Bundle pricing of inventories with stochastic demand

We consider a retailer selling a fixed inventory of two perishable products over a finite horizon. The products are sold individually or as part of a bundle. The customers arrive following a Poisson Process and each customer makes a purchasing decision based on her reservation prices of individual products: she either buys one of the individual products, buys the bundle or leaves without a purchase. Assuming that the customer reservation prices follow a bivariate distribution, we determine the optimal product and the bundle prices that maximize the expected revenue. The performances of three bundling strategies (mixed bundling, pure bundling and unbundling) under different reservation price distributions, demand arrival rates and starting inventory levels are compared. The impact of the shape of the reservation prices is also investigated by considering both the bivariate normal and bivariate gamma densities, the latter of which has not been considered before in the literature. Our numerical results based on the bivariate normal reservation prices indicate that the performances of the policies heavily depend on the parameters of the demand process and the initial inventory levels. Bundling is observed to be most effective with negatively correlated reservation prices and when the starting inventory levels are high. When the starting inventory levels of two products are equal, most of the benefits of bundling can be achieved through pure bundling in case of excess supply. However, the mixed bundling strategy clearly dominates the other two when the starting inventory levels are not equal. Our numerical results show that bundling becomes more effective and the expected revenues increase as the products become less substitutable and more complementary. We also observe that the correct modeling of the reservation price distribution is important, and the use of sub-optimal prices resulting from assuming bivariate normal density when in fact the appropriate distribution is bivariate gamma may result in significant losses in the expected revenue especially if the reservation prices are negatively correlated. Finally, the model is extended to allow for price changes during the selling horizon using a dynamic programming formulation. It is shown that offering price bundles mid–season may be a more effective mechanism than changing individual product prices.