Abstract. In recent years, most researchers and users consider profile monitoring as a good technique for single-variable or multiple-variable quality control. A profile is a relation between one response variable and independent variables controlled during the time. Process Capability Indices are measured to evaluate processes performance in producing conforming products. In this paper, the process capability and profile monitoring have been evaluated using the similarity-based technique. Curve similarity is a subcategory of pattern recognition or image processing measures the degree of similarity. Where using it profiles monitoring is reduced to monitor a random variable and simplify the calculation of process capability indices. Chebyshev distance determines the distance between each sample profile and a target profile. This approach is applied in monitoring and measuring of PCI of nonlinear vertical density profile (NVDP) and simple polynomial profile (SPP) of an automotive engine.
Keywords:
1. Introduction
Statistical process control (SPC) is a branch of statistic that measures and controls the quality of products. It should be done in 2 phases; One of them is initial basically SPC process and the Second one is assessment, measuring and detection of faulty products. A control chart is a kind of primary toll in SPC. Various type of control chart has appeared in researches such as a traditional Shewhart-type chart, cumulative sum (CUSUM) control charts, exponentially weighted moving average (EWMA), multivariate EWMA (MEWMA) and multivariate CUSUM (MCUSUM) Hoteling's . When the process is in-control we can use capability index to measuring the stability of the process.
Process capability indices (PCIs) aim to the stability of the process. For measuring these indices there are a quality character ( ) and specification limits (SLs) that include: The target value ( ), the upper specification limit ( ) and the lower specification limit ( ). The common indices discussed in the literature are , , , and . These indices proposed by Juran (1974), Hsiang and Taguchi (1985), Kane (1986), and Choi and Owen (1990), respectively. Sudhansu et al. (2010) proposed a method to generalizing PCIs. Yum and Kim (2011) review the literature of PCIs between 2000-2009.
In some situation, the quality of products can describe by the relationship between a response variable and one or multiple independent variables that called “profile”. By paper of Woodall (2007), the concept of the profile has grown and increased in the statistical topic. Depending on the application, various form of the profile has appeared: simple linear profile (SLP), multiple linear profiles (LP), polynomial profile, non-linear profile (NLP), Logistic regression profile, nonparametric profile, geometric profile, and wave profile. Maleki et al. (2018) categorized the profile monitoring to seven general profile monitoring that includes: statistical design, economic and/or economic-statistical design, parameter estimation, diagnosis, change point estimation, capability analysis and other. various statistical techniques have been used to monitoring of profiles.
Williams et al. (2007) used three types of control chart to monitoring NLP in phase I. Also, Vaghefi et al. (2009) monitored NLP in phase II using two developed two control chart. Amiri et al. (2010) investigated the second‐order polynomial profile of the automotive industry. They presented a linear mixed model method and used ‐based procedure to evaluate the stability of the process. Noorossana and Ayoubi (2012) proposed a nonparametric bootstrap control chart for monitoring SLP. Fun et al (2013) tried to present a new approach for the reflow process data that based on the sum of Sine functions. Ghahyazi et al. (2014) for the first time studied on the multistage process in phase II that produce SLP. Soleimani and Noorossana (2014) studied on the violated independency assumption in phase II when the process describes as multivariate SLP. Grasso et al. (2014) focused on reductionism of data and achieving a better comprehension of the complex process. They used principal component analysis (PCA) on the multivariate profile. Amiri et al. (2014) regard the correlation between SLP and define the product has multivariate normal quality. Also, they proposed a method using MEWMA control chart to monitor the correlated profile and multivariate quality characteristics. Amiri et al. (2015) using three methods include method, likelihood ratio test (LRT) and a -based method monitored GLM regression profiles in Phase I. The LRT method was better than two others. Aly et al. (2015) compared three approach when the process is in-control in phase II and describes with SLP. Using the standard deviation of the average run length (SDARL) metric the comparison is performed under the assumption of estimated parameters. Paynabr et al. (2016) proposed a new change-point approach for multivariate profile monitoring. The approach includes a framework for monitoring, modeling in phase-I. Huwang et al. (2016) represent the quality of process using proportional odds model. They used EWMA control charts for monitoring and simulation to demonstrating the proposed chart. They believed estimating the change point and detection of the parameters of change is the ability of the method. Atashgar and Zargarabadi (2017) studied the real case that is plastic parts manufacturing industries. In this case, there is a correlation between the response variables. They analyzed the performance of multivariate profile model using Wilks’ lambda method in phase-I. Shadman et al. (2017) generalized LPs in phase II using change point approach, rao score test, and LRT. Kalaei et al. (2018) regard cascade property in different stages of a multistage SLP process in phase I. The method reduces the complicated condition of monitoring a multistage process. Awad et al (2018) proposed an approach can detect a fault in the multivariate statistical process. Because of complexity, they used a data‐driven approach and artificial neural network (ANN). This literature was noted a small sample of research in this field. Maleki et al. (2018) prepared a bibliometric paper about profile monitoring.
Shahriari and Sarrafian (2009) tried to present PCIs for SLP. In generating the dependent variable, PCIs is first calculated in multiple points and, then, the smallest value is introduced as PCIs. This is observed to mean neglecting process capability in other infinite points of the independent variable. Hosseinifard et al. (2011) used gamma distribution for non-normal response variable and non-conforming proportion of response variable to analyze the PCI of SLP. Ebadi and Amiri (2012) proposed three methods for measuring process capability in multivariate SLPs. Hosseinifard and Abbasi (2012) used a non-conforming proportion of response variable to evaluate the PCIs. Also, in [16] process capability used for SLP. Nemati et al. (2014) suggested a functional PCI in a circular profile and, also, presented a functional technique to measure PCIs in that profile in the full range of the independent variable. The method used a reference profile, functional specification limits and functional natural tolerance limits to present a functional form of PCIs. Nemati et al. (2014) developed functional PCIs for SLP. Wang (2014) measured the process yield for SLPs with the one‐sided specification. The index used in [15] to measure PCI when the quality of the process is characterized by a logistic regression profile. In addition, approximated confidence interval based on the percentile bootstrap method. Guevara and Vargas (2015) proposed methods for measuring the capability of NLP, based on the concept of functional depth. These methods do not have distributional assumptions and extended to functional data. Wang and Tamirat (2016) proposed process yield for multivariate LPs with one-sided specification limits. Guevara and Vargas (2016) presented a method for measure the capability of multivariate NLP, based on PCA for multivariate functional data and the concept of functional depth. Wang (2016) evaluated the process yield in multivariate LPs in manufacturing processes with process yield index. The Output an exact measure for the process yield and approximate confidence interval for is constructed. Wang et al (2017) using EWMA proposed a new acceptance sampling plan with the yield index for SLP with one‐sided specifications. Chiang et al (2017) calculated the PCI for SLP using MEWMA control chart under within‐profile autocorrelation. Alevizakos et al. (2018) used the index and propounded a method to measure the PCIs in Poisson regression profile and. Aslam et al (2018) presented a new multiple dependent state repetitive sampling plan based on the yield index for linear profiles. The operating characteristic function of the proposed plan is developed for linear profiles.
Most of PCIs and methods of profile monitoring include advance statistic. Using curve similarity index (CSI) simplifies the procedure of profile monitoring and assessment of process capability. In this paper, the CSI presented through Chebyshev distance. The paper is organized as follows. Section 3 and 4 present types of distance, process capability indices. Section 5 props approach of similarity of curves in profile monitoring and PCIs. The last sections are case example and conclusion.
2- Distances
There is a topic in computational geometry that called curvature matching. Which including some of the application in image processing. The similarity of curves is evaluated by studying methods for measuring the similarity between the two curves. Similarity assessment of curves is used in various fields such as speech recognition, signature recognition, simulation of shapes, computer vision, and time series evaluation. To assess the resemblance of the curves, we use the distance curves criteria includes Fréchet distance, Minkowski distance, Hausdorff distance and etc. Minkowski distance obtains the distance between a set of points. Chebyshev distance is a kind of Minkowski distance. Points n are presented as Eq. (1) in an n-dimensional space. Minkowski distance between the points is obtained by Eq. (2).
(1)
(2)
Respecting Figure 1, for every , and , the distance turns to Manhattan, Euclidean, and Chebyshev distance, and represent a follower, respectively:
(3)
(4)
(5)
3- Process capability indices
As mentioned in literature there are various types of PCI. Common capability indices are and . The first PCI is the that measures the potential capability of the process and doesn’t regard to mean of it.
(6)
Where LSL is lower specification limit, USL is upper specification limit and is the process standard deviation. LNTL is lower natural tolerance limit and UNTL is upper natural tolerance limit . is the mostly used PCI because it compares process dispersion and tolerance range with considering the position of process mean.
(7)
Eq. (6) and Eq. (7) considered for both SLs. In some situation, we must describe and for a unique specification. So, we use and for unique specification. and defined as follow:
(8)
The and using Eq. (9) consider the indices and .
(9)
The indices have normally assumptions. Clements (1989), constable and Hobbs (1992), Pearn and Kots (1994) and Pearn and Chen (1995) developed PCI to non-normal process. The Clements methods are defined as follow:
(8)
Where:
(9)
Where percentile value of the data.
4- The Curve Similarity Approach
As mentioned in the literature various method presented for profile monitoring and there are a few types of research in the field of PCIs. In an industrial environment, the decision maker interest that uses a simple method for measuring the quality of products and stability of manufacturing. In common SPC analyzes, performed on the univariate variable. Where the control chart evaluates the univariate variable of the process is in-control or not. In traditional PCIs Each of , , , and is a point in traditional univariate process capability assessment and we use differences of these points to evaluate the capability of the process (Fig.1). But in profile case they are curves. Fig. 2 represents the linear limits in SLP for example. So, differences between lines must be considered to determine PCIs for SLP. The method transforms this complexity to traditional univariate.
Fig. 1. Specification and tolerance Limits in traditional univariate
Fig. 2. Functional Reference line and various limits in SLP.
In the curve similarity approach, the distance between the sample profile and the target profile is used as a benchmark for profile monitoring and the evaluation of PCIs. The random variable is the Chebyshev distance, which is obtained by comparing the collected points for each profile with the corresponding points of the target profile. If the Chebyshev value is zero, it means matching between all the profile points are created and the target profile. We use this type of distance when a finite set of points is presented from each curve. It is necessary the process designers provide a USL Chebyshev distance. Transformation of profile monitoring and measuring PCIs to monitoring and measuring a variable are the advantage of the curve similarity approach. Fig. 3 represent a flowchart prepared for curve similarity method.
Fig. 3. Flowchart of curve similarity approach.
5. Case example 1
In some profiles, there is a linear relationship between a response variable and one or more independent variables. The nonlinear relationship in a regression means that if differentiation is performed relative to parameters, the parameter still remains in the relationship. The overall equation of nonlinear regression model in this paper is shown in Eq. (10).
(10)
are the value of an independent variable in vector instance the vector of which is written as . is the value corresponding to and is the value of the number of independent random variable with a normal distribution having the average of zero and variance of . Young et al (1999) proposed the relationship between vertical density of wood board and different depths as a nonlinear profile in Eq. (11) and its called bathtub function A profile meter (benefiting from Laser technology) is used for standard sampling of vertical density. The device scans different depths and measures their densities.
(11)
A considerable advantage of this nonlinear model is that its parameters can be interpreted. For example, , , and show profile fitness with the center of and lower part . differences of these parameters allow the chart to approach symmetry toward . in these examples, they called the depth of location for sheet and showed it with the vector . Values of were obtained for 314 depths. In fact, the values of an independent variable are specified and equal mm for every . In other words, for every from depth toward inside the sheet, vertical density is recorded and values of form a nonlinear fitness curve. Fig. 4 shows the vertical density of the wood board produced at 314 design points in 24 profile.
Figure 4. The vertical density of 24 wood boards
Values of the vertical density of the target curve are generated by Eq. (12). Fig .5 shows the fitness of profile 1 to bathtub function.
(12)
Figure 5. profile 1 fit to bat
Chebyshev distance of nonlinear profile of 24 wood board sample is presented in Table 1.
Table 1: Chebyshev Distance of 24 sample of wood board profile
The assumption of the normality distribution of Chebyshev distance is rejected according to Fig 6. To evaluation of PCIs, the data set must follow a normal distribution. Johnson's transformation (1992) is used to find out if the process follows a particular distribution or no. when Johnson's transformation represents the process is in-control we find out the process have non-normality distribution. So, must use the Clements method to calculation PCIs.
Figure 6. Probability plot of Chebyshev distance under normality assumption
Figure 7. I-MR Chart of Chebyshev Distance after Johnson Transformation
Figure 8. Probability plot of Chebyshev distance of wood board profiles
The I-MR control chart of Chebyshev distance after Johnson transformation is shown in Fig. 7. After that the process represents is in-control the process capability is to be evaluated. We use the Clement method when the Chebyshev distance does not follow the normal distribution. At first, detects the distribution of process in the clement method. Figure 8 shows that the created Chebyshev distance of the nonlinear profile of the wood board is followed the lognormality distribution. Due to the process is in-controlled, we can calculate the process capability index. The lower specification limit and upper specification limit are 0 and 15.5. The process capability index is calculated in terms of the lognormal distribution. The two unique specification limits as follow:
The process capability indices are:
Considering the value of , which is equal to 1.03, it can be claimed that the process of producing wood board is capable of creating nonlinear profiles between the depth and vertical density of the boards. This result is consistent with the research results of Wang et al. (2014).
Figure 9. Outputs of SPP for 20 sample engines.
6. Case example 2
The general model of the second-order polynomial profile (SPP) is represented in Eq. (13).
(13)
Where, the pair observation (x_{ij},y_{ij}) is obtained in random sample in which is the design point for the explanatory variable in sample. , and are parameters of second-order polynomial profile(sample) and are random variables, independently normally distributed with mean zero and variance equal to .
Amiri et al. (2009) have proposed a case study of a second-order polynomial profile in automobile engines. In this SPP, indicator variable is RPN and the response variable is torque. The RPN values are set equal to 1500, 2000, 2500, 2660, 2800, 2940, 3500, 4000, 4500, 5000, 5225, 5500, 5775 and 6000. In this paper, we use 14 design points of 20 sample engines. the target profile defined as Eq. (14). Fig. 9 and Table 2 represent the profiles and Chebyshev distance between target profile and sample profiles, respectively.
(14)
Table 2: Chebyshev Distance of 20 sample of the automobile engine
Figure 10 shows one point is out of control and caused the process represented out of normality distribution. The reason is the profile 4 and is removed from the sample set of profiles. In Figure 11, the normal probability diagram is shown in relation to the remaining 19 profile profiles, which indicates the normal distribution is not rejected for Chebyshev distance.
Figure 10. Probability Plot of Chebyshev Distance for 20 automobile engines
Figure 11. Probability Plot of Chebyshev Distance for 19 automobile engines
Figure 12. I-MR chart of Chebyshev distance of automobile engines
The I-MR chart of Chebyshev distance represents the process is in-control and no points are out of the limits (Fig. 11). Therefore, we can calculate the PCIs and the upper and lower specification limits of the Chebyshev distance for SPP of automobile engine 0, 14. The indices for this profile defined as follow:
(8)
The and indices are:
(9)
The value of the obtained PCIs is 2.12 and 1.22 which can be claimed that the process of producing the simple polynomial profiles of the motor torque has good stability.
7. Conclusion
In this paper, the curve similarity approach was used to monitor and evaluate the process capability of producing nonlinear and polynomial profiles. The Chebyshev distance is the basis for judging for difference of each sample profile from the target profile. Two case examples have been presented on nonlinear profiles between the depth and vertical density of wood board and the simple polynomial profile of automotive engine. According to the obtained values, the processes have the proper ability to produce nonlinear profiles and good polynomial profiles.
It is suggested that the curve similarity approach with other approaches be compared in terms of accuracy in process capability. The similarity of the curves was used to profile monitoring. Therefore, it is recommended to compare the speed of operation and its accuracy with other methods of profile monitoring. Using other distances and evaluating could be the basis for future research such as the Fréchet, Manhattan and the other of Euclidean distance. The approach can be used in other profile processes such as non-linear multi-variable profiles, geometric profiles and other types of profiles.
7. Reference
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