Tractable Profit Maximization over Multiple Attributes under Discrete Choice Models

A fundamental problem in revenue management is to optimally choose the attributes of products, such that the total profit or revenue or market share is maximized. Usually, these attributes can affect both a product's market share (probability to be chosen) and its profit margin. For example, if a smart phone has a better battery, then it is more costly to be produced, but is more likely to be purchased by a customer. The decision maker then needs to choose an optimal vector of attributes for each product that balances this trade-off. In spite of the importance of such problems, there is not yet a method to solve it efficiently in general. Past literature in revenue management and discrete choice models focus on pricing problems, where price is the only attribute to be chosen for each product. Existing approaches to solve pricing problems tractably cannot be generalized to the optimization problem with multiple product attributes as decision variables. On the other hand, papers studying product line design with multiple attributes all result in intractable optimization problems. Then we found a way to reformulate the static multi-attribute optimization problem, as well as the multi-stage fluid optimization problem with both resource constraints and upper and lower bounds of attributes, as a tractable convex conic optimization problem. Our result applies to optimization problems under the multinomial logit (MNL) model, the Markov chain (MC) choice model, and with certain conditions, the nested logit (NL) model.