IEEE TRANSACTIONS ON RELIABILITY, VOL. R-22, NO. 5, DECEMBER 1973 255 Optimal Reliability Design of a System: A New Look KRISHNA BEHARI MISRA, MEMBER, IEEE, AND MILAN D. LJUBOJEVIC, STUDENT MEMBER, IEEE Abstract-The reliability literature offers an abundance of Before tradeoffs can be studied and an optimum configura- methods for the optimal design of systems under some con- tion selected, the system itself must be analyzed. Once the sys- straints. In most of the papers, the problem considered is: tem is modeled, the model can be analyzed by standard pro- given reliabilities of each constituent component and their cedures. The description and analysis of various system models constraint-type data, optimize the system reliability. This can be found in [1-4]. Next, a designer looks for ways of amounts to the assignment of optimal redundancies to each improving the system performance, i.e., in our case, reliability. stage of the system, witlh each component reliability specified. Basically, there are two ways of achieving high system relia- This is a partial optimization of the system reliability. At the bility. One is to develop highly reliable components, the other design stage, a designer has many options, e.g., component reli- is to use redundancy. ability improvement and use of redundancy. A true optimal Several authors [5-15] have considered the problem of system design explores these alternatives explicitly. Our paper optimizing system reliability with specified component relia- demonstrates the feasibility of arriving at an optimal system bilities (usually high), under given constraints. This is a partial design using the latter concept. For simplicity, only a cost optimization as it overlooks the fact that it is often very ex- constraint is used, however, the approach is more general and pensive to produce highly reliable components. The cost of can be extended to any number of constraints. A particular duplicating a component is often less than that of manufactur- cost-reliability curve is used to illustrate the approach. ing a component with reliability equal to that of the duplicated Reader Aids: subsystem. This is because the cost of producing a component Purpose: Widen state of the art increases exponentially with its reliability, and in the limiting Special math needed for explanations: Probability, Lagrange case [16] the cost of producing a 100% reliable component is multipliers infinite. For example, if a subsystem has a reliability of 0.95, Special math needed for results: Same one can obtain a reliability of 0.9975 just at twice the parts Results useful to: Theoreticjly inclined reliability engineers cost of one, by duplicating the system; if the other constraints are not violated. But the cost of producing a single subsystem I. INTRODUCTION or component of 0.9975 reliability would be enormous. There- A general system consists of many subsystems which in turn fore while designing a system, the optimization of system consist of many components. Breakdown or malfunction of reliability within the available resources can be done, consider- any one of these may cause or lead to the breakdown or failure ing both component reliability and the number of redundancies of the entire system. This in turn may result in heavy loss of as variables. money and time. Therefore, improvement of systemreliabi We demonstrate that if resources are to be best utilized one msoneyn andetimed.Thrfr,irvmnof should start with the assumption of some relationship between iS often desired. Each subsystem is considered to be essential for the overall the component-reliability and its cost and then arrive at some operational success of the mission of the system (i.e., the sys- optimal redundancy level. We use an exponential cost-reliabil- tem is partitioned into subsystems so that this is so). Mathe- ity relation for each component; the coefficients will be differ- matically this can be stated that the subsystems are operation- ent for each component. We have also assumed that there are ally in series in the system model. Evaluation of mission re- no other constraints (weight, volume, etc.) however, these can quirements is required if the necessary criteria are to be formu- be easily incorporated in the problem. The weight or volume lated for analyzing alternatives and evaluating tradeoffs. With- of a subsystem may not be related to its reliability in the same out specific knowledge of the mission requirements, realistic way as its cost. decisions on redundancy, design changes, and other aspects of Therefore, the form of constraints may be different for cost, reliability improvement can not be reached. weight, or volume. Where the weight or volume of a subsystem does not increase appreciably with its reliability the constraints would involve only redundancy as variables. Manuscript received August 7, 1972; revised February 26, 1973. K. B. Mi-sr is wihh Deatmn of Electrical~Enieeig Uni- Our aim iS to show that the system design iS not optimal versity of Roorkee, Roorkee, U.P., 1ndia, on leave at the Laboratorium without the consideration of basic cost-reliability data for the fur Reakforregelung und Anlagensicherung, Technischen Universitat subsystems or components. The design engineer should take Munchen, Reaktorstation, West Germany. M. D. Ljubojevic is with the Process Computer Group at the this into account before building a reliable system with given Mathematical Institute, Technical University of Munich, West Germany. constraints.