Optimal maintenance center inventories for fault‐tolerant repairable systems
Authors: Lawrence Sidney H. , Schaefer Margaret K. 
Published: October 2002

In many military and commercial contexts, complex equipment which is expected to perform very reliably is often designed to be fault‐tolerant, that is, able to function although some of the parts have failed. A popular fault‐tolerant design is the m‐out‐of‐n system, where there are n identical parts, at least m of which must be functional for machine operation. Complex equipment of this type often undergoes scheduled maintenance overhauls at regular intervals during which all failed components are replaced. Failure to have replacements on hand for failed parts requires emergency measures at premium cost. When repairable parts are highly reliable and expensive, both holding and shortage costs are high. A reasonable objective is to choose initial spares inventory to minimize the sum of holding costs and expected shortage costs.

We first develop a model to determine the optimal repairable parts inventory for a maintenance center servicing machines containing a single m‐out‐of‐n system. The model is then extended to handle a related problem, finding optimal maintenance center inventories for machines containing several m‐out‐of‐n systems of different parts, minimizing total expected costs subject to a constraint on total inventory investment.

We assume that there is a fleet of machines, which experience identical workloads. There is a cycle time of T days between overhauls for an individual machine. A machine arrives at the maintenance center for overhaul each day. At the overhaul, all failed parts are removed and sent to a repair shop, from which they eventually return to the maintenance center to be used again as spares. The total number of spares undergoing repair and on hand is a constant. There are no backorders; if the number on‐hand spares is insufficient to meet demand at an overhaul, a shortage penalty is assessed which depends on the number and type of spares required.

While computing holding costs is straightforward, computing expected shortage costs is more complex. Expected shortage costs are dependent upon several factors, including component failure rates, the values of m and n, part repair rates, and the initial number of spares on hand. We assume that the system of interest is well specified, so that the parameters of the model are known except for the number of initial spares of each type, which are the decision variables. We model the on‐hand inventory of each type of part as a Markov chain with the number of spares on hand at the end of each day as the states, under the assumptions that failure rates are constant and repair times follow independent exponential distributions. We then calculate the steady‐state probabilities of stockout of various numbers of spares, as a function of the initial spares inventory. The expected shortage costs for a given type of spare may then be calculated by finding the product of the penalty cost for lacking p spares and the probability of lacking p spares and summing over all possible p values.

Solutions to the problem of finding optimal initial inventory level for a machine containing a single m‐out‐of‐n system may be found easily by enumeration. Solutions to the constrained problem where the machine contains several independent m‐out‐of‐n systems, may be found by dynamic programming. Sensitivity analysis of costs to changes in the inventory investment constraint is clear, and computational effort is reasonable. A simple example is included to illustrate the solution method for both problems.