Solving a Relaxed Min-Cost Redundancy Allocation Model using Lagrange Multiplier and Newton’s Method

Redundancy allocation is a valuable technique that system engineers can use to design high level of reliability into complex systems. Broadly however, redundancy allocation problems are NP-hard. The main goal of this paper is to solve a relaxed minimum-cost problem by proving that Newton’s method finds the optimal value of the Lagrange multiplier. The paper first establishes lower and upper bounds on the optimal Lagrange multiplier, and then starting from an initial value determined by he the model’s parameters, Newton’s method finds the optimal value of the Lagrange multiplier. The paper also illustrates the method with two examples and presents a general conclusion.