Bi-Objective Optimization of Product Selection and Ranking Considering Sequential Search

Abstract

Customer choices in online retailing are often influenced by sequential search behavior. However, most existing models ignore the dynamic property of this process. To address this gap, we study a bi-objective product selection and ranking (BP-SS) problem considering sequential search, aiming to jointly optimize the expected revenue and market share. We first develop a two-stage choice model with consideration sets to capture how sequential search influences customer decision-making. Based on this model, we formulate the BP-SS problem and analyze its structural properties, including problem reformulation, unimodality analysis and ranking rules. To solve the problem efficiently, we develop a dynamic programing-based approximation algorithm. Numerical experiments demonstrate that the algorithm consistently outperforms benchmark methods, especially in large-scale scenarios. Moreover, a moderate increase in the trade-off parameter substantially improves market share without a huge revenue loss. This study offers a novel and robust framework for product selection and ranking in online retailing, offering practical insights for balancing profitability and competitive positioning.

Introduction

With the rapid growth of online retailing and the increasing diversity of product offerings, retailers are facing growing challenges in optimizing product selection and ranking strategies (Chen et al., 2025; Compiani et al., 2024). The overwhelming volume of available products often exceeds customers’ cognitive limits, making it difficult for retailers to effectively manage product visibility and influence customer decisions (Dahooie et al., 2024; Huh & Liu, 2025). Although a broader product assortment can enhance consumer appeal, it may also increase decision complexity (Liu & Li, 2024). Therefore, effective product selection and ranking strategies are essential for facilitating customer decision-making and ensuring the visibility of the most relevant products.

One critical challenge in this domain arises from customer behavior under sequential search (R. Ursu et al., 2025). Empirical evidence indicates that customers are more likely to select products positioned prominently in the lists, which underscores the importance of product ranking in purchase decisions (Compiani et al., 2024; R. M. Ursu, 2018). This behavioral pattern shows that customer choices are influenced not only by the inherent utility of products but also by their ranking in the product list. In this context, sequential search plays a crucial role in shaping purchase outcomes, so optimizing the product selection and ranking decision requires careful consideration of sequential search.

Another critical challenge pertains to the objectives used in product selection and ranking. Most existing studies focus exclusively on revenue maximization and achieve substantial advancements in the fields of marketing and operations (Caro et al., 2020; Heger & Klein, 2024). However, as market competition intensifies, relying solely on revenue objectives may overlook long-term strategic goals. Increasingly, scholars and practitioners recognize the strategic importance of market share (Maddah et al., 2021; R. Wang & Sahin, 2018). In a competitive market, increasing market share enhances customer loyalty, improves brand visibility, and secures a retailer’s market position (Tan & Guo, 2023). Therefore, it is crucial for retailers to strike a balance between short-term revenue and long-term market share when optimizing product selection and ranking decisions.

To address these challenges, it is essential to effectively capture customer choice behavior when they confront numerous product options (Tan et al., 2024). The Multinomial Logit (MNL) model is one of the most widely used tools for capturing such choice behavior, due to its strong predictive ability and interpretability (Prashker & Bekhor, 2004; Train, 2009). The MNL model is widely adopted in operations and marketing for modeling utility-maximizing choice behavior (Feng et al., 2022; Talluri & van Ryzin, 2004). Recent studies further highlight its practical relevance, particularly in real-world applications. For instance, a large-scale field experiment conducted on the Alibaba platform demonstrates the practical superiority of the MNL model over machine learning approaches in terms of financial performance (Feldman et al., 2022). Motivated by these advantages, this study adopts the MNL model to investigate product selection and ranking under sequential search.

Building on these insights, this study employs the MNL model to investigate the interaction of customer choice and develops a bi-objective product selection and ranking (BP-SS) problem considering sequential search. The aim is to jointly optimize the expected revenue and market share, providing a structured understanding of how retailers can balance short-term revenue with long-term strategic competitiveness. Specifically, this research seeks to answer three key questions: (1) How does sequential search impact customer choice behavior? (2) How can we develop a mathematical model for the BP-SS problem? (3) What efficient methods can be developed to solve the proposed problem?

To address these questions, we first adopt the MNL model and introduce a two-stage choice model with consideration sets to capture customer behavior under sequential search. Based on this model, we formulate the BP-SS problem and analyze its structural properties to support algorithmic design. Given the problem’s NP-hardness, we reformulate it and develop a dynamic programing-based approximation algorithm to obtain near-optimal solutions. Finally, extensive numerical experiments using both simulation and real-world datasets are conducted to validate the effectiveness and scalability of the proposed algorithm.

The main contributions of this paper are as follows:

1. We introduce a two-stage choice model that explicitly accounts for sequential search in customer choice.

2. We formulate the BP-SS problem that integrates both the expected revenue and market share objectives, which offers a more comprehensive performance metric for retail decision-making.

3. We analyze the structural properties of the BP-SS problem, including problem reformulation, unimodality analysis, and ranking rules.

4. We present an efficient approximation algorithm based on dynamic programing. The results show that this algorithm outperforms existing benchmarks, particularly in large-scale instances.

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 describes the two-stage choice model and formulates the BP-SS problem. Section 4 presents the problem analysis, and Section 5 introduces the proposed approximation algorithm. Section 6 reports the results of numerical experiments to evaluate algorithm performance. Finally, Section 7 shows the conclusions.

Literature Review

Sequential Search

Due to limited attention or search costs, customers tend to prioritize products that appear at the top of online listings (Agarwal et al., 2011; Rock et al., 2024). Existing studies highlight the significance of product ranking in shaping customer choice behavior. Products positioned higher on the page receive more attention and benefit from enhanced visibility, leading to an increased likelihood of selection (Chandon et al., 2009; Ghose & Yang, 2009; Santos et al., 2012; R. M. Ursu et al., 2024). This behavior is referred to as sequential search, where customers browse products sequentially and make choices based on the order of appearance (Kim et al., 2017). Sequential search implies that customer choice is influenced not only by product assortment but also by the order in which products are presented. Neglecting these factors may result in biased estimations of customer preferences (Adomavicius & Zhang, 2016; Asadpour et al., 2023; Cao & Sun, 2024; Ferreira et al., 2022).

To model customer decision-making under sequential search, researchers develop two primary frameworks. The first is the Cascade model, which characterizes customer behavior as a multi-stage process and assumes that products are viewed in a top-down sequence based on their vertical position on a webpage (Gao et al., 2022; Zhao et al., 2025). The second is the choice model with consideration sets. In this model, the selection process occurs in two stages: customers initially form a consideration set of sequentially viewed products and then make an optimal choice from this set (Aouad et al., 2021; Hauser, 2014; Heger & Klein, 2024).

The related literature suggests that the consideration set offers a robust framework for explaining heterogeneity in customer preferences, accounting for up to 80% of the variance in decision-making processes, even with rough assumptions on consideration set distributions (Manzini & Mariotti, 2014). Empirical results further validate its robustness in reflecting the effects of sequential search on customer decisions (Santos et al., 2012; R. M. Ursu et al., 2024).

Product Selection and Ranking Problem

Customer Choice Model

To capture the complexity of customer decision-making, researchers have introduced numerous customer choice models (Berbeglia et al., 2022; Demoliner & Damacena, 2019). A review of related literature identifies discrete choice models as the most popular choice models, including the MNL model (Kök & Fisher, 2007; Tan & Guo, 2023), the Nested Logit model (Hensher & Greene, 2002; Tarhini et al., 2023) and the Mixed Logit model (McFadden & Train, 2000; Tan et al., 2024).

Among these models, we focus on the MNL model due to its relevance to our study. The MNL model assumes that each product is associated with a utility consisting of a deterministic component derived from observable attributes and a random component determined by unobserved factors (Lu, 2024). Due to its computational simplicity and interpretability, the MNL model is extensively applied in disciplines, such as transportation (Deneke et al., 2024; Lo & Topaloglu, 2022), marketing (R. Wang, 2021; M. Wang et al., 2023), and product decisions (Iurasova, 2025; Strauss et al., 2018).

Bi-Objective Optimization

Most existing studies on product selection and ranking focus on expected revenue maximization. For example, Gallego et al. (2020) consider consumer heterogeneity and present a product framing and pricing model. Aouad and Segev (2021) propose a polynomial-time approximation scheme to find near-optimal ranking positions for product ranking based on vertically differentiated preferences. Lei et al. (2022) examine revenue optimization under inventory constraints.

However, these studies often neglect customer-centric objectives. In contrast, a growing body of literature addresses alternative goals, such as market share maximization. For instance, R. Wang and Sahin (2018) explore the impact of customer search cost and propose the k-quasi-attractiveness-ordered strategy to maximize market share. Asadpour et al. (2023) study the sequential submodular maximization problem in online retailing to maximize purchase engagement. Tan and Guo (2023) study a capacitated assortment problem that considers market share, and they provide a two-stage approach for deriving an optimal solution. Despite this progress, the combined optimization of revenue and market share remains underexplored in the product selection and ranking problem.

Approximation Method

Given the computational complexity of product selection and ranking problems, researchers develop various approximation algorithms. For instance, R. Wang and Sahin (2018) provide a fully polynomial time approximation algorithm to solve the product ranking problem considering consumer search cost. Gallego et al. (2020) develop an algorithm with an approximation ratio of $6/{\pi }^2$ for the product framing problem under the MNL model. Under the similar choice model assumption, Aouad and Segev (2021) introduce approximate dynamic programing to maximize the expected revenue in the product ranking decisions. Asadpour et al. (2023) present an optimal $(1-1/e)$ -approximation algorithm for the product ranking problem to maximize purchase engagement. Derakhshan et al. (2022) prove that the product ranking problem on online platforms is NP-complete, and they provide a nuanced approximation algorithm based on dynamic programing. To sum up, approximation methods are indispensable for solving the BP-SS problem, particularly given the dual complexity introduced by sequential search and bi-objective optimization.

In summary, although sequential search and product optimization are prevalent in online retailing, existing literature rarely integrates them into a unified analytical framework. This study is the first to utilize the MNL model to explore the interaction between sequential search and customer choice, while also considering the trade-offs between revenue and market share in the BP-SS problem. Compared to existing research, this work offers a more comprehensive and technically rigorous approach to product selection and ranking.

Problem Description

Figure 1 illustrates the three main stages of the methodology proposed in this study: problem description, property analysis, and approximation algorithm. The problem description phase involves a theoretical analysis of customer choice behavior utilizing consideration sets and the MNL model, which culminates in the formulation of the BP-SS model. This model seeks to optimize both expected revenue and market share. The property analysis phase transforms the BP-SS model into the RBP-SS model by partitioning product preference weights. This transformation is followed by an unimodality analysis of the bi-objective function, and the derivation of ranking rules to guarantee convergence of the solution. The final phase introduces the dynamic programing-based approximation algorithm, and its performance is validated through numerical experiments using both simulation and real-world datasets.

Figure 1.

Methodology presented in this study.

Sequential Search and Choice Model

We construct a nested list structure for consideration sets to analyze the impact of sequential search on customer choice behavior. For simplicity, we exclude costs associated with stockouts, replenishment, and procurement. We assume that customers consider only a subset of the available products before making a purchase, referred to as the consideration set (Bettman et al., 1998). This set comprises all products that a customer sequentially browses, with its size being positively correlated with the search effort exerted by the customer during the product search process (Mehta et al., 2003). Hence, products ranked higher are more likely to be included in the consideration sets.

To formalize customer behavior under sequential search, we propose a two-stage choice model, illustrated in Figure 2. In the first stage, customers sequentially browse the available product list and form consideration sets based on product rankings. In the second stage, customers make their final purchase decision from the formed consideration set. We assume that the customer choice behavior adheres to the MNL model, wherein customers select the product that maximizes utility, which depends on product attributes (e.g., price, quality) and customer preferences. Although sequential search significantly influences the composition of consideration sets in the first stage, it does not affect the preference weights assigned to products in the second stage.

Figure 2.

Illustration of two-stage choice model.

We consider a market with n candidate products and n candidate product positions. Given a product ranking $\pi :\left[n\right]$ , the set of products sold is denoted as $S_{\pi }$ , with rankings remaining constant across all consideration sets. Upon customer arrival, they sequentially browse the list of products up to position $k\in \left[n\right]$ and form a product consideration set $S_{\pi }\left(k\right)$ , which includes all products up to position k in the ranking π. If customers choose to stop browsing earlier, they leave the site without selecting any product.

The probability distribution of the consideration set is related to product rankings. Customers are categorized according to the final positions they browse. We let ${\alpha }_k>0$ denote the probability that position k is selected as the final position. The probability distribution of the product consideration sets $\{S_{\pi }\left(k\right):k\in \left[n\right]\}$ is captured by the probability vector ${\alpha }_k$ , where each component corresponds to the likelihood of selecting a specific final position.

Once a consideration set $S_{\pi }\left(k\right)$ is formed, the customer makes a purchase decision by selecting the product that maximizes their utility. In the MNL model, let $v_i$ denote the preference weight of product i, and $v_0$ denote the preference weight of the no-purchase option. We assume that $r_i$ represents the revenue associated with product i, while $r_0$ denotes the revenue of the no-purchase option.

Let m denote the position index of product i in the ranking π. Given the consideration set $S_{\pi }\left(k\right)$ , the probability $P_{\mathrm{im}}(k,S_{\pi })$ of product i in the ranking position m is expressed as

$$P_{\mathrm{im}}(k,S_{\pi })=\{\begin{array}{cc}\frac{v_{\mathrm{im}}{s}_{\mathrm{im}}}{v_0+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_{\mathrm{jm}}{s}_{\mathrm{jm}}}\hfill & \mathrm{if}(i,m)\in S_{\pi }\left(k\right)\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

(1)

where $v_{\mathrm{im}}$ denotes the preference weight of product i in position m and the denominator represents the sum of the preference weights of all products in the consideration set. We let $s_{\mathrm{im}}\in {\{0,1\}}^{n\times k}$ .

Since product preference weights are exogenous, we have $v_{\mathrm{im}}=v_i$ . Thus, the Equation 1 simplifies to

$$P_{\mathrm{im}}(k,S_{\pi })=\{\begin{array}{cc}\frac{v_i{s}_{\mathrm{im}}}{v_0+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}}\hfill & \mathrm{if}(i,m)\in S_{\pi }\left(k\right)\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

(2)

The probability of the no-purchase option is $P_0(k,S_{\pi })=v_0/(v_0+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}})$ , and the total probability of all customer choices sums to 1.

This two-stage customer choice model offers two key advantages. First, it provides a more accurate representation of customer choice behavior under sequential search and offers a better explanation of the uncertainty in the customer decision-making process. Second, it establishes a clear theoretical foundation for understanding the interaction between product choice and ranking position, which is crucial for investigating the BP-SS problem.

Model Formulation

We present the bi-objective function that considers the expected revenue and market share and then build the mathematical model for the BP-SS problem. Given the product consideration set $S_{\pi }\left(k\right)$ , the expected revenue objective is denoted as ${\overline{R}}_k\left(S_{\pi }\right)$ , and it is computed as

$${\overline{R}}_k\left(S_{\pi }\right)=\sum _{(i,m)\in S_{\pi }\left(k\right)}{r}_i{P}_{\mathrm{im}}(k,S_{\pi })$$

(3)

Equation 3 represents the traditional objective that focuses only on the expected revenue, which is insufficient in a competitive market environment. Market share also plays a pivotal role in enhancing the retailer’s competitive positioning. Therefore, we introduce market share as a second optimization objective, denoted as ${\overline{M}}_k\left(S_{\pi }\right)$ . The expected market share is computed as

$${\overline{M}}_k\left(S_{\pi }\right)=\sum _{(i,m)\in S_{\pi }\left(k\right)}{P}_{\mathrm{im}}(k,S_{\pi }).$$

(4)

Given the product consideration set $S_{\pi }\left(k\right)$ , the bi-objective function that considers the expected revenue and market share is described as ${\overline{R}}_k\left(S_{\pi }\right)+\theta {\overline{M}}_k\left(S_{\pi }\right)$ , where $\theta \left(\theta \ge 0\right)$ is a parameter that controls the trade-offs between the two objectives. The BP-SS problem is formalized as

$$(\mathbf{BP}\mathbf{-}\mathbf{SS})\underset{S_{\pi }:\left|S_{\pi }\right|\le C}{max}\{\sum _{k\in \left[n\right]}{\alpha }_k·\{{\overline{R}}_k\left(S_{\pi }\right)+\theta {\overline{M}}_k\left(S_{\pi }\right)\}\}$$

(5)

We assume that the parameter θ is exogenous and the product consideration set $S_{\pi }$ satisfies the cardinality constraint $\left|S_{\pi }\right|\le C(2\le C\le n)$ , ensuring its size remains within practical limits. The BP-SS problem seeks to optimize product selection and ranking decisions to maximize the expected revenue and market share. We focus on how variations in product selection and ranking decisions influence customer preferences, rather than the impact of external factors. To streamline the model, we assume that the product preference weight of the no-purchase option is $v_0=1$ .

Problem Analyze

We analyze the properties of the BP-SS problem to develop an efficient approximation algorithm. Since existing literature suggests that the traditional product selection problem is NP-hard even with two customer groups (Bront et al., 2009; Rusmevichientong et al., 2014), the BP-SS problem is also NP-hard. Unless P=NP, it is unlikely that an optimal solution to the BP-SS problem can be found in polynomial time. Therefore, we reformulate the problem and analyze its properties to establish a theoretical foundation for developing an efficient approximation algorithm.

Problem Reformulation

Since the BP-SS problem is NP-hard and involves large data dimensions with high computational complexity, we propose a reformulation that simplifies the BP-SS problem while minimizing the loss in the value of the bi-objective function. Specifically, the reformulation consists of two main steps: converting the raw product preference weights into structured data and refining the bi-objective function.

First, we let $\underset{¯}{v}=\underset{i\in \left[n\right]}{min}{v}_i$ and $\overline{v}={\max }_{i\in \left[n\right]}{v}_i$ denote the minimum and maximum product preference weights among all candidate products, where $\underset{\_}{v}\ge v_{\min }$ and $\underset{¯}{v}\le v_{max}$ . We let $L=\lceil {\log }_{1+e}(v_{max}/v_{min})\rceil$ , where $e\in (0,1)$ . Given an integer $l\in \left[L\right]$ , we have a product partition ${\sigma }_l=\{i\in \left[n\right]:\underset{¯}{v}{(1+e)}^{l-1}<v_i\le \underset{¯}{v}{(1+e)}^l\}$ . Thus, the preference weight $v_i$ of product i is approximated by

$$v_i=\underset{¯}{v}{(1+e)}^l$$

(6)

where $l\in \left[L\right]$ denotes the smallest integer that satisfy $v_i\le \underset{¯}{v}{(1+e)}^l$ .

The boundary of the partition is defined as ${\sigma }_L=\{i\in \left[n\right]:v_i\le \underset{¯}{v}{(1+e)}^L\}$ . Accordingly, the candidate products are divided into L partition sets. This procedure ensures that the loss in the value of the bi-objective function remains negligible within the stabilization error interval $(1+e)$ . It also reduces the size of the input data and simplifies the calculation.

Figure 3 depicts the reformulation of product preference weights and the formation of product partitions. The long vertical lines in Figure 3 represent the original preference weights of the products within the interval $(1+e)$ . The products are grouped into partition sets according to Equation 6.

Figure 3.

Illustration of product partition sets.

Then, we refine the bi-objective function. We let ${\omega }_i=r_i{v}_i$ and $\underset{¯}{\omega }=\underset{i\in \left[n\right]}{min}{\omega }_i$ , so we have ${\omega }_i=\underset{¯}{\omega }{(1+e)}^{\tau }$ , where the parameter τ denotes the largest integer satisfy ${\omega }_i\ge \underset{¯}{\omega }{(1+e)}^{\tau }$ . Given the product consideration set $S_{\pi }\left(k\right),k\in \left[{\sigma }_L\right]$ , the expected revenue objective is $R_k\left(S_{\pi }\right)=\sum _{(i,m)\in S_{\pi }\left(k\right)}{\omega }_i{s}_{\mathrm{im}}/1+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}$ and the market share objective is $M_k\left(S_{\pi }\right)=\sum _{(i,m)\in S_{\pi }\left(k\right)}{v}_i{s}_{\mathrm{im}}/1+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}$ . Therefore, the BP-SS problem is reformulated as the RBP-SS problem

$$(\mathrm{RBP}-\mathrm{SS})\underset{S_{\pi }:\left|S_{\pi }\right|\le C}{max}\{\sum _{k\in \left[n\right]}{\alpha }_k·\{R_k\left(S_{\pi }\right)+\theta M_k\left(S_{\pi }\right)\}\}.$$

(7)

The goal of this reformulation is to convert the product weights and bi-objective function data into a structured format, facilitating the design of the approximation algorithm. Since the expected revenue and market share objectives are formulated as polynomial functions with respect to $(1+e)$ , the loss in optimality of the bi-objective function is negligible. In addition, this reformulation does not affect the core properties of the BP-SS problem. The RBP-SS problem remains NP-hard, and it aims to identify the optimal products and their ranking positions to maximize the expected revenue and market share.

Unimodality Analysis

We conduct a theoretical analysis of the relationship between the ranking positions and the values of the bi-objective function. To facilitate this analysis, we define the bi-objective function $R_k\left(S_{\pi }\right)+\theta M_k\left(S_{\pi }\right)$ , which maps the ranking position $k\in \left[n\right]$ to the corresponding product consideration set $S_{\pi }\left(k\right)$ . We find that the bi-objective function $k\to R_k\left(S_{\pi }^{*}\right)+\theta M_k\left(S_{\pi }^{*}\right)$ is unimodal, as demonstrated in Lemma 1.

• Lemma 1. Given the optimal solution $S_{\pi }^{*}$ , the bi-objective function $k\to R_k\left(S_{\pi }^{*}\right)+\theta M_k\left(S_{\pi }^{*}\right)$ is unimodal. Formally, there exists a product ranking position $k\in \left[n\right]$ such that the bi-objective function $R_k\left(S_{\pi }^{*}\right)+\theta M_k\left(S_{\pi }^{*}\right)$ is non-decreasing in the interval $[1,k]$ and non-increasing in the interval $[k,n]$ .

Proof. We employ a counterfactual approach and assume that there exists a ranking position $k\in \left[n\right]$ , such that the optimal solution $S_{\pi }^{*}$ satisfies $R_k\left(S_{\pi }^{*}\right)+\theta M_k\left(S_{\pi }^{*}\right)>R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)$ and $R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)<R_{k+2}\left(S_{\pi }^{*}\right)+\theta M_{k+2}\left(S_{\pi }^{*}\right)$ . Let $i_{k+1}$ denote product i at position $k+1$ . Then, $R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)$ denotes the bi-objective function generated by adding product $i_{k+1}\in \left[n\right]\backslash S_{\pi }^{*}\left(k\right)$ to the solution $S_{\pi }^{*}\left(k\right)$ , and $R_{k+2}\left(S_{\pi }^{*}\right)+\theta M_{k+2}\left(S_{\pi }^{*}\right)$ denotes the bi-objective function generated by adding product $i_{k+2}\in \left[n\right]\backslash S_{\pi }^{*}(k+1)$ to the solution $S_{\pi }^{*}(k+1)$ . Thus, we have

$$\begin{array}{c}(R_k\left(S_{\pi }^{*}\right)+\theta M_k\left(S_{\pi }^{*}\right))-(R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right))\hfill \\ =(R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right))\hfill \\ \multicolumn{1}{c}{(\frac{1+\sum _{(j,m)\in S_{\pi }^{*}\left(k\right)}{v}_j{s}_{\mathrm{jm}}+v_{i_{k+1}}{s}_{i_{k+1}m}}{1+\sum _{(j,m)\in S_{\pi }^{*}\left(k\right)}{v}_j{s}_{\mathrm{jm}}}-1)}\\ \multicolumn{1}{c}{-\frac{{\omega }_{i_{k+1}}{s}_{i_{k+1}m}+\theta v_{i_{k+1}}{s}_{i_{k+1}m}}{1+\sum _{(j,m)\in S_{\pi }^{*}\left(k\right)}{v}_j{s}_{\mathrm{jm}}}}\\ =(R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)-({\omega }_{i_{k+1}}/v_{i_{k+1}}+\theta ))\hfill \\ \multicolumn{1}{c}{\frac{v_{i_{k+1}}{s}_{i_{k+1}m}}{1+\sum _{(j,m)\in S_{\pi }^{*}\left(k\right)}{v}_j{s}_{\mathrm{jm}}}}\end{array}.$$

According to $R_k\left(S_{\pi }^{*}\right)+\theta M_k\left(S_{\pi }^{*}\right)>R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)$ , we have ${\omega }_{i_{k+1}}/v_{i_{k+1}}+\theta <R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)$ . Similarly, according to $R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)<R_{k+2}\left(S_{\pi }^{*}\right)+\theta M_{k+2}\left(S_{\pi }^{*}\right)$ , we have ${\omega }_{i_{k+2}}/v_{i_{k+2}}+\theta >R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)$ . From this, it follows that ${\omega }_{i_{k+2}}/v_{i_{k+2}}>{\omega }_{i_{k+1}}/v_{i_{k+1}}$ .

Then, we interchange products at the $k+1\mathrm{st}$ and $k+2\mathrm{nd}$ positions in the optimal solution $S_{\pi }^{*}$ and define the new feasible solution as $S{\prime }_{\pi }$ , which satisfies the constraint

$$\begin{array}{cc}\multicolumn{1}{c}{}& R_{k+1}\left({S\prime }_{\pi }\right)+\theta M_{k+1}\left({S\prime }_{\pi }\right)\hfill \\ \multicolumn{1}{c}{}& \ge min\{{\omega }_{i_{k+2}}/v_{i_{k+2}}+\theta ,R_k\left(S_{\pi }^{*}\right)+\theta M_k\left(S_{\pi }^{*}\right)\}\hfill \\ \multicolumn{1}{c}{}& >R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right)\hfill \end{array}.$$

Thus, we have

$$\begin{array}{cc}\multicolumn{1}{c}{}& (R\left({S\prime }_{\pi }\right)+\theta M\left({S\prime }_{\pi }\right))-(R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right))\hfill \\ \multicolumn{1}{c}{}& ={\alpha }_{k+1}\{(R_{k+1}\left({S\prime }_{\pi }\right)+\theta M_{k+1}\left({S\prime }_{\pi }\right))-(R_{k+1}\left(S_{\pi }^{*}\right)+\theta M_{k+1}\left(S_{\pi }^{*}\right))\}>0\hfill \end{array}.$$

This conclusion contradicts the claim that $S_{\pi }^{*}$ is an optimal solution, so the original assumption holds. Proof ends.

The unimodal property in Lemma 1 implies that the value of the bi-objective function increases as customers begin browsing products, peaking at an optimal ranking position. Beyond this point, further browsing results in diminishing revenues as the probability of customer choice decreases with the additional products presented. This observation highlights the trade-offs between customer engagement and the efficiency of product selection. Figure 4 provides a visualization of the unimodal pattern of the bi-objective function.

Figure 4.

Unimodality of the bi-objective function.

In Figure 4, the bi-objective function exhibits a unimodal pattern, characterized by a single maximum value. This property guarantees the existence of a unique optimal solution, which greatly simplifies the decision-making process for retailers. In practice, the unimodal function ensures an optimal solution within the given constraints, enhancing the ability to model and forecast economic behavior, particularly in competitive markets. Moreover, the unimodal property supports more effective decision-making decisions such as product ranking, pricing strategies, and promotional activities. For example, when customer willingness to engage in product searches is low, retailers can implement targeted incentives, such as offering discounts or rewards for browsing a certain number of items, to enhance consumer engagement. The presence of a distinct peak in the unimodal function facilitates the identification of the most effective strategies for long-term growth and revenue maximization, ensuring a more efficient resource allocation.

Ranking Rules

We introduce specific product ranking rules to simplify the solution of the BP-SS problem. To maximize the expected market share, we employ an empirical solution that arranges products in descending order based on their preference weights. However, this weight-order policy places high-weighted products at the top positions, which results in smaller customer consideration sets that overlook high-revenue products placed at lower positions, thereby reducing the expected revenue. Similarly, to maximize the expected revenue, the empirical solution ranks products based on their expected revenues, placing high-revenue products at the top position. However, this leads to an inadequate supply of products and a decreased likelihood of customer choice.

To balance the expected revenue and market share objectives, we adopt the following product ranking rules. Let $m\left(i\right)$ denote the ranking position of product i, and let ${\rho }_i={\omega }_i+\theta v_i$ . The ranking rules are defined as follows.

• Assumption 1. For two distinct products $i,j(i\ne j)$ , (a) if $v_i=v_j$ and ${\omega }_i>{\omega }_j$ , $m\left(i\right)<m\left(j\right)$ ; (b) if ${\rho }_i={\rho }_j$ and $v_i<v_j$ , $m\left(i\right)<m\left(j\right)$ .

Based on Assumption 1, we introduce two product ranking rules to guide the solution of the RBP-SS problem. Assumption 1(a) posits that when two products have identical preference weights, the product i with higher revenue is ranked higher, that is $m\left(i\right)<m\left(j\right)$ . This reflects common practices in product ranking, where revenue is a critical determinant of product priority. Assumption 1(b) suggests that when two products have identical bi-objective function values but different preference weights, the product with a lower preference weight is ranked higher.

These ranking rules serve to simplify the computational complexity of the RBP-SS problem by structuring the ranking logic. Assuming deterministic demand and fixed preference weights, these rules provide a foundation for the design of the efficient approximation algorithm that strike a balance between maximizing the expected revenue and market share.

We then demonstrate that the optimal solution to the RBP-SS problem follows Assumption 1, as shown in Lemma 2.

• Lemma 2. The optimal solution $S_{\pi }^{*}$ to the RBP-SS problem follows Assumption 1.

Proof. We use a counterfactual approach to prove Lemma 2. First, we claim that the optimal solution $S_{\pi }^{*}$ satisfies Assumption 1(a): if $v_i=v_j$ and ${\omega }_i>{\omega }_j$ , then $m^{*}\left(i\right)<m^{*}\left(j\right)$ , where $m^{*}\left(i\right)$ denotes the optimal ranking position of product i in the solution $S_{\pi }^{*}$ . Next, we obtain a feasible solution ${S″}_{\pi }$ by swapping the positions of the products $i,j(i\ne j)$ in the solution $S_{\pi }^{*}$ . Obviously, the solution ${S″}_{\pi }$ does not follows Assumption 1(a). To compare the values of the bi-objective function between the solutions $S_{\pi }^{*}$ and ${S″}_{\pi }$ , we have

$$\begin{array}{cc}\multicolumn{1}{c}{}& (R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right))-(R\left({S″}_{\pi }\right)+\theta M\left({S″}_{\pi }\right))\hfill \\ \multicolumn{1}{c}{}& =\sum _{k=m^{*}\left(j\right)}^{m^{*}\left(i\right)}{\alpha }_k(\frac{\sum _{(l,m)\in S_{\pi }^{*}}{\omega }_l{s}_{\mathrm{lm}}+\theta v_l{s}_{\mathrm{lm}}}{1+\sum _{(l,m)\in S_{\pi }^{*}}{v}_l{s}_{\mathrm{lm}}}-\frac{\sum _{(l,m)\in {S″}_{\pi }}{\omega }_l{s}_{\mathrm{lm}}+\theta v_l{s}_{\mathrm{lm}}}{1+\sum _{(l,m)\in {S″}_{\pi }}{v}_l{s}_{\mathrm{lm}}})\hfill \\ \multicolumn{1}{c}{}& =\sum _{k=m^{*}\left(j\right)}^{m^{*}\left(i\right)}{\alpha }_k\frac{({\omega }_j{s}_{\mathrm{jm}}+\theta v_j{s}_{\mathrm{jm}})-({\omega }_i{s}_{\mathrm{im}}+\theta v_i{s}_{\mathrm{im}})}{1+\sum _{(l,m)\in S_{\pi }^{*}}{v}_l{s}_{\mathrm{lm}}}\hfill \\ \multicolumn{1}{c}{}& =\sum _{k=m^{*}\left(j\right)}^{m^{*}\left(i\right)}{\alpha }_k\frac{{\omega }_j{s}_{\mathrm{jm}}-{\omega }_i{s}_{\mathrm{im}}}{1+\sum _{(l,m)\in S_{\pi }^{*}}{v}_l{s}_{\mathrm{lm}}}<0\hfill \end{array}.$$

As $R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right)<R\left({S″}_{\pi }\right)+\theta M\left({S″}_{\pi }\right)$ , the conclusion contradicts the original claim, and the optimal solution $S_{\pi }^{*}$ must satisfy Assumption 1(a).

Similarly, we claim that the optimal solution $S_{\pi }^{*}$ satisfies Assumption 1(b): if ${\rho }_i={\rho }_j$ and $v_i<v_j$ , then $m^{*}\left(i\right)<m^{*}\left(j\right)$ . If the solution $S_{\pi }^{*}$ does not satisfy Assumption 1(b), a feasible solution can be obtained by exchanging the positions of the products $i,j(i\ne j)$ . The conclusion contradicts the original claim, and the optimal solution $S_{\pi }^{*}$ must satisfy Assumption 1(b). Proof ends.

Lemma 2 confirms that the optimal solution to the RBP-SS problem satisfies the ranking rules. These rules simplify the solution process without causing a significant loss in the value of the bi-objective function. Moreover, since products satisfying these rules correspond to approximate counterparts in the product partitions, the ranking rules offer a robust foundation for the subsequent design of the approximation algorithm with negligible loss of optimality.

Approximation Algorithm

Conceptual Framework

We present the conceptual framework of the proposed approximation algorithm for solving the RBP-SS problem. The central idea involves building an approximation function $f_{S_{\pi }}$ to approximate the bi-objective function $R_k\left(S_{\pi }\right)+\theta M_k\left(S_{\pi }\right),\forall k\in {\sigma }_l$ of the RBP-SS problem. The optimization mechanism is illustrated in Figure 5.

Figure 5.

Conceptual illustration of the approximation algorithm.

As shown in Figure 5, the abscissa denotes the ranking positions of products viewed by customers, while the ordinate denotes the values of the bi-objective function. Given a parameter $e\in (0,1)$ , the set of n candidate products is divided into L subsets. The dots represent the actual values of the bi-objective function, and the triangles represent the approximated values given by $f_{S_{\pi }}$ . Since the product positions are discrete, both the bi-objective function and the approximation function are defined over discrete domains.

The approximation function $f_{S_{\pi }}$ is constructed as a step function. Specifically, within each partition set ${\sigma }_l,\forall l\in \left[L\right]$ , the approximation function $f_{S_{\pi }}$ takes the minimum value of the bi-objective function $R_k\left(S_{\pi }\right)+\theta M_k\left(S_{\pi }\right)$ . Formally, this is expressed as $f_{S_{\pi }}\left(k\right)=min\{R_k\left(S_{\pi }\right)+\theta M_k\left(S_{\pi }\right):k\in {\sigma }_l\}$ . For example, if ${\sigma }_1=\left\{1\right\}$ and ${\sigma }_2=\{2,3\}$ in Figure 5, then we have $f_{S_{\pi }}\left(1\right)=R_1\left(S_{\pi }\right)+\theta M_1\left(S_{\pi }\right)$ and $f_{S_{\pi }}\left(2\right)=f_{S_{\pi }}\left(3\right)=R_2\left(S_{\pi }\right)+\theta M_2\left(S_{\pi }\right)$ .

Given that the bi-objective function is unimodal with respect to ranking positions, the minimum value of each partition set ${\sigma }_l$ must be at one of its boundaries. We assume that $k_{max}$ is the ranking position with the optimal value of the bi-objective function. Then, the bi-objective function is non-decreasing in the interval $[1,k_{max}]$ and non-increasing in the interval $[k_{max},n]$ . We define $g_l=min{\sigma }_l$ in the interval $[1,k_{max}]$ and define $g_l=max{\sigma }_l$ in the interval $[k_{max},n]$ . Hence, the approximation function $f_{S_{\pi }}$ is described and simplified as $f_{S_{\pi }}\left(k\right)=\{R_{g_l}\left(S_{\pi }\right)+\theta M_{g_l}\left(S_{\pi }\right),:k\in {\sigma }_l\}$ .

This approach offers several computational advantages. First, it reduces the number of function evaluations by approximating each partition using only its boundary values. Second, it alleviates the need for detailed knowledge of the underlying probability distribution of individual products, and it requires only the aggregated probabilities $\sum _{k\in {\sigma }_l}{\alpha }_k$ for each partition set.

In summary, when the number of partition sets is sufficiently large, the approximation function provides a close estimate of the original bi-objective function, as shown in Figure 5. Because approximate preference weights and revenues are expressed as integer powers of $(1+e)$ , the loss of optimality is negligible. This simplification significantly reduces the computational complexity of solving the RBP-SS problem.

Algorithm Procedure

Based on the above framework, we present the implementation steps of the approximation algorithm. The flowchart of this algorithm is shown in Figure 6.

Figure 6.

Flowchart of the approximation algorithm.

As illustrated in Figure 6, the approximation algorithm begins with data collecting. These inputs include product weights, revenues, probability distributions, and the related parameters. Following this preparatory step, the algorithm proceeds in four main steps.

• Step 1: forming partition sets. The weights are approximated according to Equation 6, and products with identical approximated weights are grouped into the same partition ${\sigma }_l=\{i\in \left[n\right]:\underset{¯}{v}{(1+e)}^{l-1}<v_i\le \underset{¯}{v}{(1+e)}^l\}$ . Then, the candidate products are grouped into $L=\lceil {\log }_{1+e}(v_{max}/v_{min})\rceil$ partition sets, where the partition boundary is ${\sigma }_L=\{i\in \left[n\right]:v_i\le \underset{¯}{v}{(1+e)}^L\}$ .

• Step 2: sorting products in descending order. After partitioning, the products within each partition set are sorted in descending order of $R_k\left(S_{\pi }\right)+\theta M_k\left(S_{\pi }\right):k\in {\sigma }_l$ .

• Step 3: solving the RBP-SS problem based on dynamic programing. Dynamic programing is employed to solve the RBP-SS problem. The solution process involves two key components: state transitions and recursive formulation. Specifically, we assume that the current state variable is $g_l\in \left[n\right]$ , and the next state variable is $g_{l+1}\in \left[n\right]$ . We then have ${\sigma }_l=[g_l,g_{l+1}-1]$ in the interval $\left[1,k_{\max }\right]$ and ${\sigma }_l=[g_{l-1}+1,g_l]$ in the interval $[k_{max},n]$ . Given the consideration set $S_{\pi ,l}$ , the function $F(g_l,g_{l+1},S_{\pi ,l})$ represents the value of the bi-objective function starting from the state variable $g_1$ to the current state variable $g_l$ . The recursive equation is described as follows:

$$F(g_l,g_{l+1},S_{\pi ,l})=max\{F(g_{l-1},g_l,S_{\pi ,l-1})+\left(\sum _{k\in {\sigma }_l}{\alpha }_k\right)·f_{S_{\pi }}\left(k\right)\},$$

(8)

where $f_{S_{\pi }}\left(k\right)=\{R_{g_l}\left(S_{\pi }\right)+\theta M_{g_l}\left(S_{\pi }\right),\forall k\in {\sigma }_l\}$ .

Equation 8 comprises two main components: the first captures the cumulative value of the bi-objective function $F(g_{l-1},g_l,S_{\pi ,l-1})$ from the state variable $g_1$ to $g_{l-1}$ , while the second represents the value of the approximation function $\left(\sum _{k\in {\sigma }_l}{\alpha }_k\right)·f_{S_{\pi }}\left(k\right)$ in the set ${\sigma }_l$ .

To ensure the existence of an optimal solution to the recursive formulation, appropriate boundary conditions must be defined. Specifically, for the initial partition set ${\sigma }_1=[g_1,g_2-1]$ , the boundary value is $F(g_1,g_2,S_{\pi ,1})=\left(\sum _{k\in {\sigma }_l}{\alpha }_k\right)·f_{S_{\pi }}\left(k\right),\forall k\in {\sigma }_1$ ; for all other sets, the function value is negative infinity. Therefore, the boundary conditions are formally expressed as follows:

$$\begin{array}{c}\multicolumn{1}{c}{F(g_1,g_2,S_{\pi ,1})=}\\ \multicolumn{1}{c}{\{\begin{array}{cc}\left(\sum _{k\in {\sigma }_1}{\alpha }_k\right)·f_{S_{\pi }}\left(k\right),\forall k\in {\sigma }_1\hfill & \mathrm{if}{\sigma }_1=[g_1,g_2-1]\hfill \\ \multicolumn{1}{c}{-\infty }& \mathrm{otherwise}\hfill \end{array}}\end{array}$$

(9)

We then discuss the constraints of the recursive equations. First, we assume that the partition sets are organized in a nested list structure. Second, the solution must satisfy the capacity constraint, denoted by $\left|S_{\pi ,l}\right|\le C$ , and the final state variable of the recursive process is represented as $g_{L+1}=g_L\le C$ . These constraints ensure the existence of an optimal solution to the RBP-SS problem and minimize the loss in the value of the bi-objective function.

Moreover, we need to determine the optimal ranking positions for the selected products. This process involves two main steps: first, sorting the L partition sets to form the initial sequence of ranking positions; second, sorting the products within each partition set according to the ranking rules. This ensures that the solution maximizes the objective function of the recursive Equation 8 and forms an optimal ranking position strategy π for the product consideration set $S_{\pi ,l}$ .

• Step 4: outputting the optimal solution of the bi-objective function $F=max\left\{F(g_l,g_{l+1},S_{\pi ,l})\right\}$ . According to the above algorithm procedure, we build a multidimensional matrix to facilitate the dynamic programing process. And then we solve the recursive equation to obtain the optimal solution $S_{\pi }$ and the corresponding value of the bi-objective function F.

The pseudocode for the approximation algorithm is presented as follows.

Approximation algorithm
Input: product information $v_i$ and $r_i$ , $\forall i\in \left[n\right]$ , probability distribution ${\alpha }_k,k\in {\sigma }_l$ , and parameters θ, C and e
Output: optimal value of the bi-objective function F
1.	initialization:set $S_{\pi ,0}=Ø$ , $F_0=0$
2.	reformulate the problem and divide products into L partition sets
3.	approximate product preference weights and sort products in descending order
4.	according to Equation 6, group products with the same approximate weights into partition set ${\sigma }_l,\forall l\in \left[L\right]$ with an interval of $(1+e)$ , where $L=\lceil {\log }_{1+e}(v_{max}/v_{min})\rceil$
5.	sort products in descending order of $R_k\left(S_{\pi }\right)+\theta M_k\left(S_{\pi }\right):k\in {\sigma }_l$
6.	solve the RBP-SS problem in the L partition sets
7.	for $l=1,\cdots ,L$ do
8.	calculate $F(g_l,g_{l+1},S_{\pi ,l})$ using Equation 8
9.	denote as a feasible solution $F(g_l,g_{l+1},S_{\pi ,l})$ if $\left|S_{\pi ,l}\right|\le C$
10.	end for
11.	$F=max\left\{F(g_l,g_{l+1},S_{\pi ,l})\right\}$

Since the running time of the approximation algorithm varies exponentially with the number of products, this algorithm qualifies as a polynomial time approximation algorithm. We then examine its approximation quality in Lemma 3.

• Lemma 3. Given a parameter $e>0$ , if the optimal value of the bi-objective function is $\{R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right)\}$ , the value of the bi-objective function $F\left(S_{\pi }\right)$ related to $S_{\pi }$ satisfies $(1-2e)\{R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right)\}\le F\left(S_{\pi }\right)\le \{R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right)\}$ .

Proof. We first examine the left-hand side of the inequality. We have

$$\begin{array}{cc}\multicolumn{1}{c}{}& F\left(S_{\pi }\right)\hfill \\ \multicolumn{1}{c}{}& =\sum _{k\in \left[n\right]}{\alpha }_k·\{R_k\left(S_{\pi }\right)+\theta M_k\left(S_{\pi }\right)\}\hfill \\ \multicolumn{1}{c}{}& =\sum _{k\in \left[n\right]}{\alpha }_k·\{\frac{\sum _{(i,m)\in S_{\pi }\left(k\right)}{\omega }_i{s}_{\mathrm{im}}}{v_0+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}}+\theta \frac{\sum _{(i,m)\in S_{\pi }\left(k\right)}{v}_i{s}_{\mathrm{im}}}{v_0+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}}\}\hfill \\ \multicolumn{1}{c}{}& \ge \sum _{k\in \left[n\right]}{\alpha }_k·\{\frac{(1-e)\sum _{(i,m)\in S_{\pi }\left(k\right)}{\omega }_i{s}_{\mathrm{im}}}{v_0+(1+e)\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}}+\theta \frac{(1-e)\sum _{(i,m)\in S_{\pi }\left(k\right)}{v}_i{s}_{\mathrm{im}}}{v_0+(1+e)\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}}\}\hfill \\ \multicolumn{1}{c}{}& =\frac{1-e}{1+e}\{R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right)\}\hfill \\ \multicolumn{1}{c}{}& \ge (1-2e)\{R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right)\}\hfill \end{array}$$

Then, we examine the right-hand side of the inequality. We have

$$\begin{array}{cc}\multicolumn{1}{c}{}& F\left(S_{\pi }\right)\hfill \\ \multicolumn{1}{c}{}& \le \sum _{k\in \left[n\right]}{\alpha }_k·\{\frac{\sum _{(i,m)\in S_{\pi }\left(k\right)}{\omega }_i{s}_{\mathrm{im}}}{v_0+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}}+\theta \frac{\sum _{(i,m)\in S_{\pi }\left(k\right)}{v}_i{s}_{\mathrm{im}}}{v_0+\sum _{(j,m)\in S_{\pi }\left(k\right)}{v}_j{s}_{\mathrm{jm}}}\}\hfill \\ \multicolumn{1}{c}{}& =\{R\left(S_{\pi }^{*}\right)+\theta M\left(S_{\pi }^{*}\right)\}\hfill \end{array}.$$

Proof ends.

Lemma 3 demonstrates that the proposed approximation algorithm achieves a guaranteed performance bound. Specifically, the solution obtained by the algorithm is within a factor of $(1-2e)$ compared to the global optimum.

According to Lemma 3, we present robust and general convergence results for the approximation algorithm. Unlike heuristic methods that rely on simulation-based validation, the proposed algorithm offers a formal theoretical guarantee on solution quality under any reformulation of the BP-SS problem. This makes it particularly advantageous in practical applications, such as retail product selection and ranking, where reliable and interpretable decision-making frameworks are essential.

Numerical Experiments

Benchmark Algorithms

To evaluate the performance of the approximation algorithm in solving the BP-SS problem, we compare it with two classical solution methods: revenue order and local search. The implementation details of these benchmark algorithms are described in the following.

Revenue order. We sort n candidate products in descending order based on their revenues. The candidate sets consist of the empty set and C subsets. In each subset, products are ranked according to their revenues. Using revenue order, the one that maximizes the bi-objective function is selected as the optimal product set. This method is straightforward to implement and relies solely on the retailer’s prior knowledge of product revenues and weights.

Local search. The local search method starts by randomly generating an initial product ranking that satisfies the given constraints. It then iteratively swaps the positions of two products to explore the neighborhood of the current solution. If the new solution generates a higher value of the bi-objective function, it replaces the current solution. This search process continues until the algorithm meets the termination condition. Specifically, we define $U=\{(i,j):1\le i\le j\le n\}$ as the set of the exchanged positions and define $S_{\pi }^{\left(k\right)}$ as a feasible solution with the $k\mathrm{th}$ iteration. Then we exchange the product positions $(i,j)\in U$ in the feasible solution. Thus, we can obtain $S_{\pi (i\leftrightarrow j)}^{\left(k\right)}$ , where $S_{\pi (i\leftrightarrow j)}^{\left(k\right)}\left(i\right)=S_{\pi }^{\left(k\right)}\left(j\right)$ and $S_{\pi (i\leftrightarrow j)}^{\left(k\right)}\left(j\right)=S_{\pi }^{\left(k\right)}\left(i\right)$ . The iterative process repeats until the bi-objective function reaches its maximum value. In addition, the algorithm terminates if the incremental ratio between the function values of two consecutive iterations is less than 0.1%.

Simulation Study

The simulation data setup is described as follows. The revenue parameters are generated as independent random numbers from a standard lognormal distribution, and the product preference weight parameters are generated as independent random numbers from a uniform distribution over [0,1]. We denote the capacity constraint as $C=15$ . The choice probabilities follow a uniform distribution of [0,1]. The probabilities satisfy $\sum _{k=1}^{15}{\alpha }_k=1$ . The trade-off parameter is denoted by $\theta =5$ and the error parameter is denoted by $e=0.4$ .

Simulation Results

We set the number of products $n\in \{30,50,300,500\}$ to simulate most of real-world scenarios. For each scenario, we generate four different cases and repeat each case ten times to compute the average value of the bi-objective function. The results are summarized in Table 1.

Table 1.

Results of the Simulation Cases.

n	Average values of the bi-objective function
	AA	LS	Gap	RO	Gap
30	6.81	6.79	0.29%	5.91	15.23%
50	7.13	7.11	0.28%	6.41	11.23%
300	8.42	8.28	1.69%	8.09	4.08%
500	8.83	8.51	3.76%	8.39	5.24%

Table 1 presents the average values of the bi-objective function obtained by the approximation algorithm and two benchmark algorithms. The second, third, and fifth columns show the average values solved by the approximation algorithm (AA), local search (LS), and revenue order (RO). The fourth and sixth columns report the gaps between the AA and LS, and between the AA and RO, respectively.

From Table 1, when the number of candidate products is small, the average values of the bi-objective function obtained by the approximation algorithm are slightly lower than those obtained by the local search, but substantially higher than those obtained by the revenue order. Notably, the performance gap between the approximation algorithm and local search increases as the number of products increases. This is attributed to the tendency of local search to converge prematurely to local optima in large-scale instances. Additionally, the approximation algorithm consistently outperforms the revenue order method, although the performance gap narrows as the product set size increases. Overall, the approximation algorithm demonstrates robust performance and scalability in addressing the BP-SS problem.

We then provide the average running time of 10 simulation instances across four cases in Table 2.

Table 2.

Average Running Time of the Simulation Cases.

n	Average running time (s)
	AA	LS	Gap
30	<1	<1	-
50	<1	<1	-
300	13.25	97.87	86.46%
500	39.51	392.43	89.93%

Table 2 reveals a significant difference in the average running times between the approximation algorithm and local search. For small-scale cases ( $n\in \{30,50\}$ ), both the approximation algorithm and local search exhibit similar average running times of less than 1 s. However, for large-scale cases ( $n\in \{300,500\}$ ), the computational efficiency of the approximation algorithm becomes superior to that of the local search. This indicates the superior scalability of the approximation algorithm in large-scale applications.

Sensitivity Analysis

Impact of Data Volume

To evaluate the impact of data volume, we compare the performance of the three algorithms across different problem sizes. Figure 7 presents the bi-objective function values for 10 random simulation cases with varying product numbers.

Figure 7.

Results of different data volumes: (a) n = 30, (b) n = 50, (c) n = 300, and (d) n = 500.

Figure 7 reveals the performance of the three algorithms in four different cases. In small-scale cases, the approximation algorithm exhibits stable and consistent performance, indicating its reliability in limited data environments. As the product number increases, the approximation algorithm maintains superior stability and effectiveness. This suggests that the approximation algorithm is well-suited to handling large-scale data and offers reliable optimization outcomes.

To further analyze the influence of data volume on the approximation algorithm, we present the function values of the three algorithms in small and large data, as shown in Figure 8.

Figure 8.

Results under small and large data.

Figure 8 shows that the performance of the revenue order method is suboptimal for both small and large data. While the local search maintains relatively strong performance in smaller datasets, its effectiveness declines sharply as the data volume increases. Conversely, the approximation algorithm sustains high performance, validating its adaptability and robustness in large-scale decision-making environments. Therefore, the proposed approximation algorithm shows better adaptability and optimization capabilities for solving the BP-SS problem.

Impact of the Trade-Off Parameter

To explore the impact of the trade-off parameter θ on the computational results, we randomly generate two instances of $n\in \{30,300\}$ . The capacity constraint is $C=10$ and the trade-off parameter is $\theta \in \{1,3,5,10,30,50\}$ . The results are summarized in Table 3.

Table 3.

Results Under Different Trade-Off Parameters.

n	θ	Ren	MS	AA	LS	Gap	RO	Gap
30	1	5.99	73.45%	6.71	6.71	0.00%	6.24	7.53%
	3	5.84	81.24%	8.27	8.24	0.36%	7.13	15.99%
	5	5.46	86.52%	9.79	9.72	0.72%	8.02	22.07%
	10	5.07	88.77%	13.95	13.79	1.16%	9.95	40.20%
	30	4.28	89.98%	31.27	30.93	1.10%	18.88	65.63%
	50	3.82	91.48%	49.56	48.84	1.47%	27.47	80.41%
300	1	7.02	61.24%	7.63	7.52	1.49%	6.12	24.71%
	3	7.01	70.07%	9.11	8.24	10.58%	6.59	38.27%
	5	6.56	75.22%	10.32	9.45	9.22%	7.05	46.40%
	10	5.63	82.77%	13.91	12.87	8.06%	9.22	50.83%
	30	5.03	85.48%	30.67	27.49	11.58%	16.90	81.51%
	50	4.23	87.65%	48.05	41.69	15.26%	22.58	112.84%

In Table 3, the first column denotes the number of products, while the second column denotes the different trade-off parameters. The third and fourth columns show the optimal values of the expected revenue (Ren) and market share (MS) obtained by the approximation algorithm. The fifth, sixth and eighth columns report the optimal bi-objective values obtained by the three algorithms (AA, LS and RO). The seventh and ninth columns represent the performance gaps between the approximation algorithm and the two benchmark algorithms.

Table 3 presents the optimal solutions of the three algorithms under different trade-off parameters. Intuitively, an increase in trade-off parameters corresponds to an increase in the expected market share but a decrease in the expected revenue. For instance, increasing the parameters (from 1 to 3) leads to a slight decrease in expected revenues (from 5.99 to 5.84) and a substantial increase in market share (from 73.45% to 81.24%). This shift results from prioritizing broader market reach, thereby replacing high-revenue items with products of greater appeal.

To visualize this relationship, Figure 9 plots the trends in the expected revenue and market share under different trade-off parameters.

Figure 9.

Revenue and market share under different trade-off parameters: (a) n = 30 and (b) n = 300.

Figure 9 illustrates the trends in the expected revenue (blue curve) and market share (orange curve) under different trade-off parameters in the two instances (n = 30 and n = 300). The results show that the expected revenue decreases and the expected market share increases as the trade-off parameter increases. This pattern holds consistently across both small and large instances. When the trade-off parameter is low, increasing it effectively boosts market share without significant revenue loss. However, when the trade-off parameter is high, further increases lead to diminishing returns in revenue, despite a slight increase in market share. This phenomenon can be attributed to intensified market competition and resource constraints, where further pursuit of market share may not translate into proportional gains.

Real-World Application

To further validate the practical effectiveness of the approximation algorithm, we apply it to a real-world dataset of apples obtained from Platform A, located in Chengdu, China. This dataset is collected from March 13 to April 13, 2023, and contains transaction records of 48 candidate apple products. Each product is described by two quantitative features: revenue (in yuan) and an estimated preference weight. The revenue values are directly extracted from the raw transaction data. To estimate the preference weights, we employ the Expectation-Maximization (EM) algorithm (Vulcano et al., 2012). Table 4 summarizes the key features of this dataset.

Table 4.

Data Description.

n	Revenue	Weight
1	7.13	1.659
2	13.84	0.946
3	19.42	0.229
4	14.99	0.670
5	13.50	1.077
6	5.97	1.898
7	13.25	1.127
8	7.17	1.636
9	16.86	0.468
10	19.22	0.258
11	13.77	1.032
12	21.86	0.144
13	16.14	0.507
14	21.80	0.155
15	24.11	0.008
16	22.02	0.137
17	16.60	0.487
18	13.00	1.212
19	13.92	0.892
20	13.09	1.157
21	7.00	1.699
22	23.13	0.047
23	18.49	0.267
24	6.13	1.823
25	14.31	0.678
26	5.52	1.910
27	20.20	0.192
28	13.62	1.063
29	6.73	1.709
30	9.72	1.571
31	23.15	0.024
32	10.50	1.558
33	23.09	0.110
34	6.68	1.818
35	17.61	0.338
36	12.86	1.245
37	15.53	0.539
38	12.64	1.326
39	15.52	0.640
40	18.46	0.304
41	11.55	1.452
42	20.12	0.215
43	12.78	1.294
44	12.80	1.262
45	17.24	0.430
46	5.35	1.932
47	11.10	1.522
48	10.35	1.567

After discussing with the platform’s operations manager, the retailer aims to select up to 12 products from the pool of 48 candidates to maximize the expected revenue and market share. To achieve this, we incorporate the capacity constraint $C=12$ to ensure that the final product assortment contains no more than 12 items. Besides, the probability distribution of the consideration sets consists of 12 independent random samples following a uniform distribution over the interval [0,1], which is normalized to ensure the total probability sums to one, that is $\sum _{k=1}^{12}{\alpha }_k=1$ .

Table 5 presents the results under different trade-off parameters in the real-world application. As shown in Table 5, the trade-off parameter governs the balance between maximizing revenue and expanding market share. As the trade-off parameter increases, the expected market share rises while the expected revenue decreases.

Table 5.

Results in the Real-World Application.

θ	Values of the bi-objective function			Optimal solution
	AA	Revenue	Market share
0	10.70	10.70	-	[10, 3, 38, 44, 34, 39, 9, 22, 2, 41, 26, 13]
1	11.41	10.69	72.74%	[10, 1, 3, 38, 44, 34, 39, 9, 22, 2, 41, 26]
3	12.87	10.68	72.74%	[10, 1, 3, 38, 44, 34, 39, 22, 9, 2, 41, 26]
5	14.34	10.64	73.92%	[10, 1, 18, 3, 38, 44, 34, 39, 22, 9, 2, 41]
10	18.04	10.59	74.46%	[10, 1, 18, 3, 24, 38, 44, 36, 12, 16, 8, 34]
30	34.24	9.77	81.56%	[46, 40, 37, 42, 43, 35, 17, 10, 19, 6, 4, 27]
50	50.55	9.76	81.57%	[46, 40, 37, 42, 43, 35, 17, 19, 6, 10, 4, 27]

The results demonstrate that as the trade-off parameter increases appropriately, the expected market share rises without incurring significant revenue loss, especially when the trade-off parameter is small. This behavior reflects the inherent tension between the two objectives. For instance, moderate increases in the trade-off parameter from 3 to 5 yield a 1.18% increase in market share achieved with a mere 0.04 unit decrease in the expected revenue. This indicates that retailers can strategically adjust their objective function to prioritize broader market appeal without sacrificing profitability.

From a managerial perspective, the findings in Table 5 underscore the practical value of the approximation algorithm in enabling data-driven product selection. The algorithm allows retailers to adjust their strategic priorities by controlling the trade-off parameter by business objectives. Moreover, the results reveal diminishing marginal returns in market share as the parameter increases, highlighting a point beyond which further resource allocation may not be cost-effective.

Therefore, in market saturation or budget limitations, retailers may prefer to maintain a balanced product portfolio that prioritizes high-revenue items while securing a market share threshold sufficient to retain competitiveness. Additionally, the algorithm provides the optimal product mix and corresponding rankings, thereby offering an actionable framework for online product assortment planning and performance management.

Discussion and Conclusions

This study addresses the bi-objective optimization problem for product selection and ranking in online retailing. Specifically, it focuses on the impact of sequential search behavior and the trade-offs between the expected revenue and market share. To address this challenge, we first construct a two-stage choice model that incorporates consideration sets and then formulate the BP-SS problem. The aim is to jointly optimize the product selection and ranking decision to achieve a balanced outcome between competing business objectives. To solve the BP-SS problem, we propose a dynamic programing-based approximation algorithm and validate its performance through extensive numerical simulations and real-world applications. The results provide valuable theoretical insights and practical guidance, laying the groundwork for further research and real-world implementations.

Theoretical Implications

From a theoretical perspective, our contributions are threefold.

First, we extend the classical MNL model by integrating it with a two-stage choice framework that includes consideration sets. This enhanced model captures more realistic consumer decision-making processes by considering its sequential and probabilistic properties. Unlike conventional MNL-based approaches, our model accommodates the sequential and dynamic nature of product visibility and consumer fatigue, thereby offering a more nuanced representation of consumer purchasing behavior.

Second, we introduce the bi-objective optimization framework that jointly considers the expected revenue and market share, which advances the literature beyond traditional single-objective models. This bi-objective structure reflects the multi-faceted nature of business goals in e-commerce, where retailers must balance immediate profitability with long-term strategic positioning. Our numerical results demonstrate that modest adjustments to the trade-off parameter can yield significant improvements in market share without significant revenue loss. This finding is consistent with and builds upon prior work such as Maddah et al. (2021), who also emphasize the value of multi-objective decision-making in retail analytics.

Third, our proposed dynamic programing-based approximation algorithm introduces an efficient and scalable approach for solving complex product selection and ranking problems. The algorithm exhibits robust performance across multiple test scenarios. For instance, in large-scale settings with 500 candidate products, it achieves an 89.93% improvement in computational efficiency over benchmark methods without compromising solution quality. This makes it a promising approach for deployment in real-time and large-scale retail environments.

Overall, these contributions offer a novel methodological and analytical framework that bridges behavioral modeling, multi-objective optimization, and algorithmic efficiency, thus advancing both the theory and application of assortment planning and product prioritization in digital commerce.

Practical Implications

From a practical perspective, the findings also offer actionable insights for practitioners operating in competitive online retail markets.

First, the bi-objective optimization framework provides a structured and data-driven approach to managing product selection and ranking. By explicitly modeling the trade-offs between short-term revenue and long-term market share, the proposed method enables retailers to tailor their assortment strategies according to dynamic business objectives. This is particularly critical in competitive, fast-paced markets where customer loyalty is low and strategic assortment decisions must balance performance with differentiation.

Second, integrating product selection with ranking optimization enables retailers to align their offerings more effectively with consumer preferences and shopping behaviors. Retailers can utilize this framework to deploy strategic positioning strategies. For example, placing high-margin or seasonal products in top-ranking positions increases exposure and conversion. Real-world practices, such as Amazon’s use of customer browsing and purchasing data to personalize assortments (Kim et al., 2017), underscore the practical relevance of such strategies. Our findings suggest that for retailers with lower market penetration, strategically ranking high-utility products alongside selectively chosen high-margin items can serve as a dual-purpose approach, which maximizes purchase likelihood while facilitating opportunities for cross-selling and upselling. This strategy fosters increased customer engagement and supports both short-term and long-term growth goals.

Third, the proposed dynamic programing-based approximation algorithm offers a flexible and computationally tractable tool for product assortment planning. Retailers can adapt this algorithm to various market contexts by incorporating business constraints and customer segmentation data. Beyond e-commerce, this framework holds promise for broader applications in domains such as healthcare, education, and public policy (Binsar & Mauritsius, 2020). For instance, in healthcare decision-making, a similar bi-objective framework could be employed to rank treatment options based on expected health outcomes and patient satisfaction, thereby improving allocation efficiency and service personalization.

Limitations and Future Research Directions

Despite the valuable contributions of this study, there are several limitations and future research directions. First, this study focuses on a specific online retailing environment and does not explore alternative approaches from cross-disciplinary perspectives. Thus, it would be interesting to apply the proposed model and algorithm to other domains, such as healthcare and education. Second, it would be interesting to incorporate varying customer segments and provide a more nuanced model of customer behaviors, like the mixed logit model or Markov model. Besides, it would be valuable to expand the scope to include other market information, such as pricing or omnichannel strategies, and provide robust optimization strategies of how this information influences product selection and ranking decisions.