The Quality Management (QM) movement, as we know it today, began in the 1920s, when Walter Shewhart of Bell Laboratories developed a system, known as Statistical Process Control (SPC) to measure variability in production systems for the purpose to diagnose problems. Later, during World War II, the US War Department hired Dr. W. Edwards Deming, a physicist and researcher at the US Census Bureau, to teach statistical process control to the defense industry. Quality control and statistical methods were considered critical factors in a successful war effort (Woodall, 2000; Michel & Fogliatto, 2002; Montgomery, 2014). Unfortunately, most companies in the United States stopped using these statistical tools after the war. US occupation forces in Japan invited Deming to help Japan with the postwar census. He was also invited to present lectures for business leaders on statistical process and quality control.

Quality is a subjective term that is related to the characteristics of the product or service that influence its ability to satisfy the stated or implied needs of users, or simply to products or services that are free from defects and deficiencies (Juran & Gryna, 1988). From a technical point of view, quality can be seen from several dimensions (Garvin, 1993), such as: reliability, durability, ease of maintenance and availability, aesthetics and design. From the point of view of organizational management, quality is a broad approach that involves principles, methods and techniques, when effectively use, reduces losses, increases productivity and, consequently, the effectiveness of organizations (Shewhart, 1931; Smith, 1947; Juran & Gryna, 1988; Kanji, 1994; Kim et al., 2003; Samohyl, 2009; Ryan, 2011; Montgomery, 2014; Toledo et al., 2013; Lizarelli et al., 2016).

As for quality planning, according to Garvin (1993), it is the process of developing actions linked to organizational strategy, including key requirements, performance indicators and operational procedures that ensure standardization and compliance with project requirements. In addition, Deming (1986), supported by the works of Shewhart, introduced, in addition to its 14 points that define the guidelines for the implementation of QM, the PDCA (Plan, Do, Check and Act) and the motivation for the use of statistical methods in support of problem solving. The PDCA is a cyclical model for planning and implementing actions aimed at solving problems and continuously improving processes.

A modern term that defines QM is Total Quality Management (TQM), which raises the scope of quality, with an emphasis on the organization's strategic aspects, and its principles are continuous improvement, involving everyone in quality problems, the use of methods and techniques in problem solving and motivational programs (Garvin, 1993). The methods for solving problems stand out in TQM, especially the SPC.

SPC is a broad set of quality tools known to industrial organizations, whose purpose is to improve processes. One way to do this is to verify that processes are operating in a state of control or meet design specifications. It is also used in initial studies of machine and equipment capability, or even in the choice of a new process, to ensure that it performs better than the old (Woodall, 1985; Baker & Brobst, 1978; Graves et al., 1999; Qin, 2003; Chakraborti, 2006; Jensen et al., 2006; Elg et al., 2008; Castagliola et al., 2009; Samohyl, 2009; Yu & Liu, 2011; Castagliola et al., 2013).

It was Shewhart and Deming who developed the first statistical tools to help correct and improve quality. Later, the Japanese began using these tools, guided by Kaoru Ishikawa, head of the Japanese Union of Scientists and Engineers (JUSE), and expanded them worldwide. For more than half a century, SPC has played a key role in controlling and improving the quality of industrial processes, initially based on Shewhart control charts. The primary issue regarding control charts is to understand the variability of a particular quality characteristic, establish process control, and promote improvement actions (Sower et al., 1994; Baker & Brobst, 1978; Graves et al., 1999; Woodall, 2000; Kim & Schniederjans, 2000; Chakraborti, 2006; Duarte & Saraiva, 2008; Ho & Trindade, 2009; Celano et al., 2013; Wiederhold et al., 2016; Aykroyd et al., 2019).

According to Woodall (1985) and Woodall & Montgomery (2014), control charts are used to distinguish between two types of variation. One is the variation of common causes, which is inherent in the process and cannot be changed without changing the process itself. The other type are special causes of variation that generate interruptions with significant effects on the process and must be removed.

The general knowledge is that the process is considered stable or In Control (IC) if the successively observed chart statistics are plotted within the control limits, that is, if the process is being influenced only by common causes. However, when chart statistics are plotted outside the control limits, this may be a sign that the process may be Out of Control (OC) and corrective action in the process may be required, as suggested by Jensen et al. (2006). When a special cause is detected, the normal action is to stop the process to eliminate it. This will bring stability to the process, but sometimes they also require financial resources, time and opportunities.

One of the limitations of Shewhart charts is that they address the problem of monitoring and controlling small batches. Another limitation is that if the purpose of a process control system is to make economic decisions about the process, it is possible to balance the consequences of two situations: taking action when it is not necessary (overreaction) versus not taking action when it is required. Thus, it is possible to make economically viable decisions about the condition of the process. According to Woodall (1985) there is a need to react, however, only when a cause has sufficient impact for it to be practical and economical to remove it to improve quality. Literature has investigated the development of economic models for process control (Woodall & Montgomery, 2014; García-Díaz & Aparisi, 2005).

Moreover, production systems have undergone significant changes. Large scale gave way to lean and customized production, bringing smaller and diversified production plans. According to Castillo et al. (1996), this trend is strongly related to the use of Just In Time (JIT) manufacturing techniques to reduce costs related to intermediate and finished product inventories.

Some doubts about how to statistically evaluate these processes through traditional methods arose from there. Strictly speaking, Shewhart control charts were proposed for monitoring large volume production processes. In these systems, the implementation of charts is not a big problem, as process information is always available, unlike JIT or job shop type production systems, for example (Hillier, 1969; Cullen & Bothe, 1989; Crowder, 1992; Sower et al., 1994; Castillo et al., 1996; Castillo & Montgomery, 1996; Khoo et al., 2005; Celano et al., 2010; Gu et al., 2011,2014; Wiederhold et al., 2016).

The first was to establish control limits to retrospectively test whether the process was under statistical control when reference samples had been collected for plotting during phase I. For each initial subgroup, the $\overline{X}$ and R observations should be plotted on the charts. If values were outside the control limits, then the subgroup should be discarded and the control limits recalculated. In the second stage, new control limits should be defined to verify the permanence of the process in this same stability condition. The probability of type I errors should also be considered at this stage.

Later, Quesenberry (1991) proposed the use of the Q chart to deal with the observed problems. However, the chart had the same problem as Hillier (1969). While the occurrence of errors type I was low, trend detection was not as satisfactory as desired.

Shepardson et al. (1992), studied the inefficiencies of the Q chart and proposed the use of a control chart based on the Kalman filter, specifically when the process standard deviation was known and the mean was not. Castillo & Montgomery (1996) proposed adaptations that allowed the use of the chart in inverse situations, that is, when the mean was known and the standard deviation of the process was not.

Wasserman (1994) proposed the use of the Exponentially Weighted Moving Average (EWMA) chart based on a first order dynamic linear model with constant variation. According to Lucas & Saccucci (1990), the interpretation and implementation of the EWMA chart was reasonably easy.

The CUSUM chart, was initially presented by Page, in England, with the purpose of quickly detecting small changes in the process (Page, 1954; Barnard, 1959; Kemp, 1961; Brook & Evans, 1972; Singh & Prajapati, 2013; Celano et al., 2012; Montgomery, 2014; Abbasi & Haq, 2019).

Hawkins & Olwell (1998) proposed the use of an adapted chart called self-starting CUSUM for a small number of subgroups. The idea of this chart was to use regular process measurements for self-tuning and maintenance. Each successive observation should be standardized using the mean and standard deviation, not from a special fit sample, but from all observations accumulated up to the time of inspection. As the process proceeded and produced new observations, the mean and standard deviation estimates the true values (Hawkins & Olwell, 1998).

Unlike Shewhart control charts and EWMA chart, the use of the CUSUM chart was considered more complex, however, its implementation became easier with the availability of numerous statistical tools and software on the market (Kemp, 1961; Brook & Evans, 1972; Hawkins & Olwell, 1998; Castagliola & Maravelakis, 2011).

Nenes & Tagaras (2007) presented charts based on Bayes' theorem for monitoring small production batches. Celano et al. (2013) proposed the use of control charts based on the t student distribution as an alternative as efficient as traditional charts with known parameters.

This work studies the proposal of Cullen & Bothe (1989). The authors presented a chart where the difference between the measure found of the quality characteristic with its nominal measure, or with a target value, should be plotted on traditional control charts $\overline{X}$ and R and called it the target value control chart.

The DNOM control chart can be used in the manufacturing process of more than one product sequentially. It represents a change in traditional CEP techniques, as it provides the means to monitor and control processes that would otherwise be considered inadequate (Cullen & Bothe, 1989; Crowder, 1992; Farnum, 1992; Sower et al., 1994; Montgomery, 2014). Montgomery (2014) presents three important conditions for the proper use of this chart:

• The variation in the process must be the same for all parts, or as close as possible;

• The procedure is more efficient when the size of the samples collected is the same for all products;

• The charts have an intuitive appeal when the nominal measure of the characteristic is its own target value.

As with the other control charts, when the statistical parameters of the process are unknown, the constructive procedure of the chart of the deviation from the nominal is carried out in two phases. In phase I, samples of the process are used to estimate the parameters and calculate the control limits used in the next phase. In phase II, also called the monitoring phase, new samples are extracted and verified. If the observations are not located within the control limits, the process is considered out of control and a probable special cause must be identified (Grant, 1965; Castillo & Montgomery, 1996; Woodall & Montgomery, 2014; Chakraborti, 2000; Jensen et al., 2006; Chakraborti, 2006; Samohyl, 2009; Ryan, 2011; Montgomery, 2014; Jones-Farmer et al., 2014).

Psarakis et al. (2014), clarify that the good performance of the control charts depends on an accurate estimate of parameters performed during phase I of the implementation of the charts. During this phase, it is important that special causes are eliminated so that samples for defining the control limits used in the monitoring phase are collected from a stable process.

Jensen et al. (2006) highlights that the use of control charts with estimated parameters is a potential weakness. The unavailability of data, as a consequence, calculation of invalid control limits, can cause a significant rate of false alarms in the process, in addition to reducing the Detection Power (PD) of the control chart. For that reason, Chakraborti et al. (2008) reinforce the importance of defining an adequate number and sample size during phase I so that the performance of the control chart is as close as possible to the ideal.

To simulate the use of the nominal deviation control chart in any production process and analyze its performance, mathematical models were developed in the Maple 2016 software. The first model was created to calculate the ARL of the control chart built with known parameters, or KK chart, seen in Appendix 1. The chart's ARL is calculated in the same way as the mean chart, as both are based on a Normal probability distribution.

If $\overline{X}\text{i}$ is calculated from a sample n = $\left\{{\text{x}}_{\text{1}}\text{,}\text{}\text{} \text{}\text{}{\text{x}}_{\text{2}}\text{,}\text{}\text{} \text{}\text{}\mathrm{..,} x_n\right\}$, the probability of error type I is α = P ($\overline{X}$ $\notin$ (LCL, UCL)) or α = 1 $\text{--}$ P ($\overline{X}$ $\in$ (LCL, UCL)). Considering known parameters, it is possible to say that the process is in control when:

Considering an out of control process $\text{}\text{}\mu \text{}\text{}\text{}\text{} \text{}\text{}\text{+}\text{}\text{} \text{}\text{}\text{}\text{}\delta \text{}\text{}\text{}\text{}\sigma \text{}\text{}{}_{\text{0}}$, if $\text{}\text{}\delta \text{}\text{}\text{}\text{} \text{}\text{}\ne \text{}\text{} \text{}\text{}\text{0}$ and k $\text{}\text{} \text{}\text{}\text{=}\text{}\text{} \text{}\text{}\text{3}/\sqrt{\text{n}}$, UCL and LCL are calculated according to Equations 2 and 3:

It is possible to write $\text{P}\left(\text{LCL}\le \overline{X}\le \text{UCL}\right)$, according to Equations 4 to 7:

If $\text{k}=\text{3}/\sqrt{\text{n}}$ e $\text{P}\left(\text{LCL}\le \overline{X}\le \text{UCL}\right)$ $\text{=}$ $\text{P}\left(\text{--} \text{3}\text{}\text{} \text{}\text{}\text{--}\text{}\text{} \text{}\text{}\text{}\text{}\delta \text{}\text{}\sqrt{\text{n}} \le \text{Z} \le \text{3}\text{}\text{} \text{}\text{}\text{--}\text{}\text{} \text{}\text{}\text{}\text{}\delta \text{}\text{}\sqrt{\text{n}}\right)$, the probability is obtained from Equation 8:

It can be write:

If the process is in control when $\delta =0$, therefore, the Run Length (RL) is geometric with probability $P \left(\delta , n\right)$, then, the ARL of control chart KK is calculated according to Equation 10 bellow:

A model was created to simulate the ARL of a deviation from nominal control chart with both parameters estimated, seen in Appendix 2. If $\left\{{\text{x}}_{\text{1}}\text{,}\text{}\text{} \text{}\text{}{\text{x}}_{\text{2}}\text{,}\text{}\text{} \text{}\text{}\mathrm{..,} x_n\right\}$, $\text{i}=\text{1}, \text{2}, \dots \text{m}$, extracted during phase I, $\text{}\text{}\mu \text{}\text{}{}_{\text{0}}$ and $\text{}\text{}\sigma \text{}\text{}{}_{\text{0}}$ can be estimated according to Equation 11:

Being:

Where:

Considering an in-control process with parameters and control limits estimated:

It is possible to say that the process is in control when:

Being:

The probability function of U is f $\left(\text{u}\right)=\text{m}\left(\text{n}-\text{1}\right) f_{{\text{x}}^{\text{}\text{} \text{}\text{}\text{2}}}\left(\text{m}\left(\text{n}-\text{1}\right)\text{u}\right)$, and if RL is geometric with probability $\text{P} \left(\text{W}, \text{U}, \Delta , \text{}\text{}\delta \text{}\text{}, \text{m}, \text{n}\right)$, the ARL of control chart UU is calculated according to Equation 17:

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