Integrated Production, EWMA Scheme, and Maintenance Policy for Imperfect Manufacturing Systems of Bolt-On Vibroseis Equipment Considering Quality and Inventory Constraints

Abstract

In recent years, the synergistic effect among production, maintenance, and quality control within manufacturing systems has garnered increasing attention in academic and industrial circles. In high-quality production settings, the real-time identification of minute process deviations holds significant importance for ensuring product quality. Traditional approaches, such as routine quality inspections or Shewhart control charts, exhibit limitations in sensitivity and response speed, rendering them inadequate for meeting the stringent requirements of high-precision quality control. To address this issue, this paper presents an integrated framework that seamlessly integrates stochastic process modeling, dynamic optimization, and quality monitoring. In the realm of quality monitoring, an exponentially weighted moving average (EWMA) control chart is employed to monitor the production process. The statistic derived from this chart forms a Markov process, enabling it to more acutely detect minor shifts in the process mean. Regarding maintenance strategies, a state-dependent preventive maintenance (PM) and corrective maintenance (CM) mechanism is introduced. Specifically, preventive maintenance is initiated when the system is in a statistically controlled state and the inventory level falls below a predefined threshold. Conversely, corrective maintenance is triggered when the EWMA control chart generates an out-of-control (OOC) signal. To facilitate continuous production during maintenance activities, an inventory buffer mechanism is incorporated into the model. Building upon this foundation, a joint optimization model is formulated, with system states, including equipment degradation state, inventory level, and quality state, serving as decision variables and the minimization of the expected total cost (ETC) per unit time as the objective. This problem is formalized as a constrained dynamic optimization problem and is solved using the genetic algorithm (GA). Finally, through a case study of the production process of vibroseis equipment, the superiority of the proposed model in terms of cost savings and system performance enhancement is empirically verified.

1. Introduction and Literature Review

Rapidly changing market demands, varying production plans in the short run, and increasing complexity of manufacturing systems have posed a challenge for improving the quality and efficiency of production processes. Production (production plan and inventory control), maintenance, and quality are three main operational areas in manufacturing. Mathematical modeling assumes a pivotal role in tackling these intricate problems, facilitating the integration of production and maintenance, implementing statistical process control, and achieving optimization (Zhang et al. [1]; Alhidairah et al. [2]; Jafari et al. [3]). In the past, many studies dealt with the optimal design of the three operation areas separately. However, this may severely affect the global optimality (Frahani and Tohidi [4]; Hafidi et al. [5]). In light of this, this research endeavors to develop an optimal integrated model for production, maintenance, and quality control. It is anticipated that, through this model, process variation can be reduced, process quality can be enhanced, the reliability of the production system can be strengthened, and production costs can be effectively saved. During this process, mathematical modeling will function as a core instrument, enabling us to precisely depict the relationships among various factors and accomplish the integrated optimization of production and maintenance. Developing an optimal integration of production, maintenance, and quality control models is the object of this study, which can reduce the process variation, improve process quality, increase the reliability of the production system, and save production costs.

In the joint design literature, some studies were developed based on the two-aspect integration of three functions (production, maintenance and quality) in manufacturing systems, for example, production and maintenance policies (Slameh and Ghattas [6], Goel, Grievink, and Weijnen [7], El-Ferik [8], Nahas [9]) and quality and maintenance (Ben-Daya and Rahim [10], Charongrattanasakul and Pongpullponsak [11], Xiang et al. [12], Duan et al. [13], Huang et al. [14]). However, the three functions interact mutually, and practitioners need to find the optimal management method to balance the three tasks by considering them simultaneously (Salmasnia et al. [15], Salmasnia et al. [16], Hadian et al. [17], Frahani and Tohidi [4], Zhang et al. [18]). Mathematical modeling is capable of integrating these three functions into a unified framework for analysis by constructing functional relationships among multiple variables. This approach enables the realization of more efficient integrated management.

During the last twenty years, many integrated models have been proposed to examine assignable causes based on Shewhart type control charts. These models demonstrate distinct characteristics in the realm of mathematical modeling. Ben-Daya and Makhdoum [19] presented a joint optimization economic production quantity (EPQ) and quality model under various PM policies, and a Shewhart $\overline{\mathrm{X}}$ chart was used to monitor the main characteristic. Rahim and Ben-Daya [20] proposed an economic production quantity and quality control design under the surveillance of an $\overline{\mathrm{X}}$ control chart. In this study, mathematical models were also utilized to ascertain the optimal production and quality control parameters. Pandey et al. [21] developed an integrated cost model for jointly optimizing the PM interval and some design parameters of the $\overline{\mathrm{X}}$ control chart, where two failure models were introduced for machine failures. By combining probability distributions and cost functions, mathematical modeling of maintenance and quality control was achieved. Pan et al. [22] introduced the EPQ model into the integrated quality control and maintenance policy, further enriching the mathematical model system of production and maintenance integration. Noting the possible presence of multiple assignable causes, Salmasnia et al. [15] proposed a joint design of production, maintenance, and control charts. In this design, a variable sampling interval scheme was employed, and the dynamic changes of the sampling interval were considered in the mathematical modeling process. Recently, Bahria et al. [23] modeled a joint maintenance and quality control strategy for which quality control was again performed by an $\overline{\mathrm{X}}$ control chart to analyze process data. Salmasnia et al. [24] provided a joint determination of production cycle length, maintenance policy, and several parameters of the control chart, where the time value of money and the stochastic shift size were considered. To keep the reliability of the system, they also adopted the strategy of non-uniform sampling. Hadian et al. [17] developed a joint model of maintenance planning, inventory, and quality control policy. Recently, Shi et al. [25] proposed an optimal model for optimizing production, maintenance, and quality control of imperfect manufacturing systems considering timely replenishment. Although they suggested that, when designing the integrated model, a proper control chart should be considered, the $\overline{\mathrm{X}}$ charting scheme is still applied in their numerical study. It is clear that the $\overline{\mathrm{X}}$ chart still stands as a popular statistical process monitoring (SPM) tool used in different integrated production, maintenance, and quality control policies.

When implementing a control chart for monitoring the non-conforming fraction, it is worth pointing out that choosing an efficient online monitoring scheme is crucial to undertake the corresponding maintenance action. The Shewhart type control charts are easy to implement and efficient in detecting large variations in a process, but they are relatively inefficient in monitoring small and moderate changes (Serel and Moskowitz [26], Ardakan, [27], Hesamian et al. [28], Sukparungsee et al. [29]). In the literature, Serel and Moskowitz [26] presented an EWMA cost minimization model, and it can be regarded as a type of economic–statistical design. Charongrattanasakul and Pongpullponsak [8] investigated the integrated system approach to model quality control and maintenance policy using the EWMA scheme. Six conditions, namely, in-control alert signal, out-of-control signal, in-control no signal, out-of-control no signal, warning limit alert signal, and warning limit no signal, are given in the integrated model. After that, an EWMA chart was adopted to combine PM and CM actions on the process. Ardakan et al. [27] provided a hybrid model to connect the control chart with the planned and reactive maintenance. Yin et al. [30] considered a production process with multiple quality characteristics, and a cost model based on a multivariate EWMA chart is established. Noman et al. [31] presented an integrated optimization method that addresses both maintenance policies and quality control practices, which investigated the optimal integrated design for maintenance and quality control, yet the production and inventory control policies are not considered. To the best of our knowledge, there is no literature on the joint design production, maintenance, and quality policy based on the EWMA scheme. Therefore, this paper considers the EWMA monitoring scheme in the optimal integrated design to increase the scheme’s sensitivity in small to moderate shifts.

In actual situations, production processes are subject to assignable causes, which mainly lead to producing defective products. A proper maintenance policy is necessary for restoring the system to a good situation or an as-good-as-new state to maintain the production process at a high quality level. However, maintenance without taking the quality control information or production requirements into consideration may increase the process variability, risk of machine failure, and production costs. The choice of maintenance is constrained by the maintenance plan, quality monitoring information, and inventory size. Some researchers investigated PM, CM, age-based maintenance policies, etc. in the integrated model based on the signal given by the control chart and the equipment state. For instance, Nourelfath et al. [32] performed an imperfect PM policy in the integrated production, maintenance, and quality model for an imperfect production process. They employed a linear relationship between the PM cost and the improvement of the conditions. Bouslah et al. [33] considered imperfect maintenance as a part of the setup activity at the beginning of each lot production and major maintenance when the proportion of defectives in a rejected lot reaches or exceeds a given threshold. They offered an integrated production, preventive maintenance, and quality control strategy. In the mathematical modeling, factors such as the threshold of the proportion of non-conforming products and the number of batches were taken into account. Salmasnia et al. [16] provided an integrated EPQ method, maintenance, and quality control policy based on a $\mathrm{V}\mathrm{P}-{\mathrm{T}}^2$ Hotelling chart, where PM and CM are conducted under different production scenorios. Rivera-Gómez et al. [34] presented a joint optimization design of production and maintenance policy, where an adaptive sampling plan and the minimal repair and PM policies are considered. Besides the above discussion, many researchers have made contributions to providing maintenance policy for improving the quality and reliability of machines (see Bahria et al. [23], Baria et al. [35], He et al. [36], Hadian et al. [17]), which can be referred to in the integrated model. Since the production capacity and machine availability are significantly influenced by the maintenance, a proper maintenance policy should be determined and adjusted according to the actual condition of the production process in the integrated production, maintenance, and quality control model.

To satisfy the continuous demand of customers, the production and inventory control policies are also of concern to managers. The inventory size is usually limited in many factories, therefore, the optimization of production scheduling and inventory control should be considered in the integrated production, maintenance and quality control model (Pan et al. [22], Fakher et al. [37], Salmasnia et al. [15], Salmasnia et al. [16], Hadian et al. [17]). In mathematical modeling, inventory levels can be described by production rate, demand rate, and time, while taking into account the constraints of upper and lower inventory limits. However, one can find that few researchers considered the viewpoint of inventory constraints. To this end, in this paper, we consider the inventory to be constrained.

In addition, most of the inventory policies considered in the integrated model are based on the assumption that the products monitored by the control chart are perfect before the control chart triggers a signal. The rejection of defective products is not clarified in the previous integrated optimal studies. The control chart may generally not reflect the OOC situation in actual production processes due to its self-limit. Bahria et al. [35] once pointed out that the number of defective items is proportional to the quantity produced before the control chart signals. Mokhtari and Asadkhani [38] proposed an economic production quantity model, and two rejection of defective items policies are compared. Their comparison results showed that the decision of removing the faulty items once per subproduction cycle would cost less than the decision of disposing of the defective items once per cycle. For some production processes, for example, the high-voltage rotor bolts’ hot upsetting process, the defective items cannot be reworked or will be reworked at a relatively larger additional cost, resulting in that the defective items are not suitable to put into inventory for sale at a later time from the economic viewpoint. Therefore, in this paper, we considered the discarding defective items strategy instead of putting defective items into inventory.

Table 1. Summary and Comparison of related contributions.

Author(s)	Type of Control Chart		Failure Mechanism	Maintenance Policy		Inventory Control
	$\overline{\mathit{X}}$	EWMA		Preventive Maintenance	Corrective Maintenance	Without Shortage	With Shortage	Reject Defective Items
Pandey et al. [21]	√		√	√	√
Charongrattanasakul, Pongpullponsak [11]		√	√	√	√
Pan et al. [22]	√		√	√	√
Xiang [9]	√		√	√	√
Yin et al. [30]		√	√	√	√
Ardakan et al. [27]		√	√	√	√
Nourelfath et al. [32]			√	√
Lin et al. [39]			√	√		√
Salmasnia et al. [15]	√		√	√	√
Fakher et al. [37]			√	√
Bahria et al. [23]	√			√	√	√	√	√
Duffuaa et al. [40]	√		√	√	√	√
Salmasnia et al. [24]	√		√	√		√
Rivera-Gómez et al. [34]				√		√		√
Hadian et al. [17]	√		√	√	√	√	√
Wan et al. [41]	√		√	√	√	√
Zhang et al. [18]			√	√	√	√		√
Lv et al. [42]			√	√	√		√
The proposed model		√	√	√	√	√	√	√

summarizes the work related to this paper. To the best of our knowledge, the existing integrated model based on the EWMA chart has not considered the inventory constraints for carrying out PM in various integrated optimal designs. To give scientific and efficient guidance for production management, we present an integrated EWMA chart to control and improve the quality of the process and reduce the number of defects. According to the information from the EWMA control chart and inventory level, a proper maintenance action is carried out to restore the machine to an as-good-as-new state. Considering conditions of shortage and without shortage and the rejection of defective items, the inventory level is evaluated. Finally, the joint design for production, EWMA control, and maintenance policy is established to bridge the research gap.

With respect to the available literature, the proposed integrated design of EWMA monitoring scheme, maintenance, and inventory policies for optimizing the production process has the following benefits:

(1)

To suit the need of monitoring moderate and small shifts, the EWMA charting scheme is adopted to replace the Shewhart $\overline{\mathrm{X}}$ charting scheme, which can better control and improve the quality of the production process.

(2)

To suit the actual conditions of factories, a preventive maintenance action is carried out to improve the machine’s state based on its usage time, comprehensively considering quality condition and inventory levels.

(3)

To accurately model the inventory, both scenarios with and without shortages are considered.

A new integrated optimal design of production, maintenance, and quality control is established for guiding the production management, where the joint design is connected by the expected total cost (ETC) per unit time in one production cycle.

The rest of this paper is organized as follows. Section 2 describes the notations, problems, and assumptions of this study. In Section 3, we briefly introduce the EWMA chart for quality control. Afterwards, in Section 4, we define three scenarios in a production cycle. Section 5 gives the cost analysis of the proposed joint design of the production, maintenance, and quality control model. Section 6 is devoted to establishing the integrated model for the production (production plan and inventory control), maintenance, and quality policy. Section 7 gives a real case study to demonstrate the implementation of the proposed integrated model. Finally, some conclusions and discussions are given in Section 8.

2. Notations, Problem Description, and Assumptions

2.1. Notations

Decision Variables	Description
$k$	Number of samplings in one production cycle
$h$	Sampling interval
$n$	Sample size
$L$	Coefficient of control limits of the EWMA chart
$r$	Smoothing parameter
Parameters	Description
$m$	Lower critical inventory level
$M$	Upper critical inventory level
$e_1$	Lower production rate
$e_2$	Upper production rate
$\lambda$	The occurrence rate of quality shifts
$u_0,\sigma$	The mean and standard deviation of the key quality characteristic
$u_1$	The mean of the key characteristic when the process shifts
$\delta$	The magnitude of shifts in process mean
$UCL$	The upper control limit
$LCL$	The lower control limit
$ARL$	Average run length
$\alpha$	The false alarm rate of the EWMA control chart
$p\left(t|\delta \right)$	$\mathrm{Probability} \mathrm{of} \mathrm{the} \mathrm{control} \mathrm{chart} \mathrm{signals} \mathrm{at} \mathrm{the} t-\mathrm{th} \mathrm{sample} \mathrm{when} \mathrm{the} \mathrm{process} \mathrm{is} \mathrm{OOC} \mathrm{with} \mathrm{shift} \delta$
τ	The average time between the last sampling taken in the IC process to the occurrence of the assignable cause
$P$	Production rate
$D$	Demand rate
$I$	Inventory level
$I_{max}$	Maximum inventory level
$C_I$	Cost of the quality loss per time unit when the process is IC
$C_O$	Cost of the quality loss per time unit when the process is OOC
$C_F$	Fixed inspection cost of sampling
$C_s$	Variable inspection cost of sampling
$C_1$	Cost of preventive maintenance (PM)
$C_2$	Cost of corrective maintenance (CM)
$C_3$	Maintenance cost when PM is replaced by CM
$C_f$	Cost of inspecting a false alarm
$S$	The setup cost per production cycle
$B$	The inventory holding cost per unit time
$B_1$	The unit shortage cost per time unit
$\beta$	The proportion of defective items produced in OOC processes
$t_1$	The duration of PM
$t_2$	The duration of CM
$t_3$	The maintenance time when PM is replaced by CM
$E\left(O\right)$	The expected cost of setup
$E\left(I\right)$	The expected inventory holding cost
$E\left(Q\right)$	The expected quality loss cost
$E\left(S\right)$	The expected sampling inspection cost
$E\left(M\right)$	The expected maintenance cost
$E\left(C_{Fa}\right)$	The expected false alarm cost
$EC$	The average total cost per production cycle
$ET$	Expected operating time per production cycle

2.2. Problem Description

In this paper, a single-unit imperfect manufacturing system is considered, where three main tasks, production and inventory scheduling, quality control, and maintenance, are involved. The key problems concerned in each part are described as follows.

(a). 

Production and inventory

We assume that one machine produces a single product in batches at rate $P$ to satisfy a constant market demand rate $D$. When $P\le D$, the products flow from the factory into the market directly. When $P>D$, demands are fully satisfied and the inventory is built up at the rate $P-D$. Similar to Salmasnia et al. [15], the production rate $P$ is set larger than the demand rate $D$ in this paper. In production–inventory control issues, a two-critical-number policy $\left(m,M\right)$ with $0\le m\le M$ for the production operational mechanism has been studied by many researchers (Gaver [43], Sobel [44], Lin [45]).

We defined $e_1$ as the production rate when the inventory level reaches its maximum $M$ and $e_2$ as the rate when the level is below the minimum inventory threshold m ($0\le e_1<e_2$). Specifically, when the inventory level rises from $m$ to $M$, the production rate switches from $e_2$ to $e_1$, and the production rate switches from $e_1$ to $e_2$ when the inventory level falls from $M$ to $m$. In this study, we consider that the production run process starts at a zero inventory level with a production rate $e_2=P$. When the inventory is at a full level, the production ends with a production rate $e_1=$ 0. In addition, to ensure the quality of products, a 100% inspection of products is performed before they flow into the market or the inventory.

(b). 

Quality control

Since the production system is imperfect, some assignable causes may occur due to the deterioration of the machine and some external environmental factors, resulting in defective (or non-conforming) items being produced. The manufacturer will give an additional cost to rework or scrap the defective products. To reduce the rate of defective products, quickly detecting the occurrence of assignable causes is needed. To this end, the EWMA control chart is adopted by monitoring the quality characteristic of products to reflect whether an assignable cause occurs.

The process operates in an in-control (IC) state when production starts, and a sampling policy with sampling interval $h$ and sample size $n$ is adopted to plot the EWMA control chart. The quality characteristic of products to be monitored is assumed to follow a normal distribution with mean $u_0$ and standard deviation $\sigma$. When an assignable cause occurs, the process mean may shift from $u_0$ to $u_1=u_0+\delta \sigma$. In this paper, the standard deviation is assumed to be known and remains always the same (Hadian et al. [17]). When the control chart signals and practitioners check that an assignable cause occurs, the production ends. Otherwise, the production will end at the time of the kth sampling when no signal is triggered.

(c). 

Maintenance

The process ends when the control chart detects an assignable cause, otherwise, the process continues with no signal alert of a process shift until the kth sampling inspection. Therefore, based on the comprehensive consideration of the production–inventory and quality control, PM will be carried out both considering the process operates after the $k$th sampling inspection and inventory level reaches the critical value $I_{max}$, where $I_{max}=kh\ast \left(P-D\right)$. It is noted that PM is carried out here to improve the machine’s condition under the constraints of inventory level. Moreover, once the control chart gives an alarm signal before the $k$th sampling inspection when an assignable cause occurs, CM action will be taken. Both PM and CM actions considered here are perfect.

A production cycle contains the production uptime for producing items and production downtime for maintenance. Two possible conditions, with shortage and without shortage, are considered in this paper as follows.

■

Without shortage: (1) Production uptime: The process starts at a zero inventory level, then, the inventory is built up to the maximum inventory level $I_{max}$ at time $kh$ or the inventory reaches the level $I^{\prime } \left(I^{\prime }{<I}_{max}\right)$ when an alarm is triggered by the control chart. (2) Production downtime: The process develops from the inventory level $I_{max}(or I^{\prime })$ to zero.

■

With shortage: (1) Production uptime: The production uptime in the case of shortage is the same as the production time in the case of without shortage. (2) Production downtime: The process develops from the inventory level $I_{max} (or I^{\prime })$ to backorder.

In the above two conditions, we assume that the maintenance duration in the condition of without shortage (with shortage) is less (longer) than the inventory depletion period, then, no shortage (shortage) will happen in one production cycle.

Moreover, the classical evolution method of the inventory level is based on the assumption that products are all perfect (or all conforming) (Pan et al. [19]). However, the control chart may be unable to reflect the OOC process due to its self-limit. Considering that the rejection of defective items can reduce the occupancy of storage space and management cost, therefore, defective products will be rejected in this paper, and the corresponding description can be found in Section 5.

Finally, the integration design of production, EWMA control, and maintenance policy is investigated in this study, and the objective is to minimize the expected total cost per unit time (ETC) under the constraints of the inventory size, the sample size, the sampling interval, the sampling number, the false alarm rate, and the run length of the OOC process and provide the optimal decision variables: sampling inspection number $k$, sampling interval $h$, sample size $n$, coefficient $L$ of control limits of the EWMA control chart, and smoothing parameter $r$.

2.3. Assumptions

The following assumptions and definitions are used to develop the proposed model:

(1)

The time to the occurrence of an assignable cause follows the exponential distribution with parameter $\lambda$ (Salmasnia et al. [16], Seif [46], Duffuaa et al. [40], Huang et al. [14]).

(2)

The PM and CM restore the production system to an as-good-as-new condition (Salmasnia et al. [15], Salmasnia et al. [16], Hadian et al. [17]).

(3)

The time of finding a false alarm, the time of sampling, and the time to validate the assignable cause can be ignored, since they are relatively small in comparison with the total operation time in one production cycle (Huang et al. [14]).

The proportion of defective items of the OOC process is assumed to be a constant. (Bahria et al. [35]).

Subsequently, the study will elaborate on the design of the EWMA control chart and its implementation within the production process. Building on this foundation, we will analyze three distinct scenarios: firstly, when an assignable cause is detected by the control chart, triggering corrective maintenance; secondly, when a drift remains undetected by the chart but is discovered during periodic preventive maintenance; and thirdly, when only routine preventive maintenance is performed without any detected drift. For each scenario, the calculation methods for various production costs—including inspection, quality control, maintenance, and downtime costs—will be derived in detail. Finally, the corresponding expected total cost per unit time (ETC) expressions will be formulated to form the basis of the integrated optimization model.

3. The EWMA Control Charting Scheme

Note that most existing studies focus on using the $\overline{X}$ control chart to reduce the complexity of designing the integrated model for production, maintenance, and quality control policy, regardless of the detection efficiency. Therefore, the EWMA charting scheme is considered in this paper.

First, the EWMA control chart is briefly introduced in this section. Assume that $\left(X_1,X_2,\dots ,X_n\right)$ is a group of random samples collected from a normally distributed process with mean $u_0$ and standard deviation $\sigma$. $\overline{X_t}$ is the sample mean, and it can be denoted as

$$\overline{X_t}={\displaystyle \frac{1}{n}}{\sum }_{j=1}^n{x}_j,j=1,\dots n, t=1,2,\dots .$$

It is clear that $\overline{X_t}~N\left(u_0,{\displaystyle \frac{{\sigma }^2}{n}}\right)$. Then, the EWMA statistic can be established as follows:

$$Z_t=r\overline{X_t}+\left(1-r\right)Z_{t-1},t=1,2,\dots ,$$

where $Z_0=u_0$, and the asymptotic upper and lower control limits of the EWMA control chart are

$$UCL=u_0+L{\displaystyle \frac{\sigma }{\sqrt{n}}}\sqrt{{\displaystyle \frac{r}{2-r}}}, \mathrm{and} LCL=u_0-L{\displaystyle \frac{\sigma }{\sqrt{n}}}\sqrt{{\displaystyle \frac{r}{2-r}}}$$

The design of an EWMA control chart consists of sample size ($n$), coefficient of control limits ($L$), smoothing parameter $\left(r\right)$. Further, the probability can be calculated by the Markov chain method. When the EWMA control chart signals at the $i$-th sample, we obtained

$$p\left(rl=i|\delta \right)=p_{initial}\left(R_{i-1}-R_i\right)\mathbf{1},$$

(1)

where $rl$ is the run length of the control chart under the shift $\delta$, $p_{initial}$ is a vector of initial probabilities, $R$ is a $\left(t-1,t-1\right)$ matrix containing transient states. 1 is the $\left(t-1\right)\times 1$ vector of unities. Detailed calculation of Equation $\left(1\right)$ can be found in the Supplementary Materials. Moreover, the expected average run length ($ARL$) can be expressed by

$$ARL=E\left(rl|\delta \right)={\sum }_{rl=1}^{\infty }rl\times p\left(rl|\delta \right).$$

Thus, the $ARL$ for the IC process (${ARL}_0$) can be calculated by setting $\delta =0$, and $ARL$ for the OOC process (${ARL}_1$) can also be computed when shift $\delta$ is not equal to zero.

4. Different Scenarios in One Production Run Process

In manufacturing processes, the system is affected by some assignable causes, and usually, the time to the occurrence of an assignable cause is assumed to follow a continuous distribution, where an exponential distribution is commonly used in modeling the occurrence of quality shifts (Salmasnia et al. [16], Seif [46], Duffuaa et al. [40], Huang et al. [14]. Note that, in Bahria et al. [35], the failure mechanism has not been considered, resulting in that the corresponding effect of the frequency of assignable causes on the production cost is not clear. Therefore, the failure mechanism is considered in this paper, and possible scenarios due to an assignable cause that happen in one production cycle are described close to real production processes. Especially, we adopted three scenarios which have been considered in Pan et al. [22], Salmasnia et al. [15], Salmasnia et al. [16], Huang et al. [14], and Hadian et al. [17].

• Scenario 1:

As shown in Figure 1, the production process starts in the IC state, and the quality shift does not occur from the beginning of the production cycle to the time when the inventory is built up to the maximum level $I_{max}$, where the number of sampling inspections $k$ is taken in this period. In this situation, PM action is carried out to guarantee the reliability of the manufacturing equipment. The occurrence probability of scenario 1 ($S_1$) is expressed as follows:

$$p\left(S_1\right)=1-F\left(kh\right)=e^{-\lambda kh}.$$

(2)

The expected IC and OOC time in $S_1$ can be obtained as

$$E\left(t_{IC}|S_1\right)=kh, E\left(t_{OOC}|S_1\right)=0.$$

(3)

and the expected total time $E\left(t|S_1\right)$ is

$$E\left(t|S_1\right)=E\left(t_{IC}|S_1\right)+E\left(t_{OOC}|S_1\right)=kh$$

(4)

• Scenario 2:

Figure 2 shows that the production process starts in the IC state and shifts to the OOC state at time $t$ between interval $\left[\left(i-1\right)h, ih\right]$ when an assignable cause occurs, and the process parameter changes from ${\mu }_0$ to ${\mu }_1$ with ${\mu }_1={\mu }_0+\delta \sigma .$ The chart can detect the shift before the end of the $k$^th sampling, that is, the control chart first gives alarm at the $\left(i+j-1\right)$^th sampling, $j=1,2,\dots ,k-i+1$. When the process shift is detected, CM is adopted to restore the machine to an as-good-as-new state. The occurrence probability of scenario 2 ($S_2$) is given as follows:

$$\begin{gathered} p\left(S_2\right)={\sum }_{i=1}^k{\sum }_{j=1}^{k-i+1}{\int }_{\left(i-1\right)h}^{ih}{\lambda e}^{-\lambda t}dtp\left(j|\delta \right) \\ ={\sum }_{i=1}^k{\sum }_{j=1}^{k-i+1}\left(e^{-\lambda \left(i-1\right)h}-e^{-\lambda ih}\right)p\left(j|\delta \right). \end{gathered}$$

(5)

The expected IC and OOC time in $S_2$ are

$$E\left(t_{IC}|S_2\right)={\displaystyle \frac{{\sum }_{i=1}^k{\sum }_{j=1}^{k-i+1}\left(e^{-\lambda \left(i-1\right)h}-e^{-\lambda ih}\right)p\left(j|\delta \right)\left(\left(i-1\right)h+\tau \right)}{P\left(S_2\right)}}$$

(6)

and

$$E\left(t_{OOC}|S_2\right)={\displaystyle \frac{{\sum }_{i=1}^k{\sum }_{j=1}^{k-i+1}\left(e^{-\lambda \left(i-1\right)h}-e^{-\lambda ih}\right)p\left(j|\delta \right)\left(jh-\tau \right)}{P\left(S_2\right)}},$$

(7)

and the expected total time $E\left(t|S_2\right)$ is

$$E\left(t|S_2\right)=E\left(t_{IC}|S_2\right)+E\left(t_{OOC}|S_2\right),$$

(8)

where $\tau$ is the average time from the last sampling to the occurrence of the shift, and according to the assumption that the time to the occurrence of an assignable cause follows the exponential distribution with parameter $\lambda$, therefore, $\tau$ is derived as follows:

$$\tau ={\displaystyle \frac{{\int }_h^{2h}\left(t-h\right)\lambda e^{-\lambda h}dt}{{\int }_h^{2h}\lambda e^{-t}dt}}={\displaystyle \frac{1-\left(1+\lambda h\right)e^{-\lambda h}}{\lambda \left(1-e^{-\lambda h}\right)}}.$$

(9)

• Scenario 3:

Figure 3 shows that the production process starts in an IC state and shifts to an OOC state at time $t$ between interval $\left[\left(i-1\right)h, ih\right]$ when an assignable cause occurs, and the process parameter changes from ${\mu }_0$ to ${\mu }_1$ with ${\mu }_1={\mu }_0+\delta \sigma$. However, due to the limits of the control chart’s capability, the monitoring scheme is unable to detect the quality shift until the end of the $k$^th sampling inspection and the inventory level reaches the maximum value $I_{max}$. The process continues with no signal alert until the PM is scheduled. But, after checking and finding the true quality shift, a CM is carried out to replace the PM for repairing the machine to an as-good-as-new state. In this case, the occurrence probability of scenario 3 ($S_3$) can be given as

$$\begin{gathered} p\left(S_3\right)={\sum }_{i=1}^k{\int }_{\left(i-1\right)h}^{ih}{\lambda e}^{-\lambda t}dt\left(1-{\sum }_{j=1}^{k-i+1}p\left(j|\delta \right)\right) \\ ={\sum }_{i=1}^k\left(e^{-\lambda \left(i-1\right)h}-e^{-\lambda ih}\right)\left(1-{\sum }_{j=1}^{k-i+1}p\left(j|\delta \right)\right). \end{gathered}$$

(10)

The expected IC and OOC time in $S_3$ are easily obtained as

$$E\left(t_{IC}|S_3\right)={\displaystyle \frac{{\sum }_{i=1}^k\left(e^{-\lambda \left(i-1\right)h}-e^{-\lambda ih}\right)\left(1-{\sum }_{j=1}^{k-i+1}p\left(j|\delta \right)\right)\left(\left(i-1\right)h+\tau \right)}{p\left(S_3\right)}}$$

(11)

and

$$E\left(t_{OOC}|S_3\right)={\displaystyle \frac{{\sum }_{i=1}^k\left(e^{-\lambda \left(i-1\right)h}-e^{-\lambda ih}\right)\left(1-{\sum }_{j=1}^{k-i+1}p\left(j|\delta \right)\right)\left(\left(i+j-1\right)h-\tau \right)}{p\left(S_3\right)}}.$$

(12)

and the expected total time $E\left(t|S_3\right)$ is

$$E\left(t|S_3\right)=E\left(t_{IC}|S_3\right)+E\left(t_{OOC}|S_3\right).$$

(13)

5. Production Cost and Production Cycle Time Analysis

The issue of production costs has long been the core concern for enterprises due to limited human, material, and financial resources, which motivates many researchers to establish an economic integration design of the production, maintenance, and quality control policy. Therefore, the production cost and production cycle time will be analyzed in this section for constructing the integrated model.

The expected production cost of one production cycle contains the expected quality loss cost, the expected sampling cost, the expected maintenance cost, the expected false alarm cost, the expected setup cost, and the expected cost of inventory control. As the production run time and the probability of each scenario are given in Section 4, the expected production cost and expected production cycle time for one production cycle will be introduced through the following expressions.

(1) 

Expected quality loss cost

The defective items (or the non-conforming products) continues even when the process is monitored by the control chart. However, when the process is in the IC state, the proportion of the defective products is very low. When a process shift occurs, the key characteristic parameters of the products may deviate from the designed value, resulting in defective products being produced. There is an obvious fact that the quality loss cost of the OOC process is higher than that of the IC process. The quality loss cost can be expressed as

$$E\left(C_Q\right)={\sum }_{i=1}^{3}E\left(C_{Qi}|S_i\right)p\left(S_i\right), i=1,2,3,$$

(14)

and

$$E\left(C_{Qi}|S_i\right)=\left\{\begin{array}{c}{C}_{I}E\left(t_{IC}|S_i\right), for i=1,\\ C_{I}E\left(t_{IC}|S_i\right)+C_{O}E\left(t_{OOC}|S_i\right), for i=2,3,\end{array}\right.$$

(15)

where $C_I$ and $C_O$ are the quality loss per time unit in the IC state and OOC state, and $C_I$ and $C_O$ can be estimated by the Taguchi quality loss function (Pan et al. [19]).

(2) 

Expected sampling cost

Sampling cost happens when the control chart plots based on the sampling data to see whether the production process is in the IC state. To compute the sampling cost in each scenario, the average number of samplings needs to be found. From Figure 1, Figure 2 and Figure 3, we obtained that the numbers of samplings in scenario 1 and scenario 3 are $k$. However, the number of samplings for scenario 2 is equal to ${\displaystyle \frac{\left(E\left(t_{IC}|S_3\right)+E\left(t_{OOC}|S_3\right)\right)}{h}}$, where a true signal is detected before the $k$th sampling. Thus, the average cost of sampling is given by

$$E\left(C_s\right)={\sum }_{i=1}^{3}E\left(C_{si}|S_i\right)p\left(S_i\right), i=1,2,3,$$

(16)

and

$$E\left(C_{si}|S_i\right)p\left(S_i\right)=\left\{\begin{array}{c}k\left(C_F+C_{S}n\right), for i=1,3,\\ \left(C_F+C_{S}n\right){\displaystyle \frac{\left(E\left(t_{IC}|S_3\right)+E\left(t_{OOC}|S_3\right)\right)}{h}}, for i=2,\end{array}\right.$$

(17)

where $\left(C_F+C_{S}n\right)$ is the summation of the fixed and variable cost of each sampling.

(3) 

Expected maintenance cost

The maintenance cost is different in three scenarios. As shown in Figure 3, the process is always in an IC state until the production stops and the PM cost will be included in scenario 1. However, in scenario 2, the process shift occurs and the control chart gives an alarm signal, therefore, the cost of CM should be considered. The maintenance cost of scenario 3 will be higher than the maintenance cost of scenario 2 because the process experiences both PM and CM actions. Therefore, the maintenance cost can be obtained as follows:

$$E\left(C_M\right)={\sum }_{i=1}^{3}E\left(C_{Mi}|S_i\right)p\left(S_i\right), for i=1,2,3,$$

(18)

where

$$E\left(C_{Mi}|S_i\right)=C_i, for i=1,2,3,$$

(19)

and $C_i$ represents the cost of maintenance in scenario $i$.

(4) 

Expected false alarm cost

A false alarm happens when the EWMA monitoring statistic exceeds the surveillance limits but the process is in the IC state. The false alarm exists because the control chart is constructed based on the theory of hypothesis testing, therefore, the cost of the false alarm should be taken into account, and it can be expressed as follows:

$$E\left(C_{Fa}\right)={\sum }_{i=1}^{3}E\left(C_{Fai}|S_i\right)p\left(S_i\right),$$

(20)

where

$$E\left(C_{Fai}|S_i\right)=\left\{\begin{array}{c}{C}_{f}k\alpha , for i=1,\\ C_f\left[{\displaystyle \frac{{\sum }_{i=1}^k\left(e^{-\lambda \left(i-1\right)h}-e^{-\lambda ih}\right)\left({\sum }_{j=1}^{k-i+1}p\left(j|u_0,\delta \right)\right)\left(i-1\right)\alpha }{p\left(S_2\right)}}\right], for i=2,\\ C_f\left[{\displaystyle \frac{{\sum }_{i=1}^k\left(e^{-\lambda \left(i-1\right)h}-e^{-\lambda ih}\right)\left(1-{\sum }_{j=1}^{k-i+1}p\left(j|u_0,\delta \right)\right)\left(i-1\right)\alpha }{p\left(S_3\right)}}\right], for i=3,\end{array}\right.$$

(21)

and $C_f$ represents the cost of inspecting a false alarm, and $\mathsf{\alpha }$ is the false alarm rate.

(5) 

Expected setup cost

The expected setup cost in one production cycle can be given as

$$E\left(C_o\right)={\sum }_{i=1}^{3}E\left(C_{oi}|S_i\right)p\left(S_i\right), for i=1,2,3,$$

(22)

where

$$E\left(C_{oi}|S_i\right)=S,$$

(23)

$S$ is the setup cost per production cycle.

(6) 

Expected inventory control cost

A machine produces items at a rate of P. The products will be 100% inspected and defective products are discarded. We assume that the proportion of average defective items of the OOC process is a constant $\beta$, where $\beta$ can be estimated from historical data (Bahria et al. [35]). However, for the IC process it is reasonably assumed that a very low proportion of defective products are produced, and we can ignore defective products. The expected average inventory control cost will be calculated for all possible conditions in one production cycle. Depending on whether the maintenance duration is less or longer than the inventory depletion period, two cases consisting of shortage and without shortage in one production process are considered, and each case under different production scenarios will be given as follows.

(a) 

Scenario 1

■

Case: without shortage:

Figure 4a shows the evolution of the average inventory level for the case without shortage in scenario 1. The expected inventory level contains two parts, $Z_1$ and $Z_2$, which are respectively the average inventory holding level during the production uptime and production downtime. The duration of PM $t_1$ is smaller than ${\displaystyle \frac{kh\left(P-D\right)}{D}}$, where ${\displaystyle \frac{kh\left(P-D\right)}{D}}$ represents the time from the inventory level $I_{max}$ to zero. Therefore, we have the following calculations:

$$t_1\le {\displaystyle \frac{kh\left(P-D\right)}{D}},$$

$$Z_1={\displaystyle \frac{\left(P-D\right){\left(kh\right)}^2}{2}}, Z_2={\displaystyle \frac{{\left[kh\left(P-D\right)\right]}^2}{2D}}.$$

(24)

The average inventory level in case (a) for scenario 1 is given by ${\mathsf{\Gamma }}_1=Z_1+Z_2$.

■

Case: with shortage

Figure 4b shows the evolution of the expected inventory level for the case of shortage in scenario 1. In this scenario, the PM period is larger than the inventory consumption period, which can be expressed by

$$t_1>{\displaystyle \frac{kh\left(P-D\right)}{D}}$$

Let ${\mathsf{\Gamma }}_1^{\prime }$ be the average inventory level. ${\mathsf{\Gamma }}_1^{\prime }$ contains the inventory holding level $Z_{11}$ and $Z_{12}$. Three functions, $Z_{11}$, $Z_{12}$, and $Z_{13}$, are given as

$$Z_{11}={\displaystyle \frac{\left(P-D\right){\left(kh\right)}^2}{2}}, Z_{12}={\displaystyle \frac{{\left[kh\left(P-D\right)\right]}^2}{2D}}, Z_{13}=D\left[t_1-{\displaystyle \frac{kh\left(P-D\right)}{D}}\right].$$

(25)

Therefore, the average inventory level in case (b) of scenario 1 is ${\mathsf{\Gamma }}_1^{\prime }=Z_{11}+Z_{12}$, and ${{\mathsf{\Psi }}_1=Z}_{13}$ denotes the average shortage level.

(b) 

Scenario 2

■

Case: without shortage:

From Figure 5a, we can see that the duration of CM $t_2$ is less than or equal to the duration of the inventory consumption, and it can be expressed by

$$t_2\le {\displaystyle \frac{E\left(t|S_2\right)\left(P-D\right)-\beta PE\left(t_{OOC}|S_2\right)}{D}},$$

where $E\left(t_{OOC}|S_2\right)$ and $E\left(t|S_2\right)$ can be found in Equations (7) and (8). Let $Z_3$ and $Z_4$ respectively be the average inventory level during the production uptime and production downtime. $Z_3$ and $Z_4$ are described as

$$Z_3={\displaystyle \frac{kh\left(P-D\right)E\left(t|S_2\right)}{2}},$$

(26)

$$Z_4={\displaystyle \frac{{\left[\left(P-D\right)E\left(t|S_2\right)-\beta PE\left(t_{OOC}|S_2\right)\right]}^2}{2D}}$$

(27)

The average inventory level is ${\mathsf{\Gamma }}_2=Z_3+Z_4$ for the case of without shortage in scenario 2.

■

Case: with shortage

For the case of shortage in scenario 2, the production process is disrupted by CM, and the period of CM $t_2$ is longer than the period of inventory consumption (see Figure 5b).

$$t_2>{\displaystyle \frac{E\left(t|S_2\right)\left(P-D\right)-\beta PE\left(t_{OOC}|S_2\right)}{D}},$$

The average inventory level $Z_{14},{ Z}_{15}$ and the average shortage level $Z_{16}$ are respectively expressed by

$$Z_{14}={\displaystyle \frac{\left(P-D\right){E\left(t|S_2\right)}^2}{2}},$$

(28)

$$Z_{15}={\displaystyle \frac{{\left[\left(P-D\right)E\left(t|S_2\right)-\beta PE\left(t_{OOC}|S_2\right)\right]}^2}{2D}},$$

(29)

$$Z_{16}=D\left[t_2-{\displaystyle \frac{\left(P-D\right)E\left(t|S_2\right)-\beta PE\left(t_{OOC}|S_2\right)}{D}}\right].$$

(30)

Finally, the average inventory level in case (b) for scenario 2 is ${\mathsf{\Gamma }}_2^{\prime }=Z_{14}+Z_{15}$, and ${{\mathsf{\Psi }}_2=Z}_{16}$ is the average shortage level.

(c) 

Scenario 3

■

Case: without shortage:

As we can see from Figure 6a, the expected inventory level in one production cycle can be divided into two parts, $Z_5$ and $Z_6$, and the maintenance duration is less than or equal to ${\displaystyle \frac{kh\left(P-D\right)-\beta PE\left(t_{OOC}|S_3\right)}{D}}$. The ${\displaystyle \frac{kh\left(P-D\right)-\beta PE\left(t_{OOC}|S_3\right)}{D}}$ denotes the inventory consumption period starting from the time that the inventory is built up to the maximum value to the end of the production cycle.

$$t_3\le {\displaystyle \frac{kh\left(P-D\right)-\beta PE\left(t_{OOC}|S_3\right)}{D}},$$

$Z_5$ and $Z_6$ are respectively obtained as

$$Z_5={\displaystyle \frac{\left(P-D\right){\left(kh\right)}^2}{2}},$$

(31)

$$Z_6={\displaystyle \frac{{\left[kh\left(P-D\right)-\beta PE\left(t_{OOC}|S_3\right)\right]}^2}{2D}}.$$

(32)

Therefore, the average inventory level in case (a) for scenario 3 is ${\mathsf{\Gamma }}_3=Z_4+Z_5$.

■

Case: with shortage

From Figure 6b, we can find that the period of maintenance $t_3$ is longer than the period of inventory consumption, where the relationship between $t_3$ and the period of the consumption of the inventory can be defined as

$$t_3>{\displaystyle \frac{kh\left(P-D\right)-\beta PE\left(t_{OOC}|S_3\right)}{D}}.$$

The equation for the average inventory level and shortage level is obtained using the following formula,

$$Z_{17}={\displaystyle \frac{\left(P-D\right){\left(kh\right)}^2}{2}},$$

(33)

$$Z_{18}={\displaystyle \frac{{\left[\left(P-D\right)kh-\beta PE\left(t_{OOC}|S_3\right)\right]}^2}{2}},$$

(34)

$$Z_{19}=D\left[t_3-{\displaystyle \frac{\left(P-D\right)kh-\beta PE\left(t_{OOC}|S_3\right)}{D}}\right].$$

(35)

Therefore, the average inventory level in case (b) of scenario 3 is ${\mathsf{\Gamma }}_3^{\prime }=Z_{15}+Z_{16}$, and ${{\mathsf{\Psi }}_3=Z}_{19}$ is the average shortage level.

Following the above analysis of the average inventory level, we define $E\left(I\right)$ as the average inventory control cost, where $E\left(I\right)$ contains the average inventory holding cost and the average shortage cost. For easy expression, we give three indicators in three scenarios as follows:

$${\varphi }_1=\left\{\begin{array}{c}1, if t_1\le {\displaystyle \frac{kh\left(P-D\right)}{D}}, \\ 0, if t_1>{\displaystyle \frac{kh\left(P-D\right)}{D}},\end{array}\right.$$

(36)

$${\varphi }_2=\left\{\begin{array}{c}1, if t_2\le {\displaystyle \frac{E\left(t|S_2\right)\left(P-D\right)-\beta PE\left(t_{OOC}|S_2\right)}{D}},\\ 0, if t_2>{\displaystyle \frac{E\left(t|S_2\right)\left(P-D\right)-\beta PE\left(t_{OOC}|S_2\right)}{D}},\end{array}\right.$$

(37)

$${\varphi }_3=\left\{\begin{array}{c}1, if t_3\le {\displaystyle \frac{kh\left(P-D\right)-\beta PE\left(t_{OOC}|S_3\right)}{D}} ,\\ 0, if t_3>{\displaystyle \frac{kh\left(P-D\right)-\beta PE\left(t_{OOC}|S_3\right)}{D}},\end{array}\right.$$

(38)

Finally, we define $E\left(I\right)$ as follows:

$$E\left(I\right)={\sum }_{i=1}^3{\varphi }_{i}B{\mathsf{\Gamma }}_{i}p\left(S_i\right)+{\sum }_{i=1}^3\left(1-{\varphi }_i\right)\left(B{\mathsf{\Gamma }}_i^{\prime }p\left(S_i\right)+B_1{\mathsf{\Psi }}_{i}p\left(S_i\right)\right).$$

(39)

(7) 

Expected production cycle time

In different scenarios, we can find that the expected production cycle time depends on the production uptime, the maintenance duration, and the period of the inventory consumption. Therefore, the expected production cycle time $ET$ can be given as

$$ET={\sum }_{i=1}^{3}E\left(T_i|S_i\right)p\left(S_i\right), for i=1,2,3,$$

(40)

where $E\left(T_i|S_i\right)$ is the average length of one production cycle in the occurrence of scenario $i$, and $E\left(T_i|S_i\right)$ can be expressed as follows:

$$E\left(T_1|S_1\right)=\left\{\begin{array}{c}kh+{\displaystyle \frac{kh\left(P-D\right)}{D}}, if t_1\le {\displaystyle \frac{kh\left(P-D\right)}{D}},\\ kh+t_1, if t_1>{\displaystyle \frac{kh\left(P-D\right)}{D}},\end{array}\right.$$

(41)

$$E\left(T_2|S_2\right)=\left\{\begin{array}{c}E\left(t|S_2\right)+{\displaystyle \frac{E\left(t|S_2\right)\left(P-D\right)-\beta PE\left(t|S_2\right) }{D}}, if t_2\le {\displaystyle \frac{E\left(t|S_2\right)\left(P-D\right)-\beta PE\left(t|S_2\right)}{D}},\\ E\left(t|S_2\right)+t_2, if t_2>{\displaystyle \frac{E\left(t|S_2\right)\left(P-D\right)-\beta PE\left(t|S_2\right)}{D}},\end{array}\right.$$

(42)

$$E\left(T_3|S_3\right)=\left\{\begin{array}{c}kh+{\displaystyle \frac{kh\left(P-D\right)-\beta Pkh}{D}}, if t_3\le {\displaystyle \frac{kh\left(P-D\right)-\beta Pkh}{D}},\\ kh+t_3, if t_3>{\displaystyle \frac{kh\left(P-D\right)-\beta Pkh}{D}}.\end{array}\right.$$

(43)

6. Integrated Model of Production, EWMA Control, and Maintenance Policy

In this section, the integrated model of the production, EWMA control, and maintenance policy will be established based on the expected total cost per time unit (ETC), namely, the ETC model. Based on the ETC model, the objective is to minimize the ETC to achieve an optimal production level under the given constraints of the inventory size, the false alarm rate, the ARL of the OOC process, the sampling interval, the sample size, and sampling number. Further, the solution method will be introduced to deal with this optimization problem.

(1) 

The objective function and constraints

The expected total cost $EC$ per production cycle consists of expected quality loss cost $E\left(Q\right)$, expected sampling inspection cost $E\left(S\right)$, expected maintenance cost $E\left(M\right)$ and expected false alarm cost $E\left(C_{Fa}\right)$, expected cost of setup cost $E\left(O\right)$, and expected inventory control cost $E\left(I\right)$, and $EC$ can be expressed by

$$EC=E\left(Q\right)+E\left(S\right)+E\left(M\right)+E\left(C_{Fa}\right)+E\left(O\right)+E\left(I\right).$$

(44)

The average expected total cost of one production cycle per time unit (ETC) is

$$ETC={\displaystyle \frac{EC}{ET}},$$

(45)

To make the integrated model close to the real production process, sampling number $k$, sampling interval $h$, sample size $n$, smooth parameter, inventory size $I$, and type I error $\alpha$ (Salmasnia et al. [16]; Zhou et al. [47]; Wan et al. [48]) and ARL of OOC processes ${ARL}_1$ are limited. Especially, the maximum value of the false alarm ${\alpha }_0$ (Salmasnia et al. [16]) and the maximum ARL of the OOC process ${ARL}_{1,max}$ (Pan et al. [22], Salmasnia et al. [15]) should be prefixed. The optimization problem of the integrated model is formulated as follows:

$$Min\left\{ETC\right\},$$

(46)

subject to

$$k_{min}\le k\le k_{max},$$

(47)

$$h_{min}\le h\le h_{max},$$

(48)

$$n_{min}\le n\le n_{max},$$

(49)

$$0<r<1,$$

(50)

$$I\le I_{max},$$

(51)

$$\alpha \le {\alpha }_0,$$

(52)

$${ARL}_1<{ARL}_{1,max},$$

(53)

$$h>0,L>0,n\in N^{+},k\in N^{+},$$

(54)

where $I=kh\left(P-D\right)$.

(2) 

Solution approach

• The complexity of the presented integrated model leads to the difficulty of solving the objective function, where both the non-linear target and non-linear constraint functions increase the complexity of the model. In the literature, many efficient algorithms were developed to solve the optimization problem when establishing an integrated design of production, maintenance, and quality control policy, for example, Pan et al. [22] used Pattern Search Tool in MATLAB 7.0. Charongrattanasakul and Pongpullponsak [11], Yin et al. [30], and Huang et al. [14] adopted the genetic algorithm (GA). Salmasnia et al. [15] and Salmasnia et al. [16] applied meta-heuristic algorithms, etc. Without loss of generality, the GA is adopted to solve the optimization problem defined in Equations (46)–(54). The solution steps using the GA are shown in

  Table 2. Genetic Algorithm for Optimal ETC.

  Step 1	Set model parameters. Specify the process parameters: quality shift rate$\lambda$$, \mathrm{initial} \mathrm{mean} u_0, \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d} \mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \sigma$, mean shift magnitude $\delta$, and other relevant values.
  Step 2	Configure GA parameters. The iteration time, the population size, the crossover fraction, and the mutation probability parameters are predetermined. In this paper, we set (iteration time, population size, crossover fraction, mutation probability) = (200,100,0.8,0.1).
  Step 3	Initialize population. The procedure starts at randomly generating 100 solutions that satisfy the constraints (Equations (47)–(54)) described in Section 6.
  Step 4	Evolute fitness.$\mathrm{For} \mathrm{each} \mathrm{individual}, \mathrm{compute} \mathrm{the} \mathrm{fitness} \mathrm{value} \mathrm{using} \mathrm{the} \mathrm{objective} \mathrm{function} ETC={\displaystyle \frac{EC}{ET}}$.
  Step 5	While stopping criterion is not met:
  	1. Select parents using roulette wheel selection based on fitness.
  	2. Perform crossover on selected parents with probaility 0.8.
  	3. Apply mutation to offspring with probability 0.1.
  	4. Evaluate new population and compute fitness for each offspring.
  	5. Replace population with new offspring.
  	6. Check stopping criterion: halt if no improvement in ETC over 50 consecutive generations.
  Step 6	Return the best solution found and its corresponding ETC value.

  :

7. Case Study and Analysis

In Section 6, an integrated model for production, EWMA control, and maintenance policy is established, and the GA solution method is also provided. To show the application of the proposed model, first, a real example of a vibroseis equipment production process is introduced. Then, a comparative study is conducted between the integrated model established using the $\overline{X}$ control chart and the proposed model. Finally, the sensitivity analysis is performed to investigate the influence of key input parameters on the objective function and decision parameters.

(1) 

Case study

In this subsection, a real example of a vibroseis equipment production process is adopted to illustrate the application of the proposed model, where the vibroseis equipment’s bolt is one of the widespread parts used in vibroseis equipment. Vibrators installed on trucks of a vibroseis continuously impact the ground to generate seismic waves. The bolts play a crucial role in fixing various structures. Therefore, the quality and lifespan of vibroseis bolts are of great importance. The hardness is one of the key quality characteristics of the vibroseis equipment’s bolt and is mainly influenced by the heating temperature. Due to some assignable causes, the heating temperature will deviate from the target value, resulting in the hardness being unqualified and further leading to an increase in the proportion of defective bolts. Therefore, the EWMA control chart is adopted to monitor the heating temperature of the high-voltage rotor bolt production process.

To provide an optimal design of production, maintenance, and quality control policy for this bolt production process and minimize the production cost, the proposed integrated model is applied. According to the historical data and the information collected from engineers of the vibroseis equipment’s bolt production process, some input parameters are given in

Table 3. Input parameters for the vibroseis equipment’s bolt production process.

Parameter	$\lambda$	$u_0$	$\sigma$	$C_F$	$C_S$	$C_I$	$t_1$	$\beta$
value	0.001/h	800 °C	1.5	30$	3$	300$	0.5h	0.2
Parameter	$C_O$	$C_f$	$C_1$	$C_2$	$C_3$	$n_{max}$	$t_2$	$k_{max}$
Value	600$	800$	2000$	4000$	3500$	20	1.0h	100
Parameter	${\alpha }_{max}$	P	D	S	B	$t_3$	$h_{max}$	$B_1$
Value	0.01	100	60	100$	0.001$	1.1h	20	0.1

.

From Table 3, we can see that the production process stays in an IC state for an average of 100 h because of the shift parameter $\lambda =0.01$. Moreover, the average magnitude of the shift when the process goes OOC is approximately 0.25 standard variance. With variables detailed and constraints specified, the ETC and decision variables $\left\{k,h,n,L,r\right\}$ can be obtained by solving Expression (47) subject to

$$\alpha \le 0.01,$$

(55)

$${ARL}_1<10,$$

(56)

$$0<r<1,$$

(57)

$$5\le n\le 20,$$

(58)

$$0\le h\le 20,$$

(59)

$$0\le k\le 100,$$

(60)

$$kh\ast \left(P-D\right)\le I_{max}=12000,$$

(61)

$$h>0,L>0,n\in N^{+},k\in N^{+}.$$

(62)

Figure 7 shows the iteration process of the GA for calculating the optimal ETC with $\delta =$ 0.25. It can be seen that the values of the ETC under iterations 146–196 are approximately the same, which means that the ETC has not been remarkably improved during iterations 146–196. Therefore, it can be concluded with high confidence that the obtained results are very close to the real optimal solution. Finally, the optimal solution of decision parameters is $\left\{k,h,n,L,r\right\}=\left\{75,4,20,1.6429,0.3952\right\}$, and the ETC is 242.6404. It is suggested by these results that, during one production cycle, 75 sampling instances are needed, and one should take one sample of size 20 every 4 h. The coefficient of the control limits of the EWMA control chart should be set at 1.6429 and the smooth parameter is recommended to be 0.3952. Moreover, the economic production quantity (EPQ) for one production cycle is $EPQ=khP=30000$, and the maximum inventory level is $E\left(I\right)=kh\left(P-D\right)=\mathrm{12,000}$, which means that the utilization rate of inventory capacity is 100%.

(2) 

Comparison

Since there is no existing design for performing preventive maintenance (PM) based on information from the EWMA control chart and inventory level, a comparative study is conducted among three models: the model without monitoring, the model using the $\overline{X}$ chart, and the proposed method. The comparison results are presented in

Table 4. The comparison results of the proposed model, the model without monitoring, and the model based on the $\overline{X}$ chart.

$\mathit{\delta }$	Model Without Monitoring		$\mathbf{Model} \mathbf{Using} \mathbf{the} \overline{\mathit{X}} \mathbf{Chart}$		The Proposed Model
	$\left[\mathit{k},\mathit{h}\right]$	ETC	$\left[{\mathit{k}}_1,{\mathit{h}}_1,{\mathit{n}}_1,{\mathit{L}}_1\right]$	ETC	$\left[\mathit{k},\mathit{h},\mathit{n},\mathit{L},\mathit{r}\right]$	ETC
0.25	[15,19.9750]	337.3120	[62,4.8185,20,1.5455]	250.7435	[75,4,20,1.6429,0.3952]	242.6404
0.5	[15,19.9998]	335.5869	[69,4.3336,20,2.1574]	235.2727	[68,4.4118,20,2.2311,0.5931]	226.8182
0.75	[22,13.6364]	335.9043	[67,4.4662,16,2.4176]	228.8077	[71,4.2254,16,2.6863,0.5738]	220.0941
1.0	[18,16.6667]	335.2473	[73,4.0978,12,2.6348]	225.4887	[78,3.8462,11,2.7786,0.4579]	216.4485
1.5	[16,18.7501]	335.0049	[79,3.7927,7,2.8231]	222.1195	[88,3.4091,12,2.8021,0.1592]	212.0050
2.0	[15,20]	334.9024	[81,3.6933,5,2.9877]	220.5056	[72,4.1667,18,2.4179,0.0421]	207.0293
3.0	[16,18.7501]	335.0047	[80,3.7500,5,2.9997]	220.0562	[79,3.7975,7,2.3978,0.0473]	203.2866

. In the model using the $\overline{X}$ chart, the assumptions, maintenance policies, and inventory control strategy are consistent with those of the proposed model using the EWMA chart. However, the decision parameters for the $\overline{X}$ control chart model include the sampling number sampling number $k_1$, sampling interval $h_1$, sample size $n_1$, and coefficient of the control limits $L_1$ of $\overline{X}$ control chart. As shown in Table 4, the proposed model is the most economical. Furthermore, the proposed model is superior to the model without monitoring.

It can also be concluded that the economic performance of the proposed integrated method based on the EWMA control chart is superior to the traditional integrated model based on the $\overline{X}$ control chart, because the ETC of our model is consistently less than the model of using the $\overline{X}$ control chart. For example, the ETC of our proposed model based on the EWMA control chart is 242.6404 when shift $\delta$ is 0.25, while the ETC of the model based on the $\overline{X}$ control chart is 250.7435.

(3) 

Sensitivity analysis

In this subsection, the effect of parameters on the ETC is investigated. To quickly capture the relationship between the input parameters and the ETC under the same shift $\delta$, the experiment design is first considered. Second, the effects of $\delta$ on the ETC and decision parameters are also given.

The orthogonal experimental design (ODE) method is a well-known experimental design method, where both the characteristics of the uniformly dispersed and symmetrical comparability exist, and the ODE can significantly reduce the number of experiments when there are many influence factors and levels. Salmasnia et al. [13] once adopted a Taguchi ODE to analyze the effect of input parameters on two comparative models. Motivated by Salmasnia et al. [13] and Wan [45], a numerical experiment using the ODE is also taken in this subsection.

For exploring the effect of eleven parameters $\{\lambda ,B,C_I,C_0,C_1,C_2,C_3, C_F,C_S,C_f,B_1\}$ on the ETC adequately, the number of parameter levels is set to 3. Detailed arrangements of experiments by ODE $L_{27}\left(3^{11}\right)$ under the case of $\delta =0.25$ are listed in

Table 5. The Orthogonal experimental $L_{27}\left(3^{11}\right)$ design for sensitivity analysis when $\delta =0.25$.

Exp.No.	$\mathit{\lambda }$	$\mathit{B}$	${\mathit{C}}_{\mathit{I}}$	${\mathit{C}}_0$	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_{\mathit{F}}$	${\mathit{C}}_{\mathit{S}}$	${\mathit{C}}_{\mathit{f}}$	${\mathit{B}}_1$	Decision Parameters	ETC
1	0.001	0.0005	150	500	1500	3000	3000	25	3	700	0.1	[32,9.375,20,1.7093,0.3505]	113.0511
2	0.001	0.0005	150	500	2000	4000	3500	30	15	800	0.5	[4,20,5,1.5440,0.0859]	117.5981
3	0.001	0.0005	150	500	2500	5000	4000	35	27	900	0.9	[5,19.9416,5,1.6281]	123.2958
4	0.001	0.001	200	600	1500	3000	3000	30	15	800	0.9	[3,19.9999,5,1.1947,0.0396]	144.3174
5	0.001	0.001	200	600	2000	4000	3500	35	27	900	0.1	[4,19.9969,5,1.2230,0.0421]	149.8485
6	0.001	0.001	200	600	2500	5000	4000	25	3	700	0.5	[35,8.5715,20,1.7263,0.3548]	150.0100
7	0.001	0.0015	250	700	1500	3000	3000	35	27	900	0.5	[3,20,5,1.2175,0.0416]	178.0378
8	0.001	0.0015	250	700	2000	4000	3500	25	3	700	0.9	[38,7.8947,5,1,0.6070]	188.7280
9	0.001	0.0015	250	700	2500	5000	4000	30	15	800	0.1	[4,19.9996,6,2.0237,0.5255]	179.5152
10	0.005	0.0005	200	700	1500	4000	4000	25	15	900	0.1	[4,19.9983,5,1.9427,0.4053]	184.9048
11	0.005	0.0005	200	700	2000	5000	3000	30	27	700	0.5	[3,16.4487,5,1.3739,0.0586]	180.7463
12	0.005	0.0005	200	700	2500	3000	3500	35	3	800	0.9	[40,7.5,20,1.2132,0.3855]	172.0500
13	0.005	0.001	250	500	1500	4000	4000	30	27	700	0.9	[39,7.6923,5,1.1119,0.4922]	210.2503
14	0.005	0.001	250	500	2000	5000	3000	35	3	800	0.1	[40,7.5,20,1.4423,0.3591]	199.7689
15	0.005	0.001	250	500	2500	3000	3500	25	15	900	0.5	[38,7.8947,5,1.0406,0.3265]	205.9329
16	0.005	0.0015	150	600	1500	4000	4000	35	3	800	0.5	[40,7.1221,20,1.3005,0.3833]	144.6626
17	0.005	0.0015	150	600	2000	5000	3000	25	15	900	0.9	[3,16.8316,5,1.2092,0.0409]	147.0074
18	0.005	0.0015	150	600	2500	3000	3500	30	27	700	0.1	[3,19.8754,5,1.1997,0.004]	152.1716
19	0.009	0.0005	250	600	1500	5000	3500	25	27	800	0.1	[40, 7.5, 5.0, 1.0, 0.6614]	239.3768
20	0.009	0.0005	250	600	2000	3000	4000	30	3	900	0.5	[40,6.8831,20,1.2637,0.3776]	212.8657
21	0.009	0.0005	250	600	2500	4000	3000	35	15	700	0.9	[40,7.5,5,1.016,0.7822]	228.5852
22	0.009	0.001	150	700	1500	5000	3500	30	3	900	0.9	[40,5.4348,20,1.2776,0.3800]	172.8708
23	0.009	0.001	150	700	2000	3000	4000	35	15	700	0.1	[40,6.440603,5,1,0.8483]	176.3145
24	0.009	0.001	150	700	2500	4000	3000	25	27	800	0.5	[40,7.296,5,1,0.8625]	188.4005
25	0.009	0.0015	200	500	1500	5000	3500	35	15	700	0.5	[40,7.5,5,1.0181,0.6538]	200.5966
26	0.009	0.0015	200	500	2000	3000	4000	25	27	800	0.9	[40,7.5,5,1.0029,0.5732]	197.0845
27	0.009	0.0015	200	500	2500	4000	3000	30	3	900	0.1	[40,7.4725,20,1.1916,0.3786]	184.0319
I	149.38	174.72	148.37	172.40	176.45	172.43	173.77	179.39	170.89	177.83	175.44	W = 526.89
II	177.50	177.52	173.73	174.32	174.44	177.45	177.69	149.06	176.09	175.86	175.43
III	200.01	174.65	204.78	180.17	176.00	177.02	175.43	174.80	179.91	173.20	176.02
R	50.64	2.88	56.41	7.77	2.01	5.02	3.91	30.33	9.02	4.63	0.59

, and corresponding results of decision parameters and the ETC are recorded. Especially, to better understand the influence of eleven input parameters on the ETC, the results of the range analysis are also summarized in the last line of Table 5. The values of I, II, and III are prepared for calculating the range value R, and W represents the summarization of I, II, and III under the same parameter. The range value R is expressed as $\mathrm{R}=\max \left\{\mathrm{I},\mathrm{I}\mathrm{I},\mathrm{I}\mathrm{I}\mathrm{I}\right\}-\mathrm{m}\mathrm{i}\mathrm{n}\left\{\mathrm{I},\mathrm{I}\mathrm{I},\mathrm{I}\mathrm{I}\mathrm{I}\right\}$ under the same parameter. We define that ${ETC}_{si}$ is the summation of ETC under the same parameter level $i$ $(i=1,2,3)$ and I, II, and III are respectively the mean of ${ETC}_{si}$. For example, the first value 149.38 for row 29 in Table 4 is the mean of ${ETC}_{s1}$ under $\lambda =$ 0.001 and $\mathrm{R}=\max \left\{\mathrm{149.38,177.50,200.01}\right\}-\mathrm{m}\mathrm{i}\mathrm{n}\left\{\mathrm{149.38,177.50,200.01}\right\}$ = 50.64.

For clear observation, the analysis of the effect of input parameters on the ETC is reported in Figure 8. Furthermore, the effects of $\delta$ on the ETC and decision parameters are plotted in Figure 9. The results of sensitivity analysis can be concluded as follows.

■

Effect of eleven key parameters on the ETC

From Figure 8, the following conclusions are obtained.

(1)

As we can see in Figure 8, the parameter $\lambda$, the quality loss per time $C_I$ in IC processes, and the cost of fixed sampling $C_F$ are three of the most effective input parameters on the ETC. Moreover, the range analysis results show that the order of three influence factors is $C_I>\lambda >C_F$ (see the last line in Table 4).

(2)

Eight parameters $B,C_o,C_1,C_2,C_3,C_s,C_{INSP},B_1$ have a smaller effect on the ETC.

(3)

Moreover, one can find that the increase in the parameters $\lambda$, $C_I$, $C_o$, and $C_s$ can lead to an increase in the ETC. This phenomenon is consistent with the common knowledge that the occurrence rate of shift $\lambda$ increases will increase the production cost per unit time. It is also obvious that increasing the quality loss cost parameter $C_I$ in the IC process, the quality loss cost parameter $C_o$ in the OOC process, and the varied sampling cost parameter $C_s$ will also increase the ETC.

(4)

It is worth pointing out that the increase in $C_f$ can reduce the ETC because the decision parameters can be adjusted to compensate the impact of the increase in the $C_f$ on ETC.

■

Effect of δ on decision parameters and the ETC

From Figure 9, we can obtain the following conclusions.

(1)

When the magnitude of shift increases, the ETC decreases.

(2)

Figure 9b shows that, as $\delta$ increases, the coefficient of the control limits of the EWMA control chart $L$ increases under moderate and small shifts $\delta \le 1.5$.

(3)

One can also conclude from Figure 9c that, as $\delta$ increases, the value of the smooth parameter $r$ increases under moderate and small shifts $\delta \le 1.5$. However, the value of the smooth parameter shows a downward trend when $\delta >1.5$.

There exists strong interaction between parameters $k$ and $h$ (see Figure 9d). The decision parameter $k$ increases (decreases), the parameter $h$ decreases (increases). It is because the process needs to satisfy the constraint $kh\left(P-D\right)\le I_{max}$ that two decision parameters $(k,h)$ are adjusted to realize the minimized ETC.

8. Conclusions

In this paper, considering production quality and inventory constraints, we developed an integrated model of the production, EWMA control, and maintenance policy. The EWMA charting scheme is adopted to guarantee the process quality, efficiently detect moderate and small shifts, and reduce the quality loss cost. Considering the inventory size of the production process is finite in practice, a limited inventory size is considered in the proposed model. From the viewpoint of taking the opportunity to improve the state of the system, the PM action is carried out both considering the IC information given by the EWMA control chart and the inventory state. Finally, the proposed integrated model is applied to a vibroseis equipment’s bolt production process, and the decision parameters are given for arranging high-quality production. The corresponding results show that the production costs for the bolt production process are greatly reduced, which demonstrates the proposed model is more economic. Therefore, the proposed model is more suitable to be recommended to production decision-makers.

Despite the above contributions, this study has several limitations that suggest valuable directions for future research. First, the model relies on the assumption of perfect detection, where faults are always identified once they occur. In practice, detection systems may exhibit varying reliability, and incorporating imperfect detection processes represents an important extension. Second, parameters such as the failure rate are treated as fixed, while in real settings they may change over time or be influenced by external factors; developing more dynamic models would enhance practical applicability. Furthermore, the current work assumes that the occurrence of an assignable cause does not affect the process variance, and the quality characteristic follows a normal distribution. Future studies could investigate joint monitoring schemes for both mean and variance under non-normal distributions. Moreover, this study focuses on a single quality characteristic and a single-machine system. Research could be extended to incorporate multiple quality characteristics and multimachine manufacturing environments, which pose greater complexity but are more representative of actual industrial systems. Finally, relaxing other simplifying assumptions, such as immediate maintenance resource availability, would further improve the model’s robustness and broaden its applicability.