On designing an assorted control charting approach to monitor process dispersion: an application to hard-bake process

The monitoring of process variability is an important feature to get optimal output from a process. Control charts are vital tools used for efficient process monitoring. The commonly used types of charts include Shewhart, cumulative sum and exponentially weighted moving average (EWMA) control charts. This study focuses on dispersion control charts using some efficient transformation for small and medium instabilities. We intend to propose an assorted method to monitor a range of disturbances in process dispersion, using the well-known max approach. We have used several measures to evaluate the suggested assorted control chart. Based on these measures, we have compared the proposed assorted method with many existing charts. The study proposal outperforms the existing counterparts in detecting various amounts of shifts in process dispersion. Finally, a real-life application of the proposed chart is demonstrated to monitor the flow width measurements in a hard-bake process.


Introduction
A series of activities that takes some inputs and results into some output is termed as a process. Each process has its own dynamics that lead to a targeted output. For instance, a manufacturing process differs from a mechanical process in terms of procedures and inputs/outputs. The stability of any process is one of the major concerns for the continuity of the process. The process stabilization is generally required immediately with the beginning of mass production in any process. Otherwise, there may be a lot of waste output from the process that leads to process deterioration. The list of such processes may include business process, health process, banking process, among others. One such process is the hard-bake process. The hardbake is a common method used to stabilize the printed features for providing optimum performance at etches for the photoresist pattern. In this process, one potential variable of interest is flow width measurements of wafers (X) that needs timely attention for effective performance of the process. We will propose an efficient monitoring mechanism of such a process in this study.
In a standard statistical process control (SPC) environment, we focus on removing the assignable cause(s) after the detection of any out of control (OoC) signals. There are many sources of variability in the processes, and SPC is a very handy tool to differentiate between natural and unnatural variations. The quality of a process is determined by different parameters such as location, shape and dispersion. The dispersion parameter is of prime importance as the stability of other parameters (like location) depends on dispersion. Generally, dispersion charts are used for two main reasons (i) if the variation in the process increases, there is a possibility that more defective units will be produced (ii) if the variation in the process decreases then more units will be near the target value and hence process capability will also increase. These changes can be quickly detected by the dispersion charts. These charts are important while interpreting the results of a location chart because they assume that standard deviation remains constant.
The Shewhart range (R) chart and the Shewhart standard deviation (S) chart are used to monitor the process variability of small subgroup sizes [1] introduced the cumulative sum (CUSUM) chart to monitor process variability among different subgroups. Many authors have evaluated the performance of CUSUM and exponentially weighted moving average (EWMA) control charts that were based on the subgroup standard deviation (cf. [2][3][4][5][6][7][8][9] and [10]). One-sided EWMA control chart based on the natural log was suggested by [11] to monitor subgroup variance [12] have proposed a CUSUM chart based on the logarithmic transformation of the subgroup variance [13] proposed a MaxMin EWMA chart to monitor process variability [14] have discussed and compared several control charts to monitor variation in the process [15] introduced a maximum (MAX) control chart to monitor the centre and spread of the variable simultaneously [16] proposed a MAX-EWMA control chart that can be used to monitor the process location and dispersion simultaneously. The chart was based on plotting the Maximum (MAX) of the standardized location and dispersion statistics against a single control limit [17] extended the idea of MAX statistic to a CUSUM control chart and showed that their proposed method works better than the MAX-EWMA [18] proposed a new two-sided S 2 chart based on logarithmic transformation for monitoring variation in the process. A new CUSUM-S 2 to monitor the process variation was proposed by [19,20] introduced some useful control charts for location based on different sampling schemes [21] used different methods to increase the sensitivity of the mixed EWMA-CUSUM control charts for the location parameter.
Inspired by these improvements and modifications, this study proposes a new assorted method that can be used to identify small, intermediate and large shifts (in process dispersion) simultaneously. The proposal will serve as a single charting structure, instead of having multiple designs, and hence it will provide ease in the implementation for practitioners.
From here on, the paper is structured as the classical charts for dispersion and some of their modifications are presented in section 2; measures used to evaluate and compare the performance of different charts are given in section 3; the proposed assorted method and its construction is discussed in section 4; section 5 provides the performance of the proposed assorted method along with its comparison with the counterparts; section 6 is about how the proposed methods are applied in a real situation; section 7 gives the concluding remarks.

The control charts for process dispersion
Shewhart, CUSUM and EWMA are the 3 classical approaches to monitor the process dispersion. Below is a brief outline of these and some other modified charts. Assume X being a random variable following a normal distribution with mean (μ 0 ) and given a standard deviation (σ 0 ) i.e. X ij ∼ N(μ 0 , (δσ 2 0 )) where i = 1, 2, 3, . . . . . . and j = 1, 2, . . . .., n. where n is the size of the i th sample. The disturbed/dislocated dispersion in a shifted process is denoted by σ 1 and is described as σ 1 = δσ 0 .

The Shewhart R chart
The Shewhart R chart (cf. [22]) is used to monitor the process variability for small group sizes (sample size less than or equal to 10). Let R 1 , R 2 , . . . . . . , R m be the sample ranges of m samples.
Define R = Wσ , where σ is the notation used for the standard deviation of the process and the standard error of R is given as where d 3 is the standard error of the statistic W. The general upper control limit (UCL) of the R chart is Similarly, the 3-sigma UCL of the R chart is The process declares an OoC if R i > UCL. where i = 1, 2, 3, . . . ..

The Shewhart S chart
The Shewhart S chart (cf. [22]) is used to monitor the standard deviation (σ ) in the process. Assume that at disposition there are m preliminary samples each of size n, and let S i be the standard deviation of the ith sample.
Then the average of m standard deviation is defined as The general and 3-sigma UCLs of the Shewhart S chart is If S i fall outside the UCL, then the process is considered as OoC.

The EWMA lnS 2 control chart
For the monitoring of process variance, [11] applied the EWMA scheme to the normal approximation of natural logarithmic S 2 σ 0 2 where σ 0 2 is the IC process variance.
To enhance the efficiency for monitoring the process variability, they re-adjust the EWMA statistic to 0 if it is less than 0. The re-adjustment of smaller EWMA statistics to 0 may improve the EWMA statistic inertia problem and increase its detection ability. They used the following EWMA statistic where EWMA 0 = ln(σ 2 0 ), λ is the smoothing constant, σ 2 0 = 1 and S 2 i is the sample variance. The upper control limit of the EWMA statistic is where L is the charting constant and 15(n − 1) 5 We have OoC signals if EWMA i is greater than UCL. [12] proposed one-sided the CUSUM lnS 2 control chart to monitor process variance. The CUSUM statistic used in their study is given by

The CUSUM lnS 2 control chart
where C 0 = u for 0≤ u < h and S 2 i is the sample variance. The OoC signal is issued as soon as C i > h. [23] proposed the CUSUM control chart based on a chisquare transformation for the monitoring of process variability. They proved that , will an approximately standard normal distribution when σ = σ 0 . Based on the said transformation, a CUSUM statistic by [14] is given as

The χ-CUSUM control chart
where C + 0 = 0 and k is the reference value. The control limit of this statistic if h.

The P σ CUSUM control chart
The inverse normal transformation was applied by [14] to design a P σ CUSUM control chart. The following statistic is used in this study is the CDF of chi-square distribution with n − 1 degrees of freedom. [1] proposed a CUSUM chart, based on the subgroup range, to monitor the process variability. The plotting statistic used in this study is

The CUSUM R control chart
The quantity k is called the reference value and h is the control limit for the chart. [2] proposed a CUSUM S control chart to monitor the process dispersion. The statistic used in this study is given as follows:

The CUSUM S control chart
where S i is the sample standard deviation, C 0 = 0 and k is the reference value. Immediate corrective action is

Performance evaluation
This section gives a brief framework of the measures used for the evaluation of proposed and other control charts under discussion. Let X be a random variable that follows a normal distribution i.e. X ij ∼ N(μ 0 , (δσ 2 0 )), i = 1, 2, . . . . . . .. and j = 1, 2, . . . ., n. Here it should be observed: δ = 1 corresponds to a situation where the process is in control (IC); δ = 1 corresponds to an OoC situation.
From the above notations, we can define the shift as δ = σ 1 σ 0 where σ 0 and σ 1 are IC and OoC standard deviations, respectively, and n is the sample size.
Using these terms, some performance measures are discussed hereafter.

Run Length (RL):
It is the number of observed subgroups before a signal is received by the chart.
(1 ≤ RL < ∞). The value of RL is desired to be large under IC situation. On the contrary, when the process is OoC, we want the value of RL to be small.
Average run length (ARL): A very well-known measure to determine the effectiveness of a chart is named ARL. It is the average number of observed subgroups before a signal is received by the chart. In addition, ARL is classified into two types, the IC ARL that is denoted by ARL 0 and the OoC ARL that is denoted by ARL 1 . ARL 0 of a chart requires to be maximized because it indicates the average number of subgroups until a false alarm is received. On the contrary ARL 1 needs to be minimized as it tells us about how quickly a chart detects the OoC situation.
Standard deviation run length (SDRL): The consistency of RL is studied by SDRL. Mathematically, Extra quadratic loss (EQL): It is the weighted average ARL of a chart in a given interval of shifts from δ min to δ max (see [24] for more details). Mathematically , EQL is defined as Sequential extra quadratic loss (SEQL): SEQL relates to a certain shift (say δ i ) of the EQL, mathematically described as where i = 2, 3, . . . . . . ., δ max (cf. [25]).
Relative average run length (RARL): It is the weighted average ARL of a chart relative to the benchmark chart in a given interval of shifts from δ min to δ max . It measures the efficiency of a chart in terms of its ARL benchmark for each value of shift (cf. [24]). Mathematically, RARL is defined as where ARL(δ) is the ARL of the chart under study and ARL benchmark (δ) is the ARL of benchmark chart for a given value of δ. A chart having a minimum EQL is generally set to be the benchmark. This implies that RARL value for the benchmark chart will always be 1 and RARL will be greater than 1 for other charts under study. Sequential relative average run length (SRARL): SRARL is the RARL of chart up to a certain value of δ (say δ i ), mathematically described as [25]).

The design structure of the one-sided S 2 − Assorted k,λ chart
The design structure of the one-sided S 2 − Assorted k,λ chart is discussed in this section to monitor process variations where the goal is to target a range of shifts (that includes small, medium and large). The proposed S 2 − Assorted k,λ is designed for upward detection in the process variability.
The sample variance is defined as We define the following statistic that may be used for the monitoring of process variance For large shifts in the process dispersion, we define a Shewhart's-type statistic U 1i as where c s is the control limit coefficient for the Shewhart control chart.
For moderate shifts in the process dispersion, we define a CUSUM type statistic U 2i as is the CUSUM statistic and h c is its control limit.
Similarly, for small shifts in the process dispersion, we define a EWMA-type statistic U 3i as where is the EWMA statistic and L e is its control limit. The value sensitivity parameter λ lies between 0 and 1. Finally, using the max approach, we define the plotting statistic of the proposed chart as U i in Equation (4) will always have a positive value. Accordingly, the control limit is set as For any subgroup if U i exceeds 1, it implies an OoC signal from the proposed assorted chart.
Justification for UCL = 1: The justification for selecting 1 as UCL is outlined below: (4)), so U i > 1 indicates the following: (1)) ⇒ the Shewhart statistic V i has exceeded its control limits that are denoted by c s ; (ii) and/or U + 2i > 1 (cf. Equation (2)) ⇒ the CUSUM statistic CUSUM i has exceeded its control limits that are denoted by h c ; (iii) and/or U 3i > 1 (cf. Equation (3)),⇒ the EWMA statistic EWMA i has exceeded its control limits that are denoted by L e ; The design parameters S 2 − Assorted are k and λ i.e. these two quantities identify how sensitive the proposed chart is for a different amount of shifts. The amount of shift and the corresponding optimal design parameters are portrayed in Table 1. After choosing the design parameters, the next task is to set the three limit coefficients (h c , L e , and c s ) so that the overall ARL of the chart is fixed at the desired level. For the said purpose we have implemented the following criteria: Objective: The objective is to minimize EQL. Constraints: The constraints are overall ARL 0 is fixed at the desired level and ARLs of three individual charts (i.e. Shewhart, CUSUM and EWMA) are equal. Mathematically, ARL 0 = such that ARL Shewhart = ARL EWMA = ARL CUSUM .
For an IC situation, we need to adjust the limits coefficients (h c , L e and c s ) of S 2 − Assorted k,λ chart under the said criteria to fix ARL 0 at 200. For the said purpose, we have chosen 15 combinations of the design parameters k and λ covering a wide range of shifts. For a fixed ARL 0 = 200 the three limit coefficients are selected in such a way that equal contribution is coming from all the three charts i.e. the individual ARL 0 values for three charts are exactly the same. Fifteen combinations of the design parameters and the corresponding control limit coefficients are given in Table 2, where the optimal choice of parameters is indicated by bold values and the overall ARL 0 is fixed at 200.
Special Cases: Following charts are a special case of the proposed S 2 − Assorted k,λ : It is to be mentioned that the current study is designed under normality; however, one may extend it for other distributional environments, such as Weibull, Burr, Power, etc. (cf. [26,27] and [28])

Performance evaluations
This section discusses the efficiency assessments and comparisons of the S 2 − Assorted k,λ chart and some existing charts. The existing charts comprise Shewhart R, Shewhart S, EWMA lnS 2 , CUSUM lnS 2 , CUSUM R, χCUSUM, P σ CUSUM, and CUSUM S charts. A number of performance measures including ARL, EQL, SEQL, RARL, and SRARL have been used. To evaluate these measures, we addressed several OoC situations in an attempt to evaluate these measures by considering variable shifts (small, moderate and large) ranging from 1 to 3. The algorithm for the computation of these measures is given as follows: (i) A random sample of size n is generated from a normal distribution with mean μ and variance σ 2 ; (ii) set the control limit coefficients using implementation criteria given in Section 4; (iii) calculate the plotting statistic U i using Equation (4); (iv) based on the choices of λ and k, execute RL operational measure using steps (i)-(iii), (cf. Tables 1 and 2); (v) calculate the ARL and SDRL by repeating step (vi) and generating a distribution of RLs; Furthermore, [25] discussed the procedure to calculate SEQL and SRARL.

Performance evaluation of the proposed S 2 − Assorted k,λ chart
This section provides the assessment of the proposed S 2 − Assorted k,λ chart in terms of measures like ARL and EQL. Many (k, λ) pairs are studied at different levels of δ 1 . The results are given in Table 3 at ARL 0 = 200. We have presented the ARL graphs in Figure 1 and Table 3. The findings support the following: • The case 10 is the optimal choice because it has a minimum ARL 1 for varying shifts (cf. Figure 1(d)) and EQL = 20.18 (cf. Table 3 Figure 1(c,d)).
The results depicted that the S 2 − Assorted 1.00,0.05 has a minimum ARL 1 .  • To get further insight, Figure 2 presents run-length curves for Case 10 (optimal choice) at different levels of shift (δ). Figure 2 shows that for the proposed S 2 − Assorted chart, the probability for shorter run lengths increases as the shift level gets higher. This helps in quicker detection with an increase in the magnitude of shifts. • The sensitivity of the proposed S 2 − Assorted k,λ chart improves with a reduction in λ at a particular selection of k and it is valid for all k attributes. • The sensitivity of the proposed S 2 − Assorted k,λ chart improves with a reduction in k at a particular selection of λ and it is valid for all values of λ (cf. Table 3).

Comparative analysis
The efficiency assessments and contrasts of the S 2 − Assorted k,λ with contending charts including Shewhart R, Shewhart S, EWMA lnS 2 , CUSUM lnS 2 , CUSUM R, χ CUSUM, P σ CUSUM, and CUSUM S are discussed in this section. The comparative assessment is based on two techniques: firstly, based on individual measures (such as ARL); secondly, based on overall measures (such as SEQL, EQL, RARL and SRARL). In addition to assessing these measures, we have discussed distinct OoC scenarios by considering the different amounts of shift (δ) (cf. Table 4) • The S 2 − Assorted 1.00,0.05 chart has minimum EQL value (i.e. 25.56), so it is considered as a benchmark chart. The EQL s of other competing charts including Shewhart R, Shewhart S, EWMA lnS 2 , CUSUM lnS 2 , CUSUM R, CUSUM S, χ CUSUM and the P σ CUSUM

An application
This section gives a real-life implementation of the proposed chart where the manufacturing of semiconductors is going on. An outline of the process related to photolithography is given and finally the application of proposed assorted and some classical methods is explained.

Hard-Bake process
The hard-bake is a well-known procedure used to stabilize the printed topographies for providing optimum performance at etches for the photoresist pattern. The temperature of hard-bake differs but always remains less than 200 C 0 depending on the resistance. The removal of the solvent is ensured by the final bake step. This improves the bonding in wet plating processes so that the resistance is increased against plasma etches. The objective is to minimize the delay between hardbake and plating so that the rehydration of the substrate can be prevented. A repetition of the hard-bake process before etch is optional if the delay after bake is more than 1 h. Sometimes in manufacturing processes, the hard-bake substrates are stored in a dry box because letting the material move from hard-bake to etch is less practical (cf. [29]). An image of a spin coating photoresist process is shown in Figure 3, where five quality characteristics of interest namely resist dispenser, photoresist, resist flies, wafer (width flow measurements) and vacuum chuck is labelled (cf. [30]).

An application of quality control
In this section, we use a dataset extracted from semiconductor manufacturing of a hard-bake process to monitor flow width measurements of wafers. In an application, it was observed that the flow width measurement of wafers is the main variable of interest in this study (cf. [22], as shown in Figure 3. The dataset consists of 25 samples of size 5 each. The application of the proposed chart on the said dataset is outlined in the following steps: First, we constructed the following control charts of hard-bake measurements process data with their respective settings (such that ARL 0 = 200) as listed below: The proposed S 2 − Assorted 1.00,0.05 chart with charting constant (h c = 2.2298, L e = 2.2100 and c s = 2.8295) and UCL = 1; The S 2 Shewhart chart with control limit coefficient (K = 3.84) and upper control limits (UCL = 0.07277); The S 2 CUSUM chart with k = 1.0, control limit coefficient (h = 1.88) and (UCL = 1.88); The S 2 EWMA chart with λ = 0.05 and L = 1.81 and UCL = 0.2898. Figure 4 portrays the graphical display of the charts under discussion on hard-bake measurements data. None of the charts shows any false alarms for the first 25 in-control samples.

Application through data perturbation
To address various possible factors of OoC situations, we manipulated the dataset by data perturbation (cf. [31] and [32]). We used a range of distortions (1.1σ , 1.5σ and 2σ ) to perturb the data and applied the proposed S 2 − Assorted 1.00,0.05 and S 2 Shewhart, S 2 CUSUM charts and S 2 EWMA charts. Figures 5-7 and Table 5 demonstrate the graphic and tabular depiction and detection abilities of the resulting charts.  The results of our finding are outlined as follows: • The S 2 − Assorted 1.00,0.05 chart detected a single OoC signal for (1.1σ ) shift at sample number 45 (cf. Figure 5). • The S 2 Shewhart and S 2 CUSUM charts did not detect any OoC signal for (1.1σ ) shift (cf. Figure 5). • The S 2 EWMA chart captured 5 OoC signals for (1.1σ ) shift at sample numbers 40-46 (cf. Figure 5).
The above analyses indicate that the proposed chart is very effective in detecting a range of shifts in the process dispersion.
The possible explanation for these OoC signals might be the inclusion of special cause(s) in the hard-bake process such as substrate preparation, photoresist coating, edge bead removal, exposure, post exposure bake, developing the image and hard bake. These special cause variations can be possibly the result of external variables including temperature, gas flow and chemical composition. Fixing the issues with these variables in a timely manner is very important because it may damage the whole process and as a result waste of time and cost etc.

Summary and conclusions
Control charts have a central position among all the tools included in the SPC tool kit. These charts are classified based on the size of the shift they target.The Shewhart R and the Shewhart S are the fundamental control charts for detecting large shifts, while the CUSUM lnS 2 and the EWMA lnS 2 charts are used for moderate and smaller shift in variation. In this study, we proposed an assorted (S 2 − Assorted k,λ ) charting mechanism to detect smaller to larger shifts in process variability as a single chart. The efficiency of the S 2 − Assorted k,λ chart is assessed by using the well-known measures such as ARL, EQL, SEQL, and RARL and compared with Shewhart R, Shewhart S, EWMA lnS 2 , CUSUM lnS 2 , CUSUM R, χCUSUM, P σ CUSUM and CUSUM S charts.
A thorough result assessment advocated that the S 2 − Assorted k,λ chart is effective for all kinds of shifts. The performance of the S 2 − Assorted k,λ control chart at k = 1.0 and λ = 0.05 is best in terms of different run length properties. Because, the S 2 − Assorted 1.00,0.05 chart has lowest EQL, it is therefore considered a benchmark chart. The RARL of contending charts are more than 1 which indicates that the performance of the S 2 − Assorted 1.00,0.05 chart is best among the Shewhart R, Shewhart S, EWMA lnS 2 , CUSUM lnS 2 , CUSUM R, χCUSUM, P σ CUSUM and CUSUM S charts. Furthermore, SEQL is calculated to investigate the performance of the aforementioned charts at different amounts of shifts and it also supports the proposed chart. The scope of the current study may be extended in other directions as well such as profiles monitoring, non-parametric design structures and multivariate control charts.