Optimal assortments of vertically differentiated products : analytical solution and properties
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This dissertation focuses on three cases of the following two stage problem in the context of multi-product inventories of vertically differentiated products. In Stage 1 of the problem, the manager determines the optimal production quantities of different products when the demands are uncertain. In Stage 2 of the problem, the demands for different products are observed. Now, the manager meets the demand of each product using the inventory of the product or by carrying out a downward substitution from the inventories of higher performance products. The manager’s objective is to maximize the expected revenue from the decisions made at the two stages collectively. The first problem addressed in this dissertation focuses on the case when different products are produced simultaneously on the same set of machines due to random variations in the manufacturing process. These systems, referred to as co-production systems, are very common in the semi- conductor industry, the textile industry and the agriculture industry. For this problem, we provide an analytical solution to the two stage problem, and discuss managerial insights that are specific to co-production systems and are not extendible to multi-item inventories of products that can be ordered or manufactured independently. The second problem addressed in this dissertation focuses on the case when different products can be ordered or manufactured independently, and no constraints to meet minimum fill rate requirements or to restrict the total inventory below a certain level are present. We present an analytical solution to this problem. The third problem addressed in this dissertation focuses on the case when different products can be ordered or manufactured independently and fill rate constraints and total inventory constraints are present. When the demands are multivariate normal, we show that this two stage problem can be reduced to a non-linear program using some new results for the determination of partial expectations. We also extend these results to higher order moments of the multivariate distribution and discuss their applications in solving some common operations management problems.