Research on maintenance optimal policy based on product quality control

Abstract

According to the correlation between product quality and equipment degradation state, an equipment maintenance policy is designed by integrating periodic equipment inspection and product quality control. In this policy, np-chart is used to monitor the abnormal shift of the product quality characteristic based on periodic inspection of the equipment. By considering the inspection result of the product quality shift and equipment degradation state, the corresponding maintenance action is chosen. Furthermore, the optimal maintenance model based on product quality control is proposed and is solved with genetic algorithm. The experimental results validated the feasibility of this model.

Introduction

In the production process, product quality is the primary evaluation factor. Many factors influence product quality, such as equipment, raw material, manufacturing method, and assembly crew.^1,2 Therefore, product quality can be improved by reducing process variation. In 1924, Shewhart developed the concept of statistical process control (SPC), which is a methodology for measuring and monitoring quality control. ³ The control chart is one such quality control tool that checks for process stability by determining whether the product quality is in an abnormal state. ⁴ Furthermore, in the modern manufacturing industry, complex manufacturing equipment can deteriorate gradually until eventual failure. To reduce the deterioration rate of the equipment, a series of equipment maintenance policies have been proposed, including time-based maintenance and condition-based maintenance.^5–7 According to the maintenance policy, predetermined maintenance actions are implemented. Although maintenance management and SPC are able to ensure the quality of a production system, they are usually treated independently. In 1995, Ben-Daya and Duffuaa ⁸ noticed that there is a correlation with product quality and maintenance activity. Since then, industry and academia recognized the relationship between product quality and maintenance. Research increasingly focused on integrated periodic equipment inspection and product quality control.

In general, x-bar chart is used to monitor the average value of the quality characteristic of produced items.^9–11 It has the advantages of simple operation and wide applicability. In addition, x-bar chart is suitable for monitoring variable data, in which each point represents the most recent data. Therefore, x-bar chart could be used to measure continuously varying quality characteristics. For counted data, attribute control charts display changes in the number of defective products, rejects, or unacceptable outcomes. Among the various types of attribute control charts, np-chart is often used as an indicator of the consistency and predictability of the level of defects in the process. It is valid as long as data are collected in subgroups that are of the same size. Recently, a few scholars introduced np-chart into the integrated research of product quality and maintenance. Wang ¹² assumes that a system is in one of the three possible states, including in-control, out-of-control, and failure. Based on this assumption, he proposed three maintenance models based on np-chart.

Although the above research noted the relationship between product quality and maintenance, they assume that equipment is the key factor influencing product quality. Thus, current maintenance policy requires that the equipment is maintained as soon as the product quality is determined to be in abnormal state.^13,14 Nevertheless, during real product processes, factors other than the equipment (e.g. raw material, manufacturing method) may cause an abnormal shift of product quality. ¹⁵ Because equipment is not the only reason for this result, the equipment can undergo excess maintenance. To avoid this dilemma, equipment inspection is performed to determine whether the equipment is stopped by a sudden breakdown.

In the manufacturing industry, periodic inspection is used to discover hidden failures. If the equipment breaks down before the scheduled maintenance, it causes great economic loss. Equipment deterioration is the major factor that causes defective product outputs. Meanwhile, the level of product quality can indirectly reflect the equipment’s deterioration state. Considering the correlation between the deterioration state of product quality and the equipment’s deterioration state, it is necessary to design a more rational maintenance policy based on periodic inspection and SPC. In this article, a maintenance policy that integrates periodic equipment inspection and product quality control is proposed. In this policy, np-control chart is used to monitor the shift of product quality and the deterioration state of the equipment is introduced to determine whether the equipment requires maintenance. Finally, a maintenance model based on product quality control is proposed and solved by genetic algorithm.

Assumption

• A deteriorating single-equipment system is considered, and the equipment deterioration behaviour is described by a stochastic process.

• Product quality inspection is performed with np-chart and n samples are taken every h hours. For np-chart, it is used to detect quality shifts by judging whether the number of defective items is larger than a specific number.

• The equipment state can be detected only by inspection, which is conducted every $\tau$ hours. The appropriate maintenance action will be performed according to the inspection results. Meanwhile, between the equipment inspections, an integrated monitoring result of the np-chart with equipment deterioration level will determine whether the equipment requires maintenance.

• When the equipment breaks down, the deterioration level is greater than the corrective maintenance (CM) threshold. Then, the equipment is replaced immediately.

Maintenance policy based on product quality control

Following these assumptions, a new maintenance policy is designed based on product quality control. First, the equipment is inspected every $\tau$ hours. Simultaneously, n samples are taken and product quality is inspected every h hours $(h<\tau )$ . The np-chart is used to identify any abnormal shift of product quality by monitoring the number of nonconforming units in a sample. If there is an alert signal from the control chart, the equipment is inspected immediately. For either periodic or alert inspections, if the equipment deterioration level x exceeds the preventative maintenance threshold $D_p$ , preventive maintenance (PM) is performed; however, no maintenance action is taken if $0⩽x<D_p$ .

In addition, CM is performed immediately if the equipment undergoes an unexpected breakdown.

In the proposed policy, maintenance actions are assumed to include PM and CM. After maintenance, the equipment state will return to ‘good as new’.

According to this maintenance policy, Figure 1 shows three different scenarios in production.

Figure 1.

Three scenarios of the maintenance model.

Scenario 1

The equipment is inspected every $\tau$ hours if the control chart generates no alarm signal within $\tau$ hours. During the periodic equipment inspection, the corresponding maintenance action is taken. PM is performed if the equipment deterioration x exceeds the preventative maintenance threshold $D_p$ ; however, no maintenance action is taken if $0⩽x<D_p$ .

Scenario 2

If the $i\mathrm{th}$ product quality inspection result shows that the product quality has become abnormal before the time to perform periodic maintenance, the equipment is inspected immediately. If the equipment deterioration level x exceeds the preventative maintenance threshold $D_p$ , PM is performed; however, no maintenance action is taken if $0⩽x<D_p$ .

Scenario 3

If the equipment suddenly breaks down during the manufacturing process, CM is implemented immediately.

Optimal maintenance model based on product quality control

According to the above assumptions and maintenance policy, minimizing the average cost rate is made the aim of the maintenance model. Therefore, the objective function is

$$minE(P)=min\frac{E(C)}{E(T)}$$

(1)

The value $\tau$ is chosen to be an equipment inspection interval and h is chosen to be a quality sampling interval. The relationship $\tau =\mathrm{kh}$ is established by letting k represent the number of quality samples within an equipment inspection interval. Because the maintenance actions bring the equipment back to ‘good as new’, the maintenance can be modelled as a renewal process. Using renewal reward theory, the long-run average cost per unit time can be expressed as the expected cost per cycle divided by the expected time of a cycle. The expected average cost per unit time is

$$E(P)=\underset{t\to +\infty }{lim}\frac{E(C(t))}{t}=\frac{E(C(\tau ))}{E(T(\tau ))}$$

(2)

The expected cycle cost and cycle cost of each scenario are expressed as follows.

Scenario 1 $(S_1)$

In Scenario 1, the control chart shows that product quality has not shifted to an abnormal state in $\tau$ hours of production. The precondition of Scenario 1 is that the equipment does not experience sudden failure within $\tau$ hours. The inspection error of the control chart includes type I and type II errors. Type I error refers to the control chart signalling a false alarm when the product quality is in the normal state. Type II error refers to the control chart signalling no alarm despite an abnormal shift of product quality. Although the control chart does not detect any abnormal signal, the state of real product quality exists in two possible scenarios ( $S_{11}$ and $S_{12}$ ) taking into consideration the type I error of the control chart:

• 1.  $S_{11}$ : The product quality is in the normal state within the interval between two equipment inspections and the control chart has no alert signal.

In $S_{11}$ , its average cost and average time are

$$E(C|S_{11})=\mathrm{kn}{C}_{\mathrm{sam}}+C_{\mathrm{mi}}+C_{\mathrm{pm}}\underset{D_p}{\overset{D_f}{\int }}{f}^{(\tau )}(x)\mathrm{dx}$$

(3)

$$E(T|S_{11})=\tau$$

(4)

In view of the precondition of Scenario 1, the probability of this scenario is

$$\begin{array}{c}\multicolumn{1}{c}{P(S_{11})=\{[1-G(\mathrm{kh})]-\sum _{i=1}^k{(1-\alpha )}^{i-1}\alpha [1-G(\mathrm{ih})]\}}\\ \multicolumn{1}{c}{\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\end{array}$$

(5)

• 2.  $S_{12}$ : The product quality shifts to an abnormal state within the interval between two equipment inspections, but the control chart has no alert signal.

In $S_{12}$ , a misdetection occurs in the control chart and the shift of product quality is not identified. Its average cost and average time are

$$E(C|S_{12})=\mathrm{kn}{C}_{\mathrm{sam}}+C_{\mathrm{mi}}+C_{\mathrm{pm}}\underset{D_p}{\overset{D_f}{\int }}{f}^{(\tau )}(x)\mathrm{dx}$$

(6)

$$E(T|S_{12})=\tau$$

(7)

The probability of this scenario is

$$P(S_{12})=[G(kh)-\sum _{i=1}^k\sum _{j=1}^i{(1-\alpha )}^{j-1}(1-{\beta }^{i-j})\beta \underset{(i-1)h}{\overset{\mathrm{ih}}{\int }}g(t)\mathrm{dt}\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}$$

(8)

Scenario 2 $(S_2)$

In Scenario 2, the control chart shows that the abnormal shift of product quality occurs at the $j\mathrm{th}$ product quality inspection, which is before the time for periodic maintenance. Analogous to $S_1$ , the precondition of Scenario 2 is that the equipment does not experience sudden failure within $\tau$ hours. Taking into consideration the type II error of the control chart, the state of the real product quality exists two possible scenarios ( $S_{21}$ and $S_{22}$ ):

• 1.  $S_{21}$ : The product quality is in an abnormal state within the interval between equipment inspections and an alert signal occurs in the control chart. In $S_{21}$ , its average cost and average time are

$$E(C|S_{21})=\mathrm{in}{C}_{\mathrm{sam}}+C_{\mathrm{mi}}+C_{\mathrm{pm}}\underset{D_p}{\overset{D_f}{\int }}{f}^{(\mathrm{ih})}(x)\mathrm{dx}$$

(9)

$$E(T|S_{21})=\underset{0}{\overset{\mathrm{kh}}{\int }}\mathrm{tg}(t|\mathrm{kh})\mathrm{dt}+\mathrm{hAR}{L}_1-\delta$$

(10)

where $\mathrm{AR}{L}_1$ represents the average run length during which the product quality is in an abnormal state, and

$$\mathrm{AR}{L}_1=\frac{1}{1-\beta }=\frac{1}{1-\sum _{x=0}^d\left(\begin{array}{c}n\hfill \\ \multicolumn{1}{c}{x}\end{array}\right)p_1^x{(1-p_1)}^{n-x}}$$

(11)

and $\delta$ is the duration between the time when the abnormal shift of product occurs and when the control chart monitors this shift, so

$$\delta =\sum _{i=1}^k\underset{(i-1)h}{\overset{\mathrm{ih}}{\int }}(t-\mathrm{ih})g(t|\mathrm{kh})\mathrm{dt}$$

(12)

The probability of this scenario is

$$\begin{array}{c}\multicolumn{1}{c}{P(S_{21})=[\sum _{i=1}^k\sum _{j=1}^i{(1-\alpha )}^{j-1}(1-{\beta }^{i-j})\beta \underset{(i-1)h}{\overset{\mathrm{ih}}{\int }}g(t)\mathrm{dt}]}\\ \multicolumn{1}{c}{\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\end{array}$$

(13)

• 2.  $S_{22}$ : The product quality is normal within the interval between equipment inspections period, but there is an alert signal in the control chart. In $S_{22}$ , there is a false alarm signal in the control chart. Its average cost and average time are respectively

$$E(C|S_{22})=\mathrm{in}{C}_{\mathrm{sam}}+C_{\mathrm{mi}}$$

(14)

$$E(T|S_{22})=\mathrm{ih}$$

(15)

The probability of this scenario is

$$P(S_{22})=\{\sum _{i=1}^k{(1-\alpha )}^{i-1}\alpha [1-G(\mathrm{ih})]\}\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}$$

(16)

Scenario 3

In Scenario 3, the equipment has stopped because of a sudden breakdown. The probability of this scenario is

$$P(S_3)=\underset{0}{\overset{\tau }{\int }}\underset{D_f}{\overset{+\infty }{\int }}{f}^{(t)}(x)\mathrm{dxdt}$$

(17)

Its average cost and time are respectively

$$E(C|S_3)=C_{\mathrm{cm}}$$

(18)

$$E(T|S_3)=t$$

(19)

According to the cost analysis of all scenarios, the total average cost and total average time are

$$\begin{array}{c}\multicolumn{1}{c}{E(C(\tau ))=E(C|S_1)P(S_1)+E(C|S_2)P(S_2)+E(C|S_3)P(S_3)}\\ \multicolumn{1}{c}{=E(C|S_{11})P(S_{11})+E(C|S_{12})P(S_{12})+E(C|S_{21})P(S_{21})+E(C|S_{22})P(S_{22})+E(C|S_3)P(S_3)}\\ \multicolumn{1}{c}{=[\mathrm{kn}{C}_{\mathrm{sam}}+C_{\mathrm{mi}}+C_{\mathrm{pm}}\underset{D_p}{\overset{D_f}{\int }}{f}^{(\tau )}(x)\mathrm{dx}]}\\ \multicolumn{1}{c}{\{[1-G(\mathrm{kh})]-\sum _{i=1}^k{(1-\alpha )}^{i-1}\alpha [1-G(\mathrm{ih})]\}\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\\ \multicolumn{1}{c}{+[\mathrm{kn}{C}_{\mathrm{sam}}+C_{\mathrm{mi}}+C_{\mathrm{pm}}\underset{D_p}{\overset{D_f}{\int }}{f}^{(\tau )}(x)\mathrm{dx}]}\\ \multicolumn{1}{c}{[G(\mathrm{kh})-\sum _{i=1}^k\sum _{j=1}^i{(1-\alpha )}^{j-1}(1-{\beta }^{i-j})\beta \underset{(i-1)h}{\overset{\mathrm{ih}}{\int }}g(t)\mathrm{dt}]\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\\ \multicolumn{1}{c}{+\{\sum _{i=1}^k\sum _{j=1}^i[\mathrm{in}{C}_{\mathrm{sam}}+C_{\mathrm{mi}}+C_{\mathrm{pm}}\underset{D_p}{\overset{D_f}{\int }}{f}^{(\mathrm{ih})}(x)\mathrm{dx}]{(1-\alpha )}^{j-1}(1-{\beta }^{i-j})\beta \underset{(i-1)h}{\overset{\mathrm{ih}}{\int }}g(t)\mathrm{dt}\}}\\ \multicolumn{1}{c}{\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}+\{\sum _{i=1}^k(\mathrm{in}{C}_{\mathrm{sam}}+C_{\mathrm{mi}}){(1-\alpha )}^{i-1}\alpha [1-G(\mathrm{ih})]\}\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\\ \multicolumn{1}{c}{+\underset{0}{\overset{\tau }{\int }}{C}_{\mathrm{cm}}\underset{D_f}{\overset{+\infty }{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\end{array}$$

(20)

$$\begin{array}{c}\multicolumn{1}{c}{E(T(\tau ))=E(T|S_1)P(S_1)+E(T|S_2)P(S_2)+E(T|S_3)P(S_3)}\\ \multicolumn{1}{c}{=E(T|S_{11})P(S_{11})+E(T|S_{12})P(S_{12})+E(T|S_{21})P(S_{21})+E(T|S_{22})P(S_{22})+E(T|S_3)P(S_3)}\\ \multicolumn{1}{c}{=\tau \{[1-G(\mathrm{kh})]-\sum _{i=1}^k{(1-\alpha )}^{i-1}\alpha [1-G(\mathrm{ih})]\}\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\\ \multicolumn{1}{c}{+\tau [G(\mathrm{kh})-\sum _{i=1}^k\sum _{j=1}^i{(1-\alpha )}^{j-1}(1-{\beta }^{i-j})\beta \underset{(i-1)h}{\overset{\mathrm{ih}}{\int }}g(t)\mathrm{dt}]\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\\ \multicolumn{1}{c}{+[\underset{0}{\overset{\mathrm{kh}}{\int }}\mathrm{tg}(t|\mathrm{kh})\mathrm{dt}+\mathrm{hAR}{L}_1-\delta ][\sum _{i=1}^k\sum _{j=1}^i{(1-\alpha )}^{j-1}(1-{\beta }^{i-j})\beta \underset{(i-1)h}{\overset{\mathrm{ih}}{\int }}g(t)\mathrm{dt}]}\\ \multicolumn{1}{c}{\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}+\{\sum _{i=1}^k\mathrm{ih}{(1-\alpha )}^{i-1}\alpha [1-G(\mathrm{ih})]\}\underset{0}{\overset{\tau }{\int }}\underset{0}{\overset{D_f}{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\\ \multicolumn{1}{c}{+\underset{0}{\overset{\tau }{\int }}t\underset{D_f}{\overset{+\infty }{\int }}{f}^{(t)}(x)\mathrm{dxdt}}\end{array}$$

(21)

The goal of the maintenance policy is to minimize the average cost rate. The objective function is shown in equation (2). According to the defined hypothesis, there are five factors influencing the average cost rate: the quality sample size $(n)$ , the quality sampling interval $(h)$ , the acceptable number of defective units $(d)$ , the equipment inspection interval $(\tau )$ , and the PM threshold $(D_p)$ . Thus, these factors $(n,h,d,\tau ,D_p)$ are used as decision variables in the proposed model.

Simulation and analysis

Deterioration is regarded as a time-dependent stochastic process. The deterioration process is assumed to evolve as a gamma stochastic process. Therefore, the random increment follows a gamma probability density, ¹⁶ with shape parameter v and scale parameter u

$$\mathrm{Ga}(\mathrm{\Delta }x|v,u)=\frac{u^v}{\mathrm{\Gamma }(v)}\mathrm{\Delta }{x}^{v-1}·\exp (-u\mathrm{\Delta }x)I_{(0,+\infty )}(\mathrm{\Delta }x)$$

(22)

where $\mathrm{\Gamma }(v)={\int }_0^{+\infty }\mathrm{\Delta }{x}^{v-1}{e}^{-\mathrm{\Delta }x}d\mathrm{\Delta }x$ is the gamma function.

Let T be the time at which the deterioration level of the equipment reaches w; it is defined as $T=inf(t>0:x(t)⩾w)$ . The probability distribution function of reaching the deterioration threshold w at T is

$$H_T(t)=P(T⩽t)=P(x(T)⩾w)=\underset{w}{\overset{+\infty }{\int }}{f}^{(t)}(x)\mathrm{dx}$$

(23)

Using the gamma function, the probability distribution function becomes

$$H_T(t)=\underset{w}{\overset{+\infty }{\int }}{f}^{(t)}(x)\mathrm{dx}=\frac{\mathrm{\Gamma }(\mathrm{vt},\mathrm{uw})}{\mathrm{\Gamma }(\mathrm{vt})}$$

(24)

where $\mathrm{\Gamma }(\mathrm{vt},\mathrm{uw})={\int }_{\mathrm{uw}}^{+\infty }{\gamma }^{\mathrm{vt}-1}{e}^{-\gamma }d\gamma$ .

The number of defective units is assumed to follow a binomial distribution and the np-chart is applied to estimate the shift of product quality. The probability of type I error $\alpha$ is

$$\alpha =1-\sum _{x=0}^d\left(\begin{array}{c}n\hfill \\ \multicolumn{1}{c}{x}\end{array}\right)p_0^x{(1-p_0)}^{n-x}$$

(25)

The probability of type I error β is

$$\beta =1-\sum _{x=0}^d\left(\begin{array}{c}n\hfill \\ \multicolumn{1}{c}{x}\end{array}\right)p_1^x{(1-p_1)}^{n-x}$$

(26)

The time of the product quality shift follows a Weibull distribution, and its probability density function $g(t)$ is

$$g(t)={\lambda }^{\nu }\nu t^{\nu -1}{e}^{{(-\lambda t)}^{\nu }}$$

(27)

The Weibull cumulative distribution function $G(t)$ is

$$G(t)=1-e^{-{(\lambda t)}^{\nu }}$$

(28)

Numerical simulation

To validate the feasibility of the proposed model, the range of the decision variables is given as $5⩽n⩽50$ , $1⩽h⩽4$ , $1⩽d⩽n$ , $5⩽\tau ⩽40$ , and $0⩽D_p⩽D_f$ .

The following parameters are fixed. The shape parameter and the scale parameter are set as $(v,u)=(1,1)$ to calibrate the gamma deterioration process. The CM threshold is set as $D_f=50$ . The time of product quality shift follows a Weibull distribution with $(\lambda ,\upsilon )=(0.05,2)$ . Let the defective rate while product quality is in the normal state $(p_0)$ be 0.01, and the defective rate while product quality is in the abnormal state $(p_1)$ be 0.08. The costs of sampling and charting, equipment inspection, preventative maintenance, and CM are $C_{\mathrm{sam}}=10$ , $C_{\mathrm{mi}}=200$ , $C_{\mathrm{pm}}=1000$ , and $C_{\mathrm{cm}}=5000$ , respectively.

Genetic algorithm was introduced by John Holland at the University of Michigan. Its idea is to simulate the mechanics of natural evolution and genetics. ¹⁷ It is known that genetic algorithm has the characteristic of its potential for parallelization and global optimization. Since genetic algorithm is first proposed as an optimization tool for maintenance scheduling activities by Munoz, it began to be widely applied in maintenance field. Recently, genetic algorithm has been proved to be an effective approach for maintenance optimization problems.^18–20 In this article, genetic algorithm is used to solve the proposed model. Let the population size be 20, the crossover probability $(p_c)$ be 0.7, the mutation probability $(p_m)$ be 0.15, and the maximum number of generations be 200. The algorithm was implemented in MATLAB 6.0 and the tests were run 20 times. The simulation result is shown in Table 1.

Table 1.

Optimal objective function value and the best values of five variables.

		The proposed model
The objective function value		21.3124
The decision variables	$n^{*}$	6
	$h^{*}$	3.9818
	$d^{*}$	2
	${\tau }^{*}$	19.9090
	$D_p^{*}$	49.8502

Table 1 shows the optimal objective function value and the best values of the five variables as produced by genetic algorithm.

Figure 2 shows that the average cost rate decreased as the number of generations increased. The optimal maintenance solution was found by genetic algorithm within 200 generations.

Figure 2.

Convergence of the average cost rate over generations of genetic algorithm.

Figure 3 shows the effects of variable on the average cost rate. Figure 3(a)–(e) suggests that the five decision variables $(n,h,d,\tau ,D_p)$ have non-negligible impact on the optimal cost rate.

Figure 3.

Effects of variables on the average cost rate: (a) effect of n on the average cost rate, (b) effect of h on the average cost rate, (c) effect of d on the average cost rate, (d) effect of τ on the average cost rate and (e) effect of D_p on the average cost rate.

Sensitivity analysis

To evaluate the influence of every cost parameter and the defective rate in the result of the proposed model, sensitivity analysis is listed in Tables 2–6.

Table 2.

Simulation result with different values of $C_{\mathrm{cm}}$ .

No.	$C_{\mathrm{mi}}$	$C_{\mathrm{pm}}$	$C_{\mathrm{cm}}$	$P_0$	$P_1$	n	h	d	$\tau$	$D_p$	$minE(P)$
1	200	1000	1000	0.01	0.08	6	3.9818	2	19.9090	49.8502	21.3124
2	200	1000	5000	0.01	0.08	6	3.9802	2	19.9075	49.6787	21.3147
3	200	1000	10000	0.01	0.08	6	3.9811	2	19.9056	49.4158	21.3154

Table 3.

Simulation result with different values of $C_{\mathrm{pm}}$ .

No.	$C_{\mathrm{mi}}$	$C_{\mathrm{pm}}$	$C_{\mathrm{cm}}$	$P_0$	$P_1$	n	h	d	$\tau$	$D_p$	$minE(P)$
1	200	1000	5000	0.01	0.08	6	3.9802	2	19.9075	49.6787	21.3147
2	200	3000	5000	0.01	0.08	6	3.9828	2	19.9139	49.7053	21.3170
3	200	5000	5000	0.01	0.08	6	3.9807	2	19.9035	49.9068	21.3173

Table 4.

Simulation result with different values of $C_{\mathrm{mi}}$ .

No.	$C_{\mathrm{mi}}$	$C_{\mathrm{pm}}$	$C_{\mathrm{cm}}$	$P_0$	$P_1$	n	h	d	$\tau$	$D_p$	$minE(P)$
1	200	1000	5000	0.01	0.08	6	3.9802	2	19.9075	49.6787	21.3147
2	500	1000	5000	0.01	0.08	6	3.9945	2	23.9669	49.4472	38.6268
3	1000	1000	5000	0.01	0.08	6	3.9979	2	27.9854	49.5177	58.0948

Table 5.

Simulation result with different values of $P_0$ .

No.	$C_{\mathrm{mi}}$	$C_{\mathrm{pm}}$	$C_{\mathrm{cm}}$	$P_0$	$P_1$	n	h	d	$\tau$	$D_p$	$minE(P)$
1	200	1000	5000	0.01	0.08	6	3.9802	2	19.9075	49.6787	21.3147
2	200	1000	5000	0.05	0.08	6	3.9806	2	19.9032	49.3296	20.6450
3	200	1000	5000	0.08	0.08	9	3.9977	2	27.9839	49.5660	19.1969

Table 6.

Simulation result with different values of $P_1$ .

No.	$C_{\mathrm{mi}}$	$C_{\mathrm{pm}}$	$C_{\mathrm{cm}}$	$P_0$	$P_1$	n	h	d	$\tau$	$D_p$	$minE(P)$
1	200	1000	5000	0.01	0.01	6	3.9814	2	19.9070	49.9625	21.6162
2	200	1000	5000	0.01	0.03	6	3.9812	2	19.9111	49.9591	21.5771
3	200	1000	5000	0.01	0.08	6	3.9802	2	19.9075	49.6787	21.5747

Table 2 shows that with the increase in the CM cost, the optimal PM threshold will decrease and the optimal equipment inspection interval will shorten. This implies that if the CM cost increases, the PM threshold should be reduced or the equipment inspection interval should be reduced. This method is helpful to reduce the probability of equipment failure.

From Table 3, the PM threshold will increase with an increase in the PM cost. This indicates that if the PM cost increases, the PM threshold should be increased to reduce the frequency of equipment maintenance.

In Table 4, the equipment inspection interval will lengthen with an increase in the equipment inspection cost. This indicates that if the equipment inspection cost increases, longer the equipment inspection interval should be extended to reduce the frequency of equipment inspection.

From Table 5, the quality inspection interval will lengthen with an increase in $P_0$ . Because $P_0$ is related to the probability of a false alarm, the quality inspection interval should be extended to improve the accuracy of the control chart if $P_0$ increases.

Table 6 shows that the quality inspection interval will shorten with an increase in $P_1$ . Because $P_0$ is related to the probability of misdetection, the quality inspection interval should be shortened to reduce the probability of misdetection in the control chart if $P_1$ increases.

Application example

The proposed model was applied to an automotive part manufacturing company. This company produces plastic car parts, such as auto clips and plastic rivet. They are produced by an injection moulding machine, which operates continuously. In the manufacturing processes, n samples were taken and product quality was inspected every h hours. The deterioration of the injection moulding machine occurred over time and its state was inspected periodically. As the major component, the screw was usually used to determine the deterioration state of the injection moulding machine. ²¹ The normal clearance between the screw and barrel of this injection moulding machine was limited to the range $(0.15,0.3)$ . The screw should be replaced if the clearance between the screw and barrel exceeds 0.3. Thus, the CM threshold $D_f$ is set at 0.3. As the clearance between the screw and barrel increased, screw wear became increasingly serious. Therefore, the wear behaviour of the screw was assumed to be described by a Gamma process with scale parameter $v=0.001$ and shape parameter $u=1.5$ . Because the wear of the screw could influence product quality, set the parameters of the Weibull distribution $(\lambda ,\upsilon )$ be $(0.005,0.8)$ according to the wear process of the machine. The defective rate while product quality was in the normal state $p_0$ is set at 0.01, and the defective rate while product quality was in the abnormal state $p_1$ is set at 0.03 from the records of machine runs. Likewise, the relevant costs are set at $C_{\mathrm{sam}}=0.001$ , $C_{\mathrm{mi}}=10$ , $C_{\mathrm{pm}}=1000$ , and $C_{\mathrm{cm}}=3000$ . The following limits were imposed on the decision variables: $50⩽n⩽400$ , $1⩽h⩽24$ , $1⩽d⩽n$ , $2⩽\tau ⩽7$ , and $0.15⩽D_p⩽D_f$ , which were determined by the normal clearance between the screw and barrel.

With the above parameters and using genetic algorithm to solve the proposed model, the minimum average cost rate was 0.3376. In the actual production environment, 300 samples were taken and product quality was inspected every $12\mathrm{h}$ . The clearance between the screw and barrel was checked every 3 months. Using these operational parameters, the minimum average cost rate is 0.7563. For comparison to the actual production process, the optimum solution is $(n^{*},h^{*},d^{*},{\tau }^{*},D_p^{*})=(382,23.8139,1,5,0.2946)$ with the proposed model. The optimal solution obtained by the proposed model is significantly better than the solution for the actual production process. The results may provide guidance for the manager to redesign the inspection plan to control product quality and make maintenance decisions.

Conclusion

In the manufacturing process, PM is necessary to reduce equipment failure and improve the utilization rate of equipment. According to the correlation between product quality and maintenance, a maintenance policy based on product quality control is proposed in this article. Integrated with the theory of SPC and equipment maintenance, np-chart is used to monitor the shift of product quality in this maintenance policy. The appropriate maintenance action is then decided by combining the monitoring result of product quality and the inspection result of the equipment state. Thus, an optimal maintenance model based on product quality control is proposed using renewal reward theory. This model is solved by genetic algorithm. The simulation analysis of the proposed maintenance model shows that the model is helpful for guiding the operations manager in making suitable maintenance decision. The proposed maintenance policy and model can assist both researchers and machine manufacturers in finding the optimum solution for improving production quality and machine utilization. This research is suitable for application to large production processes using deteriorating single-equipment deteriorating systems. In the future, owing to the increasing complexity of production systems, this research will be extended to the integrated approach of multivariate SPC and maintenance decisions for degrading multi-component systems.