Augmented Lagrangian Approach to the Newsvendor Model with Component Commonality

Component commonality is a well-known approach in manufacturing, where the same components are used for multiple products. It has been implemented by many established companies such as Airbus, Kodak, Toyota, etc. We consider a standard two-product inventory model with a common component. The demands for the products are independent random variables. Instead of the usual approach to minimize the total shortage quantity, we propose to minimize the total shortage cost. The resulting problem is a non-convex nonlinear mathematical program. We illustrate the use of a primal-dual proximal method to solve this problem by obtaining numerically the optimal allocations of components. In particular, we show that a higher unit shortage cost induces a higher allocation.

A numerical solution to Monge’s problem with a Finsler distance as cost

Monge’s problem with a Finsler cost is intimately related to an optimal ow problem. Discretization of this problem and its dual leads to a well-posed finite-dimensional saddle-point problem which can be solved numerically relatively easily by an augmented Lagrangian approach in the same spirit as the Benamou–Brenier method for the optimal transport problem with quadratic cost. Numerical results validate the method. We also emphasize that the algorithm only requires elementary operations and in particular never involves evaluation of the Finsler distance or of geodesics.