Optimization Procedure for the Analysis of Coherent Structures

The determination of the reliability level at which to manufacture the components of a coherent structure so that the system reliability h(p) is at a certain level and the overall system cost is minimized is considered. The cost of utilizing component ci at reliability level pi, Ci(pi), is assumed to be a convex increasing function of pi with a continuous first derivative and Ci´(qi)>0 where qi is the lower bound on the reliability level for component ci. Since for most coherent structures the constraint set defines a nonconvex set, any mathematical programming procedure blindly applied to the program converges to a local optimum rather than a global optimum. However, in certain cases, the global optimum can be found for the series and parallel (SP) type of systems. The key to the solution is to optimize each module separately and then to substitute a component for each module where the cost function for the component is the value of the objective function for the module. As long as the cost function for each module maintains the convexity property with In R or In(1 - R) as the argument (R being the reliability of the module), the optimization procedure can continue and a global optimum found.