(657g) Industrial-Scale Selective Maintenance Optimization Using Bathtub-Shaped Failure Rates

Chemical plants should be ideally robust and reliable in continuous operation. Unexpected component failures at the plant may cause costly disruptions to the operation. In order to avoid disruptions, the operators of the plant schedule major shutdowns, enabling maintenance operations to be conducted for the components (e.g., electrical drives, pumps and fans) of the plant. As these shutdowns are expensive, both in terms of direct maintenance costs and lost production time, the maintenance operations that are performed during a shutdown should be carefully selected. Such decision-making is challenging because a modern industrial plant may consist of hundreds - or even thousands - of individual components with various levels of criticality.

Selective maintenance, first introduced by Rice et al. [1], aims at finding the optimal subset maintenance actions to be performed for a multicomponent system. The objective is to maximize the reliability of the system for the next operation window, subject to maintenance duration and/or cost constraints, or vice versa. Selective maintenance has been applied to various fields, ranging from aircraft maintenance (in between flight missions) to maintenance shutdowns of industrial plants [2-6]. The connecting factor in these applications is that the system has predefined operating windows, and maintenance actions can only be conducted in between the windows.

The component lifetimes in selective maintenance literature are commonly assumed to follow either the exponential or Weibull distributions [7]. In the case of the former, the underlying assumption is that the failure rates are constant. Thus, only corrective maintenance actions are sensible; the replacement of a functioning component would have no influence on the system reliability. The Weibull distribution, on the other hand, can be used to describe components with increasing, constant or decreasing failure rates. However, the distribution is not suitable for modeling non-monotone failure rates. In reality, many engineering components have a non-monotone bathtub-shaped failure rate, i.e. a combination of decreasing infant mortality rate, constant random failure rate and an increasing degradation. A wide range of parametric distributions have been proposed in the literature to model bathtub shaped failure rates (see, for example, references [8-11]).

In a recent review paper, Cao et al. [7] stress the lack of data-driven approaches in selective maintenance literature. For the operators of the plant, the starting point for selective maintenance is typically some, perhaps limited dataset of component lifetimes. However, in the corresponding literature, the aspect of data availability is often omitted, and the starting point is typically defined as a given lifetime distribution with arbitrarily chosen parameters. Therefore, we link the statistical analysis of component lifetime data to the selective maintenance. As the starting point, we use two open-source lifetime datasets with bathtub-shaped failure rate distributions [12-13].

When considering only a single maintenance break, the selective maintenance decision-making can be formulated as a mixed-integer nonlinear programming (MINLP) problem. The mathematical expression of the reliability of a serial-parallel system involves products of decision variables, which typically results in a non-convex MINLP problem. Recently, Ye et al. [14] presented a convexified form of the reliability algebraic term. However, instead of selective maintenance, their work considered the reliability design of a new chemical plant. The convexified model is guaranteed to find the global optimum with a non-global MINLP solver. The authors showed that the solution time of their convexified model, using the non-global solver DICOPT [15], was around half of that of the nonconvex model, using the global solver BARON [16], for an example problem containing 42 binary variables.

In order to improve the efficiency of selective maintenance optimization for industrial-scale problems, while still guaranteeing the optimality of the solution, we present two, concurrently applicable, improvements to the efficiency of MINLP based optimization (our target problem size is 500 system components). First, we modify the aforementioned convexification of the reliability expression by Ye et al. [14] into our selective maintenance optimization model. We formulate the convexified reliability expressions for the following maintenance options: 1) only replacement, and 2) replacement or minimal repair. Second, our statistical analysis shows that the component-specific reliability is reduced if the age of the component and the next planned operation window are within certain limits. This reduction is caused by the infant mortality period of new components. We preclude component replacements in such cases by variable pre-assignments, which reduces the size of the decision space.

References:

[1] Rice, W. F., Cassady, C. R., & Nachlas, J. A. (1998). Optimal maintenance plans under limited maintenance time. In Proceedings of the seventh industrial engineering research conference, 1–3.

[2] Cassady, C. R., Murdock Jr, W. P., & Pohl, E. A. (2001a). Selective maintenance for support equipment involving multiple maintenance actions. European Journal of Operational Research, 129, 252–258.

[3] Cassady, C. R., Pohl, E. A., & Paul M., W. (2001b). Selective maintenance modeling for industrial systems. Journal of Quality in Maintenance Engineering, 7, 104–117.

[4] Rajagopalan, R., & Cassady, C. R. (2006). An improved selective maintenance solution approach. Journal of Quality in Maintenance Engineering, 12, 172–185.

[5] Lust, T., Roux, O., & Riane, F. (2009). Exact and heuristic methods for the selective maintenance problem. European Journal of Operational Research, 197, 1166–1177.

[6] Amaran, S., Sahinidis, N. V., Sharda, B., Morrison, M., Bury, S. J., Miller, S., & Wassick, J. M. (2015). Long-term turnaround planning for integrated chemical sites. Computers & Chemical Engineering, 72, 145–158.

[7] Cao, W., Jia, X., Hu, Q., Zhao, J., & Wu, Y. (2018). A literature review on selective maintenance for multi-unit systems. Quality and Reliability Engineering International, 34, 824–845.

[8] Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated weibull family for analyzing bathtub failure-rate data. IEEE transactions on reliability, 42, 299–302.

[9] Xie, M., Tang, Y., & Goh, T. N. (2002). A modified weibull extension with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 76, 279–285.

[10] Sarhan, A. M., & Apaloo, J. (2013). Exponentiated modified weibull extension distribution. Reliability Engineering & System Safety, 112, 137–144.

[11] Jiang, R. (2013). A new bathtub curve model with a finite support. Reliability Engineering & System Safety, 119, 44–51.

[12] Wang, F. K. (2000). A new model with bathtub-shaped failure rate using an additive burr xii distribution. Reliability Engineering & System Safety, 70, 305–312.

[13] Meeker, W. Q., & Escobar, L. A. (1998). Statistical methods for reliability data. John Wiley & Sons.

[14] Ye, Y., Grossmann, I. E., & Pinto, J. M. (2018). Mixed-integer nonlinear programming models for optimal design of reliable chemical plants. Computers & Chemical Engineering, 116, 3–16.

[15] Viswanathan, J., & Grossmann, I. E. (1990). A combined penalty function and outer-approximation method for MINLP optimization. Computers & Chemical Engineering, 14, 769–782.

[16] Tawarmalani, M., & Sahinidis, N. V. (2005). A polyhedral branch-and-cut approach to global optimization. Mathematical Programming, 103, 225–249.