Transshipment of Inventories: Dual Allocations vs. Transshipment Prices

Abstract

We study a newsvendor game with transshipments, in which n retailers face a stochastic demand for an identical product. Before the demand is realized, each retailer independently orders her initial inventory. After the demand is realized, the retailers select an optimal transshipment pattern and ship residual inventories in order to meet residual demands. Unsold inventories are salvaged at the end of the period. We compare two methods for distribution of residual profit -- transshipment prices (TP) and dual allocations (DA) -- that were previously analyzed in literature. Transshipment prices are selected a priori, before the demand is known, while dual allocations, which are obtained by calculating the dual prices for the transshipment problem, are calculated ex post, after observing the true demand. We first study the conditions for the existence of the Nash equilibria under DA, and then compare the performance of the two methods and show that neither allocation method dominates the other. Our analysis suggests that DA may yield higher efficiency among ``more asymmetric'' retailers, while TP works better with retailers who are alike'', but the difference in profits does not seem significant. We also link expected dual prices to TP, and use those results to develop heuristics for TP with more than two symmetric retailers. For general instances with more than two asymmetric retailers, we propose a TP agreement which uses a neutral central depot to coordinate the transshipments (TPND). While DA in general outperform TPND in our numerical simulations, its ease of implementation makes TPND an attractive alternative to DA when the efficiency losses are not significant (e.g., high critical fractiles or lower demand variances).