Integrated Optimization Model for Maintenance Policies and Quality Control Parameters for Multi-Component System

The practical applications of integrated maintenance policies and quality for a multicomponent system are more complicated, still rare, and incomplete to meet the requirements of Industry 4.0. Therefore, this work aims to extend the integration economic model for optimizing maintenance policies and quality control parameters by incorporating the Taguchi loss function for a multi-component system. An optimization model is developed based on preventive maintenance, corrective maintenance policies, and quality control parameters with the CUSUM (Cumulative Sum) chart, which is widely used for detecting small shifts in the process mean. The model was developed to minimize the expected total cost per unit of time and to obtain the optimal values of decision variables: the size of samples, sample frequency, decision interval, coefficient of the CUSUM chart, and preventive and corrective maintenance intervals. The solution steps were employed by selecting a case study in the Alahlia Mineral Water Company (AMWC). Then, the design of experiments based on one-factor-at-a-time was used to evaluate the effect of selected decision variables on the expected total cost. Finally, sensitivity analysis was performed on the selected decision variables to demonstrate the robustness of the developed model. A predictive maintenance plan was developed based on the optimal value of preventive maintenance interval, and the results showed that the performance of the maintenance plan realizes the full potential of the integrated model. In addition, the case study results indicate that the extended integrated model for multicomponent is the new standard for the quality production of multi-component systems in future works.


Introduction
Manufacturers face rising obstacles in today's highly competitive business climate, as well as the need for high levels of customer satisfaction. Therefore, they strive to increase production performance by reducing breakdowns and quality losses in order to maximize profit and preserve competitive advantages. Throughout manufacturing cycles, systems or equipment wear down or degrade due to use and age. Moreover, systems might malfunction at any time, which can impede production. Therefore, maintenanceincluding inspection, repair, and replacement-is essential for boosting systems' availability and ensuring continuous production. Based on how they intend to handle failures and maintenance, there are two types of maintenance policies: preventive and corrective [1]. In practical applications, corrective maintenance (CM) is used to return equipment to its operational state after failure; as a result, this kind frequently results in severe delays and substantial financial losses. To reduce the likelihood of equipment failure or degradation, preventive maintenance (PM) is carried out at predetermined intervals or in accordance with predetermined criteria. These plans come with expenses, including those for samples, downtime, repairs, and inspections, among others. After that, integrating these two maintenance policies will result in cost savings.
Moreover, a good PM policy will lower the breakdown rates and process variances and maintain the expected productivity of the machine [2,3], which improves the quality levels. Therefore, quality control and maintenance policies are linked to cost savings and improving production performance. Furthermore, this will be ineffective when PM is performed on a single component (or machine) in a complicated production system. Hence, PM tasks should be performed on multi-component systems. Performing group PM tasks can significantly reduce total downtime and associated costs, such as labor costs, lost productivity due to machine breakdowns, and other production losses [4].
Since systems have many interrelated components, modeling and optimizing maintenance plans must consider the interactions between these connected components. These interactions also offer the chance for collective maintenance, which might lead to cost savings. Several approaches and models have been studied for maintenance optimization in multi-component systems [2,[5][6][7][8][9][10]. However, there are few approaches (or models) for maintenance optimization in real-world multi-component systems. In earlier work by Dekker and Roelvink [11], a heuristic replacement strategy was utilized, applicable only if a fixed component category had been replaced. In another study, Huang [12] presented a new method based on the cost boundary condition for replacement scheduling and inferred that their approach is more effective. Similarly, Samrout et al. [13] presented an ant colony optimization method to minimize the PM cost of the multi-component system with constant failure rates. Moreover, Laggoune et al. [14] proposed a PM plan method for a multi-component system with random failures. Recently, the literature has reported integration methodologies for the multi-component system [15]. Magnanini and Tolio [16] proposed a mathematical model for preventive maintenance to minimize maintenance, inventory, and backlog costs. Furthermore, Andreas et al. [17] developed a holistic approach based on reinforcement learning algorithms to optimize maintenance plans to improve resource utilization in manufacturing systems.
Integrating quality control and maintenance policy models has been gaining popularity recently among researchers and industrial companies for improving product quality and reducing maintenance costs [18]. Several studies have been performed on integrating quality control and maintenance policy, and it was suggested that process quality control and machine maintenance should be examined simultaneously for cost-effectiveness. Integrating quality control and maintenance choices has been proven to considerably lower total production costs for single-machine units and multi-stage production systems. The studies by [19][20][21][22][23] showed that combining SPC (Statistical Process Control), maintenance policy, and production planning in the industrial system can significantly reduce costs. From a historical viewpoint, researchers began constructing combined economic models of control charts and maintenance utilizing X-bar control charts because of the association between quality control and maintenance [24][25][26][27]. Other studies by Baker [28] and Montgomery and Heikes [29] were conducted on the non-exponential probability distribution function of the in-control period. Consequently, Lorenzen and Vance [30] developed a unified model that is flexible enough to apply to a situation where production continues or ceases during the search or repair time. In addition, a study by Tagaras [24] integrated an economic model for maintenance procedures, process control, and other design parameters. After that, Rahim [25] developed a model for designing the optimal parameters of an X-bar control chart and PM periods for a manufacturing system with an increasing failure rate that follows the Weibull distribution.
Over time, researchers have paid greater attention to exponentially weighted moving average (EWMA) control charts than X-bar charts [31][32][33]. For example, Sultana et al. [34] developed an economic model using an EWMA control chart based on the VSIFT (variablesampling interval with fixed time) sampling policy and preventive maintenance. However, many studies currently prefer CUSUM over X-bar and EWMA charts. Moreover, CUSUM responded quickly and accurately when identifying shifts within one or two standard deviations [35].
Shrivastava et al. [36] developed a new model to combine PM and SPC-based CUSUM charts and determined the decision variables optimization, which effectively minimizes the total cost per unit of time. In addition, Li et al. [37] introduced a new integrated model for joint CUSUM control chart parameters and an age-based PM policy. They used a recursive algorithm and multiphase system to model each phase. Saha [38] recently integrated a new economic model for maintenance policy and quality control. Using the CUSUM control chart, the researcher monitored the process's mean and variance. Taguchi's quadratic loss function and a modified linear loss function were utilized to calculate the in-control and outof-control processes. Farahani et al. [39] integrated a model to optimize statistical quality control policy: sample size, sampling interval, control limit, and preventive maintenance policy. The proposed model was developed based on a continuous Markov chain and − X Chart to minimize the cost per unit of time. Regarding the multi-component system, works based on integrated maintenance policy and quality control are limited. Only one study integrated the quality and PM policy for the multi-component system by Al-Shayea et al. [40]. They utilized an EWMA chart to present a new model for integrated PM and SPC for a multi-component system.
It is evident from the literature mentioned above that the practical applications of integrated maintenance policies and quality for a multi-component system are more complicated and still rare and incomplete. Although a multi-component system provides a precious and comprehensive analysis that readily minimizes the overall maintenance costs and sustains high-quality products, very limited studies were conducted regarding the integrated quality and PM models for the multi-component system. Furthermore, multicomponent systems require a complicated procedure that demands additional data than conventional methods. Moreover, this recent work requires a deep understanding and analysis of the small shift process. It is an ideal system for investigating a wide range of failure rates. Therefore, there is a need to develop a new model for integrating the CUSUM chart and PM policy for multi-component systems, which considers the limits and gaps in the previous studies. This research aims to extend the developed model by Pandey et al. [41] for integrating maintenance policy and quality control chart for the multi-component system using the CUSUM chart incorporating the Taguchi loss function.
The structure of this paper is as follows. The problem description is provided in Section 2. Section 3 describes the research methodology and explains the formulation of a multi-component system cost model. The selected case study is described in Section 4 to show how the constructed model may be used. Section 5 presents the results of the design of experiments and the sensitivity analysis to study the effects of decision variables on the expected cost. Finally, Section 6 presents the conclusions inferred from the research and provides future research directions.

Problem Description
Maintenance of multi-component systems is essential to ensure optimal functioning and longevity. Multi-component systems comprise multiple interconnected and interdependent parts, and failure of any one part can have significant consequences for the entire system. According to a study by Kamal, Arumugam, and Yousuf [42], "neglecting the maintenance of multi-component systems can lead to catastrophic consequences, including system failure, loss of production, and environmental disasters." Therefore, the maintenance of multi-component systems is critical for ensuring their reliable and safe operation, minimizing downtime, and maximizing productivity and profitability. In this paper, to improve the efficiency of such systems as well as their production quality, we considered the integration of statistical process control and maintenance policies for a multi-component system.
To address the quality issue, we have assumed that, at any period, the production process can be generally divided into two states. The first one is an in-control state which indicates that the production is smoothly running with no problems during a specific period. The second stage is the out-of-control state which happens after a certain amount of time due to machine (components or parts) wear or worker error, and products with quality issues are produced during this state. In addition, to address the machine down-times issue, we assume that downtimes can be categorized into the following two classes, according to their consequence on the production process [36]: -Type 1 (F 1 ) is related to the mechanical failure of the machines in the system. -Type 2 (F 2 ) is quality-related and is observed when the production process goes into an out-of-control state. When such failures are observed, an immediate shutdown occurs, and all corrective actions are carried out to restore the process to its normal operation (i.e., the in-control state). However, the process may also worsen due to external causes, such as operator mistakes, bad quality parts, environmental effects, etc. In this case, the process is reset to the in-control state.
The process was monitored to observe any occurrence of either type of failure (F 1 and F 2 ) or any external causes (E). Therefore, these types of maintenance and quality planning failures must be considered. An optimization model is developed to generate integrated maintenance plans and quality control parameters for a real production system consisting of N components (i.e., a multi-component system). The following assumptions are considered while developing the optimization model: - Each automatic machine can process only one part at a time, which imposes a single characteristic to quality (CTQ). -Failure modes (F 1 and F 2 ) are independent. Failure reports from the company's records were used to obtain these probabilities. - The required resources to detect, maintain and restore the process are always available, so no waiting times are considered. Figure 1 illustrates the adopted methodology to develop the proposed integrated optimization model. The proposed model integrates maintenance plans and quality control parameters to help improve the quality of the production operation in a real multi-component production system. The following steps briefly summarize the adopted methodology: -

Proposed Integrated Optimization Model
Step 1: Defining the problem. The performance of the manufacturing system is significantly impacted by the breakdown of machines with multi-component. - Step 2: Select the quality control chart and maintenance policy to develop the model. - Step 3: Select the production system. - Step 4: Monitor the selected machine with multi-component using a CUSUM chart. - Step 5: Monitor the failure and repair rate. In this step, the data related to the mean time between failure and the mean time to repair all selected components were gathered and fitted for suitable distributions. - Step 6: Develop the integrated model based on CM, PM intervals, and CUSUM chart parameters. - Step 7: Solve the developed integrated maintenance policy and quality control mathematical model. - Step 8: Sensitivity analyses were conducted to illustrate the robustness of the developed model due to the stochastic nature of the problem under investigation. - Step 9: Discuss the obtained results.  In the developed optimization model, we have suggested using the CUSUM control chart for detecting minor shifts in the process mean (i.e., normal state) during operation. The CUSUM control chart has been proven effective in such cases (i.e., analyzing quality characteristics of a product or monitoring failures that cause the process mean to shift during operation) [36][37][38], which motivates us to use it in our integrated model. For more details about the CUSUM control chart calculations, the reader is referred to [43]. The notations used in the developed model with their descriptions are listed in the appendix (under the List of Abbreviations section). However, we will define all the notations used in the text after the proposed model (i.e., objective function and constraints). The following stochastic and non-linear optimization model is proposed to solve the considered problem: In the developed optimization model, we have suggested using the CUSUM control chart for detecting minor shifts in the process mean (i.e., normal state) during operation. The CUSUM control chart has been proven effective in such cases (i.e., analyzing quality characteristics of a product or monitoring failures that cause the process mean to shift during operation) [36][37][38], which motivates us to use it in our integrated model. For more details about the CUSUM control chart calculations, the reader is referred to [43]. The notations used in the developed model with their descriptions are listed in the appendix (under the List of Abbreviations section). However, we will define all the notations used in the text after the proposed model (i.e., objective function and constraints). The following stochastic and non-linear optimization model is proposed to solve the considered problem: where E[TC] (M * Q)CUSUM is the expected total cost per unit of time function, which is a stochastic and non-linear function that includes maintenance (corrective and preventive) and quality (using CUSUM control charts) related costs. E[TC] (M * Q)CUSUM can be calculated using Equations (7)- (25). While n, h, d, k, and T PM i are the decision variables representing sample size, sampling frequency, control limit coefficient, decision interval, and preventive maintenance interval, respectively. The model will be solved to identify the optimal values for n, h, d, k, and T PM i while minimizing the total expected cost per unit of time (E[TC] (M * Q)CUSUM ). The model introduces the lower and upper bounds for each decision variable using constraints (2)- (6). The model will also predict the preventive maintenance interval, as shown in the case study section (Section 4).
As mentioned above, the expected total cost function per hour (E[TC] (M * Q)CUSUM ) consists of corrective maintenance cost (C CM ), preventive maintenance cost (C PM ), and cost of process quality loss ([CQ] P−F ). Thus, it is calculated as follows: According to Pandey et al. [41], the corrective and preventive maintenance cost can be estimated as follows: are the expected number of corrective and preventive maintenance actions to be performed on component i, P is the production rate (Carton/h), C PL is the cost of production lost (USD/h), and C L is the cost of maintenance labor (USD/h). The expected number of corrective and preventive maintenance actions (n CM i and n PM i ) are estimated, based on the assumption that the component i failures follow a Weibull distribution, as follows [44,45]:  (11) where α i and γ i are the ith component's shape and scale parameters of the Weibull distribution, and t e is the evaluation time. The preventive maintenance interval (T PM i ) can also be estimated as follows [44]: Al-Shayea et al. [40] extended equations 10, 11 and 12 for multicomponents. Finally, the expected costs associated with process quality loss during the evaluation period ([CQ] P−F ) are calculated using the following equation: where E C p is the expected cost of process quality, and t c is the length of time between sequential in-control periods of a process cycle. On the one hand, E C p includes the operating cost during the in-control and out-of-control states, which consists of the following cost functions: • E C f : the expected cost of false alarms, which includes the cost of both investigating and analyzing the false alarms, and it is given as where C f is the cost associated with false alarms, t f is the time required to investigate false alarms, and S is the expected number of samples. At the same time, the process is in the in-control state, and ARL 0 is the average run length when the process is in the in-control state. Both S and ARL 0 can be obtained as follows [41]: where h is the sampling frequency (the decision variable from Equation (3)) and λ is the total process failure rate due to external (λ 1 ) and machine degradation (λ 2 ), where λ = λ 1 + λ 2 .
where k is the control limit coefficient of the CUSUM chart, and b = d + 1.661, where d is the critical threshold associated with the CUSUM chart.
• E[C s ]: the expected sampling cost per cycle, which can be calculated as follows [45]: where F is the fixed cost, V is the variable cost, n is the sample size (the decision variable from Equation (2)), ARL MC 1 represents the average run length during a period of out-ofcontrol operation due to machine failure, ARL E 1 is the average run length during an out-ofcontrol period caused by external reasons, τ represents the average elapsed time between the last sample before an assignable cause and its occurrence when the maintenance and quality policies are linked, τ is also estimated as h/2, and t s is the time required for sampling and chart plotting. Similar to ARL 0 obtained in Equation (16), ARL MC 1 and ARL E 1 are calculated using the following equation: where δ is the process magnitude, which has the value δ MC for ARL MC : the quality loss per unit of time in the control state, which is calculated using the Taguchi loss function (TLF), and is given as [45] E where A is the scrap or rework cost, R is the percentage of non-conforming units in an in-control state, and ∆ is a tolerance factor.
• E[L out ] MC : the quality loss per unit time when the process is in an out-of-control state due to machine degradation, which is also calculated using the Taguchi loss function (TLF) and is given as [45] E where f ( ) is the normal density function of a quality characteristic in an out-of-control state, and R δ MC probability of non-conforming items being produced as a result of machine failure.
• E[L out ] E : the quality loss per unit of time when the process is in an out-of-control state due to external factors (E), which is also calculated using the Taguchi loss function (TLF) and is given as [45] where R δ E probability of non-conforming items being produced due to external factors.
• E[C r ]: the expected cost of detecting and repairing the process due to external causes (E).
where C S is the cost of resetting the process, and t r is the expected time to reset the process.
• E[C CM ] F 2 : the expected cost of resetting and restoring the process (through CM) after a downtime of Type 2 (F 2 ), which is calculated as follows: where MT F 2 CM is the time required for corrective repairs, and FC CM p is the fixed cost for the conducted corrective repairs.
Thus, E C p is obtained as follows: On the other hand, t c includes time spent in control, time spent out of control, and time spent resetting processes or restoring machines. According to Pandey et al. study [45], the expected cycle length (t c ) can be calculated as follows: where t a is the time required to determine the occurrence of assignable causes. Based on the developed integrated optimization model introduced above in Equations (1)- (25), an optimal predictive maintenance plan will be developed to reduce defects and improve productivity. Figure 2 shows a flowchart for this process, which will result in developing a predictive maintenance policy based on the optimized maintenance interval for each machine (or component), i.e., T PM i . and improve productivity. Figure 2 shows a flowchart for this process, which will result in developing a predictive maintenance policy based on the optimized maintenance interval for each machine (or component), i.e., .

Case Study
An automatic filling system is part of a fully automatic water bottling production line that is installed and operates in the National Mineral Water Company (AMWC). As shown in Figure 3, an automatic filling system is a filling machine consisting of the main components: Fill box flux, Flowmeter, Valve battery, Seals and O-rings for filling heads. Those components have been determined based on each component's importance and relation to this study. In this study, the filling machine represents a real-life application that demonstrates the suitability of the extended integration model. Maintenance and quality are two major issues in the company. The number of failures increased due to insufficient maintenance plans for detecting and preventing the failure of the main components. In addition, those failures influence product quality. However, the maintenance of this automated machine is very complex and needs a model for reducing the number of failures and enhancing product quality. This study aims to develop an integrated maintenance policy and quality model to overcome these issues.

Case Study
An automatic filling system is part of a fully automatic water bottling production line that is installed and operates in the National Mineral Water Company (AMWC). As shown in Figure 3, an automatic filling system is a filling machine consisting of the main components: Fill box flux, Flowmeter, Valve battery, Seals and O-rings for filling heads. Those components have been determined based on each component's importance and relation to this study. In this study, the filling machine represents a real-life application that demonstrates the suitability of the extended integration model. Maintenance and quality are two major issues in the company. The number of failures increased due to insufficient maintenance plans for detecting and preventing the failure of the main components. In addition, those failures influence product quality. However, the maintenance of this automated machine is very complex and needs a model for reducing the number of failures and enhancing product quality. This study aims to develop an integrated maintenance policy and quality model to overcome these issues.

Results
Failure data for machine components are collected and analyzed using the ReliaSoft Weibull++6 software to fit the distribution for each component. The analysis results show that each component failure can be modeled by a Weibull distribution. In addition, the Weibull distribution parameters (shape and scale parameter) for each component failure were used to calculate the number of failures and the preventive maintenance interval. Table 1 shows the Weibull parameters of the filling machine components. The filling machine runs in three shifts per day, 8 h per shift, for six days per week. The machine fills 330 mL water bottles with a process mean of 327 mL. and a process standard deviation of 1.5. The size of the shift that is due to external reasons is δ E = 1, and the shift due to machine failure is δ m/c = 0.8, which happens randomly and causes the process means to shift from µ 0 to (µ 0 + δ). The process quality monitoring is being monitored by using a CUSUM chart analysis. At this stage, the sought-after critical to quality characteristic (CTQ) is the shortage in filling the bottle's water volume. According to the adapted quality control policy, samples of filled water bottles were taken from the production line every hour (h), inspected by the naked eye, confirmed, and measured using a liquid measuring cup. The CUSUM chart is implemented as shown in Figure 4. It is evident that there is a shift in the process mean. This shift in the process is due to failure in the Seals and O-rings of the filling head (component number one) and valve battery (component number three). To solve this issue in the quality acceptance and to eliminate the process shifts, the PM must be scheduled and periodically replace the O-rings, change the O-rings of the valve battery, or change the card and check the inner springs of the valve battery.
The initial values of the necessary parameters of the process quality control chart to develop the integrated model are presented in Table 2.
The developed model is composed of mathematical equations to find the optimal values of sample frequency, sample size, the width of the control chart limit, and the preventive maintenance interval while minimizing the total expected cost of the maintenance and quality control. A computer programming code of the integrated model has been developed using MATLAB2020a. An optimal set of decision variables that minimizes the expected total cost per unit of time for the system (E[TC] (M * Q)CUSUM ) are presented in Table 3.  The initial values of the necessary parameters of the process quality control chart to develop the integrated model are presented in Table 2. The developed model is composed of mathematical equations to find the optimal values of sample frequency, sample size, the width of the control chart limit, and the preventive maintenance interval while minimizing the total expected cost of the maintenance and quality control. A computer programming code of the integrated model has been developed using MATLAB2020a. An optimal set of decision variables that minimizes the expected total cost per unit of time for the system ( [ ] ( * ) ) are presented in Table 3.

Design of Experiments Based on One-Factor-at-a-Time
The one-factor-at-a-time approach was used to study the influence of the addressed decision variables on the expected total cost. Figure 5a illustrates the impact of sample frequency (h) on the anticipated total cost. Other parameters, decision interval, sample size, and preventive maintenance interval, were kept constant. It can be observed that the sampling frequency increases from 1 to 5, and the expected total cost slightly decreases and increases with the increase in h. Pandey et al. [45] found that the total cost decreases by reducing the sampling frequency from 2 samples per second to 3.5 samples per second and increases with increasing sample frequency. Figure 5b depicts the anticipated total cost decrease with the increase in decision interval (d). This can be explained by the fact that the increase in interval (d) will decrease the ARL, which will reduce the probabilities of repair, rejection, and out-of-control costs. Figure 5c shows the expected cost variations with the change in the control limit coefficient (k). It shows that the cost remains constant and started to rise with a value of 0.6 and higher for the control limit coefficient (k). Since the standard deviation value is fixed, increasing the control limit coefficient (k) value will increase the region between the control limits, which, in turn, increases the possibility of a poor product being accepted. Thus, rejection, repair, and out-of-control costs also decrease. Regarding the effect of preventive maintenance interval on the total cost, Figure 5d shows how the total cost decreases with the increase in PM interval until it reaches a certain point where the cost goes up again. That increase in cost is because longer PM intervals come with a higher probability of failure, which results in more costs (CM and replacement costs). The point with the minimum maintenance cost is the optimal PM interval. and increases with the increase in h. Pandey et al. [45] found that the total cost decreases by reducing the sampling frequency from 2 samples per second to 3.5 samples per second and increases with increasing sample frequency. Figure 5b depicts the anticipated total cost decrease with the increase in decision interval (d). This can be explained by the fact that the increase in interval (d) will decrease the ARL, which will reduce the probabilities of repair, rejection, and out-of-control costs. Figure 5c shows the expected cost variations with the change in the control limit coefficient (k). It shows that the cost remains constant and started to rise with a value of 0.6 and higher for the control limit coefficient (k). Since the standard deviation value is fixed, increasing the control limit coefficient (k) value will increase the region between the control limits, which, in turn, increases the possibility of a poor product being accepted. Thus, rejection, repair, and out-of-control costs also decrease. Regarding the effect of preventive maintenance interval on the total cost, Figure  5d shows how the total cost decreases with the increase in PM interval until it reaches a certain point where the cost goes up again. That increase in cost is because longer PM intervals come with a higher probability of failure, which results in more costs (CM and replacement costs). The point with the minimum maintenance cost is the optimal PM interval. Therefore, the obtained optimal ranges of the decision variables of the developed maintenance and quality control model are summarized in Table 4. In addition, the obtained optimal range of CUSUM chart parameters will be used to develop the CUSUM chart for monitoring product quality. Therefore, the obtained optimal ranges of the decision variables of the developed maintenance and quality control model are summarized in Table 4. In addition, the obtained optimal range of CUSUM chart parameters will be used to develop the CUSUM chart for monitoring product quality.

Sensitivity Analysis
The sensitivity analysis issue is crucial and has been used for estimating the effects of the cost variables and process over the selected range on the integrated model of preventive policy and statistical quality control chart. In addition, this analysis further illustrates the relationship between CUSUM parameters and preventive maintenance, as well as the performance of the integrated model [39]. A sensitivity analysis was conducted on the importance of cost and process variables. Table 5 presents the selected variables and their range. Level 1 is the basic level employed to solve the integrated model for the secreted case study. Levels 2 and 3 were chosen based on the company allowance policies, representing the increase in the primary level by +10% (level 2) and +20% (level 3). After identifying the range, the developed model was solved according to the presented values of each variable to calculate the total cost per unit of time. The obtained results illustrate that the expected total cost per unit of time values are close to the optimal values found earlier in a similar setting. Consequently, the results show that the optimum design of the integrated maintenance policy and quality control chart model based on the CUSUM chart is generally robust to errors in variable estimation. The analysis shows that the proposed integrated model is robust to errors in variable estimations up to (+20%).

Preventive Actions to Reduce the Defects
More frequent defects are analyzed and evaluated periodically, and proper preventive actions are taken to keep these defects at a minimum. A policy is set to monitor the components in the filling stage and implement a preventive maintenance interval (T PM i ) to predict future failures. The policy calls for each component to be checked against all possible failures in a very cost and time-effective way so that the filling process runs smoothly and the system is more available for production. Therefore, machine breakdown is much less, and the cost is reduced.
Based on this study, the developed predictive maintenance introduces preventive maintenance intervals (T PM i ) with a timely schedule for preventive maintenance jobs for the N-components in the filling machine to spot any possible failure or defect. These preventive maintenance intervals should take place every 800 h. Figure 6 shows the improvement in total cost per hour. The total cost per hour is only USD 81.5479, which is much less than USD 163.12 by approximately 50% of lost production per hour due to product defects and the waste of production time during the repair of these defects. Therefore, the results indicate that the performance of the developed integrated model realizes the full potential of the maintenance model. ventive maintenance intervals should take place every 800 h. Figure 6 shows the improvement in total cost per hour. The total cost per hour is only USD 81.5479, which is much less than USD 163.12 by approximately 50% of lost production per hour due to product defects and the waste of production time during the repair of these defects. Therefore, the results indicate that the performance of the developed integrated model realizes the full potential of the maintenance model. The results obtained from the CUSUM chart were used to optimize the preventive maintenance interval ( ). Table 6 lists the developed solutions based on the optimized values for the main causes that should be implemented to reduce defects.  The results obtained from the CUSUM chart were used to optimize the preventive maintenance interval (T PM i ). Table 6 lists the developed solutions based on the optimized T PM i values for the main causes that should be implemented to reduce defects.

Conclusions
This study proposed an extended integrated economic model for maintenance policies (maintenance interval) and CUSUM chart parameters for the multi-component system. This integration model is used to predict and optimize the preventive maintenance interval, sampling interval, sample frequency, sample size, and control limit chart to minimize the total cost per unit of time. The case study illustrates the problem, and sensitivity analysis demonstrates the dependence between quality control parameters and maintenance. The model has the following potential utilizations: -Managers can control the quality of produced units and monitor the production line and its different states using the proposed solution steps. - The proposed methodology helps to identify the optimal preventive maintenance interval needed to improve production output and minimize downtime with enhanced product quality. -A maintenance plan for a multi-component system was developed based on the optimal value of preventive maintenance interval. The result showed that the total cost was reduced by approximately 50% compared with the current system.
This study shows that maintenance and process quality control based on the CUSUM chart is a more effective and powerful integrated model for improving product quality and process performance. This study could be expanded to include the integrated quality control and reliability-centered maintenance policy using the mixed EWMA and CUSUM control charts to detect a shift in the process mean.

Acknowledgments:
The authors would like to acknowledge the support provided by Taiz University.

Conflicts of Interest:
The authors declare no conflict of interest. Failure rate due to an external factor λ 2

List of Abbreviations
Failure rate due to machine degradation αi The shape parameter of Weibull distribution for ith component γi Average run length in-control state (average number of samples taken before a false alarm occurs) ARL 1 Average run length in an out-of-control state ARL E