On a number theoretic property of optimal maintenance grouping

Abstract

In this paper we consider the problem of preventive maintenance of a failure prone system, for which a number of maintenance actions has to be executed on a regular basis. For each action i the frequency is prescribed. Between consecutive actions of type i there is an integer interspacing of T(i) time units. The set-up costs are activity dependent. The set-up structure is supposed to be tree-like and additive over the set-up nodes involved in the action or group of actions. Hence, for different activities with common setup nodes joint execution leads to set-up costs reduction. The question is how the actions should be arranged in time in order to exploit this set-up costs reduction effect maximally. It is shown that the time averaged set-up costs are minimal if a main peak clustering property is satisfied: all maintenance actions are combined at one moment in time. Intuitively, this property is appealing, but it asks for some interesting and non-trivial applications of number theory and inductive reasoning, to prove it