Efficient Phase II Monitoring Methods for Linear Profiles Under the Random Effect Model

A profile is a functional relationship between two or more variables used to monitor the process performance and its quality. Sometimes, the aforementioned relationship is linear or nonlinear depending upon the situation. A monitoring method based on the linear profiles is known as linear profiling which is commonly used due to its simplicity and efficacy. Linear profiling methods have been studied by many researchers with a fixed effect model. However random effect model provides a more suitable interpretation as compared to the fixed effect model under different real-time monitoring methods. Therefore in this article, we are intended to propose a linear profiling EWMA method (<inline-formula> <tex-math notation="LaTeX">${\mathrm {EWMA}}_{x \mathrm {[R]}}$ </tex-math></inline-formula>-3 chart) and <inline-formula> <tex-math notation="LaTeX">${\mathrm {MEWMA}}_{x \mathrm {[R]}}$ </tex-math></inline-formula> chart based on the random effect model using different ranked set sampling techniques such as ranked set sampling (RSS), extreme RSS (ERSS), median RSS (MRSS), double RSS (DRSS), double ERSS (DERSS) and double MRSS (DMRSS). The ranked set sampling (RSS) schemes are not only cost-effective method but also an efficient mechanism as compared to simple random sampling. A designed simulation study used Average Run Length (ARL) as an evaluation measure to witness the detection ability of newly offered <inline-formula> <tex-math notation="LaTeX">${\mathrm {EWMA}}_{x \mathrm {[R]}}$ </tex-math></inline-formula>-3 chart, <inline-formula> <tex-math notation="LaTeX">${\mathrm {MEWMA}}_{x\mathrm {[R]}}$ </tex-math></inline-formula> chart and existing <inline-formula> <tex-math notation="LaTeX">${\mathrm {EWMA}}_{x \mathrm {[SRS]}}$ </tex-math></inline-formula>-3 chart. The extensive simulation showed that the proposed <inline-formula> <tex-math notation="LaTeX">${\mathrm {EWMA}}_{x \mathrm {[R]}}$ </tex-math></inline-formula>-3 chart and <inline-formula> <tex-math notation="LaTeX">${\mathrm {MEWMA}}_{x \mathrm {[R]}}$ </tex-math></inline-formula> chart have superiority to detect faults in the process compared to a competitive counterpart. The results are further justified with real data application related to a combined cycle power plant.


I. INTRODUCTION
Product quality and cost are the most significant consumer preferences in this modern era.This is the reason why most of the production companies engaged in enhancing product quality with the minimum cost of production, in competing with other companies.To achieve this goal, a product should be free from all defects, and these features of a product can be made through effective monitoring of the process.These processes occur with certain variations such as nonassignable and assignable causes of variation.The first one is the natural or integral part of the process that cannot be eliminated from the process.However, the second one damages the process unnaturally and needs intensive care through different statistical tools to deal with it.Statistical tools under the The associate editor coordinating the review of this manuscript and approving it for publication was Zhaojun Li .umbrella of Statistical Process Control (SPC) are used for the enhancement of process quality by minimizing the assignable cause of variations Mahmood and Xie [1].A control chart introduced by Shewhart [2] is a primary tool of the SPC toolkit which provides a graphical outlook of the process quality.The control chart designed with the upper and lower specification limits to decide whether the process is under the In-Control (IC) or Out-Of-Control (OOC) state.
In recent times, the profile monitoring has drawn significant consideration from the researchers in which the quality characteristics of interest are defined by a regression model.Such as the density of a wood board is dependent upon the depth of board that is fixed.This functional relationship can be represented by a profile model.This linear profiling approach also terms as monitoring method of the process through regression.Commonly, Phase II methods are applied to distinguish shifts in process parameters defined under the linear profiles model through different performance measures to run the process smoothly.Initially, Kang and Albin [3] proposed methods based on combined EWMA and R chart in which error term was examined through the EWMA control chart whereas the dispersion of error terms was observed through the R chart, and Hoteling's T 2 charts were designed to monitor the intercept and slope of the linear profiles model.Reference [3] control charting structures are not able to properly advise the OOC parameters of linear profiles.To resolve this problem, Kim et al. [4] recommended a methodology established on the transformed simple linear profile model.The proposed EWMA-3 structure consists of separate control limits and test statistics for the error variance as well as for regression parameters.The findings of the EWMA-3 structure were more illustratable as compared to ordinary EWMA chart for linear profiles monitoring.Noorossana et al. [5] proposed a Multivariate CUSUM (MCUSUM/R) chart for the profiles monitoring.Zou et al. [6] and Mahmoud et al. [7] suggested control chart structures based on change-point techniques and Gupta et al. [8] made a comparison of the methods used by [4] and Croarkin and Varner [9].Automated control charts of periodic residuals of simple linear profiles were introduced by Zou et al. [10].The linear profiles model with autocorrelation problem was argued by Jensen et al. [11].Zhang et al. [12] recommended a likelihood ratio-based charting structure for simple linear profiles and Saghaei et al. [13] proposed a technique based on CUSUM control chart.Mahmoud et al. [14] highlighted a linear profiling study when the sample size is at most two in the process, and a method based on the likelihood ratio test for change point models was introduced by Yeh and Zerehsaz [15].The idea of considering the parametric uncertainty in fixed effect model was well addressed by Abbas et al. [16] using Bayesian approach.The parametric uncertainty in profile model is further investigated through Double EWMA and CUSUM charts in Phase II (cf.[17], [18]).Mahmoud et al. [19] proposed several alternative methods based on the EWMA structure which are comparatively efficient than the existing EWMA-3 structure.Recently, Saeed et al. [20] provided a charting structure based on the progressive linear profile statistics which is effective to monitor small-to-moderate shifts in the parameters of simple linear profiles.
The aforementioned literature and references therein have studied profiling defined by a fixed effect model.These kinds of literature assumed specified values for explanatory variables.However, there are certain situations where this assumption may not hold.It is quite common in regression model applications where the level of the predictors in a process cannot be controlled and are variable in nature.These profiles models are usually referred as the random effect models Greene [21].For example Atmosphere Pressure (Y ) is strongly affected by Wind Speed (X ).The functional relationship among these quality characteristics can be better represented by a profile model Abbasi et al. [22].As the wind speed may vary at different sampling intervals so modeling through fixed effect model is not an appropriate choice and may produce misleading results.The suitable option for such situation and situations similar to it is random effect models.Sampson [23] study cases of random effect regression models in simple as well as in multivariate situations.Noorossana et al. [24] considered a case study of tape thickness that is dependent upon four random locations.This functional relationship is efficiently modeled through random effect model.Abbas et al. [25] explored the case in Bayesian perspective for the efficient monitoring of random effect models.There is lots literature available in which researchers used random effect model to represent the profile model for the monitoring of process parameters (cf.[26]- [28]).
The aforementioned studies have used Simple Random Sampling (SRS) strategy to draw random observations from a normal distribution for the case of fixed and random effect models.As time proceeds and development occurred in sciences, new sampling schemes were introduced and effectively used in the different dimension of social and natural sciences.These new sampling schemes not only enhance the literature but also improve the efficiency of experimental results.The notion of Ranked Set Sampling (RSS) was first familiarized by McIntyre [29] to estimate the grazing land and crops and modifications of RSS were provided by Takahasi and Wakimoto [30].Likewise, other fields of sciences, the RSS schemes and its modified versions are extensively and effectively used in SPC literature (cf.[31]- [34]).The literature suggested above and references therein efficiently used RSS techniques to make proposed control charts more proficient and more reliable.In this study, we have investigated randomness in the independent variable using different RSS techniques.This study investigated the properties of random effect model after the modification of control limits coefficients, choice of sampling distribution for dependent and independent variables using RSS.Three separate EWMA control charts are designed under RSS techniques to monitor the process parameters of profiles model defined by a random effect model.
The rest of the article outlined follows: Section 2 provides the estimation process of the linear profiles model under RSS schemes.Section 3 presents the proposed and competing for control charting structures.Section 4 demonstrates the detailed simulation setup and comparative analysis is reported in section 5.The real data application is presented in section 6, while conclusions and recommendations are made in section 7.

II. SIMPLE LINEAR PROFILES USING RSS STRATEGIES
In this section, we will describe the structure of the ranked set strategies used in the stated proposal.Further, the derivation of understudy profile model is discussed using the RSS strategies.

A. RANKED SET SAMPLING SCHEMES
The RSS scheme was first proposed by [29], and reforms on RSS were provided by [30].The structure of RSS schemes work such as n 2 elements are selected, and n sets of n random VOLUME 7, 2019 samples are constructed.The selected units are ranked in each individual set with the help of auxiliary variable, personal inspection or judgment without any actual measurement.The smallest ranked element is selected from the preliminary set, the second least ranked element from the subsequent set, and then the third set used to pick the third least ranked element.This procedure keeps going on until the highest ranked element is obtained from the last set.In this way, n elements are selected from n sets and if the above-mentioned procedure is repeated l times than n * lelements are obtained, and the sampling scheme is named as RSS.Practically these strategies come up with tangible benefits as Dell and Clutter [35] illustrated that mean estimator of RSS is more efficient compared to SRS even if there are errors in the ranking.Further Stokes [36] proved that the estimated variance of RSS is more competent as compared to variance of SRS.
Median Ranked Set Sampling (MRSS) is an extended form of RSS introduced by Muttlak [37].In MRSS n 2 samples are selected from the target population at random and arranged into n sets just like RSS, every set having n units.All the observations in each set are organized on visual scrutiny or with the help of the auxiliary variable.For the case of odd, we choose n+1 2 th ranked element from every set or select the median unit from each set.For the even sample size, we choose n 2 th ranked elements from the first n 2 sets and n+1 2 th ranked elements from the last n 2 sets.In result, n units are obtained through the aforementioned procedure, and if the above-mentioned procedure is recurrent l times than n * l units are obtained from MRSS.
Extreme Ranked Set Sampling (ERSS) is another type of RSS proposed by Samawi et al. [38].For the case of ERSS, n 2 random units are arranged into n sets, where all sets consist of n samples.All n observations are arranged in each set on visual judgment.For the case of odd set size, choose the smallest ranked elements from first n−1 2 sets; the largest ranked elements from last the n−1 2 sets and the median ranked element from n+1 2 set.For the case of even set size, we choose the least ranked element from first half sets, and the largest ranked element from last half sets to obtain the sample of n observations.The aforementioned procedure is repeated l times to get n * l units from ERSS.
Al-Saleh and Al-Kadiri [39] proposed another sampling technique similar to RSS, where n 3 sampling units are selected and divided into n sets each set consisting of n 2 sampling units known as Double Ranked Set Sampling (DRSS).These sampling units are arranged in each set with respect to the auxiliary variable.The n 2 sampling units are selected from the total of n 3 sampling units using RSS technique.Again n sets are formed from these n 2 units, and n units are chosen.The process may be repeated l times to obtain n * l observations from DRSS.The similar extension of MRSS and ERSS can also be obtained and named by Double Median Ranked Set Sampling (DMRSS), and Double Extreme Ranked Set Sampling (DERSS) respectively.For a detailed understanding, see [33] and [40].

B. ESTIMATION OF LINEAR PROFILES MODEL UNDER RSS SCHEMES
This section describes the basic framework of the linear profiles model under RSS schemes.The least-square method is applied to obtain the estimates of intercept, slope and errors variance, which are further used to design control charting structures under RSS.The simple linear profiles model for the j th order and k th cycle using RSS techniques can be defined as: where  1) can be transformed to obtain the independent estimators for intercept, slope and errors variance as follows: where The average of the predictor can be obtained by . This transformed model is further used for the simulation study and comparative analysis.For the case of perfect ranked set sampling, the estimated slope coefficient for the j th profile and the kth cycle is obtained by and the variances of slope and intercept parameters are . Further, the estimated slope is standardized to control the variability factor due to the random X , which leads to fix the control limits of the slope parameter.Then the standardized slope coefficient is defined as , and the resultant standardized slope follows a standard normal distribution.The mean square error defined as is the unbiased estimator of error variance in which e [i]jk is the i th ordered error term from the k th cycle and j th profile for [R] technique and defined as

III. LINEAR PROFILE MONITORING METHODS
In this section, control charting structures of competing and proposed linear profile methods are presented by considering the random effect of the predictor variable.

A. EWMA-3 CONTROL CHART UNDER SRS
Reference [24] proposed three independent EWMA statistics for the intercept, slope and error variance monitoring separately for the case of random X.In their study, random samples are drawn from a normal distribution using the SRS scheme.The EWMA statistics are defined as follows: where, The smoothing constant θ must lies between 0 and 1 (i.e., 0 ≤ θ ≤ 1).The corresponding control limits of the EWMA statistics for intercept, slope and errors variance are given as follows: where, 5   (cf.[41]).

B. EWMA-3 CONTROL CHART UNDER RSS SCHEMES
In the case of the random effect model, the EWMA-3 chart under different ranked set schemes is designed for the intercept, slope and errors variance monitoring.This study has considered the case of perfect ranked set schemes in which main and auxiliary variables are perfectly correlated with ρ = 1.The memory type structure considered here is represented by the notation EWMA x[R] − 3 control chart.Mathematically it can be proved that the variance of the intercept transforms from after breaking the condition of the fixed random variable.Then the EWMA I [j][R] statistic for j th profile while monitoring of intercept and its corresponding LCL and UCL using different [R] techniques are: where, Now for the monitoring of slope coefficients, it can be seen that the EWMA S[j][R] statistic depends over explanatory variable X .Under the scenario of random X the EWMA S[j][R] will lead to variable control limit as it will depend on a new sample, each time the process is repeated and the values of EWMA S[j](R) will not match.To overcome this problem, standardized slope estimators are used.The EWMA S[j][R] statistic for j th profile and its corresponding LCL and UCL for the monitoring of slope are defined as: where, EWMA S[R] (0) = 0.The process defined under stable condition till the EWMA S While discussing the randomness of the explanatory variable of the profile model, it has no effect on EWMA E[j][R] statistic, so the control limits and test statistic wouldn't be changed for errors variance monitoring defined for the j th profile as: ) Var ln MSE j (10) where, EWMA E[R] (0) = ln σ 2 (0) .The under study process assumed to be in-control as long as the In this subsection, we have considered a general case of profiles monitoring and assumed an in-control general profiles VOLUME 7, 2019 model when explanatory variables are not fixed and defined as: where C = C (1) , C (2) , C (3) , . . ., C (k) is the vector of regression coefficients of k-dimension.The represented error terms j are identically and independently distributed with a multivariate vector of standard normal distribution.In this model X j is provided in the form of (1, X * j ) when X * j is orthogonal to 1 on 1 in the situation of n j -variates.We have further assumed that X j is the multivariate normally distributed vector of n j -variates with mean vector µ X and variance-covariance matrix of σ 2 X .Zou et al. [42] assumed a general profiles model with fixed X j 's and designed a MEWMA [SRS] chart to monitor simultaneously, the m + 1 regression parameters that includes m coefficients and the errors variances under SRS.We have further extended MEWMA scheme of [42] when the explanatory variables are random using different imperfect and perfect ranked set schemes (i.e., RSS, MRSS, ERSS) defined with a notation [R].Now from the model in Equation ( 11), we defined a transformation under ranked set schemes as: and where ) is defined as the inverse of a multivariate standard normal cumulative function, while G (.; υ) is the represented chi-square distribution with pre-specified υ degrees of freedom.The standardization of regression coefficients in Equation (12) and the transformation of the process standard deviations in Equation ( 13) allows accommodating the effect of random explanatory variables and sampling size (n) selection.

Now define a statistic ω
, that combines the estimated regression coefficients and standard errors and form a (m+1) variate of random vector.When the process is in-control the statistic ω j multivariate normally distributed that has zero mean vector and variance-covariance matrix H = X T j X j 0 0 1 .
Then we can define a MEWMA x[R] statistic under ranked set schemes [R] as: where, Z [R]0 represents the preliminary vector of (m+1)dimension.The value of smoothing constant λ can be anywhere between 0 and 1.The upper control limit for the MEWMA x[R] chart for an out-of-control situation can be defined as: The value of the control limit coefficient L [R] is adjusted to obtain a pre-specified in-control average run length (ARL 0 ).

IV. SIMULATION SCHEME AND PERFORMANCE MEASURE
This section discusses the simulative work, and comparative analysis of the control charting structures based on simple and ranked set sampling schemes.2. Subsequently, parameters have been estimated.In the next phase, all the test statistics are computed on the subject of each control chart with θ = 0.2 as smoothing constant.The shifts denoted by ϕ are incorporated in the process parameters (i.e., intercept, slope and error variance) in terms of σ units as: A 0 toA 0 + (ϕ I * σ ), A 1 toA 1 + (ϕ S * σ ), and σ toϕ E * σ .The magnitude of shifts in intercept is taken as 0.2-2 with a jump of 0.2, and for the slope, 0.025-0.25 are taken with a jump of 0.025.In the error variance, shifts are taken as 1.2-3 by a 0.2 shift difference.The average run length (ARL) is the performance measure used in this study for the evaluation of charts where ARL is defined as the mean points falling inside the specification limits before a point falls outside of specification limits (cf.[18], [22], [43]).Further, simulation study having 10,000 iterations is carried out to draw the findings of the stated proposal.To obtain the overall in-control ARL (ARL 0 ) 200, the control limit coefficients for each chart are adjusted on the ARL 0 = 600, while for the overall ARL 0 = 370, individual control limit coefficients are set on the ARL 0 = 1110 (cf.Table 1).

V. COMPARATIVE ANALYSIS
This subsection explains the simulation results and finding of existing and competing for profile monitoring methods.The shifts are initiated in the procedure to check the detection ability of the control charts.These shifts are the illustration of the change in profile parameters in any manufacturing process.We have computed OOC ARL's (ARL 1 ) for the simple and different ranked set sampling techniques to evaluate the performance of EWMA   Thus, a control chart with smaller ARL on the small shifts in intercept, slope and error variance will be considered as the most efficient control chart.The ARL results are reported in Tables 2-4 for the monitoring of intercept, slope and errors variance under SRS and RSS schemes.The impact of ARL performance on the proposed and competing charts with accounting the shifts in intercept, slope and errors variance are described in the following lines.

A. SHIFTS IN INTERCEPT PARAMETER
The intercept term in the profile model is the conditional mean of the response variable when X=0. Taking into account a shift in intercept is vital, as a shift in intercept means changing the origin point of the regression line.Now, considering the case of shifts in intercept, it is seen that at the first shift ϕ I = 0.2 when ARL 0 = 200, the ARL 1 values for EWMA that is based on SRS.Among these ranked set schemes, the EWMA x[DMRSS] -3 chart constructed by using DMRSS scheme showed the best performance as compared to others (cf.Table 2).It continues with the best performance at different shift values, but the magnitude of difference in ARL is small at larger shift values.

B. THE SHIFT IN SLOPE PARAMETER
The slope parameter is vital as it explains the rate of change in response with respect to a unit change in the independent variable.This mean introducing a shift in slope will change the original rate of change in response that ends up with a false decision.For the case of slope shifts, results are provided in Table 3

C. ERROR VARIANCE SHIFTS
The basic assumption of error terms in the regression model the normality with mean zero and fixed variance.This is the scenario of process under IC state, and this assumption is severally affected after shifts in error variance.This means a change in error variance will change the parameter of the regression model.For the case of shifts in error variance, results of the linear profile monitoring methods are reported in Table 4.In the presence of a 1.2σ shift in the errors, it is seen that the EWMA x[DERSS] -3 chart showed the smallest ARL 1 's around 13.2 at the fixed ARL 0 = 200.In conclusion, it is observed that all the proposed EWMA x[R] -3 charts outperform the EWMA x[SRS] -3 chart.

D. JOINT SHIFTS IN INTERCEPT AND SLOPE
The efficiency of control charts using RSS techniques for the case of joint shifts in the intercept and slope have also been reviewed at ARL 0 = 200 (cf.Tables 5-11).The ARL 1 is observed as 105.4,66.5, 66, 60.5, 32.6, 45, and 24.5 for the EWMA DERSS] -3 and EWMA x[DMRSS] -3 charts respectively, when slope is shifted around 0.025 and intercept is shifted almost 0.2..This indicates a significant improvement in the performance of control charts at small shifts with the use of perfect ranked set sampling schemes, particularly using MDRSS strategy.For the EWMA x[SRS] -3 chart, it is seen that the ARL 1 = 5.3 for the pair of shifts (0.025, 2) while ARL 1 is reported as 8.8 for the shifted pairs (0.25, 0.2).The EWMA x[RSS] -3 chart provides ARL 1 = 3.1 and 5.1 at (0.025, 2) and (0.25, 0.2) pairs of shifts in the slope and intercepts, respectively.At (0.025, 2) and (0.25, 0.2) shifted pairs, results showed that the EWMA x[ERSS] -3 chart provides ARL 1 = 3.3 and 5.4, while the EWMA x[MRSS] -3 chart have ARL 1 equals to 2.9 and 4.6.These findings provide the evidence that the linear profile monitoring method under MRSS scheme is slightly better than the ERSS, RSS and SRS schemes.The EWMA x[DRSS] -3 and EWMA x[DERSS] -3 charts offers ARL 1 = 2.1 and 3.1, and ARL 1 = 2.5 and 3.9 respectively, at shifted pairs (0.025, 2) and (0.25, 0.2).The EWMA x[DMRSS] -3 chart offer the best performance among all with ARL 1 = 1.8 and 2.5 at (0.025, 2) and (0.25, 0.2) pairs of shifts in slope and intercept respectively.Overall significant improvement in the detection ability is seen with the proposed EWMA x[R] -3 charts because ARL 1 is reduced from 105.4 to 24.5 at the shifted pair (0.025, 0.2) (cf.Tables 5 and 11).

E. OVERALL
The ARL curves for the EWMA x[SRS] -3, EWMA x[R] -3 charts are presented in Figures 1-6.Under the scenario of shifts in intercept and slope, the EWMA x[DMRSS] -3 have lower ARL curve as compared to all other charts, which is the evidence of its superiority.The pattern of the ARL curves also indicates that when there are shifts in errors variance, the EWMA x[DERSS] -3 chart is on the lower side.This shows a better performance of EWMA x[DERSS] -3 chart compared to all competing charts.The amount of difference is high at a smaller shift while this difference is low at the large shift values among existing and competing charts.12-14).

VI. A REAL DATA APPLICATION
To highlight the importance of the stated proposal, we have applied proposed charts on the dataset related to combined cycle power (CCP) plant.A CCP plant given in Figure 7 comprises steam turbines, gas turbines, and heat recovery steam generators.In the mechanism of a CCP plant, gas generators are used to generate electrical power, and waste heat of the exhaust gases are further utilized by steam generators to produce electricity.
In this study, we are using a dataset reported by Tüfekci [44].A CCP plant with electricity generating capacity 480 MW was designed with 1 × 160 MW ABB steam turbine, 2 × 160 MW ABB 13E2 gas turbines, and 2× dual heat recovery steam generators.In a CCP plant, the main load is dependent on the gas turbine which is sensitive to the ambient conditions such as atmospheric pressure (AP), ambient temperature (AT), Vacuum (V) and relative humidity (RH).Reference [44] used ambient conditions as explanatory vari-  ables and full load electrical power output (PE) as a dependent variable.On the bases of 9568 data points, a possible subset regression is used to find the significant explanatory variables for the dependent variable PE.Although, all ambient conditions play a role in the PE, AT is the most influential factor and most widely studied about gas turbines.Reference [44] reported that there exists −0.95 correlation between PE and the AT.Further, which can be interpreted as: if AT increase a unit ( • C), then PE reduced to 2.1713 MW.This model has R 2 = 89.89but might have the problem of mild normality (cf. Figure 7).Hence, we used only the first 2500 data points and obtained the following model, PE = 497.362− 2.1860AT (17) which can be interpreted as: if AT increase a unit ( • C), then PE reduced to 2.1860 MW.This model has R 2 = 90.06,PP-plot is plotted in Figure 8 and Anderson-Darling test with the summary numbers given as AD = 0.736 and P-value = 0.055.From this analysis, we conclude that there is no strong evidence against normality (cf. Figure 8).Therefore, we are using model 12 as an in-control model.The implementation of EWMA x[SRS] -3, EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 charts under SRS, DMRSS and DERSS, respectively on the real data set is described as follows: Step 1: For the analysis, estimates of the simple linear profile parameters are obtained using five random data points (n = 5) drawn by SRS, DMRSS and DERSS schemes.Further, these estimates are computed for a large number of time (10 6 iterations) using extensive Monte Carlo simulations.Hence, the average and standard deviation of the estimates with respect to sampling schemes are obtained and reported in Table 15.
Step 2: For the analysis, we have fixed the overall ARL 0 = 200 to obtain the charting constants of EWMA x[SRS] -3, EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 charts.These constants are computed by extensive Monte Carlo simulations (10 6 iterations).The resulting control limits are given in Table 15.
Further, the following profiles are used to introduce several amounts of shifts in terms of σ PE = 16.58.

VII. CONCLUSIONS AND RECOMMENDATIONS
This study presented a new linear profiles monitoring method for random effect model with the application of various ranked set schemes like RSS, MRSS, ERSS, DRSS, DMRSS, and DERSS.The structure is designed on the bases of three independent EWMA x[R] -3 charts to monitor the process parameters of random effect model such as intercept, slope and errors variance.A comprehensive analysis based on simulation and real data application is carried for the comparison of proposed EWMA x[R] -3 charts and existing EWMA x[SRS] -3 chart.The simulation study and practical application provide evidence that the recommended charts have higher detection ability as compared to the existing chart while monitoring shifts in the intercept and slope.For the case of errors variance monitoring, the EWMA x[DERSS] -3 charts come up with the best detection ability.The significance in performance is high at smaller shifts and its magnitude decrease with the increase in shift values.Further, the joint shifts in the slope and intercept of the transformed model are incorporated, which indicates a significant improvement in the performance ability of linear profiling structures.Overall, findings of this study divulge that the ranked set sampling schemes improve the detection capability of control charts for the monitoring of linear profiles model.The scope of the current research can be extended and enhanced with the inclusion of Bayesian techniques, the run-rule schemes both in classical and Bayesian setup.Further the current simple linear random model can be extended for multivariate cases and of course extension in a nonlinear profiles model.
Study of joint shifts is important as the change in the origin of the regression model will mislead in terms of rate of change.The combine shifts are introduced for all possible combinations of intercept and slope shifts using RSS sampling techniques.The purpose is to observe detection ability of the EWMA x[SRS] -3, EWMA x[RSS] -3, EWMA x[ERSS] -3, EWMA x[MRSS] -3, EWMA x[DRSS] -3, EWMA x[DERSS] -3 and EWMA x[DMRSS] -3 charts by taking joint shifts into account.The findings are described in the following lines:

F
. EVALUATION MEWMA x[R] CHARTS This subsection describes the performance of newly designed MEWMA x[R] charts using the ranked set schemes of RSS, MRSS, and ERSS at different settings of correlation coefficients among main and auxiliary variable.The MEWMA chart under SRS represented by MEWMA x[SRS] chart, while under RSS, MRSS and ERSS are represented by MEWMA x[RSS] , MEWMA x[MRSS] , and MEWMA x[ERSS] charts respectively.The value of smoothing is chosen as

•
For the detection of shifts in the variance of disturbance term, we multiply 25 sets of PE with 0.1σ PE and the resulting 25 profiles with index 176 to 200 for EWMA x[SRS] -3, EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 charts are portrayed in Figures 9-11.For EWMA x[SRS] -3, EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 charts the number of OOC profiles with their respective indices are given in Table 16.When there is a shift in intercept parameter, the results revealed that EWMA x[SRS] -3 chart offers 5 OOC signals while EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 charts alarms 18 and 19 OOC signals, respectively.In the presence of shifts in slope parameter, EWMA x[SRS] -3 chart signaled no OOC points while EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 signaled 1 and 15 OOC signals, respectively.Further, when the joint shift is introduced in intercept and slope, EWMA x[SRS] -3 chart signaled 14 OOC points while EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 signaled 12 and 55 OOC signals, respectively.Moreover, a similar pattern is also observed when shifts are introduced in error variance.Hence, the EWMA x[DERSS] -3 has a better detection ability relative to EWMA x[SRS] -3, EWMA x[DMRSS] -3 charts.The findings of the real case study also showed the evidence that the EWMA x[DMRSS] -3 chart performed well while considering shifts in intercept and slope and EWMA x[DERSS] -3 has a better detection ability in case of shifts in error variance.

VOLUME 7
, 2019 TAHIR MAHMOOD received the B.S. degree (Hons.) in statistics from the Department of Statistics, University of Sargodha, Sargodha, Pakistan, in 2012, and the M.S. degree in applied statistics from the Department of Mathematics and Statistics, KFUPM, in April 2017.He is currently pursuing the Ph.D. degree with the Department of System Engineering and Engineering Management, City University of Hong Kong, Hong Kong.He was a Teaching Assistant with the Department of Statistics, University of Sargodha, from 2012 to 2015.His current research interests include statistical process control and linear profile monitoring.MUHAMMAD RIAZ received the Ph.D. degree in statistics from the Institute of Business and Industrial Statistics, University of Amsterdam, The Netherlands, in 2008.He is currently a Professor with the Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.His current research interests include statistical process control, nonparametric techniques, experimental designs, and computational statistics (focusing code development).148296 VOLUME 7, 2019 jk denotes the i th ordered observation from the k th cycle and j th profile using different ranked set sampling techniques denoted by [R] such as RSS, MRSS, ERSS, DRSS, DMRSS, and DERSS.The errors follow a standard normal distribution, and the explanatory variable follows a normal distribution with mean µ x[R] and variance σ 2 x[R].The model provided in Equation ( The original model in Equation 1 takes the intercept and slope as 3 and 2, respectively, while the transformed model presented in Equation 2 takes intercept as 13 and slope remain the same.The i.i.d.errors are generated from a standard normal distribution, and the explanatory variable X ∼ N (mean = 5, variance = 5/3).At first, n 2 bivariate random numbers are generated for errors and explanatory variables through different RSS techniques when n=4.The main variables are ranked with respect to auxiliary variables, and four values are selected one in each cycle.The values of the response variable are generated after using Equation

TABLE 1 .
Coefficients of control limits under SRS and RSS schemes.
under RSS, ERSS, MRSS, DRSS, DERSS, DMRSS schemes) charts.For comparison purposes, the focus remains on the small shifts because the proposed and competing for control charts are efficient for the small shifts in a process.VOLUME 7, 2019

TABLE 5 .
Performance comparison of EWMA x[SRS] -3 for the joint shift in intercept and slope at ARL 0 =200.

TABLE 6 .
Performance comparison of EWMA x[R] -3 for the joint shift in intercept and slope at ARL 0 =200.

TABLE 7 .
Performance comparison for combine shift in intercept and slope at ARL 0 = 200 using ERSS.

TABLE 8 .
Performance comparison for combine shift in intercept and slope at ARL 0 =200 using MRSS.

TABLE 9 .
Performance comparison for combine shift in intercept and slope at ARL 0 =200 using DRSS.

TABLE 10 .
Performance comparison for combine shift in intercept and slope at ARL 0 =200 using DERSS.

TABLE 11 .
Performance comparison for combine shift in intercept and slope at ARL 0 =200 using DMRSS.

TABLE 12 .
Performance comparison of MEWMA x[SRS] and MEWMA x[R] charts for intercept shift using imperfect ranked set schemes.

TABLE 13 .
Performance comparison of MEWMA x[SRS] and MEWMA x[R] charts for slope shift using imperfect ranked set schemes.

TABLE 14 .
Performance comparison of MEWMA x[SRS] and MEWMA x[R] charts for errors variance shift using imperfect ranked set schemes.

TABLE 15 .
Estimates of the simple linear profile parameters with control charting constant.

TABLE 16 .
Number of OOC profiles with respect to profile indices.

•
For the shift in intercept, we have added 0.25σ PE in the values of PE, and the resulting 25 profiles with index 101 to 125 for EWMA x[SRS] -3, EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 charts are portrayed in Figures 10-12.
• For the joint shifts in intercept and slope, we have added 0.25σ PE in the values of PE and multiplied 0.075σ PE with the values of AT, and the resulting 25 profiles with index 151 to 175 for EWMA x[SRS] -3, EWMA x[DMRSS] -3 and EWMA x[DERSS] -3 charts are portrayed in Figures 9-11.