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Assortment optimization under the Sequential Multinomial Logit Model

https://doi.org/10.1016/j.ejor.2018.08.047Get rights and content

Highlights

  • We studied a choice model that generalizes the Multinomial Logit Model.

  • The choice model is a special case of the perception adjusted Luce model (PALM).

  • We consider the assortment optimization problem under this new choice model.

  • We extend the heuristic of revenue ordered to account for different levels.

  • We prove that this new heuristic is optimal for the choice model studied.

Abstract

We study the assortment optimization problem under the Sequential Multinomial Logit (SML), a discrete choice model that generalizes the Multinomial Logit (MNL). Under the SML model, products are partitioned into two levels, to capture differences in attractiveness, brand awareness and, or visibility of the products in the market. When a consumer is presented with an assortment of products, she first considers products in the first level and, if none of them is purchased, products in the second level are considered. This model is a special case of the Perception-Adjusted Luce Model (PALM) recently proposed by Echenique et al. (2018). It can explain many behavioral phenomena such as the attraction, compromise, similarity effects and choice overload which cannot be explained by the MNL model or any discrete choice model based on random utility. In particular, the SML model allows violations to regularity which states that the probability of choosing a product cannot increase if the offer set is enlarged.

This paper shows that the seminal concept of revenue-ordered assortment sets, which contain an optimal assortment under the MNL model, can be generalized to the SML model. More precisely, the paper proves that all optimal assortments under the SML are revenue-ordered by level, a natural generalization of revenue-ordered assortments that contains, at most, a quadratic number of assortments. As a corollary, assortment optimization under the SML is polynomial-time solvable.

Introduction

The assortment optimization problem is a central problem in revenue management, where a firm wishes to offer a set of products with the goal of maximizing the expected revenue. This problem has many relevant applications in retail and revenue management (Kök, Fisher, & Vaidyanathan, 2005). For example, a publisher might need to decide the set of advertisements to show, an airline must decide which fare classes to offer on each flight, and a retailer needs to decide which products to show in a limited shelf space.

The first consumer demand models studied in revenue management were based on the independent demand principle. This principle stated that customers decide beforehand which product they want to purchase: If the product is available, they make the purchase and, otherwise, they leave without purchasing. In these models, the problem of finding the best offer set of products is computationally simple, but this simplicity comes with an important drawback: These models do not capture the substitution effects between products. That is, they cannot model the fact that, when a consumer cannot find her/his preferred product, she/he may purchase a substitute product. It is well-known that choice models that incorporate substitution effects improve demand predictions (Newman, Ferguson, Garrow, Jacobs, 2014, Ratliff, Venkateshwara Rao, Narayan, Yellepeddi, 2008, van Ryzin, Vulcano, 2015, Talluri, Van Ryzin, 2004, Vulcano, van Ryzin, Chaar, 2010). One of the most celebrated discrete choice models is the Multinomial Logit (MNL) (Luce, 1959, McFadden, 1974). Under the MNL model, the assortment problem admits a polynomial-time algorithm (Talluri & Van Ryzin, 2004). However, the model suffers from the independence of irrelevant alternatives (IIA) property (Ben-Akiva & Lerman, 1985) which says that, when a customer is asked to choose among a set of alternatives S, the ratio between the probability of choosing a product x ∈ S and the probability of choosing y ∈ S does not depend on the set S. In practice, however, the IIA property is often violated. To overcome this limitation, more complex choice models have been proposed in the literature such as the Nested Logit model (Williams, 1977), the latent class MNL (Greene & Hensher, 2003), the Chain model (Blanchet, Gallego, & Goyal, 2016), and the exponomial model (Alptekinoğlu, Semple, 2016, Daganzo, 1979). All these models satisfy the following property: The probability of choosing an alternative does not increase if the offer set is enlarged. Despite the fact that this property (known as regularity) appears natural, it is well-known that it is sometimes violated by individuals (Debreu, 1960, Tversky, 1972a, Tversky, 1972b, Tversky, Simonson, 1993). Recently, there have been efforts to develop discrete choice models that can explain complex choice behaviours such as the violation of regularity, one of the most prominent examples is the perception-adjusted Luce model (PALM) (Echenique, Saito, & Tserenjigmid, 2018).

While the PALM and the nested logit are both conceived as sequential choice processes, they have important differences. Probably the most important difference is that the nested logit model belongs to the family of random utility models (RUM)1, and therefore can not accommodate regularity violations. On the other hand the PALM does not belong to the RUM class, and allows regularity violations as well as choice overload. In terms of the choice process, in the nested logit model customers first select a nest, and then a product within the nest. In the perception-adjusted Luce’s model products are separated by preference levels, so when a customer is offered a set of products, she first chooses among the offered products belonging to the lowest available level, and if none of them are chosen then she selects among the next available level, and keeps repeating this process until no more levels are available or until a purchase is made.

In this paper, we study the assortment optimization problem for a two-stage discrete choice model model that generalizes the classical Multinomial Logit model. This model, which we call the Sequential Multinomial Logit (SML) for brevity, is a special case of the recently proposed model known as the perception-adjusted Luce model (PALM) (Echenique et al., 2018). In the SML model, products are partitioned a priori into two sets, which we call levels. This product segmentation into two levels can capture different degree of attractiveness. For example, it can model customers who check promotions/special offers first before considering the purchase of regular-priced products. It can also model consumer brand awareness, where customers first check products of specifics brands before considering the rest. Finally, the SML can model product visibilities in a market, where products are placed in specific positions (aisles, shelves, web-pages, etc.) that induce a sequential analysis, even when all the products are at sight. Our main contribution is to provide a polynomial-time algorithm for the assortment problem under the SML and to give a complete characterization of the resulting optimal assortments.

A key feature of the PALM and the SML, is their ability to capture several effects that cannot be explained by any choice model based in random utility (such as for example the MNL, the mixed MNL, the Markov chain model, and the stochastic preference models). Examples of such effects include attraction, (Doyle, O’Connor, Reynolds, & Bottomley, 1999), the compromise effect (Simonson & Tversky, 1992), the similarity effect (Debreu, 1960, Tversky, 1972b), and the paradox of choice (also known as choice overload) (Chernev, Böckenholt, Goodman, 2015, Haynes, 2009, Iyengar, Lepper, 2000, Schwartz, 2004). These effects are discussed in the next section. In particular, the SML allows for violations of regularity. There are very few analyses of assortment problems under a choice model outside the RUM class.

Our algorithm is based on an in-depth analysis of the structure of the SML. It exploits the concept of revenue-ordered assortments that underlies the optimal algorithm for the assortment problem under the MNL. The key idea in our algorithm is to consider an assortment built from the union of two sets of products: A revenue-ordered assortment from the first level and another revenue-ordered assortment from the second level. Several structural properties of optimal assortments under the SML are also presented.

The heuristic of revenue-ordered assortments, consists in evaluating the expected revenue of all the assortments that can be constructed as follows: fix threshold revenue ρ and then select all the products with revenue of at least ρ. This strategy is appealing because it can be applied to assortment problems for any discrete choice model. In addition, it only needs to evaluate as many assortments as there are different revenues among products. In a seminal result, Talluri and Van Ryzin (2004) showed that, under the MNL model, the optimal assortment is revenue-ordered. This result does not hold for all assortment problems however. For example, revenue-ordered assortments are not necessarily optimal under the MNL model with capacity constraints (Rusmevichientong, Shen, & Shmoys, 2010). Nevertheless, in another seminal result, Rusmevichientong et al. (2010) showed that the assortment problem can still be solved optimally in polynomial time under such setting.

Rusmevichientong and Topaloglu (2012) considered a model where customers make choices following an MNL model, but the parameters of this model belong to a compact uncertainty set, i.e., they are not fully determined. The firm wants to be protected against the worst-case scenario and the problem is to find an optimal assortment under these uncertainty conditions. Surprisingly, when there is no capacity constraint, the revenue-ordered strategy is optimal in this setting as well.

There are also studies on how to solve the assortment problem when customers follow a mixed Multinomial Logit Model. Bront, Méndez-Díaz, and Vulcano (2009) showed that this problem is NP-hard in the strong sense using a reduction from the minimum vertex cover problem (Garey & Johnson, 1979). Méndez-Díaz, Miranda-Bront, Vulcano, and Zabala (2014) proposed a branch-and-cut algorithm to solve the optimal assortment under the Mixed-Logit Model. An algorithm to obtain an upper bound of the revenue of an optimal assortment solution under this choice model was proposed by Feldman and Topaloglu (2015). Rusmevichientong, Shmoys, Tong, and Topaloglu (2014) showed that the problem remains NP-hard even when there are only two customers classes.

Another model that attracted researchers attention is the nested logit model (Williams, 1977). Under the nested logit model, products are partitioned into nests, and the selection process for a customer goes by first selecting a nest, and then a product within the selected nest. It also has a dissimilarity parameter associated with each nest that serves the purpose of magnifying or dampening the total preference weight of the nest. For the two-level nested logit model, Davis, Gallego, and Topaloglu (2014) studied the assortment problem and showed that, when the dissimilarity parameters are bounded by 1 and the no-purchase option is contained on a nest of its own, an optimal assortment can be found in polynomial time; If any of these two conditions is relaxed, the resulting problem becomes NP-hard, using a reduction from the partition problem (Garey & Johnson, 1979). The polynomial-time solution was further extended by Gallego and Topaloglu (2014), who showed that, even if there is a capacity constraint per nest, the problem remains polynomial-time solvable. Li, Rusmevichientong, and Topaloglu (2015) extended this result to a d-level nested logit model (both results under the same assumptions over the dissimilarity parameters and the no-purchase option). Jagabathula (2014) proposed a local-search heuristic for the assortment problem under an arbitrary discrete choice model. This heuristic is optimal in the case of the MNL, even with a capacity constraint.

Wang and Sahin (2018) has studied the assortment optimization in a context in which consumer search costs are non-negligible. The authors showed that the strategy of revenue-ordered assortments is not optimal. Another interesting model sharing similar choice probabilities to those of the PALM, is the one proposed in Manzini and Mariotti (2014) which is based on consider first and choose second process. Echenique et al. (2018) showed that the PALM and the model by Manzini and Mariotti are in fact disjoint. The assortment optimization problem under the Manzini and Mariotti model has been recently studied by Gallego and Li (2017), where they show that revenue-ordered assortments strategy is optimal. Another choice model studied is the negative exponential distribution (NED) model (Daganzo, 1979), also known as the Exponomial model (Alptekinoğlu & Semple, 2016) in which customer utilities follow negatively skewed distribution. Alptekinoğlu and Semple (2016) proved that when prices are exogenous, the optimal assortment might not be revenue-ordered assortment, because a product can be skipped in favour of a lower-priced one depending on the utilities. This last result differs from what happens under the MNL and the Nested Logit Model (within each nest). Another recently proposed extension to the Multinomial Logit Model, is the General Luce Model (GLM) (Echenique & Saito, 2018). The GLM also generalizes the MNL and falls outside the RUM class. In the GLM, each product has an intrinsic utility and the choice probability depends upon a dominance relationship between the products. Given an assortment S, consumers first discard all dominated products in S and then select a product from the remaining ones using the standard MNL model. Flores, Berbeglia, and Van Hentenryck (2017) studied the assortment optimization problem under the GLM. The authors showed that revenue ordered assortments are not optimal, and proved that the problem can still be solved in polynomial time.

Recently, Berbeglia and Joret (2017) studied how well revenue-ordered assortments approximate the optimal revenue for a large class of choice models, namely all choice models that satisfy regularity. They provide three types of revenue guarantees that are exactly tight even for the special case of the RUM family. In the last few years, there has been progress in studying the assortment problem in choice models that incorporate visibility or position biases. In these models, the likelihood of selecting an alternative not only depends on the offer set, but also in the specific positions at which each product is displayed (Abeliuk, Berbeglia, Cebrian, Van Hentenryck, 2016, Aouad, Segev, 2015, Davis, Gallego, Topaloglu, 2013, Gallego, Li, Truong, Wang, 2016).

We mentioned that PALM and SML are able to accommodate many effects that can’t be explained by models the RUM class. We briefly describe each one of them in the following paragraphs.

The attraction effect stipulates that, under certain conditions, adding a product to an existing assortment can increase the probability of choosing a product in the original assortment. We briefly describe two experiments of this effect. Simonson and Tversky (1992) considered a choice among three microwaves x, y, and z. Microwave y is a Panasonic oven, perceived as a good quality product, and z is a more expensive version of y. Product x is an Emerson microwave oven, perceived as a lower quality product. The authors asked a set of 60 individuals (N=60) to choose between x and y; they also asked another set of 60 participants (N=60) to choose among x, y, and z. They found out that the probability of choosing y increases when product z is shown. This is a direct violation of regularity, which states that the probability of choosing a product does not increase when the choice set is enlarged, as described by McCausland and Marley (2013). Another demonstration of the attraction effect was carried by Doyle et al. (1999) who analyzed the choice behaviour of two sets of participants (N=70 and N=82) inside a grocery store in the UK varying the choice set of baked beans. To the first group, they showed two types of baked beans: Heinz baked beans and a local (and cheaper) brand called Spar. Under this setting, the Spar beans was chosen 19% of the time. To the second group, the authors introduced a third option: a more expensive version of the local brand Spar. After adding this new option, the cheap Spar baked beans was chosen 33% of the time. It is worth highlighting that the choice behaviour in these two experiments cannot be explained by a Multinomial Logit Model, nor can it be explained by any choice model based on random utility.

The compromise effect (Simonson & Tversky, 1992) captures the fact that individuals are averse to extremes, which helps products that represent a “compromise” over more extreme options (either in price, familiarity, quality, ...). As a result, adding extreme options sometimes leads to a positive effect for compromise products, whose probabilities of being selected increase in relative terms compared to other products. This phenomenon violates again the IIA axiom of Luce’s model and the regularity axiom satisfied by all random utility models (Berbeglia & Joret, 2016). Again, the compromise effect can be captured with the PALM.

The similarity effect is discussed in Tversky (1972b), elaborating on an example presented in Debreu (1960): Consider x and z to represent two recordings of the same Beethoven symphony and y to be a suite by Debussy. The intuition behind the effect is that x and z jointly compete against y, rather than being separate individual alternatives. As a result, the ratio between the probability of choosing x and the probability of choosing y when the customer is shown the set {x, y} is larger than the same ratio when the customer is shown the set {x, y, z}. Intuitively, z takes a market share of product x, rather than a market share of product y.

Finally, the choice overload effect occurs when the probability of making a purchase decreases when the assortment of available products is enlarged. To our knowledge, the first paper that shows the empirical existence of choice overload is written Iyengar and Lepper (2000). In their experimental setup, customers are offered jams from a tasting booth displaying either 6 (limited selection) or 24 (extensive selection) different flavours. All customers were given a discount coupon for making a purchase of one of the offered jams. Surprisingly, 30% of the customers offered the limited selection used the coupon, while only 3% of customers offered the extensive selection condition used the coupon. Another studies of choice overload are in 401(k) plans Sethi-Iyengar, Huberman, and Jiang (2004), chocolates (Chernev, 2003b), consumer electronics (Chernev, 2003a) and pens (Shah & Wolford, 2007). For a more in depth discussion of this effect the reader is referred to Schwartz (2004). Readers are also referred to Chernev et al. (2015) for a review and meta-analysis on this topic.

Section snippets

Problem formulation

This section presents the sequential Multinomial Logit Model considered in this paper and its associated assortment optimization problem. Let X be the set of all products and x0 be the no-choice option (x0 ∉ X). Following Echenique et al. (2018), each product x ∈ X is associated with an intrinsic utility u(i) > 0 and a perception priority level l(x) ∈ {1, 2}. The idea is that customers perceive products sequentially, first those with priority 1 and then those with priority 2. This perception

Properties of optimal assortments

In this section we derive properties of the optimal solutions to the assortment problem under the SML. These properties are extensively used in the proof of our main result (Theorem 1) in Section 4. We establish bounds on the following: any product offered on any optimal solution, and also the assortments considered on an optimal solution on each level. We assume a set of products X=X1X2 and use the following notationsU(S)=xSu(x),α(S)=xSu(x)r(x)xSu(x)=xSu(x)r(x)U(S)andλ(Z,S)=U(Z)U(S)+u0

Optimality of revenue-ordered assortments by level

This section proves that optimal assortments under the SML are revenue-ordered by level, generalizing the traditional results for the MNL (Talluri & Van Ryzin, 2004). As a corollary, the optimal assortment problem under the SML is polynomial-time solvable.

Definition 1 Revenue-ordered assortment by level

Denote by Nij the set of all products in level i with a revenue of at least rij (1 ≤ j ≤ mi) and fix Ni0= by convention. A revenue-ordered assortment by level is a set S=N1j1N2j2X for some 0 ≤ j1 ≤ m1 and 0 ≤ j2 ≤ m2. We use A to denote the

Numerical experiments

In this section, we analyse numerically the performance of revenue-ordered assortments (RO) against our proposed strategy (ROL) by varying the number of products, the distribution of revenues and utilities in each level, and the utility of the outside option. In our experiments with up to 100 products, we found that the optimality gap can be as large as 26.319%.

Each family or class of instances we tested is defined by three numbers: the number of products in the first and second level (n1, n2),

Conclusion and future work

This paper studied the assortment optimization problem under the Sequential Multinomial Logit (SML), a discrete choice model that generalizes the multinomial logit (MNL). Under the SML model, products are partitioned into two levels. When a consumer is presented with such an assortment, she first considers products in the first level and, if none of them is appropriate, products in the second level. The SML is a special case of the Perception Adjusted Luce Model (PALM) recently proposed by

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