Replacement policies for a parallel system with shortage and excess costs

• Excess cost, which involves a waste of operation, is introduced.
• Random mean time 1/θ of replacement is taken into consideration for replacement policies.
• Random number N of units is taken into consideration for replacement policies.
• Number N of working times for a parallel system with n units is discussed.
• Number n of available units for a scheduled replacement time T is discussed.

It has been assumed in most maintenance models that (i) the number of units for a parallel system can be predefined precisely, (ii) maintenance cost after failure should be avoided, and (iii) age replacement can always be performed at its optimized times. However, these assumptions are challenged from practical perspectives in this paper. For this purpose, shortage and excess costs, which claim that replacement done too early before failure involves a waste of operation, are introduced into replacement models, such as replacement plans with time T or distribution G(t) for constant n or random N of units. Furthermore, the number N of working times and the number n of available units for a scheduled replacement time T are discussed. For each model, the expected replacement cost for one cycle and expected replacement cost rate for a long run are optimized, and numerical examples are given.