Joint decision of condition-based maintenance and production scheduling for multi-component systems

Abstract

In this study, system maintenance and production scheduling are jointly decided to solve the problems of resource idleness and time cost increase due to system maintenance in the processing of production scheduling. Based on the previous research on the integration of scheduling and maintenance of single machine and multi machine, considering the deterioration and economic dependence of each component in the system, a joint strategy of multi component condition-based maintenance combined with production scheduling is formulated. All the maintenance combinations and probability calculation formulas in the production scheduling process are developed using the deterioration state space partition method, and the probability density function and its numerical solution method are derived. A joint decision model is established to minimize the total weighted expected completion time. Finally, taking the KS5 adjustable multi axis tapping machine as an example, numerical experiments were conducted to verify the correctness of the proposed strategy and the established model.

Introduction

Production scheduling plays an important role in the manufacturing industry. It mainly studies the problem of rationally allocating limited resources to different processing jobs within a certain period of time. ¹ In conventional scheduling, it is often assumed that the system is continuously available. However, in a real production scheduling process, the system deteriorates owing to the influence of the processing object and external environment. When the degradation reaches a certain degree, it leads to system failure and the corresponding production downtime. ² Preventive maintenance (PM) activities can restore system performance and ensure smooth production scheduling. However, frequent PM will increase the total cost and reduce the enterprise profit. Therefore, the conflict between maintenance and scheduling should be effectively resolved, and the concept of conducting appropriate PM in the production scheduling process is attracting increasing research interest.

In recent years, many scholars have studied the joint optimization problem of system maintenance and production scheduling. Duffuaa et al. ³ formulated a hybrid maintenance strategy of PM and corrective maintenance (CM) for a single equipment, and established a joint model aiming at minimizing the total maintenance cost. Cui ⁴ also for a single machine, considering the influence of fault uncertainty, in order to minimize the weighted sum of quality robustness and solution robustness, an integrated model of production scheduling and maintenance is established. Liu et al. ⁵ considered the degradation state and virtual service life of a system; the authors established an integrated model with the minimum total expected cost as the goal. Tambe and Kulkarni ⁶ studied the same joint optimization problem as Liu et al, ⁵ and compared the joint optimization model with the model developed using conventional methods. Abdelrahim and Vizvári ⁷ proposed a branch and bound algorithm to optimize the model. Pan et al. ⁸ introduced the effective life and the remaining life to describe the degradation degree of the machine. Pandey et al. ⁹ and Hadidi et al. ¹⁰ considered the integration of system maintenance and production scheduling, established a comprehensive model of single machine scheduling and maintenance. Wu et al. ¹¹ and Lee and Cehn ¹² used the same type of parallel machine as the research object, and optimized the maximum completion time as the goal to establish an integrated model of parallel scheduling and maintenance. Several studies^13–17 have established an integrated model of production scheduling and maintenance for job shop, flow shops, and flexible job shops.

In the previous researches on the integration of scheduling and maintenance, the system is often regarded as a whole, such as the joint study of maintenance and single-machine, parallel-machine and multi-machine scheduling, and the complex structure of the system is often ignored. However, in the mechanical manufacturing industry, many mechanical equipment contain multiple components, such as a group of drilling machine tools composed of multiple bits in a production workshop. Excessive wear or failure of any drill bit will affect the quality of the product and indirectly affect the smooth progress of the scheduling job. Therefore, it is of great significance to study the deterioration status of each component in the system and arranging appropriate maintenance activities for multiple components.

As a result of technological advancement, condition based maintenance (CBM) is gradually used in practice to arrange maintenance activities according to the degradation state of the system.^18–20 An opportunistic maintenance (OM) strategy based on the actual deterioration state of components was first proposed by Wijnmalen and Hontelez. ²¹ Castanier et al. ²² considered a two-component series system as the research object, assuming that when the degradation state of the component is greater than the predefined fault threshold, the component will not fail immediately but continues to degrade to the next detection or maintenance time. Hosseini et al. ²³ developed a greedy heuristic local search algorithm based on multi-component maintenance scheduling. Zhu et al. ²⁴ proposed a new CBM strategy for multi-component systems with continuous random deterioration. Zhou et al. ²⁵ proposed an opportunistic PM scheduling algorithm based on dynamic programing for multi-component series systems. Zhang and Zeng ²⁶ focused on multi-component systems under continuous conditions, and established a long-term cost rate model for the same multi-component systems.

The remainder of this paper is structured as follows. Section 2 describes the system characteristics, and the joint strategy. Section 3 introduces the joint optimization model. In Section 4, the degradation state space partition of multi-component systems is described, and the probability and probability density of maintenance requirement combination are derived. Section 5 describes the numerical analysis and example verification. Finally, Section 6 presents a summary of the main conclusions of the study and future outlook.

System description

Problem description and assumptions

The research object is an identical multi-component series production system, the deterioration state of which is detectable. The degradation of each component will affect the scheduling of the entire system. Conversely, choosing different maintenance activities and job sequences will also influence the degradation of each component. Therefore, it is particularly important to propose a reasonable joint strategy of maintenance and scheduling. The author compares the research of this paper with the existing research, and summarizes it in Table 1.

Table 1.

A characteristic comparison of the models.

References	Multi-component maintenance decision		Production scheduling decision	Joint decision of production scheduling and system maintenance	Joint decision of maintenance and production scheduling for multi-component system
	TBM	CBM
Castanier et al. 22	✓
Wu et al. 11		✓
Alaswad and Xiang 18		✓
Hong et al. 19		✓
Qi et al. 20		✓
Zhu et al. 24		✓
Meng et al. 17			✓
Kang and Subramaniam 1				✓
Duffuaa et al. 3				✓
Cui 4				✓
Liu et al. 5				✓
This paper		✓			✓

Suppose there are $n$ scheduled jobs to be processed, where the processing time of job $j$ is $p_j$ , the weight is ${\omega }_j$ , $j=1,2,...,n$ . The processing time of the job in position $i$ is $p_{\left[i\right]}$ , the weight is ${\omega }_{\left[i\right]}$ , $i=1,2,...,n$ ; after the ith job is completed, the deterioration state of component $m$ is $z_{\left[i\right]}^m$ . The general failure of each component in the process of scheduled job processing is of the form of soft failure; that is, the current job can still be processed, and the soft failure is rectified after completion of the current job. ²⁴

The system is evaluated after each job is completed, and the corresponding maintenance activities are arranged according to the inspection situation. Before scheduling the first job, component $m$ is in a brand-new state. The degradation increment $\mathrm{\Delta }{z}_{\left[i\right]}^m=z_{\left[i\right]}^m-z_{[i-1]}^m$ is supposed to be nonnegative, stationary, and statistically independent. It follows the stochastic distribution with the probability density function as $f\left(z\right)$ . Consequently, the distribution of increments in $t$ units of time is $f^{\left(t\right)}\left(z\right)$ , where $f^{\left(t\right)}\left(z\right)$ is the tth convolution of $f\left(z\right)$ .

Joint strategy

We set the OM threshold $D_O$ , PM threshold $D_P$ , and CM threshold $D_F$ to satisfy $0<D_O<D_P<D_F$ . A joint strategy for CBM and production scheduling were developed for a multi-component system. The specific strategy is described below.

When the ith scheduled job is completed:

1. If the degradation state $z_{\left[i\right]}^m$ of any component $m$ satisfies $D_F<z_{\left[i\right]}^m<\infty$ , CM is carried out for the component, and the maintenance time is $t_{\mathrm{CM}}$ .

2. If the degradation state $z_{\left[i\right]}^m$ of any component $m$ satisfies $D_P<z_{\left[i\right]}^m<D_F$ , PM is carried out for the component, and the maintenance time is $t_{\mathrm{PM}}$ ;

3. While any component $m$ is undergoing PM or CM, if a component $d(d\ne m)$ deteriorates and satisfies $D_O<z_{\left[i\right]}^d<D_P$ , the component $d$ will simultaneously be maintained opportunistically, and the maintenance time of OM is the same as that of PM.

4. If the deterioration state of any component is lower than $D_P$ , no maintenance will be carried out, and the next scheduled job is continued.

After any component is maintenance, its state is restored to the “as-good-as-new” condition. Considering that the failure of components will cause serious system damage and render maintenance difficult, it is generally considered that $\mathrm{CM}$ takes more time than $\mathrm{PM}$ . Therefore, when multiple components undergo maintenance simultaneously, the total maintenance time is calculated based on the component for which the maintenance takes the longest. With the three-component systems taken as an example, the joint decision diagram of maintenance and scheduling is shown in Figure 1.

Figure 1.

Schematic diagram of joint decision of condition-based maintenance and production scheduling of three-component systems.

In Figure 1, $p_{[1]}~p_{[n]}$ and $C_{[1]}~C_{[n]}$ respectively represent the processing time and completion time of the 1st to the $n\mathrm{th}$ scheduled job.

At time $C_{[1]}$ , the first scheduled job is completed, and the deterioration states of the three components are less than $D_O$ , and no component needs maintenance.

At time $C_{[2]}$ , the deterioration value of components 1 and 3 are in the normal operation area, and the deterioration value of component 2 is in the OM area. However, no other component provides opportunities for maintenance at this time; therefore, no components are under maintenance.

At time $C_{[3]}$ , the deterioration value of component 2 is in the $\mathrm{PM}$ area, so $\mathrm{PM}$ is performed on it. The deterioration values components 1 and 3 are in the $\mathrm{OM}$ area. Since component 2 provides opportunities for maintenance, three components undergo maintenance simultaneously. The process of analysis at other times is similar, and it will not be described too much.

Joint decision model

The scheduling problem studied herein is the “non-preemptive scheduling” problem. ⁵ The decision variable x_ij of the joint decision model is expressed as given in equation (1).

$$x_{\mathrm{ij}}=\{\begin{array}{c}1\mathrm{the}i\mathrm{th}\mathrm{processing}\mathrm{job}\mathrm{is}\mathrm{job}\mathrm{j}\hfill \\ \multicolumn{1}{c}{0\mathrm{else}}\end{array}i,j=1,2,...,n$$

(1)

$$p_{\left[i\right]}=\sum _{j=1}^n{p}_j{x}_{\mathrm{ij}}$$

(2)

$${\omega }_{\left[i\right]}=\sum _{j=1}^n{\omega }_j{x}_{\mathrm{ij}}.$$

(3)

Equation (2) represents the processing time of the job in position $i$ , whereas equation (3) represents the weight of the job in position $i$ . The joint decision model of condition-based maintenance and production scheduling for a multi-component series system is established as follows:

$$min\sum _{i=1}^n{\omega }_{\left[i\right]}E\left(C_{\left[i\right]}\right)$$

(4)

$$\mathrm{s}.\mathrm{t}.\sum _{i=1}^n{x}_{\mathrm{ij}}=1j=1,2,...,n$$

(5)

$$\sum _{j=1}^n{x}_{\mathrm{ij}}=1i=1,2,...,n$$

(6)

$$x_{\mathrm{ij}}=0,1i,j=1,2,...,n$$

(7)

$$0<D_O<D_P<D_F.$$

(8)

Equation (4) represents the optimization objective to minimize the total weighted expected completion time of processing jobs. Equations (5)–(7) represent the single-machine scheduling constraints of the objective function. The single-machine system can only arrange one processed job or maintenance activity simultaneously, and a certain processed job can only be scheduled in one location at a certain time. Equation (8) represents the value range of maintenance decision variables.

$C_a{P}_b{O}_h$ indicates that during a maintenance activity, the number of components for $\mathrm{CM}$ is $a$ , the number of components for $\mathrm{PM}$ is $b$ , the number of components for $\mathrm{OM}$ is $h$ . $E\left(C_{\left[i\right]}\right)$ includes two parts; the first part is the total processing time of job 1 to job $i$ , and the second part is the system maintenance time before the $i\mathrm{th}$ processed job. The specific expression is given in equation (9).

$$E\left(C_{\left[i\right]}\right)=\sum _{k=1}^i\{p_{\left[k\right]}+\sum _{1\le a+b+h\le M}(I_{(a>0)}{t}_{\mathrm{CM}}+I_{(a=0\&b>0)}{t}_{\mathrm{PM}}+I_{[(a>0\mathrm{or}b>0)\&h>0]}{t}_d(D_P-D_O))P_{C_a{P}_b{O}_h\left[k\right]}\}$$

(9)

where $p_{\left[k\right]}$ is the processing time of the $k\mathrm{th}$ job in the job sequence. When there are multiple components requiring simultaneous maintenance, the maintenance time is calculated according to the longest time required; considering this, an indicator function $I_A\left(x\right)$ is introduced. When there are components requiring maintenance, $I$ is taken as 1, that is, $I_{(a>0)}{t}_{\mathrm{CM}}$ is the $\mathrm{CM}$ time of the component, and $I_{(a=0\&b>0)}{t}_{\mathrm{PM}}$ is the $\mathrm{PM}$ time of the component;

As the deterioration state of the components during $\mathrm{OM}$ did not meet the threshold of $\mathrm{PM}$ – that is, the components underwent maintenance in advance. The cost is converted into the time domain to reflect the loss in the objective function, and the time penalty of unit deterioration is $t_d$ . Therefore, $I_{[(a>0\mathrm{or}b>0)\&h>0]}{t}_d(D_P-D_O)$ is the time penalty given.

Degradation state space partition and probability calculation

State space partition and probability

Figure 2 shows the degradation process of component $m$ and the partition of maintenance areas. According to the maintenance threshold line, the deterioration process of components is divided into normal operation area, $\mathrm{OM}$ area, $\mathrm{PM}$ area, and $\mathrm{CM}$ area.

Figure 2.

Schematic diagram of the deterioration process of component $m$ .

Zhang and Zeng ²⁶ proposed a degradation state space partition based on periodic maintenance. The present study also applies this idea and combines it with the production scheduling problem; we propose the degradation state partition and probability calculation formula of a single component, two-component, and three-component system when the $i\mathrm{th}$ scheduled job is completed. To simplify the main text, the process of induction from a single-component system, two-component system, to a three-component system is explained in Appendix A.

In a production system composed of $M$ components, as the maintenance nature of each component is identical, it is only necessary to consider how many components need to undergo maintenance simultaneously, and what type of maintenance is needed. $P_{C_a{P}_b{O}_h\left[i\right]}^M$ represents the probability of maintenance demand combination $C_a{P}_b{O}_h$ in $M$ components when the $i\mathrm{th}$ scheduled job is completed. The probability value is related to the degradation state of the system when the $i\mathrm{th}$ scheduled job is completed. $Z=(z_1,z_2,...,z_M)$ is used to indicate the system state $(z_{\left[i\right]}^1,z_{\left[i\right]}^2,...z_{\left[i\right]}^M)$ when the $i\mathrm{th}$ scheduled job is completed. The probability of the maintenance requirement combination $C_a{P}_b{O}_h$ in $M$ components can be further expressed as $P_{C_a{P}_b{O}_h\left[i\right]}^M\left(Z\right)$ . $U_{s\left[i\right]}(0\le s\le M)$ indicates that $s$ of the $M$ components are awaiting OM when the $i\mathrm{th}$ scheduled job is completed, which satisfies $C_0{P}_0{O}_0=\bigcup _{s=0}^M{U}_{s\left[i\right]}$ .

Therefore, the probability $P_{C_a{P}_b{O}_h\left[i\right]}^M$ calculation of the occurrence of maintenance combination $C_a{P}_b{O}_h$ in a system consisting of $M$ identical components, when the $i\mathrm{th}$ scheduled job is completed, is expressed as in equations (10)–(14).

• (1) When $a+b+h=0$ , no components for maintenance.

$$\begin{array}{c}\multicolumn{1}{c}{P_{C_0{P}_0{O}_0\left[i\right]}^M\left(Z\right)=P_{U_0\left[i\right]}^M\left(Z\right)+P_{U_1\left[i\right]}^M\left(Z\right)}\\ +...+P_{U_{M-1}\left[i\right]}^M\left(Z\right)+P_{U_M\left[i\right]}^M\left(Z\right)\hfill \\ =P_{C_0{P}_0{O}_0\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{U_0\left[i\right]}^1(z_M)\hfill \\ +P_{C_0{P}_0{O}_0\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{U_1\left[i\right]}^1(z_M)\hfill \end{array}$$

(10)

where

$$P_{U_0\left[i\right]}^M(Z)=P_{U_0\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{U_0\left[i\right]}^1(z_M)$$

(11)

$$\begin{array}{c}\multicolumn{1}{c}{P_{U_K\left[i\right]}^M(Z)=P_{U_K\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{U_0\left[i\right]}^1(z_M)}\\ +P_{U_{K-1}\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{U_1\left[i\right]}^1(z_M)\hfill \\ (0<K<M)\hfill \end{array}$$

(12)

$$P_{U_M\left[i\right]}^M(Z)=P_{U_{M-1}\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{U_1\left[i\right]}^1(z_M).$$

(13)

• (2) When $1\le a+b+h\le M$ , maintenance for at least one component is needed.

$$\begin{array}{c}\multicolumn{1}{c}{P_{C_a{P}_b{O}_h\left[i\right]}^M(Z)=I_{(a+b\ge 1,h\ge 0,1\le a+b+h\le M)}{P}_{C_a{P}_b{O}_h\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{U_0\left[i\right]}^1(z_M)}\\ +I_{(a+b\ge 1,h\ge 1,2\le a+b+h\le M)}{P}_{C_a{P}_b{O}_{h-1}\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{U_1\left[i\right]}^1(z_M)\hfill \\ +I_{(b\ge 1,a+b\ge 2,h\ge 0,2\le a+b+h\le M)}{P}_{C_a{P}_{b-1}{O}_h\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{C_0{P}_1{O}_0\left[i\right]}^1(z_M)\hfill \\ +I_{(a=1,b=0,h\ge 0,1\le a+b+h\le M)}{P}_{U_h\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{C_0{P}_1{O}_0\left[i\right]}^1(z_M)\hfill \\ +I_{(a\ge 1,a+b\ge 2,h\ge 0,2\le a+b+h\le M)}{P}_{C_{a-1}{P}_b{O}_h\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{C_1{P}_0{O}_0\left[i\right]}^1(z_M)\hfill \\ +I_{(a=1,b=0,h\ge 0,1\le b+h\le M)}{P}_{U_h\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})*P_{C_0{P}_1{O}_0\left[i\right]}^1(z_M)\hfill \end{array}$$

(14)

Derivation of joint probability density function

The joint probability density function of the system is a product of the probability density function of the degradation state of each component expressed as equation (15).

$$\mathrm{\pi }(z_{\left[i\right]}^1,z_{\left[i\right]}^2,...z_{\left[i\right]}^M)=\mathrm{\Omega }\left(z_{\left[i\right]}^1\right)\mathrm{\Omega }\left(z_{\left[i\right]}^2\right)...\mathrm{\Omega }\left(z_{\left[i\right]}^M\right)$$

(15)

To obtain the probability density function of the system when the $i\mathrm{th}$ scheduled job is completed, it is necessary to consider the possible maintenance requirements of all components when the $(i-1)\mathrm{th}$ scheduled job is completed.

Case 1: The remaining $M-1$ components were not maintenance. This indicates that the degradation state of these components is in the range of $(0,D_P)$ when the (i − 1)th scheduled job is completed, and its maintenance probability is expressed as given in equation (16).

$$\begin{array}{c}{P}_{C_0{P}_0{O}_0\left[i\right]}^{M-1}(z_1,z_2,...,z_{M-1})\hfill \\ ={\int }_0^{D_P}{\int }_0^{D_P}...{\int }_0^{D_P}\mathrm{\pi }(z_{\left[i\right]}^1,z_{\left[i\right]}^2,...z_{\left[i\right]}^{M-1}){\mathrm{dz}}_{\left[i\right]}^1{\mathrm{dz}}_{\left[i\right]}^2...{\mathrm{dz}}_{\left[i\right]}^{M-1}\hfill \end{array}$$

(16)

We assumed that the degradation state of component $m$ is $z_{[i-1]}^m$ before the completion of the (i − 1)th scheduled job (i.e. before the ith scheduled job starts). After the system runs for $P_{\left[i\right]}$ time units, the degradation state of component $m$ reaches $z_{\left[i\right]}^m$ . The possible maintenance situation of component $m$ when the (i − 1)th scheduled job is completed is given below based on the above analysis.

• 1. The probability that component $m$ is not maintenance when the (i − 1)th scheduled job is completed is ${\int }_0^{min(z_{\left[i\right]}^m,D_P)}\mathrm{\Omega }\left(z_{[i-1]}^m\right){\mathrm{dz}}_{[i-1]}^m$ . Therefore, the probability density function is expressed as given in equation (17).

$$\mathrm{\Omega }\left(z_{\left[i\right]}^m\right)={\int }_0^{min(z_{\left[i\right]}^m,D_p)}\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}(z_{\left[i\right]}^m-z_{[i-1]}^m){\mathrm{dz}}_{[i-1]}^m.$$

(17)

• 2. When the (i − 1)th scheduled job is completed, component $m$ is arranged for maintenance, and the maintenance probability is ${\int }_{D_P}^{\infty }\mathrm{\Omega }\left(z_{[i-1]}^m\right){\mathrm{dz}}_{[i-1]}^m$ . After maintenance, the degradation state value returned to 0. Hence, the probability density function is expressed as equation (18).

$$\mathrm{\Omega }\left(z_{\left[i\right]}^m\right)={\int }_{D_P}^{\infty }\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}\left(z_{\left[i\right]}^m\right){\mathrm{dz}}_{[i-1]}^m.$$

(18)

In summary, when the (i − 1)th scheduled job is completed, the remaining $M-1$ components are not maintenance, and the probability density function is expressed in equation (19).

$$\begin{array}{c}{\mathrm{\Omega }}_1\left(z_{\left[i\right]}^m\right)=\hfill \\ {\int }_0^{min(z_{\left[i\right]}^m,D_p)}\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}(z_{\left[i\right]}^m-z_{[i-1]}^m){\mathrm{dz}}_{[i-1]}^m\hfill \\ +{\int }_{D_P}^{\infty }\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}\left(z_{\left[i\right]}^m\right){\mathrm{dz}}_{[i-1]}^m\hfill \end{array}$$

(19)

Case 2: At least one of the remaining $M-1$ components has undergone maintenance. The analysis process is the same as case 1, and the probability density function of component $m$ running for $P_{\left[i\right]}$ time units to reach $z_{\left[i\right]}^m$ is expressed in equation (20).

$$\begin{array}{c}{\mathrm{\Omega }}_2\left(z_{\left[i\right]}^m\right)=\hfill \\ {\int }_0^{min(z_{\left[i\right]}^m,D_O)}\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}(z_{\left[i\right]}^m-z_{[i-1]}^m){\mathrm{dz}}_{[i-1]}^m\hfill \\ +{\int }_{D_O}^{\infty }\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}\left(z_{\left[i\right]}^m\right){\mathrm{dz}}_{[i-1]}^m.\hfill \end{array}$$

(20)

According to the above analysis, the probability density function is expressed as equation (21).

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Omega }\left(z_{\left[i\right]}^m\right)=P_{C_0{P}_0{O}_0\left[i\right]}^{M-1}*{\mathrm{\Omega }}_1\left(z_{\left[i\right]}^m\right)+(1-P_{C_0{P}_0{O}_0\left[i\right]}^{M-1})*{\mathrm{\Omega }}_2\left(z_{\left[i\right]}^m\right)}\\ =P_{C_0{P}_0{O}_0\left[i\right]}^{M-1}*[{\int }_0^{min(z_{\left[i\right]}^m,D_p)}\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}(z_{\left[i\right]}^m-z_{[i-1]}^m){\mathrm{dz}}_{[i-1]}^m+{\int }_{D_P}^{\infty }\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}\left(z_{\left[i\right]}^m\right){\mathrm{dz}}_{[i-1]}^m]+\hfill \\ (1-P_{C_0{P}_0{O}_0\left[i\right]}^{M-1})*[{\int }_0^{min(z_{\left[i\right]}^m,D_O)}\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}(z_{\left[i\right]}^m-z_{[i-1]}^m){\mathrm{dz}}_{[i-1]}^m+{\int }_{D_O}^{\infty }\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}\left(z_{\left[i\right]}^m\right){\mathrm{dz}}_{[i-1]}^m]\hfill \end{array}$$

(21)

Numerical solution

Equation (21) is combined and simplified to obtain equation (22).

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Omega }\left(z_{\left[i\right]}^m\right)=[{\int }_0^{min(z_{\left[i\right]}^m,D_O)}\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}(z_{\left[i\right]}^m-z_{[i-1]}^m){\mathrm{dz}}_{[i-1]}^m+{\int }_{D_O}^{\infty }\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}\left(z_{\left[i\right]}^m\right){\mathrm{dz}}_{[i-1]}^m]}\\ +P_{C_0{P}_0{O}_0\left[i\right]}^{M-1}*[{\int }_{min(z_{\left[i\right]}^m,D_O)}^{min(z_{\left[i\right]}^m,D_p)}\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}(z_{\left[i\right]}^m-z_{[i-1]}^m){\mathrm{dz}}_{[i-1]}^m-{\int }_{D_O}^{D_P}\mathrm{\Omega }\left(z_{[i-1]}^m\right)f^{\left(p_{\left[i\right]}\right)}\left(z_{\left[i\right]}^m\right){\mathrm{dz}}_{[i-1]}^m]\hfill \end{array}$$

(22)

To simplify the expression, let $g\left(x\right)=f^{\left(p_{\left[i\right]}\right)}\left(x\right)$ , and perform a variable substitution for equation (22). Equation (23) can be obtained after further simplification without distinguishing specific components.

$$\begin{array}{c}\mathrm{\Omega }\left(z_{\left[i\right]}\right)=[{\int }_0^{min(z_{\left[i\right]},D_O)}\mathrm{\Omega }\left(z_{[i-1]}\right)g(z_{\left[i\right]}-z_{[i-1]}){\mathrm{dz}}_{[i-1]}+{\int }_{D_O}^{\infty }\mathrm{\Omega }\left(z_{[i-1]}\right)g\left(z_{\left[i\right]}\right){\mathrm{dz}}_{[i-1]}]\hfill \\ +{\left[{\int }_0^{D_P}\mathrm{\Omega }\left(z_{\left[i\right]}\right){\mathrm{dz}}_{\left[i\right]}\right]}^{M-1}*[{\int }_{min(z_{\left[i\right]},D_O)}^{min(z_{\left[i\right]},D_p)}\mathrm{\Omega }\left(z_{[i-1]}\right){\mathrm{dz}}_{[i-1]}g(z_{\left[i\right]}-z_{[i-1]})-{\int }_{D_O}^{D_P}\mathrm{\Omega }\left(z_{[i-1]}\right)g\left(z_{\left[i\right]}\right){\mathrm{dz}}_{[i-1]}]\hfill \end{array}$$

(23)

$${\int }_a^{b}f\left(x\right)\mathrm{dx}=\sum _{s=1}^{n}f\left({\epsilon }_s\right)\mathrm{\Delta }{x}_s$$

(24)

Referring to the solution methods by Fredholm integral formulas, the orthogonal approximation rule of equation (24) is applied to equation (23), and $\infty$ in equation (23) is truncated using $D_{max}$ . The approximate equation can be expressed as given in equation (25).

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Omega }\left(\mathrm{ih}\right)=[h\sum _{j=\mathrm{s}}^r\mathrm{\Omega }\left(\mathrm{jh}\right)g\left(\mathrm{ih}\right)+h\sum _{j=1}^{min(i,s)}\mathrm{\Omega }\left(\mathrm{jh}\right)g(\mathrm{ih}-\mathrm{jh})]}\\ +{\left[h\sum _{i=1}^p\mathrm{\Omega }\left(\mathrm{ih}\right)\right]}^{M-1}.\hfill \\ [h\sum _{min(i,s)}^{min(i,p)}\mathrm{\Omega }\left(\mathrm{jh}\right)g(\mathrm{ih}-\mathrm{jh})-h\sum _{j=\mathrm{s}}^{\mathrm{p}}\mathrm{\Omega }\left(\mathrm{jh}\right)g\left(\mathrm{ih}\right)]\hfill \\ \mathrm{where}x=\mathrm{ih},y=\mathrm{jh},z=\mathrm{kh},D_O=\mathrm{sh},\hfill \\ D_P=\mathrm{ph},D_F=\mathrm{qh},D_{max}=\mathrm{rh}\hfill \end{array}$$

(25)

Rewrite the required probability density function into a matrix form and express it as equation (26).

$$\mathrm{\Omega }=h{G}_1+h{G}_2+{\left[h{G}_3\right]}^{M-1}.(h{G}_4-h{G}_5).$$

(26)

The definition of $G_1,G_2,...$ is presented in Appendix B. At this point, the numerical approximate solution of $\mathrm{\Omega }\left(z_{\left[i\right]}\right)$ can be obtained by solving the linear equations. The probability density function of the system can be calculated using $\mathrm{\pi }(z_{\left[i\right]}^1,z_{\left[i\right]}^2,...z_{\left[i\right]}^M)=\underset{m=1}{\overset{M}{\Pi }}\mathrm{\Omega }\left(z_{\left[i\right]}^m\right)$

Numerical experiments

Let us consider the KS5 type adjustable multi-axis tapping machine as an example; it contains several identical and independent taps with the same distribution. Zhang and Zeng ²⁶ and van Noortwijk ²⁷ mention that Gamma processes are often used to characterize continuous wear processes such as wear, fatigue, and a degrading health index. The degradation of multiple taps in the tapping machine conforms to the degradation process of non-negative increment – stable and monotonically increasing. It can be approximately characterized by the Gamma process, so it is assumed that the degradation increment is a Gamma distribution. Any tap can undergo continuous wear accumulation degradation during the scheduled process. Let the degradation follow a Gamma process with $\alpha =3.5$ and $\beta =0.25$ . Set the CM threshold $D_F$  = 10, the component penalty time given for OM $t_d=1$ , the PM time $t_{\mathrm{PM}}=3$ , the CM time $t_{\mathrm{CM}}=10$ , and the total number of components $M=5$ . The following analysis is performed for the built model in terms of model accuracy, validity, and sensitivity of the corresponding parameters.

A genetic algorithm was used to optimize the joint model. The operation parameters were defined as follows: population size: 25; crossover probability: 0.8; mutation probability: 0.1; generation gap: 0.8; maximum genetic generation: 100; and number of simulations: 20.

Design of genetic algorithm

The design of genetic algorithm coding

The decision variables of production scheduling and maintenance threshold are uniformly coded into one-dimensional fixed length chromosome with growth degree of $n+2$ , where 1 to n bits represent the processing sequence of single machine scheduling, and the last two bits represent preventive maintenance threshold and opportunistic maintenance threshold respectively. The coding diagram is shown in Figure 3:

Figure 3.

Encoding of genetic algorithm.

For the processing sequence, the natural number encoding of $[1,n]$ is used. The number in the encoding represents job $j$ , and the corresponding position of the number represents the position $i$ of job $j$ in the scheduling sequence composed of $n$ jobs. The codes of preventive maintenance threshold and opportunistic maintenance threshold are real numbers in 1 bit $[0,D_F]$ and $[0,D_P]$ respectively.

Design of genetic operators

• (1) Selection operator: select chromosome by roulette.

• (2) Crossover operator:

Create a random number from the interval (0, 1). If the random number is less than the set crossover probability, randomly select two chromosomes and perform the crossover operation.

For the first to nth bits, the single-point crossover method is adopted. For the last two preventive maintenance thresholds and opportunistic maintenance thresholds, arithmetic crossing method is used. For $D_P$ , suppose there are two individuals $D_{\mathrm{PA}\prime }$ and $D_{\mathrm{PB}\prime }$ . After arithmetic crossover, two new individuals are generated as $D_{\mathrm{PA}}$ and $D_{\mathrm{PB}}$ . The crossover process is shown in equation (27).

$$\{\begin{array}{c}{D}_{\mathrm{PA}}=\mu D_{\mathrm{PB}\prime }+(1-\mu )D_{\mathrm{PA}\prime }\hfill \\ \multicolumn{1}{c}{D_{\mathrm{PB}}=\mu D_{\mathrm{PA}\prime }+(1-\mu )D_{\mathrm{PB}\prime }}\end{array}$$

(27)

For $D_O$ , the crossover process is shown in equation (28).

$$\{\begin{array}{c}{D}_{\mathrm{OA}}=\mu D_{\mathrm{OB}\prime }+(1-\mu )D_{\mathrm{OA}\prime }\hfill \\ \multicolumn{1}{c}{D_{\mathrm{OB}}=\mu D_{\mathrm{OA}\prime }+(1-\mu )D_{\mathrm{OB}\prime }}\end{array}$$

(28)

where µ is a constant in the range of (0, 1). The diagram of cross operation is shown in Figure 4.

• (3) Mutation operator:

Figure 4.

The crossover operation of genetic algorithm.

Create a random decimal from the interval (0, 1). If the decimal is lower than the set mutation probability, the current chromosome will be mutated.

Optimization process of genetic algorithm

The algorithm flow is shown in Figure 5:

Figure 5.

Flow chart of genetic algorithm.

Correctness analysis

In this paper, $\mathrm{\Omega }\left(z_{\left[i\right]}^m\right)$ is defined as the probability density function of the deterioration state of component $m$ when the ith scheduled job is completed. Its integral in the entire domain should be 1 based on the normalization of the probability density function. Table 2 presents the calculation results of the probability density function integral $\Psi ={\int }_0^{\infty }\mathrm{\Omega }\left(z_{\left[i\right]}^m\right)d{z}_{\left[i\right]}^m$ for various values of the degradation parameter $\alpha ,\beta$ , and the values of $\Psi$ are close to 1. The instability of the results is mainly caused by errors in the calculation process; however, the overall trend is unchanged.

Table 2.

values corresponding to different degradation parameters.

$\alpha$	$\beta$	$\Psi$
2.5	0.25	0.993561
3.5	0.25	0.994255
4.5	0.25	0.996251
3.5	0.20	0.994230
3.5	0.30	0.994471
3.5	0.40	0.996237

For the multi-component system with $M$ identical multi components considered in this study, the system joint probability density function satisfies $\mathrm{\pi }(z_{\left[i\right]}^1,z_{\left[i\right]}^2,...z_{\left[i\right]}^M)=\mathrm{\Omega }\left(z_{\left[i\right]}^1\right)\mathrm{\Omega }\left(z_{\left[i\right]}^2\right)...$ $\mathrm{\Omega }\left(z_{\left[i\right]}^M\right)$ , and since $\mathrm{\Omega }\left(z_{\left[i\right]}^1\right)=\mathrm{\Omega }\left(z_{\left[i\right]}^2\right)=...=\mathrm{\Omega }\left(z_{\left[i\right]}^M\right)={\int }_0^{\infty }\mathrm{\Omega }\left(z_{\left[i\right]}^m\right)d{z}_{\left[i\right]}^m\approx 1$ , the existence equation (29) holds.

$$\begin{array}{c}{\int }_0^{\infty }...{\int }_0^{\infty }\mathrm{\pi }(z_{\left[i\right]}^1,z_{\left[i\right]}^2,...z_{\left[i\right]}^M){\mathrm{dz}}_{\left[i\right]}^1{\mathrm{dz}}_{\left[i\right]}^2...{\mathrm{dz}}_{\left[i\right]}^M\hfill \\ ={\int }_0^{\infty }...{\int }_0^{\infty }\mathrm{\Omega }\left(z_{\left[i\right]}^1\right)\mathrm{\Omega }\left(z_{\left[i\right]}^2\right)...\mathrm{\Omega }\left(z_{\left[i\right]}^M\right){\mathrm{dz}}_{\left[i\right]}^1{\mathrm{dz}}_{\left[i\right]}^2...{\mathrm{dz}}_{\left[i\right]}^M\hfill \\ \approx 1\hfill \end{array}$$

(29)

That is, the integral of $\mathrm{\pi }(z_{\left[i\right]}^1,z_{\left[i\right]}^2,...z_{\left[i\right]}^M)$ in the entire domain is approximately 1, which satisfies the general properties of the probability density function. Furthermore, it is verified that the probability density function of the deterioration state of component $m$ and its numerical solution derived in this paper are accurate when the ith scheduled job is completed

Effectiveness analysis

In the previous joint research on maintenance and production scheduling, the maintenance strategy combining PM and CM was often used. OM was not considered during the entire scheduling process, and this strategy was defined as a joint strategy 1. The joint strategy of multi-component condition-based maintenance and production scheduling established in this paper is defined as joint strategy 2. The comparison of the two strategies is shown in Table 3.

Table 3.

Comparison of two joint strategies.

Joint strategyState of deterioration	Strategy 1 (predecessors)	Strategy 2 (this article)
$({\mathrm{D}}_{\mathrm{F}},\infty )$	CM	CM
$({\mathrm{D}}_{\mathrm{P}},{\mathrm{D}}_{\mathrm{F}})$	PM	PM
$({\mathrm{D}}_{\mathrm{O}},{\mathrm{D}}_{\mathrm{P}})$	No maintenance	OM
$(0,{\mathrm{D}}_{\mathrm{O}})$	No maintenance	No maintenance

Comparison of results of two joint strategies for the same job scale

In order to compare the advantages and disadvantages of the two strategies and show the effectiveness of the model built in the joint strategy 2 of introducing OM in this paper, the processing time and weights of scheduled jobs that match the degradation parameters in the Gamma distribution are randomly generated as shown in Table 4. Set it to obey the uniform distribution of 1∼7 and 1∼10 respectively, and carry out numerical experiments under the same and different job scales.

Table 4.

Scheduling job processing time and its corresponding weight.

$j$	$p_j$	${\omega }_j$	$j$	$p_j$	${\omega }_j$
1	3.15	5	6	4.54	9
2	6.98	8	7	1.98	4
3	4.14	6	8	5.47	7
4	2.13	4	9	1.99	3
5	6.27	1	10	4.25	9

Taking the processing time and the corresponding weight of the scheduled job in Table 4, the results obtained after the optimization of the two joint strategies are shown in Table 5.

Table 5.

Optimization results corresponding to the two joint strategies.

Joint strategies	Job sequence	$D_{\mathrm{p}}$	$D_{\mathrm{o}}$	Objective
Strategy 1	3-7-9-2-4-10-1-8-6-5	6.98	—	414.36
Strategy 2	9-7-1-5-10-4-6-2-3-8	5.78	3.69	341.25

According to the experimental results in Table 5, the objective obtained by using joint strategy 2 in this paper is 341.25, which is 73.11 lower than the objective of 414.36 obtained by joint strategy 1, and the weighted expected total completion time is significantly reduced, which proves the effectiveness of joint strategy 2.

The optimization process of genetic algorithm based on joint strategy 2 is shown in Figure 6.

Figure 6.

Schematic diagram of genetic algorithm optimization process.

Comparison of results of two joint strategies under different job scales

In order to show that the model established in this paper is still effective under different job scales, the number of components is set as $M=5$ . The comparison results of the two joint strategies under different job scales are shown in Table 6.

Table 6.

Optimization results under different job scales.

Job scale	Objective		Decrease value
	Strategy 1	Strategy 2
30 items	1243.18	998.54	244.64
50 items	2382.47	1921.37	461.10
100 items	3847.24	2920.11	927.13

According to the data in the table, it can be seen intuitively that as the scale of the job gradually increases, the total completion time decreases gradually. When there are 30 scheduled jobs, the optimization result of modeling according to joint strategy 1 is 1243.18, and the optimization result of the model built according to joint strategy 2 under the same conditions is 998.54, which is 244.64 lower than strategy 1. It can be seen that the objective of the optimization model built with joint strategy 2 is better, and when the scale of the job is expanded, the objective reduction effect is more significant.

At the same time, the author gives the trend graph of the optimization time of genetic algorithm to the change of job scale when the number of components is constant, as shown in Figure 7.

Figure 7.

The impact of the change in the number of jobs on the running time.

Comparison of the results of the two joint strategies under different number of components

When the job scale is set as $n=10$ and the number of components is taken as $M=3$ , $M=5$ , and $M=8$ , numerical experiments are carried out under two different joint strategies.

Table 7 shows the objectives corresponding to the two joint strategies under different number of components. When $M=3$ , the objective obtained by using joint strategy 2 compared with joint strategy 1 is reduced to 32.17. When the number of components in the system gradually increases to 5 and 8, the reduction value also increases to 73.11 and 218.27. That is, as the number of components increases, the total weighted expected completion time obtained by using joint strategy 2 is shorter than that of joint strategy 1, and the objective is better, which proves the effectiveness of joint strategy 2 and the model built in this paper.

Table 7.

Optimization results of two joint strategies under different number of components.

Joint strategiesNumber of components	Strategy 1 (predecessors)	Strategy 2 (this article)	Decrease value
3	326.27	294.1	32.17
5	414.36	341.25	73.11
8	724.25	505.98	218.27

The author gives a trend graph of the optimization time of genetic algorithm with the number of components when the job scale is unchanged, as shown in Figure 8.

Figure 8.

The impact of the change in the number of components on the running time.

Sensitivity analysis

In the numerical experiments, different parameter values will have different effects on the results of the objective function. The sensitivity of the model is analyzed from the following aspects: ratio $t_{\mathrm{PM}}/t_{\mathrm{CM}}$ of PM time to CM time, and component degradation speed $\alpha$ .

Influence of the change in $t_{\mathrm{PM}}/t_{\mathrm{CM}}$ on experimental results

Set the job scale $n=10$ , the number of components $M=5$ , the scheduled job processing time, and the corresponding weights are consistent with Table 4. The results are presented in Table 8.

Table 8.

Effect of $t_{\mathrm{PM}}/t_{\mathrm{CM}}$ on optimization results.

$t_{\mathrm{PM}}/t_{\mathrm{CM}}$	$D_P$	$D_O$	Job sequence	Objective
1/10	4.08	2.98	9-10-7-3-4-6-8-2-1-5	347.56
5/10	6.23	4.57	2-3-10-4-6-5-7-9-1-8	412.38
9/10	8.97	5.64	5-1-9-8-2-4-10-3-7-6	436.24

When the ratio of PM time to CM time is 1/10, that is, the time spent on PM is much less than the time of CM, and the obtained PM threshold is small, and a lower PM threshold will increase the probability of PM of components, and then improve the availability of the system.

When the ratio of PM time to CM time increases to 5/10 and 9/10, the objective also increases to 412.38 and 436.24. Especially, when the ratio is 9/10, the time of PM and CM is close, and the PM threshold is increased to 8.97, which reduces the frequency of PM and adjusts the job sequence accordingly, which makes the experimental results better.

Effect of $\alpha$ on experimental results

The parameter $\alpha$ represents the degradation speed of the component. To show that the built model can still be applied to different equipment with different degradation speeds, the values of this parameter are 2.5, 3.5, and 4.5 under the condition that other parameters remain unchanged. The optimization results are presented in Table 9.

Table 9.

Effect of $\alpha$ on optimization results.

$\alpha$	$D_P$	$D_O$	Job sequence	Objective
2.5	5.74	4.23	6-2-3-7-1-4-10-8-9-5	287.24
3.5	5.28	3.84	9-5-7-8-3-6-2-4-10-1	367.28
4.5	4.86	3.09	1-9-8-2-10-3-5-6-4-7	454.79

A comparison of the optimization results corresponding to different values of $\alpha$ indicates that when $\alpha$ changes from 2.5 to 4.5, the objective changes from 287.24 to 454.79, an increase of 167.55, that is, with the increase in $\alpha$ , the component degradation speed increases. The probability of maintenance increases, leading to an increase in the weighted expected completion time.

Conclusion

This study focused on the problem of idle resources and time cost increase caused by system maintenance in the process of production scheduling. We investigated the joint decision-making of condition-based maintenance and production scheduling for multi-component systems. For the multi-component system with economic relevance, an integrated condition-based maintenance strategy combining OM, PM, and CM was adopted. The maintenance combinations of the system in the scheduling process were expressed using the degraded state space partition method, the general probability calculation formula under the corresponding situation was obtained, and the probability density function was derived. A joint optimization model of multi-component condition-based maintenance and production scheduling was established to minimize the total weighted expected completion time. Finally, a numerical analysis of the model was carried out, and the effectiveness of the model was verified by comparing the optimization results of different joint strategies and sensitivity analysis of parameters.

The research presented in this paper is mainly aimed at joint modeling and optimization of condition-based maintenance and production scheduling of the same multi-component system. The strategy can be extended to different multi-component maintenance.