Assortment Planning Under the Multinomial Logit Model with Totally Unimodular Constraint Structures

We consider constrained assortment problems assuming that customers select according to the multinomial logit model (MNL). The objective is to find an assortment that maximizes the expected revenue per customer and satisfies a set of totally unimodular constraints. We show that this fractional binary problem can be solved as an equivalent linear program. We use this result to solve five classes of practical assortment optimization and pricing models under MNL, including (1) assortment models with various bounds on the cardinality of the assortment, (2) assortment models where we need to decide the display location of the selected products, (3) pricing models with a finite menu of possible prices, (4) quality consistent pricing models where the prices of the products have to follow a specified quality ordering, (5) assortment models with precedence constraints. We show that all of these classes of problems can be solved as linear programs. In some instances, constraints can be combined as long as total unimodularity is preserved. In addition, we show how the results extend to a larger class of attraction choice models that avoid some of the shortcomings of MNL. The problem of selecting assortments to maximize expected profits or expected welfare arises in a variety of industries ranging from transportation and retailing to travel and leisure. There is a growing concern in these industries to find the right set of products given that offering additional products cannibalizes demand for existing products, while the demand lost from excluding products can be partially recaptured among the remaining products. In revenue management, for example, airlines have developed fare admission control policies assuming that demand for different fares are independent. This assumption was tenable when fares were sufficiently differentiated in terms of price and restrictions, but clearly does not hold in the current environment where fares are far more similar and demand substitution and cannibalization are prevalent. For airlines, it is essential to update their pricing and admission control models to include the possibility of demand substitution among fare classes. This is also an important problem in retailing where substitution between products may occur based on prices, product attributes and display locations. Assortment problems often have associated constraints that make them more challenging. In this paper, we consider constrained assortment optimization problems assuming customers choose according to the multinomial logit model (MNL) or according to other more general attraction choice models. The objective is to maximize the expected revenue obtained from each customer and the constraints on the offered assortment can be captured by a totally unimodular constraint matrix. We formulate this constrained assortment problem as a fractional program with binary decision variables to model the inclusion of products in the assortment. Our main result shows that we can transform this fractional program into a linear program where the integrality constraints can be relaxed because of the totally unimodular nature of the constraints. This result allows us to formulate and solve a variety of practical assortment and pricing problems, the majority of which were not known to be tractable in the literature. Main Contributions. Let N be the product consideration set from which we want to select an assortment S ⊂ N to offer to customers. An assortment can be identified by its incident vector x = {xj : j ∈ N} ∈ {0, 1}|N |, where xj = 1 if j ∈ S and xj = 0 if j / ∈ S. The set of feasible assortments is given by F = {x ∈ {0, 1}|N | : ∑ j∈N aij xj ≤ bi ∀ i ∈ M} for a totally unimodular matrix [aij ]i∈M,j∈N . Given feasible product offer decisions x ∈ F , each customer chooses among the offered products according to MNL. The objective is to choose x ∈ F , a feasible set of products to offer, to maximize the expected revenue obtained from each customer. Although this assortment problem is a fractional program with binary decision variables, our main result shows that it can directly be solved as a linear program. Building on our main result, we show how to solve five classes of practical assortment and pricing problems. First, we work with assortment problems under MNL where there are cardinality constraints on the assortment. Rusmevichientong, Shen and Shmoys (2010) consider assortment problems with a limit on the total number of offered products. Their approach generates candidate assortments and checks the performance of the candidates, whereas we give a direct linear programming formulation. Our approach also extends to more general cardinality constraints. In