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Author: Zhen Chen
Publisher:
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Languages : en
Pages : 0
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Book Description
Assortment optimization is a fundamental problem in revenue management, in which the objective usually is to select a subset of products to offer to customers in order to maximize expected revenue or profit. However, business practices often involve multiple, and potentially conflicting goals. In this work, we propose a general framework and a novel reformulation method for solving multi-objective assortment optimization problems. Specifically, we consider problems with a separable sum of multiple convex objective functions on linear combinations of choice probabilities, and we present a reformulation that effectively "linearizes" the problem. We prove that the reformulated problem is equivalent to the original problem and that it leads to a unified solution approach to multi-objective assortment optimization problems in various contexts. We show that the approach encompasses a wide range of operational objectives, such as risk, customer utility, market share, costs with economies of scale, and dualized convex constraints. We first illustrate our approach with the multinomial logit model without any constraints or with allowance for totally unimodular constraints. We further show that our framework leads to tractable solutions under the nested logit model and the Markov chain choice model. Together with large-scale numerical experiments to demonstrate the efficiency and practicality of our methods, we highlight that our work provides a powerful and flexible tool for solving multi-objective assortment problems, which arise frequently in practical revenue management settings.
Author: Zhen Chen
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Languages : en
Pages : 0
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Book Description
Assortment optimization is a fundamental problem in revenue management, in which the objective usually is to select a subset of products to offer to customers in order to maximize expected revenue or profit. However, business practices often involve multiple, and potentially conflicting goals. In this work, we propose a general framework and a novel reformulation method for solving multi-objective assortment optimization problems. Specifically, we consider problems with a separable sum of multiple convex objective functions on linear combinations of choice probabilities, and we present a reformulation that effectively "linearizes" the problem. We prove that the reformulated problem is equivalent to the original problem and that it leads to a unified solution approach to multi-objective assortment optimization problems in various contexts. We show that the approach encompasses a wide range of operational objectives, such as risk, customer utility, market share, costs with economies of scale, and dualized convex constraints. We first illustrate our approach with the multinomial logit model without any constraints or with allowance for totally unimodular constraints. We further show that our framework leads to tractable solutions under the nested logit model and the Markov chain choice model. Together with large-scale numerical experiments to demonstrate the efficiency and practicality of our methods, we highlight that our work provides a powerful and flexible tool for solving multi-objective assortment problems, which arise frequently in practical revenue management settings.
Author: Yan Liu
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Languages : en
Pages : 0
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We analyze the assortment optimization problem faced by a monopolistic firm selling n substitutable products in both store and online channels. In our problem, a subset of products (the assortment) are offered in store while all products are sold online. Compared to purchasing in store, purchasing online would lead to an additional cost to customers in the form of a delivery fee or waiting cost. Interestingly, we find that the introduction of online channel hurts the firm and renders the firm to offer more products in store. However, with capacity constraint in store, the online channel, which enables customers to purchase high profit-margin products that are removed out of the assortment due to capacity limit, might benefit the firm.
Author: Basak Bebitoglu
Publisher:
ISBN:
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Languages : en
Pages : 30
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Book Description
We study the assortment optimization problem in an online setting where a retailer uses multiple distribution centers to fulfill customer orders. Due to space, handling or other constraints, each distribution center can carry up to a pre-specified number of products. It is assumed that each distribution center is primarily responsible for a geographical region whose customers' choice is governed by a separate multinomial logit model. A distribution center can satisfy the demand from other regions, but this incurs an additional shipping cost for the retailer. The problem for the retailer is to determine which products to carry in each of its distribution centers and which products to offer for sale in each region so as to maximize its expected profit (revenue minus the shipping costs). We first show that the problem is NP-complete. We develop a conic quadratic mixed integer programming formulation and suggest a family of valid inequalities to strengthen this formulation. Numerical experiments show that our conic approach, combined with valid inequalities over-perform the mixed integer linear programming formulation and enables us to solve moderately sized instances optimally. We also study the effect of various factors such as the strength of the outside option, capacity constraint and shipping cost on company's profitability and assortment selection. Finally, we study the effect of not allowing cross-shipments or not considering them in assortment decisions and show that these may lead to substantial losses for an online retailer.
Author: Carlos Coello Coello
Publisher: Springer Science & Business Media
ISBN: 0387367977
Category : Computers
Languages : en
Pages : 810
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Book Description
This textbook is a second edition of Evolutionary Algorithms for Solving Multi-Objective Problems, significantly expanded and adapted for the classroom. The various features of multi-objective evolutionary algorithms are presented here in an innovative and student-friendly fashion, incorporating state-of-the-art research. The book disseminates the application of evolutionary algorithm techniques to a variety of practical problems. It contains exhaustive appendices, index and bibliography and links to a complete set of teaching tutorials, exercises and solutions.
Author: Antoine Désir
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Languages : en
Pages : 0
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Assortment optimization is an important problem that arises in many practical applications such as retailing and online advertising. In this problem, the goal is to select a subset of items that maximizes the expected revenue in the presence of (1) the substitution behavior of consumers specified by a choice model, and (2) a potential capacity constraint bounding the total weight of items in the assortment. The latter is a natural constraint arising in many applications. We begin by showing how challenging these two aspects are from an optimization perspective. First, we show that adding a general capacity constraint makes the problem NP-hard even for the simplest choice model, namely the multinomial logit model. Second, we show that even the unconstrained assortment optimization for the mixture of multinomial logit model is hard to approximate within any reasonable factor when the number of mixtures is not constant.In view of these hardness results, we present near-optimal algorithms for the capacity constrained assort- ment optimization problem under a large class of parametric choice models including the mixture of multinomial logit, Markov chain, nested logit and d-level nested logit choice models. In fact, we develop near-optimal algorithms for a general class of capacity constrained optimization problems whose objective function depends on a small number of linear functions. For the mixture of multinomial logit model (resp. Markov chain model), the running time of our algorithm depends exponentially on the number of segments (resp. rank of the transition matrix). Therefore, we get efficient algorithms only for the case of constant number of segments (resp. constant rank). However, in light of our hardness result, any near-optimal algorithm will have a super polynomial dependence on the number of mixtures for the mixture of multinomial logit choice model.
Author: Qingwei Jin
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Languages : en
Pages : 0
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We study assortment optimization problems under multinomial logit choice model with two tree structured consideration set models, i.e., the subtree model and the induced paths model. In each model, there are multiple customer types and each customer type has a different consideration set. A customer of a particular type only purchases product within his consideration set. The tree structure means all products form a tree with each node representing one product and all consideration sets are induced from this tree. In the subtree model, each consideration set consists of products in a subtree and in the induced paths model, each consideration set consists of products on the path from one node to the root. All customers make purchase decisions following the same multinomial logit choice model except that different customer types have different consideration sets. The goal of the assortment optimization is to determine a set of products offered to customers such that the expected revenue is maximized. We consider both unconstrained problem and capacitated problem. We show that these problems are all NP-hard problems and propose a unified framework, which captures the tree structure in both models, to design fully polynomial time approximation schemes (FPTAS) for all these problems. Besides, we identify a special case under the induced paths model, showing that it can be solved in $O(n)$ operations.
Author: Heng Zhang
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Languages : en
Pages : 0
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We consider an assortment optimization problem where the customer may purchase multiple products and possibly more than one unit of each product purchased. We adopt the customer consumption model based on the Multiple-Discrete-Choice (MDC) model proposed by Huh and Li (2021). We identify conditions under which the profit-ordered sets are optimal. Without these conditions, we show that assortment optimization is NP-hard. Furthermore, we prove that a generalization of the profit-ordered sets achieves an approximation guarantee of 1/2. We also present an algorithm that computes an epsilon-optimal solution to the assortment problem in running time polynomial in 1/epsilon and the problem input size, once we impose the mild technical assumption that model parameters are bounded.
Author: Elaheh Fata
Publisher:
ISBN:
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Languages : en
Pages : 62
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Book Description
We consider an assortment optimization problem where a customer chooses a single item from a sequence of sets shown to her, while limited inventories constrain the items offered to customers over time. In the special case where all of the assortments have size one, our problem captures the online stochastic matching with timeouts problem. For this problem, we derive a polynomial-time approximation algorithm which earns at least 1-ln(2-1/e), or 0.51, of the optimum. This improves upon the previous-best approximation ratio of 0.46, and furthermore, we show that it is tight. For the general assortment problem, we establish the first constant-factor approximation ratio of 0.09 for the case that different types of customers value items differently, and an approximation ratio of 0.15 for the case that different customers value each item the same. Our algorithms are based on rounding an LP relaxation for multi-stage assortment optimization, and improve upon previous randomized rounding schemes to derive the tight ratio of 1-ln(2-1/e).
Author: Chenhao Wang
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Languages : en
Pages : 0
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Although assortment optimization has been extensively studied, not much is known about how it is affected by advertising. In this paper, we address this gap by considering a novel joint advertising and assortment optimization problem. To capture the effect of advertising in the context of assortment planning, we assume that one can increase the preference weight of a product by advertising it, and the degree of improvement is decided by the effectiveness of advertising, which could be product-specific, and the amount of advertising efforts allocated to that product. Given budget constraints on advertising, our objective is to find a solution, which is composed of an advertising strategy and an assortment of products, that maximizes the expected revenue. We analyze the structural properties of this problem and derive effective solutions under different settings. If there is no capacity constraint on the number of products displayed to consumers, we show that revenue-ordered assortments still maintain optimality, and we leverage this result to derive an optimal solution. For the cardinality constrained case, it is difficult to solve the optimization problem directly; therefore, we show by relaxation that a near-optimal solution can be found efficiently.
Author: Jacob Feldman
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Languages : en
Pages : 0
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We study assortment optimization problems under a natural variant of the multinomial logit model where the customers are willing to focus only on a certain number of products that provide the largest utilities. In particular, each customer has a rank cutoff, characterizing the number of products that she will focus on during the course of her choice process. Given that we offer a certain assortment of products, the choice process of a customer with rank cutoff k proceeds as follows. The customer associates random utilities with all of the products as well as the no-purchase option. She ignores all alternatives whose utilities are not within the k largest utilities. Among the remaining alternatives, the customer chooses the available alternative that provides the largest utility. Under the assumption that the~utilities follow Gumbel distributions with the same scale parameter, we provide a recursion to compute the choice probabilities. Considering the assortment optimization problem to find the revenue-maximizing assortment of products to offer, we show that the problem is NP-hard and give a polynomial-time approximation scheme. Since the customers ignore the products below their rank cutoffs in our variant of the multinomial logit model, intuitively speaking, our variant captures choosier choice behavior than the standard multinomial logit model. Accordingly, we show that the revenue-maximizing assortment under our variant includes the revenue-maximizing assortment under the standard multinomial logit model, so choosier behavior leads to larger assortments offered to maximize the expected revenue. We conduct computational experiments on both synthetic and real datasets to demonstrate that incorporating rank cutoffs can yield better predictions of customer choices and yield more profitable assortment recommendations.
