Michael Krause

Maintenance Policies for Monotonic Multi-State Networks

PhD studentMichael Krause
Research areaPreventive and Condition-Based Maintenance Planning


Almost all technical systems deployed as production assets in industry and service companies are subject to wear and tear in use. In many cases, the reliability, availability, lifetime, value, and capacity of the systems can be
improved through appropriate maintenance measures. Thus, issues of maintenance planning are of major importance in their industrial application and in scientific research. The total cost for the upkeep and repair of the fixed assets of German companies amounts to about 20 billion euro p.a. The marked significance of maintenance applies especially to enterprises in primary industry such as steel mills, where more than a quarter of the personnel is employed in maintenance. The annual maintenance costs for the technical facilities and machines of the German steel industry amount to between 4% and  6% of the total gross tangible fixed assets of the companies. Contemporary facility management conceives of maintenance as a main proactive process, which makes an essential contribution to the total value added of the enterprise. The optimization of maintenance policies enables the reduction of total expenditures and reduces depreciation of the facility through wear and tear. The enhancement of product quality and upgrading of system utilization allow for an improvement on the revenue side.

In the wake of increased customer orientation, automation, and growing complexity of value-added processes, the maintenance of technical systems in these latter years has again come under closer scrutiny both in industry and as the object of scientific research. Since the beginning of the 1960s thousands of treatises on the modeling and solution of maintenance planning problems have been published.

The maintenance of technical systems is to ensure that the functional capability is upheld or, in the event of failure, restored. Monotonic multi-state networks consist of several components (modularity). The maintenance of one component does not impair the performance of the system (monotonicity), and beside the "(perfectly) functioning" and "failed" states still, possibly infinitely many, further states may exist for each of the components (multi-state system).

Exploring the available literature has revealed that none of the treatises on multi-state networks is unreservedly suitable to the optimization of condition-based maintenance policies. Our research aims at closing this gap. We assume that wear and tear of the individual components follow given stochastic processes and that the condition of the system uniquely depends on the states of its components. Moreover, we suppose that a maintenance budget is given and that we are able to specify the opportunity cost for the deterioration of the system performance due to wear. We are searching for a most effective component-specific maintenance policy, which minimizes the cost incurred by performance losses of the system over a finite planning horizon, having to keep within constraints of budget.

Initially, a dynamic programming formulation was developed for the case of multi-state networks with deterministic wear processes of the components. It was demonstrated how for a two-period planning horizon an closed-form solution can be calculated by applying the Bellman recursion method. The nonlinear optimization problem at the first level is solved by exploiting the Karush-Kuhn-Tucker conditions. The general variant of the problem is formulated as a stochastic dynamic program. A method of simulation-based approximate dynamic programming (ADP) with look-up tables was implemented and validated on test instances. When comparing the results to the closed-form solutions it turned out that the ADP method is reliably capable of generating near-optimum solutions for the special case of deterministic wear processes.