Technical Note—Capacitated Assortment Optimization: Hardness and Approximation

Désir, Antoine; Goyal, Vineet; Zhang, Jiawei

doi:10.1287/opre.2021.2142

Cited by 52 publications

(13 citation statements)

References 25 publications

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“…Finally, discrete choice models have been a fundamental building block in many research topics in revenue management Fisher 2007, Bernstein et al 2015), and it continues to be an active area of research. In particular, the assortment optimization problem was addressed under the LC-MNL model (Méndez-Díaz et al 2014, Rusmevichientong et al 2014, Désir et al 2022,…”

Section: Overview Of the Discrete Choice Modelsmentioning

confidence: 99%

A Nonparametric Stochastic Set Model: Identification, Optimization, and Prediction

Chen¹,

Митрофанов²

2023

Preprint

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The identification of choice models is crucial for understanding consumer behavior and informing marketing or operational strategies, policy design, and product development. The identification of parametric choicebased demand models is typically straightforward. However, nonparametric models, which are highly effective and flexible in explaining customer choice, may encounter the challenge of the dimensionality curse, hindering their identification. A prominent example of a nonparametric model is the ranking-based model, which mirrors the random utility maximization (RUM) class and is known to be nonidentifiable from the collection of choice probabilities alone. Our objective in this paper is to develop a new class of nonparametric models that is not subject to the problem of nonidentifiability. Our model assumes bounded rationality of consumers, which results in symmetric demand cannibalization and intriguingly enables full identification. Additionally, our choice model demonstrates competitive prediction accuracy compared to the state-of-the-art benchmarks in a real-world case study, despite incorporating the assumption of bounded rationality which could, in theory, limit the representation power of our model.In addition, we tackle the important problem of finding the optimal assortment under the proposed choice model. We demonstrate the NP-hardness of this problem and provide a fully polynomial-time approximation scheme through dynamic programming. Additionally, we propose an efficient estimation framework using a combination of column generation and expectation-maximization algorithms, which proves to be more tractable than the estimation algorithm of the aforementioned ranking-based model.

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Section: Overview Of the Discrete Choice Modelsmentioning

confidence: 99%

A Nonparametric Stochastic Set Model: Identification, Optimization, and Prediction

Chen¹,

Митрофанов²

2023

Preprint

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“…Since all RUM can be approximated by the LC-MNL our results imply that over that class of discrete choice models, the bound R ≤ mR * is arbitrarily close for all m ≤ n. Since the MNL is a regular model, the bound for the previous section applies, so we conclude that for all RUMs, the bound R ≤ min(m, n)R * is arbitrarily close. It is worth to mention that similarly to the TAOP, the RAOP under the LC-MNL is NP-hard [Désir et al, 2020] 1 .…”

Section: Bounds For Models That Satisfy the Monotone Utility Propertymentioning

confidence: 99%

“…In their study of the TAOP under the LC-MNL,Désir et al [2020] considered its continuous relaxation and showed that there is no efficient algorithm that can achieve an approximation factor guarantee of at least O( 1 m 1−δ ) for any constant δ > 0 unless N P ⊂ BP P .…”

mentioning

confidence: 99%

The Refined Assortment Optimization Problem

Berbeglia,

Flores,

Gallego

2021

Preprint

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We introduce the refined assortment optimization problem where a firm may decide to make some of its products harder to get instead of making them unavailable as in the traditional assortment optimization problem. Airlines, for example, offer fares with severe restrictions rather than making them unavailable. This is a more subtle way of handling the trade-off between demand induction and demand cannibalization. For the latent class MNL model, a firm that engages in refined assortment optimization can make up to min(n, m) times more than one that insists on traditional assortment optimization, where n is the number of products and m the number of customer types. Surprisingly, the revenue-ordered assortment heuristic has the same performance guarantees relative to personalized refined assortment optimization as it does to traditional assortment optimization. Based on this finding, we construct refinements of the revenue-order heuristic and measure their improved performance relative to the revenue-ordered assortment and the optimal traditional assortment optimization problem. We also provide tight bounds on the ratio of the expected revenues for the refined versus the traditional assortment optimization for some well known discrete choice models.

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“…v k ′ j µ LR (j, S * )p j using bounds from ( 5) and (7). We have where δ 2 := ǫα max i∈[n] { h, u i }.…”

Section: Fptas For the Generalized Markov Chain Model With Low Rank M...mentioning

confidence: 99%

“…+ α p = 1), and v k ∈ Q n + for all k ∈ [p] denote the MNL parameters for segment k. However, even for a mixture of MNL model with two segments, Rusmevichientong et al [17] show that the assortment optimization problem is NP-hard. Moreover, Désir et al [7] show that it is hard to approximate within a factor better than Ω(n 1−ǫ ) in general. So the mixture of MNL model is quite intractable.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Markov Chain Model to Capture Dynamic Preferences and Choice Overload

Goutam,

Goyal,

Soret

2019

Preprint

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Assortment optimization is an important problem that arises in many practical applications such as retailing and online advertising where the goal is to find a subset of products from a universe of substitutable products that maximize a seller's expected revenue. The demand and the revenue depend on the substitution behavior of the customers that is captured by a choice model. One of the key challenges is to find the right model for the customer substitution behavior. Many parametric random utility based models have been considered in the literature to capture substitution. However, in all these models, the probability of purchase increases as we add more options to the assortment. This is not true in general and in many settings, the probability of purchase may decrease if we add more products to the assortment, referred to as the choice overload. In this paper we attempt to address these serious limitations and propose a generalization of the Markov chain based choice model considered in Blanchet et al. [2]. In particular, we handle dynamic preferences and the choice overload phenomenon using a Markovian comparison model that is a generalization of the Markovian substitution framework of [2]. The Markovian comparison framework allows us to implicitly model the search cost in the choice process and thereby, modeling both dynamic preferences as well as the choice overload phenomenon. We consider the assortment optimization problem for the cases of our generalized Markov chain model when the underlying Markov chain is rank-1 (this is a generalization of the Multinomial Logit model) and when it has a low rank (this is a generalization of the mixture of MNLs model). We show that the assortment optimization problem under these models is NP-hard and also present a fully polynomial-time approximation scheme (FPTAS) in both these cases.

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Technical Note—Capacitated Assortment Optimization: Hardness and Approximation

Cited by 52 publications

References 25 publications

A Nonparametric Stochastic Set Model: Identification, Optimization, and Prediction

A Nonparametric Stochastic Set Model: Identification, Optimization, and Prediction

The Refined Assortment Optimization Problem

A Generalized Markov Chain Model to Capture Dynamic Preferences and Choice Overload

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