For single period inventory models with normally distributed, correlated individual demands we examine the problem of minimizing the cost of inventory centralization as a function of the covariance matrix. In a stable centralized setting there are no incentives for any party to break-away -- referred to as nonempty core conditions. For the allocated benefits in inventory centralization, nonempty core conditions are always satisfied. In this paper we discuss a step by step greedy optimization procedure which computes an optimal centralization solution. The procedure manipulates the correlations without changing the mean or the variance at each store. We do not just accept that in the centralized setting the parties are better-off but for the first time provide the analysis of how to maximize their collective benefits.