached statistical methods led several researchers to focus on soft computing approach to monitoring the processes. Zorriassatine et al. [48], Hwarng [49], Guh [50], Hwarng [51] investigated the process monitoring using a model based on an ANN where the processes involving more than one variable are considered. Furthermore Guh and Shiue [52] approached the same condition using the decision tree. In the manufacturing cases, quality engineers in practice to control and improve a process might consider a sequence stages corresponding to a production line. The common condition can also be found in service systems. A process including several stages to produce an output is referred to as the multistage system. Multistage systems should be controlled with a different approach comparing with the case that the quality specification(s) is controlled only at one stage. Recently controlling of the multistage systems is investigated by some authors. Shi and Zhou [53] and Liu [54] reviewed the methods used to control and diagnose the variation propagates itself to other stages in a multistage system. Although one can find in literature different proposed control charts to monitor the processes involving one or more variables, the essential point here is that these control charts— however sensitive it is—are not capable of detecting the real time when the change is first manifested in the process. In other words, the time when the control chart gives an out-ofcontrol signal is not the real time that the process starts to be affected with a sustained special cause or a special cause set. The time by which the chart gives an out-of-control signal only indicates the presence of special causes of variation in the process. This means that before the chart triggers the signal of the out-of-control condition, a change has taken place in the process and the chart could not detect it instantly. The time when the process is affected by a sustained special cause(s) and inevitably departures to an out-of-control condition is referred to as the change point. Change point is concerned with the real time that a change manifests in the process. Identification of the change point plays an essential role to route when quality engineers start a root-cause analysis to detect and eliminate the special causes affected the process. In the case of sensitive to the change, the time of the triggered

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out-of-control signal may be close to the change point. However, in the case insensitive to the change, the expected signal time would be far away from the change point. Based on the results reported in literature, control charts are addressed as incapable tools to identify the change point efficiently. Hence, because the out-of-control signal should not be close in some sense, relative to sensitivity of the chart, to the value of the change point then the signal is a bias estimator for the change point. Identifying the change point will help revealing the source(s) of the change when quality engineers focus on the root-cause analysis. The knowledge of the change point would provide a good start to investigate the special causes and can lead to reduce the error of incorrect identification of the source (s) responsible to the out-of-control condition. This paper includes a review of literature that considers the mean change point when univariate and multivariate processes are approached along with their statistical and soft computing approaches. In the next section, the change point definitions along with its different features and confidence region are introduced. Mean change point schemes for univariate and multivariate environments are provided in Sections 3 and 4, respectively. Finally, our concluding remarks are discussed in Section 5.

2 Change point definition Pyzdek [1] discussed three basic properties of a distribution when a process is investigated. Pyzdek [1] addressed the location, the spread and the shape of a distribution. Spread refers to the variance and the standard deviation of the distribution, while the shape refers to the pattern followed by the process distribution, and finally the location refers to the parameters of the distribution followed by the process, such as the mean. The three properties are shown graphically in Fig. 2. Now, assume X 1 ; X 2 ; X 3 ; . . . ; X t ; . . . ; X T is a sequence of independent subgroup averages observed from a process. It is

LOCATION

SPREAD

SHAPE Fig. 2 Three potential change in a process

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assumed that the random variable follows a known density function f0 ðX ; Q0 Þ, where Q0 refers to the parameter space. It is also assumed that the process is in-control till time t, then after time t a change affects the process and the process shifts to an out-of-control condition and it remains in the new condition until the chart signals later on time T under the assumption that this signal is not a false alarm. Hence, X tþ1 ; X tþ2 ; . . . ; X T are the observations from the process affected by a special cause(s). In this case, the time point t indicates the time when the change first takes place in the process. In a process where more than one variable are involved, and the process is measured by a joint level of p correlated quality characteristics, we assume X 1 ; X 2 ; . . . ; X t ; X tþ1 ; . . . ; X T to be independent observation vectors where Xi follows a known p-variate distribution. If the probability distribution changes from f0 ðX ; Q0 Þ to f1 ðX ; Q1 Þ after the time when a change manifests itself in the process, i.e. after time t, then at least one of the following cases occurs: 1. The known probability distribution of the random variable after the time t not changes but the parameters are not the same. It means f0 0f1 but Q0 6¼ Q1 . 2. The probability distribution after the time t changes and the different parameters corresponding to the probability distributions are known. The concept is provided when f0 ≠f1. The stated cases in literature are referred to as the change point and the time t is denoted by the change point. The delay time of the out-of-control signal after a real change in the process is shown in Fig. 3. Change point identification makes an opportunity for quality engineers to do effectively a rootcause analysis to identify and eliminate the special cause.

For example, assume that the quality characteristic of interest in a univariate process follows the normal distribution with mean μ0 and variance σ20 when the process is in-control. If after an unknown time t, a special cause of variation affects the process mean or the process variance or both leading to a changed value μ1 and/or σ21 , respectively, then it is concluded that the process departs to an out-of-control condition at time t. In other words, if T refers to the current subgroup and that t ≥ T then it is concluded that the process is operating in an in-control condition and the parameter(s) has not changed yet. This condition can be stated as follows: xi N ðμ0 ; σ20 Þ; i ¼ 1; 2; . . . ; t: : xi N ðμ1 ; σ21 Þ; i ¼ t þ 1; t þ 2; . . . ; T

The illustrative example given by Samuel et al. [55] is helpful in recognizing the conceptual meaning of the change point. They considered a process of forged piston ring where the ring’s outer diameter is the quality characteristic that should be monitored. In this numerical example, the ring’s outer diameter is 100 mm. Then, when the process is incontrol, the mean of the quality characteristic is equal to 100. The process standard deviation is σ05 and the subgroup size is n 04. In this case, UCL and LCL of the Shewhart control chart are equal to 107.5 and 92.5, respectively. The averages of the subgroups are shown in Table 1. Table 1 shows that X 27 > UCL and thus Shewhart X control chart indicates an out-of-control condition at T027. Samuel et al. [55] using their proposed estimator that will be discussed later, showed that the change has occurred at time t016. The addressed time leads the quality engineers to examine the log books and records for special cause that may have occurred between the change point (t015) and the subgroup 16.

Change detection time

Change point

μ = 10 Fig. 3 The delay time when the signal appears

ð1Þ

μ = 10.5

Int J Adv Manuf Technol (2013) 64:1663–1683 Table 1 Subgroup averages of the change point example

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Xi

2.1 Change point features

1 2 3 4 5 6 7 8 9 10 11 12 13 14

100.450 97.450 102.450 100.675 98.550 97.950 102.950 98.825 102.325 103.075 99.600 98.825 100.425 96.075

15 16 17 18 19 20 21 22 23 24 25 26 27

101.225 103.075 101.925 101.350 103.575 102.925 100.675 100.600 105.125 106.700 96.050 102.175 107.900

The parameter(s) of a process may drift due to different types of deterioration. Accordingly, different types of change points appear in the process. For example, if the process is initially in an in-control condition and after an unknown point of time the value of the parameter(s) of the process, say the mean, suddenly shifts from μ0 to μ1 and remains as it is without any further change until the special cause is detected and eliminated, then this is referred to as a step change point (see Fig. 1). Assume that δ denotes a step shift in the process mean then d ¼ ðμ μ0 Þ=σx is the standardized value of the step shift in the mean. Here, the pffiffiffi standard deviation is denoted by σx ¼ σ= n where n is the number of samples. A step change may occur, for example, when the process tool suddenly breaks down. Several researchers like Samuel et al. [55], Pignatiello and Simpson [58], Pignatiello and Samuel [59], and Noorossana et al. [60] have investigated the step change point in their research (Fig. 4). Step change is only one of the change types may affect a process. Owing to some types of disturbance, a parameter(s) of a process may gradually drift. In reality, a process might be influenced by a change that follows a gradient shape. Assume that we have a normally distributed process that is initially in an in-control condition with known parameters, say mean μ and variance σ2. In this case, assume that the independent observations come from a normal process with known μ0 and σ20 when the process is in an in-control condition but after an unknown time t, the parameter μ0 starts to change with slop equal to β and thus the process departs to an out-of-control condition. Hence, the parameter μ0 changes over the time as follows:

Subgroup i

Obviously, a good estimate of the change time will lead the engineers to start with a good searching point of the special cause and then eliminate the cause effectively. In other words, knowing the real time when the process has changed would simplify the search for the special cause and it would be found more quickly. Therefore, in the case where a change affects the process, an important index for a control chart performance is how quickly the chart reacts when a change takes a place in the process. Here, the out-of-control average run length (ARL) is commonly used to assess how quickly the chart response to the change manifested in a process. Here, if E (T) denotes the time when the control chart signals an outof-control condition, and the real change is at time t, then it will be: EðT Þ ¼ ARL þ t:

ð2Þ

One can refer to Basseville and Nikiforov [56] and Csorgo and Horvath [57] as one of the best references considering the change point and its related issues.

ð3Þ

μi ¼ μ0 þ bði tÞ; i ¼ 1; 2; . . . ; t; t þ 1; t þ 2; . . . ; T :

In this case, since the trend of the process is linearly affected, then the real time when the process starts to change is referred to as a linear change point. Here, at the times i ¼ 1; 2 ; . . . ; t process is in-control condition but at the times i ¼ t þ1; t þ 2; . . . ; T the process has shifted to an out-of-control

μ = μ1 = μ0 + δ ∗σ 0

n

μ = μ0

τ Fig. 4 Step change

T

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condition. In this case the process variance is not changed and there are two unknown parameters, i.e., β and t. A linear change can be, for example, due to a tool wear or when leakage of a pipe gradually increases in a chemical process. Figure 5 shows the linear change in a process graphically. Linear change point has been investigated by several researchers such as Perry and Pignatiello [61], and Atashgar and Noorossana [62]. The process parameters may also experience multiple step changes or combined step and linear changes over the time period that the chart is incapable to signal the out-of-control condition. When one could inference a change manifested itself to the process in the way of non-decreasingly the change is referred to as the isotonic change. An isotonic change includes a step change corresponding to an enhanced step or multi-step change in which case the steps corresponds to the enhanced steps, or linear trend when its slope belongs to a positive value and combined change without decreasing behavior. It means that if, for example, the mean of a normally distributed process affected by an isotonic change, then the change of the mean parameter will be found as follows: μi μi1 ; i ¼ t þ 2; . . . ; T :

ð4Þ

In the case where δi is the magnitude of the mean step change in different change times t, then the following relationship indicates the isotonic change in the mean. d i d i1 ; i ¼ t þ 2; . . . ; T :

ð5Þ

Consequently, an isotonic change leads one to make an inference that is based on the prior knowledge of to not decrease the value of process parameter after at least an enhancement. The isotonic change could be found in many industrial applications such as dose, temperature, time and so on. In another example, assume we have quality

characteristics in a production line that depends on the raw materials, human attention, and the precision of the machine. If the three influential factors increasingly change at different times, then this will lead to a multi-step or combined step-linear change in the change of isotonic. The isotonic change type is graphically shown in Fig. 6. A process also might experience a non-increasing deterioration. This type of behavior may appear when one or more influential process input random variables change in a linear trend with negative slope, or change in a stepwise reduction way at one or more times. When a non-increasing change is experienced at different times, then this is referred to as an antitonic change. An antitonic change in a normally distributed process mean should be in the following form: μi μi1 ; i ¼ t þ 2; . . . ; T :

ð6Þ

Most of the change point procedures proposed in literature assume that the underlying process change is exactly known. For example, the proposed procedures introduced in the prior references are developed and evaluated assuming the parameter(s) change type follows a linear or a step change shape. However, in reality, it is rare to have a prior knowledge of the change type. When a procedure is designed assuming the change of the process parameter(s) is not known before and then the parameter(s) change is of isotonic, antitonic or combined change type, then it is considered as a monotonic change. In the monotonic change type, a non-decreasing or a non-increasing or a combined isotonic–antitonic change may appear at different times in the process until the out-of-control condition is detected using a specified scheme. Considering a p-variate process with a monotonic change, one of the quality characteristics may change with an isotonic behavior, for example, while the others experience antitonic change type at the same time or at different times while the process experiences an out-ofcontrol condition. Monotonic change point is investigated by several authors such Perry et al. [63,64], Noorossana and Shadman [65] and Atashgar and Noorossana [66]. 2.2 Phase of control charting

μ

μi =μ0+β (i −τ )

μ =μ 0 τ

Fig. 5 Linear trend in a process mean

T

When a process is statistically controlled and the change point identification is considered, the knowledge of the process parameters lead the quality engineers to the phase of consideration, i.e., phases I and II. The two phases are discussed by Woodall [67]. Phase II refers to the case when the in-control parameters are known. It is also called the prospective problem. In this phase it is assumed that the process parameters are accurately estimated or known. However, in the initial state of process monitoring, there is a phase that one does not know the process parameters when the process assumed is in-control condition and it is needed


(

X ~ N μ0,σ0

μ

n

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)

(

X i ~ N μi ,σ0

n

)

μτ +1 > μ0 and μi ≥ μi −1 for i =τ + 2, ,T

μ = μ0

unknown behavior

τ

T

Fig. 6 Isotonic change in process mean

to estimate the parameters. In this case, it refers to analysis of phase I or retrospective problem and also named historical analysis. In phase I, quality engineers should detect the special cause of variation along with estimating the process parameters when the process is in an in-control condition. Sullivan [68] stated that the retrospective analysis closely resembles the change point issue. In phase I, the major task of the quality engineers is to estimate the change point t and the parameters μ0 and σ20 as preliminary values, under the assumption that the process follows the normal distribution. In phase II, when the change point is considered, the major task will be estimating the time when the preliminary values of the parameters changes to μ1 and σ21 , respectively. 2.3 Confidence region When the change point analysis is considered and a procedure is proposed, an important issue that helps in analyzing the performance of the procedure is referred to as the confidence region. Constructing a confidence region for the time when the process changes will help quality engineers to find a meaningful set as a starting point to search the process log books and record for the special cause. Meeker and Escobar [67] reported that the approach of confidence region in statistics is developed based on asymptotic theory. However, there are small interval between the change point and the out-of-control signal time then the approximation based on the asymptotic may not be suitable. Confidence region provides a search window that involves a set of possible change points to quickly identify the special cause. The set covers the possible change time with an assumed level of confidence. The window set value could lead

quality engineers to doing their search effectively focusing on the change point set candidates. It seems that the window involving the set can greatly enhance the probability of identifying the special cause when the root-cause analysis is approached. Pignatiello and Samuel [59] used the likelihood function to obtain a confidence region for the process change point. Box and Cox [69] approached to obtain a confidence set (CS) using the log likelihood function. The CS is obtained using the following equation: ð7Þ indicates the maximum value of log likewhere lihood function, is the t value that maximizes the log likelihood function, and loge LðtÞshows the value of the log likelihood function at time t as follows: T t pffiffiffiffiffi P P Loge LðtÞ ¼ T loge ð 2p σ x Þ 2σn 2 ½ x2i 2μ0 xi 0

i¼1

i¼1

ð8Þ

2

þtμ20 ðT tÞxT ;t : Assuming that k1 and k2 are two constants determined by the subgroup averages, and then logeL(t) will be calculated as follows: loge LðtÞ ¼ k1

n ½k2 ðT tÞðxT;t μ0 Þ2 : 2σ20

ð9Þ

To obtain a 100(1−α)% confidence region Box and Cox [69] also calculated the positive constant D as follows: D¼

1 2 c : 2 1;a

ð10Þ

Siegmund [70] developed a 100(1−α)% confidence set for the change point where normally distributed the process

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mean used asymptotic theory based on the log likelihood function. Siegmund [71] calculated D as follows:

point t, when a signal STþ > hþ appears using the following equation.

D ¼ loge ½1 ð1 aÞ1=2 :

STþ ¼ maxf0; ZT k þ þ STþ1 g;

ð11Þ

Perry and Pignatiello [61] used several values of D for the confidence set providing an extended selection when more accuracy is desired. They proposed D00.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, and 3. The CS equation means where the difference between log likelihood function and maximum log likelihood function is less than D; the value of t corresponding to the difference is included in the confidence region. To show the performance of the confidence sets, the coverage probability average versus the cardinality sets graph is used. Substantially, a suitable graph should lead one for a given coverage probability smaller confidence sets. In another word, preferred graph shows increasing the average of coverage probability along with decreasing the average size of cardinality. Perry et al. [63] stated that using maximum likelihood estimator (MLE) to identify the change point in a process allows readily available to confidence set. The availability makes a distinct advantage for MLE versus CUSUM and EWMA procedures.

3 Mean change point for univariate processes Literature of the change point problems often refers to the retrospective analysis for a number of observations. Moreover, most of the researchers in literature approached to develop their proposed change point schemes for the time when quality engineers have received before an out-ofcontrol signal using a specified control chart. In other words, most of the proposed change point schemes will activate after the time a chart such the Shewhart’s control chart signals an out-of-control condition. Hence, a procedure that has the ability to identify the change point and simultaneously perform the signal of the out-of-control condition along with additional knowledge such detecting the change direction or diagnose the change feature is considered as an attractive scheme in theory and real practice. The origin of the change point problem first back to Page [10] when he addressed the identification of the sub samples by considering the change in a sequential setting. He also addressed the problem in his other scientific reports such Page [72–74]. Page [75] used a sequential probability ratio test (SPRT) of H0 : μ ¼ μ0. In this research, the starting time of the last SPRT or the point of the rejection test is considered as the change point. In other words, the author used the test to identify the starting point of the rejection test as the change point. The estimator proposed by Page [10] is used to identify the increase in the process mean, i.e. change

ZT ¼

X μ0 σX

:

ð12Þ

Now assume that δc is a specific standardized shift value, then the alternative hypothesis will be: H1 : μ ¼ μc :

ð13Þ

It is important to note that the SPRT proposed by Page [10,75] considers the process when it is in an in-control or an out-of-control condition. Therefore, the pre-specified condition is important here or the model will not be able to find the change point inherently. In 1994, Srivastava [76] investigated the estimator proposed by Page [10] and concluded that it is biased to the change point of a random variable. The extension of applying control charts to statically control a process and the importance of finding a solution to an effective root-cause analysis when a process departs to an out-of-control condition motivated the researchers to focus on the change point problem. Hinkley [77] focused on the change point in a sequence of random variables, where the continuous variable follows the normal distribution and the change occurs in the mean. In the same year, Hinkley and Hinkley [78] discussed the problem of inference about the point when the case of zero–one variable is the request from likelihood point of view. Pettitt [79] investigated a conditional test of no change against a change that takes place in the process and compared the results with the likelihood ratio test. Also Quandt [80–82] investigated two phases regression so-called switching. Among the solutions proposed to identify the change point is to find the maximum likelihood as the best approach comparing with the others. For over 10 years, the maximum likelihood procedure has always played an important role in the change point estimation where a univariate or a multivariate process is considered. Samuel et al. [55] investigated the change point problem in the case of the mean step change in which the change manifests itself in a univariate process and the Shewhart’s chart signals an out-of-control condition before. In the scientific report, they proposed an estimator to identify the change point using the maximum likelihood. Samuel et al. [55] first considered the Shewhart’ control chart using a numerical example that relates to a piston ring production line considered in Section 2 of this paper. They assumed that the considered process follows the normal distribution and it is initially in an in-control condition. In the research of Samuel et al. [55] the observations were produced with a known mean and variance. Then, they assumed a step


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ð14Þ

manifests itself in the process. The proposed estimator is usable when a chart such as Shewhart X signals an out-ofcontrol condition. They proposed the following estimator to identify the mean linear change point where indicates the maximum likelihood estimate of the last observation number when the process is in an in-control condition. Here, t is the value of t that maximizes the log likelihood function.

Here, T is the time when Shewhart’s chart signals an outof-control condition, μ0 belongs to the period of the time when the process is in an in-control condition, and

ð17Þ

change manifesting itself to the univariate process. As they showed, their proposed MLE is capable to effectively estimate the step change point when the change takes place in the process. Based on the proposed estimator, it is required to find t where 0≤t UCLEWMA , the change point would be estimated by, b t EWMA ¼ maxft : wt μ0 g:

ð22Þ

Nishina [88] also defined a run for EWMA and defined the change point as the time when the run starts to reject. A run is supposed to be a sequence of signals that appear continuously on one side of the center line of the EWMA chart. Nishina [88] reported that CUSUM and EWMA have the same performance; however the CUSUM scheme is superior when the change point is investigated. Pignatiello and Simpson [58] proposed a model based on likelihood ratio to identify the step change point when the mean shifts regardless of the magnitude of the shift manifests in the process. They used a likelihood ratio for a mean process where it follows the normal distribution. They investigated a tool that statistically detects the out-of-control condition along with the identification of the step change point. Pignatiello and Simpson [58] reported the performance of the proposed model compared with several CUSUM schemes using the Monte Carlo simulation technique and concluded that the performance of the proposed model is better than any of the CUSUM charts in the terms of ARL. Then, Perry et al. [64] also investigated the process fraction nonconforming and proposed an estimator to detect the time when a step change takes a place in the process. The proposed estimator can be used after a p or np chart signals an out-of-control condition. In this research, they assumed that the observations follow the binomial distribution where p ¼ p0 and the parameter is a known value, but after the change point the unknown magnitude δ affects the parameter and shifts it, where p ¼ p1 and p1 ¼ dp0 . Let be the unbiased statistic to estimate the parameter, where

indicates the subgroup fraction

nonconforming. Here Di denotes the number of nonconforming unit in the ith subgroup and ni denotes the ith subgroup


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Table 3 Results reported by Pignatiello and Samuel [59] Schemes

bt MLE bt CUSUM bt EWMA

δ δ00.5

δ01

δ01.5

δ02

δ03

Biased (overestimate) Biased (overestimate) Biased (overestimate)

Unbiased Unbiased Unbiased

Unbiased Biased (underestimate) Biased (underestimate)



size. They also used the np chart and the Monte Carlo simulation to analyze the performance of their proposed estimator. In the analytic study section, they considered different changes and included an example to show the simplicity of the proposed procedure and its accuracy. Hawkins et al. [89] focused on the mean parameter of a normal process and proposed a model to allow testing the step change in the process along with estimating the change point assuming that the parameters of the process are unknown, i.e., phase I of statistical process control. They focused on the change when a sustained special cause affects the process assuming the variance value does not change but the mean value changes after the change point. They claimed that their proposed model can help quality engineers when the change manifested in the process follows the isolated non-sustained special causes. They discussed that by using their proposed model quality engineers allow progressing seamlessly from the phase I data set gathering through phase II. Therefore, the model can help quality engineers to statistically control a process without requiring the knowledge corresponding to the parameter(s) of an in-control condition and then it allows ignoring the phase I settings. Hawkins et al. [89] discussed another benefit for the proposed change point model where it is compatible with Shewhart chart and the chart is a choice where isolated non-sustained special cause is potentially possible. Holbert [90] proposed a model assuming normality and uniform prior distribution over the time of the observation with fixed sample size based on the single break switching regression and showed a variety examples corresponding to economics and biology. The authors proposed the Bayesian estimator for the change point considering both step and linear change point. The authors used the Bayesian approach based on two phase linear regression model. The two phase linear regression model is defined as follows: yt ¼

a1 þ b1 xt þ "t ; t ¼ 1; 2; . . . ; m a2 þ b2 xt þ "t ; t ¼ m þ 1; m þ 2; . . . n;

ð23Þ

where xt is not a stochastic explanatory variable, εt is the 6 ða2 ; b 2 Þ random errors following N ð0; σ2 Þ , ða1 ; b1 Þ ¼ denotes the regression parameters and m addresses the time when after xm a shift could be found in the process.

The literature on the change point also addresses that the product partition model can provide a way for the researchers where the change point is investigated. Hartigan [91] introduced a product partition model (PPM) to analyze the change point problem. In this approach, the time related to the change point is considered as a random variable. Later the PPM has also been considered to analyze the clustering and outlier detection by several researchers. Barry and Hartigian [92] and Crowley [93] used the PPM to identify the change point when the special cause affects the mean of a process with the normal distribution. Quintana and Iglesias [94] using the PPM proposed their model to detect outliers in a linear disturbance when the random variable follows a normal distribution. In this report, they presented a decisiontheoretic formulation to identify the change point where one can find here an important extension related to the PPM. Quintana et al. [95] also applied the PPM to analyze a model of measurement error approached clustering method. Loschi and Cruz [96] provided a model extending PPM to obtain the posterior distributions for the situations and the change point numbers as well as the posterior distribution corresponding to the change that takes place. Fearnhead [97] and Fearnhead and Liu [98] investigated multiple change point problem and proposed algorithms based on filtering as an approach to obtain the posterior. Since the PPM considers the number of blocks as a random variable, Rosangela et al. [99] proposed the linear regression models to analysis of multiple change point using the PPM. To obtain posterior parameters distribution in the regression model, they modified the algorithm proposed by Barry and Hartigan [92]. They also discussed in this research the advantages that using PPM approach can be provided. They reported that their proposed method prepares additional information from the posterior distribution other than the posterior means. Turner et al. [100] proposed a FORTRAN program to detect shifts in the mean, standard deviation or both when the process follows the normal distribution for phase I of statistical process control. The program is developed from a likelihood ratio test. They claimed that the proposed program is powerful compared to an X-chart when it used alone or combined with an MR chart. The procedure LRT, named by the authors, consists of control charting and diagnostic

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purpose uses likelihood ratio test. Although the proposed procedure is sensitive to a step shift, they claimed that it could be applied when multi shifts appear in the process parameters. The importance of the change point leads different authors to use different ways that might provide the change point knowledge. Hierarchical method is also presented in literature as one of the ways that can help the authors in analyzing the time when the process shifts to an out-ofcontrol condition. Sullivan [68] addressed when a multiple step shifts affects a process, the X-chart and CUSUM chart may fail to identify the presence of the shifts for the stage 1 or retrospective analysis of SPC. To overcome this weakness, Sullivan [68] proposed a new method to detect effectively a single or multiple shifts induced to the process. The proposed scheme can show the number of the shifts appeared in the process mean. The author also investigated the methods proposed by McGee and Carleton [101], Hawkins [102], Dempster et al. [103], and Vostikova [104] and analyzed the hierarchical method and segmentation used to identify the change point. McGee and Carleton [101] approached hierarchical clustering and standard regression theory and investigated multiple change point model as follows: 0

yi ¼ Xi bj þ "i ; Tj1 < i Tj ; j ¼ 1; 2; :::; ðR þ 1Þ;

ð25Þ

i ¼ 1; 2; . . . ; m: where yi indicates the value of ith observation, Xi is a p predictor variables vector, βj indicates a parameter vector, Tj is one of the R change points, and m is the observation number. Here, T0 and TR+1 0m. Moreover, in this algorithm the error follows the standard normal distribution, i.e., mean zero and variance one. Hawkins [102] also generalized the model proposed by McGee and Carleton [101]. The hierarchical clustering could be expressed as a specialized case of the expected maximization (EM) algorithm introduced by Dempster et al. [101]. Recently, Habibi [105] investigated the change point problem using bootstrap methods. The author discussed that the common marginal distribution of bootstrap samples includes a mixture of two distributions before and after the change. Then he focused on the change point issue as a mixture modeling problem. Then using the EM algorithm, Habibi [105] proposed his model to estimate the parameters of mixture distributions. EM is a method of finding the maximum likelihood when the given data is not completed or includes missing values. It leads to two main application of EM. The first is when one can find limitations in the observations process and then the data indeed includes missing values. The second leads to the condition when optimizing the likelihood function is not analytically tractable. The author discussed that the EM algorithm consists of

two steps, E-step and M-step. Habibi [105] in a sequence with a change point could derive the bootstrap samples and since the common marginal distribution consists of two mixture distributions, he estimated the mixture distribution parameters (Dempster et al. [103]). He believes that using mixture distributions is relatively much easier compared with using the change point problems. Vostikova [104] proposed a method to detect a single or multiple changes using binary segmentation. It seems that using the proposed method the data may not segmented at a location properly. Although one can find different methods in control charting arena using soft computing in literature, especially artificial neural network discussed in Section 1, soft computing approach is less applied to identify the change point in univariate process. In literature it can be found that Ghazanfari et al. [106] focused on clustering soft computing to identify the change point. The authors approached the change point problem using clustering technique when Shewhart control chart is used in a univariate process. Their proposed model does not depend on the known parameters or an assumed known distribution for the process. Moreover, the model proposed by Ghazanfari et al. [106] can also be used for both phase I and phase II of SPC to estimate the change point. They approached parametric and sequential methods to illustrate their proposed model and clustered the data into two groups, i.e., the observation data before the time of the change and the observation data after the time of the change. To analyze the proposed model comparing the other existence model, Ghazanfari et al. [106] developed three criteria leading to a detailed discussion.

4 Mean change point for multivariate processes Let a process includes independent observations X1, X2,…, Xt , Xt+1,…, XT where vector Xi includes two or more variables and follows a joint probability distribution. As discussed before, t is the real time when the process shifts to an out-of-control condition and T is the time when the chart signals the out-of-control condition. In the case when a p-variate process is investigated and a special cause(s) affects the process parameter(s), the signal of an out-ofcontrol condition leads the quality engineers to consider a root-cause analysis. Here, suppose X ¼ ðX1 ; X2 ; . . . ; Xp Þ0 is a p×1 random vector whose jth element indicates the jth quality specification of interest. In the case when X follows, for example, the normal distribution, the parameters of the joint probability distribution include the mean vector μ ¼ ðμ1 ; μ2 ; . . . ; μp Þ0 and ∑ matrix, i.e., Np ðμ μ; SÞ. The case stated above indicates that the process only experiences a change point; however a process might experience the case when the special cause(s) affects more than


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one quality specification in different times, i.e., different change points. In another words t 1 ; t 2 ; . . . ; t p for the p different quality characteristics may appear in the process. In many applications the first change is the most important and in which an alarm is needed, and then usually one concentrates on the time as follows: t min ¼ minft 1 ; . . . ; t p g

ð26Þ

Where an investigated multivariate process indicates an array and the change point is considered, the case is referred to as the single-path change point. However, in the case of the data that consist several arrays corresponding to the population of the sample space and the change point is considered, then the case is referred to as the multi-path change point. The multi-path change point has been investigated by several authors such Joseph [107], Joseph and Wolfson [108–112] and Asgharian and Wolsfon [113]. In this paper, the singlepath change point is focused and the literature related to the multi-path change point is not provided in this review. In the case of the mean change point when the single-path change point is considered, Nedumaran et al. [114] investigated the mean step change in a process with p correlated quality specifications. They assumed that the process follows the normal distribution and the special cause only affects the mean vector. In this paper they first discussed the monitoring of the multivariate process using χ2 control chart when several correlated characteristics simultaneously are studied. Here, the statistic χ2 is defined as follows: 0 c 2 ¼ n X μ0 S1 ðX μ0 Þ: ð27Þ They assumed that the process initially follows a normal distribution with known μ and Σ parameters. Hence when the mean vector from in-control value of μ 0 μ0 shifts to an unknown value μ 0 μ1 and c 2T exceeds the UCL of the χ2 one could conclude that the process experiences a step change after some unknown time t assuming 0 t T 1. Based on the assumed case Nedumaran et al. [114] proposed an estimator to change point identification using the maximum likelihood as follows: _

t ¼ arg max Mt ; t ¼ 0; 1; . . . ; T 1:

10,000 repetition of the simulation study in terms of ARL and standard error reported by them indicates that the proposed change point estimator on average is close to the actual change point regardless the magnitude of the shift. The authors investigated the multivariate processes where the considered mean vector includes p02, 5, and 10 elements. However, the reported simulation results indicate that when the shift magnitude is equal or more than 2 the estimated change points address underestimate values. Nedumaran et al. [114] also reported an empirical distribution of the estimated change point but did not consider confidence region when they focused on evaluating the proposed estimator. Nedumaran et al. [114] did not discuss the performance of the estimator when the process includes different correlation coefficients corresponding to the covariance matrix. They evaluated the proposed estimator assuming a constant and known variance-covariance matrix without stating the assumed covariance values. They also did not evaluate the proposed estimator when the sample size is changed. On the other hand, Noorossana et al. [115] investigated the change point in a multivariate environment with the same condition considered by Nedumaran et al. [114] but they focused on the process when the change linearly affects the mean vector of a normally distributed process. Noorossana et al. [115] proposed the estimator to identify the linear change point using the maximum likelihood. The authors also analyzed their proposed MLE using the Monte Carlo simulation. In a linear trend, it is assumed that the mean vector function is as follows: μi ¼ μ0 þ ði tÞb ; i ¼ t þ 1; t þ 2; . . . ; T

where b is a p×1 vector with the jth element referring to the linear trend of the jth quality specification. It is obvious that the scalar t and the vector b are the unknown parameters that correspond to the latest observed subgroup when the process is in an in-control condition and the linear trend vector, respectively. One can estimate the b vector using the following equation: T P

ð28Þ

In the above equation, t shows the value of t for which Mt attains its maximum and is calculated as: X1 ðX t ;T μ0 Þ: ð29Þ Mt ¼ ðT tÞð X t ;T μ0 Þ 0 Nedumaran et al. [114] considered an illustrative example and evaluated their proposed estimator using the Monte Carlo simulation when the change starts to affect the process from subgroup 101. In other word the simulated process mean is shifted from μ0 to μ1 after 100 subgroups produced from an in-control condition. The results corresponding to a

ð30Þ

b^ ¼

i¼tþ1 T P i¼tþ1

ði tÞ ðX i μ0 Þ:

ð31Þ

ði tÞ2

Noorossana et al. [115] proposed the time of the change point when linear trend affects the mean vector as follows:

ð32Þ

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where X i is a p×1 vector that refers to the ith subgroup. Noorossana et al. [115] evaluated the proposed estimator where μ0 00 and the covariance matrix is as follows: 2 3 1 0:5 ::: 0:5 6 0:5 1 ::: 0:5 7 7 S0 ¼ 6 ð34Þ 4 ::: ::: ::: ::: 5: 0:5 0:5 ::: 1 The authors assumed that the linear change manifested in the simulated process mean from the 51st subgroup and then they reported the results of the performance of their proposed estimator with 10,000 repetitions. The reported results led the authors to conclude that the proposed estimator has a high performance to identify the change point when different change magnitudes linearly take place in a multivariate process. Although Noorossana et al. [115] did not analyze their proposed MLE in details; Arbabzadeh [116] in her dissertation addressed the details of the discussion related to the estimator proposed by Noorossana et al. [115]. Arbabzadeh [116] reported the performance of the proposed estimator compared with other schemes such Nedumaran et al. [114] when a linear change induced to the process mean. When multivariate problem is considered, the directionally invariance issue is an important property when MCI and χ2 control charts are investigated. Based on the property, the performance of the ARL corresponding to the control chart only via the decentralized parameter l addresses the out-ofcontrol mean vector and out-of-control covariance matrix. The decentralized parameter l is calculated as follows, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l¼

nðμ μ μ0 Þ0S1 ðμ μ μ0 Þ:

ð35Þ

When the linear trend is considered then replacement μ1 ¼ μ0 þ ði tÞ b with μ leads to the following, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð36Þ li ¼ nb 0 S1 b ði tÞ: In this case the decentralized parameter varies during the time and when the change follows linear trend the variation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P gradient is a ¼ nb 0 1 0 b . The equation leads to an identical ARL performance for all b vectors with equal a value. For example in a bivariate process with the following parameters, the vectors including b 1 ¼ ½0:01; 0:010 , b 2 ¼ ½0:01; 00 and b3 ¼ ½0:01; 00 based on the directionally invariance property, since for them, i.e. b1, b2, and b3 the values of a are the same, ARL performance is identical when each of the vectors appears in the process. 1 0:5 μ0 ¼ ½0; 00 ; S0 ¼ : ð37Þ 0:5 1 Noorossana et al. [115], considering the directionally invariance property reported the evaluation of the accuracy and

precision for their proposed estimator. Arbabzadeh [116] analyzed different ways to detecting an out-of-control condition and concluded that the proposed estimator is superior comparing with the step change estimator proposed by Nedumaran et al. [114] when linear trend is considered. Arbabzadeh [116] analyzed the proposed model after signaling an out-of-control condition using χ2, MCI, and MEWMA control charts. However, neither the Arbabzadeh [116] estimator nor Nedumaran et al. [114] estimator is a built-in control chart. The incapability leads one to use another control chart when one of the estimators is used. Arbabzadeh [116] also investigated the confidence sets and reported an analysis in details. For further details about the estimator, the reader is directed to Arbabzadeh [116]. Both the estimators proposed by Nedumaran et al. [114] and Noorossana et al. [116] can be used by quality engineers when the parameters of the process are known, i.e., phase II of the statistical process control. Zamba and Hawkins [117] investigated the change point problem in multivariate environment when a sustained step change takes place in the mean vector as one of the cases might appear in the process. They assumed that the covariance matrix does not change during the time the process is studied. They focused on using a sequential estimation method in the sake of removing the boundary that could be found between phases I and II of process controlling where the parameters are unknown for the quality engineers. Zamba and Hawkins [117] generalized the likelihood ratio test to estimate the change point in the mean of p-variate normal data. Hawkins and Qiu [118] also investigated the same condition when the knowledge corresponding to the parameters is imperfect and the environment is a univariate process. On the other hand, an important problem especially is in the case that the dimension of the correlated quality specifications tended to be of a highdimension. Several ways proposed in literature to overcome the problem of high-dimension such as principal component analysis method. Li et al. [119] considered the highdimensional multivariate environment when multiple change point may be detected without assuming a pre-determined distribution for a process. They developed a model using a decision tree based on a supervised learning solution and compared the proposed procedure with the MEWMA. In the paper it is not allowed to find how the learning is conducted and thus one is not able to use the procedure simply. In which the supervised learning is approached, such the Li et al. [119]’s paper, one should be able to compare the different dimensions of the supervision of the model to lead a minimum error rate when the result of the trained model is analyzed. Moreover, there is no evidence that the authors assumed a specified covariance matrix when the multivariate process with related variables is studied. By using artificial neural network approach Ahmadzadeh and Noorossana [120] proposed a procedure to identify the mean change point in a multivariate process. The authors’


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Table 4 Detailed specifications of the networks Network

No. Of hidden layer

No. Of hidden layer neurons

No. Of output layer neurons

2 1 1

8 11 12

3 1 1

A B C

procedure can be activated when the MEWMA chart detects an out-of-control condition before. Later, Ahmadzadeh [121] also discussed again the model proposed by Ahmadzadeh and Noorossana [120]. As stated before when a control chart addresses a special cause(s) that has manifested itself in the process and the process shifted to an out-of-control condition the quality engineers initiate a root-cause analysis to identify and then eliminate the cause(s) responsible. In the case where a multivariate environment is studied, a root-cause analysis is more complicated compared with a univariate process. In the case of multivariate process with related quality specifications quality engineers can only experience a root-cause analysis effectively when a scheme allows them to identify (1) out-of-control condition, (2) variable(s) responsible to the out-of-control condition, (3) change point and (4) in an ideal condition it allows them to determine the change direction(s), all simultaneously. When a p-variate process shifts to an out-of-control condition several authors including Blazek et al. [122], Jackson [123], Funchs and Benjamin [124], Subramanyam and Houshmand [125], Mason et al. [126,127], Atienza et al. [128], Maravelakis et al. [129], Niaki and Abbasi [130], Aparisi et al. [131], Guh [50], Guh and Shiue [52], Hwanrg [51] and Atashgar and Noorossana [132] have focused on the interpretation of the diagnostic analysis and the identification of the source responsible.

Transfer function for hidden layer

Transfer function for output layer

Training algorithm

Generalization feature

Tan-sigmoid Tan-sigmoid Tan-sigmoid

Log-sigmoid Log-sigmoid Log-sigmoid

Trainbfg Trainbfg Trainbfg

Early stopping Early stopping Early stopping

In the case of real, each of the diagnostic analysis, the control chart and the change point estimator, alone, cannot lead the practitioners to an effective root-cause analysis when the multivariate process in an out-of-control condition is considered. In other words, the schemes studied before in literature cannot provide all the four stated needs corresponding to the knowledge that should be applied by practitioners, at the same time. In the case of real, using several schemes is impossible for quality engineers or it makes a complex condition for quality engineers to analyze the root-cause(s) of the out-of-control condition. To provide simultaneously all the four knowledge Atashgar and Noorossana [62] focused on the mean change point along with the diagnostic analysis and detection of an outof-control condition to propose a model based on artificial neural network. The ANN model proposed by Atashgar and Noorossana [62] approached the supervised learning and can provide a comprehensive model when a linear change affects the mean vector of a bivariate process. The model developed by them includes three networks as shown in Table 4. In this paper, Atashgar and Noorossana [62] introduced a new approach to help increasing the capability of the training and then allow increasing the accuracy of the performance of the ANN model so-called subinterval approach by the authors. They trained the proposed model followed with a supervised learning approach based on subinterval training as shown in Table 5:

Table 5 Subintervals and training iterations for networks A, B and C

Network A

Network B

Network C

No.

Subinterval

No. of combinations

No. of iterations

Total

1 2 3

ð0:75 x1 0:75Þ _ ð0:75 x2 0:75Þjðx1 ; x2 6¼ 0Þ ðx1 0; 75 _ x1 0:75Þ ^ ðx2 0:75 _ x2 0:75Þjðx1 ; x2 6¼ 0Þ ðx1 ¼ 0Þ _ ðx2 ¼ 0Þ

2,940 1,156 128

20 11 200

58,800 12,716 25,600

4 Total 1 2 3 Total 1 2 3 Total

In-control

1 4,225 12 52 1 65 3,900 196 1 4,097

8,000

8,000 105,116 3,000 44,200 17,000 64,200 85,800 11,760 65,000 162,560

x1 1:5 _ x1 1:5 1:5 x1 1:5 In-control

ð1:25 x1 1:25Þ _ ð1:25 x2 1:25Þ ð1:25 x1 _ x1 1:25Þ ^ ð1:25 x2 _ x2 1:25Þ In-control

250 850 17,000 22 60 65,000

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Table 6 Detail specifications of ANNs Network

A B C D E F

No. Of hidden layer

No. Of input layer neurons

No. Of hidden layer neurons

No. Of output layer neurons

2 2 2 2 1 1

24 24 24 24 24 24

16 14 30 30 31 38

1 2 4 4 1 1

Later, Noorossana et al. [60] assuming a step change type proposed an ANN including three modules of networks using supervised learning which helps to detect an out-ofcontrol condition, provide a diagnostic analysis which allows identifying the quality specification responsible for the initiation fault, and estimate the change point along with recognizing the direction of the change(s), all simultaneously. Noorossana et al. [60] also used subinterval approach to develop their proposed model. They claimed that the proposed model is more accurate compared with the model proposed by Atashgar and Noorossana [66] when a step change appears in the multivariate process. Table 6 shows the specifications and the structure of training for the networks. The reader is directed to the text of the paper for the details of the specifications of the networks. Although most of the proposed models described before, have been evaluated using the Monte Carlo simulation technique, Noorossana et al. [60] evaluated their proposed model using different case study examples including production examples along with the Monte Carlo simulation. They investigated a particular grade of lumber production and a body production line of a car manufacturing factory to monitor the specification of gaps. Gap is defined as the Fig. 7 Position of gaps on car (source: Noorossana et al. [60])

Transfer function hidden layer

Transfer function output layer

Training algorithm

Generalization feature

Tan-sigmoid Tan-sigmoid Tan-sigmoid Tan-sigmoid Tan-sigmoid Tan-sigmoid

Log-sigmoid Log-sigmoid Log-sigmoid Log-sigmoid Log-sigmoid Log-sigmoid

Trainbfg Trainlm Trainbfg Trainlm Trainbfg Trainbfg

Early Early Early Early Early Early

stopping stopping stopping stopping stopping stopping

distance between bonnet and fenders, lid trunk and frame sides, front doors and fenders, front doors and frame sides, and finally, rare doors and frame sides. Figure 7 shows the situation of the gaps. The values related to the gaps between the lid trunk with right frame side and the lid trunk with left frame side are reported in millimeters in phase I of statistical process control as: " μ0 ¼

# 3:6086 0:0602 ;S ¼ 0:0356 3:6686

0:0356 1 ;R ¼ 0:0528 0:6308

0:6308 : 1

ð38Þ They showed that how the change point helps practitioners to find the source(s) influenced the process, using the proposed model when different step change values affect the mean parameter of the process. As stated before, they also considered the production line of a particular lumber with the mean vector and covariance matrix parameters corresponding to the stiffness and bending strength in unit of 1 b/sq inch measured in phase I as follows: " # 265 100 66 1 0:6 ; SL ¼ μL ¼ ð39Þ ; RL ¼ 66 121 0:6 1 470

Gaps are controlled in the line production

Int J Adv Manuf Technol (2013) 64:1663–1683 Table 7 Performance evaluation of Noorossana et al. [60] and Niaki and Abbasi [30] models

No.

1 2 3 4 5 6 7 8 9

Shift

(2σ,0) (0,2σ) (2σ,2σ) (2.5σ,0) (0,2.5σ) (2.5σ,2.5σ) (3σ,0) (0,3σ) (3σ,3σ)

1679

Error rate %

Improving percentage

Niaki and Abbasi [30] model

Noorossana et al. [60] model

15.4 12.6 13.8 9.8 11.6 10.8 7.0 14.6 4.6

5.9296 2.5840 0.2000 3.7864 2.7808 0.2000 4.1728 3.3792 0.2000

Noorossana et al. [60] compared the performance results of their proposed model with the model proposed by Niaki and Abbasi [130] as shown in Table 7. Much of the literature on the change point estimator is directed toward a specified change type, i.e. step change and

159.7139 387.6161 6,800.0000 158.8210 317.1461 5,300.0000 67.7531 332.0549 2,200.0000

linear trend. However, in the case of real one may not have a prior knowledge of the change type. Atashgar and Noorossana [66] without assuming a specified change type proposed an ANN with supervised learning based on networks of artificial neural network which helps to identify all of the: (1) out-of-

Table 8 Summarized capability of schemes on the literature Capability Out-ofcontrol signal Approach

Soft computing

Univariate

Hwarng and Hubele [17]

√

Cheng [19]

√

Guh and Tannock [22]

√

Diagnostic analysis

Step

Step

Linear

Monotonic

Guh [50]

√

Hwarng [50]

√

√

Guh and Shiue [52]

√

√

√

Atashgar and Noorossana [62]

√

Noorossan et al. [60]

√

√

√

√

√

√

√

√

√

Shewhart [2]

√

CUSUM

√

EWMA

√ √ √

Noorossana and Shadman [65] T2 Hotelling [24]

√

MCUSUM

√

MEWMA

√

√

√

√

Mason [126,127] Nedumaran et al [114].

√

√

Perry and Pignatiello [61] Multivariate

√

√

Atashgar and Noorossana [66]

Ahmadzadeh [120,121] Univariate

Monotonic

√

Aparisi et al. [131]

Statistical

Linear

√

Ghazanfri et al. [106] Multivariate

Change point estimation

√ √

Noorossana et al [115]. Zamba and Hawkins [117]

√

Li et al.[119]

√

1680

control condition, (2) the quality specification(s) responsible to the out-of-control condition, (3) the change point and (4) the direction(s) of the change(s), at the same time. The proposed scheme provides the capability of identifying all required knowledge leading to an effective root-cause analysis; all simultaneously, when an unknown change type belongs to a monotonic change affects the mean vector of the process that follows the normal distribution. They analyzed the proposed model using the Monte Carlo simulation. Hence the proposed scheme provides the capability of identifying all required knowledge, at the same time, leading to an effective root-cause analysis; when quality engineers do not know the change type. The models proposed by Atashgar and Noorossana [66] and Noorossana et al. [60] reported the results under a bivariate process and they did not discuss about the processes include more dimensions. The most important schemes and their capabilities discussed through literature are summarized in Table 8.


statistical and the soft computing approaches. In this paper, the analytical discussion and the comparison of the capabilities of the proposed schemes led us to summarize the capabilities of the schemes. The analysis is shown that one cannot find a comprehensive scheme approached statistically to use on multivariate processes. A comprehensive scheme is defined as the scheme that can identify (1) outof-control condition, (2) change point, (3) variable(s) responsible to the out-of-control condition, and (4) direction of the change(s), all simultaneously without assuming the change type that might manifest itself in the process. Hence, a comprehensive scheme approached statistically which requires any assumption regarding the change type is desirable. In this desired case, the type of the change should belong to a family of monotonic change type. However, among the soft computing approach, one can refer to only an ANN comprehensive scheme reported only where the dimension is not high. Hence, the procedures based on the soft computing approach to monitor multivariate processes must be further investigated.

5 Conclusion Control charts can only help in detecting the out-of-control conditions and commonly, relative to the change magnitude, signal after a few times a change takes place in the process. Identification of the change point using the proposed schemes in literature can help quality engineers to find the real time when the change has manifested in the univariate process leading to an effective root-cause analysis. However, when an environment with several related quality specifications is analyzed; it is more complex compared with a simple case of a process with one quality specification. When a multivariate process is focused on the change point analysis anole, it cannot lead the quality engineers to an effective root-cause analysis. In the multivariate processes the information including (1) the out-of-control condition, (2) the change point, (3) the variable(s) responsible to the out-of-control condition, and (4) the direction of the change (s) are crucial and play an essential role for a root-cause analysis. Hence the quality engineers can route effectively the analysis to eliminate the source(s) of contributing of the unnatural condition when all the information supply the quality engineers at the same time. Moreover, most of the schemes related to the control charts, the diagnostic analysis procedures and the change point estimators, proposed in literature are directed to monitor the process assuming a predefined change type (especially step change). However, different change types might take place in the process parameter(s) and then it is not practical to assume a specified change type a prior. In this paper, it was reviewed the research literature on the mean change point focused in the univariate or the multivariate processes. This review provides both the

Acknowledgments The author is grateful to Dr. Mahmoud A Mahmoud for precious comments before submitting the paper.

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