In many cases, redundant systems are beset by both independent and dependent failures. Ignoring dependent variables in MTBF evaluation of redundant systems hastens the occurrence of failure, causing it to take place before the expected time, hence decreasing safety and creating irreversible damages. Common cause failure (CCF) and cascading failure are two varieties of dependent failures, both leading to a considerable decrease in the MTBF of redundant systems. In this paper, the alpha-factor model and the capacity flow model are combined so as to incorporate CCF and cascading failure in the evaluation of MTBF of a 2-out-of-3 repairable redundant system. Then, using a transposed matrix, the MTBF function of the system is determined. Due to the fact that it is difficult to estimate the independent and dependent failure rates, industries are interested in considering uncertain failure rates. Therefore, fuzzy theory is used to incorporate uncertainty into the model presented in this study, and a nonlinear programming model is used to determine system's MTBF. Finally, in order to validate the proposed model, evaluation of MTBF of the redundant system of a centrifugal water pumping system is presented as a practical example.

The nature of electro-optical equipment in various industries and the pursuit of the goal of reducing costs demand high reliability on the part of electro-optical systems. In this respect, reliability improvement could be addressed through a reliability allocation problem. Subsystem reliability must be increased such that the requirements as well as defined requisite functions are ensured in accordance with the designers' opinion. This study is an attempt to develop a multiobjective model by maximizing system reliability and minimizing costs in order to investigate design phase costs as well as production phase costs. To investigate reliability improvement feasibility in the design phase, effective feasibility factors in the system are used and the sigma level index is incorporated in the production phase as the reliability improvement difficulty factor. Thus, subsystem reliability improvement priorities are taken into consideration. Subsystem dependency degree is investigated through the design structure matrix and incorporated into the model's limitation together with modified criticality. The primary model is converted into a single-objective model through goal programming. This model is implemented on electro-optical systems, and the results are analyzed. In this method, reliability allocation follows two steps. First, based on the allocation weights, a range is determined for the reliability of subsystems. Afterward, improvement is initiated based upon the costs and priorities of subsystem reliability improvement. Keywords Reliability allocation Á Design and production cost Á Reliability improvement difficulty Á Sigma level Á Design structure matrix (DSM)