It is well recognized that using the hot standby redundancy provides fast restoration in the case of failures. However the redundant elements are exposed to working stresses before they are used, which reduces the overall system reliability. Moreover, the cost of maintaining the hot redundant elements in the operational state is usually much greater than the cost of keeping them in the cold standby mode. Therefore, there exists a tradeoff between the cost of losses associated with the restoration delays and the operation cost of standby elements. Such a trade-off can be obtained by designing both hot and cold redundancy types into the same system. Thus a new optimization problem arises for the standby system design. The problem, referred to in this work as optimal standby element distributing and sequencing problem (SE-DSP) is to distribute a fixed set of elements between cold and hot standby groups and select the element initiation sequence so as to minimize the expected mission operation cost of the system while providing a desired level of system reliability. This paper first formulates and solves the SE-DSP problem for 1-out-of-N: G heterogeneous non-repairable standby systems. A numerical method is proposed for evaluating the system reliability and expected mission cost simultaneously. This method is based on discrete approximation of time-to-failure distributions of the system elements. A genetic algorithm is used as an optimization tool for solving the formulated optimization problem. Examples are given to illustrate the considered problem and the proposed solution methodology. number of elements in the system number of HS elements index of the element initiated after i −1 failures r.v. representing the time-to-failure of element j scale and shape parameters of element j with Weibull failure distribution probability that element j fails in time interval i after its initiation cost (per time unit) of keeping element j in hot standby (or operation) mode cost (per time unit) of keeping element j in cold standby mode startup cost of cold standby element j startup cost of hot standby element j mission time system reliability expected mission cost random cumulative working time of elements s(1), s(2), …, s(i): X i = min τ , ∑ j = 1 i T s ( j ) ) Pr{X j =Δi} number of considered time intervals during the mission duration of each time interval total cost of using HS element j failed or switched off at time t total cost of using CS element j initiated at time t 0 and failed or switched off at time t cumulative distribution function probability density function probability mass function random variable genetic algorithm hot standby cold standby standby element distributing and sequencing problem 
Cold vs. hot standby mission operation cost minimization for 1-out-of-N systems