In this paper, we consider a Markov chain choice model with single transition. In this model, customers arrive at each product with a certain probability. If the arrived product is unavailable, then the seller can recommend a subset of available products to the customer and the customer will purchase one of the recommended products or choose not to purchase with certain transition probabilities. The distinguishing features of the model are that the seller can control which products to recommend depending on the arrived product and that each customer either purchases a product or leaves the market after one transition. We study the assortment optimization problem under this model. Particularly, we show that this problem is generally NP-Hard even if each product could only transit to at most two products. Despite the complexity of the problem, we provide polynomial time algorithms for several special cases, such as when the transition probabilities are homogeneous with respect to the starting point, or when each product can only transit to one other product. We also provide a tight performance bound for revenue-ordered assortments. In addition, we propose a compact mixed integer program formulation that can solve this problem of large size. Through extensive numerical experiments, we show that the proposed algorithms can solve the problem efficiently and the obtained assortments could significantly improve the revenue of the seller than under the Markov chain choice model.
In the study of assortment optimization problem, an important class of assortment is called the revenue-ordered assortment, in which the assortment is selected according to the revenue order of each product. Revenue-ordered assortment has been shown to be optimal in many assortment optimization problems or has a worst-case performance guarantee (see, e.g., Talluri2004 , rusmevichiengtong_robust , rusmevichiengtong_mmnl , wang_wang ). Indeed, as shown in @cite_12 , the optimal revenue-ordered assortment achieves a worst-case ratio of the optimal revenue under a class of regular discrete choice models''. In this paper, we show that the revenue-ordered assortment is optimal under a special case of the MCST model, and has a worst-case performance guarantee under the general case. Furthermore, in our numerical experiments, we show that the revenue-ordered assortment generally achieves good performance.
@cite_12
2105562458 2493059808 2420698967 2117923605
We study a class of assortment optimization problems where customers choose among the offered products according to the nested logit model. There is a fixed revenue associated with each product. The objective is to find an assortment of products to offer so as to maximize the expected revenue per customer. We show that the problem is polynomially solvable when the nest dissimilarity parameters of the choice model are less than one and the customers always make a purchase within the selected nest. Relaxing either of these assumptions renders the problem NP-hard. To deal with the NP-hard cases, we develop parsimonious collections of candidate assortments with worst-case performance guarantees. We also formulate a convex program whose optimal objective value is an upper bound on the optimal expected revenue. Thus, we can compare the expected revenue provided by an assortment with the upper bound on the optimal expected revenue to get a feel for the optimality gap of the assortment. By using this approach, our computational experiments test the performance of the parsimonious collections of candidate assortments that we develop. We consider an online assortment optimization problem, where in every round, the retailer offers a K-cardinality subset (assortment) of N substitutable products to a consumer, and observes the response. We model consumer choice behavior using the widely used multinomial logit (MNL) model, and consider the retailer's problem of dynamically learning the model parameters, while optimizing cumulative revenues over the selling horizon T. Formulating this as a variant of a multi-armed bandit problem, we present an algorithm based on the principle of "optimism in the face of uncertainty." A naive MAB formulation would treat each of the N choose K possible assortments as a distinct "arm", leading to regret bounds that are exponential in K. We show that by exploiting the specific characteristics of the MNL model it is possible to design an algorithm with O(√NT) regret, under a mild assumption. We demonstrate that this performance is nearly optimal, by providing a (randomized) instance of this problem on which any online algorithm would incur at least ΩOmega(√NT K) regret. Assortment planning is an important problem that arises in many industries such as retailing and airlines. One of the key challenges in an assortment planning problem is to identify the “right” model for the substitution behavior of customers from the data. Error in model selection can lead to highly suboptimal decisions. In this paper, we consider a Markov chain based choice model and show that it provides a simultaneous approximation for all random utility based discrete choice models including the multinomial logit (MNL), the probit, the nested logit and mixtures of multinomial logit models. In the Markov chain model, substitution from one product to another is modeled as a state transition in the Markov chain. We show that the choice probabilities computed by the Markov chain based model are a good approximation to the true choice probabilities for any random utility based choice model under mild conditions. Moreover, they are exact if the underlying model is a generalized attraction model (GAM) of which the MNL model is a special case. We also show that the assortment optimization problem for our choice model can be solved efficiently in polynomial time. In addition to the theoretical bounds, we also conduct numerical experiments and observe that the average maximum relative error of the choice probabilities of our model with respect to the true probabilities for any offer set is less than 3 where the average is taken over different offer sets. Therefore, our model provides a tractable approach to choice modeling and assortment optimization that is robust to model selection errors. Moreover, the state transition primitive for substitution provides interesting insights to model the substitution behavior in many real-world applications. We consider an assortment optimization problem where a retailer chooses an assortment of products that maximizes the profit subject to a capacity constraint. The demand is represented by a multinomial logit choice model. We consider both the static and dynamic optimization problems. In the static problem, we assume that the parameters of the logit model are known in advance; we then develop a simple algorithm for computing a profit-maximizing assortment based on the geometry of lines in the plane and derive structural properties of the optimal assortment. For the dynamic problem, the parameters of the logit model are unknown and must be estimated from data. By exploiting the structural properties found for the static problem, we develop an adaptive policy that learns the unknown parameters from past data and at the same time optimizes the profit. Numerical experiments based on sales data from an online retailer indicate that our policy performs well.