Three-Sided Fuzzy Stable Matching Problem Based on Combination Preference

Abstract

Previous studies, constrained by the overly rigid stability requirements, often fail to adapt to complex systems and struggle to identify stable outcomes that align with the practical context of multi-agent resource allocation. To address the three-sided matching problem in complex socio-technical and business management systems, this paper proposes a fuzzy stable matching method for three-sided agents under a framework of combinatorial preference relations, integrating network and decision theory. First, we construct a membership function to measure the degree of preference satisfaction between elements of different agents, and then define the concept of fuzzy stability. By incorporating preference satisfaction, we introduce the notion of fuzzy blocking strength and derive the generation conditions for blocking triples and fuzzy stability under the fuzzy stable criterion. Furthermore, we abstract the three-sided matching problem with combined preference relations into a shortest path problem. Second, we prove the equivalence between the shortest path solution and the stable matching outcome. We adopt Dijkstra’s algorithm for problem-solving and derive the time complexity of the algorithm under the pruning strategy. Finally, we apply the proposed model and algorithm to a case study of project assignment in software companies, thereby verifying the feasibility and effectiveness of this three-sided matching method. Compared with existing approaches, the fuzzy stable matching method developed in this study demonstrates distinct advantages in handling preference uncertainty and system complexity. It provides a more universal theoretical tool and computational approach for solving flexible resource allocation problems prevalent in real-world scenarios.

1. Introduction

With the increasingly complex demands for multilateral resource allocation in complex systems such as healthcare systems, educational management systems, and labor market systems—key areas at the intersection of socio-technical and business management systems—the problem of three-sided stable matching plays a pivotal role in systemic optimization and decision-making [1,2]. For instance, in the matching of resident physicians, doctors not only need to weigh the reputation of the hospital but also assess the development prospects of specific departments. In educational enrollment, the matching of students, supervisors, and research projects involves reconciling multiple factors such as academic directions and resource adaptation. In the labor market, the interaction among job seekers, employers, and positions also involves a preference-driven dynamic. At this point, the traditional two-sided matching theory falls short in such scenarios because it cannot describe the combination preference dependency relationship of the three participants [3]. There is an urgent need for more refined modeling and solution methods.

The consideration of combinatorial preference [4] makes the three-sided matching problem more challenging. In real-world scenarios, agents’ choices are often based on a holistic evaluation of attribute combinations rather than independent ranking. For example, doctors may develop a nonlinear preference for the “hospital-department” combination rather than comparing hospitals or departments alone. When students choose a supervisor, they may take into account both the supervisor’s research direction and project funding situation at the same time. This combinatorial dependency relationship leads to an exponential growth of the preference space, which requires a redefinition of core concepts such as the criteria for blocking triples and stability conditions in traditional stable matching theory. This study proposes fuzzy stability. By quantifying the satisfaction degree of the subject with the matching through the membership function, the classic “non-blocking pair” stability condition is extended to “acceptable blocking intensity”, thereby defining fuzzy stable matching—that is, the matching state where the “deviation intention” of all potential blocking pairs does not exceed the system’s tolerance threshold. This paradigm shift has led the matching model to shift from pursuing absolute stability to seeking flexible, satisfaction-driven stability, laying a theoretical foundation for dealing with complex matching problems in the real-world, where information is incomplete and preference expressions are ambiguous.

Existing studies often rely on heuristic algorithms to solve large-scale three-sided matching problems in complex systems, but they present significant limitations for systemic optimization: on the one hand, heuristic methods rely on random search strategies, which can produce unstable matching outcomes, compromising the reliability of the matching system; on the other hand, they lack strict theoretical guarantees of stability, which may affect the fairness and acceptability of systemic decision-making, which leads to unstable matching results and makes it difficult to ensure repeatability. Furthermore, the stochastic optimization process may fall into a local optimum, especially in high-dimensional combinatorial preference spaces, where convergence efficiency degrades considerably. Therefore, how to improve computational efficiency while ensuring stability has become one of the key challenges in the research of three-sided matching.

To overcome the randomness defect of the existing heuristic algorithms, a deterministic solution method based on graph theory and network optimization is proposed. By transforming the three-sided matching problem into a shortest path problem under multiple constraints and using the deterministic optimization technique of Dijkstra’s algorithm, the uniqueness and repeatability of the matching results are guaranteed. This framework needs to address the problem of high-dimensional state space caused by combinatorial preferences and design an efficient graph structure representation method to reduce computational complexity. Aiming at the rigid limitations of preference expression in traditional stable matching, the theory of fuzzy mathematics is introduced to describe the satisfaction of the matching subject with the matching result. By defining the fuzzy membership function to quantify the compatibility degree of the three-sided matching and constructing the fuzzy blocking triplet determination condition, more practical, flexible matching can be achieved in an uncertain environment. Combined with the multi-objective optimization method, a dynamic trade-off between matching quality and stability can be realized.

Based on the three-sided stable matching problem of combined preference, this paper aims to propose a new matching method that takes into account both computational efficiency and stability. Specific contributions include (1) rigid stability constraints for complex matching systems, where the membership function is utilized to provide the fuzzy blocking strength and fuzzy stability conditions under the formal definition; (2) transform the combined preference into the shortest path problem in graph theory, and construct a fuzzy stable matching model under combined preference in combination with fuzzy satisfaction; and (3) design efficient algorithms to solve stable matching, analyze their computational complexity, feasibility and effectiveness, and provide scalable matching methods for traditional matching problems.

The remaining parts of this study are arranged as follows: The second part conducts a literature review of the existing related work in this study. The third part formally describes the three-sided stable matching problem under combined preference in combination with graph theory, presents the relevant concepts and theories of fuzzy stable matching, and constructs the three-sided stable matching model under combined preference. The fourth part presents the problem-solving method in combination with Dijkstra’s algorithm. The fifth part assesses the feasibility and effectiveness of the method in combination with specific calculation examples. The sixth part summarizes the full text.

2. Related Work

Gale and Shapley proposed the theory of two-sided stable matching in 1962 [5], and constructed the basic framework of stable matching through the deferred acceptance algorithm, proving the existence of stable solutions. However, when extended to three-sided matching, the difficulty of the problem increases significantly. The three-sided stable matching problem was first formally posed by Knuth [4] in 1976. He first raised the question of whether the two-sided stable matching problem could be extended to the three-sided case—that is, the three-sided stable matching problem. Regarding the stability issue of three-sided matching, many scholars have conducted relevant discussions. Yang and Zhao [6] constructed a mathematical model based on maximum cardinality and stable matching, considering the psychological characteristics of incomplete acceptance and limited compromise of the subject, and proposed a search optimization three-sided stable matching algorithm under limited compromise conditions. Yang and Zhao [7] proposed a matching algorithm based on the preference order threshold constraint condition, provided the relevant definitions of three-sided unidirectional non-cyclic matching and its stability, and established a mathematical model that meets the system stability requirements in the one-to-one case. Cseh et al. [8] demonstrated that the five two-sided matching models still maintained stability after being converted to three-sided matching models. Existing studies have provided a theoretical basis and methodological support for solving the three-sided matching problem in reality. However, most of these studies are limited to matching problems with cyclic preferences, which also restricts their application and promotion by the preference relationship between the subjects. Teng et al. [9] hold that the three-sided stable matching problem should meet three conditions: Firstly, the matching subject should have three sides. Secondly, any one of the three parties has a certain evaluation of the other two. Finally, all three sides of the subject should be rational utility maximizers. Guo et al. [10] formulated the selection process among the three parties involved in emergency rescue as a three-sided matching problem, constructed a multi-attribute comprehensive preference score matrix, and established a three-sided matching satisfactory and stable model under the background of the initial stage of emergency rescue with the goals of system stability and maximizing total preference utility. So far, whether stable results of three-sided matching problems under various preference assumptions exist remains an indispensable topic of discussion for subsequent scholars.

Knuth [4] first proposed the concept of combinatorial preference in his research, arguing that if each subject has a certain preference for other subjects in a paired manner, then this form of preference relationship structure is considered to have the characteristics of combinatorial preference. In the case of combined preference, if the preference of the first subject within the binary group is prioritized for sorting, and only when there is no difference in the preferences of the first subject, and then the preference of the second subject within the group is used for sorting, such preferences are called dictionary-based preferences [11]. For the three-sided combinatorial preference structure, Alkan [12] and Ng and Hirschberg [13] independently established that whether there exists a stable matching result is an NP-hard problem. Danilov [14] provided sufficient conditions for the existence of a stable matching in multilateral systems. Mordig et al. [11] proved that stable three-sided matching always exists for dictionary-based preference structures and proposed an innovative solution method based on educational background. They modified the new algorithm proposed in this study by applying the two-sided Gale–Shapley algorithm to the many-to-many two-sided market, and finally applied it to the three-sided matching problem. Feng et al. [15] separated the dictionary-based preferences at the beginning and the end, and solved the regained preferences using the Top Trading Cycles (TTC) algorithm, obtaining more reasonable results.

However, all the above-mentioned studies are based on the assumption that preference relations are completely known and precisely comparable, making it difficult to deal with the widespread uncertainty and fuzziness in real matching. For this reason, scholars have begun to introduce the theory of fuzzy sets into matching problems for research. Preference quantification studies the choice tendencies of individuals or groups. Traditional methods are based on classical set theory, while fuzzy set theory [16] better handles fuzzy preferences in reality through membership functions. In recent years, the application of fuzzy sets in preference modeling has been continuously deepened. Shivani et al. [17] utilized nonlinear membership functions to characterize decision preferences in waste management; Buyukozkan et al. [18] combined spherical fuzzy sets to solve the problem of renewable energy selection; Pinki et al. [19] fused fuzzy multi-objective programming to optimize transportation systems; Chai et al. [20] adopted interval type-2 fuzzy sets to enhance the accuracy of selecting battery suppliers for electric vehicles. Chang et al. [21] utilize the Takagi–Sugeno (T-S) fuzzy model technique for piecewise linear weighting to represent nonlinear models, which are used to represent vehicle-mounted network systems. Liu et al. [22] constructed a secure cascaded output feedback control framework based on the T-S fuzzy model for nonlinear networked cascade control systems under spoofing attacks. These studies have expanded the application of fuzzy set theory in matching problems, providing effective support for complex decision-making and system optimization.

In recent years, machine learning and heuristic algorithms have been introduced into matching modeling, such as learning implicit preferences through neural networks [23], designing a trainable neural network to address the challenges brought by severe outliers in matching problems [24], and utilizing the mechanism of deep state learning networks to transform matching states into potential canonical forms [25]. The differential evolution algorithm [26], the ASCIID swarm algorithm [27], and other methods are introduced to solve the matching problem. In addition, multi-criteria decision analysis methods, such as TOPSIS and AHP, are also used to quantify multi-dimensional preferences [28]. The shortest path algorithm is also widely applied in matching problems, especially in graph-based matching models, where the matching problem can be transformed into a network flow problem, and the shortest path is used to optimize the matching cost or benefit [29]. In two-sided matching, the shortest path can be used to calculate the weight sum of the optimal matching. For instance, Gale and Shapley’s stable matching algorithms can optimize matching stability by constructing preference graphs and finding the shortest path. In task allocation problems, the shortest path algorithm can efficiently solve the minimum cost matching [30]. Gao et al. [31] described stereo matching as an energy minimization problem that can be solved by searching for the shortest path in a directed graph. Their method outperforms dynamic programming and graph cutting in both accuracy and speed.

There are still several limitations in the existing research on matching problems: (1) Traditional matching models overly rely on the rigid stability assumption. This strict constraint condition is difficult to adapt to the incomplete information and dynamically changing environment that are widespread in real scenarios, resulting in insufficient robustness of the algorithm. (2) The three-sided matching problem under combinatorial preference involves the pairing preference of one matching agent for the other two matching agents, and is limited by fuzzy and uncertain evaluations. However, existing research lacks a modeling framework that integrates combinatorial preference and fuzzy quantization. (3) Although mainstream heuristic algorithms can enhance computational efficiency, they have problems such as large fluctuations in solution quality and uncontrollable convergence, which seriously affect the reliability of the matching system. Therefore, starting from the above deficiencies, this study proposes a solution method using fuzzy quantization under combined preference, and designs a matching method in combination with the relevant knowledge of graphs to enhance the reliability and practicability of the matching system.

3. Problem Modeling

3.1. Formal Description

The three-sided matching problem can be formally described as follows: there are three disjoint finite principal sets and their elements $A=\left\{a_1,a_2,\dots ,a_m\right\}$, $B=\left\{b_1,b_2,\dots ,b_n\right\}$, $C=\left\{c_1,c_2,\dots ,c_p\right\}$ (m, n, p are all positive integers), whose triples are denoted as $(a_i,b_j,c_k)\in A\times B\times C=\mathsf{\Omega }$ ($i=\left\{1,\dots ,m\right\}$, $j=\left\{1,\dots ,n\right\}$, $k=\left\{1,\dots ,p\right\}$). The matching M is a subset of triples, and any two triples $(a_i,b_j,c_k)$, $(a_{i^{\prime }},b_{j^{\prime }},c_{k^{\prime }})\in M$ satisfied $i\ne i^{\prime },j\ne j^{\prime },k\ne k^{\prime }$ (that is, each subject appears in at most one triplet).

If the subject preferences are all given in the form of subject pairs formed by one subject to the other two, then such preferences are called combinatorial preferences [4] (as shown in Figure 1). That is, element a_i in subject A has a certain preference for the tuple $\left(b_j,c_k\right)$ in subject $B\times C$, element b_j in subject B has a certain preference for the tuple $\left(a_i,c_k\right)$ in subject $A\times C$, and element c_k in subject C has a certain preference for the tuple $\left(a_i,b_j\right)$ in subject $A\times B$. In the case of combined preference, if the preference of the first subject within the binary group is prioritized for sorting, and only when there is no difference in the preferences of the first subject, and then the preference of the second subject within the group is used for sorting, such preferences are called dictionary-based preferences.

3.2. Stability Conditions

Matching M is stable if and only if there are no-blocking triples.

Definition 1 [4].

For a three-sided match M with a combinatorial preference structure, if there exists a matching group $\left(a_i,b_j,c_k\right)\in \mathsf{\Omega }-M$  such that the subjects $a_i,b_j,c_k$ in the group, respectively, satisfy the following conditions:

(1)

For $a_i$, satisfy $\left(b_j,c_k\right){\succ }_{a_i}{M}_{A\to B\times C}\left(a_i\right)$;

(2)

For $b_j$, satisfy $\left(a_i,c_k\right){\succ }_{b_j}{M}_{B\to A\times C}\left(b_j\right)$;

(3)

For $c_k$, satisfy $\left(a_i,b_j\right){\succ }_{c_k}{M}_{C\to A\times B}\left(c_k\right)$;

Then $\left(a_i,b_j,c_k\right)$ is called the blocking group of M.

Theorem 1. 

When the preference structure of a three-sided matching problem satisfies the combinatorial preference, if the following constraints are simultaneously satisfied:

$$\begin{array}{l}{x}_{a_i\to b_j\times c_k}+{\displaystyle {\sum }_{b_{j^{\prime }}\times c_{k^{\prime }}:M_{a_i\to b_{j^{\prime }}\times c_{k^{\prime }}}>M_{a_i\to b_j\times c_k}}{x}_{a_i\to b_{j^{\prime }}\times c_{k^{\prime }}}\ge 1}\\ y_{b_j\to a_i\times c_k}+{\displaystyle {\sum }_{a_{i^{\prime }}\times c_{k^{\prime }}:M_{b_j\to a_{i^{\prime }}\times c_{k^{\prime }}}>M_{b_j\to a_i\times c_k}}{y}_{b_j\to a_{i^{\prime }}\times c_{k^{\prime }}}\ge 1}\\ z_{c_k\to a_i\times b_j}+{\displaystyle {\sum }_{a_{i^{\prime }}\times b_{j^{\prime }}:M_{c_k\to a_{i^{\prime }}\times b_{j^{\prime }}}>M_{c_k\to a_i\times b_j}}{z}_{c_k\to a_{i^{\prime }}\times b_{j^{\prime }}}\ge 1}\end{array}$$

(1)

Then a stable match can be generated. Among them, match M is a matrix composed of 0 and 1, where x, y, and z, respectively, represent mismatch or match, taking values of 0 or 1. $a_i\to b_j\times c_k$ denotes that the preferred direction of $a_i$ is $b_j\times c_k$.

Proof. 

Sufficiency proof.

Suppose the constraint conditions hold, but the match M is unstable—that is, there exists a blocking triplet $(a_i,b_j,c_k)$ such that

$a_i$ prefers $(b_j,c_k)$ over the current match, that is $x_{a_i}\to b_j\times c_k=0$, and the summation term is 0 (there is no better match). This contradicts the first constraint $x_{a_i\to b_j\times c_k}+{\displaystyle {\sum }_{b_{j^{\prime }}\times c_{k^{\prime }}:M_{a_i\to b_{j^{\prime }}\times c_{k^{\prime }}}>M_{a_i\to b_j\times c_k}}{x}_{a_i\to b_{j^{\prime }}\times c_{k^{\prime }}}\ge 1}$.

The proof of $y_{b_j\to a_i\times c_k}+{\displaystyle {\sum }_{a_{i^{\prime }}\times c_{k^{\prime }}:M_{b_j\to a_{i^{\prime }}\times c_{k^{\prime }}}>M_{b_j\to a_i\times c_k}}{y}_{b_j\to a_{i^{\prime }}\times c_{k^{\prime }}}\ge 1}$ and $z_{c_k\to a_i\times b_j}+{\displaystyle {\sum }_{a_{i^{\prime }}\times b_{j^{\prime }}:M_{c_k\to a_{i^{\prime }}\times b_{j^{\prime }}}>M_{c_k\to a_i\times b_j}}{z}_{c_k\to a_{i^{\prime }}\times b_{j^{\prime }}}\ge 1}$ is the same.

Therefore, the assumption does not hold, and the match M is stable.□

Proof of necessity. 

Let M be stable, but a certain constraint does not hold (for example, for a_i):

There exists $(b_j,c_k)$ such that $x_{a_i\to b_j\times c_k}+{\displaystyle {\sum }_{b_{j^{\prime }}\times c_{k^{\prime }}:M_{a_i\to b_{j^{\prime }}\times c_{k^{\prime }}}>M_{a_i\to b_j\times c_k}}{x}_{a_i\to b_{j^{\prime }}\times c_{k^{\prime }}}=0}$. That is, $a_i$ does not match $(b_j,c_k)$ and does not match any better combination. Therefore, $a_i$ prefers $(b_j,c_k)$ rather than the current match.

If b_j and c_k are not matched to a better combination either, then $(a_i,b_j,c_k)$ is a blocking triplet, which contradicts stability. Therefore, the constraint must hold true. □

Meanwhile, in this study, a fuzzy stability threshold is introduced into the trilateral matching problem under combinatorial preference, aiming to quantify the satisfaction degree of the matching scheme through the membership degree of fuzzy mathematics, thereby addressing the issue where traditional strict stability may be too rigid or difficult to achieve.

In the subject set $A=\left\{a_1,a_2,\dots ,a_m\right\}$, the subject a_i generates a preference order for the elements in $B=\left\{b_1,b_2,\dots ,b_n\right\}$ according to the arrangement rule from the most preferred to the least preferred, and assigns the preference order value $0,\dots ,n-1$ in sequence to represent the grade of the subject elements in B in subject a_i’s preference order, denoted by $ran{k}_{a_{i}B}$. The preference order processing method for other preference directions is the same.

Let $ran{k}_{a_i{b}_j}$ represent the grade of the subject b_j in the subject a_i preference order. Then, the satisfaction of subject a_i with the matching triplet $\left(a_i,b_j,c_k\right)$ under the combined preference can be expressed as the degree to which $(b_j,c_k)$ approaches the highest preference level in the a_i preference order—that is, the membership degree ${\mu }_{a_i}(b_j,c_k)$:

$${\mu }_{a_i}(b_j,c_k)=1-{\displaystyle \frac{{\gamma }_1\cdot ran{k}_{a_i{b}_j}+{\gamma }_2\cdot ran{k}_{a_i{c}_k}}{{\gamma }_1\cdot \max ran{k}_{a_{i}B}+{\gamma }_2\cdot \max ran{k}_{a_{i}C}}}$$

(2)

where ${\gamma }_1$ and ${\gamma }_2$ are the primary priority weights. When ${\gamma }_1={\gamma }_2$, the triple performance is not a dictionary type combination preference, on the other hand, the preference for dictionary type combination. It is not difficult to see that ${\mu }_{a_i}(b_j,c_k)\in [0,1]$, for matching triples $\left(a_i,b_j,c_k\right)$, ${\mu }_{b_j}(a_i,c_k)\in [0,1]$ represents b_j’s satisfaction with the combination $\left(a_i,c_k\right)$, and ${\mu }_{c_k}(a_i,b_j)\in [0,1]$ represents c_k’s satisfaction with the combination $\left(a_i,b_j\right)$. The closer the membership degree is to 0, the lower the preference intensity is; that is, the less satisfied the matching subject is with the current matching (among which, the default membership degree of the unmatched element is 0—that is, the most dissatisfied).

Traditional weak stability requires complete non-blocking, but in actual situations, the matching subject also allows for a certain degree of non-satisfaction. Now, a fuzzy stability is constructed to represent this degree. Define the fuzzy blocking strength $\beta (a_i,b_j,c_k)$ of triple $(a_i,b_j,c_k)$ for matching M as

$$\underset{(a_i,b_j,c_k)\notin M}{\beta (a_i,b_j,c_k)}=\min \left(\begin{array}{l}max(0,{\mu }_{a_i}(b_j,c_k)-{\mu }_{a_i}(M(a_i))),\\ max(0,{\mu }_{b_j}(a_i,c_k)-{\mu }_{b_j}(M(b_j))),\\ max(0,{\mu }_{c_k}(a_i,b_j)-{\mu }_{c_k}(M(c_k)))\end{array}\right)$$

(3)

where M (a_i) represents the current matching combination of a_i in M.

Definition 2. 

Sets a threshold $\tau \in \left[0,1\right)$, stating that the match M is $\tau$-fuzzily stable if

$$\forall (a_i,b_j,c_k)\notin M,\beta (a_i,b_j,c_k)\le \tau$$

(4)

Meanwhile

$$\tau <\underset{(a_i,b_j,c_k)\in M}{min}({\mu }_{a_i}(M(a_i)),{\mu }_{b_j}(M(b_j)),{\mu }_{c_k}(M(c_k))).$$

(5)

where when $\tau =0$, it degenerates into the traditional weak stability (completely non-blocking triad); when $\tau >0$, slight blocking is allowed, but the intensity does not exceed $\tau$.

This next section focuses on the core connotation of the “fuzzy” concept—that is, allowing a slight disruption not exceeding the threshold τ, rather than pursuing a rigid stability with absolute no-blocking. The larger the τ value is, the higher the tolerance for fuzziness becomes, and the stronger the flexibility of matching becomes; the smaller the τ value is, the lower the tolerance for fuzziness becomes, and the more similar the matching is to the traditional rigid stability. The following clarifies the linguistic definition of “fuzzy” and clearly distinguishes it from “no-blocking”:

$$\tau \in \left\{\begin{array}{l}\left[0.15,0.25\right],\mathrm{Strong} \mathrm{fuzzy} \mathrm{tolerance}\\ \left[0.08,0.15\right),\mathrm{Moderate} \mathrm{fuzzy} \mathrm{tolerance}\\ \left(0,0.08\right),\mathrm{Weak} \mathrm{fuzzy} \mathrm{tolerance}\\ \left\{0\right\},\mathrm{No} \mathrm{fuzzy} \mathrm{tolerance}\end{array}\right.$$

(6)

Here are examples to illustrate the fuzzy blocking strength and $\tau$-fuzzy stability.

Example 1. 

Suppose there is a three-sided matching problem, where $A=\left\{a\right\}$, $B=\{b_1,b_2\}$, $C=\{c_1,c_2\}$. The preference order of subject a is as follows:

• A to B: $b_2\prec b_1$, that is $ran{k}_{aB}(b_1)=0,ran{k}_{aB}(b_2)=1$;
• a to C: $c_2\prec c_1$, that is $ran{k}_{aC}(c_1)=0,ran{k}_{aC}(c_2)=1$.

Set weights ${\gamma }_1={\gamma }_2=0.5$, then for a, the membership degree of the combination $\left(b_j,c_k\right)$ is

$${\mu }_a\left(b_j,c_k\right)=1-{\displaystyle \frac{0.5\times ran{k}_{aB}\left(b_j\right)+0.5\times ran{k}_{aC}\left(c_k\right)}{0.5\times 1+0.5\times 1}}=1-\left(0.5\times ran{k}_{aB}\left(b_j\right)+0.5\times ran{k}_{aC}\left(c_k\right)\right)$$

The calculation shows

$$\begin{array}{l}{\mu }_a\left(b_1,c_1\right)=1-\left(0.5\times 0+0.5\times 0\right)=1\\ {\mu }_a\left(b_1,c_2\right)=1-\left(0.5\times 0+0.5\times 1\right)=0.5\\ {\mu }_a\left(b_2,c_1\right)=1-\left(0.5\times 1+0.5\times 0\right)=0.5\\ {\mu }_a\left(b_2,c_2\right)=1-\left(0.5\times 1+0.5\times 1\right)=0\end{array}$$

Similarly, calculate the membership degree of each element in B and C, where the preference order of each element is $c_2{\prec }_{b_1}{c}_1,c_2{\prec }_{b_2}{c}_1,b_2{\prec }_{c_1}{b}_1,$ $b_2{\prec }_{c_2}{b}_1$. Suppose the current match is $M=\left\{\left(a,b_2,c_2\right)\right\}$, then for a, b₂, c₂, the current matching membership degrees are all 0.

Now consider the fuzzy blocking strength of the unmatched triplet $\left(a,b_1,c_1\right)$. This match has a membership degree of 1 for a, b₁, c₁. Now calculate the fuzzy blocking strength $\beta (a,b_1,c_1)$ of the triplet $\left(a,b_1,c_1\right)$:

$$\beta (a,b_1,c_1)=\min \left(\begin{array}{l}max(0,{\mu }_a(b_1,c_1)-{\mu }_a(M(a))),\\ max(0,{\mu }_{b_1}(a,c_1)-{\mu }_{b_1}(M(b_1))),\\ max(0,{\mu }_{c_1}(a,b_1)-{\mu }_{c_1}(M(c_1)))\end{array}\right)=\min \left(\begin{array}{l}max(0,1-0),\\ max(0,1-0),\\ max(0,1-0)\end{array}\right)=1$$

Now, according to the range of values for τ as defined in Definition 2, it can be known that regardless of how τ is taken, there is always $\beta (a,b_1,c_1)>\tau$. Therefore, $M=\left\{\left(a,b_2,c_2\right)\right\}$ does not constitute τ-fuzzy stability.

3.3. Graph-Based Modeling

This section, in combination with the relevant knowledge of graphs, models the three-sided matching process with a combined preference structure as a graph path search problem, as shown below. The matching ends after all the elements in either subject are matched, so there are $\min \left\{m,n,p\right\}$ columns in Figure 2, S is the starting point, T is the end point, and M_i is the matching triple.

A directed graph G is an ordered tuple (N, E), denoted as $G=\left(N,E\right)$, where $N=\left\{n_1,\dots ,n_n\right\}$ ($n\ge 1$) is the set of points of G, and each point represents a matching triplet $(a_i,b_j,c_k)$, $E=\left\{e_{pq}\right\}$ is the edge set of G, e_pq is an ordered tuple $\left\{n_p,n_q\right\}$, if $e_{pq}=\left\{n_p,n_q\right\}$, then it is said that e_pq connects n_p and n_q, and points n_p and n_q are called the endpoints of e_pq. In the search process of the graph, the matching triples $(a_i,b_j,c_k)$ and $(a_{i^{\prime }},b_{j^{\prime }},c_{k^{\prime }})$ represented by any two points on each feasible path, satisfied $i\ne i^{\prime },j\ne j^{\prime },k\ne k^{\prime }$, ensure the uniqueness of each subject match. The weight of the edge $w_{e_{pq}}$ represents the preference deviation degree of the point n_q in $e_{pq}=\left\{n_p,n_q\right\}$. Thus, the graph model of three-sided $\tau$-fuzzy stable matching problem can be obtained as follows:

$$\begin{array}{l}\min \left({\displaystyle \sum _{e_{pq}\in path}{w}_{e_{pq}}\cdot x_{pq}}\right)\\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{e\in out(p)}{x}_{pq}-{\displaystyle \sum _{e\in in(p)}{x}_{pq}}}=\left\{\begin{array}{l}1,p=s,\\ -1,p=t,\\ 0,\mathrm{Others}.\end{array}\right.\\ \\ x_{pq}=\left\{\begin{array}{l}1,{\mathrm{If} \mathrm{edge} \mathrm{e}}_{pq} \mathrm{is} \mathrm{selected} \mathrm{in} \mathrm{the} \mathrm{optimal} \mathrm{solution},\\ 0,\mathrm{Others}.\end{array}\right.\\ \underset{(a_i,b_j,c_k)\notin path}{\beta (a_i,b_j,c_k)}\le \tau .\end{array}\right.\end{array}$$

(7)

where the weight of the edge $\underset{e_{pq}\in path}{w_{e_{pq}}}=1-{\displaystyle \frac{1}{3}}\left({\mu }_{a_i}+{\mu }_{b_j}+{\mu }_{c_k}\right)$. The s and t, respectively, represent the starting point and the ending point in the shortest path diagram. The threshold of the fuzzy blocking intensity $\tau <\underset{(a_i,b_j,c_k)\in path}{min}({\mu }_{a_i}(M(a_i)),{\mu }_{b_j}(M(b_j)),{\mu }_{c_k}(M(c_k)))$, out (p) represents the set of edges leaving point p, and in (p) represents the set of edges entering point p. When $\tau =0$, the optimization model for the three-sided weakly stable matching problem can be obtained:

$$\begin{array}{l}\min \left({\displaystyle \sum _{e_{pq}\in path}{w}_{e_{pq}}\cdot x_{pq}}\right)\\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{e\in out(p)}{x}_{pq}-{\displaystyle \sum _{e\in in(p)}{x}_{pq}}}=\left\{\begin{array}{l}1,p=s,\\ -1,p=t,\\ 0,Others.\end{array}\right.\\ x_{a_i\to b_j\times c_k}+{\displaystyle {\sum }_{b_{j^{\prime }}\times c_{k^{\prime }}:M_{a_i\to b_{j^{\prime }}\times c_{k^{\prime }}}>M_{a_i\to b_j\times c_k}}{x}_{a_i\to b_{j^{\prime }}\times c_{k^{\prime }}}\ge 1},\\ y_{b_j\to a_i\times c_k}+{\displaystyle {\sum }_{a_{i^{\prime }}\times c_{k^{\prime }}:M_{b_j\to a_{i^{\prime }}\times c_{k^{\prime }}}>M_{b_j\to a_i\times c_k}}{y}_{b_j\to a_{i^{\prime }}\times c_{k^{\prime }}}\ge 1},\\ z_{c_k\to a_i\times b_j}+{\displaystyle {\sum }_{a_{i^{\prime }}\times b_{j^{\prime }}:M_{c_k\to a_{i^{\prime }}\times b_{j^{\prime }}}>M_{c_k\to a_i\times b_j}}{z}_{c_k\to a_{i^{\prime }}\times b_{j^{\prime }}}\ge 1},\\ \\ x_{pq}=\left\{\begin{array}{l}1,{\mathrm{If} \mathrm{edge} \mathrm{e}}_{pq} \mathrm{is} \mathrm{selected} \mathrm{in} \mathrm{the} \mathrm{optimal} \mathrm{solution},\\ 0,Others.\end{array}\right.\end{array}\right.\end{array}$$

(8)

This section has established a complete mathematical model of trilateral fuzzy stable matching under combined preference through formal description, stability condition definition, and graph model construction, providing a theoretical basis for subsequent algorithm design and solution.

4. Theory and Algorithm

4.1. Theoretical Analysis

Theorem 2 (Relationship between the Shortest Path Solution and the Stable Matching Result).

Let the directed graph G (V, E) of the three-sided matching problem be given, where the vertices $v\in V$ represent matching triples and the edges $e_{v{v}^{\prime }}\in E$ represent matching operations under the constraint conditions from point v to v’. If the weight of the shortest path from the initial state v_S to the final state v_M is bounded, then, according to the matching M non-blocking triplet corresponding to the shortest path obtained in Model (7), stability is satisfied.

Proof. 

Suppose there exists a blocking triplet $\left(a_i,b_j,c_k\right)\notin M$, such that ${\mu }_{a_i}\left(b_j,c_k\right)>{\mu }_{a_i}\left(M\left(a_i\right)\right)$ (the same applies to the other two preferred directions).

If $\left(a_i,b_j,c_k\right)$ is added to the matching and the relevant triples in the original matching are replaced, a new path P′ can be obtained, which has a lower total weight and contradicts the shortest path of the original one. Therefore, the matching M corresponding to the shortest path must be a non-blocking triplet.□

Definition 3 (Kernel). 

The kernel of a three-sided stable matching problem is the set of all stable matches.

Theorem 3 (Sufficient Conditions for Nuclear Non-emptiness). 

Let the combination preference of the tripartite subjects satisfy the following conditions:

①

Transitivity

For any subject $a_i\in A$, if $\left(b_1,c_1\right){\succ }_{a_i}\left(b_2,c_2\right)$ and $\left(b_2,c_2\right){\succ }_{a_i}\left(b_3,c_3\right)$, then $\left(b_1,c_1\right){\succ }_{a_i}\left(b_3,c_3\right)$ (the preference relation for subjects B and C directions under similar conditions also holds).

②

Non-conflict

The preference conflict of the sum of any two triples $\left(a_i,b_j,c_k\right)$ and $\left(a_p,b_q,c_r\right)$ can be eliminated through a finite number of adjustments.

Then the kernel of the three-sided matching problem is non-empty.

Proof. 

Consider the matching state as the vertices of the graph, the state transition as the edges, and the non-empty kernel as equivalent to the existence of an endpoint in the graph. Transitivity ensures that the path is acyclic and can guide the partial order direction of the state graph, enabling the algorithm to converge in order. Non-conflict ensures that the path terminates at the endpoint.□

Theorem 4 (Nuclear Existence under Fuzzy Preference). 

If the membership function is fuzzy, ${\mu }_{a_i}\left(b_j,c_k\right)$ satisfied the following:

①

Continuity: Continuous in the space of B × C.

②

Uniqueness: There exists a unique combination of the maximum membership degree.

Then there exists a $\tau$-fuzzy stable match M such that for all unmatched triples $\left(a_i,b_j,c_k\right)$, there is

$$min({\mu }_{a_i}(b_j,c_k),{\mu }_{b_j}(a_i,c_k),{\mu }_{c_k}(a_i,b_j))\le \tau$$

(9)

Proof. 

Let $\mathsf{\Omega }$ be the set of all possible fuzzy matches, where each match M satisfies that for each agent $a_i\in A$, and there exists a matching triplet $\left(a_i,b_j,c_k\right)$ such that ${\mu }_{a_i}\left(b_j,c_k\right)\ge \tau$ (similar conditions also hold for the preference relation in the directions of agent B and C). $\mathsf{\Omega }$ has the following properties:

①

Non-empty: At least a match exists $M\ne \varnothing$.

②

Closed interval: Due to the bounded degree of membership ($\mu \in \left[0,1\right]$). Therefore, $\mathsf{\Omega }$ is a closed interval and bounded.

For the current match M, define $\mathsf{\Phi }\left(M\right)$ as the new match updated by the following rules:

①

Find all the triples $\left(a_i,b_j,c_k\right)$ that satisfy;

②

Replace each blocking triple with the conflicting triples in the current match;

③

Keep the non-blocking part unchanged.

According to Definition 2, it can be known that the matching $\mathsf{\Phi }\left(M\right)$ ultimately generated by the above calculation process does not have blocking triples, and the membership function is continuous, the maximum membership combination is unique, and the matching process will not fall into an infinite loop. The algorithm eventually converges to a stable match M; that is, at this time, M is $\tau$-fuzzily stable.□

4.2. Algorithm Process

According to the three-sided matching diagram of the shortest path shown in Figure 2, it can be known that the solution algorithm needs to solve the non-negative-weight graph under the single-source shortest path and can ensure that the solution result is the optimal solution. Therefore, taking the Dijkstra algorithm as the core framework and combining the matching unique feature design pruning strategy, an improved shortest path algorithm is proposed.

①

Input: Three-sided subject sets A, B, C and their respective combined preferences (or fuzzy preference relationships).

②

Construction diagram:

Vertex set: All possible matching states (including unmatched states and matched states).

Edge set: Matching operations that satisfy constraint conditions.

③

Weight allocation: Calculate the preference deviation degree of each edge.

④

Solve the shortest path: Use Dijkstra’s algorithm.

⑤

Output: The matching scheme corresponding to the shortest path.

4.3. Algorithm Complexity

Theorem 5. 

If the sets of three subjects are, respectively, $A=\left\{a_1,a_2,\dots ,a_m\right\}$, $B=\left\{b_1,b_2,\dots ,b_n\right\}$, $C=\left\{c_1,c_2,\dots ,c_p\right\}$ (m, n, and p are all positive integers), then the computational complexity of the algorithm is O (mnp²).

Proof. 

As shown in Figure 2, starting from point S, each time a node is passed, mnp possible paths need to be traversed. However, due to the limitation of the matching condition, that is, the same element can only appear once on the same path, and each time a node is passed, the feasible choice of one element will be reduced in the next node. The above description can be formalized as follows: Starting from point S, the first optional path is mnp, the second optional path is (m − 1) (n − 1) (p − 1), and so on. Let $\min (m,n,p)=p$, then until all elements in one of the subjects A, B, C have completed matching, the number of algorithm calculations is

$${\displaystyle \sum _{k=0}^{p-1}(m-k)(n-k)(p-k)}.$$

because

$$(m-k)(n-k)(p-k)=mnp-k(mn+np+pm)+k^2(m+n+p)-k^3$$

then

$${\displaystyle \sum _{k=0}^{p-1}(m-k)(n-k)(p-k)}={\displaystyle \sum _{k=0}^{p-1}mnp}-{\displaystyle \sum _{k=0}^{p-1}k(mn+np+pm)}+{\displaystyle \sum _{k=0}^{p-1}{k}^2(m+n+p)}-{\displaystyle \sum _{k=0}^{p-1}{k}^3}$$

and because

$${\displaystyle \sum _{k=0}^{p-1}{k}^0}=p$$

$${\displaystyle \sum _{k=0}^{p-1}k}={\displaystyle \frac{p(p-1)}{2}}$$

$${\displaystyle \sum _{k=0}^{p-1}{k}^2}={\displaystyle \frac{(p-1)p(2p-1)}{6}}$$

$${\displaystyle \sum _{k=0}^{p-1}{k}^3}={\left({\displaystyle \frac{p\left(p-1\right)}{2}}\right)}^2$$

Therefore, by substituting the original formula, we can obtain

$${\displaystyle \sum _{k=0}^{p-1}(m-k)(n-k)(p-k)}=mn{p}^2-{\displaystyle \frac{(mn+np+pm)(p^2-p)}{2}}+{\displaystyle \frac{(m+n+p)(2p^3-3p^2+p)}{6}}-{\displaystyle \frac{{(p^2-p)}^2}{4}}$$

Now, based on the growth trend of the algorithm with the input scale m, n, p, the highest order mnp² is retained, and the time complexity of the algorithm can be obtained as O (mnp²).□

As shown in Figure 2, the three-sided matching problem involves mnp² nodes. Now, the time complexity of some algorithms of the same type is compared, as shown in

Table 1. Comparison of algorithm time complexity.

Algorithm Name	Time Complexity
Dijkstra algorithm	$O\left(V^2\right)\to O\left(m^2{n}^2{p}^4\right)$
Bellman–Ford	$O\left(V\cdot E\right)\to O\left(mn{p}^2\cdot E\right)$, E represents the number of sides
Floyd–Warshall	$O\left(V^3\right)\to O\left(m^3{n}^3{p}^6\right)$
Simplex method	$O\left(2^V\right)\to O\left(2^{mn{p}^2}\right)$

Note: In three-sided matching, the dimension of the state space is determined by the number of three-sided subjects. Theoretically, the number of states is all possible matching combinations, and its theoretical state scale is equivalent to all possible matching combinations, expanding exponentially with the increase in the number of subjects. This has become the core bottleneck restricting the efficiency of the algorithm. The method proposed in this section addresses this issue. For the same matched element (i.e., the same $i/j/k$), it only retains the state with the optimal preference deviation and eliminates the inferior states to achieve the purpose of pruning. This mechanism reduces the total number of states to be processed by screening valid states, alleviates the computational pressure brought by the high-dimensional state space, and enhances the feasibility of the algorithm in medium- and large-scale matching scenarios. It is worth noting that, as the core of the algorithm still relies on the traversal idea of Dijkstra’s algorithm, when dealing with ultra-large-scale matching problems (such as the number of three subjects each exceeding several hundred levels), the traversal logic itself may still lead to an increase in time complexity. It is necessary to further break through the performance boundary by combining strategies such as state compression and hierarchical optimization. However, compared with traditional methods, this algorithm achieves the maximization of state redundancy while ensuring the search ability for the optimal solution through pruning strategies, and to a certain extent, balances the solution accuracy and efficiency.

.

This section not only theoretically proves the rationality of the proposed model and the properties of the solution but also designs a solution algorithm based on Dijkstra and analyzes its complexity, ensuring the rigor and feasibility of the method.

5. Experiment and Analysis

5.1. Example Description

Consider a case where a software development company, denoted as H, initiates five key commercial projects ($C_1,C_2,\dots ,C_5$). Due to the long development cycle and high technical complexity of these projects, once each project manager and development engineer is matched with a certain project, they will be fully committed and cannot participate in other projects. Therefore, how to efficiently and fairly complete the three-sided matching of project manager–development engineer–project has become an important task for the company’s human resources department.

This matching involves 15 development engineers ($A_1,A_2,\dots ,A_{15}$) and 12 project managers ($B_1,B_2,\dots ,B_{12}$). Each project manager is responsible for architecture design, code review, and team management, while the development engineers are in charge of specific functional implementation. The core objective of matching is as follows:

Direction alignment: Each project should be completed by a project manager and a development engineer whose expertise and professionalism align with the project’s requirements.

Ability matching: Project managers hope to collaborate with engineers who have solid technical skills and efficient communication. Development engineers tend to join projects at the forefront of technology with great growth potential, or work with project managers who have strong leadership capabilities.

Preference optimization: Each project manager and development engineer has their own preferred projects, and each project also has a preference for capabilities and professional directions. The matching should respect these preferences as much as possible and minimize forced assignments.

Stability: Each project manager and development engineer can only be assigned to one project to avoid resource dispersion. After the matching is completed, there should be no unmatched “project manager-development engineer–project” combination that would satisfy all three parties more; otherwise, the matching would be unstable.

The mutual preference orders among the three agent sets are numerically specified in

Table 2. Preferences of development engineers for project managers.

Number	B1	B2	B3	B4	B5	B6	B7	B8	B9	B10	B11	B12
A1	7	2	9	5	11	0	10	3	6	1	8	4
A2	4	10	1	8	6	11	5	2	0	3	9	7
A3	11	5	0	7	9	3	8	1	4	10	2	6
A4	6	9	3	10	2	7	1	11	5	8	4	0
A5	8	1	11	4	0	6	9	5	10	2	7	3
A6	3	7	10	2	8	1	6	0	11	4	5	9
A7	0	6	5	9	3	10	2	7	8	11	1	4
A8	10	4	7	1	5	8	0	9	3	6	11	2
A9	5	0	8	3	1	4	11	10	2	7	6	9
A10	2	11	6	0	7	9	4	8	1	5	3	10
A11	9	3	4	11	10	2	7	6	0	1	8	5
A12	1	8	2	6	4	5	3	11	9	0	10	7
A13	4	2	9	8	6	11	10	0	7	3	5	1
A14	7	5	1	10	9	0	4	2	6	8	11	3
A15	10	7	3	4	11	8	5	9	2	0	6	1

,

Table 3. The project preferences of development engineers.

Number	C1	C2	C3	C4	C5
A1	3	1	4	0	2
A2	2	0	1	4	3
A3	4	3	0	2	1
A4	1	4	2	3	0
A5	0	2	3	1	4
A6	2	1	4	0	3
A7	3	0	1	4	2
A8	4	2	0	3	1
A9	1	3	2	0	4
A10	0	4	3	2	1
A11	2	0	4	1	3
A12	3	1	0	4	2
A13	4	3	1	2	0
A14	1	2	3	0	4
A15	0	4	2	3	1

,

Table 4. Project managers’ preference for development engineers.

Number	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10	A11	A12	A13	A14	A15
B1	8	3	12	6	14	1	10	5	0	9	7	11	4	13	2
B2	11	4	7	0	13	2	14	9	6	1	12	5	10	8	3
B3	5	10	1	14	3	8	12	7	11	4	0	13	6	2	9
B4	2	9	6	13	10	5	1	14	8	3	11	0	7	12	4
B5	7	0	13	4	12	9	3	10	2	14	5	8	11	1	6
B6	14	6	4	11	1	12	7	3	9	8	2	10	0	5	13
B7	1	12	10	5	8	13	4	2	14	7	9	3	6	0	11
B8	9	7	2	12	0	11	6	13	3	5	14	4	8	10	1
B9	13	5	0	8	4	7	11	1	12	10	3	2	9	6	14
B10	3	14	9	10	6	0	8	12	5	2	1	7	13	4	11
B11	10	1	14	7	11	4	0	8	13	6	4	9	2	3	5
B12	6	8	3	1	5	14	13	0	10	11	12	6	4	9	7

,

Table 5. Project managers’ preferences for the project.

Number	C1	C2	C3	C4	C5
B1	3	1	0	4	2
B2	2	4	1	0	3
B3	0	2	4	3	1
B4	4	3	2	1	0
B5	1	0	3	2	4
B6	2	1	0	4	3
B7	3	4	1	0	2
B8	0	3	4	2	1
B9	1	2	3	4	0
B10	4	0	2	1	3
B11	2	3	1	0	4
B12	3	0	4	1	2

,

Table 6. Project preference for development engineers.

Number	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10	A11	A12	A13	A14	A15
C1	8	3	12	6	14	1	10	5	0	9	7	11	4	13	2
C2	11	4	7	0	13	2	14	9	6	1	12	5	10	8	3
C3	5	10	1	14	3	8	12	7	11	4	0	13	6	2	9
C4	2	9	6	13	10	5	1	14	8	3	11	0	7	12	4
C5	7	0	13	4	12	9	3	10	2	14	5	8	11	1	6

and

Table 7. Project preference for project managers.

Number	B1	B2	B3	B4	B5	B6	B7	B8	B9	B10	B11	B12
C1	7	2	9	5	11	0	10	3	6	1	8	4
C2	4	10	1	8	6	11	5	2	0	3	9	7
C3	11	5	0	7	9	3	8	1	4	10	2	6
C4	6	9	3	10	2	7	1	11	5	8	4	0
C5	8	1	11	4	0	6	9	5	10	2	7	3

. A higher numerical value indicates a stronger preference.

5.2. Case Analysis

For the above calculation example, the following analysis is conducted:

Unique match: Each project manager, development engineer, and each project can be included in at most one matched triple, meaning that once matched, it cannot be adjusted.

Development engineer preference: Each development engineer ranks the “project manager + project” combination. For example, A₁ may be more inclined towards (B₃,C₁) rather than (B₅,C₂).

Project manager preferences: Each project manager ranks the “development engineer + project” combination. For instance, B₃ might prefer to collaborate with A₇ on C₁ rather than A₂ on C₂.

Project preference: Each project ranks the combination of “project manager + development engineer”. For example, C₁ may prefer the collaboration between A₇ and B₃ rather than that between A₂ and B₃.

Stability condition: After matching is completed, there are no unmatched combinations $\left(A^{\prime },B^{\prime },C^{\prime }\right)$, such that the following occurs:

①

Development engineer $A^{\prime }$ prefers $\left(B^{\prime },C^{\prime }\right)$ rather than the current match;

②

Project manager $B^{\prime }$ prefers $\left(A^{\prime },C^{\prime }\right)$ rather than the current match;

③

Project $C^{\prime }$ is not occupied or leans more towards $\left(A^{\prime },B^{\prime }\right)$.

Based on the above analysis, it can be known that the problem in this context at this time can be transformed into a trilateral stable matching problem with combinatorial preferences.

Let ${\gamma }_1={\gamma }_2=0.5$, $\tau =0.2$; the matching results can be obtained as shown in

Table 8. The optimal results of trilateral matching for item matching.

Matching Triples	Satisfaction of the Three Sides	Overall Satisfaction
$\left(A_3,B_5,C_1\right)$	$A_3=0.773,B_5=0.667,C_1=0.840$	0.747
$\left(A_4,B_8,C_4\right)$	$A_4=0.800,B_8=0.667,C_4=0.880$	0.782
$\left(A_7,B_2,C_2\right)$	$A_7=0.267,B_2=0.889,C_2=0.880$	0.679
$\left(A_5,B_1,C_5\right)$	$A_5=0.667,B_1=0.778,C_5=0.720$	0.721
$\left(A_9,B_7,C_3\right)$	$A_9=0.733,B_7=0.722,C_3=0.680$	0.712

:

5.3. Algorithm Effectiveness Analysis

①

Verification of the matching effect under dictionary-based preference

In actual situations, there exist dictionary-based preferences for matching subjects, such as the following: When development engineers select the project manager–project matching combination, they will give priority to projects that are challenging or more proficient, and then consider the cooperating project managers. At this time, ${\gamma }_1,{\gamma }_2$ in Equation (2) is no longer equal. To ensure that the algorithm can adapt to the changing needs of different matching subjects and avoid the weight parameters not playing a role in the operation, the value is now changed ${\gamma }_1,{\gamma }_2$ for re-matching. The results are as follows in

Table 9. Optimal matching results under dictionary preference.

$\mathbf{Value} \mathbf{of} {\mathit{\gamma }}_1,{\mathit{\gamma }}_2$	Matching Triples	Satisfaction of the Three Sides	Overall Satisfaction
${\gamma }_1=0.9,{\gamma }_2=0.1$	$\left(A_3,B_1,C_1\right)$	$A_3=0.903,B_1=0.777,C_1=0.766$	0.815
	$\left(A_4,B_8,C_4\right)$	$A_4=0.893,B_8=0.769,C_4=0.861$	0.841
	$\left(A_{11},B_4,C_2\right)$	$A_{11}=0.864,B_4=0.708,C_2=0.774$	0.782
	$\left(A_{10},B_{12},C_5\right)$	$A_{10}=0.786,B_{12}=0.700,C_5=0.869$	0.785
	$\left(A_9,B_7,C_3\right)$	$A_9=0.883,B_7=0.900,C_3=0.708$	0.831
${\gamma }_1=0.1,{\gamma }_2=0.9$	$\left(A_3,B_7,C_1\right)$	$A_3=0.723,B_7=0.540,C_1=0.814$	0.693
	$\left(A_2,B_6,C_4\right)$	$A_2=0.787,B_6=0.640,C_4=0.549$	0.659
	$\left(A_{10},B_2,C_2\right)$	$A_{10}=0.787,B_2=0.540,C_2=0.717$	0.681
	$\left(A_5,B_{11},C_5\right)$	$A_5=0.702,B_{11}=0.740,C_5=0.575$	0.672
	$\left(A_6,B_{12},C_3\right)$	$A_6=0.745,B_{12}=0.800,C_3=0.460$	0.668
${\gamma }_1=0.7,{\gamma }_2=0.3$	$\left(A_3,B_5,C_1\right)$	$A_3=0.730,B_5=0.764,C_1=0.817$	0.770
	$\left(A_4,B_8,C_4\right)$	$A_4=0.854,B_8=0.727,C_4=0.870$	0.817
	$\left(A_{11},B_4,C_2\right)$	$A_{11}=0.753,B_4=0.691,C_2=0.748$	0.731
	$\left(A_{10},B_{12},C_5\right)$	$A_5=0.652,B_1=0.855,C_5=0.748$	0.751
	$\left(A_9,B_7,C_3\right)$	$A_9=0.820,B_7=0.827,C_3=0.695$	0.781
${\gamma }_1=0.3,{\gamma }_2=0.7$	$\left(A_3,B_5,C_1\right)$	$A_3=0.738,B_5=0.514,C_1=0.866$	0.706
	$\left(A_4,B_8,C_4\right)$	$A_4=0.721,B_8=0.571,C_4=0.891$	0.728
	$\left(A_9,B_7,C_2\right)$	$A_9=0.721,B_7=0.857,C_2=0.361$	0.647
	$\left(A_5,B_1,C_5\right)$	$A_5=0.689,B_1=0.657,C_5=0.689$	0.678
	$\left(A_6,B_{12},C_3\right)$	$A_6=0.738,B_{12}=0.857,C_3=0.471$	0.688

.

In the three-sided matching problem of development engineer–project manager–project, the adjustment of the weight coefficients γ₁ and γ₂ will directly affect the final matching result. As γ₁ increases, the weights of the subject terms related to γ₁ in the numerator and denominator increase, and the results tend to choose the combination that is more in line with the preference of the subject terms related to γ₁. As γ₂ increases, the weights of the principal terms related to γ₂ in the numerator and denominator also increase. The results tend to choose the combination that is more suitable for the principal terms related to γ₂. The above experimental results can reflect this changing rule to a certain extent. Therefore, in the actual matching process, the weights can be flexibly adjusted according to the needs of the principal, and a reasonable matching strategy can be formulated for the matching principal.

②

Robustness testing

Robustness testing aims to verify the stability and reliability of the algorithm in test scenarios that include error data handling, abnormal situation handling, and invalid operations.

Table 10. Results of algorithm robustness test.

Test Type	Test Case Design	Detection Index	Reference Value for Normal Scenarios	Measured Values of Abnormal Scenarios
Null value input	The preference matrix is either none or an empty list	1. Matching completion rate 2. Time-consuming calculation	1.100% 2.1.8 s	1.100% 2.1.82 s
Non-numerical data	The preference matrix contains strings or Boolean values	Average satisfaction	0.728	0.725
Negative value preference	The preference value appears negative	Average satisfaction	0.728	--
Shortage of project managers	There are 5 projects but only 3 project managers	Matching completion rate	100%	60%
There is an overstaffing of development engineers	15 development engineers are competing for two projects	The distribution rate of the top two satisfaction levels	100%	100%

below presents the specific implementation strategies and test results of the systematic testing.

This three-sided matching algorithm performs well in the robustness test, demonstrating good stability and adaptability. The algorithm can effectively handle various types of abnormal inputs, including null values (throwing a clear exception and automatically completing the null value with the minimum preference value of the subject), non-numeric values (the same as the null value handling method), and negative value preferences (the algorithm is applicable to non-negative values), ensuring that the system can still operate stably under extreme conditions. Meanwhile, in the context of resource competition, the algorithm still generates reasonable matching results and can reliably support complex business matching requirements.

③

Generalization analysis of models and algorithms

To verify the applicability of the three-sided matching logic and algorithm in different scenarios, the following explanations are provided in three areas: manufacturing production scheduling, educational resource allocation, and medical team formation.

• Case 1: Production line Scheduling for new energy Vehicle Manufacturers in the manufacturing industry

A certain new energy vehicle manufacturer has initiated a project for five brand-new power battery production lines. Each production line requires one production line supervisor and several core technical workers. The production line supervisor is responsible for process design and quality control, while the technical workers are responsible for equipment debugging and standardized operations. Once a match is made, the supervisor and the workers must follow up on the production line throughout the process and are not allowed to provide cross-line support.

Ability matching: Supervisors tend to choose workers with high operational proficiency and low tolerance for errors. Both supervisors and workers are more willing to join production lines with high levels of automation and complete safety guarantees. Meanwhile, workers are more inclined to follow supervisors with flexible management styles. The production line has a preference for capacity targets and requires supervisors and workers with high professional capabilities and judgment.

Stability: Avoid the situation where an unmatched combination of supervisors, workers, and production lines leads to a significant increase in efficiency or satisfaction for either party.

• Case 2: Resource Allocation of an Educational Group

A certain educational group plans to open new classes in five new campuses. Each class needs to be matched with one head teacher and two subject backbone teachers. The head teacher is responsible for class management and communication between home and school, while key teachers are responsible for curriculum design and teaching implementation. Once personnel are assigned, they must be fixed in the classes of the same campus and are not allowed to substitute teachers across campuses.

Ability matching: Head teachers tend to choose teachers with strong collaboration and good classroom control. Head teachers and teachers are more willing to join campuses with a solid student base and abundant teaching and research resources. Meanwhile, teachers are more inclined to follow head teachers with high communication efficiency. The campus has a preference for an overall improvement in teaching quality and requires class advisors and teachers who meet both the geographical location and teaching ability requirements.

Stability: Avoid significant improvement in students’ academic performance or teacher–student satisfaction caused by an unmatched homeroom teacher–class combination.

• Case 3: Establishment of a multidisciplinary diagnosis and Treatment Team in a tertiary Hospital

A tertiary hospital has initiated five Multi-Disciplinary Team (MDT) diagnosis and treatment projects for major diseases. Each project requires one leading expert and three core doctors from different departments. The leading expert is responsible for formulating the diagnosis and treatment plan and coordinating among multiple departments, while the core doctor is responsible for specialized diagnosis and treatment. Once the team is formed, it needs to follow up on the patients of this project throughout the process and cannot provide support across projects.

Ability matching: Leading experts tend to choose doctors with rich clinical experience and strong academic collaboration. Both experts and doctors are more willing to join projects with abundant case resources and sufficient research support. Doctors are more inclined to follow leading experts with significant influence in the industry. The project has a preference for diagnosis and treatment models (surgery first/targeted therapy first), and requires experts and doctors who can accurately grasp the diagnosis and treatment rhythm and correctly estimate the complexity of cases.

Stability: Avoid significant improvements in patient treatment outcomes or team diagnosis and treatment efficiency due to an unmatched lead expert–doctor–project combination.

All the above cases involve three-sided matching problems, and the problem structure conforms to the combinatorial preference relationship. The matching goals are all satisfactory and stable matches, which have the same logic as the case in Section 5.1. Therefore, the three-sided matching model and algorithm can be used for a solution, further confirming the generalization of the model and algorithm in different management problems.

Through detailed examples of project allocation in software companies, this section verifies the effectiveness, robustness, and generalization ability of the proposed method in generating stable matching schemes, adapting to different preference structures, and dealing with abnormal inputs.

6. Conclusions

In view of the incompatibility between the rigid stability requirements of traditional three-sided matching and the adaptability of complex systems, this study proposes a three-sided matching method based on fuzzy stability theory. By constructing a rank-weighted membership function (integrating combinatorial preference weights and normalized preference ranks) to quantify the preference satisfaction between agents, we define fuzzy blocking pairs, fuzzy stability conditions, and stability transition rules. We further transform the three-sided matching problem into a shortest path problem, prove the equivalence between the shortest path solution and stable matching outcomes, and improve the efficiency of Dijkstra’s algorithm via a pruning strategy. Verified by practical project allocation cases in software companies, the method not only generates fuzzy stability-compliant matching schemes but also achieves excellent performance in subject satisfaction and computational efficiency.

Theoretically, we establish a fuzzy stability framework for combinatorial preference-based complex three-sided matching systems, and improve matching quality and stability by integrating the shortest path transformation. Practically, the introduced fuzzy blocking strength enhances the flexibility and practicality of matching results, which can be applied to multi-party matching scenarios such as software project team formation, providing effective solutions for person–job matching and resource scheduling in complex decision-making environments.

The core premise of the model is that the subject’s preference is a static value. In actual matching, the subject’s preference may change due to various factors, but the current model is unable to capture such dynamic changes. Future research can focus on designing preference update functions, quantifying dynamic factors such as time dimensions and environmental feedback into preference adjustment coefficients, and embedding them in membership functions to achieve real-time dynamic updates of preferences. Meanwhile, future research will also optimize the algorithm through state compression and phased solution strategies to better meet the requirements of large-scale three-sided stable matching.