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Register a new userAn Exact Method for (Constrained) Assortment Optimization Problems with Product Costs
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We study the problem of optimizing assortment decisions in the presence of product-specific costs when customers choose according to a multinomial logit model. This problem is NP-hard and approximate solutions methods have been proposed in the literature to obtain both primal and dual bounds in a tractable manner. We propose the first exact solution method for this problem and show that provably optimal assortments of instances with up to one thousand products can be found, on average, in about two tenths of a second. In particular, we propose a bounding procedure based on the approximation method of Feldman and Topaloglu (2015a) to provide tight primal and dual bounds at a fraction of their computing times. We show how these bounds can be used to effectively identify an optimal assortment. We also describe how to adapt our approach for handling cardinality constraints on the size of the assortment or space/resource capacity constraints.
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This paper considers a general convex constrained problem setting where functions are not assumed to be differentiable nor Lipschitz continuous. Our motivation is in finding a simple first-order method for solving a wide range of convex optimization problems with minimal requirements. We study the method of weighted dual averages (Nesterov, 2009) in this setting and prove that it is an optimal method.
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In this paper we present a new algorithmic realization of a projection-based scheme for general convex constrained optimization problem. The general idea is to transform the original optimization problem to a sequence of feasibility problems by iteratively constraining the objective function from above until the feasibility problem is inconsistent. For each of the feasibility problems one may apply any of the existing projection methods for solving it. In particular, the scheme allows the use of subgradient projections and does not require exact projections onto the constraints sets as in existing similar methods. We also apply the newly introduced concept of superiorization to optimization formulation and compare its performance to our scheme. We provide some numerical results for convex quadratic test problems as well as for real-life optimization problems coming from medical treatment planning.
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.
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