On the complexity of a maintenance problem for hierarchical systems

TL;DR

This paper proves that the Frequency-Constrained Maintenance Jobs ($FCMJ$) problem is integer-factorization hard, even for a very simple hierarchical system with two components in a single module. The authors construct a polynomial-time reduction from integer factorization to a tailored $FCMJ$ instance, using a two-step argument to force a specific cycle time and then extract a nontrivial divisor from the remaining variable. As a corollary, they derive the first hardness result for Levi et al.'s modular maintenance scheduling problem, connecting the difficulty to fundamental number-theoretic structure via the $\mathrm{lcm}$-based module cost. The result implies that, unless integer factorization is in P, there is no general polynomial-time algorithm for $FCMJ$, even in small hierarchical settings, highlighting the necessity of heuristic or problem-specific approaches for practical maintenance planning.

Abstract

We prove that a maintenance problem on frequency-constrained maintenance jobs with a hierarchical structure is integer-factorization hard. This result holds even on simple systems with just two components to maintain. As a corollary, we provide a first hardness result for Levi et al.'s modular maintenance scheduling problem (Naval Research Logistics 61, 472-488, 2014).

Figures (1)

• Figure 1: An example of the FCMJ model. One module $0$ contains two components $1$ and $2$ with cycle limits $f_1=5$ and $f_2=6$. The diagram depicts a feasible schedule where components have cycle times $q_1=4$ and $q_2=6$. For a given module or component $j$, the alternating gray and white areas represent time intervals between maintenances. The setup cost $K_j$ is incurred whenever $j$ is maintained.

Theorems & Definitions (4)

• Theorem 2.1
• Lemma 3.1
• proof
• Theorem 3.1