Convex Optimization of Centralized Inventory Operations

Given a finite set of outlets with joint normally distributed demands and identical holding and penalty costs, inventory centralization induces a cooperative cost allocation game with nonempty core. It is well known that for this newsvendor inventory setting the expected cost of centralization can be expressed as a constant multiple of the standard deviation of the joint distribution. The lowering of the centralized cost without changing the mean and variance of demand at each outlet corresponds to a semidefinite optimization problem. This paper establishes a closed-form optimal solution of the semidefinite program and a core allocation of the cost at optimality. The issue of cost (and benefit) allocation separate from the optimization is also studied and it is shown that an exponential-size linear program can be approximated by a polynomial-size second-order program.