A condition-based maintenance policy for multi-component systems with a high maintenance setup cost

Condition-based maintenance (CBM) is becoming increasingly important due to the development of advanced sensor and information technology, which facilitates the remote collection of condition data. We propose a new CBM policy for multi-component systems with continuous stochastic deteriorations. To reduce the high setup cost of maintenance, a joint maintenance interval is introduced. With this interval and the control limits of components as decision variables, we develop a model for the minimization of the average long-run maintenance cost rate of the systems. Moreover, a numerical study of a production system consisting of a large number of non-identical components is presented, including a sensitivity analysis. Finally, our policy is compared to a failure-based policy and an age-based policy, in order to evaluate the cost-saving potential.

Authors and Affiliations

1. Integration Engineering, BHS Business Unit, Vanderlande Industries B.V., Vanderlandelaan 2, 5466 RB, Veghel, The Netherlands

   Qiushi Zhu

2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Zhong Guan Cun east road 55, Haidian district, Beijing, 100190, China

   Hao Peng

3. Department of Industrial Engineering, Eindhoven University of Technology, P.O.Box 513, 5600 MB, Eindhoven, The Netherlands

   Geert-Jan van Houtum

Corresponding author

Correspondence to Hao Peng.

Appendix A: Description of two comparison policies

1.1 Failure-based policy

When the degradation of one component \(X_i(t)\) in the system reaches \(H_i\), a CM action is taken. For each component \(i\in I\), the failure-based policy implies that there is no PM action taken, so that no control limit \(C_i\) is set on the degradation before \(H_i\) is reached (see Fig. 1). Or equivalently, \(C_i=H_i\). The optimization algorithm of our model in Sect. 3.2 remains unchanged in essence. Equations (6), (7), (11), (13) and (12) are derived as follows,

1.2 Age-based policy

Unlike the failure-based policy, PM actions are taken at joint maintenance time point according to a threshold \(A_i\) on the age of component i. It is almost the same as our proposed policy in Sect. 2, except the ages of components are observed, instead of the condition. Notice the assumptions in Sect. 2.2 are also valid. Since PM actions are taken at a joint maintenance time point to save setup costs, the decision variable \(A_i\) should be a multiple of \(\tilde{\tau }\), i.e., \(A_i=k_i \tilde{\tau }\). Hence, if there is no failure before \(A_i\), a PM action will be performed at \(A_i\) which is also a joint maintenance point. Otherwise, if there is a failure, a CM action will be performed at the next closest joint maintenance point, similarly to the maintenance policy proposed in this paper. The optimization algorithm of this age-based policy is also similar to the one proposed in Sect. 3.2, except Eqs. (11)–(13) are derived as follows,

and \(f_{T_{H_i}}(x)\) is the probability density function of the failure time [\(C_i=H_i\) in Eq. (14)]. Notice that the distribution of the failure time is the same as the distribution of the passage time of \(H_i\), because a soft failure occurs when the degradation process crosses the threshold \(H_i\).

Appendix B: The average cost rate of single component over two decision variables \(C_i\) and \(\tau \)

To show how the objective function varies with two decision variables \(C_i\) and \(\tau \), we plot the the average cost rate of component 1, which is a function of both \(C_1\) and \(\tau \), in Fig. 5 as an example.

Average cost rate on component 1 over \(\tau \) and \(C_1\) (left 3D plot; right contour plot)

Full size image

Appendix C: Optimization algorithm

The procedure of the nested enumeration algorithm can be summarized in Algorithm 1.

Notice different grid sizes can be used for optimizing \(C_i\) and \(\tau \) , which will also affect the computational duration. In this paper, we use the grid size \(H_i/500\) and \(M_{\tau }/500\) for \(C_i\) and \(\tau \) respectively. The upper bound \(M_{\tau }\) is a very large value (at least larger than \(\max _{i\in I}\{G_i\}\).). In this paper, we choose \(M_{\tau }=300 \) days.

To determine the grid sizes for \(C_i\) and \(\tau \) on \(H_i\) and \(M_{\tau }\), we suggest the decision makers first to select a large grid size, e.g., \(H_i/100\) and \(M_{\tau }/100\), in order to have a brief overview of the objective function. Then if the company is sensitive to the cost difference between different sub-optimal solutions incurred by the grid sizes, a smaller grid size can be set for searching the optimal \(C_i\) and \(\tau \). Notice that while changing the grid sizes, we should also observe the changes of the objective function. If the objective function is not sensitive to the changes of the grid sizes, we can stop further decreasing the grid sizes.

Appendix D: Computation performance

Instead of optimizing \(C_i(\tau )\) and \(\tau \) simultaneously, we used a nested approach. Namely, we (i) optimize \(C_i(\tau )\) for each component under a given \(\tau \) and then (ii) optimize \(\tau \) for the system. The motivation of such a decomposition is to reduce the computation time of large-scale problems. When the amount of components in a system is large, the solution space of decision variables increases dramatically, also known as “curse-of-dimensionality”.

For example, a system consisting of two components (\(i\in \{1,2\}\)) is considered in our optimization model. For each component, we have to optimize the \(C_i(\tau ) \in (0, H)\). Suppose we discretise the degradation range (0, H) into 10 grids with a grid size H / 10. The size of the solution space \((C_1,C_2)\) at a given \(\tau \) value is \(10^2\). In the case of this two-component system, it is plausible to optimize \(\tau \) and \(C_i\) simultaneously. However, if a system consists of 50 components, then the size of its solution space will be \(10^{50}\) under each given \(\tau \), which is nearly impossible to solve within a short period. Therefore, it is not efficient to optimize \(\tau \) and \(C_i\) simultaneously. To solve such a large-scale problem within a reasonable computation time, we propose a nested approach to decompose the problem at system level into component level (see Sect. 3.2). This approach will reduce the solution space to \(10\times 50\) under a given \(\tau \). Regarding the numerical example in Sect. 4, the code is built in MATLAB with the runtime of \(4.6\times 10^3\) seconds (by a computer with a 2.5 GHz processor and 4 G RAM).

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