High Reliability at Minimum Cost

This paper investigates the minimum cost of improving the reliability of complex technical systems. The two major methods to improve reliability are redesigning the system for higher reliability or providing redundant components to replace failed elements. The costs of redesign for reliability or adding redundancy are estimated. The most cost-effective combination for high reliability can be identified.

The cost of increasing the intrinsic reliability of a system can be modeled as cost proportional to 1/(system failure rate)a, where the exponent “a” measures the difficulty of increasing reliability. The “a” exponent can vary from 0.25 to about 2.5. Operational reliability can also be increased by using redundant systems. The failure rate for N parallel redundant units is (system failure rate)N. The cost of redundancy is N times the system cost. The total redundant system cost is proportional to N/(system failure rate)a.

The cost of redundancy increases as N gets larger, but larger N allows a higher system failure rate, which reduces the system design cost. There is a certain N, a certain level of redundancy, that has the minimum cost to achieve the required overall redundant system failure rate. The minimum cost for the redundant system is achieved at the optimum level of redundancy. The N for minimum cost is equal to -a ln (redundant system failure rate). The minimum cost of the N redundant systems is proportional to N * (original system failure rate)a. The optimum redesigned individual system failure rate is proportional to exp (-1/a), so the greater the difficulty, the higher the optimum individual system failure rate. Increasing the intrinsic reliability of a system encounters diminishing returns and at some point it becomes more cost-effective to add redundancy. The difficulty of increasing intrinsic system reliability determines the optimum design for high reliability at minimum cost.