A Heuristic Approach to Competitive Facility Location via Multi-View K-Means Clustering with Co-Regularization and Customer Behavior

Abstract

Solving competitive facility location problems can optimize market share or operational efficiency in environments where multiple firms compete for customer attention. In such contexts, facility attractiveness is shaped not only by geographic proximity but also by customer preference characteristics. This study presents a novel heuristic framework that integrates multi-view K-means clustering with customer behavior modeling reinforced by a co-regularization mechanism to align clustering results across heterogeneous data views. By jointly exploiting spatial and behavioral information, the framework clusters customers and facilities into meaningful market segments. Within each segment, a bilevel optimization model is applied to represent the sequential decision-making of competing entities—where a leader first selects facility locations, followed by a reactive follower. An empirical evaluation on a real-world dataset from San Francisco demonstrates that the proposed approach, using optimal co-regularization parameters, achieves a total runtime of approximately 4.00 s—representing a 99.34% reduction compared to the full CFLBP-CB model (608.58 s) and a 99.32% reduction compared to a genetic algorithm (585.20 s). Concurrently, it yields an overall profit of 16,104.17, which is an approximate 0.72% increase over the Direct CFLBP-CB profit of 15,988.27 and is only 0.21% lower than the genetic algorithm’s highest profit of 16,137.75. Moreover, comparative analysis reveals that the proposed multi-view clustering with co-regularization outperforms all single-view baselines, including K-means, spectral, and hierarchical methods. This superiority is evidenced by an approximate 5.21% increase in overall profit and a simultaneous reduction in optimization time, thereby demonstrating its effectiveness in capturing complementary spatial and behavioral structures for competitive facility location. Notably, the proposed two-stage approach achieves high-quality solutions with significantly shorter computation times, making it suitable for large-scale or time-sensitive competitive facility planning tasks.

1. Introduction

The competitive facility location problem (CFLP) focuses on determining optimal sites for new facilities with the objective of maximizing the market share acquired by these facilities (refer to survey papers [1,2,3]). Various models addressing this problem can be categorized based on the nature of competition, the characteristics of the location space, and the behavior of customers.

The location space in the CFLP can take several forms: it may be discrete, with new facilities selected from a finite set of candidate sites [4]; structured as a network, allowing facility placement at network nodes or even along edges [5]; or continuous, permitting facilities to be established at any point within a defined region [6].

Customer behavior is another crucial factor in these models. Various customer response patterns are typically captured using attraction functions, which quantify how customers are influenced by competing facilities. The most widely used customer choice mechanisms are known as the proportional and binary rules [7]. Under the proportional rule, customers distribute their patronage among all facilities based on the relative attractiveness of each (see, for example, [6,8,9]). In contrast, the binary rule assumes that each customer chooses to patronize only the most attractive facility [4,10,11]. While these rules account for much of the observed customer behavior in competitive location models, in some cases, variations or combinations may be needed to better capture real-world complexities.

Furthermore, competition in facility location models can be classified as static, with foresight, or dynamic. In static competition, existing facilities owned by competitors are already present, and they do not adjust their positions when a new firm enters the market [12]. In the foresight scenario, potential competitors are expected to enter the market soon after the new facilities are established—the leader first selects locations to maximize market share, anticipating subsequent facility placements by the follower [13]. In dynamic competition, firms continually reassess and update their facility locations over time [14].

Among these types of competition, dynamic competition is particularly notable for the strategic interplay between rival firms, which is effectively modeled using bilevel optimization frameworks. In such models, a leader first makes facility location decisions to maximize market share or profitability, and a follower then responds by optimizing its own placements [15]. Bilevel modeling enables a more realistic representation of competitive dynamics compared to single-level formulations, making it particularly suitable for problems involving interdependent decision structures. To support complex decision-making in large-scale competitive facility location problems—particularly those involving extensive customer or facility datasets—clustering techniques such as K-means clustering, known for their simplicity and widespread use, are frequently employed. These methods partition the data into k clusters characterized by high intra-cluster similarity and low inter-cluster similarity, thereby simplifying the problem structure and enhancing computational efficiency.

Conventional K-means clustering is inherently constrained by its single-view assumption, treating all features with equal importance through uniform weighting. This simplification often fails to reflect the realities of complex datasets, where feature dimensions typically contribute unequally to the clustering structure. As a result, applying equal weights can obscure meaningful patterns and compromise clustering quality—particularly when some attributes are more informative than others. In the context of the CFLP, standard K-means clustering is commonly applied using either spatial coordinates or customer preferences, or by concatenating them into a single feature vector. Such an approach disregards the distinct nature and relative importance of each data view; yet both facilities and customers are defined by multiple complementary attributes. Overlooking this multi-view structure can result in suboptimal cluster formations and, consequently, inferior location decisions.

To overcome these limitations, multi-view clustering has emerged as a powerful paradigm in machine learning, enabling the integration of multiple heterogeneous data views into a unified clustering framework [16]. Multi-view clustering techniques enable a more comprehensive understanding of the fundamental framework of massive datasets, resulting in it being useful in image analysis, bioinformatics, and social network research. The initial research on multi-view clustering was originally performed by Bickel and Scheffer [17]. Throughout the span of two decades, various types of multi-view clustering techniques were provided considering the findings of that remarkable research [18].

Given its widespread adoption and computational efficiency, K-means clustering serves as a practical foundation for implementing multi-view clustering with co-regularization. This approach jointly leverages multiple feature spaces and encourages consistency across them through co-regularization mechanisms. While multi-view clustering has demonstrated strong empirical performance in various domains, its application to competitive facility location problems remains relatively unexplored, presenting a novel and promising direction for research.

This study addresses an important research gap by introducing a novel approach that integrates multi-view K-means clustering into a bilevel facility location framework. In particular, we incorporate geographic data, the preference views of customers, and facility data to create high-quality, behavior-aware clusters that serve as the basis for hierarchical optimization. This integration improves clustering quality and interpretability in several key ways:

• It captures richer customer and facility representations by jointly leveraging geographic and preference data.
• It reveals latent patterns that are not discernible within any single view—for example, clusters with distinct preference profiles despite spatial proximity.
• It enforces alignment between views through co-regularization, thereby yielding clusters that are meaningful across multiple data dimensions.
• It provides robustness against noise or missing values in individual views by using complementary information from others.
• It supports decision interpretability by producing clusters grounded in both geographic and behavioral factors.

This research introduces a novel heuristic framework for the competitive facility location problem with customer behavior (CFLP-CB) by integrating multi-view K-means clustering with co-regularization to ensure consistency across multiple data views. By clustering facilities and customers based on heterogeneous feature spaces and applying bilevel optimization within each cluster, the framework effectively captures leader–follower dynamics while maintaining strategic realism and computational tractability. The proposed methodology is validated on a real-world dataset from San Francisco, incorporating synthetic behavioral data to reflect diverse customer preferences. Empirical results show that this approach not only significantly reduces computation time but also improves market share performance when compared to baseline K-means clustering combined with bilevel optimization. Overall, the framework provides a scalable, interpretable, and data-driven solution for the CFLP in complex, multi-attribute decision environments.

This research work is systematically organized to reflect the development of this approach: Section 2 reviews the relevant literature, identifying a gap in current methods where multi-view behavioral integration remains underexplored. Section 3 presents the problem definition and the proposed bilevel optimization model, which includes a behavior-aware attraction function that jointly considers spatial proximity and preference alignment. Section 4 describes the algorithmic framework, detailing the multi-view K-means clustering method with inter-view co-regularization and the bilevel optimization process for facility assignment. Section 5 presents a detailed case study of a retail franchise expansion in San Francisco where clustering is based on geographic and preference features. Section 6 discusses the strategic implications of the results, and Section 7 concludes the work with insights and directions for future research, such as incorporating real customer preference data and exploring advanced multi-objective extensions.

2. Related Work

The facility location problem has long been studied in both cooperative and competitive contexts. In competitive environments, firms must strategically select facility locations while anticipating the responses of their competitors. These interdependent decisions are best modeled using bilevel optimization frameworks, where a leader makes the first move and a follower responds. This sequential structure captures the asymmetric information, anticipatory behavior, and competitive dynamics inherent in real-world market entry.

The roots of competitive location modeling can be traced to Hotelling’s seminal 1929 work [19], which applied game theory to the analysis of spatial competition. Over the decades, this concept was extended through the development of leader–follower models, most notably the Stackelberg framework [20], which formalized sequential decision-making in competitive environments. A recent comprehensive survey by Drezner and Eiselt [21] offers a detailed classification of competitive location models, distinguishing between game-theoretic approaches (which jointly consider location and pricing decisions) and operations research models (which focus on optimizing new facility placement among existing competitors). Their review highlights the integration of realistic features such as facility attractiveness, customer choice behavior, price sensitivity, and market uncertainty. They emphasize that future research should continue to address challenges posed by customer heterogeneity, demand elasticity, and behavioral attraction rules, particularly in data-driven environments.

Building on these foundations, several studies have proposed bilevel models that incorporate increasingly realistic representations of customer decision behavior. For example, Biesinger et al. [22] formulated six bilevel models that reflect combinations of binary, proportional, and mixed attraction mechanisms under both essential and nonessential demand conditions. They solved these models using a hybrid evolutionary algorithm integrated with mixed-integer linear programming (MILP) evaluation techniques. Similarly, Casas-Ramírez et al. [23] proposed a hybrid heuristic approach to efficiently solve the bilevel p-median problem with ordered customer preferences, demonstrating improved solution quality and scalability compared to traditional exact methods. The key characteristics of recent competitive facility location studies are summarized in

Table 1. Summary of recent studies on the competitive facility location problem (CFLP).

Study	Objective Function	Solution Method	Application Domain
Drezner (2014)  [1]	Maximize market share with location and quality differentiation	Bilevel model with best-response dynamics	Retail competition with quality-sensitive customers
Biesinger et al. (2017) [22]	Maximize market share with different types of customer behavior rules	MILP	Retail competition
Casas-Ramírez et al. (2017) [23]	Minimize the total cost between facility and customer	Reformulation of bilevel optimization to single-level optimization and heuristics	Retail competition, service networks
Kochetov et al. (2018) [24]	Maximize market share under proportional customer behavior	Bilevel MINLP, heuristics	Service networks
Rahmani et al. (2021) [25]	Maximize expected profit under demand uncertainty and competition	Branch-and-cut algorithm	Inventory and distribution systems
Beresnev et al. (2022) [26]	Maximize leader’s profit in a two-stage bilevel model	MILP reformulation to compute upper bound	Competitive retail location
Latifi et al. (2022) [27]	Maximize follower’s market share with discrete facility attractiveness and foresight	Bilevel integer programming with exact algorithm and dominance rules	Competitive retail location planning
Yu et al. (2022) [28]	Maximize leader’s market share under demand uncertainty	Two-stage robust optimization	Retail facility planning
Parvasi et al. (2023) [29]	Maximize domestic firm’s profit via price-setting competition	Bilevel game-theoretic optimization	International retail price competition
Zhou et al. (2023) [30]	Minimize total routing cost under soft customer clustering	Bilevel memetic algorithm with savings heuristic and local search	Logistics and vehicle routing
Calvete et al. (2024) [31]	Maximize net profit and total customer preference	Multi-objective optimization	Supply chain planning
Lin et al. (2024) [32]	Maximize own revenue considering nested customer preferences	Bilevel optimization, nested logit model	Retail stores, parcel lockers, park-and-ride stations
Legault et al. (2025) [33]	Maximize expected market share under random utility-based customer choice	Submodular optimization, simulation-based reformulation	Retail location with choice uncertainty

to highlight their objectives, solution methods, and application domains.

These studies highlight both the expressiveness of bilevel optimization for modeling strategic competition and the computational burden it imposes when applied to large-scale, real-world datasets. To reduce this complexity, researchers have increasingly employed clustering-based heuristics, particularly K-means clustering, to partition the problem into smaller subproblems. While effective at improving tractability, most K-means-based models rely on a single view of the data, typically spatial proximity, and fail to incorporate multidimensional characteristics such as customer preferences, behavior, or demographics.

To address this limitation, the machine learning literature offers multi-view clustering (MVC)—a family of techniques designed to integrate heterogeneous data representations into a cohesive clustering framework. As reviewed by Zhou et al. [16], MVC methods such as co-regularized spectral clustering, graph learning, and contrastive representation learning have demonstrated strong performance in domains like bioinformatics, image recognition, and document analysis. These methods improve robustness, accuracy, and interpretability by leveraging both shared and unique features across views. Despite their maturity in data science applications, MVC techniques remain largely unexplored in operations research, particularly for CFLPs involving strategic interactions and customer diversity.

This research addresses that gap by proposing a novel heuristic method that integrates multi-view K-means clustering with inter-view co-regularization and a bilevel optimization process for facility assignment in CFLPs. The proposed approach first clusters customers and facilities using both geographic and preference data, and then applies bilevel optimization within each cluster to model leader–follower dynamics. By unifying modern clustering techniques with strategic optimization, this method offers a scalable, data-aware, and behaviorally realistic framework for location planning in competitive, real-world environments.

3. Problem Statement and Mathematical Model

3.1. Motivating Example

To motivate the creation of the proposed mathematical formulation, we begin with a simplified example that illustrates the core structure of the CFLP as a bilevel optimization problem.

Example 1. 

Consider a competitive market in which two firms seek to find optimal locations to serve customer demand. Figure 1 illustrates the data for this scenario: blue circles represent customer demand points, while red circles denote the set of candidate facility sites from which both firms can choose to establish their facilities. In this situation, the first firm to select its locations is designated as the leader, and the subsequent firm is referred to as the follower.

In the CFLP, formulated as a bilevel optimization problem, optimal facility locations are determined by maximizing a profit function based on customer demands for both the leader and follower. As an example, if both firms establish two facilities, the classical CFLP computes placement by minimizing costs, depicted by the shortest dashed lines in Figure 1b. These lines represent customer–facility assignments driven by objectives like market share, accessibility, or total travel cost. In contrast, our proposed method, illustrated in Figure 1a, first applies a clustering technique to group customer demands. Optimal facility locations are then selected from candidate sites within each cluster, subsequently enabling profit function computation.

3.2. Proposed Bilevel Optimization Model with Preference-Aware Attraction

This subsection introduces the notational framework adopted in this study and formally defines the CFLBP-CB. A complete list of notations used throughout the manuscript is provided in the Notation and Definition section. The CFLBP-CB addresses a market environment in which a new entrant, referred to as the leader, seeks to establish p facilities within a region already occupied by an incumbent competitor, regarded as the follower, who is allowed to respond optimally by locating r additional facilities. Both the leader and the follower select facilities from a predetermined set of potential locations, with neither party having fixed facility placements at the outset. Customers in this study are assumed to exhibit a proportional behavior rule, which means their demand is allocated among all available facilities.

The mathematical formulation of the CFLP under proportional customer behavior, originally introduced by Suárez-Vega et al. [34], presumes that customers distribute their demand among all accessible facilities in accordance with the proportional attractiveness of each facility, as depicted in Figure 1b. Kochetov et al. [24] further examined a specific variant characterized by proportional essential customer behavior and developed a bilevel mixed-integer nonlinear programming (MINLP) framework to model the hierarchical, sequential decision-making process involving a market-entering leader and a competitive follower. Motivated by the studies [24,34], we propose an enhanced bilevel model that integrates a preference-based dimension into the customer–facility assignment mechanism. To do this, we define $p_{ij}$ as a similarity parameter representing the preference alignment between facility i and customer j. This parameter is incorporated into the attraction coefficient $v_{ij}$, enabling more accurate capture of the behavioral complexity in customer–facility relationships. As a result, the proposed formulation simultaneously considers spatial proximity and individual customer preferences. The proposed bilevel optimization model for the CFLBP-CB is detailed as follows.

Leader’s Problem (Upper Level):

$$\begin{array}{cc}\hfill \underset{x}{max}& \sum _{j\in J}{w}_j·{\displaystyle \frac{{\sum }_{i\in I}{v}_{ij}{x}_i}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i^{*}}}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \sum _{i\in I}{x}_i=p\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & x_i\in \{0,1\},\forall i\in I\hfill \end{array}$$

(3)

Here, $y^{*}=(y_1^{*},y_2^{*},\dots ,y_m^{*})$ represents the optimal solution to the follower’s problem, defined below.

Follower’s Problem (Lower Level):

$$\begin{array}{cc}\hfill \underset{y}{max}& \sum _{j\in J}{w}_j·{\displaystyle \frac{{\sum }_{i\in I}{v}_{ij}{y}_i}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \sum _{i\in I}{y}_i=r\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & y_i\in \{0,1\},\forall i\in I\hfill \end{array}$$

(6)

The objective functions (1) and (4) maximize the market share (i.e., fulfilled demand) for the leader and follower, respectively. The customer attraction coefficient is allocated based on the preference similarity coefficient, defined as follows:

$$v_{ij}={\displaystyle \frac{p_{ij}}{d_{ij}+ϵ}}={\displaystyle \frac{{\prod }_{m\in M}min\left\{1,\frac{{\mathrm{Preference}}_{m,i}}{{\mathrm{Preference}}_{m,j}}\right\}}{d_{ij}+ϵ}}$$

(7)

where M denotes the set of preference dimensions (e.g., product categories, service features), and ${\mathrm{Preference}}_{m,i}$ and ${\mathrm{Preference}}_{m,j}$ represent the value of the mth preference dimension for facility i and customer j, respectively. Specifically, ${\mathrm{Preference}}_{m,i}$ denotes the facility’s capability to serve the mth preference dimension of customer j, while ${\mathrm{Preference}}_{m,j}$ reflects the importance or demand level of the mth preference dimension from the customer’s perspective. This formulation ensures that $p_{ij}$ captures the behavioral alignment between customer needs and facility offerings across all dimensions, while normalizing to avoid overweighting any single preference component. Here, $d_{ij}$ is the distance between facility i and customer j, $p_{ij}$ is the preference similarity ratio, and $ϵ$ is a small positive constant used to avoid division by zero.

Constraints (2) and (5) ensure that the leader and follower open exactly p and r facilities, respectively. In the lower-level problem, the leader’s facility decisions $x_i$ are treated as fixed parameters.

3.3. Linear Reformulation of the Follower Problem

To efficiently solve the lower-level problem, we adopt a linearization strategy originally proposed by Kochetov et al. [24], which reformulates the nonlinear objective using a set of auxiliary variables. Specifically, two new types of variables are introduced: $z_j$, which represents the inverse of the total attraction experienced by customer j, and $y_{ij}$, which denotes the portion of demand from customer j served by the follower’s facility at location i. These are defined as follows:

$$\begin{array}{cc}\hfill z_j& ={\displaystyle \frac{1}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i}}\forall j\in J\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill y_{ij}& =w_j{z}_j{v}_{ij}{y}_i\forall i\in I,j\in J\hfill \end{array}$$

(9)

To avoid confusion, we clarify the roles of decision variables $y_i$ and $y_{ij}$. The binary variable $y_i$ indicates whether the follower opens a facility at location i, while the continuous variable $y_{ij}$ represents the portion of customer j’s demand allocated to facility i, conditional on $y_i=1$. The auxiliary variable $z_j$ serves as the reciprocal of the denominator in the original fractional objective, capturing the total attraction from both leader and follower facilities. These transformations enable a linear representation of the original nonlinear model, facilitating its reformulation as a mixed-integer linear program (MILP) for the follower’s subproblem given fixed decisions $x_i$ from the leader.

The resulting MILP is expressed as follows:

$$\begin{array}{cc}\hfill max& \sum _{j\in J}\sum _{i\in I}{y}_{ij}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \sum _{i\in I}{y}_{ij}+w_j{z}_j\sum _{i\in I}{v}_{ij}{x}_i\le w_j\forall j\in J\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & y_{ij}\le w_j{y}_i\forall i\in I,j\in J\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & y_{ij}\le w_j{v}_{ij}{z}_j\le y_{ij}+W(1-y_i)\forall i\in I,j\in J\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & y_{ij}\ge 0,z_j\ge 0\forall i\in I,j\in J\hfill \end{array}$$

(14)

Constraints (11) ensure that the total fulfilled demand from customer j (by both the leader and the follower) does not exceed $w_j$. Constraints (12) restrict $y_{ij}$ to zero when the follower does not open a facility at location i, while (13) enforces consistency with the nonlinear relation in (9). A sufficiently large constant W ensures that these constraints are valid for all feasible solutions.

To ensure the validity of the linearization, a sufficiently large constant W must be selected such that constraints (13) correctly enforce the conditional structure of Equation (9). Specifically, W must satisfy the following:

$$W\ge \underset{j\in J}{max}{w}_j·\underset{i\in I,j\in J}{max}{v}_{ij}·\underset{j\in J}{max}{z}_j$$

(15)

where the term ${max}_j\left(z_j\right)$ is bounded above by ${max}_j\left(1/{\sum }_{i\in I}{v}_{ij}{x}_i\right)$, as implied by constraint (11). This ensures that the upper bound on $y_{ij}$ is valid when $y_i=0$ and sufficiently tight when $y_i=1$. Choosing W conservatively guarantees that the MILP formulation remains equivalent to the original nonlinear model defined by Equations (1)–(4).

3.4. Theoretical Equivalence of the Reformulated Model

To ensure that the MILP, as (10)–(14), accurately represents and finds the optimal solution for the follower’s problem in (4)–(6), we provide the following theoretical results.

Lemma 1. 

Let $(y_i^{*},y_{ij}^{*},z_j^{*})$ be an optimal solution to the MILP defined by constraints (10)–(14) for a fixed leader decision variable $x={\left(x_i\right)}_{i\in I}\in {\{0,1\}}^{\left|I\right|}$. Then, for every customer $j\in J$, the auxiliary variable $z_j^{*}$ satisfies

$$z_j^{*}={\displaystyle \frac{1}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i^{*}}}.$$

Proof. 

From constraint (11), we have, for each $j\in J$,

$$\sum _{i\in I}{y}_{ij}^{*}+w_j{z}_j^{*}\sum _{i\in I}{v}_{ij}{x}_i\le w_j.$$

From constraints (12) and (13), we also have

$$\begin{array}{c}\hfill y_{ij}^{*}=w_j{v}_{ij}{z}_j^{*}{y}_i^{*},\forall i\in I,j\in J.\end{array}$$

(16)

Substituting this into the inequality (16) gives

$$\begin{array}{cc}\hfill \sum _{i\in I}{w}_j{v}_{ij}{z}_j^{*}{y}_i^{*}+w_j{z}_j^{*}\sum _{i\in I}{v}_{ij}{x}_i& \le w_j\hfill \\ \hfill w_j{z}_j^{*}\left(\sum _{i\in I}{v}_{ij}{y}_i^{*}+\sum _{i\in I}{v}_{ij}{x}_i\right)& \le w_j.\hfill \end{array}$$

Dividing both sides by $w_j>0$, we obtain

$$z_j^{*}\left(\sum _{i\in I}{v}_{ij}{x}_i+\sum _{i\in I}{v}_{ij}{y}_i^{*}\right)\le 1,$$

which implies

$$z_j^{*}\le {\displaystyle \frac{1}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i^{*}}}.$$

To complete the proof, we show that this inequality holds with equality at optimality. As the MILP objective is

$$\sum _{j\in J}\sum _{i\in I}{y}_{ij}^{*}=\sum _{j\in J}{w}_j{z}_j^{*}\sum _{i\in I}{v}_{ij}{y}_i^{*},$$

which is increasing in $z_j^{*}$, if $z_j^{*}$ were strictly less than the upper bound above, then increasing $z_j^{*}$ slightly would improve the objective without violating constraint (11), contradicting the optimality. Therefore, the inequality must be tight:

$$z_j^{*}={\displaystyle \frac{1}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i^{*}}},\forall j\in J.$$

   □

Theorem 1. 

Let $x={\left(x_i\right)}_{i\in I}\in {\{0,1\}}^{\left|I\right|}$ be a fixed leader decision satisfying constraint (2). Suppose that $(y_i^{*},y_{ij}^{*},z_j^{*})$ is an optimal solution to the MILP formulation defined by (10)–(14), along with the facility count constraint ${\sum }_{i\in I}{y}_i=r$ and integrality condition $y_i\in \{0,1\}$. Then, the binary vector $y^{*}=\left(y_i^{*}\right)$ is an optimal solution to the original nonlinear follower problem:

$$\begin{array}{cc}\hfill \underset{y\in {\{0,1\}}^{\left|I\right|}}{max}& \sum _{j\in J}{w}_j·{\displaystyle \frac{{\sum }_{i\in I}{v}_{ij}{y}_i}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i}}\hfill \end{array}$$

(17)

subject to the facility constraint ${\sum }_{i\in I}{y}_i=r$.

Proof. 

Let $x={\left(x_i\right)}_{i\in I}\in {\{0,1\}}^{\left|I\right|}$ be the fixed leader decision, and let $(y_i^{*},y_{ij}^{*},z_j^{*})$ be an optimal solution to the MILP formulation given by (10)–(14). Our goal is to show that the binary vector $y^{*}=\left(y_i^{*}\right)$ is an optimal solution to the original nonlinear follower problem.

• Step 1: Recovering the structure of the nonlinear objective from MILP constraints.

From constraint (13), for all $i\in I$, $j\in J$, we know that $y_{ij}\le w_j{v}_{ij}{z}_j{y}_i$. At optimality, this inequality is tight when $y_i^{*}=1$, giving the following:

$$y_{ij}^{*}=w_j{v}_{ij}{z}_j^{*}.$$

When $y_i^{*}=0$, the constraint implies $y_{ij}^{*}\le 0$, and, since $y_{ij}^{*}\ge 0$, we have the following:

$$y_{ij}^{*}=0.$$

Thus, in both cases, we can write the following:

$$y_{ij}^{*}=w_j{v}_{ij}{z}_j^{*}{y}_i^{*},\forall i\in I,j\in J.$$

(18)

Now substitute (18) into constraint (11):

$$\begin{array}{cc}\hfill \sum _{i\in I}{y}_{ij}^{*}+w_j{z}_j^{*}\sum _{i\in I}{v}_{ij}{x}_i& =\sum _{i\in I}{w}_j{v}_{ij}{z}_j^{*}{y}_i^{*}+w_j{z}_j^{*}\sum _{i\in I}{v}_{ij}{x}_i\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & =w_j{z}_j^{*}\left(\sum _{i\in I}{v}_{ij}{x}_i+\sum _{i\in I}{v}_{ij}{y}_i^{*}\right).\hfill \end{array}$$

(20)

Substituting Equation (20) into constraint (11) yields the following:

$$z_j^{*}\le {\displaystyle \frac{1}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i^{*}}}.$$

(21)

As $z_j^{*}$ appears positively in the objective function (and the problem is a maximization), and according to Lemma 1, any solution where inequality (21) is not tight would contradict the optimality. Thus, at optimality, the following applies:

$$z_j^{*}={\displaystyle \frac{1}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i^{*}}},\forall j\in J.$$

(22)

• Step 2: Establishing optimality via contradiction.

Assume, for the sake of contradiction, that $y^{*}$ is not an optimal solution to the nonlinear follower problem. Then, there exists another feasible vector $\widehat{y}\in {\{0,1\}}^{\left|I\right|}$ such that ${\sum }_{i\in I}{\widehat{y}}_i=r$ and

$$\sum _{j\in J}{w}_j·{\displaystyle \frac{{\sum }_{i\in I}{v}_{ij}{\widehat{y}}_i}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{\widehat{y}}_i}}>\sum _{j\in J}{w}_j·{\displaystyle \frac{{\sum }_{i\in I}{v}_{ij}{y}_i^{*}}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{y}_i^{*}}}.$$

Following the same reasoning as in Step 1, one could construct variables ${\widehat{y}}_{ij}=w_j{v}_{ij}{\widehat{z}}_j{\widehat{y}}_i$, where

$${\widehat{z}}_j={\displaystyle \frac{1}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{\widehat{y}}_i}}.$$

This implies that $(\widehat{y},{\widehat{y}}_{ij},{\widehat{z}}_j)$ is feasible for the MILP, and it yields an objective value:

$$\sum _{j\in J}\sum _{i\in I}{\widehat{y}}_{ij}=\sum _{j\in J}{w}_j·{\displaystyle \frac{{\sum }_{i\in I}{v}_{ij}{\widehat{y}}_i}{{\sum }_{i\in I}{v}_{ij}{x}_i+{\sum }_{i\in I}{v}_{ij}{\widehat{y}}_i}},$$

which is strictly greater than the MILP objective at $(y_i^{*},y_{ij}^{*},z_j^{*})$, contradicting the optimality of the original MILP solution.

Therefore, our assumption must be false. Hence, $y^{*}$ is an optimal solution to the nonlinear follower problem.    □

3.5. Numerical Illustration of Preference Similarity $p_{ij}$

As the proposed CFLBP-CB model introduces a new parameter, $p_{ij}$, the following example illustrates its conceptual role and practical interpretation.

Example 2. 

Consider a simplified scenario with three potential facilities and four customers. Each customer has a unit demand and expresses preferences across two dimensions, representing demand for product categories 1 and 2, respectively. Each facility, in turn, is characterized by capability scores in the same two dimensions, indicating how well it can fulfill the corresponding customer demand.

Applying the definition of the preference similarity ratio $p_{ij}$ given in Equation (7) to the data in

Table 2. Customer preferences and facility capabilities. Dashes (—) indicate values that are not applicable.

	Demand		Supply
ID	Product Category 1	Product Category 2	Product Category 1	Product Category 2
	(${\mathbf{Preference}}_{\mathbf{1},\mathit{j}}$)	(${\mathbf{Preference}}_{\mathbf{2},\mathit{j}}$)	(${\mathbf{Preference}}_{\mathbf{1},\mathit{i}}$)	(${\mathbf{Preference}}_{\mathbf{2},\mathit{i}}$)
Customer 1	1.0	1.0	—	—
Customer 2	0.8	0.9	—	—
Customer 3	1.0	0.7	—	—
Customer 4	0.9	0.9	—	—
Facility 1	—	—	0.9	0.7
Facility 2	—	—	0.8	0.9
Facility 3	—	—	1.0	1.0

, we compute $p_{ij}$ for each customer–facility pair. For example, the value for Customer 1 and Facility 1 is calculated as follows:

$$p_{11}=min\left\{1,{\displaystyle \frac{0.9}{1.0}}\right\}·min\left\{1,{\displaystyle \frac{0.7}{1.0}}\right\}=0.9·0.7=0.63.$$

The complete matrix of preference similarity ratios is shown in

Table 3. Preference similarity ratio matrix $p_{ij}.$.

Facility∖Customer	Customer 1	Customer 2	Customer 3	Customer 4
Facility 1	0.6300	0.7778	0.9000	0.7778
Facility 2	0.7200	1.0000	0.8000	0.8889
Facility 3	1.0000	1.0000	1.0000	1.0000

.

The results presented in Table 3 illustrate the degree of alignment between customer preferences and facility capabilities across all customer–facility pairs. A higher value of $p_{ij}$ indicates a stronger match between a customer’s demand and a facility’s supply in terms of the two product categories. Notably, Facility 3 achieves a perfect similarity score of $p_{ij}=1.0$ for all customers, reflecting its ability to fully satisfy customer preferences in both dimensions. In contrast, Facility 1 and Facility 2 exhibit varying degrees of mismatch depending on the customer. For example, Facility 1 yields the lowest similarity score with Customer 1 ($p_{11}=0.63$), due to relatively weaker supply in both product categories compared to the customer’s demand. Meanwhile, Facility 2 achieves a perfect match with Customer 2 ($p_{22}=1.0$) but provides only partial alignment with other customers.

These results highlight the importance of incorporating the preference similarity ratio $p_{ij}$ in the CFLBP-CB model. By quantifying the compatibility between customer demand and facility supply, $p_{ij}$ directly influences customer attraction and allocation decisions in the bilevel framework. Facilities with higher $p_{ij}$ values are more likely to attract demand, reinforcing the model’s ability to account for heterogeneous preferences and supply heterogeneity in competitive location decisions.

4. Algorithmic Framework

This section outlines the detailed procedures for the clustering process and the optimization process. Algorithm 1 is used to partition a large set of candidate facility locations and customers into coherent clusters. Once this clustering is complete, Algorithm 2 is applied to determine the optimal facility locations within each cluster.

4.1. Multi-View K-Means Clustering with Co-Regularization

To address the heterogeneity in customer and facility features, we propose a generalized multi-view K-means clustering framework that integrates multiple data views—such as spatial coordinates and preference characteristics—into a unified clustering model. Motivated by Kumar et al. [35], our method applies co-regularization directly to the cluster assignment level, a departure from the original approach, which regularizes the agreement between eigenvector embeddings. By doing so, we make the approach more scalable and interpretable for structured data settings, such as those involving customer and facility attributes in the CFLBP-CB.

Algorithm 1: Generalized multi-view K-means clustering with co-regularization

Input: $X^{\left(1\right)},X^{\left(2\right)},\dots ,X^{\left(V\right)}$: V views of the data, each $X^{\left(v\right)}\in {\mathbb{R}}^{n\times d_v}$;  K: Number of clusters;  ${\lambda }_v$: Co-regularization penalty for each view v;  max_iter: Maximum number of iterations;  Output: Final unified cluster assignments $\widehat{L}$ and centroids ${\mu }_k^{\left(v\right)}$ for each view v.

Algorithm 2: Generalized optimization assignment with minimum facility constraint

Input  : Final cluster labels $\widehat{L}$ from Algorithm 1;       Dataset D with facility and customer data;       View matrices $X^{\left(1\right)},X^{\left(2\right)},\dots ,X^{\left(V\right)}$;       Optimization model $\mathtt{CFLBP}\text{-}\mathtt{CB}$;       Required minimum number of facilities per cluster m.  Output:  For each cluster k, the optimal facility locations assigned to the leader and follower, denoted as $f_k^{\left(\mathrm{leader}\right)}$ and $f_k^{\left(\mathrm{follower}\right)}$, respectively.

The silhouette method [36], introduced by Rousseeuw (1987), is a widely used technique for evaluating the effectiveness of clustering methods by calculating the silhouette coefficient. This coefficient measures the appropriateness of assigning a data point to a cluster considering both its cohesion within the cluster and its separation from other clusters. The silhouette coefficient $s_i$ for a data point i is defined as follows:

$$s_i={\displaystyle \frac{b_i-a_i}{max(a_i,b_i)}}$$

where

• $a_i$ is the average distance from point i to all other points in the same cluster;
• $b_i$ is the average distance from point i to all points in the nearest neighboring cluster.

The silhouette coefficient ranges from −1 to 1. A value close to 1 indicates that the point is well-matched to its own cluster and poorly matched to neighboring clusters. A value close to 0 suggests that the point is on or near the boundary between two clusters. A value close to −1 implies that the point is likely misclassified, as it is closer to a neighboring cluster than to its own.

The clustering process of Algorithm 1, which is introduced in this work, follows the steps below:

• Step 1: Initialization

In the initialization step, for each view v, cluster labels are randomly assigned to each sample. Specifically, for each view v, the cluster labels $L^{\left(v\right)}$ are initialized randomly from the set ${\{1,\dots ,K\}}^n$, where each sample i in view v is assigned a random cluster label between 1 and K.

After initializing the cluster labels, the next task is to compute the initial centroids. For each view v and each cluster k, the centroid ${\mu }_k^{\left(v\right)}$ is computed by averaging the data points that have been assigned to that cluster. The centroid is given by the following formula:

$${\mu }_k^{\left(v\right)}={\displaystyle \frac{1}{|L_k^{\left(v\right)}|}}\sum _{x_i\in L_k^{\left(v\right)}}{X}_i^{\left(v\right)}$$

where $L_k^{\left(v\right)}$ represents the set of data points assigned to cluster k in view v, and $X_i^{\left(v\right)}$ is the feature vector of sample i in view v.

• Step 2: Iterative clustering process

In this step, the algorithm iteratively updates cluster assignments and centroids. It consists of two main subprocesses: the assignment step and the centroid update step. These subprocesses are repeated until convergence is achieved.

• Step 2.1: Assignment step  

In the assignment step, for each view v, each data point i is assigned to the cluster that minimizes a penalized distance. The distance between the data point $x_i^{\left(v\right)}$ and the centroid ${\mu }_k^{\left(v\right)}$ of cluster k in view v is first calculated as follows:

$$D_k={\parallel x_i^{\left(v\right)}-{\mu }_k^{\left(v\right)}\parallel }^2$$

Next, for each other view $u\ne v$, a co-regularization penalty is applied. If the sample is assigned to a different cluster in view u, the distance is increased by ${\lambda }_v$, where ${\lambda }_v$ is the co-regularization penalty for view v:

$$\mathrm{if}{L}_i^{\left(u\right)}\ne k:D_k\leftarrow D_k+{\lambda }_v$$

This penalty encourages consistency across views by discouraging assignments that conflict between views. After calculating the distances for all clusters, the data point i is assigned to the cluster that minimizes $D_k$ across all views:

$$L_i^{\left(v\right)}\leftarrow arg\underset{k}{min}{D}_k$$

This ensures that each sample is assigned to the closest cluster in each view while respecting the co-regularization penalty.

• Step 2.2: Centroid update step  

After the assignment step, the centroids are recomputed for each view and cluster. For each view v and each cluster k, the new centroid ${\mu }_k^{\left(v\right)}$ is calculated as the mean of the data points assigned to the cluster:

$${\mu }_k^{\left(v\right)}={\displaystyle \frac{1}{|L_k^{\left(v\right)}|}}\sum _{x_i\in L_k^{\left(v\right)}}{X}_i^{\left(v\right)}$$

This ensures that the centroids reflect the updated assignments. After updating the centroids, if the cluster assignments do not change compared to the previous iteration, the algorithm terminates, indicating convergence. If the assignments have changed, the algorithm proceeds to the next iteration.

• Step 3: Post-processing (majority voting and silhouette tie-breaking)  

After the algorithm converges, post-processing is performed to finalize the cluster assignments. Specifically, the silhouette score is computed for each sample i in each view v to assess the clustering quality. The silhouette score $s_i^{\left(v\right)}$ for a sample i in view v is computed based on the cluster assignments $L^{\left(v\right)}$ and is defined as follows:

$$s_i^{\left(v\right)}=\mathrm{silhouette}(x_i,L^{\left(v\right)})={\displaystyle \frac{b_i^{\left(v\right)}-a_i^{\left(v\right)}}{max\left(a_i^{\left(v\right)},b_i^{\left(v\right)}\right)}}$$

(23)

where $\mathrm{silhouette}(·)$ denotes the silhouette coefficient of the data point $x_i$ with respect to the cluster labels $L^{\left(v\right)}$. Here, $a_i^{\left(v\right)}$ is the average distance from sample i to all other samples in the same cluster in view v, and $b_i^{\left(v\right)}$ is the average distance from sample i to all samples in the nearest neighboring cluster in view v.

Next, majority voting is applied across all views. For each sample i, the number of votes for each cluster label k is counted across all views. The number of votes for cluster k is given by the following:

$${\mathrm{votes}}_k=\sum _{v=1}^V\mathbb{I}(L_i^{\left(v\right)}=k)$$

where $\mathbb{I}$ is the indicator function, which counts the votes for each label. If there is a tie in the majority vote, meaning multiple clusters have the same number of votes, the tie is resolved by selecting the cluster with the highest silhouette score among the tied labels. The final cluster assignment for sample i is then given by the following:

$${\widehat{L}}_i\leftarrow arg\underset{k}{max}\left\{s_i^{\left(v\right)}\right|v=1,\dots ,V\}$$

This ensures that, in the case of ties, the cluster with the best local clustering quality across views is selected.

The proposed clustering algorithm is presented in detail in Algorithm 1.

Example 3. 

Consider a dataset consisting of three data points, where each sample $x_i\in {\mathbb{R}}^4$ is composed of two distinct views: spatial coordinates and preference attributes. Specifically, let each point $x_i=(x_i^{\left(1\right)},x_i^{\left(2\right)})$, where $x_i^{\left(1\right)}\in {\mathbb{R}}^2$ denotes the spatial view and $x_i^{\left(2\right)}\in {\mathbb{R}}^2$ denotes the preference view. The dataset is defined as follows:

$$x_1=(0,0,1.00,0.00),x_2=(0,1,0.15,1.00),x_3=(10,10,0.10,1.00).$$

The objective is to partition the data into $K=2$ clusters. We compare the outcomes of (i) traditional K-means clustering applied to the concatenated features in ${\mathbb{R}}^4$, and (ii) the proposed multi-view K-means clustering algorithm with co-regularization. For the multi-view approach, the following co-regularization penalty parameters are used:

$${\lambda }^{\left(1\right)}=5.0(\mathit{spatial}\mathit{view}),{\lambda }^{\left(2\right)}=0.1(\mathit{preference}\mathit{view}).$$

As a baseline, we first present the clustering result obtained by applying the traditional K-means algorithm.

Step 1: Initialization. Let the initial cluster centroids be selected as follows:

$${\mu }_1^{\left(0\right)}=x_1=(0,0,1.00,0.00),{\mu }_2^{\left(0\right)}=x_3=(10,10,0.10,1.00).$$

Step 2: Assignment. To determine the cluster membership for each data point, we compute the squared Euclidean distance from each point to the initial centroids:

$$\parallel x_1-{\mu }_1^{\left(0\right)}{\parallel }^2=0,\parallel x_1-{\mu }_2^{\left(0\right)}{\parallel }^2=201.81,\parallel x_2-{\mu }_1^{\left(0\right)}{\parallel }^2=2.7225,{\parallel x_2-{\mu }_2^{\left(0\right)}\parallel }^2=181.0025,$$

$$\parallel x_3-{\mu }_1^{\left(0\right)}{\parallel }^2=201.81,{\parallel x_3-{\mu }_2^{\left(0\right)}\parallel }^2=0.$$

Each data point is then assigned to the cluster whose centroid it is closest to. Consequently, the initial cluster assignments are

$$C_1^{\left(1\right)}=\{x_1,x_2\},C_2^{\left(1\right)}=\left\{x_3\right\},$$

where $C_1^{\left(1\right)}$ and $C_2^{\left(1\right)}$ denote Clusters 1 and 2 determined by the initial centroids, respectively.

Step 3: Centroid update. The cluster centroids are then updated by computing the mean of the data points assigned to each cluster. Specifically, the updated centroids are given by the following:

$${\mu }_1^{\left(1\right)}={\displaystyle \frac{1}{2}}(x_1+x_2)=(0,0.5,0.575,0.5),{\mu }_2^{\left(1\right)}=x_3=(10,10,0.10,1.00).$$

Step 4: Reassignment check. To determine whether the cluster assignments have changed, we recompute the squared Euclidean distances from each data point to the updated centroids:

$$\parallel x_1-{\mu }_1^{\left(1\right)}{\parallel }^2=0.6806,\parallel x_1-{\mu }_2^{\left(1\right)}{\parallel }^2=201.81,{\parallel x_2-{\mu }_1^{\left(1\right)}\parallel }^2=0.6806$$

$$\parallel x_2-{\mu }_2^{\left(1\right)}{\parallel }^2=181.05\parallel x_3-{\mu }_1^{\left(1\right)}{\parallel }^2=190.73,{\parallel x_3-{\mu }_2^{\left(1\right)}\parallel }^2=0.$$

Since each point remains closest to its previously assigned centroid, no reassignment occurs. Thus, the algorithm has converged. Therefore, the final cluster assignment obtained by the traditional K-means algorithm is

$$C_1^{\left(2\right)}=\{x_1,x_2\},C_2^{\left(2\right)}=\left\{x_3\right\}.$$

We next present the clustering result obtained by applying the proposed multi-view K-means clustering algorithm.

Step 1: Initial assignments. We begin by specifying the initial cluster assignments for each view. Let

$$L^{\left(1\right)}=[1,1,2],L^{\left(2\right)}=[1,2,2],$$

where $L^{\left(1\right)}$ and $L^{\left(2\right)}$ denote the cluster labels in Views 1 and 2, respectively. This indicates that, in the spatial view ($v=1$), data points $x_1$ and $x_2$ are initially assigned to Cluster 1, while $x_3$ is assigned to Cluster 2. Simultaneously, in the preference view ($v=2$), $x_1$ is assigned to Cluster 1, and both $x_2$ and $x_3$ are assigned to Cluster 2.

Step 2: Compute view-specific centroids. Based on the initial cluster assignments, we compute the centroids separately for each view by averaging the data points assigned to each cluster:

$$\begin{array}{cccc}\hfill {\mu }_1^{\left(1\right)}& ={\displaystyle \frac{1}{2}}(x_1^{\left(1\right)}+x_2^{\left(1\right)})=[0,0.5],\hfill & \hfill {\mu }_2^{\left(1\right)}& =x_3^{\left(1\right)}=[10,10],\hfill \\ \hfill {\mu }_1^{\left(2\right)}& =x_1^{\left(2\right)}=[1.00,0.00],\hfill & \hfill {\mu }_2^{\left(2\right)}& ={\displaystyle \frac{1}{2}}(x_2^{\left(2\right)}+x_3^{\left(2\right)})=[0.125,1.00].\hfill \end{array}$$

Step 3: Assignment computation for $x_2$.

View 1 (Spatial):

$$\begin{array}{cc}\hfill \parallel x_2^{\left(1\right)}-{\mu }_1^{\left(1\right)}{\parallel }^2& =0.25+{\lambda }^{\left(1\right)}=5.25,\hfill \\ \hfill \parallel x_2^{\left(1\right)}-{\mu }_2^{\left(1\right)}{\parallel }^2& =181+0=181.\hfill \end{array}$$

View 2 (Preference):

$$\begin{array}{cc}\hfill \parallel x_2^{\left(2\right)}-{\mu }_1^{\left(2\right)}{\parallel }^2& =1.7225+0=1.7225,\hfill \\ \hfill \parallel x_2^{\left(2\right)}-{\mu }_2^{\left(2\right)}{\parallel }^2& =0.000625+{\lambda }^{\left(2\right)}=0.100625.\hfill \end{array}$$

Step 4: View-wise reassignment.

View 1: $D_1^{\left(1\right)}=5.25<D_2^{\left(1\right)}=181\Rightarrow L_2^{\left(1\right)}=1$;

View 2: $D_2^{\left(2\right)}=0.100625<D_1^{\left(2\right)}=1.7225\Rightarrow L_2^{\left(2\right)}=2$.

Step 5: Unified clustering decision. Since the views disagree on the assignment of $x_2$, we apply majority voting. As labels $L_2^{\left(1\right)}=1,L_2^{\left(2\right)}=2$ have the same number of votes, we need to apply silhouette scores of $x_2$ separately in each view to assign the final cluster. By using the definition of the silhouette scores, we obtain $s_2^1=0.926$ and $s_2^2=0.962$. This implies that $x_2$ is assigned to Cluster 2. In terms of $x_1,x_3$, these two points have the same label across the two views, and thereby the majority vote $x_1$ is in Cluster 1 and $x_1$ is in Cluster 2. Then, the final cluster label is

$$\widehat{L}=[1,2,2].$$

4.2. Optimization Assignment

The generalized optimization assignment with a minimum facility constraint (see Algorithm 2) is designed to optimize the CFLBP-CB across multiple clusters while ensuring that each cluster contains at least the specified minimum number of facilities. The procedure begins by initializing the unified set of K clusters obtained from Algorithm 1. In this phase, the algorithm operates in two main subprocesses: cluster-wise optimization, where facility assignments are optimized within each cluster, and updating unassigned facilities, where any remaining facilities are allocated to satisfy the required constraints.  

Step 1: Cluster-wise optimization  

The first step involves optimizing the assignment of facilities and customers within each cluster. For each cluster c, the customers $C_c$ and facilities $F_c$ associated with that cluster are extracted. If the number of facilities in the cluster is less than the required minimum m, the algorithm skips the optimization for that cluster. Otherwise, the optimization model is executed using the bilevel optimization problem, the CFLBP-CB. Once the CFLBP-CB model is solved, the facilities within the cluster are marked as used by setting the Used flag in the dataset. Finally, the cluster c is appended to the list of assigned clusters.  

Step 2: Update unassigned facilities  

After the cluster-wise optimization, the algorithm updates the list of unassigned clusters by removing the clusters that have already been assigned. It then identifies the unused facilities that have not been assigned to any cluster. For each unused facility $f_i$, the algorithm temporarily assigns it to each unassigned cluster and computes the silhouette scores for all views. The silhouette score is used to assess the quality of the facility’s assignment to each cluster. The facility is then assigned to the cluster that yields the highest silhouette score, ensuring that the facility is placed in the most appropriate cluster based on both cohesion and separation. The updated cluster assignments for the facilities are then reflected in the dataset. Finally, Step 1 (cluster-wise optimization) is repeated.

The details of the generalized optimization assignment algorithm with the minimum facility constraint can be found in Algorithm 2.

5. Case Study

The experiments are conducted on a CPU with a maximum clock speed of 5.00 GHz. The K-means algorithm is custom implemented, while spectral and hierarchical clustering methods are sourced from Scikit-learn 1.6.1, benefiting from optimized code. The optimization model is formulated in Pyomo 6.9.2 and solved using MindtPy bundled with Pyomo 6.9.2 with the Outer Approximation (OA) strategy, employing GLPK 5.0 for mixed-integer subproblems and IPOPT 3.14.17 for nonlinear ones. These open-source solvers are selected for their accessibility and reproducibility. However, GLPK is less efficient than commercial solvers for large MILPs, and IPOPT may face challenges with nonconvex or nonsmooth problems. Additionally, the OA strategy can be computationally intensive due to repeated MILP-NLP solving cycles.

To evaluate the effectiveness of the proposed heuristic framework, we conduct a case study using a real-world dataset provided by the PySAL spatial analysis library (available at https://pysal.org/spopt/notebooks/facloc-real-world.html, accessed on 1 August 2024) The dataset includes geographic information for potential facility sites and customer demand sites distributed throughout the city of San Francisco. This urban-scale scenario provides a realistic testbed for assessing competitive facility location strategies under spatial and behavioral considerations.

The details of candidate facility locations and customer sites are summarized in

Table 4. Candidate facility locations in the city of San Francisco ¹.

Candidate Location	Latitude	Longitude
Location 1	37.7724	–122.5100
Location 2	37.7538	–122.4889
⋮	⋮	⋮
Location 16	37.7971	–122.3989

¹ Source: PySAL facility location dataset.

and

Table 5. Customer demand sites in the city of San Francisco ¹.

Customer ID	Latitude	Longitude	Population
1	37.6508	−122.4887	4135
2	37.6600	−122.4835	4831
⋮	⋮	⋮	⋮
205	37.7823	−122.4165	8188

¹ Source: PySAL facility location dataset.

, respectively.

5.1. Franchise Expansion into New Territories

This case study simulates a retail expansion scenario in which a parent company (the leader) opens five company-owned stores, and then franchisees (the followers) open five additional stores in response. The objective is to optimize joint market coverage and profitability while avoiding cannibalization between company-owned and franchised outlets. Based on this setup and using the real-world dataset summarized in Table 4 and Table 5, the case study involves $\left|I\right|=16$ candidate facility locations and $\left|J\right|=205$ customer sites distributed across the city of San Francisco.

In this setting, we assume two customer preference dimensions. The first preference ($Preferenc{e}_1$) is Price Sensitivity, which reflects the extent to which a customer values affordability. The second preference ($Preferenc{e}_2$) is Service Expectation, which captures the importance a customer places on high service quality. The customer attraction coefficient is then defined as

$$v_{ij}={\displaystyle \frac{p_{ij}}{d_{ij}+ϵ}}={\displaystyle \frac{min\left\{1,\frac{Preferenc{e}_{1,i}}{Preferenc{e}_{1,j}}\right\}\times min\left\{1,\frac{Preferenc{e}_{2,i}}{Preferenc{e}_{2,j}}\right\}}{d_{ij}+ϵ}},$$

(24)

where $i\in \{1,2,\dots ,16\}$ represents candidate facility locations and $j\in \{1,2,\dots ,205\}$ represents customer sites.

5.2. Results

To address the experimental setup described in Section 5.1, we utilize the proposed two-stage solution framework. In the first stage, clustering is performed using Algorithm 1, which employs a behavior-aware multi-view K-means approach with co-regularization to ensure consistency across different data views. In the second stage, Algorithm 2 is applied to solve the CFLBP-CB within each cluster, effectively capturing the leader–follower dynamics characteristic of competitive decision-making.

5.2.1. Clustering Results

To implement Algorithm 1, which performs behavior-aware multi-view K-means clustering with co-regularization, we set the number of clusters to $K=5$ and define two data views ($V=2$), corresponding to geographic and preference information. Since explicit customer preference data are not available in the original dataset, we synthetically generate behavioral attributes to construct $Preferenc{e}_{1,i}$, $Preferenc{e}_{2,i}$, $Preferenc{e}_{1,j}$, and $Preferenc{e}_{2,j}$. The co-regularization penalties are configured as ${\lambda }_1=0.025$ for the spatial view and ${\lambda }_2=0.075$ for the behavioral view, ensuring that both data sources contribute appropriately to the clustering process.

The initial outcomes of the multi-view K-means clustering are depicted in Figure 2. The left subplot displays clustering based on geographic proximity (X–Y), while the right subplot shows clustering based on preference features (Preferences 1–2). In both subplots, points are colored according to their assigned clusters: blue for Cluster 0, green for Cluster 1, brown for Cluster 2, gray for Cluster 3, and cyan for Cluster 4. The centroids of each cluster are indicated by markers in the same color as their respective clusters. The geographic clusters in the left subplot demonstrate strong spatial compactness, particularly for Cluster 2. In contrast, clusters in the preference space (right subplot) are more dispersed and overlapping, highlighting greater variability in these attributes. While noticeable differences exist between cluster assignments in the spatial and behavioral views, the multi-view K-means clustering approach introduces a degree of alignment, providing a solid foundation for further cluster integration.

To further explain the discrepancies between views, Figure 3 illustrates inconsistencies in cluster assignments, where black-circled points denote customers whose groupings differ between geographic and preference perspectives. Such mismatches, arising when spatially close customers exhibit distinct behavioral profiles or vice versa, underscore the challenge of simultaneously achieving spatial and behavioral consistency within a single clustering solution. This observation motivates a refinement stage, which integrates the two views using majority voting and silhouette scores to produce a more unified clustering outcome, as presented in Figure 4. Here, cluster membership is indicated by color, with consensus achieved through the aforementioned refinement process. The unified results highlight that even customers within the same geographic cluster can exhibit distinct behavioral preferences, reflecting real-world complexities in competitive facility location problems. To quantify these unified clusters,

Table 6. Number of customers and facilities assigned to each cluster.

Cluster	Number of Customers	Number of Facilities
0	43	2
1	35	2
2	32	5
3	44	5
4	51	2

reports the balanced distribution of customers and facilities after refinement, confirming the effectiveness of the proposed clustering strategy (Algorithm 1). Collectively, Figure 2, Figure 3 and Figure 4 and Table 6 demonstrate that our multi-view K-means clustering framework, incorporating co-regularization and refinement, achieves superior inter-view consistency and improved intra-cluster compactness compared to single-view approaches. This unified clustering not only resolves ambiguous assignments effectively but also provides a stable and interpretable input structure for subsequent facility location optimization, ensuring alignment between spatial accessibility and customer preferences.

5.2.2. Bilevel Optimization Results

To determine the optimal facility locations for the parent company (leader) and its franchisees (follower), we apply Algorithm 2 to the unified clusters generated by Algorithm 1. The CFLBP-CB model is solved within each cluster using Algorithm 2, with the parameters set to max_iter = 100, $ϵ=1\times {10}^{-6}$, and a required minimum of $m=2$ candidate sites per cluster (since each cluster must have at least two candidate sites, one for the leader and one for the follower).

Table 7. Overall, leader, and follower profits for each cluster.

Clustering	Overall Profit	Leader Profit/Leader Market Share	Follower Profit	Follower Market Share
	Equation (1) + Equation (10)	Equation (1)	Equation (10)	Equation (4)
Cluster 0	3673.64	2394.47	1279.17	1905.52
Cluster 1	3208.62	2421.07	787.55	1078.93
Cluster 2	2152.80	1650.62	603.42	1650.62
Cluster 3	3075.48	2430.24	645.24	1969.76
Cluster 4	4307.10	2740.41	1566.69	2359.59
Total profit	16,417.64
Time (s)
Cluster	0.1708
Optimize	2.9801

summarizes the resulting performance metrics, including overall profit, leader profit, follower profit, leader market share, and follower market share, for Clusters 0 through 4.

The results demonstrate that profit outcomes vary notably across clusters, influenced by the spatial and behavioral characteristics of customers. For example, Cluster 4 achieves the highest overall and leader profits (4307.10 and 2740.41, respectively), indicating a strong alignment with customer preferences or a larger customer base, followed closely by Cluster 0. In contrast, Clusters 2 and 3 exhibit comparatively lower overall and follower profits, suggesting either smaller populations or less favorable positioning for the follower. These outcomes highlight that leader profit tends to remain relatively stable and substantial across clusters, whereas follower profit is more sensitive to competitive facility placement, as reflected by the lower values in Clusters 1, 2, and 3. This pattern is characteristic of competitive facility location problems solved via bilevel optimization. Note that the follower market share, shown in the last column of Table 7, is effectively the complement of the leader’s market share.

Figure 5 illustrates the distribution of customer clusters and the corresponding optimal facility locations under two distinct views: geographic features (X–Y) and preference features (Preferences 1–2). Each color signifies a customer cluster, and the triangles mark the selected facility locations that are optimized to capture the most market share within each cluster. The figure demonstrates that facilities are generally placed in areas with higher customer densities, particularly in clusters where customers are more tightly grouped. There is clear alignment between customer distributions and optimal facility sites, indicating the effectiveness of the multi-view K-means clustering approach in supporting profitable facility placement. This geographic and preference alignment underpins the profit patterns observed in Table 7, emphasizing that strategic clustering and facility placement are key drivers of market success in competitive location problems.

5.2.3. Sensitivity of Multi-View K-Means Clustering to Co-Regularization Parameters

To evaluate the robustness of the proposed framework, we conduct a sensitivity analysis by varying the co-regularization parameters ${\lambda }_1$ and ${\lambda }_2$, which control the influence of spatial and behavioral views, respectively, during the multi-view K-means clustering stage.

Table 8. Sensitivity of overall profit with respect to co-regularization parameters ${\lambda }_1$ and ${\lambda }_2$.

Co-Regularization Parameters		Overall Profit
${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$
0.025	0	14,577.37
0.005	0.045	15,493.84
0.0025	0.105	15,279.92
0	0	15,812.44

reports the overall profit values obtained from the subsequent bilevel optimization for each tested parameter configuration.

The results demonstrate that, when ${\lambda }_1=0.025$ and ${\lambda }_2=0$, the clustering process is biased toward spatial alignment and disregards behavioral consistency, yielding the lowest overall profit of 14,577.37. In contrast, when the behavioral view is given moderate influence (${\lambda }_1=0.005$, ${\lambda }_2=0.045$), the profit increases substantially to 15,493.84, indicating the value of incorporating customer preferences in the clustering stage. However, pushing the behavioral influence further (${\lambda }_2=0.105$) while keeping spatial influence minimal (${\lambda }_1=0.0025$) results in a slightly lower profit (15,279.92), suggesting that excessive bias toward one view can reduce overall performance.

Notably, the highest overall profit of 15,812.44 is achieved when both co-regularization terms are set to zero (${\lambda }_1=0$, ${\lambda }_2=0$), implying that independent clustering of views followed by decision-level integration (through majority voting and silhouette-based refinement) may be sufficient to produce a well-aligned and effective cluster structure. These findings emphasize that, while co-regularization can help enforce alignment across data views, it must be applied judiciously. Over-regularization may constrain the natural clustering tendencies of individual views and diminish optimization outcomes. A balanced or even unpenalized approach, in combination with robust post-clustering refinement, may offer the best trade-off in practice.

5.2.4. Parameter Tuning

It is evident from Table 8 that the choice of co-regularization parameters significantly influences the achievable overall profit. To further illustrate and generalize the effect of varying ${\lambda }_1$ and ${\lambda }_2$ across a broader parameter space, we provide a surface plot in Figure 6.

Figure 6 presents a 3D surface plot that depicts the relationship between the co-regularization parameters ${\lambda }_1$ and ${\lambda }_2$ and the overall profit attained by the proposed framework. The plot reveals that the profit varies nonlinearly across the parameter space, with several local maxima and minima. Notably, the highest overall profit is observed for specific combinations of moderate values of both ${\lambda }_1$ and ${\lambda }_2$, rather than at their extreme ends. This underscores the importance of carefully tuning the balance between spatial and behavioral regularization during the clustering process. In addition, regions with steep gradients indicate that the model’s performance can be sensitive to small changes in these parameters, underscoring the importance of careful parameter selection. Employing strategies such as grid search can help identify the values of $\lambda$ that yield the highest overall profit, thereby ensuring stable model performance.

5.2.5. Results from Single-View Clustering

Table 9. Profit outcomes of the bilevel optimization model under three single-view clustering algorithms (K-means, spectral, hierarchical) applied to different input feature views: geographic only (2D), preference only (2D), and combined (4D). Each entry includes clustering and optimization time, as well as per-cluster and total profits for leader and follower facilities.

Clustering Feature View	Clustering Algorithm	Time (s)		Overall	Leader	Follower
		Cluster	Optimize	Profit	Profit	Profit
Geographic Only (2D)	K-means
Cluster 0				2487.71	1646.93	840.78
Cluster 1				4492.30	3514.33	977.96
Cluster 2				1748.42	1109.41	639.01
Cluster 3				4717.91	2945.07	1772.84
Cluster 4				2116.53	1503.62	612.91
Total profit		0.0349	3.1019	15,562.87	10,719.37	4843.50
Preference Only (2D)	K-means
Cluster 0				4115.54	3336.64	778.90
Cluster 1				2587.65	2342.99	244.66
Cluster 2				2695.60	2229.22	466.38
Cluster 3				3894.70	2351.02	1543.69
Cluster 4				2267.66	1607.27	660.39
Total profit		0.0604	4.6191	15,561.16	11,867.13	3694.03
Geographic + Preference (4D)	K-means
Cluster 0				3706.87	3031.13	675.74
Cluster 1				3746.44	2993.62	752.82
Cluster 2				2554.85	1895.87	658.98
Cluster 3				1944.56	1016.50	928.06
Cluster 4				2452.30	1783.51	668.79
Total profit		0.0665	4.9162	14,405.02	10,720.63	3684.39
Geographic Only (2D)	Spectral
Cluster 0				4418.97	3640.63	778.34
Cluster 1				3622.13	2663.59	958.54
Cluster 2				2617.91	1620.06	997.84
Cluster 3				1988.18	1241.54	746.64
Cluster 4				2348.86	1888.13	460.73
Total profit		0.0363	4.6648	14,996.05	11,053.95	3942.10
Preference Only (2D)	Spectral
Cluster 0				5996.14	5313.10	683.04
Cluster 1				2422.52	2061.96	360.57
Cluster 2				2437.39	1972.63	464.76
Cluster 3				2137.91	1384.10	753.81
Cluster 4				2610.02	1707.89	902.14
Total profit		0.0330	6.3096	15,603.99	12,439.67	3164.32
Geographic + Preference (4D)	Spectral
Cluster 0				2259.98	1645.98	614.00
Cluster 1				2555.68	2189.37	366.32
Cluster 2				4686.46	3531.55	1154.91
Cluster 3				3180.11	1974.97	1205.14
Cluster 4				2046.26	1486.62	559.64
Total profit		0.0330	6.3096	15,603.99	12,439.67	3164.32
Geographic Only (2D)	Hierarchical
Cluster 0				4336.97	3398.04	938.92
Cluster 1				3559.18	2648.60	910.57
Cluster 2				2545.23	1745.23	800.00
Cluster 3				1663.02	1055.61	607.41
Cluster 4				3402.49	2345.99	1056.50
Total profit		0.0034	2.7864	15,292.71	10,979.30	4313.41
Preference Only (2D)	Hierarchical
Cluster 0				3576.64	2175.02	1401.62
Cluster 1				2784.32	1921.26	863.06
Cluster 2				3359.66	2748.92	610.74
Cluster 3				4460.88	3824.70	636.19
Cluster 4				1311.06	1102.98	208.09
Total profit		0.0035	4.6121	15,376.47	11,534.48	3841.98
Geographic + Preference (4D)	Hierarchical
Cluster 0				3789.64	2875.07	914.56
Cluster 1				2553.21	1930.72	622.49
Cluster 2				4141.97	2942.29	1199.68
Cluster 3				2587.48	2101.52	485.96
Cluster 4				1692.95	1267.86	425.09
Total profit		0.0035	4.5028	14,765.25	11,117.48	3647.77

reports the bilevel optimization outcomes across three single-view clustering algorithms—K-means, spectral, and hierarchical—each applied to three types of input feature spaces: geographic only (2D), preference only (2D), and combined geographic–preference (4D). For each configuration, five clusters are formed, with corresponding leader and follower profits, total profit, and computation times for the clustering and optimization steps. This comparative analysis reveals how clustering strategy and feature representation influence profitability and leader–follower dynamics.

K-means clustering applied to geographic features yields the highest total profit (15,562.87), with Cluster 3 producing the greatest cluster-level return (4717.91), including the top follower profit (1772.84). Preference-based clustering results in a comparable total profit (15,561.16), though it displays stronger leader dominance in several clusters, suggesting reduced market competition due to finer preference-based segmentation. In contrast, the combined 4D feature clustering underperforms (14,405.02), indicating that richer input dimensions may introduce redundancy or fragmentation, thereby reducing clustering coherence and overall efficiency.

Spectral clustering on preference features achieves the highest total profit among all configurations (15,603.99), with substantial leader gains (12,439.67), particularly in Cluster 0 (5313.10). Geographic-based spectral clustering performs slightly worse (14,996.05), though it still offers effective spatial partitioning. Interestingly, spectral clustering with combined features also reaches 15,603.99, suggesting that this method better handles high-dimensional data compared to K-means clustering by exploiting latent structures through spectral embedding.

Hierarchical clustering demonstrates superior computational efficiency, with clustering times consistently under 0.004 s. The preference-only variant yields the highest profit (15,376.47), followed by the geographic-only variant (15,292.71) and, lastly, the 4D view (14,765.25). Notably, hierarchical clustering leads to more balanced follower profits across clusters. In the preference-only model, follower profits are relatively high and evenly distributed (e.g., 1401.62 in Cluster 0 and 902.14 in Cluster 4), reflecting improved market competition.

Spectral clustering applied to preference features yields the highest total profit (15,603.99), demonstrating its superior capability in segmenting customers according to preference-related characteristics. K-means clustering with geographic features yields a comparable profit (15,562.87), while offering strong spatial cohesion and accessibility. Hierarchical clustering stands out for its computational efficiency—with clustering times consistently below 0.004 s—and for producing a more balanced profit distribution between leaders and followers, as indicated by the narrower profit gaps across clusters. In contrast, clustering with combined geographic and preference features (4D) consistently underperforms across all methods, emphasizing the limitations of traditional algorithms in handling high-dimensional feature spaces. These findings underscore the importance of aligning clustering techniques with the underlying data structure to optimize performance in competitive facility location settings.

To support these findings, Figure 7, Figure 8 and Figure 9 illustrate the optimal facility placements derived from each single-view clustering method—including K-means, spectral, and hierarchical clustering—applied to three different input feature sets: geographic (2D), preference (2D), and combined (4D). Each subfigure presents two projections, the spatial view (left) and the preference view (right), enabling visualization of how clusters formed in one feature space manifest in the other.

Figure 7 depicts optimal facility placements derived from single-view K-means clustering using three input feature sets: geographic (2D), preference (2D), and a combined 4D representation.

• Subplot (a): Geographic features (2D). Clustering based on geographic attributes yields compact and well-separated spatial clusters. Facility placements (triangles) are centrally located within their respective clusters, supporting strong geographic accessibility. However, the corresponding preference view exhibits substantial overlap among clusters, indicating poor alignment with customer preferences.
• Subplot (b): Preference features (2D).Clustering in the preference domain produces distinct and well-separated clusters, with facility locations closely aligned with preference centers. Nevertheless, the geographic projection reveals spatial dispersion and overlap, which may hinder efficient service delivery and cost-effective deployment.
• Subplot (c): Combined geographic and preference features (4D). The integration of both feature types leads to incoherent clustering in the spatial view, with facilities misaligned and positioned far from cluster centers. While the preference view maintains some alignment, overall cluster quality deteriorates.

Figure 8 presents the optimal facility placements obtained from single-view spectral clustering applied to the geographic (2D), preference (2D), and combined (4D) feature sets.

• Subplot (a): Geographic features (2D). Clustering based on geographic attributes results in well-separated spatial clusters, with facilities (triangles) embedded within each group, supporting accessibility. However, the corresponding preference view reveals significant cluster mixing, indicating poor alignment with customer behavioral attributes.
• Subplot (b): Preference features (2D). Clustering in the preference domain produces clearly defined clusters and well-aligned facilities, demonstrating effective segmentation. Yet the spatial projection shows considerable dispersion and overlap, reducing efficiency in physical service delivery.
• Subplot (c): Combined geographic and preference features (4D).The combined feature input yields scattered and incoherent clusters in both spatial and preference views, with facilities often located far from cluster centers, suggesting poor integration of the two feature spaces.

Figure 9 illustrates the optimal facility placements derived from single-view hierarchical clustering applied to three distinct input feature sets:

• Subplot (a): Geographic features (2D).The clustering results exhibit compact and spatially coherent groups. Facilities (triangles) are centrally located within clusters, enhancing spatial accessibility. However, in the corresponding preference space, clusters display substantial overlap, reflecting poor alignment with customer preferences.
• Subplot (b): Preference features (2D). The preference view reveals reasonably well-separated clusters, and most facility placements align with the cluster centers. An exception is Cluster 4, where the facility is not centrally located, indicating inconsistency. Meanwhile, the spatial view shows scattered and overlapping clusters, undermining geographic efficiency.
• Subplot (c): Combined geographic and preference features (4D). Clusters appear poorly defined in both spatial and preference views. Facility placements are inconsistently positioned and often located far from the respective cluster centers, reducing both behavioral targeting and spatial coverage.

These findings highlight a fundamental limitation shared across single-view clustering methods (K-means, spectral, and hierarchical): optimizing in one feature space often compromises performance in the other. While each method offers distinct advantages—graph-based similarity in spectral clustering, interpretability in hierarchical clustering, and spatial cohesion in K-means clustering—their effectiveness in competitive facility location depends critically on feature-space alignment. The performance degradation observed in high-dimensional or misaligned settings underscores the importance of consistent and carefully selected input features to ensure both spatial accessibility and behavioral segmentation.

5.2.6. Results from Multi-View K-Means Clustering

Table 10. Bilevel optimization results under different co-regularization settings $({\lambda }_1,{\lambda }_2)$ using multi-view clustering.

Co-Regularization Parameters		Cluster	Leader Profit	Follower Profit
${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$
0.0225	0.075	Cluster 0	2632.18	1389.18
		Cluster 1	2565.93	831.55
		Cluster 2	1557.11	600.06
		Cluster 3	1879.81	547.71
		Cluster 4	2740.41	1566.69
		Total profit	16,310.62
		Time (s)
		Cluster	0.1255
		Optimize	3.0761
0.015	0.120	Cluster 0	1993.43	1020.81
		Cluster 1	2899.40	873.65
		Cluster 2	2571.18	395.45
		Cluster 3	1776.11	700.12
		Cluster 4	2644.41	1444.39
		Total profit	16,318.94
		Time (s)
		Cluster	0.0915
		Optimize	3.4215
0.025	0.015	Cluster 0	2444.47	574.85
		Cluster 1	2037.71	1016.45
		Cluster 2	2725.75	1107.30
		Cluster 3	1318.40	726.50
		Cluster 4	2501.47	1338.96
		Total profit	15,791.85
		Time (s)
		Cluster	0.1001
		Optimize	3.4982
0.000	0.000	Cluster 0	2505.52	909.47
		Cluster 1	3522.01	1365.49
		Cluster 2	1711.16	869.78
		Cluster 3	2151.39	949.43
		Cluster 4	1192.56	635.64
		Total profit	15,812.44
		Time (s)
		Cluster	0.1750
		Optimize	3.5193

reports the bilevel optimization results under various co-regularization settings $({\lambda }_1,{\lambda }_2)$ in the multi-view K-means clustering framework. Each setting reflects a different weighting between geographic $\left({\lambda }_1\right)$ and preference $\left({\lambda }_2\right)$ views, influencing the resulting cluster structures and facility placement performance.

The configuration $({\lambda }_1=0.015,{\lambda }_2=0.120)$ yields the highest total profit of 16,318.94, indicating that strong emphasis on preference alignment, when moderately balanced with spatial structure, can enhance market segmentation and profitability. Similarly, the setting $({\lambda }_1=0.0225,{\lambda }_2=0.075)$ achieves a comparably high profit of 16,310.62 with shorter clustering time (0.1255 s), and demonstrates well-distributed leader and follower profits across clusters—particularly a strong performance in Cluster 4.

By contrast, overly imbalanced or weak regularization settings lead to lower performance. For instance, $({\lambda }_1=0.025,{\lambda }_2=0.015)$, which places less emphasis on preference consistency, yields a reduced total profit of 15,791.85. Likewise, the absence of regularization $({\lambda }_1=0,{\lambda }_2=0)$ results in 15,812.44, suggesting that uncoordinated clustering across views weakens the model’s ability to align customer segments effectively.

Overall, the results emphasize the importance of balanced co-regularization in multi-view clustering. Moderate integration of geographic and preference features supports more coherent cluster structures, which translate into improved leader–follower profitability and efficient computation times. These findings highlight the need for careful tuning of regularization parameters in competitive facility location models.

5.2.7. Comparison Between Single-View and Multi-View K-Means Clustering Results

To evaluate the effectiveness of the proposed multi-view clustering framework with co-regularization, we compare its performance against single-view clustering methods—K-means, spectral, and hierarchical—applied to geographic, preference, and combined (4D) feature sets. The single-view results indicate that geographic-only and preference-only clustering yield the highest total profits (15,562.87 and 15,561.15, respectively), with relatively short clustering times (0.03–0.06 s) but longer optimization times (3.10–6.50 s). In contrast, using a combined 4D input leads to lower total profit (14,405.02) and the longest optimization time (up to 6.50 s), suggesting that naïvely merging features can dilute distinct structural information and degrade performance.

In comparison, the multi-view K-means clustering approach preserves the distinctiveness of geographic and preference views while promoting alignment between them. As shown in

Table 11. Profit and runtime comparison between single-view and multi-view clustering methods.

Clustering Feature View/ Co-Regularization Parameters	Clustering Algorithm	Time (s)		Overall	Leader	Follower
		Cluster	Optimize	Profit	Profit	Profit
Geographic Only (2D)	Single-view	0.0349	3.1019	15,562.87	10,719.37	4843.50
Preference Only (2D)	K-means clustering	0.0604	4.6191	15,561.16	11,867.13	3694.03
Geographic + Preference (4D)		0.0665	4.9162	14,405.02	10,720.63	3684.39
Geographic Only (2D)	Single-view	0.0363	4.6648	14,996.05	11,053.95	3942.10
Preference Only (2D)	Spectral clustering	0.0330	6.3096	15,603.99	12,439.67	3164.32
Geographic + Preference (4D)		0.0330	6.3096	15,603.99	12,439.67	3164.32
Geographic Only (2D)	Single-view	0.0034	2.7864	15,292.71	10,979.30	4313.41
Preference Only (2D)	Hierarchical clustering	0.0035	4.6121	15,376.47	11,534.48	3841.98
Geographic + Preference (4D)		0.0035	4.5028	14,765.25	11,117.48	3647.77
$({\lambda }_1,{\lambda }_2)=(0.025,0.075)$	Multi-view	0.1708	2.9801	16,417.64	11,535.57	4882.07
$({\lambda }_1,{\lambda }_2)=(0.0225,0.075)$	K-means clustering	0.1255	3.0761	16,310.62	11,375.44	4935.18
$({\lambda }_1,{\lambda }_2)=(0.015,0.120)$		0.0915	3.4215	16,318.94	11,884.53	4434.41
$({\lambda }_1,{\lambda }_2)=(0.025,0.015)$		0.1001	3.4982	15,791.85	11,027.80	4764.05
$({\lambda }_1,{\lambda }_2)=(0.000,0.000)$		0.1750	3.5193	15,812.44	11,082.64	4729.80

, several co-regularization settings outperform all single-view models in terms of total profit. For example, the configuration $({\lambda }_1=0.025,{\lambda }_2=0.075)$ achieves the highest profit (16,417.64), with a clustering time of 0.1708 s and an optimization time of 2.9801 s. Similarly, other configurations, such as $({\lambda }_1=0.0225,{\lambda }_2=0.075)$ and $({\lambda }_1=0.015,{\lambda }_2=0.120)$, yield profits exceeding 16,300 with comparably short runtimes. Even simple settings such as $({\lambda }_1=0.000,{\lambda }_2=0.000)$ outperform the best single-view baseline, demonstrating robustness across parameter choices. It is worth noting that these co-regularization parameters $({\lambda }_1,{\lambda }_2)$ can be selected through strategies such as grid search, as discussed in Section 5.2.4.

Notably, multi-view methods generally require longer clustering times (0.09–0.17 s) due to co-regularization but achieve shorter optimization times (2.98–3.52 s), resulting in competitive or superior overall runtimes compared to single-view methods.

These findings emphasize some key advantages of the proposed framework:

• Multi-view clustering effectively integrates complementary spatial and behavioral patterns, yielding better facility–customer groupings for optimization.
• Co-regularization provides a tunable mechanism to balance intra-view fidelity and inter-view consistency, improving clustering quality and optimization outcomes.
• Despite slightly increased clustering overhead, the total computation time is reduced due to more efficient optimization on coherent clusters.

In summary, multi-view clustering with co-regularization not only improves profit and solution quality in competitive facility location problems but also enhances computational efficiency by reducing downstream optimization time.

6. Discussion

6.1. Effectiveness of Multi-View Clustering

As shown in Table 9, the multi-view K-means clustering approach with co-regularization consistently outperforms single-view methods in terms of overall profit. While single-view clustering using either geographic or preference features can yield competitive results, the multi-view method achieves superior outcomes by preserving the structural integrity of each view and promoting inter-view consistency. The subsequent refinement step—incorporating majority voting and silhouette scores—further enhances cluster coherence and interpretability.

6.2. Insights from Co-Regularization Sensitivity

Sensitivity analysis of the co-regularization parameters ${\lambda }_1$ (geographic view) and ${\lambda }_2$ (preference view) indicates that balanced inter-view alignment is crucial for solution quality. Configurations favoring preference alignment while preserving spatial coherence (e.g., ${\lambda }_1=0.025$, ${\lambda }_2=0.075$) yield the highest profit. In contrast, settings with no regularization or strong bias toward one view result in suboptimal or unbalanced clustering. These findings underscore the importance of tuning co-regularization to effectively integrate spatial and preference features in practice.

6.3. Comparison of Computational Efficiency with Bilevel Optimization

From a computational complexity standpoint, the efficiency of the proposed framework can be approximated by

$$\mathcal{O}\left(T(n+m)VK(d+V)+{(n+m)}^{2}dV+\left(\left({\displaystyle \genfrac{}{}{0pt}{}{n/k}{1}}\right)·\left({\displaystyle \genfrac{}{}{0pt}{}{n/k-1}{1}}\right)·k·\left({\displaystyle \frac{n}{k}}·{\displaystyle \frac{m}{k}}+T_{\mathrm{solve}\left(10\right)}\right)\right)\right),$$

(25)

where the first component, $T(n+m)VK(d+V)+{(n+m)}^{2}dV$, captures the computational cost of the multi-view clustering process. This term grows polynomially with respect to the number of customers n, candidate facilities m, feature views V, and clusters K. The second component,

$$\left({\displaystyle \genfrac{}{}{0pt}{}{n/k}{1}}\right)·\left({\displaystyle \genfrac{}{}{0pt}{}{n/k-1}{1}}\right)·k·\left({\displaystyle \frac{n}{k}}·{\displaystyle \frac{m}{k}}+T_{\mathrm{solve}\left(10\right)}\right),$$

(26)

represents the computational effort required for solving the CFLBP-CB bilevel optimization problem within each cluster. This can be algebraically reformulated as follows:

$${\displaystyle \frac{n}{k}}·\left({\displaystyle \frac{n}{k}}-1\right)·k·\left({\displaystyle \frac{n}{k}}·{\displaystyle \frac{m}{k}}+T_{\mathrm{solve}\left(10\right)}\right)\approx {\displaystyle \frac{n^{3}m}{k^3}}+{\displaystyle \frac{n^2}{k}}{T}_{\mathrm{solve}\left(10\right)},$$

(27)

which reveals its polynomial nature in terms of n, m, and k. As $n\to \infty$, the asymptotic behavior of the overall runtime in Equation (25) becomes

$$\mathcal{O}\left({\displaystyle \frac{n^{3}m}{k^3}}\right),$$

(28)

demonstrating that the optimization step increasingly dominates the total computation time in large-scale instances, while the clustering components remain comparatively lightweight.

In contrast, the computational complexity of directly solving the full CFLBP-CB bilevel optimization model is substantially higher and can be estimated as

$$\mathcal{O}\left((nm+T_{\mathrm{solve}\left(10\right)})·\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)·\left({\displaystyle \genfrac{}{}{0pt}{}{n-k}{k}}\right)\right),$$

(29)

which grows exponentially with the number of selected facilities k. This expression arises from evaluating all possible combinations of selecting k leader facilities from n facility location candidates and, subsequently, k follower facilities from the remaining $n-k$ locations.

To illustrate the exponential nature of this term, we apply a standard upper bound on the binomial coefficient:

$$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\le {\left({\displaystyle \frac{ne}{k}}\right)}^k,$$

where e is the base of the natural logarithm. Substituting this into the expression for the full bilevel optimization complexity, we obtain the following approximation:

$$T_{\mathrm{solve}\left(10\right)}·{\left({\displaystyle \frac{ne}{k}}\right)}^k·{\left({\displaystyle \frac{e(n-k)}{k}}\right)}^k.$$

When $k\le n$, the term $(n-k)$ can be approximated by n, yielding the following:

$$T_{\mathrm{solve}\left(10\right)}·{\left({\displaystyle \frac{ne}{k}}\right)}^{2k}.$$

This form clearly illustrates the exponential growth with respect to the number of selected facilities k. Consequently, as $n\to \infty$, the asymptotic behavior of the overall runtime in Equation (29) becomes

$$\mathcal{O}\left({\left({\displaystyle \frac{ne}{k}}\right)}^{2k}\right).$$

(30)

This exponential behavior arises even under fixed binary leader and follower decisions ($x_i,y_i$), indicating the inherent intractability of directly solving the full CFLBP-CB model for large-scale instances.

Therefore, by comparing Equations (28) and (30), it is evident that the proposed framework incurs a significantly lower computational cost than directly solving the full CFLBP-CB model. This comparison reinforces the practical efficiency and scalability of the clustering-based decomposition strategy for addressing large-scale competitive facility location problems.

6.4. Evaluating Clustering-Induced Profit Bias

To assess the potential bias introduced by restricting demand allocation within clusters, we compare the profit outcomes generated by the proposed framework with those obtained by evaluating its facility placement decisions under the full CFLBP-CB model. As shown in

Table 12. Evaluation of the top-10 profit rankings when applying the optimal facility placements from the proposed framework to the full CFLBP-CB model.

Co-Regularization Parameters	Consistency	Silhouette		Cohesion		Profit
		X–Y	Pref 1–2	X–Y	Pref 1–2	Framework	Full Model
$(0.0050,0.1500)$	221	0.3505	–0.0773	0.7393	1.5014	15,504.54	16,104.18
$(0.0175,0.0000)$	165	0.0208	0.4059	0.6674	2.0014	14,969.93	16,071.82
$(0.0100,0.1350)$	197	0.2074	0.0408	0.7008	1.6573	14,694.02	15,907.97
$(0.0050,0.0900)$	120	0.3951	0.1509	0.7825	1.8579	15,638.27	15,882.79
$(0.0075,0.0750)$	117	0.3374	0.1909	0.7744	1.8952	15,559.72	15,882.79
$(0.0100,0.0600)$	126	0.1821	0.2854	0.7203	1.9347	15,742.97	15,835.59
$(0.0250,0.0750)$	205	0.0073	0.1872	0.6565	1.8469	16,417.64	15,770.30
$(0.0125,0.0150)$	103	0.1765	0.3914	0.715	1.997	15,548.17	15,758.24
$(0.0125,0.1500)$	220	0.0274	–0.0785	0.6563	1.5269	14,760.96	15,751.88

, the highest test-set profit under the full model (16,104.18) is achieved using the co-regularization parameters $({\lambda }_1,{\lambda }_2)=(0.0050,0.1500)$. This setting differs from the framework-optimal parameters $(0.0250,0.0750)$, which produce the highest profit within the clustering-based model but yield a lower profit of 15,770.30 when assessed on the full model.

While the configuration $({\lambda }_1,{\lambda }_2)=(0.0050,0.1500)$ achieves superior performance when evaluated under the full CFLBP-CB model and exhibits strong consistency along with a high silhouette score in the spatial (X–Y) view, it simultaneously yields a negative silhouette score ($-0.0773$) in the preference view. As shown in the preference-space projection in Figure 10a, the resulting clusters appear to be behaviorally fragmented, with significant overlap among different cluster labels. This indicates that, although the spatial structure is well-formed, the behavioral segmentation is weak, suggesting that customers with divergent preferences are grouped together due to spatial proximity.

On the other hand, considering the quality of clustering across the parameter configurations presented in Table 12, the best overall clustering structure is obtained using the co-regularization parameters $({\lambda }_1,{\lambda }_2)=(0.0050,0.0900)$, as illustrated in Figure 10b. This setting yields more coherent clusters in both the spatial and preference views, reflecting a better balance between geographic proximity and behavioral similarity. These results highlight the importance of carefully selecting co-regularization parameters in accordance with the decision-maker’s priorities—whether optimizing for global performance or ensuring high-quality, behaviorally consistent segmentation.

The correlation analysis between objective values obtained from the proposed clustering-based model and those from the full CFLBP-CB model reveals varying degrees of consistency across evaluation criteria. As shown in

Table 13. Correlation between objective values obtained from the proposed framework and the full CFLBP-CB model.

Evaluation Criteria	Correlation Coefficient
Overall objective	0.2271
Leader profit	0.5868
Follower profit	0.2506
Follower market share	0.5868

, the overall objective exhibits a weak correlation ($r=0.2271$), suggesting potential shifts in global profit rankings under the full model. The follower’s profit shows a moderate correlation ($r=0.2506$), reflecting sensitivity to inter-cluster competition. In contrast, stronger correlations are observed for the leader’s profit and the follower’s market share ($r=0.5868$), indicating that strategic positioning and market influence are relatively well preserved.

These findings imply that, while the clustering-based framework may introduce bias at the aggregate level, it remains effective for subproblems involving individual stakeholders. In particular, when the decision-maker’s focus is on leader profitability or follower market share, the proposed method offers outcomes that are reasonably aligned with those of the full model.

6.5. Comparative Metaheuristic: Genetic Algorithm

A genetic algorithm (GA) is implemented using the DEAP 1.4.3 library with the following parameter configuration: The evolutionary process spans 10 generations, including the initial generation, maintaining a constant population size of 20 individuals. Genetic operators are subsequently applied, with the crossover probability set to 0.8 and the mutation probability set to 0.2. Additionally, an individual bit-flip probability of 0.05 is used for mutation, and a tournament size of 3 is employed for parent selection. Crucially, a dedicated repair function is integrated into the evaluation process to ensure that all generated individuals strictly adhere to problem-specific constraints. These constraints mandate that the leader opens exactly p facilities ($\sum x_i=p$), the follower opens exactly r facilities ($\sum y_i=r$), and no single facility location is simultaneously opened by both players ($x_i+y_i\le 1$ for all locations i). This comprehensive parameter configuration collectively controls the balance between the exploration of the search space and its exploitation by the GA, thereby aiming to identify optimal facility locations.

As presented in

Table 14. Genetic algorithm optimization progress.

Generation	Evaluations	Average Overall Profit	Maximum Overall Profit	Time (Seconds)
0	20	14,941.60	15,595.04	91.77
1	18	15,264.77	15,661.82	89.97
2	16	15,424.87	15,991.54	80.94
3	19	15,565.02	15,991.54	93.35
4	13	15,756.81	16,028.84	66.15
5	18	15,930.98	16,099.76	89.54
6	15	15,963.93	16,104.40	73.48
7	14	15,971.32	16,104.40	69.96
8	12	16,030.23	16,104.40	58.89
9	15	16,039.75	16,137.75	74.06

, the GA demonstrates a clear convergence behavior, with both the average and maximum overall profit steadily increasing across the nine generations. The algorithm commences with an average overall profit of 14,941.60 and converges to an average of 16,039.75, while the maximum overall profit reaches 16,137.75 by the final generation. The computation time per generation varies, generally ranging from approximately 66 to 94 s.

6.6. Comparative Evaluation of Optimization Methods

The results presented in

Table 15. Profit and runtime comparison between the proposed framework with co-regularization $({\lambda }_1,{\lambda }_2)$ and the baseline CFLBP-CB model. Underlined values indicate results that are directly comparable across methods.

Solution Framework	Co-Regularization	Time (s)		Overall	Leader	Follower	Follower
		Cluster	Optimize	Profit	Profit	Profit	Market Share
Multi-view framework	(0.0250, 0.0750)	0.1708	2.9801	16,417.64	11,535.57	4882.07	8964.43
Multi-view solution on full model				15,770.30	11,390.20	4380.10	9109.80
Multi-view framework	(0.0050, 0.1500)	0.1042	3.8963	15,504.54	11,055.31	4449.22	9444.69
Multi-view solution on full model				16,104.17	11,798.48	4305.69	8701.51
Direct CFLBP-CB			608.5756	15,988.27	11,254.59	4733.68	9245.41
GA
gen 6			585.2000	16,104.40	11,594.58	4509.82	8905.42
gen 9			788.1100	16,137.75	11,872.01	4265.73	8627.98

compare the performance of three distinct solution approaches to the CFLBP-CB: the proposed multi-view clustering-based framework, direct optimization of the full bilevel CFLBP-CB model, and a metaheuristic method based on a GA. The proposed framework, which integrates Algorithms 1 and 2, emphasizes computational efficiency and interpretability by conducting localized optimization within spatially and behaviorally coherent clusters. In contrast, the full bilevel model provides a benchmark for exact global optimization, while the GA serves as a heuristic baseline, exploring the solution space more broadly through a global search.

Table 15 clearly demonstrates the substantial computational efficiency of the proposed multi-view framework. Achieving solutions within a total runtime of approximately 4 s—comprising less than 0.2 s for the clustering stage and around 3 s for the optimization phase—this approach significantly outperforms existing methods. Specifically, it yields an approximate 99.34% reduction in runtime compared to the Direct CFLBP-CB model, which requires 608.5756 s for optimization. Furthermore, its efficiency is also pronounced compared to the GA, demonstrating an approximate 99.32% reduction when benchmarked against the 585.2 s required by the GA to first surpass the proposed method’s overall profit (at generation 6).

In terms of solution quality, the multi-view framework demonstrates varying overall profit levels when evaluated under the full model, dependent on the co-regularization parameters. Specifically, an overall profit of 15,770.30 is achieved with $({\lambda }_1,{\lambda }_2)=(0.0250,0.0750)$, while a higher profit of 16,104.17 is obtained with $({\lambda }_1,{\lambda }_2)=(0.0050,0.1500)$. This variability highlights the influence of parameter selection on the solution’s overall performance. At its optimal configuration, employing co-regularization parameters of $({\lambda }_1,{\lambda }_2)=(0.0050,0.1500)$, the proposed framework yields an overall profit of 16,104.17. This result proves to be remarkably competitive with the GA, exhibiting a marginal difference of approximately 0.21% when compared to that of the GA’s highest achieved profit of 16,137.75. Furthermore, this proposed framework’s overall profit of 16,104.17 represents an approximate 0.72% increase compared to the Direct CFLBP-CB model, which yields a profit of 15,988.27.

Although the GA achieves a slightly higher profit, this advantage comes at the cost of significantly longer runtime. The proposed framework, by contrast, combines computational efficiency with interpretable, behaviorally consistent clustering, making it a scalable and practical solution for real-world decision environments that require both performance and speed.

6.7. Implications for Competitive Facility Planning

The proposed framework offers a scalable and interpretable solution for data-driven facility location in competitive environments. By incorporating a bilevel optimization model within each cluster, it captures realistic leader–follower dynamics, allowing dominant providers to anticipate and respond strategically to competitors. The attraction coefficients $v_{ij}$ integrate spatial proximity and customer preferences, aligning facility placements with diverse customer segments.

While the preference attributes used in this study are synthetic, they serve as a proof of concept for behavior-aware optimization. In practical applications, these features can be replaced with empirical preference or behavioral data (e.g., purchasing history, stated preferences), enabling the framework to support real-world competitive planning where customer behavior significantly influences service demand.

6.8. Limitations and Future Work

While the synthetic generation of preference attributes allows for simulating realistic customer heterogeneity, future work should incorporate actual demographic, socioeconomic, or transaction-level data to enhance empirical realism. Additionally, extending the current bilevel framework to include capacity constraints, pricing decisions, or stochastic demand, alongside exploring advanced multi-view clustering techniques, would significantly improve its real-world applicability.

7. Conclusions and Future Work

This paper introduces a novel heuristic framework for solving the competitive facility location problem with customer behavior (CFLBP-CB), integrating behavior-aware multi-view K-means clustering with co-regularization and bilevel optimization. Addressing key limitations of traditional location models, the proposed method accounts for both spatial proximity and customer behavioral preferences, thereby enabling more nuanced and effective facility placement decisions in competitive settings. The proposed framework adopts a two-stage structure. In the first stage, multi-view clustering with co-regularization is employed to partition the market based on both geographic and behavioral features. In the second stage, a bilevel optimization model is applied within each cluster to capture the hierarchical decision-making dynamics between the leader and the follower.

Our empirical evaluation, using a real-world San Francisco dataset, demonstrates the substantial efficacy of the proposed two-stage approach. It significantly reduces computation time by over 99.34% (from 608.58 to 4.00 s) and improves overall profit by approximately 0.72% (from 15,988.27 to 16,104.17) compared to directly solving the full bilevel optimization model. Furthermore, its integrated multi-view clustering with co-regularization outperforms all single-view baselines (K-means, spectral, and hierarchical), yielding an approximate 5.21% increase in overall profit and reduced optimization time by effectively capturing complementary spatial and behavioral structures. Notably, when benchmarked against the GA metaheuristic, our proposed method demonstrates a highly competitive overall profit. At its optimal co-regularization setting, an overall profit of 16,104.17 is achieved, which is only 0.21% lower than the highest recorded profit of the GA (16,137.75). This compelling combination that can deliver high-quality solutions while demanding significantly fewer computational resources renders the proposed approach exceptionally well-suited for large-scale or time-sensitive competitive facility planning tasks.

A sensitivity analysis provides further insights, underscoring the critical importance of carefully tuning the co-regularization parameters to achieve an optimal balance between spatial coherence and behavioral alignment. The analysis also indicates that, while refinement steps (e.g., majority voting and silhouette evaluation) can correct certain misalignments, strategic parameter selection fundamentally enhances both clustering quality and subsequent optimization outcomes.

Several avenues for future research are identified to further advance this framework. These include incorporating authentic real-world customer behavior data and extending the bilevel CFLBP-CB model to encompass additional decision variables like service capacities or pricing strategies. Furthermore, more advanced multi-view clustering approaches and alternative clustering techniques—such as spectral and hierarchical clustering—could be explored for their compatibility with co-regularization frameworks. Beyond these extensions, the developed framework holds potential for broader application in diverse domains, such as healthcare access, retail expansion, and public service planning, thereby broadening its relevance in evolving market conditions.