Reliability Optimization of a Series-Parallel System

A mathematical model is formulated for optimizing the reliability of a system subject to given linear constraints; the system has several stages in series; each stage has parallel redundancy to improve the reliability. Part I shows a new way to transform the model of constrained optimization to a saddle point problem by using Lagrange multipliers. Conditions are derived for maximizing the reliability function; Newton´s method is used to solve the resulting multidimensional nonlinear algebraic equations. Further modifications are provided to avoid inverting the large Jacobian matrices; therefore this method is practical for large systems. Part II shows how to transform the model of constrained optimization to a multistage decision process and uses the Maximum principle to arrive at the optimal decision. This approach is easy to apply, formulate, and program. The solution can be obtained without fear of nonconvergence (very often experienced with earlier methods) besides providing considerable saving in computer time. Design alternatives can be easily considered.