1. INTRODUCTION
Reliability in quality, as one of the most significant aspects of product/service quality assurance, has long been a universal challenge and will remain so in the 21st century (Qazani et al., 2019;Rjoub et al., 2017;Singh, 2021;Afanasyev et al., 2021). Indeed, because of the increasing complexity of manufacturing systems, the cost of product warranty is increasing (Singh and Dhamija, 2019;Ab Yajid, 2020). Low reliability severely affects the bottom line of business and hinders their ability to gain and maintain market share (Yousif et al., 2020). Therefore, in order to achieve continuous market presence and a lasting brand, it is crucial for businesses to alleviate the concerns of customers about whether their products will work as intended over a certain period of time (Azzawi, 2021;Zainal et al., 2021;Haq et al., 2021). Using any system without due attention to its reliability may increase the likelihood of sudden failures. In many industries (e.g. aircraft manufacturing), such failure can have profound consequences in terms of economic performance, loss of life, political implications, sense of safety, and loss of prestige (Mitra and Khan, 2021). For example, the Chernobyl incident in 1986 killed 31 employees, causes serious illness in 200 people, and caused economic losses to the tune of about three billion dollars. The explosion of Challenger spacecraft in the same year, the failure of shuttle spacecraft in 2003, the repeated oil spills off the coast of the United States, the more recent explosion of the Fukushima nuclear power plant in Japan, and many smaller incidents that occur every year around the world all highlight the importance of the subject of system reliability and safety (O’Connor and Kleyner, 2012). For a product to have the specifications desired by the customer, it must be produced by a stable or reproducible process. In other words, the production process must have minimum variability within the target range (Dan et al., 2021;Josiah et al., 2021).
The concepts of reliability and reliability monitoring have long been a subject of interest to engineers and academics. Reliability monitoring can be done through control charting in two modes: monitoring quality characteristics and monitoring failure times. One of the statistical tools available for monitoring process stability is the control charts of Shewhart (1931). Notable works in the field of reliability monitoring based on failure times include the control chart procedures proposed by Xie et al. (2002) for the monitoring of timetoevent for r number of failures. This method was also extended to the monitoring of timetoevent for a fixed number of failures. One of the advantages of this method is the ability to detect process improvements in highly reliable environments. In another study, Khoo and Xie (2009) proposed a control charting scheme for the monitoring of regularly maintained systems by examining the timebetweenevents variable. Using this chart, it is possible to decide whether a system needs more regular maintenance. In this study, the failure time distribution was estimated as an exponential distribution where the average failure rate depends on the maintenance interval. Sego et al. (2009) developed a tool for monitoring patient survival times in health service organizations like hospitals. For this purpose, they proposed a cumulative sum (CUSUM) chart of total survival time with risk adjustment for the monitoring of continuous timetoevent variables. They also investigated the effect of this chart using the cardiac surgery data and made some suggestions regarding the amount of historical data need to build a proper chart. In another study, Sürücü and Sazak (2009) modeled the timebetweenevents of Ebola using a three parameter Weibull distribution where parameters are unknown and must be estimated by a robust estimation method. They also extended the proposed method to the monitoring of the cumulative time elapsed between r failures using chisquare and normal moment approximations. Zhang et al. (2011) developed an economic model for the monitoring of timebetweenevents in cases where failure data follow an exponential distribution. Their method strives to provide a more realistic picture of the situation by considering the randomness of process shifts. The numerical evaluations of these researchers showed that considerable false alarm detection power and excellent incontrol stability make the proposed control chart suitable for monitoring large process shifts, where it can deliver significant time and cost saving. Considering the limitations of previous methods for the monitoring of rightcensored failure time data in terms of the distribution of observed failure times and censoring times, Li and Kong (2015) presented a generalized control chart that can work with a variety of distributions. More specifically, they developed a statistical method for the monitoring of failure time data in the presence of random right censoring, which employs onesample nonparametric rank tests without making any assumption about the distribution of data. In a study by Faraz et al. (2015), they developed Shewharttype z and S^{2} control charts for monitoring the reliability of processes that follow the Weibull distribution, and specifically monitoring the shifts in the scale and shape parameters of these processes. They argued that shifts in Weibull parameters leads to a variety of changes in the mean and variance of the normal variables, which can be properly detected in the mentioned charts. The advantage of this method is the ease of use while offering excellent process monitoring power. This method can also be applied to any other distribution. These researchers demonstrated the effectiveness of their method by applying it to a real dataset with the help of software applications. While most of the existing charts have been designed for the monitoring of a single process parameter such as mean or variance, recently, researchers have developed a number of charts for the simultaneous monitoring of the mean and variance of processes that follow a normal distribution (McCracken et al., 2013;Qazani et al., 2020). In some processes, mean and variance of the process shift simultaneously, and this is important because a shift in variance can affect the control limits of the mean chart. Therefore, it is better to monitor both parameters simultaneously. In such cases, instead of analyzing mean and variance in two different charts, it is more practical to use a single chart for both parameters. Gan et al. (2004) proposed a control charting scheme for the simultaneous monitoring of mean and variance of such processes. In this charting scheme, which is a combination of X and S charts, if the process parameters are unknown, at least 100 incontrol samples need to be collected and the control limits of the chart need to be determined accordingly. Given that location and dispersion parameters also play an important role in the monitoring of processes, it might be necessary to carefully track changes in their behavior. In a recent study, Zafar et al. (2018) proposed a parametric charting scheme called MaxP chart for the joint monitoring of location and dispersion parameters, which is based on progressive mean with max statistic. In this study, researchers made extensive comparisons between the proposed charting scheme and several existing schemes with the assumption of normality of quality characteristics. These researchers analyzed the performance of schemes using run length properties including average run length, standard deviation of run length, relative average run length, and a performance comparison index. The results of this study showed that the MaxP charting scheme performs relatively better in detecting shifts in process parameters (Goli et al., 2019;Golubev et al., 2021;Munawir et al., 2021).
In the majority of aforementioned control charting schemes, reliability assessments are performed through separate monitoring of variables of failure time distributions and plotting them between statistical control limits. This study aims to contribute to the simultaneous monitoring of these variables. The paper present new control charts for reliability monitoring in cases where failure times follow normal and lognormal distributions. These charts are plotted such that one can determine whether the process is in control or out of control based on the mean and standard deviation of failure times. After the simulation, we assess the validity of the proposed charts in terms of average run length (ARL).
2. METHODOLOGY
A normal distribution is a twoparameter distribution with parameters (μ, σ), where μ is the mean and σ is the standard deviation. The probability density function and the cumulative distribution function of this distribution are as follows (Mitra and Khan, 2021):
In many practical situations, failure rate of components or parts can be described by a normal distribution. For example, most mechanical components that are exposed to periodic and repetitive loads exhibit normally distributed failure rates because of fatigue (O’Connor and Kleyner, 2012). Another important distribution in the field of failure time modeling is the lognormal distribution. This distribution is commonly used to model the failure times of reliability tests. The lognormal distribution is closely related to the normal distribution. If T follows the lognormal distribution, then will follow the normal distribution.
Because of this relationship, lognormal failure times can be easily converted to normal failure times. Therefore, in the following, the control chart is developed for normal distribution. To monitor lognormal failure times, we first convert them to normal.
Considering the extensive use of this function in reliability and failure probability calculations, we can convert the normal distribution of interest into the standard normal form by changing the variable $\text{z}=(\text{x}\text{\mu})/\text{\sigma}$. Essentially, the standard normal distribution is the same as the normal distribution but with parameters (μ = 0, σ = 1). The probability distribution function and the cumulative distribution function of this distribution are as follows (Mitra and Khan, 2021):
2.1 Development of Normal Distribution Control Charts
In this section, we try to develop a control chart based on the mean and standard deviation of failure times using the logic outlined below. This chart will have an acceptance region and a rejection region. If the mean and standard deviation of the failure times are within the acceptance region, then there is no evidence of deviation in the distribution of failure times. Otherwise, the distribution of failure times may be deviated. Suppose variable X follows the normal distribution with mean μ and standard deviation σ and the distribution parameters are known ( X~ N(μ,σ2) ).
LSL and USL are the lower and upper specification limits of the normal distribution chart, and θ is the type1 error. These parameters are plotted in Figure 1.
If f(x) is the probability density function and F(x) is the cumulative distribution function of the normal distribution, we have:
The parameter θ (type1 error) is given by:
where Φ is the standard normal cumulative distribution function. If μ and σ are independent, each given amount of nonconformance or error (θ) can be produced from an infinite number of (μ, σ) combinations that satisfy the following condition.
In this equation, if there is a shift in mean (standard deviation), one can find a standard deviation (mean) that would produce the same θ. In other words, as long as the shift in one of the parameters of the failure time distribution coincides with an appropriate change in the other parameter, the process can still be considered incontrol. In Figure 2, this chart is plotted for the case where $\text{\theta}=0.05,\hspace{0.17em}\text{USL}=3,\hspace{0.17em}\text{LSL}=2.2$.
In Figure 2, the points positioned on the curve have a nonconformance of θ, the points below the curve have a nonconformance of less than θ, and the points above the curve have a nonconformance of greater than θ. Therefore, if the mean and standard deviation of failure times are such that the combination falls below the curve, it can be concluded that failure times have a favorable distribution. Now, we try to express this chart in the form of specific formulations. As shown Figure 3, it is assumed that the edges have a fixed slope.
To obtain the equation of the line , we consider a special case where the mean of failure times is such that the probability of failure times being greater than USL is almost zero. Figure 4 shows an example of this situation.
In Figure 4, θ is given by:
where has a normal distribution with zero mean and unit variance. Now, we try to use Equation (10) to determine the equation of the line between μ and σ. After applying a few simple mathematical operations on Equation (10), we arrive at:
According to Equation (11), the slope m_{1} is:
Similarly, to obtain the equation of the other edge line, we consider a special case the probability of failure times being less than LSL is almost zero. In this case, we will have:
where $\text{U},\text{N}(0,1)$ follows a normal distribution with zero mean and unit variance. Now, we use Equation (13) to determine the equation of the line between μ and σ. With a few simple mathematical operations, we arrive at:
According to Equation (14) the slope m_{2} is
The two obtained equations intersect at the midpoint of the distance between LSL and USL, where $\text{\mu}=(\text{USL}+\text{LSL})/2$. According to Equations (11) and (14), the standard deviation equivalent to this mean is:
The edges and their intersection are shown in Figure 5.
In Figure 5, ${\text{\sigma}}_{\text{L}}$ or the maximum standard deviation of the chart can be calculated based on the midpoint of the interval asad. At this point, θ is evenly distributed over the range outside the specification limits, i.e.:
Thus, by substituting $\text{\mu}=\frac{\text{USL}+\text{LSL}}{2}$ into the above equations, we arrive at:
In the following, we obtain the coordinates of the intersection of the line σ_{L} with the edges of the chart. Substituting σ_{L} into Equations (11) and (14) gives the horizontal coordinates of the intersection of the line σ_{L} with the edges based on the notations provided in Figure 6:
As shown in Figure 6, these points form a trapezoid with σ_{L}. Here, to develop an easy to draw control chart, we approximate the chart of Figure 3 with this trapezoid. Given the small difference between these two charts (see Figure 5), the approximation error can be safely ignored.
In the following sections, we will analyze the performance of the developed chart.
2.2 Failure Time Control Limits
Equations (10) and (13) can be used to obtain the upper and lower limits of failure time based on μ and θ. In other words, having these equations and knowing the mean and the nonconformance level, the upper and lower limits of failure time can be obtained as follows:
2.3 Development of Lognormal Distribution Control Chart
Given the relationship between the normal and lognormal distributions, the lognormal failure time data can be easily converted to normal. Thus, to monitor lognormal failure times, we first use a logarithmic conversion method to convert them into normal failure time data and then use the chart developed for normal distribution.
2.4 Performance Evaluate of the Proposed Chart
We use the simulation method to evaluate the performance of the proposed chart. This evaluation is performed in terms of average run length (ARL), which refers to the average number of observations before reaching the first outofcontrol observation. We also use the standard deviation of run length (SDRL), which measures the accuracy of computational results. A small ARL in phase zero indicates that the process is outofcontrol and a large ARL means that the process is incontrol. For this purpose, phases zero and one are performed as follows:
Phase Zero: In this phase, first the upper and lower limits of failure times must be obtained based on target μ and θ using Equations (21) and (22). Then, the θ level of the production process should be obtained through simulation. For this purpose, based on the obtained limits, we need to obtain the θ level that provides a certain amount of ARL. In other words, assuming that the process is in control, we must obtain the θ level based on which the process will be identified as incontrol in terms of ARL. Then, having this θ value and Equations (18) to (20), the zero phase control chart can be drawn as shown in Figure 6.
Phase one: In this phase, the failure times are monitored based on the chart obtained in phase zero. To evaluate the performance of the proposed chart, we apply some shift to the failure time distribution parameters and assess the chart’s ability to detect them. Provided in the following is a brief description of the steps followed to conduct the simulation based on the following variables.

n = Sample size

N = Number of data required to estimate failure time distribution parameters

ITN = Number of simulation iterations

RN_{ITN} = Run number in iteration ITN

MITN = Maximum number of simulation iterations

Assign values to n, N, and MITN.

Set ITN = 1.

Generate N random failure times based on the failure time distribution parameters (incontrol).

Apply shifts to the mean or standard deviation of the failure time distribution.

Set RN = 1.

Generate n failure times based on the new failure time distribution (outofcontrol).

Select N last failure times out of N+n exiting failure times.

Obtain the mean $(\text{\mu '})$ and standard deviation $({\text{\sigma}}^{\text{'}})$ of the data selected in Step 7

Use Equations (18) to (20) to check whether or not the point obtained in Step 8 is within the acceptance region of the chart. If the point is in the acceptance area, set ${\text{RN}}_{\text{ITN}}={\text{RN}}_{\text{ITN}}+1$ and return to Step 6, otherwise go to the next step.

If ITN < MTN, then set ITN = ITN + 1 and return to Step 5, otherwise obtain the ARL using the following equation:
3. NUMERICAL RESULTS
In this section, we examine the performance of the proposed control chart using the following parameter values:
Therefore, the upper and lower limits of failure times are calculated as follows:
For phase zero, we obtain the θ value of the process based on ARL=370. For example, the ARL values for different θ values are given in Table 1. According to Table 1, the minimum θ value that satisfies the desired ARL level is approximately θ=0.0253.
Figure 7 shows all the points obtained in one iteration of the simulation. For example, in this iteration, the run length is 352.
In phase one, the selected θ level is used apply various shifts to mean and standard deviation. The ARL changes resulting from these shifts are given in Table 2. The ARL and SDRL values obtained for different sample sizes (n) are provided in Table 2.
Considering that the target mean and standard deviation are μ = 50 and σ = 4, the results obtained by applying shifts indicate the following:

When the mean goes below the target value, the average run length decreases.

When the mean goes above the target value, the average run length decreases.

When the standard deviation goes below the target value, the average run length increases.

When the standard deviation goes above the target value, the average run length decreases.
As the normal control chart indicates, any shift in the mean away from the target value (increase or decrease) corresponds to moving out of the trapezoid, which means the process is outofcontrol. As a result, ARL, which is the average number of first outofcontrol observations, decreases. An increase in the standard deviation also corresponds to moving out of the trapezoid, which decreases the ARL value. However, when we decrease the standard deviation for a constant mean, the points still fall inside the normal trapezoid, which means the process is in control. Therefore, when the standard deviation decreases, ARL increases. The trends of ARL changes for different shifts in mean and standard deviation and for different sample sizes are illustrated in Figures 8 and 9.
As these Figures 8 and 9 show, there is a direct relationship between the sample size and the average run length. In both cases of constant mean and constant standard deviation, ARL decrease with increasing sample size.
4. CONCLUSION
This study presented a new control chart for the monitoring of normal and lognormal failure times. The proposed chart allows for simultaneous monitoring of shifts in both mean and standard deviation of normal and lognormal distributions. With this chart, one can easily monitor the probability distribution of failure times based on their mean and standard deviation. The performance of the proposed chart was evaluated through simulation in terms of average run length. The proposed approach can be used to determine the uniformity of reliability of supply and production processes, help manufacturers assess the reliability of their products over time, and prevent the shipment of products that are substandard from the reliability standpoint. The proposed chart is based on the assumption that the failure time distribution parameters are determined. Thus, in future studies, these parameters can be considered unknown and predicted using various estimators. Also, while this study assumed that the products will fail in a single mode, future studies may expand the work by considering several failure modes. It is also possible to develop similar charts for monitoring the parameters of other probability distributions.