# A New Efficient Method to find a Suboptimal Allocation of Components in a Series-Parallel Structure

### Systems Research Institute, Polish Academy of Sciences, Poland.

#### ABSTRACT

A method of optimizing the allocation of two-state independent components in a series-parallel structure is presented. Such a structure is composed of parallel substructures arranged in series and is considered operable if at least one component in each substructure is operable. The components are assumed to have different failure probabilities, so the system reliability depends on where they are located in the structure. The optimal allocation minimizes the system failure probability or, alternatively, maximizes its reliability. Interestingly enough, while the optimal components allocation problem for a parallel-series structure (i.e. series substructures arranged in parallel) has a well-known simple solution, the same does not hold for a series-parallel one. The considered problem has been investigated by several researchers who proposed quite elaborate solutions. This paper presents a recently developed, simple and efficient procedure finding a nearly optimal allocation, and sometimes the optimal one. The presented approach is based on a theorem specifying a threshold value that cannot be exceeded by a series-parallel system's reliability. Starting from some random allocation and using pairwise interchanges of components between the parallel substructures, the algorithm finds successive allocations that yield system reliabilities oscillating towards the value specified by the above theorem. In this way a suboptimal (or, many a time, optimal) allocation is obtained. An important feature is that the method's accuracy is expressed by the easy-to-compute upper bound of the difference between the optimal reliability and the obtained suboptimal value. The performed tests show that the method allows to find a (sub)optimal solution in a relatively small number of steps. Illustrative examples are given that demonstrate the method's modus operandi.

Keywords: Series-parallel structure, Reliability, Components independence, Pairwise interchange, Optimal/suboptimal components allocation.