Operations management under paradox of choice

Abstract

This paper examines how assortment size influences consumer purchasing behavior and retailer profits, focusing on the paradox of choice (PoC), where overly limited or excessive options can diminish purchase intent. Although PoC effects are well-documented in behavioral research, their impact on assortment and pricing strategies in retail remains underexplored. To bridge this gap, we formalize PoC mathematically by modeling the utility of the no-purchase option as a U-shaped function of assortment size. We integrate this PoC framework into several discrete choice models, using the multinomial logit (MNL) model as the focal model. By decomposing the assortment problem into subproblems with fixed assortment sizes, we develop efficient solutions for the assortment optimization and the joint assortment and price optimization problems under the MNL with PoC model. Additionally, we compare the optimal solutions of the classical MNL model and the MNL model with the PoC effect through theoretical analysis and numerical experiments, offering managerial insights to guide firms in optimizing their assortments. We further extend our approach to derive optimal assortment and pricing solutions for the nested logit model employing a linear programming approach. Moreover, we devise a fully polynomial time approximation scheme for the assortment optimization under the mixture of MNL model with the PoC effect. This study advances the integration of behavioral insights into discrete choice models, illustrating how retailers can optimize product assortments and prices by balancing variety with cognitive effects linked to assortment size.

1. Introduction

How does the size of a retailer’s product assortment impact consumer purchasing behavior and, consequently, the retailer’s profit? The first question has been extensively explored in marketing and psychology, revealing that the benefits of a larger assortment are neither linear nor straightforward. Early studies indicate that expanding the assortment enhances the likelihood of consumers finding desired products and promotes efficient shopping experiences. However, the well-known jam study by Iyengar and Lepper (2000) demonstrated a contrary effect: consumers were more likely to make a purchase when presented with six types of jam rather than 24. This illustrates the choice overload phenomenon, where larger assortments can paradoxically lead to fewer purchases (Scheibehenne et al., 2010).

The relationship between assortment size and purchase likelihood is postulated to follow an inverted U-shaped curve. For example, Shah and Wolford (2007) demonstrate that the likelihood of purchase first increases and then decreases as the number of options grows from 2 to 20, as illustrated in Figure 1. This contradiction, where a moderate increase in options can be beneficial, but an excessive number can deter purchase decisions, is known as the paradox of choice (Schwartz, 2004). Hereinafter, we refer to the paradox of choice as PoC. Despite the critical implications of PoC on consumer behavior, the second question mentioned at the outset, namely how assortment size influences a retailer’s profit, has not been thoroughly investigated. Assortment optimization, which aims to maximize a firm’s profit or revenue by choosing the right product mix, fundamentally relies on accurate demand modeling. Traditional demand models, especially the discrete choice models (DCMs), have gained popularity since the seminal work of Talluri and Van Ryzin (2004), and have been pivotal in revenue management by integrating customer preferences into demand forecasts. Yet, these models typically suggest that more options lead to higher total purchase probability, a premise contradicted by the PoC.

Figure 1.

Proportion of participants purchasing pens across assortment sizes in a study by Shah and Wolford (2007).

Our paper bridges a significant gap by providing a formal mathematical definition of the PoC effect, focusing specifically on how assortment size influences purchase intent. We isolate the effect of assortment size from other influencing factors, such as product attributes, by defining PoC solely in terms of the utility of the no-purchase option. Specifically, we model this utility as a U-shaped function of assortment size: it initially decreases (a region we term choice underload) and subsequently increases (referred to as choice overload) as the number of available options grows. This functional form captures two distinct and competing psychological mechanisms. In the underload region, the benefits of providing a more comprehensive selection dominate: as the number of options increases, the assortment better caters to variety-seeking behaviors, lowering the relative utility of the no-purchase option. However, as the assortment continues to grow, the negative effect of choice overload eventually overtakes these variety benefits, causing the utility of the no-purchase option to rise. By endogenizing these psychological factors into the no-purchase utility, we provide a parsimonious bridge between behavioral observations and structural choice modeling.

Then, by integrating this PoC framework into several common DCMs, including the multinomial logit (MNL) model, the nested logit (NL) model, and the mixture of multinomial logit (MMNL) model, we examine how PoC affects firms’ optimal operational management decisions, thereby linking assortment size directly to profit outcomes.

The main findings of our research are outlined below:

MNL model: By decomposing the assortment optimization problem under the MNL with PoC model into subproblems, we show that each subproblem can be solved efficiently by evaluating $O(n^2)$ assortments, where $n$ represents the total number of available products. We compare the optimal assortment structures under the classical MNL and MNL with PoC models, offering valuable insights for firms to adjust assortments in response to the PoC effect. Furthermore, we show that both the joint assortment and pricing problem and the conversion rate problem, which aims to maximize the total purchase probability, can be efficiently resolved. Additionally, we establish that the maximum likelihood function of our model is concave under certain nonrestrictive assumptions, ensuring efficient parameter estimation. Our numerical studies indicate a substantial improvement in model fit when PoC is incorporated into the MNL model.

NL model: We demonstrate that the assortment optimization problem can be efficiently solved through a linear programming approach involving $m+1$ variables and $O(m{n}^3)$ constraints, where $m$ denotes the number of nests and $n$ is the number of products in each nest. Furthermore, we show that the joint assortment and price optimization problem involving $p$ products in each nest, each having $b$ distinct price points, can be transformed into a linear problem with $m+1$ variables and $O(m{b}^2{p}^3)$ constraints. This linear program can also be used to solve the joint assortment and pricing problem with a cardinality constraint under the NL model.

MMNL model: We present a fully polynomial time approximation scheme (FPTAS) for the assortment optimization problem under the MMNL with PoC model, given a constant number of mixtures. That is, for any $ϵ>0$ , our approximation algorithm has a runtime that is polynomial in the input size and $1/ϵ$ and an approximation guarantee of $1-ϵ$ .

The rest of this paper is organized as follows: Section 2 reviews the relevant literature, highlighting our contributions in each research area. Section 3 introduces our framework for incorporating PoC into existing DCMs, with a focus on the MNL model. Section 4 develops the optimal solution to the assortment optimization problem under the MNL with PoC model. Section 5 studies the joint assortment and price optimization problem under the MNL with PoC model. Section 6 details parameter estimation for the MNL with PoC model and presents a numerical study comparing its performance to the model without PoC. Section 7 extends this approach to the assortment and price optimization problem under the NL and the assortment optimization problem under the MMNL models. Finally, Section 8 concludes the paper with future research directions. All proofs are provided in the online Appendix.

2. Literature review

In this section, we first survey the literature on the effect of assortment size on consumer behavior, and then discuss how assortment problems utilize DCMs for demand modeling, focusing on select studies that closely align with our research objectives.

2.1. Size effect of assortments

The influence of assortment size on consumer behavior has been well-documented. Early economics research posits that larger product assortments generally benefit consumers by increasing the likelihood of finding the desired items (Baumol and Ide, 1956). Subsequent marketing and consumer behavior studies have reinforced this perspective, showing that larger assortments not only facilitate more efficient shopping experiences (Betancourt and Gautschi, 1990) but also cater to variety-seeking behaviors and evolving consumer tastes; see Kahn (1995) for a detailed review. These findings support the notion that a larger assortment typically enhances total purchase probability, aligning well with the characteristics of traditional DCMs.

However, large assortments do not always lead to positive outcomes. The well-known jam study by Iyengar and Lepper (2000) exemplifies the choice overload phenomenon, where consumers presented with six types of jam were more likely to make a purchase than those offered 24 types. The authors posit that too many options may overwhelm consumers, increasing the decision-making effort and potentially leading to avoidance or reduced satisfaction. Furthermore, a larger assortment could diminish the appeal of the chosen option, as regret over the unchosen options can detract from the enjoyment of the selection. Building on the abovementioned literature, some studies suggest that consumer preference for larger assortments may be subject to diminishing marginal returns (Chernev, 2003b). As the assortment size increases, the cognitive effort required to evaluate additional options eventually outweighs the marginal benefits they provide, and the likelihood of purchase begins to decline (Reutskaja and Hogarth, 2009). This phenomenon of choice overload has been consistently observed in various product categories, from chocolates to mutual funds (Berger et al., 2007; Chernev, 2003a; Huberman et al., 2007). For comprehensive reviews of papers on choice overload, refer to Chernev et al. (2015) and Scheibehenne et al. (2010).

Schwartz (2004) describes this phenomenon as the paradox of choice (PoC). Note that PoC suggests an inverted U relationship between assortment size and purchase probability, which is empirically supported by Shah and Wolford (2007). Our study builds on these insights by incorporating PoC into assortment optimization, capturing how assortment size itself affects customer purchase behavior, and providing strategic guidance to enhance both profitability and consumer satisfaction.

2.2. Assortment optimization under discrete choice models

Assortment optimization involves selecting a set of products to offer to customers. Central to this process is demand modeling, which estimates the likelihood of customers choosing products from an assortment. Since the seminal work of Talluri and Van Ryzin (2004), DCMs have been widely used in revenue management to integrate customer preferences into demand forecasting.

The widely adopted MNL model, introduced by Luce (1960) and McFadden (1973), stands as the most popular DCM for its tractability in solving assortment optimization problems. Talluri and Van Ryzin (2004) illustrate that the optimal assortment under the MNL model can be found efficiently, as it consists of a set of the most profitable products, often referred to as a revenue-ordered set. Rusmevichientong et al. (2010) demonstrate that with a cardinality constraint, the optimal assortment under the MNL model deviates from being revenue-ordered, yet the problem can still be solved efficiently.

One notable limitation of the MNL model is its independence of irrelevant alternatives (IIA) property, where the addition of an item uniformly decreases the likelihood of selecting all other items. IIA can lead to inaccurate demand estimations (Train, 2009). To address this concern, researchers have considered assortment optimizations under alternative choice models that offer greater flexibility. These include the NL model (Davis et al., 2014; Li et al., 2015), the mixture of logit model (Rusmevichientong et al., 2014), and the Markov chain choice model (Blanchet et al., 2016). For a detailed overview of DCMs and their applications in assortment problems, readers can refer to Kök et al. (2015) and Wang (2021).

Recently, there has been growing interest in incorporating specific customer behaviors, such as reference prices (Wang, 2018) and contextual influences (Najafi et al., 2026), into DCMs to address assortment and price optimization problems. Despite these advancements, only a handful of studies reflect PoC as a by-product of their formulation, while even fewer explicitly model it in assortment optimization. For example, the general attraction model in Gallego et al. (2015) captures the choice underload effect by allowing the attraction of the no-purchase option to increase as the assortment size shrinks. Jiang et al. (2025) explore the assortment problem under a focal Luce model (FLM), where “focal” or star products in an assortment are overevaluated by customers. They demonstrate that when the no-purchase option is the focal product, the FLM model can capture the choice overload effect. Similarly, Flores et al. (2019) introduce a sequential multinomial logit (SML) model, where customers are offered a second assortment if they decline to purchase from the first. They show that SML can also capture choice overload. In comparison, Goutam et al. (2019) explicitly incorporate choice overload into a Markov choice model and show that the associated assortment optimization problem is NP-hard. The authors develop an FPTAS under reasonable assumptions. Nonetheless, most of these studies primarily demonstrate choice overload, overlooking the nuanced effects of smaller assortments in diminishing consumer purchase intentions.

In contrast, our study offers a holistic framework that incorporates both the positive and negative implications of PoC within a general discrete choice framework. Using several modified DCMs, we further characterize and develop efficient algorithms for the optimal solutions to the associated assortment. These implementations highlight the adaptability and effectiveness of our approach.

3. PoC framework

In this section, we introduce a framework for integrating PoC into DCMs. We will primarily focus on the MNL model to illustrate our approach.

Let $\mathcal{N}=\{1,\dots ,n\}$ denote the product universe. Each product $i\in \mathcal{N}$ provides utility $U_i$ to customers. We denote the no-purchase option as product $0$ . Let ${\mathcal{N}}^{+}:=\mathcal{N}\cup \{0\}$ . Consistent with the random utility maximization (RUM) model, we assume that for each $i\in {\mathcal{N}}^{+}$ , $U_i=u_i+{\epsilon }_i,$ where $u_i$ is the deterministic component and ${\epsilon }_i$ represents the stochastic error.

Under the RUM framework, when offered an assortment $S\subseteq \mathcal{N}$ , the probability of a customer choosing an option $i$ to maximize the customer’s utility is given by

$$\begin{array}{r}{\mathcal{P}}_i(S)=\text{Prob}(U_i\ge U_j,\mathrm{\forall }j\in S^{+},j\ne i),\mathrm{\forall }i\in S^{+},\end{array}$$

where $S^{+}:=S\cup \{0\}$ .

Under the MNL model, it is further assumed that $\{{\epsilon }_i:i\in {\mathcal{N}}^{+}\}$ are i.i.d. random variables with a Gumbel distribution specified by $\mathsf{G}\mathsf{u}\mathsf{m}\mathsf{b}\mathsf{e}\mathsf{l}\mathsf{(}\mathsf{0}\mathsf{,}\mathsf{1}\mathsf{)}$ , giving rise to the following choice probabilities:

$$\begin{array}{r}{\mathcal{P}}_i(S)=\frac{\exp (u_i)}{\exp (u_0)+\sum _{j\in S}\exp (u_j)},\mathrm{\forall }i\in S^{+}.\end{array}$$

(1)

In the common DCMs, the value of $u_0$ is assumed to be independent of $S$ . As a result, a larger assortment always increases the total likelihood of purchases (i.e., ${\mathcal{P}}_0(S_1)>{\mathcal{P}}_0(S_2)$ for any $S_1\subset S_2$ ), rendering the traditional DCMs inadequate for addressing the PoC phenomenon.

To integrate the concept of PoC, we posit that the utility of no-purchase depends on two key factors: the number of options available in an offered assortment $S$ (i.e., $|S|$ ) and a “golden number” ( $B$ ). This golden number represents the assortment size that balances the benefits of variety and the costs of cognitive burden. When the assortment is small ( $|S|<B$ ), adding options reduces the no-purchase utility by improving perceived variety and fit, reflecting choice underload. Beyond the threshold $B$ , additional options increase the cognitive burden of evaluating alternatives, making choice overload more pronounced as $|S|$ increases. Hereinafter, we denote the no-purchase utility by $u_0(|S|,B)$ . We formalize this assumption as follows:

Assumption 1

There exists a golden number $B\ge 0$ such that $u_0(|S|,B)$ weakly decreases with $|S|<B$ (i.e., choice underload) and weakly increases for $|S|>B$ (i.e., choice overload). Moreover, $u_0(B,B)=0$ , and $u_0(A,B)>0$ for any $A\ne B$ .

In Assumption 1, $u_0(B,B)=0$ is introduced solely to facilitate direct comparison with the classical MNL model, where the utility of the no-purchase option is usually normalized to 0. Relaxing it does not affect the model’s results or applicability.

We also remark that the standard MNL model inherently captures the effect that adding more products increases the probability of purchase, as a larger assortment raises the chance of a good match. Although this can partially reflect choice underload, it does not fully capture it, as the effect depends solely on aggregate attractiveness in MNL. If underload effects depend on other factors, such as assortment size, then MNL cannot represent them accurately, and further modifications are needed. For example, Shah and Wolford (2007) show that smaller assortments can deter purchase in ways that MNL alone cannot explain. In Section 6.2, we further examine this setting and show empirically that the standard MNL framework fails to reproduce the underload effect, whereas the proposed PoC model with the U-shaped no-purchase utility succeeds.

A key modification we adopt in the PoC model is to let the no-purchase utility depend directly on assortment size. The U-shaped specification of $u_0(|S|,B)$ in Assumption 1 provides a parsimonious way to operationalize the PoC effect within a DCM framework, capturing both choice overload and choice underload. It is also flexible, encompassing a wide range of functions. This allows us to modify the choice probabilities in (1) to incorporate PoC as follows:

$$\begin{array}{rl}{\mathcal{P}}_i(S,B)& =\frac{\exp (u_i)}{\exp (u_0(|S|,B))+\sum _{j\in S}\exp (u_j)}\\ & =\frac{a_i}{a_0(|S|,B)+\sum _{j\in S}{a}_j},\mathrm{\forall }i\in S^{+},\end{array}$$

(2)

where $a_i=\exp (u_i)$ for $i\in S$ and $a_0(|S|,B)=\exp (u_0(|S|,B))$ . In particular, $a_i$ represents the attractiveness of product $i\in S^{+}$ .

The U-shaped $u_0(|S|,B)$ induces a corresponding U-shaped pattern in the no-purchase probability ${\mathcal{P}}_0(S,B)$ , as illustrated in Example 1 and Figure 2. This implies that the shape of total purchase probability is consistent with the findings in Shah and Wolford (2007) (Figure 1).

Figure 2.

Illustration of the no-purchase utility and the no-purchase probability in Example 1.

Example 1

Let $B=3$ and $a_i=1$ for all $i\in \mathcal{N}$ . Additionally, let $a_0(|S|,B)=1+(B-|S|{)}^2$ , which is equivalent to $u_0(|S|,B)=\ln (1+(B-|S|{)}^2)$ , satisfying Assumption 1. Then, for any $S\subseteq \mathcal{N}$ we have

$$\begin{array}{r}{\mathcal{P}}_0(S,B)=\frac{a_0(|S|,B)}{a_0(|S|,B)+\sum _{i\in S}{a}_i}=\frac{1+(3-|S|{)}^2}{(1+(3-|S|{)}^2)+|S|}.\end{array}$$

Due to the U-shaped utility of the no-purchase option, incorporating PoC into DCMs can violate the regularity condition (i.e., the choice probability for any existing product will not increase when the assortment is expanded) within the RUM framework. Nonetheless, incorporating PoC may preserve some structural properties of the underlying choice model. For instance, in the MNL model, the IIA property (where the relative probability of choosing one option over another is unaffected by the presence or absence of other alternatives) continues to hold. Importantly, our PoC framework can be readily integrated into other DCMs that already mitigate IIA. We discuss some of these models in Section 7.

4. Assortment optimization under PoC

Suppose that each product $i\in \mathcal{N}$ generates unit profit $r_i$ . In assortment optimization, a firm selects an offer set $S\subseteq \mathcal{N}$ to maximize the aggregate expected profit. Let ${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,B)$ denote the expected profit generated by assortment $S$ under the MNL with PoC model, with a reference threshold $B$ . Then, we have

$$\begin{array}{r}{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,B)=\sum _{i\in S}{\mathcal{P}}_i(S,B)r_i=\sum _{i\in S}\frac{a_i{r}_i}{a_0(|S|,B)+\sum _{i\in S}{a}_i}.\end{array}$$

(3)

Let $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ denote the optimal assortment under the modified MNL model with the PoC effect. Formally,

$$\begin{array}{r}(\mathbf{\text{MNL with PoC}})S_{\mathsf{P}\mathsf{o}\mathsf{C}}:=\underset{S\subseteq \mathcal{N}}{\mathrm{argmax}}{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,B).\end{array}$$

(4)

Talluri and Van Ryzin (2004) show that the optimal assortment under the classical MNL model is a revenue-ordered set comprising all products whose profits are higher than a certain threshold. Below, Example 2 shows that this property does not hold when considering the PoC phenomenon. That is, when the PoC effect occurs, the optimal assortment may no longer be revenue-ordered. Example 2

Consider an assortment of three products with attractiveness values $a_1=1$ , $a_2=2$ , and $a_3=4$ and profits $r_1=9$ , $r_2=8$ , and $r_3=7$ . Let $B=1$ . Suppose that the attractiveness of the no-purchase option is $a_0(|S|,B)=\exp (0.3(B-|S|{)}^2)$ in the presence of PoC. The attractiveness of the no-purchase option without PoC is equal to 1, which also equals $a_0(B,B)$ . The optimal assortment under the MNL model includes all three products. In contrast, the optimal solution under the MNL with PoC model is not revenue-ordered, as it only contains products 2 and 3. Let ${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{M}\mathsf{N}\mathsf{L}}(S)$ denote the expected profit of an assortment $S$ under the MNL model. Table 1 shows the expected profits of $\{1,2,3\}$ and $\{2,3\}$ for each model.

Table 1.

Expected profit comparison for Example 2.

$S$	${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{M}\mathsf{N}\mathsf{L}}(S)$	${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,B)$
{1,2,3}	6.625	5.135
{2,3}	6.285	5.980

In Example 2, $B=1$ , implying that with the PoC effect, any addition to a nonempty assortment increases the no-purchase utility. ¹ Therefore, an optimal assortment must balance the trade-off between enhancing the overall attractiveness and the toll that additional products take on the no-purchase utility due to choice overload. Product 1, despite having the highest profit, is excluded from the optimal assortment under PoC because its attractiveness fails to outweigh the negative impacts of an increased assortment size. The optimal assortment size under the PoC model is smaller than that in the model without PoC considerations. This reduction in assortment size to alleviate choice overload seems intuitive. However, we will later show in Proposition 1 that the size of the optimal assortment can indeed increase in the presence of choice overload. Thus, assortment optimization under PoC is worth further investigation.

Example 2 further illustrates that ignoring the PoC effect can result in significant profit losses for firms. Specifically, ${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(\{2,3\})$ exceeds ${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(\{1,2,3\})$ by 16%. This observation motivates the need for an efficient method to identify the optimal assortment under PoC. Thus, our subsequent efforts are dedicated to efficiently determining the optimal assortment in the presence of the PoC effect. En route to this goal, we transform our problem into $n$ subproblems and design an algorithm for each subproblem. We define subproblems as follows:

Definition 1 (Sub-Problem $k$ )

Let $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ denote a profit-maximizing assortment with cardinality $k$ under the PoC model. Subproblem $k$ , denoted as $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ , is the problem that aims to find $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ . That is,

$$\begin{array}{r}{S}_{\mathsf{P}\mathsf{o}\mathsf{C}}(k):=\underset{S\subseteq \mathcal{N}:|S|=k}{\mathrm{argmax}}{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,B).\end{array}$$

(5)

Solving the subproblems $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ for all feasible values of $k$ allows us to recover $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ , the optimal assortment under the MNL with PoC model. In the next sections, we discuss how $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ can be efficiently found in two scenarios: (1) all products have a uniform profit, and (2) products can have different profits.

4.1. Special case: Identical product profits

Conversion rate captures the percentage of customers who make a purchase or take a desired action. In many cases, the primary objective of a business is to boost the customer conversion rate rather than directly increasing revenue. For instance, in targeted display advertising, the goal is to identify the best opportunities to display a banner ad to an online user who is most likely to take a desired action, such as purchasing a product or signing up for a newsletter (Lee et al., 2012). In such scenarios, the assortment optimization problem under PoC differs from Problem (4), as all products have the same profits. Thus, without loss of generality, we assume $r_i=1$ for all $i\in \mathcal{N}$ . To determine the optimal assortment that maximizes the conversion rate, one can solve the subsequent problem:

$$\begin{array}{r}{S}_{\mathsf{P}\mathsf{o}\mathsf{C}}=\underset{S\subseteq \mathcal{N}}{\mathrm{argmax}}(\sum _{i\in S}{\mathcal{P}}_i(S,B)).\end{array}$$

(6)

Since $\sum _{i\in S}{\mathcal{P}}_i(S,B)=1-{\mathcal{P}}_0(S,B)$ , increasing the conversion rate is equivalent to reducing the probability of the no-purchase option. Thus, we can simplify Problem (6) to:

$$\begin{array}{r}{S}_{\mathsf{P}\mathsf{o}\mathsf{C}}=\underset{S\subseteq \mathcal{N}}{\mathrm{argmin}}{\mathcal{P}}_0(S,B).\end{array}$$

(7)

To address Problem (7), we first examine the structure of the optimal assortment for each $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ :

$$\begin{array}{rl}{S}_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)& =\underset{S\subseteq \mathcal{N},|S|=k}{\mathrm{argmax}}\frac{\sum _{i\in S}{a}_i}{a_0(k,B)+\sum _{i\in S}{a}_i}\\ & =\underset{S\subseteq \mathcal{N},|S|=k}{\mathrm{argmin}}\frac{a_0(k,B)}{a_0(k,B)+\sum _{i\in S}{a}_i}.\end{array}$$

In the above formulation, the only component dependent on the offered assortment is $\sum _{i\in S}{a}_i$ . Therefore, given any $k$ , the optimal assortment for $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ contains the $k$ products with the highest utilities. Using this notion, we define utility-ordered sets below.

Definition 2 (Utility-Ordered Sets)

Suppose products are indexed in a decreasing order of their utilities, that is, $a_1\ge a_2\ge \cdots \ge a_n$ . For each $k\in \{1,\dots ,n\}$ , the $k$ -th utility-ordered set, denoted by $S_k^u$ , includes products ranging from $1$ to $k$ , that is, $S_k^u=\{1,2,\dots ,k\}$ .

Using Definition 2, the following theorem characterizes the optimal assortment for Problem (7).

Theorem 1

There exists a $k\ge B$ such that the optimal assortment of the conversion rate problem under the PoC effect is a utility-ordered set $S_k^u$ .

Theorem 1 indicates that a firm aiming to maximize the conversion rate should consider offering an assortment of the most attractive products to customers, and the size of that assortment should be no less than the golden number $B$ . This result follows from the property that adding a product to an assortment whose size is less than $B$ strictly improves the conversion rate. Next, we show how the assortment optimization problem with the PoC effect can be tackled when the profits of the products can take arbitrary values.

4.2. General case: Heterogeneous product profits

In this section, we develop an efficient algorithm to find the optimal solution, $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ , for subproblem $k$ for the general case where products have heterogeneous profits. Our algorithm is inspired by the polynomial-time algorithm StaticMNL developed by Rusmevichientong et al. (2010) for the capacitated assortment optimization problems under the MNL model.

Figure 3.

A visual representation of Algorithm 1 applied to a setting with three products, as demonstrated in Example 3.

For notational brevity, let

$$\begin{array}{r}{\lambda }_k^{*}={\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k))=\frac{\sum _{i\in S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)}{a}_i{r}_i}{a_0(k,B)+\sum _{j\in S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)}{a}_j}.\end{array}$$

(8)

From (8), ${\lambda }_k^{*}$ satisfies the following equation:

$$\begin{array}{r}{\lambda }_k^{*}=max\{\lambda \in \mathbb{R}:\underset{S\subseteq \mathcal{N},|S|=k}{max}\sum _{i\in S}\frac{a_i(r_i-\lambda )}{a_0(k,B)}\ge \lambda \}.\end{array}$$

(9)

Equation (9) allows us to define the following two functions that fundamentally drive our algorithm. For any $\lambda \in \mathbb{R}$ , define functions $h_i:\mathbb{R}\to \mathbb{R}$ , for $i\in \mathcal{N}$ , and $L_k:\mathbb{R}\to \{S\subseteq \{1,\dots ,n\}:|S|=k\}$ as follows:

$$\begin{array}{r}{h}_i(\lambda ):=a_i(r_i-\lambda ),\end{array}$$

and

$$\begin{array}{r}{L}_k(\lambda ):=\underset{S\subseteq \mathcal{N},|S|=k}{\mathrm{argmax}}\sum _{i\in S}{h}_i(\lambda ),\end{array}$$

(10)

where we break the ties arbitrarily. Compared to the reformulated capacitated MNL problem in Rusmevichientong et al. (2010), the cardinality constraint in (10) is strict. Moreover, since $a_0(k,B)$ is a constant for a given $k$ , we can equivalently write $L_k(\lambda )={\mathrm{argmax}}_{S\subseteq \mathcal{N},|S|=k}\sum _{i\in S}{h}_i(\lambda )/a_0(k,B)$ . Consequently,

$$\begin{array}{r}{\lambda }_k^{*}=max\{{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(L_k(\lambda )):\lambda \in \mathbb{R}\}.\end{array}$$

To find an optimal assortment of Problem (5), it suffices to enumerate $L_k(\lambda )$ for all values of $\lambda \in \mathbb{R}$ . Through geometric arguments, we show that the collection of assortments corresponding to $\{L_k(\lambda ):\lambda \in \mathbb{R}\}$ contains $O(n^2)$ sets, where $n$ is the number of items in $\mathcal{N}$ . Once $\{L_k(\lambda ):\lambda \in \mathbb{R}\}$ is established, ${\lambda }_k^{*}$ can be found in polynomial time.

Geometrically, each function $h_i(\cdot )$ corresponds to a straight line. The maximum number of intersection points among the $n$ lines $h_1(\cdot ),\dots ,h_n(\cdot )$ is $(\genfrac{}{}{0ex}{}{n}{2})\in O(n^2)$ . By equation (10), at any $\lambda \in \mathbb{R}$ , $L_k(\lambda )$ corresponds to the top $k$ lines among $h_1(\lambda ),\dots ,h_n(\lambda )$ . We begin by sorting the $O(n^2)$ intersection points of the lines $h_1(\cdot ),\dots ,h_n(\cdot )$ according to their $x$ -coordinates, where each $x$ -coordinate represents a value of $\lambda$ and the corresponding $y$ -coordinates give the values of $h_1(\lambda ),\dots ,h_n(\lambda )$ . Consider two consecutive intersection points in this ordering. Since the relative order of the lines $h_1(\cdot ),\dots ,h_n(\cdot )$ (from top to bottom) can only change at an intersection, it follows that for any $\lambda$ strictly between these two points, the ordering of the values $h_1(\lambda ),\dots ,h_n(\lambda )$ remains fixed. Consequently, the set $L_k(\lambda )$ also remains unchanged. Figure 3(a) depicts the intersection points for an example with three products. Therefore, to enumerate $L_k(\lambda )$ for all $\lambda \in \mathbb{R}$ , it suffices to enumerate all of the intersection points of $h_1(\cdot ),\dots ,h_n(\cdot )$ . The geometric algorithm, described in Algorithm 1, is grounded on this observation. Next, we discuss this algorithm in detail.

For all $1\le i<j\le n$ , where $a_i\ne a_j$ , let $x(i,j)$ denote the $x$ -coordinate of the intersection point between lines $h_i(\cdot )$ and $h_j(\cdot )$ , that is,

$$\begin{array}{r}{h}_i(x(i,j))=h_j(x(i,j))\iff x(i,j)=\frac{a_i{r}_i-a_j{r}_j}{a_i-a_j}.\end{array}$$

If $a_i=a_j$ , then, without loss of generality, set $x(i,j)=-\mathrm{\infty }$ . For ease of exposition, we incorporate a nonrestrictive assumption similar to the one in Rusmevichientong et al. (2010).

Assumption 2 (Unique Intersections)

The intersection points are unique; that is, for any $(i,j)\ne (i^{\prime },j^{\prime })$ , $x(i,j)\ne x(i^{\prime },j^{\prime })$ . ²

Order the intersection points by their $x$ -coordinates, and let $x(i_{\gamma },j_{\gamma })$ represent the $x$ -coordinate of the $\gamma$ -th intersection point. Let $\varphi$ denote the total number of intersections. By Assumption 2, we have

$$\begin{array}{rl}-\mathrm{\infty }& =x(i_0,j_0)<x(i_1,j_1)<x(i_2,j_2)<\cdots \\ & <x(i_{\varphi },j_{\varphi })<x(i_{\varphi +1},j_{\varphi +1})=+\mathrm{\infty },\end{array}$$

where the two endpoints $x(i_0,j_0)$ and $x(i_{\varphi +1},j_{\varphi +1})$ are added to aid in exposition.

For any $\gamma =0,\dots ,\varphi$ and any $\lambda \in (x(i_{\gamma },j_{\gamma }),x(i_{\gamma +1},j_{\gamma +1}))$ , the order of lines $h_1(\lambda ),\dots ,h_n(\lambda )$ remains unchanged. Let ${\mathit{\sigma }}^{\gamma }=({\sigma }_1^{\gamma },\dots ,{\sigma }_n^{\gamma })$ denote the order of these lines from top to bottom, that is, $h_{{\sigma }_1^{\gamma }}(\lambda )>h_{{\sigma }_2^{\gamma }}(\lambda )>\cdots >h_{{\sigma }_n^{\gamma }}(\lambda )$ for all $\lambda \in (x(i_{\gamma },j_{\gamma }),x(i_{\gamma +1},j_{\gamma +1}))$ . Note that $x(i_{\gamma },j_{\gamma })$ is the intersection point of lines $h_{i_{\gamma }}(\cdot )$ and $h_{j_{\gamma }}(\cdot )$ . Thus, the order of $h_{i_{\gamma }}(\cdot )$ and $h_{j_{\gamma }}(\cdot )$ swaps across $x(i_{\gamma },j_{\gamma })$ , while the ordering of the other lines remains unchanged. Therefore, ${\mathit{\sigma }}^{\gamma }$ can be obtained from ${\mathit{\sigma }}^{\gamma -1}$ efficiently by swapping $i_{\gamma }$ and $j_{\gamma }$ . Lastly, note that ${\mathit{\sigma }}^0$ is the ordering of product attractiveness $a_i$ ’s in a decreasing manner.

Let $L_k^{\gamma }=\{{\sigma }_1^{\gamma },\dots ,{\sigma }_k^{\gamma }\}$ , corresponding to the first $k$ elements in ${\mathit{\sigma }}^{\gamma }$ . Equivalently, $L_k^{\gamma }$ is the optimal solution to equation (10) for all $\lambda \in (x(i_{\gamma },j_{\gamma }),x(i_{\gamma +1},j_{\gamma +1}))$ and it contains $k$ items with the highest values of $h_i(\lambda )$ ’s.

The following example demonstrates how Algorithm 1operates.

Example 3

Consider a setting with three products, with associated lines $h_1(\lambda )=a_1(r_1-\lambda )$ , $h_2(\lambda )=a_2(r_2-\lambda )$ , and $h_3(\lambda )=a_3(r_3-\lambda )$ , for all $\lambda \in \mathbb{R}$ , see Figure 3(a). Let the products be arranged such that $a_1<a_2<a_3$ , resulting in three intersection points. For subproblem $2$ , that is, when the size of the offered assortment is limited to 2, Algorithm 1’s output is presented in Figure 3(b).

Theorem 2 establishes the running time and correctness of Algorithm 1.

Theorem 2

Algorithm 1 has a running time of $O(n^2)$ and returns $O(n^2)$ assortments, where $n$ is the number of products. Moreover, $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)={\mathrm{argmax}}_{\gamma =0,1,\dots ,\varphi }{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(L_k^{\gamma })$ and ${\lambda }_k^{*}=max\{{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(L_k^{\gamma }):\gamma =0,1,\dots ,\varphi \}$ .

We remark a key distinction between Algorithm 1, which helps find the optimal solutions of the subproblems for the MNL with PoC model, and the StaticMNL algorithm for the capacitated MNL model by Rusmevichientong et al. (2010). In each interval $(x(i_{\gamma },j_{\gamma }),x(i_{\gamma +1},j_{\gamma +1}))$ , the StaticMNL algorithm only considers the lines that have nonnegative values (i.e., $h_i(\lambda )\ge 0)$ . In contrast, we need to consider exactly the top $k$ lines in each interval, regardless of the signs of their values. This is because in $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ , the capacity constraint is strictly binding, and we need exactly $k$ products in the optimal solution. When it is necessary to include some products $i$ with negative $h_i(\cdot )$ values, our algorithm prioritizes the products with the least negative impact on profit.

Moreover, it is possible for $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ , the optimal solution to $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ , to contain products with negative $h_i(\cdot )$ values. This is more common when $k<B$ , that is, when there is choice underload. In such a case, adding additional products (even with negative $h_i(\cdot )$ values) not only reduces the attractiveness of the no-purchase option but also improves the total attractiveness of the offer set, thereby jointly leading to a higher expected profit.

In sum, Algorithm 1 allows us to solve Problem (5), that is, the cardinality-constrained subproblem $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ for each feasible assortment size $k$ , and then select the $k$ that yields the maximum revenue. The resulting $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ is the optimal assortment for Problem (4), the original assortment optimization problem.

4.3. Comparison with the MNL model

In this section, we perform a comparative analysis between the optimal assortments and profits under the classical MNL model and the MNL with PoC model.

To facilitate the comparison, we introduce a few new definitions. Let $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ and ${\lambda }_{\mathsf{M}\mathsf{N}\mathsf{L}}$ represent the optimal assortment and its expected profit under the MNL model (i.e., without accounting for the PoC phenomenon), respectively. Note that in the MNL model, the attractiveness of the no-purchase option is fixed and equal to $a_0(B,B)=1$ . As shown by Rusmevichientong et al. (2010), we have ${\lambda }_{\mathsf{M}\mathsf{N}\mathsf{L}}=max\{\lambda \in \mathbb{R}:\underset{S\subseteq \mathcal{N}}{max}\sum _{i\in S}{a}_i(r_i-\lambda )\ge \lambda \}$ . Let $k^{*}$ denote the cardinality of the optimal assortment for the PoC problem, that is, $k^{*}:={\mathrm{argmax}}_{k\le n}{\lambda }_k^{*}$ , where $n$ is the total number of products. It is evident that $S_{\mathsf{P}\mathsf{o}\mathsf{C}}=S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k^{*})$ , where $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ denotes the optimal assortment under the MNL with PoC model. Finally, for $\lambda \in \mathbb{R}$ , let $H(\lambda )$ denote the set of lines $h_i(\lambda )=a_i(r_i-\lambda )$ , $i\in \mathcal{N}$ , with nonnegative values at $\lambda$ , formally, $H(\lambda ):=\{i\in \mathcal{N}:a_i(r_i-\lambda )\ge 0\}$ .

Recall that Assumption 1 implies $a_0(|S|,B)\ge 1$ for any $S$ , where equality holds only when $|S|=B$ . The purchase probabilities therefore satisfy ${\mathcal{P}}_i(S,B)\le {\mathcal{P}}_i(S)$ for all $i\in S$ . As a result, ${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,B)\le {\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{M}\mathsf{N}\mathsf{L}}(S)$ for any $S$ , and it follows that ${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S_{\mathsf{P}\mathsf{o}\mathsf{C}},B)\le {\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{M}\mathsf{N}\mathsf{L}}(S_{\mathsf{M}\mathsf{N}\mathsf{L}})$ . That is, the classical MNL model overestimates the total expected profits whenever PoC is the actual demand mechanism. In addition, the optimal solutions under both models (i.e., with and without PoC) are the same if and only if the size of the optimal assortment in the absence of PoC coincides with $B$ .

Next, we aim to compare $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ and $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ for different values of $B$ . Proposition 1 presents our key results comparing the structure of the optimal assortments of the MNL model ( $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ ) and the MNL with PoC model ( $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ ) depending on the values of $B$ and $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|$ . This comparison is made under three scenarios, where the size of $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ is equal to, smaller than, or larger than the golden number $B$ .

Proposition 1

• (i)

If $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|=B$ , then $S_{\mathsf{P}\mathsf{o}\mathsf{C}}=S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ .

(ii)

If $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|<B$ , then $k^{*}\le B$ and $S_{\mathsf{M}\mathsf{N}\mathsf{L}}\subseteq S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ .

(iii)

If $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|>B$ , then $k^{*}\ge B$ .

Proposition 1 demonstrates that $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ (i.e., the optimal assortment without PoC) serves as an important anchor point for that under PoC. Specifically, the optimal assortments are identical if their size is exactly $B$ . However, if $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|$ is less than $B$ , then the optimal assortment under PoC contains (weakly) more products than $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ . This is because when $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|<B$ , adding more products not only reduces the attractiveness of the no-purchase option but also improves the total attractiveness of all the other options, thereby improving the total expected profit. Indeed, Proposition 1(ii) shows that $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ is a superset of $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ . That is, managerially, the firm should consider adding more products to $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ , which serves as a good starting point.

But how many more products should be added? Once the size of the assortment reaches the golden number $B$ , one may expect competing effects from the increases in the attractiveness of the no-purchase option and the total attractiveness of the products, and it becomes uncertain whether continuing to add more products is beneficial. Proposition 1(ii) indeed provides a definite answer to this question. It shows that when $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|<B$ , the size of $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ should not exceed $B$ . That is, the firm should add at most $B-|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|$ products to the optimal assortment without PoC.

In contrast, when $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|>B$ , Proposition 1(iii) shows that the size of the optimal assortment under PoC should also be weakly larger than the golden number. In this case, adding more products intensifies choice overload, whose impact eventually overshadows the increase in the total attractiveness of the products in the offer set. One may expect $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ to contain fewer products than or even be a subset of $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ when the latter’s size exceeds $B$ . However, Example 4 shows that this is not necessarily the case.

Example 4

Suppose that there are three products with attractiveness values $a_1=1$ , $a_2=1.01$ , and $a_3=5$ and unit profits $r_1=1.001$ , $r_2=0.99$ , and $r_3=0.6$ . Let the attractiveness of the no-purchase option under the PoC model be $a_0(|S|=1,B=1)=1$ , $a_0(|S|=2,B=1)=2.5$ , and $a_0(|S|=3,B=1)=2.99$ . Table 2 shows that $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ , the optimal assortment under the MNL model, includes the first and second products. In contrast, $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ , the optimal solution with PoC, includes all three products. Thus, if the size of $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ exceeds $B$ , then $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|$ is not necessarily larger than the size of $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ .

Table 2.

Comparison of the expected profits under the classical MNL and the MNL with PoC models in Example 4.

	Profit under	Profit under MNL
Assortment	classical MNL model	with PoC model
{1}	0.5005	0.5005
{2}	0.4974	0.4974
{3}	0.5000	0.5000
{1,2}	0.6647	0.4436
{1,3}	0.5715	0.4707
{2,3}	0.5705	0.4700
{1,2,3}	0.6243	0.5046

In Example 4, $S_{\mathsf{M}\mathsf{N}\mathsf{L}}=\{1,2\}$ already has a size exceeding $B$ . Under PoC, adding more products to $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ increases the attractiveness of the no-purchase option and aggravates choice overload. Intuitively, one might expect the expansion of $S_{\mathsf{M}\mathsf{N}\mathsf{L}}$ to be undesirable. However, product 3 is sufficiently attractive that its inclusion boosts the total attractiveness of the purchase options far more than the increase in no-purchase attractiveness. As a result, the incremental expected revenue from product 3 more than compensates for the loss due to choice overload, leading to the optimal assortment $S_{\mathsf{P}\mathsf{o}\mathsf{C}}=\{1,2,3\}$ .

In sum, Example 4 demonstrates that, counterintuitively, expanding an assortment that already triggers choice overload can still improve the firm’s profit. This result is consistent with the findings in Jiang et al. (2025) that the size of the optimal assortment may expand in the presence of choice overload. Our analysis specifies the conditions under which it is beneficial to have a larger (smaller) assortment when compared to the golden number $B$ . Moreover, Example 4 confirms that the classical MNL model overestimates profits. By treating the no-purchase utility as a constant and ignoring the PoC effect, the MNL baseline overstates the purchase probabilities for assortments with size different from $B$ , leading to an inflated projection of total expected revenue.

Proposition 1 directly leads to the following corollary, which compares choice probabilities and consumer surplus between the classical MNL model and the MNL with PoC model. ³

Corollary 1

When $|S_{\mathsf{M}\mathsf{N}\mathsf{L}}|\le B$ , the probability of choosing any product $i\in \mathcal{N}$ is weakly lower under the classical MNL model than under the MNL with PoC model, and consumer surplus is greater under the MNL with PoC model.

5. Joint assortment and price optimization

In this section, we study the optimal pricing strategy in the presence of the PoC effect. Instead of solely focusing on pricing with a predetermined assortment, we explore joint optimization of assortment and pricing. Note that when the assortment to be offered is known, its cardinality is fixed. This leads to a constant attractiveness of the no-purchase option for a given golden number $B$ . In this case, the joint problem under the PoC is equivalent to the pricing problem under the standard MNL model. Hence, we concentrate on the more interesting problem where the firm must make simultaneous decisions on product offerings and their respective prices.

To address this problem, we first introduce a few notations. Let the price and the cost of a product $i\in \mathcal{N}$ be denoted by $p_i$ and $c_i$ , respectively. For each product $i\in \mathcal{N}$ , its price $p_i$ is determined by the firm, while $c_i$ is an exogenous parameter. Thus, the profit for product $i$ is $r_i=p_i-c_i$ . Additionally, we assume that the utility of product $i$ , denoted by $U_i$ , depends on the price of this product. Consistent with the definition of $U_i$ in Section 3, we have $U_i(p_i)=u_i(p_i)+{ϵ}_i$ , where the utility function $u_i(p_i)$ is deterministic and it depends on the price of product $i$ , and ${ϵ}_i$ represents the stochastic error of the utility. Moreover, we assume that ${ϵ}_i$ are i.i.d. random variables with Gumbel distributions specified by $\mathsf{G}\mathsf{u}\mathsf{m}\mathsf{b}\mathsf{e}\mathsf{l}(0,1)$ . To capture the PoC effect, the utility of the no-purchase option when the assortment $S$ is offered to customers whose golden number is $B$ is $U_0=u_0(|S|,B)+{ϵ}_0$ , where $u_0(|S|,B)$ is the deterministic component responsible for capturing the PoC effect (see Assumption 1) and ${ϵ}_0$ is its stochastic error.

By redefining $U_i$ , the choice probabilities in (2) also become a function of $p_i$ . That is,

$$\begin{array}{r}{\mathcal{P}}_i(S,{\mathbf{\text{p}}}_{\mathsf{S}},B)=\frac{\exp (u_i(p_i))}{\exp (u_0(|S|,B))+\sum _{i\in S}\exp (u_i(p_i))},\mathrm{\forall }i\in S^{+},\end{array}$$

where $S\subseteq \mathcal{N}$ is the offered assortment and ${\mathbf{\text{p}}}_{\mathsf{S}}:=(p_i{)}_{i\in S}$ is the respective price vector. Similarly, the profit function of equation (3) transforms into

$$\begin{array}{r}{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,{\mathbf{\text{p}}}_{\mathsf{S}},B)=\sum _{i\in S}(p_i-c_i){\mathcal{P}}_i(S,{\mathbf{\text{p}}}_{\mathsf{S}},B).\end{array}$$

Note that ${\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,{\mathbf{\text{p}}}_{\mathsf{S}},B)$ is influenced by both the assortment of products offered, $S$ , and the prices of those products, ${\mathbf{\text{p}}}_{\mathsf{S}}$ . The joint assortment and pricing problem under the PoC effect can be formally defined as

$$\begin{array}{r}(S_{\mathsf{P}\mathsf{o}\mathsf{C}},{\mathbf{\text{p}}}_{{\mathsf{S}}_{\mathsf{P}\mathsf{o}\mathsf{C}}}^{*}):=\underset{S\subseteq \mathcal{N},{\mathbf{\text{p}}}_{\mathsf{S}}}{\mathrm{argmax}}{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,{\mathbf{\text{p}}}_{\mathsf{S}},B),\end{array}$$

(11)

where $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ represents the optimal assortment of Problem (11) and ${\mathbf{\text{p}}}_{{\mathsf{S}}_{\mathsf{P}\mathsf{o}\mathsf{C}}}^{*}$ denotes the optimal prices of the products within this assortment. With a slight abuse of notation, $S_{\mathsf{P}\mathsf{o}\mathsf{C}}$ denotes the optimal assortment for the joint assortment and pricing problem under the PoC effect throughout this section.

Recall the notion of subproblems introduced in Section 4, where we solved Problem (4) by decomposing it into $n$ subproblems that have a strict constraint on the size of the offer set. This decomposition, in turn, leads to a constant value for the attractiveness of the no-purchase option. Similarly, to tackle Problem (11), we decompose the problem into several subproblems, which aim to find $(S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k),{\mathbf{\text{p}}}_{{\mathsf{S}}_{\mathsf{P}\mathsf{o}\mathsf{C}}\mathsf{(}\mathsf{k}\mathsf{)}}^{*})$ , slightly abusing notations. In this section, $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ represents an assortment with a cardinality of $k$ and ${\mathbf{\text{p}}}_{{\mathsf{S}}_{\mathsf{P}\mathsf{o}\mathsf{C}}\mathsf{(}\mathsf{k}\mathsf{)}}^{*}$ denotes the vector of optimal prices of the products in assortment $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ . Together, they maximize the profit in the joint price and assortment optimization problem under the PoC effect. Subproblem $k$ , denoted as $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ , refers to the problem of determining $(S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k),{\mathbf{\text{p}}}_{{\mathsf{S}}_{\mathsf{P}\mathsf{o}\mathsf{C}}\mathsf{(}\mathsf{k}\mathsf{)}}^{*})$ , that is,

$$\begin{array}{r}(S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k),{\mathbf{\text{p}}}_{{\mathsf{S}}_{\mathsf{P}\mathsf{o}\mathsf{C}}\mathsf{(}\mathsf{k}\mathsf{)}}^{*}):=\underset{S\subseteq \mathcal{N}:|S|=k,{\mathbf{\text{p}}}_{\mathsf{S}}}{\mathrm{argmax}}{\mathsf{R}\mathsf{e}\mathsf{v}}_{\mathsf{P}\mathsf{o}\mathsf{C}}(S,{\mathbf{\text{p}}}_{\mathsf{S}},B).\end{array}$$

(12)

The attractiveness of the no-purchase option in $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ (i.e., $a_0(k,B)$ ) is a constant number. Therefore, Problem (12) can be viewed as the joint assortment and price optimization problem under the MNL model, where the size of the offered assortment has to be $k$ . A closely related problem to $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ is addressed by Wang (2012), who considers the joint assortment and pricing problem under the MNL model, with a constraint on the maximum size of the assortment. Corollary 1 of Wang (2012) demonstrates that if prices are decision variables, then an assortment of the full capacity size should be offered to customers, and prices are set accordingly. Therefore, the problem addressed in Wang (2012) is identical to Problem (12), and we can utilize their solution to solve $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ . The solution of Wang (2012) relies on some regularity assumptions on the utility function, presented below.

Assumption 3

• (i)

For each product $i\in \mathcal{N}$ , $\underset{p_i\to \mathrm{\infty }}{lim}(p_i-c_i)e^{u_i(p_i)}=0$ .

(ii)

For each product $i\in \mathcal{N}$ , $\mathrm{\partial }{u}_i(p_i)/\mathrm{\partial }{p}_i$ exists and is negative for all $p_i\in \mathbb{R}$ .

(iii)

For each product $i\in \mathcal{N}$ , ${\mathrm{\partial }}^2{u}_i(p_i)/\mathrm{\partial }{p}_i^2$ exists and is nonpositive for all $p_i\in \mathbb{R}$ .

Assumption 3(i) ensures that $\underset{p_i\to \mathrm{\infty }}{lim}(p_i-c_i){\mathcal{P}}_i(S,{\mathbf{\text{p}}}_{\mathsf{S}},B)=0$ . This is consistent with the pricing literature that assumes that an infinite price completely diminishes the demand (Cohen, 1977; Gallego and Van Ryzin, 1997). Assumption 3(ii) guarantees that the probability of selecting each product decreases with its own price and increases with other products’ prices in the assortment (Wang, 2012). Assumption 3(iii) is consistent with the notion of risk aversion, a widely applied concept in psychology, economics, and finance (Pratt, 1978). In essence, a risk-averse individual prefers to receive a certain amount of money equivalent to the predicted value of a risk, rather than taking the risk itself (Menezes and Hanson, 1970). This behavior is captured by a decreasing concave utility function, which precisely reflects the properties imposed by Assumptions 3(ii) and (iii), since diminishing marginal utility implies that the utility lost from a potential price increase outweighs the utility gained from an equivalent decrease in price. Proposition 2 shows that subproblem $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ can be solved efficiently under Assumptions 3.

Proposition 2

Under Assumption 3, $\mathsf{S}\mathsf{u}\mathsf{b}(k)$ can be reduced to the problem of finding the unique fixed point of $\mathcal{H}(\theta )=\underset{|S|=k}{max}\sum _{i\in S}{\tilde{h}}_i(\theta ),$ where $\mathcal{H}(\theta )$ is decreasing in $\theta$ . The total number of intersections of functions ${\tilde{h}}_i(\theta )$ and ${\tilde{h}}_j(\theta )$ for all $i,j\in \mathcal{N}$ and $i\ne j$ , is in $O(n^2)$ , where $n=|\mathcal{N}|$ . Moreover, there exists an algorithm for finding the unique fixed point in $O(\log n)$ time.

Proof Proof Sketch:

We provide an overview of the proof of the proposition; the details can be found in online Appendix A. Consider an assortment $S\subseteq \mathcal{N}$ with $|S|=k$ as a solution to Problem (12). We aim to determine the prices that maximize the firm’s profit for assortment $S$ , as formulated in Problem (13). We only consider prices that yield a positive market share for each product in the assortment, and then optimize profit across all possible assortments:

$$\begin{array}{rl}\underset{{\mathbf{\text{p}}}_{\mathsf{S}}}{max}& \mathsf{R}\mathsf{e}\mathsf{v}(S,{\mathbf{\text{p}}}_{\mathsf{S}},B)\\ \text{s.t.}& {\mathcal{P}}_i(S,{\mathbf{\text{p}}}_{\mathsf{S}},B)>0\mathrm{\forall }i\in S.\end{array}$$

(13)

Applying the Karush–Kuhn–Tucker (KKT) conditions to Problem (13), Wang (2012) shows that Assumption 3 implies a one-to-one relationship between price $p_i$ and parameter $\theta$ for each product $i\in \mathcal{N}$ , where $\theta :=p_i-c_i+1/u_i^{\mathrm{\prime }}(p_i)$ and $u_i^{\mathrm{\prime }}(p_i)=\mathrm{\partial }{u}_i(p_i)/\mathrm{\partial }{p}_i$ . Defining $f_i(\theta )$ as the price of product $i$ corresponding to $\theta$ and letting ${\tilde{h}}_i:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be defined as ${\tilde{h}}_i(\theta )=-e^{u_i(f_i(\theta ))}/(a_0(k,B)u_i^{\mathrm{\prime }}(f_i(\theta )))$ for all $i\in S$ , where ${\mathbb{R}}_{+}$ denotes the set of positive real numbers, we can express the KKT conditions as:

$$\begin{array}{r}\theta =\sum _{i\in S}{\tilde{h}}_i(\theta ).\end{array}$$

(14)

Wang (2012) shows that, under Assumption 3, ${\tilde{h}}_i(\theta )$ is decreasing in $\theta$ , and for any given $S$ , there exists a unique solution to (14), denoted by ${\theta }_S^{*}$ , which corresponds to the optimal profit for Problem (13). Thus, Problem (12) can be reformulated as selecting an assortment $S$ that maximizes the solution to (14), that is:

$$\begin{array}{r}max\{\theta \in {\mathbb{R}}_{+}:S\subseteq \mathcal{N},|S|=k\text{ and }\theta =\sum _{i\in S}{\tilde{h}}_i(\theta )\}.\end{array}$$

(15)

Defining $\mathcal{H}(\theta )=\underset{|S|=k}{max}\sum _{i\in S}{\tilde{h}}_i(\theta )$ , Problem (15) is equivalent to finding the unique fixed point of $\mathcal{H}(\theta )$ , that is, the solution to $\theta =\mathcal{H}(\theta )$ . Theorem 1 of Wang (2012) establishes that under Assumption 3, $\mathcal{H}(\theta )$ is strictly decreasing in $\theta$ with a unique fixed point ${\theta }_k^{*}$ . The assortment $S_{\mathsf{P}\mathsf{o}\mathsf{C}}(k)$ corresponding to ${\theta }_k^{*}$ is optimal for Problem (12).

To solve Problem (15), we develop an efficient algorithm that avoids enumerating all $(\genfrac{}{}{0ex}{}{n}{k})$ possible subsets of $\mathcal{N}$ . Similar to the approach in Section 4.2, we identify intersection points where ${\tilde{h}}_i(\theta )={\tilde{h}}_j(\theta )$ for each pair $i,j\in \mathcal{N}$ with $i\ne j$ . These intersection points partition the positive real line. Wang (2012) shows that under Assumption 3, each pair ${\tilde{h}}_i(\theta )$ and ${\tilde{h}}_j(\theta )$ intersects at most once, yielding at most $(\genfrac{}{}{0ex}{}{n}{2})\in O(n^2)$ intersections. Between any two consecutive intersections, the ordering of $\{{\tilde{h}}_i(\theta ){\}}_{i\in \mathcal{N}}$ remains fixed, allowing us to compute $\mathcal{H}(\theta )$ within each interval using Algorithm 1. Since $\mathcal{H}(\theta )$ is decreasing in $\theta$ , binary search efficiently identifies the interval containing the fixed point ${\theta }_k^{*}$ . The binary search procedure follows the assortment-optimization-MNL algorithm of Wang (2012). With $O(n^2)$ intersection points, the algorithm runs in $O(\log n)$ time.

We note that although $h_i(\cdot )$ lines, used to solve the assortment optimization problem under MNL with PoC in Section 4.2, and the function ${\tilde{h}}_i(\cdot )$ , used to solve the joint problem under PoC, may appear notationally similar, they serve fundamentally different purposes. The distinction stems from the increased complexity of the joint problem. Specifically, $h_i(\cdot )$ ’s are linear functions used to geometrically enumerate viable candidates for the optimal assortment under MNL with PoC. In contrast, the ${\tilde{h}}_i(\cdot )$ functions are nonlinear in price and utility, and are used to identify the unique fixed point of $\mathcal{H}(\theta )$ for a given offered assortment size in the joint optimization problem.

To illustrate the application of Proposition 2, we consider the following simple example, inspired by Example 1 of Wang (2012).

Example 5

Consider two products, indexed by $1$ and $2$ , with utility of product $i$ given by $u_i(p_i)={\alpha }_i-{\beta }_i{p}_i$ . Here, ${\alpha }_i$ denotes the intrinsic utility of product $i$ , $p_i$ the price, and ${\beta }_i$ the price sensitivity. Define ${\hat{\alpha }}_i={\alpha }_i-{\beta }_i{c}_i-\log ({\beta }_i)-1$ . Then, by definition, ${\tilde{h}}_i(\theta )=e^{{\hat{\alpha }}_i-{\beta }_i\theta }/a_0(k,B)$ .

Set ${\beta }_1=0.2$ , ${\beta }_2=0.09$ , ${\hat{\alpha }}_1=2.4$ , and ${\hat{\alpha }}_2=2.2$ . Moreover, for the classical MNL model, set $a_0=1$ ; for the MNL with PoC model, set $B=1$ , $a_0(|S|=1,B=1)=1$ , and $a_0(|S|=2,B=1)=3$ . Figure 4 illustrates the fixed-point characterization underlying Proposition 2. In each panel, we plot the functions ${\tilde{h}}_1(\theta )$ and ${\tilde{h}}_2(\theta )$ , along with the function $\mathcal{H}(\theta )$ and the $45\circ$ line. By Proposition 2, the intersection of $\mathcal{H}(\theta )$ with the $45\circ$ line determines the fixed point ${\theta }^{*}$ , corresponding to the optimal profit.

Figure 4.

Comparison of optimal profits in Example 5 under the classical MNL and the MNL with PoC models.

Under the MNL model, ${\tilde{h}}_1(\theta )=e^{2.4-0.2\theta }$ and ${\tilde{h}}_2(\theta )=e^{2.2-0.09\theta }$ . In this case, the optimal assortment includes both products (Wang, 2012); that is, $\mathcal{H}(\theta )={\tilde{h}}_1(\theta )+{\tilde{h}}_2(\theta )$ . As shown in Figure 4(a), the intersection of $\mathcal{H}(\theta )$ and the $45\circ$ line gives ${\theta }^{*}=\mathcal{H}({\theta }^{*})=7.27$ , which is the optimal profit under MNL. For the MNL with PoC model, we first solve Problem (12) for $k=1$ and $k=2$ . When $k=1$ , ${\tilde{h}}_1(\theta )=e^{2.4-0.2\theta }$ , ${\tilde{h}}_2(\theta )=e^{2.2-0.09\theta }$ , and $\mathcal{H}(\theta )=max\{{\tilde{h}}_1(\theta ),{\tilde{h}}_2(\theta )\}$ . That is, $\mathcal{H}(\theta )$ corresponds to the upper envelope of the functions ${\tilde{h}}_i(\theta )$ . As shown in Figure 4(b), the fixed point is ${\theta }^{*}=\mathcal{H}({\theta }^{*})={\tilde{h}}_2({\theta }^{*})=5.50$ . Thus, the optimal assortment for $k=1$ contains only product $2$ with profit $5.50$ . When $k=2$ , ${\tilde{h}}_1(\theta )=e^{2.4-0.2\theta }/3$ , ${\tilde{h}}_2(\theta )=e^{2.2-0.09\theta }/3$ , and $\mathcal{H}(\theta )={\tilde{h}}_1(\theta )+{\tilde{h}}_2(\theta )$ . Figure 4(c) shows that ${\theta }^{*}=\mathcal{H}({\theta }^{*})=3.84$ . Comparing the profit values across $k=1$ and $k=2$ , the optimal solution under the MNL with PoC model is to offer only product 2, and the resulting profit is 5.50.

Example 5 shows that the PoC effect reduces the optimal assortment size relative to the classical MNL model, under which the optimal assortment always includes all products (Wang, 2012). It also continues to highlight that the classical MNL model overstates purchase probabilities for assortments larger than B, leading to inflated revenue projections. This is consistent with the findings in Example 4.

6. Model estimation and numerical studies

This section describes the estimation procedure for the proposed PoC model and examines its numerical performance. We first present an efficient method for estimating an MNL with PoC model using the maximum likelihood estimation in Section 6.1. We then evaluate the performance of the estimated PoC model relative to the classical MNL through three numerical studies. To maintain focus, we present in the main text the baseline setting in which all products are priced identically, corresponding to the theoretical framework in Section 4.1. Using synthetic data generated based on Shah and Wolford (2007), we evaluate model fit and compare optimal assortments. The remaining studies, which consider heterogeneous product prices with data generated under the MMNL model, are deferred to online Appendix B. Specifically, online Appendix B.1 examines model fit and optimal profit under assortment optimization, while Appendix B.2 studies joint assortment and price optimization.

Although our estimation and comparison focus on the MNL with PoC model, the framework naturally extends to more general models such as the NL and the MMNL models. We refer the readers to Li et al. (2015) and Train (2009) for NL estimation and McFadden and Train (2000) and Train (2009) for MMNL estimation.

6.1. Concavity of log-likelihood function

The modified MNL with PoC model has a parameter $u_j$ for each option $j\in \mathcal{N}$ , a golden number $B$ , and $u_0(|S|,B)$ . We assume that $u_0(|S|,B)$ conforms with Assumption 1. Let $\mathit{u}=(u_1,\dots ,u_n)$ denote the vector of utilities for products $1,\dots ,n$ .

We describe the setup for the parameter estimation problem. Consider a dataset that contains the offered assortments and the choices of $T$ customers. For each customer $t=1,\dots ,T$ , let $S^t$ denote the offered assortment and $c_t\in S^t\cup \{0\}$ denote the customer’s choice. We assume that customers make decisions according to the modified MNL with PoC model with parameters $(\mathit{u},u_0(|S^t|,B))$ and that the choices of customers are independent of each other. Then, the likelihood function of our PoC model can be written as

$$\begin{array}{r}L(\mathit{\rho },B)=\prod _{t=1}^T\frac{\exp (u_{c_t})}{\exp (u_0(|S^t|,B))+\sum _{i\in S^t}\exp (u_i)},\end{array}$$

where $\mathit{\rho }$ is a vector of length $2n$ , whose first $n$ elements are the estimates for $u_i$ for $i\in \mathcal{N}$ , and the second $n$ elements are the estimates for $u_0(k,B)$ for $k\in \{1,\dots ,n\}$ . We denote the $m$ -th element of $\mathit{\rho }$ by ${\rho }_m$ . Therefore, the log-likelihood function is

$$\begin{array}{r}LL(\mathit{\rho },B)=\sum _{t=1}^T[u_{c_t}-\log (\exp (u_0(|S^t|,B))+\sum _{i\in S^t}\exp (u_i))].\end{array}$$

(16)

Then, the parameter estimation problem can be formulated as

$$\begin{array}{rl}\underset{\mathit{\rho },B}{max}& LL(\mathit{\rho },B)\\ \text{s.t.}& u_0(k-1,B)\le u_0(k,B)\mathrm{\forall }k>B\\ & u_0(k,B)\ge u_0(k+1,B)\mathrm{\forall }k<B\\ & u_0(B,B)=0,\end{array}$$

(17)

where the constraints follow from Assumption 1. Note that we do not assume any specific functional form for the no-purchase option. Because of the linear constraints, Problem (17) is also convex (Boyd and Vandenberghe, 2004). Thus, we can solve Problem (17) for each $B\in \{0,1,\dots ,n\}$ and select the solution with the highest log-likelihood value. We present this result formally in Proposition 3.

Proposition 3

For a given $B$ , Problem (17) is a convex optimization problem with linear constraints.

Proposition 3 indicates that we can calibrate $\mathit{\rho }$ efficiently. However, overfitting issues may arise because, for $u_0(k,B)$ alone, the model has $n$ parameters (for $1\le k\le n$ ) to be estimated, whereas there is only one parameter for the no-purchase utility in the MNL model. To alleviate the overfitting issues, we can assume a functional form for the utility of the no-purchase option. Next, we introduce two functional forms $u_0(\cdot ,B)$ in the PoC model and show the concavity of the log-likelihood function under these two functional shapes.

The first functional form is a quadratic utility function, which is widely used in various disciplines and is proven to be mathematically tractable (Luan et al., 2020). Specifically, we assume that the utility of the no-purchase option when offering $S\subseteq \mathcal{N}$ to the customer follows

$$\begin{array}{r}{u}_0(|S|,B)={\zeta }_1(B-|S|{)}^2+u_0(B,B).\end{array}$$

(18)

The second functional form is a piece-wise linear utility function, employed by Wang (2018) to integrate prospect theory into the MNL model. We thus assume that the utility of the no-purchase option when offering $S\subseteq \mathcal{N}$ to the customer follows

$$\begin{array}{r}{u}_0(|S|,B)={\zeta }_2|(B-|S|)|+u_0(B,B).\end{array}$$

(19)

Equation (18) (resp. (19)) defines the no-purchase option’s utility, addressing overfitting by using only three parameters, $u_0(B,B)$ , $B$ , and ${\zeta }_1$ (resp. ${\zeta }_2$ ). This equation, with $u_0(B,B)=0$ and ${\zeta }_1>0$ (resp. ${\zeta }_2>0$ ), satisfies Assumption 1. It can be shown that to estimate the parameters of $u_0(|S|,B)$ , at least three assortment sizes are needed. The $u_0(B,B)=0$ constraint is nonrestrictive as utility values are relative, and it can be achieved by subtracting $u_0(B,B)$ from all utilities. Thus, this constraint does not need to be imposed on the log-likelihood function. In addition, it is also unnecessary to enforce the nonnegativity constraint for ${\zeta }_1$ (resp. ${\zeta }_2$ ). If ${\zeta }_1$ (resp. ${\zeta }_2$ ) is found to be nonpositive, then it indicates that the special function form for the model is not appropriate. The following result shows that the log-likelihood function is concave under these functional forms.

Corollary 2

For a given $B$ , if the utility of the no-purchase option follows equation (18) or (19), then $LL(\mathit{\rho },B)$ is concave.

In sum, the introduced functional forms serve as examples to mitigate the overfitting issue due to the utility of the no-purchase option under the PoC model. They are consistent with Assumption 1 and ensure the concavity of the log-likelihood function. Importantly, the PoC model is flexible and can incorporate alternative functional forms that meet the same criteria.

Lastly, although our estimation and comparison focus on the MNL with PoC model, the framework naturally extends to more general models such as the NL and the MMNL models. We refer the readers to Li et al. (2015) and Train (2009) for NL estimation and McFadden and Train (2000) and Train (2009) for MMNL estimation. The intuition for these extensions lies in the flexible specification of the no-purchase utility. Specifically, the structure defined in Problem (17) can be embedded into the lower-level nests of an NL model to capture category-specific choice overload. Similarly, for the MMNL model, the PoC framework can be applied to each customer segment. By allowing segment-specific parameters for the U-shaped no-purchase utility, we account for heterogeneity in variety-seeking and choice overload across different segments.

6.2. Numerical studies

In this section, we simulate both product utilities and customer choices following the structure of Shah and Wolford (2007). We begin by outlining their experimental design and then describe how our simulation framework replicates it.

Shah and Wolford’s (2007) experiment examined how assortment size influences purchase behavior. The products consisted of 20 roller-ball pens differing in appearance, feel, and mechanism, and the participants were undergraduate students. First, 20 participants rated each pen on a scale from 1 (highly undesirable) to 10 (highly desirable). Each pen was rated twice by each participant in a random order, yielding 40 ratings per pen. These ratings were then averaged to obtain a mean desirability score for each pen. The ranking of pens based on the desirability score served as the basis for constructing the offer sets used in the experiment.

The main experiment consisted of 10 offer sets, with assortment sizes increasing in increments of two from 2 to 20 pens. The smallest set included the top-ranked pen (1st) and the pen ranked 11th. Each subsequent set retained all pens from the previous set and added two new pens: the next-highest-ranked pen not yet included and the pen ranked 10 places lower in the desirability ranking. For instance, the second set contained the 2nd- and 12th-ranked pens in addition to those in the first set, and the third set added the 3rd- and 13th-ranked pens, and so forth. Each assortment was presented to 10 students who had not participated in any previous step, resulting in 100 students across the 10 sets. All pens were priced identically, and the proportion of participants making a purchase at each assortment size is reported in Figure 1.

We assume a setting with 20 distinct products, corresponding to the 20 pens in Shah and Wolford’s (2007) experiment. To match the pretest rating procedure, we generate synthetic desirability scores for each product as if they were rated twice by 20 individuals, producing 40 ratings per product. Each rating is drawn independently from a discrete uniform distribution on $\{1,2,\dots ,10\}$ , matching the 1–10 scale used in the original study. The 40 ratings for each product are averaged to obtain a mean desirability score for that product. Let the mean desirability score of product $i\in \{1,2,\dots ,20\}$ be denoted by ${\eta }_i^{\mathit{\text{SW}}}$ and the products be labeled in decreasing order of mean desirability score, that is, ${\eta }_1^{\mathit{\text{SW}}}\ge {\eta }_2^{\mathit{\text{SW}}}\ge \cdots \ge {\eta }_{20}^{\mathit{\text{SW}}}$ .

To generate the deterministic part of the utility of product $i$ , denoted by $u_i^{\mathit{\text{SW}}}$ , the mean desirability scores are linearly rescaled so that the minimum equals $u_{min}$ and the maximum equals $u_{max}$ , where $u_{min}\in \{-0.5,0,0.5\}$ and $u_{max}\in \{1,1.5,2\}$ . Formally, we have

$$\begin{array}{r}{u}_i^{\mathit{\text{SW}}}=\frac{u_{max}-u_{min}}{{\eta }_{max}^{\mathit{\text{SW}}}-{\eta }_{min}^{\mathit{\text{SW}}}}({\eta }_i^{\mathit{\text{SW}}}-{\eta }_{min}^{\mathit{\text{SW}}})+u_{min},\end{array}$$

(20)

where ${\eta }_{min}^{\mathit{\text{SW}}}$ and ${\eta }_{max}^{\mathit{\text{SW}}}$ denote the minimum and maximum of the mean desirability scores across all products. Each pair $(u_{min},u_{max})$ defines a distinct scenario, corresponding to a specific rescaling configuration used to generate product utilities. This results in a total of nine scenarios. The resulting scaled product utilities serve as inputs for simulating sales data under the classical MNL model.

To generate sales data for each scenario, we replicate the offer set construction procedure from Shah and Wolford’s (2007) experiment. Let $S_k^{\mathit{\text{SW}}}$ denote the assortment of size $2k$ constructed according to their procedure, where $k\in \{1,2,\dots ,10\}$ . The $k$ -th assortment is defined as $S_k^{\mathit{\text{SW}}}=\{1,\dots ,k\}\cup \{10+1,\dots ,10+k\}$ , which contains the top $k$ products and the products ranked exactly ten places lower in the original ordering.

Following Shah and Wolford (2007), we assume that when offering assortment $S_k^{\mathit{\text{SW}}}$ , the customer can choose an item from $S_k^{\mathit{\text{SW}}}\cup \{0\}$ , where $0$ denotes the no-purchase option. The no-purchase option for assortment $S_k^{\mathit{\text{SW}}}$ has a deterministic utility denoted by $u_{0k}^{\mathit{\text{SW}}}$ . This utility is set to replicate the observed no-purchase frequency in the original experiment by Shah and Wolford (2007) (see Figure 1). Let $p_{0k}^{\mathit{\text{SW}}}$ denote the empirical no-purchase probability for assortment $S_k^{\mathit{\text{SW}}}$ . Then, from the choice probability of the PoC model:

$$\begin{array}{r}{u}_{0k}^{\mathit{\text{SW}}}=\ln \left(\frac{p_{0k}^{\mathit{\text{SW}}}}{1-p_{0k}^{\mathit{\text{SW}}}}\sum _{j\in S_k^{\mathit{\text{SW}}}}{e}^{u_j^{\mathit{\text{SW}}}}\right).\end{array}$$

(21)

Equation (21) ensures that the probability of choosing the no-purchase option equals $p_{0k}^{\mathit{\text{SW}}}$ under MNL.

For each scenario (i.e., each $(u_{min},u_{max})$ pair), we generate 100 independent simulations by resampling synthetic desirability scores and computing product utilities using equation (20), as well as no-purchase utilities using equation (21). In each simulation, we generate 2,500 transactions for each assortment $S_k^{\mathit{\text{SW}}}$ . Here, a transaction refers to an offered assortment and the corresponding customer choice, generated according to the classical MNL model. Since $k\in \{1,2,\dots ,10\}$ , each simulation corresponds to a dataset consisting of 25,000 customer transactions.

For each simulation $i\in \{1,\dots ,100\}$ , we compute the log-likelihood for the calibrated classical MNL model, denoted as $L{L}_{\text{MNL}}^i$ . Subsequently, we estimate and report the log-likelihood of our proposed model, denoted as $L{L}_{\mathsf{P}\mathsf{o}\mathsf{C}}^i$ . In our PoC model, we assume $u_0(|S|,B)={\zeta }_1(B-|S|{)}^2$ , for some ${\zeta }_1>0$ , which aligns with Assumption 1 and preserves the concavity of the log-likelihood function by Corollary 2.

Given that the PoC model includes the additional parameters ${\zeta }_1$ and $B$ , we adopt the Akaike information criterion (AIC) to control for overfitting, as done by Şimşek and Topaloglu (2018) and Li et al. (2015). AIC provides a standard method to control for overfitting by penalizing model complexity. Formally, AIC is defined as $2\nu -2LL$ , where $\nu$ denotes the number of model parameters and $LL$ is the log-likelihood of the fitted model (Akaike, 1998). By penalizing excessive complexity, that is, adding more parameters to the model, AIC discourages models that overfit the data to improve the log-likelihood.

We first specify the number of parameters $\nu$ for the classical MNL and PoC models. For the classical MNL, ${\nu }_{\text{MNL}}=20$ , reflecting one utility parameter per product; the no-purchase option is normalized to zero utility and is not counted as a parameter. For the PoC model, the total number of parameters is ${\nu }_{\text{PoC}}=22$ , including 20 utility parameters for the 20 products and two parameters, ${\zeta }_1$ and $B$ , used to define the no-purchase utility $u_0(|S|,B)={\zeta }_1(B-|S|{)}^2$ . Consistent with the classical MNL, the PoC model sets the baseline utility of the no-purchase option at $u_0(B,B)=0$ , estimating only ${\zeta }_1$ and $B$ to capture its variation. Thus, in the $i$ -th simulation, the AIC for the classical MNL model is $AI{C}_{\text{MNL}}^i=2\times 20-2L{L}_{\text{MNL}}^i$ , while for the PoC model it is $AI{C}_{\mathsf{P}\mathsf{o}\mathsf{C}}^i=2\times 22-2L{L}_{\mathsf{P}\mathsf{o}\mathsf{C}}^i$ .

To quantify the improvement of the PoC model over the classical MNL, we calculate the average percentage improvement in the log-likelihood, denoted by $AP{I}_{LL}$ :

$$\begin{array}{rl}AP{I}_{LL}& =\frac{1}{\kappa }\sum _{i=1}^{\kappa }\frac{|L{L}_{\text{MNL}}^i|-|L{L}_{\mathsf{P}\mathsf{o}\mathsf{C}}^i|}{|L{L}_{\text{MNL}}^i|}\times 100\\ & =\sum _{i=1}^{100}\frac{|L{L}_{\text{MNL}}^i|-|L{L}_{\mathsf{P}\mathsf{o}\mathsf{C}}^i|}{|L{L}_{\text{MNL}}^i|},\end{array}$$

where $\kappa =100$ is the number of simulations. We define the average percentage improvement in AIC as

$$\begin{array}{r}AP{I}_{AIC}=\sum _{i=1}^{100}\frac{AI{C}_{\text{MNL}}^i-AI{C}_{\mathsf{P}\mathsf{o}\mathsf{C}}^i}{AI{C}_{\text{MNL}}^i}.\end{array}$$

(22)

Absolute values are not required here, as AIC is always positive by construction. A smaller AIC value corresponds to improved model fit. Thus, a positive value for $AP{I}_{LL}$ or $AP{I}_{AIC}$ reflects an average improvement of the PoC model over the classical MNL model.

The results of this numerical study are summarized in Table 3, where each row corresponds to a scenario. The first two columns of Table 3 list the values of $u_{min}$ and $u_{max}$ , which define the range of product utilities for each scenario. Columns 3 and 4 report the average log-likelihoods across 100 simulations for the classical MNL and PoC models, denoted by $L{L}_{\text{MNL}}$ and $L{L}_{\mathsf{P}\mathsf{o}\mathsf{C}}$ , respectively. Columns 5 and 6 report the average AIC values for the classical MNL and PoC models, providing complementary measures for assessing both pure model fit and complexity-penalized performance. Columns 7 and 8 report the average percentage improvements in the log-likelihood and AIC across 100 simulations, denoted by $AP{I}_{LL}$ and $AP{I}_{AIC}$ , respectively. These metrics highlight the consistent performance advantage of the PoC model over the classical MNL model. Notably, the improvement becomes more pronounced as the PoC effect intensifies, that is, when the utility of the no-purchase option increases relative to the product utilities. For a fixed $u_{min}$ , a higher $u_{max}$ generates products with greater utilities, thereby reducing the relative influence of the no-purchase option on the choice probabilities, defined in equation (2). Across all scenarios, the values of $AP{I}_{LL}$ and $AP{I}_{AIC}$ are similar. This is because incorporating the PoC effect into the classical MNL model adds only two parameters, resulting in a minimal increase in model complexity. Therefore, we choose to only report $AP{I}_{AIC}$ for the remaining numerical studies presented in online Appendices B.1 and B.2.

Table 3.

Comparison of the MNL and PoC models for different utility scenarios.

$u_{min}$	$u_{max}$	$L{L}_{\mathsf{M}\mathsf{N}\mathsf{L}}$	$L{L}_{\mathsf{P}\mathsf{o}\mathsf{C}}$	$AI{C}_{\mathsf{M}\mathsf{N}\mathsf{L}}$	$AI{C}_{\mathsf{P}\mathsf{o}\mathsf{C}}$	$AP{I}_{LL}$	$AP{I}_{AIC}$	$\overline{|S_{\mathsf{P}\mathsf{o}\mathsf{C}}|}$	$\overline{B}$	Time ${}_{\mathsf{M}\mathsf{N}\mathsf{L}}$	Time ${}_{\mathsf{P}\mathsf{o}\mathsf{C}}$
$-0.5$	1	$-45,656.37$	$-43,700.21$	91,352.73	87,444.42	4.28	4.27	8.16	5.73	0.204	7.157
$-0.5$	1.5	$-44,957.88$	$-43,107.22$	89,955.76	86,258.44	4.12	4.11	8.11	5.94	0.177	7.282
$-0.5$	2	$-44,096.14$	$-42,359.12$	88,232.28	84,762.23	3.94	3.93	8.13	6.26	0.176	6.908
0	1	$-46,273.72$	$-44,219.09$	92,587.44	88,482.18	4.44	4.43	8.22	5.54	0.128	6.955
0	1.5	$-45,613.67$	$-43,673.33$	91,267.34	87,390.66	4.25	4.24	8.11	5.75	0.149	6.964
0	2	$-44,858.15$	$-43,019.27$	89,756.30	86,082.53	4.10	4.09	8.05	5.95	0.161	6.852
0.5	1	$-46,635.59$	$-44,461.74$	93,311.18	88,967.47	4.66	4.65	8.20	5.29	0.107	6.833
0.5	1.5	$-46,234.85$	$-44,192.84$	92,509.70	88,429.68	4.42	4.41	8.24	5.48	0.122	6.830
0.5	2	$-45,648.43$	$-43,693.98$	91,336.85	87,431.96	4.28	4.27	8.12	5.73	0.143	6.847

Beyond predictive accuracy, the simulations also highlight behavioral differences in assortment selection. With equal product prices, the classical MNL model consistently recommends the full assortment of 20 products. In contrast, the PoC model often selects a smaller subset. Column 9 reports the average optimal assortment size under the PoC model, $\overline{|S_{\mathsf{P}\mathsf{o}\mathsf{C}}|}$ , which is approximately 8. This demonstrates that not all products should be offered in the presence of PoC, even when prices are identical. This result aligns with Shah and Wolford (2007) (Figure 1), where purchase probability peaked at an assortment size between 9 and 11. ⁴ Finally, Column 10 reports the average estimated “golden number” of products, $\overline{B}$ . Across scenarios, $\overline{|S_{\mathsf{P}\mathsf{o}\mathsf{C}}|}>\overline{B}$ , consistent with Proposition 1(iii).

Finally, Columns 11 and 12 report the average parameter estimation times for the classical MNL and PoC models, denoted by Time $\text{MNL}$ and Time $\text{PoC}$ , respectively. All computations were conducted on a 2020 MacBook Air (Apple M1, 16 GB memory) using MATLAB R2022a. On average, the PoC model requires approximately 47 times more computation time than the classical MNL model. This difference arises from three factors. First, when $n=20$ , the PoC estimation requires solving $n+1=21$ independent parameter estimation problems, one for each $B\in \{0,1,\dots ,n\}$ , whereas the MNL model requires only one. Second, the no-purchase utility is fixed at 0 in the MNL model but must be computed for each transaction in the PoC model as $u_0(|S|,B)={\zeta }_1(B-|S|{)}^2$ . Third, the PoC model has 22 parameters versus 20 in the MNL model; although modest, this increase expands the Hessian matrix from $20\times 20$ to $22\times 22$ , further increasing computational cost.

7. Choice models beyond MNL

Despite its usefulness in modeling customer choice, the MNL model inherently presents some limitations. One key limitation is the IIA property (Ben-Akiva and Lerman, 1985), which implies that adding a new product to an offer set reduces the demand for all existing products by the same proportion. The IIA property may result in biased estimations of choice probabilities. To address this limitation, more complex choice models with a greater number of parameters have been considered. For example, the NL model was first presented by Williams (1977) and later expanded by McFadden (1980). The MMNL model was introduced by Boyd and Mellman (1980) and Cardell and Dunbar (1980). Train (2009) shows that both of these choice models are consistent with RUM, while also mitigating the IIA limitation.

In this section, we showcase the applicability of the PoC framework by integrating the PoC effect into the NL and MMNL models. We address the assortment optimization and the joint assortment and price optimization problems for the NL model, and the assortment optimization problem for the MMNL model in the presence of PoC.

7.1. The NL model

The NL model extends the MNL framework by grouping similar products into nests, where nests represent different attributes, such as distinct product categories, sales channels, or retail stores (Gallego and Topaloglu, 2014). Under the NL model, products are grouped into nonoverlapping nests. A consumer first chooses a nest of products and then makes a choice among the products within the chosen nest (Davis et al., 2014). This nesting structure provides a more flexible and realistic representation of consumer choices, especially when alternatives vary in similarity, and can effectively mitigate the restriction of IIA.

We develop algorithms for the assortment optimization and the joint assortment and price optimization problems under the NL with PoC model. Note that although we use continuous prices for the MNL model in previous sections, we adopt a discrete-price formulation in the NL extension as a solution approach. This does not materially restrict the model. In particular, Mirzaee et al. (2025) show that, for a given $ϵ>0$ , one can construct a discrete set of candidate prices for each product such that the optimal solution of the resulting discrete problem achieves at least a $(1-ϵ)$ fraction of the optimal revenue of the original continuous-price problem under the NL model. The size of this discrete price set is polynomial in the input size and $1/ϵ$ . Moreover, this discretization enables a linear programming reformulation with a polynomial number of variables and constraints. Hence, the discrete-price formulation serves as a theoretically justified solution approach rather than a modeling restriction.

We provide an overview of these algorithms below, with further details available in online Appendix C. Gallego and Topaloglu (2014) study the assortment optimization problem under the NL model, subject to various types of constraints on the offer set within each nest. We build our solutions for optimization problems under PoC based on their framework. Specifically, assuming that the utility of the no-purchase option in each nest exhibits the PoC effect and using Theorem $2$ of Gallego and Topaloglu (2014), we show that the optimal solution to the assortment optimization problem under the NL with PoC model can be found by solving a linear program (LP). This LP yields the maximum obtainable expected profit among feasible candidate assortments for a nest with size $k$ , where $k\in \{0,\dots ,n\}$ and $n$ is the number of available products in each nest. Note that if the strict constraint is imposed on the assortment size in a nest, then the attractiveness of the no-purchase option in that nest has a fixed value. Finally, we use the result of Gallego and Topaloglu (2014), who show that if there is a cardinality constraint on assortments offered in each nest, then candidate assortments can be found efficiently by using the StaticMNL algorithm developed by Rusmevichientong et al. (2010). The following proposition formally presents the result of the assortment optimization problem under the NL with PoC model.

Proposition 4

The optimal solution to the assortment optimization problem under the NL with PoC model can be found by solving an LP with $m+1$ decision variables and $O(m{n}^3)$ constraints, where $m$ is the number of nests and $n$ is the number of products available in each nest.

We address the joint assortment and price optimization problem under the NL with PoC model by converting it into a constrained assortment optimization problem. We do this by generating multiple copies of each product, each with a different profit level. The main idea here is to transform the joint assortment and price optimization problem into an assortment optimization problem by constraining the offered assortments to ensure that each product is offered at most at one profit level. This conversion simplifies the problem to deciding which copies of each product should be offered if that product is to be included in the offer set. Thus, using discrete profits, rather than the continuous pricing discussed in Section 5, allows us to employ the aforementioned LP to solve the joint assortment and price optimization problem under PoC. The following proposition formally addresses the joint optimization problem under the NL with PoC model.

Proposition 5

Suppose that in each nest, there are $p$ different products, each with $b$ distinct profit levels. The optimal solution to the joint assortment and price optimization problem under the NL with PoC model can be found by solving an LP with $m+1$ decision variables and $O(m{b}^2{p}^3)$ constraints, where $m$ is the number of nests.

7.2. The MMNL model

Another DCM that addresses the IIA limitation of MNL is the MMNL model. The MMNL model allows richer substitution patterns by assuming the existence of $M$ distinct customer segments, each with its own product preferences. The firm is unaware of the segment of any arriving customer. Within each segment, customer choices follow the MNL model.

The MMNL model is consistent with RUM, and any choice model derived from the random utility model can be approximated as closely as needed by a mixture of a finite, though unknown, number of MNL models (McFadden and Train, 2000). However, Rusmevichientong et al. (2014) demonstrate that the assortment optimization problem becomes NP-hard under the MMNL model even with two mixtures. Désir et al. (2022) show that for any constant $\delta >0$ , there exists no approximation algorithm for the assortment optimization problem under the MMNL model with an approximation guarantee better than $O(1/M^{1-\delta })$ , unless NP $\subseteq$ BPP, where BPP stands for bounded-error probabilistic polynomial time.

Given that under MMNL, each segment of customers makes choices according to the MNL model, we can incorporate the PoC effect into the MMNL model by applying Assumption 1 to the no-purchase option and study the resulting assortment optimization problem. To effectively address the complexities inherent in the MMNL model, along with those introduced by the PoC effect, we need a systematic approach to manage consumer heterogeneity while simplifying assortment decisions.

For a fixed number of mixtures, $M$ , we design an FPTAS for the assortment optimization problem under the MMNL with PoC model. In short, for an $ϵ>0$ , the proposed FPTAS approximates the assortment optimization problem by discretizing the space of item attributes. Specifically, we construct a grid with step size $(1+ϵ)$ over both the attractiveness values and the product of attractiveness and profit for each item in each mixture. This discretization reformulates the assortment optimization problem as a multidimensional knapsack problem. We then develop a dynamic programming-based approximation algorithm for this knapsack problem, which yields a solution that closely approximates the maximum expected profit for the assortment optimization problem under the MMNL with PoC model. The details of the FPTAS algorithm are available in online Appendix D. Proposition 6 presents the algorithm’s performance guarantee.

Proposition 6

If the number of mixtures, $M$ , is constant, then there exists an FPTAS for the assortment optimization problem under the MMNL with PoC model with an approximation guarantee of $(1-ϵ)$ and a running time of $O(\log (n{v}_{max}{)}^{2M}\cdot (n^{2M+3}/{ϵ}^{6M}))$ , where $n$ is the number of products and $v_{max}=max\{v_{max,z}/v_{min,z}:z\in \{1,\dots ,2M\}\}$ . Moreover, for each $i\in \{1,\dots ,n\}$ and $m\in \{1,\dots ,M\}$ , we have $v_{i,m}=a_{mi}\cdot r_i$ and $v_{i,m+M}=a_{mi}$ , where $a_{mi}$ represents the attractiveness of product $i$ for a customer in segment $m$ and $r_i$ denotes the corresponding profit. Also, $v_{min,z}=min\{v_{i,z}:i\in \mathcal{N}\}$ and $v_{max,z}=max\{v_{i,z}:i\in \mathcal{N}\}$ , for each $z\in \{1,\dots ,2M\}$ .

8. Conclusion

The PoC effect has been extensively studied in behavioral research. It suggests that assortments that are either too small or too large can lower consumer purchase intent. This phenomenon creates a nonmonotone relationship between the size of the assortment and the likelihood of making a purchase. Despite its importance, PoC has not been systematically incorporated into operational revenue management decisions, leaving a gap in understanding its impact on profit-maximizing strategies for assortment and pricing. Conventional DCMs often assume that larger assortments lead to higher purchase probabilities, an assumption that contradicts PoC effects.

Our research provides a novel framework that integrates PoC into DCMs, bridging a critical gap in the literature and offering valuable insights for businesses seeking to balance assortment variety with consumer purchasing behavior. We first demonstrate the effectiveness of this framework within the MNL model, employing an efficient geometric algorithm to identify optimal assortment structures. Our results reveal that optimal assortment selection and pricing strategies must consider the cognitive effects associated with the PoC effect, which classical models typically overlook. Furthermore, we extend our approach to the NL and the MMNL models. For the NL model, we develop an exact algorithm to solve the assortment and joint assortment and pricing problems. Additionally, we design an FPTAS for the MMNL model with a constant number of mixtures, accommodating complex assortment scenarios while maintaining computational efficiency.

Our work also presents an efficient calibration method based on the maximum likelihood estimation, enabling the practical application of our models with empirical data. This contribution not only substantiates our theoretical framework, but also empowers retailers to make data-informed decisions that align with real-world consumer behavior. Lastly, the PoC framework could potentially be integrated into other behavioral choice models, such as search cost (Wang and Sahin, 2018) and consider-then-choose models (Wang, 2022), to capture distinct behavioral mechanisms simultaneously.

Future research could explore expanding this framework to multiperiod settings and dynamic assortment contexts, where assortment choices evolve. Additionally, investigating the PoC effects across broader product categories and consumer segments could yield insights into how assortment and pricing strategies may be tailored to different market needs, further enhancing the applicability of PoC-informed revenue management strategies.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478261454489 - Supplemental material for Operations management under paradox of choice

Supplemental material, sj-pdf-1-pao-10.1177_10591478261454489 for Operations management under paradox of choice by Milad Mirzaee, Elaheh Fata and Guang Li in Production and Operations Management