Revenue Management Under the Markov Chain Choice Model

We consider revenue management problems when customers choose among the offered products according to the Markov chain choice model. In this choice model, a customer arrives into the system to purchase a particular product. If this product is available for purchase, then the customer purchases it. Otherwise, the customer transitions to another product or to the no purchase option, until she reaches an available product or the no purchase option. We consider three classes of problems. First, we study assortment problems, where the goal is to find a set of products to offer so as to maximize the expected revenue from each customer. We give a linear program to obtain the optimal solution. Second, we study single resource revenue management problems, where the goal is to adjust the set of offered products over a selling horizon when the sale of each product consumes the resource. We show how the optimal set of products to offer changes with the remaining resource inventory and establish that the optimal policy can be implemented through protection levels. Third, we study network revenue management problems, where the goal is to adjust the set of offered products over a selling horizon when the sale of each product consumes a combination of resources. A standard linear programming approximation of this problem includes one decision variable for each subset of products. We show that this linear program can be reduced to an equivalent one with a substantially smaller size. We give an algorithm to recover the optimal solution to the original linear program from the reduced linear program. The reduced linear program can dramatically improve the solution times for the original linear program. Incorporating customer choice behavior into revenue management models has been seeing increased attention. Traditional revenue management models assume that each customer arrives into the system with the intention of purchasing a certain product. If this product is available for purchase, then the customer purchases it. Otherwise, the customer leaves without a purchase. In reality, however, there may be multiple products that serve the needs of a customer and a customer may observe the set of available products and make a choice among them. This type of customer choice behavior is even more prevalent today with the common use of online sales channels that conveniently bring a variety of options to customers. When customers choose among the available products, the demand for a particular product naturally depends on what other products are made available to the customers, creating interactions between the demands for the different products. When such interactions exist between the demands for the different products, finding the right set of products to offer to customers can be a challenging task. In this paper, we consider revenue management problems when customers choose among the offered products according to the Markov chain choice model. In the Markov chain choice model, a customer arriving into the system considers purchasing a product with a certain probability. If this product is available for purchase, then the customer purchases it. Otherwise, the customer transitions to another product with a certain probability and considers purchasing the other product, or she transitions to the no purchase option with a certain probability and leaves the system without a purchase. In this way, the customer transitions between the products until she reaches a product available for purchase or she reaches the no purchase option. We consider three fundamental classes of revenue management problems when customers choose under the Markov chain choice model. In particular, we consider assortment optimization problems, revenue management problems with a single resource and revenue management problems over a network of resources. We proceed to describing our contributions to these three classes of problems. Main Contributions. First, we consider assortment optimization problems. In the assortment optimization setting, there is a revenue associated with each product. Customers choose among the offered products according to the Markov chain choice model. The goal is to find a set of products to offer so as to maximize the expected revenue obtained from each customer. We relate the probability of purchasing different products under the Markov chain choice model to the extreme points of a polyhedron (Lemma 1). Although the assortment optimization problem is inherently a combinatorial optimization problem, we use the relationship between the purchase probabilities and the extreme points of a polyhedron to give a linear program that can be used to obtain the optimal set of products to offer (Theorem 2). We show a useful structural property of the optimal assortment, which demonstrates that as the revenues of the products increase by the same amount, the optimal assortment to offer becomes larger (Lemma 3). This property becomes critical when we study the optimal policy for the single resource revenue management problem. Second, we consider revenue management problems with a single resource. In this setting, we need to decide which set of products to make available to customers dynamically over a