# Optimal capacity design under k-out-of-n and consecutive k-out-of-n type probabilistic constraints

## Abstract

We formulate and solve probabilistic constrained stochastic programming problems, where we prescribe lower and upper bounds for k-out-of-n and consecutive-k-out-of-n reliabilities in the form of probabilistic constraints. Problems of this kind arise in many practical situations where stockout in a number of consecutive or a certain fraction of all periods causes irreparable damage. Examples are determination of reservoir capacities used for irrigation or capital reserves of banks. The optimal solution is typically obtained by the solution of a“reliability equation” involving discrete or continuous random variables. The reliability here is approximated by the use of binomial Boolean probability bounds. For k-out-of-n reliability, the properties of Gamma distribution are used to approximate the reliability equation. For the consecutive k-out-of-n case, Binomial probability bounds (S1,S2,S3 sharp lower bounds, Hunter’s upper bound and Cherry tree upper bound) are used to create lower and upper bounds for the reliability constraint and therefore to solve the reliability equation. A bi-section algorithm is finally applied to determine the optimal capacity level.