Capacitated Facility Location with Preferences

Updated 2 August 2025

The CFLCP is a model that integrates facility capacity constraints with strict customer preference rankings to optimize cost and satisfaction.
It introduces stability notions—customer, pairwise, and cyclic-coalition—ensuring allocation fairness and achieving Pareto optimal outcomes.
Computational experiments show that reduced MILP formulations enhance efficiency and robustness in solving the allocation problem under capacity limits.

The Capacitated Facility Location Problem with Customer Preferences (CFLCP) generalizes the classical capacitated facility location problem by incorporating a model of individual (or ordinal) customer preferences for facilities. In the CFLCP, facilities are subject to explicit capacity constraints, and customer assignments must respect both these limits and each customer’s strict ranking or preference profile over open facilities. The objective is typically to minimize a total cost function (including facility opening and assignment costs), maximize aggregate preference satisfaction under constraints, or to guarantee various forms of stability and fairness in the assignment, often captured through rigorous notions of allocation stability and Pareto optimality.

1. Formal Problem Definition and Generalizations

Let $I$ be a set of customers and $J$ a set of candidate facilities, each with opening cost $f_j \geq 0$ and (finite) capacity $c_j$ . Each customer $i \in I$ possesses a strict ranking or scoring $P_i$ over the facilities $J$ . The classic CFLCP requires:

Choosing a subset of facilities $S \subseteq J$ to open, obeying cardinality, budget, or facility-specific opening constraints.
Assigning each customer $i$ $i$ to an open facility $j$ $j$ such that:
- No facility serves more than $c_j$ customers: $\sum_{i \in I} x_{ij} \leq c_j$ ,
- Each customer is assigned to exactly one facility, or possibly not assigned in variants with partial coverage.

The assignment is typically encoded with binary variables $x_{ij} \in \{0,1\}$ , denoting whether customer $i$ is assigned to facility $j$ , and $y_j \in \{0,1\}$ , denoting whether facility $j$ is open.

Preference information can be incorporated in two principal ways:

By modifying assignment costs: $g_{ij} = \mathrm{transportation~cost} + \mathrm{preference~penalty}$ .
By formulating explicit constraints to maximize global (aggregated or lexicographic) preference satisfaction or to enforce certain stability notions in feasible allocations (Domínguez et al., 29 Jul 2025).

2. Stability Notions for Preference-Constrained Assignment

The introduction of capacities means that not all customers can be assigned to their top-ranked facility. To mitigate inefficiencies or perceived unfairness, the literature on CFLCP defines increasingly strong notions of stability:

Customer Stable: No assigned customer can improve (i.e., be assigned to a more preferred, open facility that is not full) via unilateral deviation.
Pairwise Stable: No pair of customers can swap their assigned facilities and both strictly improve in preference.
Cyclic-Coalition Stable (Pareto Optimal): No coalition (“cycle”) of customers can permute assignments so that all strictly improve; equivalent to achieving a Pareto optimal matching as defined in the capacitated house allocation literature.

The paper (Domínguez et al., 29 Jul 2025) gives precise mixed-integer linear programming (MILP) formulations for each of these stability concepts, allowing computational algorithms to search directly for globally (Pareto) optimal stable matchings under capacity.

3. Mathematical Formulations and Algorithmic Approaches

A. Customer Stable Formulation

$\begin{align*} &\min_{x, y} \quad \sum_{j\in J} f_j y_j + \sum_{i\in I} \sum_{j\in J} g_{ij} x_{ij} \ &\text{subject to} \ &\quad \sum_{j\in J} x_{ij} = 1 && \forall i\in I, \ &\quad \sum_{i\in I} x_{ij} \leq c_j y_j && \forall j\in J, \ &\quad x_{ij} \leq y_j && \forall i\in I, j\in J, \ &\quad c_j y_j \leq c_j \sum_{j'\in J : j'\preceq_i j} x_{ij'} + \sum_{i'\in I\setminus\{i\}} x_{i'j} && \forall i, j. \end{align*}$

B. Pairwise and Cyclic-Coalition Stable Formulations

The pairwise stable model augments the above with constraints blocking all beneficial swaps; the cyclic-coalition model requires permutation-indexed allocation variables to eliminate all possible blocking coalitions (Domínguez et al., 29 Jul 2025). Each variant includes a reduced formulation variant (e.g., $(CFLCP)_{CS}^R$ , $(CFLCP)_{PW}^R$ , $(CFLCP)_{CC}^R$ ) to reduce variable count and allow more LP relaxation, which is critical for practical scalability.

C. Additional Pareto Matching Formulations

A novel MILP model is provided for generating Pareto optimal matchings of maximum cardinality in the Capacitated House Allocation problem by adjusting facility opening variables and objective functions appropriately (Domínguez et al., 29 Jul 2025).

4. Computational Implications and Model Efficiency

Extensive computational experiments (Domínguez et al., 29 Jul 2025) show reduced formulations (those using auxiliary variables and relaxed integrality) are both tighter in LP relaxation and dramatically more efficient: they solve more medium-size instances within set time limits, require fewer branch-and-bound nodes, and yield lower optimality gaps compared to non-reduced models.

Allocations computed under cyclic-coalition stability (Pareto optimal) frequently outperform earlier, preference-sum maximization approaches (e.g., classic bilevel assignment models) in terms of global customer satisfaction, sometimes reducing assignment cost by up to 17% over comparable benchmark solutions.

Total unimodularity in parts of the constraint matrices allows further relaxation of integrality (yielding LP relaxations with integer solutions), pointing to the possibility of efficient algorithms even at scale, given careful model design.

CFLCP bridges classic facility location models and discrete allocation/matching theory:

The Simple Plant Location Problem with Order (SPLPO) is recovered as a special case (no capacities).
The Pareto stability requirement is equivalent to matchings in the capacitated house allocation problem with ordered preferences.
Mechanism design frameworks (e.g., Generalized Median Mechanisms for one-dimensional location) motivate alternative mechanisms for selecting facility sites and assignments under both preference and incentive constraints (Aziz et al., 2018, Auricchio et al., 21 Apr 2024).
Multiobjective and multiagent models—employing evolutionary heuristics, preference elicitation, and robust ordinal regression—address the trade-off between cost minimization and preference satisfaction in high-dimensional and user-interactive settings (Barbati et al., 2022).

Cutting-edge algorithmic directions include:

LP and MIP formulations with preference-weighted assignment costs or constraints (Aardal et al., 2013, Kao, 2021).
Mechanism design and strategyproofness for anonymous, truthful assignment and approximation ratio bounds (Aziz et al., 2018, Auricchio et al., 21 Apr 2024).
Multiobjective and metaheuristic algorithms that interactively incorporate decision maker or customer preference feedback in the search for Pareto superior solutions (Barbati et al., 2022).

6. Applications and Practical Significance

The CFLCP is relevant in diverse practical settings where resources (school seats, hospital beds, housing assignments) are allocated under capacity and user preference constraints. The modeling focus on stability and Pareto optimality is directly motivated by the need to guarantee fairness (e.g., in school choice or admissions) and avoid allocations that could plausibly be improved upon by groups of stakeholders—crucial in high-stakes or regulated environments.

Empirical evidence (Domínguez et al., 29 Jul 2025) demonstrates that careful model design addressing both preference and feasibility is required to ensure allocations that are both efficient and robust to individual or coalition deviation.

Real-world deployment further requires models to scale and to support multiple fairness, efficiency, and stability objectives—suggesting both the necessity for algorithmic innovation and the potential of hybrid methods integrating exact and metaheuristic approaches.

7. Limitations and Open Challenges

While exact MILP formulations achieve strong theoretical guarantees for moderate-size problems, scalability to large-scale allocations with detailed preference data and complex facility structures remains challenging.

Moreover, incorporating rich or multiattribute preferences, as well as dynamic or online allocation, requires still more flexible and scalable algorithmic approaches. Trade-off analysis between efficiency (total cost) and satisfaction (preference realization) is statistical and context dependent, as demonstrated by comparative computational studies (Blanco et al., 28 Dec 2024).

Finally, while advanced notions of stability such as cyclic-coalition stability (Pareto optimality) are fair and robust, their computation may require sophisticated permutation-indexed models or complex constraint structures; relaxing or approximating these notions for scalability without sharply compromising stability remains an important direction for future research.

The CFLCP stands at the intersection of facility location, discrete allocation, and matching under resource constraints, offering a rich suite of model-based, algorithmic, and fairness-theoretic tools for the equitable and efficient allocation of capacity-limited resources with individualized customer or agent preferences.