Approximation Schemes for Capacity-Constrained Assortment Optimization Under the Nested Logit Model PDF Download
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Author: Danny Segev
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Languages : en
Pages : 31
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The main contribution of this paper resides in proposing a novel approximate dynamic programming approach for capacitated assortment optimization under the Nested Logit model in its utmost generality. Specifically, we show that the optimal revenue can be efficiently approached within any degree of accuracy through purely combinatorial techniques, synthesizing ideas related to continuous dynamic programming, state space discretization, and sensitivity analysis of modified revenue functions. These developments allow us to devise the first fully polynomial-time approximation scheme in this context, thus resolving fundamental open questions posed in previous papers.
Author: Danny Segev
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Languages : en
Pages : 31
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Book Description
The main contribution of this paper resides in proposing a novel approximate dynamic programming approach for capacitated assortment optimization under the Nested Logit model in its utmost generality. Specifically, we show that the optimal revenue can be efficiently approached within any degree of accuracy through purely combinatorial techniques, synthesizing ideas related to continuous dynamic programming, state space discretization, and sensitivity analysis of modified revenue functions. These developments allow us to devise the first fully polynomial-time approximation scheme in this context, thus resolving fundamental open questions posed in previous papers.
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: Sumit Kunnumkal
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Languages : en
Pages : 0
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We consider the assortment optimization problem under the nested logit model and obtain new bounds on the gap between the optimal expected revenue and an upper bound based on a certain continuous relaxation of the assortment problem. Our bounds can be tighter than the existing bounds in the literature and provide more insight into the key drivers of tractability for the assortment optimization problem under the nested logit model. Moreover, our bounds scale with the nest dissimilarity parameters and we recover the well-known tractability results for the assortment optimization problem under the multinomial logit model when all the nest dissimilarity parameters are equal to one. We extend our results to the cardinality constrained assortment problem where there are constraints that limit the number of products that can be offered within each nest.
Author: Laurent Alfandari
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Languages : en
Pages : 39
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We study the problem of finding an optimal assortment of products maximizing the expected revenue, in which customer preferences are modeled using a Nested Logit choice model. This problem is known to be polynomially solvable in a specific case and NP-hard otherwise, with only approximation algorithms existing in the literature. For the NP-hard cases, we provide a general exact method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. Contrary to the existing literature, in which assumptions are imposed on either the structure of nests or the combination and characteristics of products, no assumptions on the input data are imposed, and hence our approach can solve the most general problem setting. We show that the parameterized subproblem of the fractional programming scheme, which is a binary highly non-linear optimization problem, is decomposable by nests, which is a main advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we identify those products that are undoubtedly beneficial to offer, or not, which can significantly reduce the problem size. In the second stage, we design a tailored Branch-and-Bound algorithm with problem-specific upper bounds. Numerical results show that the approach is able to solve assortment instances with up to 5,000 products per nest. The most challenging instances for our approach are those in which the dissimilarity parameters of nests can be either less or greater than one.
Author: Knut Haase
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Languages : en
Pages : 26
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We present the constrained assortment optimization problem under the mixed logit model (MXL) with design options and deterministic customer segments. The rationale is to select a subset of products of a given size and decide on the attributes of each product such that a function of market share is maximized. The customer demand is modeled by MXL. We develop a novel mixed-integer non-linear program and solve it by state-of-the-art generic solvers. To reduce variance in sample average approximation systematic numbers are applied instead of pseudo-random numbers. Our numerical results demonstrate that systematic numbers reduce computational effort by 70%. We solve instances up to 20 customer segments, 100 products each with 50 design options yielding 5,000 product-design combinations, and 500 random realizations in under two minutes. Our approach studies the impact of market position, willingness-to-pay, and bundling strategies on the optimal assortment.
Author: Jacob Feldman
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Languages : en
Pages : 22
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We study the space constrained assortment optimization problem under the paired combinatorial logit choice model. The goal in this problem is to choose a set of products to make available for purchase with the intention of maximizing the expected revenue from each arriving customer. Each offered product occupies a specific amount of space and there is a limit on the space consumed by all of the offered products. The purchasing decision of each customer is governed by the paired combinatorial logit choice model. We provide the first efficient constant factor approximation for this problem.
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.
Author: Ruxian Wang
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Languages : en
Pages : 7
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We consider an assortment and price optimization problem where a retailer chooses an assortment of competing products and determines their prices to maximize the total expected profit subject to a capacity constraint. Customers' purchase behavior follows the multinomial logit choice model with general utility functions. This paper simplifies it to a problem of finding a unique fixed point of a single-dimensional function and visualizes the assortment optimization process. An efficient algorithm to find the optimal assortment and prices is provided.
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: Laurent Alfandari
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Languages : fr
Pages :
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