A stable matching method for technology trading with intuitionistic fuzzy multi-attribute information

Abstract

Technology trading matching facilitates quicker solution-finding for technology demanders and expedites the transformation of scientific and technological achievements. Yet, unstable matchings often lead traders to renounce existing contracts, sidestep trading intermediaries, and resort to private transactions. This results in inefficient trading mechanisms and market disarray. To ensure a stable and mutually satisfactory match for both suppliers and demanders, we propose a stable two-sided matching decision-making method that incorporates intuitionistic fuzzy multi-attribute information. Initially, we introduce an intuitionistic fuzzy TOPSIS approach to compute the comprehensive satisfaction of both suppliers and demanders by aggregating intuitionistic fuzzy information across various attributes. Subsequently, we design a multi-objective optimization model that weighs both stability and satisfaction to determine the ideal technology trading pairs. We conclude with a real-world example that demonstrates the proposed method’s application, and its effectiveness is corroborated through sensitivity and comparative analyses.

Keywords

Technology trading two-sided matching stable matching intuitionistic fuzzy sets

1 Introduction

The trajectory of the Chinese economy is transitioning from rapid growth to a phase characterized by high-quality development. Consequently, the developmental paradigm has evolved from being factor-driven and investment-centric to being anchored in innovation. Elevating the capacity for scientific and technological innovation and ardently pursuing an innovation-led developmental approach has ascended to the forefront of national strategy. Technology trading, often referred to as the “last mile” in the technological innovation journey, is pivotal in refining the economic structure, forging new engines for economic progression, and establishing fresh competitive edges [1, 2]. The technology trading market acts as a conduit between technology suppliers and demanders, functioning as a facilitator, binder, and enabler for the seamless transfer and metamorphosis of scientific and technological accomplishments [3]. The holistic services rendered by this market adeptly address challenges such as information asymmetry, communication barriers between technology suppliers and demanders, elevated transactional expenses, and further propel the transmutation and industrialization of scientific and technological breakthroughs [4 –6]. However, in the evolution of the technology trading market, the persistent issues of suboptimal efficiency and a diminished success rate in matching supply with demand have been significant impediments to the transformation of scientific and technological outputs [7, 8]. As the magnitude of technology trading amplifies, the imperative for precise and efficient matching between suppliers and demanders becomes increasingly paramount.

Given the pressing practical demand for technology trading matching, it has garnered significant interest in both industrial and academic circles. Several technology trading platforms, such as Yet2, have established technology databases and devised matching algorithms to expedite solution discovery for clients through specialized technology search and matching techniques [9]. A number of decision-making methods have been introduced to match the suppliers and demanders in technology trading. In a study by Liu and Li on the matching of technical knowledge, both parties provided multi-attribute evaluation data using crisp numbers. They then formulated a decision model grounded in prospect theory and grey relational analysis, taking into account psychological behaviors [10]. Chen et al. introduced a fuzzy axiom design approach to determine the two-sided matching degree using linguistic data in the context of knowledge service matching. They also developed a multi-objective optimization model aimed at optimizing the collective satisfaction of both suppliers and demanders [11]. Recognizing that technology suppliers and demanders might offer evaluations in the form of real numbers, interval numbers, or linguistic data for varying attributes, Wu et al. employed prospect theory and evidence theory. Their approach first aggregated two-sided preferences into intuitionistic fuzzy numbers and then computed transaction price satisfaction as a reflection of two-sided cooperative inclinations in the subsequent phase [12]. Both studies [13] and [14] delved into the technological knowledge-matching challenge, where exact satisfaction were directly provided by the two parties involved. However, while the former integrated the network collaboration effect into the overall satisfaction metric, the latter introduced a novel network value assessment technique to gauge platform advantages. These scholarly contributions underscore the efficacy of the two-sided matching decision-making approach in technology trading. Nonetheless, existing research on technology trading matching still exhibits certain gaps and limitations.

Firstly, the inherent intangibility of technology, coupled with the uncertainty surrounding the value of technological goods, the limited disclosure of technical specifics, and cognitive constraints, makes it arduous for participants in the technology trading market to furnish precise evaluative data [15]. Moreover, agents engaged in technology trading seldom experience absolute satisfaction or dissatisfaction with their counterparts. When assessing a particular attribute, a technology supplier (or demander) might harbor feelings of satisfaction or dissatisfaction towards a demander (or supplier). Yet, there often exists an ambiguous degree of satisfaction, stemming from incomplete information that remains elusive to pinpoint. Given that Intuitionistic Fuzzy Sets (IFS) are adept at encapsulating scenarios characterized by satisfactory, unsatisfactory, and uncertain information [16 –19], it becomes more pragmatic for technology trading agents to provide intuitive fuzzy evaluations across diverse attributes. Consequently, integrating the satisfaction levels of both suppliers and demanders, grounded in intuitive fuzzy multi-attribute evaluative infromation, emerges as a pressing challenge that technology trading matching must address.

The Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) stands as a preeminent method for tackling multi-attribute decision-making (MADM) problems. It was pioneered by Hwang and Yoon [20–21]. At its core, the TOPSIS approach ranks alternatives by computing their proximity to an ideal solution. The method’s commendable efficacy in addressing real-world dilemmas has piqued the interest of numerous scholars, leading to a proliferation of research in this domain. This has given rise to a myriad of extended methods, including but not limited to, fuzzy TOPSIS and Pythagorean fuzzy TOPSIS [22, 23]. These methods have found applicability in diverse arenas cover assignment of products to storage locations [24], green supplier selection [25], risk assessment of landslide hazards [26], and evaluating customers’ exposure degree to COVID-19 in restaurants[27]. Predominantly, extant research on TOPSIS and its derivative methods has been channeled towards the selection within MADM alternatives. To date, there has been no attempts to examine the intuitionistic fuzzy TOPSIS for addressing the satisfaction assessment in two-sided matching.

Secondly, the stability of technology trading matching has often been overlooked, despite its significance as a pivotal criterion in assessing the measuring of two-sided matching schemes. A matching scheme of technology trading is stable when there are no unmatched pair who would both rather be matched to each other than their current partners. If an unmatched pair of technology trading conduct transactions that are superior to their current partners, they have the incentive to break the current trading contract and engage in private transactions, resulting in the ineffectiveness of the matching mechanism for technology trading intermediaries. Therefore, the unstable matching mechanism often ends up failing in practical applications [28, 29]. Although the stable matching has been investigated in many application fields, such as Hospitals/residents problem [30], school choice [31], resource matching in cloud manufacturing [32] and personnel allocation of software project [33]. As far as we know, no previous study has investigated the matching stability between supply and demand in technology trading.

The aforementioned analysis underscores the significance of addressing the technology trading matching challenge, especially when factoring in intuitionistic fuzzy multi-attribute information and the stability of trading participants. This research introduces a stable matching method tailored for suppliers and demanders within the technology trading. Specifically, to more accurately convey the preference information of two-side agents in technology trading, we employ intuitionistic fuzzy multi-attribute information to describe the evaluation information for the opposite side. By synergizing the TOPSIS approach with IFS theory, we design an intuitionistic fuzzy TOPSIS method tailored for evaluating the satisfaction levels of technology trading agents. Concurrently, to deter participants from reneging on technology trading agreements in pursuit of more favorable potential partners, we advocate for a stable matching scheme in technology trading. Such a scheme, which garners universal approval and ensures sustained implementation, is integral to the trading ecosystem. To this end, we infuse stable matching theory into our technology trading matching paradigm and formulate an optimization model that harmoniously balances trader satisfaction with the stability of matching schemes. The salient contributions of this study can be delineated as follows:

In the face of complex and uncertain decision-making landscapes, where technology demanders and suppliers proffer intuitionistic fuzzy multi-attribute information, we innovatively introduce the intuitionistic fuzzy TOPSIS approach to measure the satisfaction of technology trading agents.

Addressing the prevalent challenge of low receptivity towards unstable and suboptimal matching outcomes, we design a multi-objective optimization model. This model judiciously weighs both stability and satisfaction, aiming to obtain the optimal matching pairs of suppliers and demanders within the technology trading arena.

The merits of our proposed trading matching method have been rigorously validated through both sensitivity and comparative analyses.

The structure of this paper is delineated as follows: Section 2 reviews the related concepts and theory. In section 3, the two-sided matching problem between suppliers and demanders of technology trading is illustrated. The satisfaction of suppliers and demanders is calculated based on the intuitionistic fuzzy TOPSIS method and the matching optimization model of technology trading is formulated in section 4. A pragmatic example is furnished to illustrate decision process and the merits of our proposed approach in Section 5. Finally, the paper summarizes the concluding remarks and discusses the future research focus.

2 Preliminaries

This section encapsulates the fundamental concepts related to Intuitionistic Fuzzy Sets (IFSs) pertinent to this paper.

Definition 1 [34]: Given a fixed set X, an intuitionistic fuzzy set (IFS) A in X is characterized as: $A = {< x, μ_{A} (x), v_{A} (x) > | x \in X}$ (1)

Here, the function μ_A : X → [0, 1] and ν_A : X → [0, 1] respectively denote the degree of membership and non-membership of element x ∈ X to the set A, subject to the constraint: $0 ⩽ μ_{A} (x) + ν_{A} (x) ⩽ 1, \forall x \in X$ (2)

The term π_A (x) =1 - μ_A (x) - ν_A (x), ∀x ∈ X represents the degree of indeterminacy of x to A, also known as the hesitancy degree. Evidently, 0 ⩽ π_A (x) ⩽1. Notably, when π_A (x) =0, an IFS A simplifies to a fuzzy set A.

For IFS A and specific elements ∀x ∈ X, Triples < x, μ_A (x) , ν_A (x) > are called intuitionistic fuzzy numbers (IFN) [35]. An IFN is usually abbreviated as α = (μ_α, ν_α), 0 ⩽ μ_α + ν_α ⩽ 1, μ_α ⩾ 0, ν_α ⩾ 0.

Definition 2 [34]: If α₁ = (μ_α₁, v_α₁) and α₂ = (μ_α₂, v_α₂) are two IFNs, then

α₁ + α₂ = μ_α₁ + μ_α₂ - μ_α₁ μ_α₂, v_α₁v_α₂;

α₁ · α₂ = μ_α₁ μ_α₂, v_α₁ + v_α₂ - v_α₁v_α₂;

$λ α_{1} = (1 - (1 - μ_{α_{1}},^{λ}, v_{α_{1}}^{λ})$ ;

${α_{1}}^{λ} = (μ_{α_{1}}^{λ}, 1 - (1 - v_{α_{1}})^{λ})$ .

Fig. 1

The two-sided matching problem in technology trading.

Distance measures serve as pivotal metrics for discerning the disparities between IFSs. Wang proposed an axiomatic definition for this measure:

Definition 3 [36]: Let d be a mapping d : IFSs (X) × IFSs (X) → [0, 1]. If an intuitionistic fuzzy set A ∈ IFSs (X) and B ∈ IFSs (X) satisfy the following properties, then d (A, B) is the distance measure of IFSs A and B.

0 ⩽ d (A, B) ⩽1;

d (A, B) =0 if and only if A = B;

d (A, B) = d (B, A);

if A ⊆ B ⊆ C, A, B, C ∈ IFSs (X), then d (A, C) ⩾ d (A, B) and d (A, C) ⩾ d (B, C).

Several definitions of the distance measure between IFSs have been presented in the current literature to calculate the deviation between two IFSs [37 –41]. The normalized Euclidean distance presented by Szmidt and Kacprzyk has some good geometric properties and is widely used. The calculation formula for normalized Euclidean distance is as follows:

Definition 4 [38]: Let A = {< x, μ_A (x) , v_A (x) > |x ∈ X} and B = < x, μ_B (x) , v_B (x) > |x ∈ X} be two IFSs in X = {x₁, x₂, ⋯ , x_n}, then normalized Euclidean distance between A and B is $d (A, B) = \sqrt{\frac{1}{2 n} \sum_{i = 1}^{n} [{(μ_{A} (x_{i}) - μ_{B} (x_{i}))}^{2} + {(ν_{A} (x_{i}) - ν_{B} (x_{i}))}^{2} + {(π_{A} (x_{i}) - π_{B} (x_{i}))}^{2}]}$ (3)

3 Description of technology trading matching

For a technology trading market, there are three elements involved in technology trading activities, namely, technology suppliers, technology demanders, and intermediaries. The technology suppliers refer to the technology developer, inventor, or owner, and the technology demanders refer to the recipient and user of the technology, which is the party implementing the technology introduction. As a provider of matching services in technology trading, an intermediary can be an offline entity intermediary or an online technology trading platform. To facilitate the elaboration of the technology trading matching problem to be studied, an illustration is designed, as shown in Fig. 1.

It can be seen that this paper studies the technology trading matching problem where technology demanders and suppliers have multiple-attribute evaluation information. The two-way arrow line indicates that technology traders interact with intermediaries, such as submitting technology demand information, technology achievement information, etc. The one-way arrow line represents the recommendation result of the technology trading matching provided by the intermediary based on the two-sided matching method.

The configuration of the problem addressed in this paper is delineated as follows. Consider an instance I pertaining to two-sided matching within the technology trading milieu. This instance encompasses two distinct and finite sets: a set S = {S₁, S₂, ⋯ , S_m} of suppliers and a set of D = {D₁, D₂, ⋯ , D_n} of demanders. The set of S = {S₁, S₂, ⋯ , S_m} epitomizes the technology suppliers, with an associated set of technologies represented as T = {T₁, T₂, ⋯ , T_m}. Here, S_i denotes the ith technology supplier and T_i is the ith technology that needs to be traded by S_i, i = 1, 2, ⋯ , m. Conversely, the set of D = {D₁, D₂, ⋯ , D_n} encapsulates the technology demanders, where D_j is the jth technology demander,j = 1, 2, ⋯ , n.

The satisfaction of suppliers and demanders is measured by the multi-attribute evaluation. Let C = {C₁, C₂, . . . , C_f} be the evaluation attributes sets of supplies, where C_p is the pth attribute, p = 1, 2,

⋯, f. E = {E₁, E₂, . . . , E_h} is the set of attributes for assessing suppliers, where E_q is the qth attribute, q = 1, 2, ⋯ , h. $W_{i} = (w_{i}^{1}, w_{i}^{2}, . . ., w_{i}^{f})^{T}$ and $V_{j} = (v_{j}^{1}, v_{j}^{2}, . . ., v_{j}^{h})^{T}$ are the weight vector of attributes provided by supplier S_i and demander D_j, respectively. $w_{i}^{p}$ is the weight of attribute C_p and $v_{j}^{q}$ denotes the weight of attribute E_q, $0 \leq w_{i}^{p} \leq 1$ , $\sum_{p = 1}^{f} w_{i}^{p} = 1$ , $0 \leq v_{j}^{q} \leq 1$ , $\sum_{q = 1}^{h} v_{j}^{q} = 1$ .

The intuitionistic fuzzy preference information matrix of supplier S_i is $A_{i} = [a_{ij}^{p}]_{n \times f}$ , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n, p = 1, 2, ⋯ , f. For matrix $A_{i} = [a_{ij}^{p}]_{n \times f}$ , $a_{ij}^{p} = (μ_{a_{ij}^{p}}, v_{a_{ij}^{p}})$ is the IFN evaluation information for attribute C_p provided by S_i, where $μ_{a_{ij}^{p}}$ and $v_{a_{ij}^{p}}$ respectively represent the satisfaction and dissatisfaction of S_i to D_j for attribute C_p. $B_{j} = [b_{ij}^{q}]_{m \times h}$ denotes the intuitionistic fuzzy preference information matrix provided by demander D_j, i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n, q = 1, 2, ⋯ , h. $b_{ij}^{q} = (μ_{b_{ij}^{q}}, v_{b_{ij}^{q}})$ is the IFN evaluation information for attribute E_q given by D_j, where $μ_{b_{ij}^{q}}$ and $v_{b_{ij}^{q}}$ respectively represent the satisfaction and dissatisfaction of D_j to technology T_i for attribute E_q.

For the two-sided matching of technology trading, a reasonable assumption is m, n ≥ 2. We also assume that each technology T_i can be traded with at most one demander and each technology demander D_j seeks to acquire a single technology at most. Thus, this study focuses on the one-to-one two-sided matching between suppliers and demanders of technology trading.

The key research question of this study is how to obtain the optimal scheme of technology trading matching, that is, to find the matching pairs (S_i, D_j) of suppliers and demanders from sets S and D that satisfy the stability and maximize satisfaction as much as possible based on the intuitionistic fuzzy preference information matrix $A_{i} = [a_{ij}^{p}]_{n \times f}$ , $B_{j} = [b_{ij}^{q}]_{m \times h}$ , weight vectors W_i and V_j.

4 A Stable matching decision method for technology trading

This section introduces a stable matching decision-making method tailored for technology trading, incorporating intuitionistic fuzzy multi-attribute information. We initially compute the satisfaction levels of suppliers and demanders utilizing the intuitionistic fuzzy TOPSIS approach. Subsequently, we formulate a multi-objective optimization model, emphasizing both stability and satisfaction, to identify the optimal matching pairs between suppliers and demanders.

4.1 Aggregate satisfaction for suppliers and demanders

The essence of TOPSIS can be summarized as follows. For a given set of alternatives, one posits the existence of a positive ideal solution A⁺ and a negative ideal solution A^-. The distance S⁺ between each alternative A and the positive ideal solution A⁺ and the distance S^- between the negative ideal solution A^- are determined, respectively. The alternative closest to the positive ideal solution A⁺ and farthest from the negative ideal solution A^- is the optimal solution [42]. This paper presents an intuitionistic fuzzy TOPSIS method for calculating the satisfaction of suppliers and demanders by combining the IFS theory with the TOPSIS method. The specific steps of the method are shown as follows.

(1) Calculate the weighted intuitionistic fuzzy preference information matrix. $R_{i} = [r_{ij}^{p}]_{n \times f}$ is the weighted intuitionistic fuzzy preference matrix of supplier S_i for attribute C_p, where $r_{ij}^{p} = (μ_{r_{ij}^{p}}, v_{r_{ij}^{p}})$ . Similarly, $T_{j} = [t_{ij}^{q}]_{m \times h}$ is denoted as the weighted intuitionistic fuzzy preference matrix of demander D_j for attribute E_q, where $t_{ij}^{q} = (μ_{t_{ij}^{q}}, v_{t_{ij}^{q}})$ . $r_{ij}^{p}$ and $t_{ij}^{q}$ can be derived as $r_{ij}^{p} = w_{i}^{p} a_{ij}^{p} = w_{i}^{p} (μ_{a_{ij}^{p}}, v_{a_{ij}^{p}}) = (1 - (1 - μ_{a_{ij}^{p}})^{w_{i}^{p}}, ν_{a_{ij}^{p}}^{w_{i}^{p}})$ (4) $t_{ij}^{q} = v_{j}^{q} b_{ij}^{q} = v_{j}^{q} (μ_{b_{ij}^{q}}, v_{b_{ij}^{q}}) = (1 - (1 - μ_{b_{ij}^{q}})^{v_{j}^{q}}, ν_{b_{ij}^{q}}^{v_{j}^{q}})$ (5) (2) Determine the intuitionistic fuzzy positive and negative ideal solutions. $N_{S_{i}}^{+}$ and $N_{S_{i}}^{-}$ are the positive and negative ideal solutions of supplier S_i for evaluation attributes, respectively. For demander D_j, $N_{D_{j}}^{+}$ and $N_{D_{j}}^{-}$ are the positive and negative ideal solutions of attributes. $N_{S_{i}}^{+}$ , $N_{S_{i}}^{-}$ , $N_{D_{j}}^{+}$ and $N_{D_{j}}^{-}$ are defined as follows: $N_{S_{i}}^{+} = ((μ_{S_{i} 1}^{+}, v_{S_{i} 1}^{+}), (μ_{S_{i} 2}^{+}, v_{S_{i} 2}^{+}), \dots (μ_{S_{i} f}^{+}, v_{S_{i} f}^{+}))$ $N_{S_{i}}^{-} = ((μ_{S_{i} 1}^{-}, v_{S_{i} 1}^{-}), (μ_{S_{i} 2}^{-}, v_{S_{i} 2}^{-}), \dots (μ_{S_{i} f}^{-}, v_{S_{i} f}^{-}))$ $N_{D_{j}}^{+} = ((μ_{D_{j} 1}^{+}, v_{D_{j} 1}^{+}), (μ_{D_{j} 2}^{+}, v_{D_{j} 2}^{+}), \dots (μ_{D_{j} h}^{+}, v_{D_{j} h}^{+}))$ $N_{D_{j}}^{-} = ((μ_{D_{j} 1}^{-}, v_{D_{j} 1}^{-}), (μ_{D_{j} 2}^{-}, v_{D_{j} 2}^{-}), \dots (μ_{D_{j} h}^{-}, v_{D_{j} h}^{-}))$ where $(μ_{S_{i} p}^{+}, v_{S_{i} p}^{+}) = (max_{1 \leq j \leq n} {μ_{r_{ij}^{p}}}, min_{1 \leq j \leq n} {v_{r_{ij}^{p}}})$ (6) $(μ_{S_{i} p}^{-}, v_{S_{i} p}^{-}) = (min_{1 \leq j \leq n} {μ_{r_{ij}^{p}}}, max_{1 \leq j \leq n} {v_{r_{ij}^{p}}})$ (7) $(μ_{D_{j} q}^{+}, v_{D_{j} q}^{+}) = (max_{1 \leq i \leq m} {μ_{t_{ij}^{q}}}, min_{1 \leq i \leq m} {v_{t_{ij}^{q}}})$ (8) $(μ_{D_{j} q}^{-}, v_{D_{j} q}^{-}) = (min_{1 \leq i \leq m} {μ_{t_{ij}^{q}}}, max_{1 \leq i \leq m} {v_{t_{ij}^{q}}})$ (9)

(3) Calculate the Euclidean distance between the two-side agents of technology trading. Let $d_{S_{i}} (D_{j}, N_{S_{i}}^{+})$ be the Euclidean distance between evaluation information $r_{ij}^{p} = (μ_{r_{ij}^{p}}, v_{r_{ij}^{p}})$ and the positive ideal solution $N_{S_{i}}^{+}$ . Let $d_{S_{i}} (D_{j}, N_{S_{i}}^{-})$ be the Euclidean distance between $r_{ij}^{p} = (μ_{r_{ij}^{p}}, v_{r_{ij}^{p}})$ and the negative ideal solution $N_{S_{i}}^{-}$ . $d_{S_{i}} (D_{j}, N_{S_{i}}^{+})$ and $d_{S_{i}} (D_{j}, N_{S_{i}}^{-})$ are formulated as follows: $\begin{matrix} d_{S_{i}} (D_{j}, N_{S_{i}}^{+}) = \\ \sqrt{\frac{1}{2 f} \sum_{p = 1}^{f} [{(μ_{r_{ij}^{p}} - μ_{S_{i} p}^{+})}^{2} + {(v_{r_{ij}^{p}} - v_{S_{i} p}^{+})}^{2} + {(π_{r_{ij}^{p}} - π_{S_{i} p}^{+})}^{2}]} \end{matrix}$ (10) $\begin{matrix} d_{S_{i}} (D_{j}, N_{S_{i}}^{-}) = \\ \sqrt{\frac{1}{2 f} \sum_{p = 1}^{f} [{(μ_{r_{ij}^{p}} - μ_{S_{i} p}^{-})}^{2} + {(v_{r_{ij}^{p}} - v_{S_{i} p}^{-})}^{2} + {(π_{r_{ij}^{p}} - π_{S_{i} p}^{-})}^{2}]} \end{matrix}$ (11) where ${\begin{matrix} π_{r_{ij}^{p}} = 1 - μ_{r_{ij}^{p}} - v_{r_{ij}^{p}} \\ π_{S_{i} p}^{+} = 1 - μ_{S_{i} p}^{+} - v_{S_{i} p}^{+} \\ π_{S_{i} p}^{-} = 1 - μ_{S_{i} p}^{-} - v_{S_{i} p}^{-} \end{matrix}$ .

Corresponding, the Euclidean distances of evaluation information $t_{ij}^{q} = (μ_{t_{ij}^{q}}, v_{t_{ij}^{q}})$ (i = 1, 2, ⋯ , m) with the positive ideal solution $N_{D_{j}}^{+}$ , and negative ideal solution $N_{D_{j}}^{-}$ are indicated by $d_{D_{j}} (S_{i}, N_{D_{j}}^{+})$ and $d_{D_{j}} (S_{i}, N_{D_{j}}^{-})$ , respectively, which are calculated as follows: $\begin{matrix} d_{D_{j}} (S_{i}, N_{D_{j}}^{+}) = \\ \sqrt{\frac{1}{2 h} \sum_{q = 1}^{h} [{(μ_{t_{ij}^{q}} - μ_{D_{j} q}^{+})}^{2} + {(v_{t_{ij}^{q}} - v_{D_{j} q}^{+})}^{2} + {(π_{t_{ij}^{q}} - π_{D_{j} q}^{+})}^{2}]} \end{matrix}$ (12) $\begin{matrix} d_{D_{j}} (S_{i}, N_{D_{j}}^{-}) = \\ \sqrt{\frac{1}{2 h} \sum_{q = 1}^{h} [{(μ_{t_{ij}^{q}} - μ_{D_{j} q}^{-})}^{2} + {(v_{t_{ij}^{q}} - v_{D_{j} q}^{-})}^{2} + {(π_{t_{ij}^{q}} - π_{D_{j} q}^{-})}^{2}]} \end{matrix}$ (13) where ${\begin{matrix} π_{t_{ij}^{q}} = 1 - μ_{t_{ij}^{q}} - v_{t_{ij}^{q}} \\ π_{D_{j} q}^{+} = 1 - μ_{D_{j} q}^{+} - v_{D_{j} q}^{+} \\ π_{D_{j} q}^{-} = 1 - μ_{D_{j} q}^{-} - v_{D_{j} q}^{-} \end{matrix}$ .

(4) Calculate the relative closeness coefficient of suppliers and demanders. The relative closeness coefficient α_ij between the demander D_j evaluated by supplier S_i and the positive ideal solution $N_{S_{i}}^{+}$ is denoted by $α_{ij} = \frac{d_{S_{i}} (D_{j}, N_{S_{i}}^{-})}{d_{S_{i}} (D_{j}, N_{S_{i}}^{+}) + d_{S_{i}} (D_{j}, N_{S_{i}}^{-})}$ (14)

Likewise, the relative closeness coefficient β_ij between the supplier S_i evaluated by demander D_j and the positive ideal solution $N_{D_{j}}^{+}$ can be calculated by $β_{ij} = \frac{d_{D_{j}} (P_{i}, N_{D_{j}}^{-})}{d_{D_{j}} (P_{i}, N_{D_{j}}^{+}) + d_{D_{j}} (P_{i}, N_{D_{j}}^{-})}$ (15)

The greater the relative closeness coefficient α_ij of the supplier S_i in technology trading, the more satisfied S_i is with the demander D_j. Similarly, the greater the relative closeness coefficient β_ij of the demander D_j, the more satisfied D_j is with the supplier S_i. According to the analyses above, the overall satisfaction degree α_ij of the supplier S_i to demander D_j and the overall satisfaction degree β_ij of the demander D_j to technology T_i can be obtained. On this basis, the overall satisfaction matrix of suppliers and demanders can be constructed as ∂ = [α_ij] _m×n and ℘ = [β_ij] _m×n respectively.

4.2 Develop and solve the stable matching model

As a third party in technology trading, an important service provided by intermediaries is to search for the optimal matching pairs of suppliers and demanders based on the evaluation information. The satisfaction of two-side agents and the stability of matching pairs are regarded as indispensable criteria for measuring the quality of matching schemes [43–44], and they are also the primary considerations for constructing the optimization model in this paper. Let x_ij be a binary decision variable, where x_ij = 1 indicates that the supplier S_i and demander D_jare matched to form a matching pair or x_ij = 0 otherwise. The optimization model of technology trading matching can be established as: $\max Z_{1} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} α_{ij} x_{ij}$ (16a) $\max Z_{2} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} β_{ij} x_{ij}$ (16b) $s . t . \sum_{j = 1}^{n} x_{ij} \leq 1, i = 1, 2, \dots, m$ (16c) $\sum_{i = 1}^{m} x_{ij} \leq 1, j = 1, 2, \dots, n$ (16d) $\begin{array}{l} \sum_{β i h < α i j} x_{i h} + \sum_{β k < β i j} x_{k j} + x_{i j} \geq 1, \\ i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{array}$ (16e) $x_{ij} \in {0, 1}, i = 1, 2, \dots, m; j = 1, 2, \dots, n$ (16f)

In the above model (16a)–(16f), the objective function (16a) is to maximize the satisfaction of all suppliers. The objective function (16b) stands for maximizing the satisfaction of all demanders. Constraint (16c) shows that each technological achievement can be traded with at most one demander. Constraint (16d) indicates that each technology demander can purchase at most one technological achievement. Equation (16e) is a stable matching constraint for both sides of the technology trading.

The model formulated in Equations (16a)–(16f) is a multi-objective optimization model. Since the scales of the objective function (16a) and (16b) are the same, the model can be solved by the linear weighted method. The multi-objective model composed of (16a)–(16f) is transformed into a single-objective model as follows: $\max Z = \sum_{i = 1}^{m} \sum_{j = 1}^{n} (w_{1} α_{ij} + w_{2} β_{ij}) x_{ij}$ (17a) $s . t . \sum_{j = 1}^{n} x_{ij} \leq 1, i = 1, 2, \dots, m$ (17b) $\sum_{i = 1}^{m} x_{ij} \leq 1, j = 1, 2, \dots, n$ (17c) $\begin{array}{l} \sum_{β i h < α i j} x_{i h} + \sum_{β k < β i j} x_{k j} + x_{i j} \geq 1, \\ i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{array}$ (17d) $x_{ij} \in {0, 1}, i = 1, 2, \dots, m; j = 1, 2, \dots, n$ (17e) $w_{1} + w_{2} = 1, 0 \leq w_{1}, w_{2} \leq 1$ (17f)

For the optimization model (17a)–(17f), w₁ and w₂ are the weight coefficients of objective functions Z₁ and Z₂ respectively. The w₁ and w₂ are usually determined by technology trading intermediaries. If w₁ > w₂, it indicates that the supply side is more important, and the intermediary will make matching decisions that are more beneficial to the suppliers. If w₁ < w₂, it means that the intermediary pays more attention to the needs of the demand side, and the matching result is more in line with the interests of demanders. w₁ = w₂ shows that the intermediary regards both sides of the technology trading as equally important. The model (17a)–(17f) is not a complicated mathematical model. Accordingly, many optimization software packages, such as Cplex, Lingo, etc., can be employed to solve the linear programming model.

4.3 Decision process of technology trading matching

In summary, the algorithm step for addressing the technology trading matching problem with intuitionistic fuzzy multi-attribute information is presented as follows.

Step 1: Obtain the intuitionistic fuzzy preference information matrix $A_{i} = [a_{ij}^{p}]_{n \times f}$ of the technology supplier A_i and the intuitionistic fuzzy preference information matrix $B_{j} = [b_{ij}^{q}]_{m \times h}$ of the technology demander B_j.

Step 2: Calculate the weighted intuitionistic fuzzy preference matrix $R_{i} = [r_{ij}^{p}]_{n \times f}$ and $T_{j} = [t_{ij}^{q}]_{m \times h}$ using Equations (4) and (5).

Step 3: Determine the intuitionistic fuzzy positive and negative ideal solutions $N_{S_{i}}^{+}$ , $N_{S_{i}}^{-}$ , $N_{D_{j}}^{+}$ and $N_{D_{j}}^{-}$ using Equations (6)–(9).

Step 4: Calculate the Euclidean distance $d_{S_{i}} (D_{j}, N_{S_{i}}^{+})$ , $d_{S_{i}} (D_{j}, N_{S_{i}}^{-})$ , $d_{D_{j}} (S_{i}, N_{D_{j}}^{+})$ and $d_{D_{j}} (S_{i}, N_{D_{j}}^{-})$ using Equations (10)–(13).

Table 1
Evaluation information provided by suppliers

D ₁ D ₂ D ₃ D ₄ D ₅ D ₆ D ₇ D ₈

S ₁ C ₁ (0.3,0.5) (0.8,0.1) (0.3,0.6) (0.2,0.6) (0.8,0.1) (0.5,0.3) (0.3,0.4) (0.5,0.2)

C ₂ (0.2,0.2) (0.4,0.6) (0.5,0.1) (0.2,0.5) (0.6,0.3) (0.3,0.6) (0.8,0.0) (0.4,0.4)

C ₃ (0.5,0.5) (0.7,0.2) (0.4,0.6) (0.2,0.2) (0.3,0.3) (0.9,0.1) (0.0,0.3) (0.5,0.5)

C ₄ (0.4,0.6) (0.8,0.1) (0.4,0.2) (0.3,0.5) (0.5,0.3) (0.7,0.2) (0.8,0.2) (0.6,0.2)

S ₂ C ₁ (0.3,0.7) (0.4,0.4) (0.3,0.7) (0.3,0.0) (0.4,0.6) (0.5,0.1) (0.4,0.6) (0.3,0.6)

C ₂ (0.4,0.5) (0.5,0.0) (0.5,0.2) (0.6,0.2) (0.2,0.4) (0.4,02) (0.7,0.2) (0.6,0.1)

C ₃ (0.7,0.2) (0.6,0.4) (0.8,0.0) (0.5,0.2) (0.4,0.3) (0.3,0.2) (0.8,0.2) (0.3,0.3)

C ₄ (0.6,04) (0.3,0.6) (0.6,0.4) (0.8,0.2) (0.5,0.5) (0.6,0.1) (0.6,0.1) (0.7,0.0)

S ₃ C ₁ (0.8,0.2) (0.8,0.0) (0.4,0.4) (0.4,0.6) (0.5,0.2) (0.3,0.3) (0.5,0.4) (0.6,0.1)

C ₂ (0.6,0.4) (0.0,0.3) (0.9,0.1) (0.3,0.5) (0.8,0.2) (0.5,0.3) (0.4,0.5) (0.8,0.0)

C ₃ (0.5,0.4) (0.5,0.4) (0.2,0.6) (0.2,0.6) (0.7,0.0) (0.8,0.2) (0.3,0.3) (0.7,0.3)

C ₄ (0.5,0.0) (0.8,0.2) (0.5,0.2) (0.5,0.3) (0.6,0.2) (0.4,0.4) (0.5,0.5) (0.3,0.4)

S ₄ C ₁ (0.7,0.2) (0.6,0.2) (0.6,0.2) (0.4,0.0) (0.5,0.1) (0.6,0.4) (0.4,0.4) (0.5,0.5)

C ₂ (0.0,0.4) (0.5,0.1) (0.7,0.1) (0.8,0.2) (0.4,0.2) (0.3,0.2) (0.6,0.1) (0.5,0.2)

C ₃ (0.8,0.2) (0.7,0.2) (0.3,0.6) (0.4,0.4) (0.6,0.1) (0.5,0.1) (0.9,0.1) (0.4,0.6)

C ₄ (0.6,0.0) (0.3,0.5) (0.5,0.4) (0.3,0.4) (0.8,0.2) (0.3,0.3) (0.1,0.3) (0.4,0.2)

S ₅ C ₁ (0.4,0.5) (0.7,0.1) (0.3,0.3) (0.5,0.1) (0.6,0.3) (0.5,0.5) (0.2,0.6) (0.6,0.1)

C ₂ (0.6,0.3) (0.5,0.1) (0.7,0.1) (0.6,0.1) (0.3,0.6) (0.2,0.6) (0.3,0.5) (0.5,0.2)

C ₃ (0.6,0.2) (0.5,0.4) (0.8,0.2) (0.3,0.2) (0.5,0.4) (0.6,0.0) (0.5,0.1) (0.3,0.2)

C ₄ (0.5,0.5) (0.7,0.2) (0.9,0.0) (0.6,0.2) (0.1,0.2) (0.7,0.2) (0.6,0.2) (0.5,0.2)

Step 5: Calculate the relative closeness coefficient α_ij and β_ij using Equations (14)–(15).

Step 6: Build the multi-objective optimization model (16a)–(16f) of technology trading matching based on the overall satisfaction matrix ∂ = [α_ij] _m×n and ℘ = [β_ij] _m×n.

Step 7: Transform the multi-objective optimization model (16a)–(16f) into the single-objective model (17a)–(17f) using the linear weighted method.

Step 8: Solve the single-objective model (17a)–(17f) to obtain the optimal matching scheme of technology trading.

5 Numerical example

Table 2
Evaluation information provided by demanders

D ₁ D ₂ D ₃ D ₄ D ₅ D ₆ D ₇ D ₈

S ₁ E ₁ (0.5,0.3) (0.4,0.6) (0.7,0.2) (0.3,0.3) (0.4,0.4) (0.7,0.3) (0.8,0.0) (0.3,0.4)

E ₂ (0.6,0.1) (0.2,0.2) (0.4,0.3) (0.6,0.4) (0.7,0.2) (0.3,0.3) (0.5,0.5) (0.6,0.1)

E ₃ (0.8,0.1) (0.2,0.5) (0.6,0.2) (0.4,0.4) (0.3,0.4) (0.7,0.2) (0.5,0.4) (0.6,0.3)

E ₄ (0.8,0.0) (0.6,0.4) (0.9,0.1) (0.5,0.4) (0.3,0.5) (0.5,0.2) (0.2,0.6) (0.7,0.1)

E ₅ (0.5,0.3) (0.2,0.6) (0.7,0.2) (0.4,0.3) (0.3,0.2) (0.1,0.5) (0.2,0.7) (0.6,0.2)

S ₂ E ₁ (0.6,0.2) (0.4,0.5) (0.6,0.1) (0.8,0.2) (0.5,0.5) (0.4,0.6) (0.3,0.7) (0.2,0.8)

E ₂ (0.7,0.3) (0.3,0.3) (0.7,0.2) (0.5,0.3) (0.3,0.4) (0.5,0.0) (0.3,0.3) (0.4,0.5)

E ₃ (0.7,0.3) (0.8,0.2) (0.6,0.0) (0.5,0.2) (0.4,0.5) (0.2,0.2) (0.1,0.6) (0.4,0.3)

E ₄ (0.6,0.0) (0.7,0.3) (0.4,0.2) (0.7,0.3) (0.8,0.2) (0.2,0.0) (0.5,0.3) (0.0,0.3)

E ₅ (0.6,0.3) (0.5,0.0) (0.3,0.2) (0.2,0.1) (0.6,0.0) (0.5,0.2) (0.6,0.1) (0.2,0.4)

S ₃ E ₁ (0.5,0.4) (0.6,0.0) (0.2,0.7) (0.0,0.3) (0.6,0.0) (0.4,0.3) (0.5,0.5) (0.6,0.4)

E ₂ (0.4,0.0) (0.2,0.3) (0.6,0.4) (0.5,0.3) (0.5,0.2) (0.1,0.1) (0.4,0.3) (0.7,0.1)

E ₃ (0.5,0.5) (0.2,0.8) (0.7,0.2) (0.3,0.0) (0.4,0.4) (0.2,0.5) (0.4,0.3) (0.1,0.4)

E ₄ (0.3,0.3) (0.1,0.3) (0.2,0.6) (0.4,0.0) (0.6,0.1) (0.5,0.5) (0.4,0.6) (0.7,0.3)

E ₅ (0.4,0.3) (0.4,0.0) (0.2,0.2) (0.6,0.1) (0.5,0.3) (0.3,0.3) (0.3,0.5) (0.8,0.2)

S ₄ E ₁ (0.5,0.2) (0.3,0.3) (0.0,0.3) (0.1,0.6) (0.2,0.4) (0.5,0.4) (0.4,0.0) (0.2,0.3)

E ₂ (0.5,0.1) (0.4,0.4) (0.3,0.6) (0.6,0.3) (0.2,0.3) (0.2,0.4) (0.4,0.3) (0.5,0.3)

E ₃ (0.6,0.3) (0.5,0.5) (0.4,0.2) (0.1,0.3) (0.3,0.7) (0.6,0.4) (0.6,0.2) (0.5,0.0)

E ₄ (0.2,0.3) (0.5,0.2) (0.6,0.1) (0.7,0.3) (0.4,0.0) (0.0,0.2) (0.8,0.1) (0.6,0.1)

E ₅ (0.3,0.3) (0.2,0.4) (0.5,0.5) (0.2,0.7) (0.4,0.6) (0.2,0.8) (0.6,0.3) (0.4,0.4)

S ₅ E ₁ (0.8,0.2) (0.5,0.1) (0.4,0.2) (0.3,0.3) (0.4,0.0) (0.0,0.4) (0.3,0.3) (0.5,0.5)

E ₂ (0.5,0.4) (0.2,0.4) (0.6,0.1) (0.4,0.4) (0.2,0.2) (0.5,0.3) (0.2,0.5) (0.1,0.4)

E ₃ (0.5,0.3) (0.2,0.0) (0.4,0.3) (0.6,0.2) (0.2,0.4) (0.7,0.3) (0.5,0.5) (0.1,0.3)

E ₄ (0.0,0.3) (0.2,0.4) (0.6,0.4) (0.5,0.5) (0.6,0.4) (0.2,0.5) (0.1,0.4) (0.2,0.6)

E ₅ (0.2,0.4) (0.3,0.3) (0.5,0.2) (0.6,0.3) (0.1,0.7) (0.5,0.4) (0.4,0.0) (0.5,0.4)

		D ₁	D ₂	D ₃	D ₄	D ₅	D ₆	D ₇	D ₈
S ₁	E ₁	(0.5,0.3)	(0.4,0.6)	(0.7,0.2)	(0.3,0.3)	(0.4,0.4)	(0.7,0.3)	(0.8,0.0)	(0.3,0.4)
	E ₂	(0.6,0.1)	(0.2,0.2)	(0.4,0.3)	(0.6,0.4)	(0.7,0.2)	(0.3,0.3)	(0.5,0.5)	(0.6,0.1)
	E ₃	(0.8,0.1)	(0.2,0.5)	(0.6,0.2)	(0.4,0.4)	(0.3,0.4)	(0.7,0.2)	(0.5,0.4)	(0.6,0.3)
	E ₄	(0.8,0.0)	(0.6,0.4)	(0.9,0.1)	(0.5,0.4)	(0.3,0.5)	(0.5,0.2)	(0.2,0.6)	(0.7,0.1)
	E ₅	(0.5,0.3)	(0.2,0.6)	(0.7,0.2)	(0.4,0.3)	(0.3,0.2)	(0.1,0.5)	(0.2,0.7)	(0.6,0.2)
S ₂	E ₁	(0.6,0.2)	(0.4,0.5)	(0.6,0.1)	(0.8,0.2)	(0.5,0.5)	(0.4,0.6)	(0.3,0.7)	(0.2,0.8)
	E ₂	(0.7,0.3)	(0.3,0.3)	(0.7,0.2)	(0.5,0.3)	(0.3,0.4)	(0.5,0.0)	(0.3,0.3)	(0.4,0.5)
	E ₃	(0.7,0.3)	(0.8,0.2)	(0.6,0.0)	(0.5,0.2)	(0.4,0.5)	(0.2,0.2)	(0.1,0.6)	(0.4,0.3)
	E ₄	(0.6,0.0)	(0.7,0.3)	(0.4,0.2)	(0.7,0.3)	(0.8,0.2)	(0.2,0.0)	(0.5,0.3)	(0.0,0.3)
	E ₅	(0.6,0.3)	(0.5,0.0)	(0.3,0.2)	(0.2,0.1)	(0.6,0.0)	(0.5,0.2)	(0.6,0.1)	(0.2,0.4)
S ₃	E ₁	(0.5,0.4)	(0.6,0.0)	(0.2,0.7)	(0.0,0.3)	(0.6,0.0)	(0.4,0.3)	(0.5,0.5)	(0.6,0.4)
	E ₂	(0.4,0.0)	(0.2,0.3)	(0.6,0.4)	(0.5,0.3)	(0.5,0.2)	(0.1,0.1)	(0.4,0.3)	(0.7,0.1)
	E ₃	(0.5,0.5)	(0.2,0.8)	(0.7,0.2)	(0.3,0.0)	(0.4,0.4)	(0.2,0.5)	(0.4,0.3)	(0.1,0.4)
	E ₄	(0.3,0.3)	(0.1,0.3)	(0.2,0.6)	(0.4,0.0)	(0.6,0.1)	(0.5,0.5)	(0.4,0.6)	(0.7,0.3)
	E ₅	(0.4,0.3)	(0.4,0.0)	(0.2,0.2)	(0.6,0.1)	(0.5,0.3)	(0.3,0.3)	(0.3,0.5)	(0.8,0.2)
S ₄	E ₁	(0.5,0.2)	(0.3,0.3)	(0.0,0.3)	(0.1,0.6)	(0.2,0.4)	(0.5,0.4)	(0.4,0.0)	(0.2,0.3)
	E ₂	(0.5,0.1)	(0.4,0.4)	(0.3,0.6)	(0.6,0.3)	(0.2,0.3)	(0.2,0.4)	(0.4,0.3)	(0.5,0.3)
	E ₃	(0.6,0.3)	(0.5,0.5)	(0.4,0.2)	(0.1,0.3)	(0.3,0.7)	(0.6,0.4)	(0.6,0.2)	(0.5,0.0)
	E ₄	(0.2,0.3)	(0.5,0.2)	(0.6,0.1)	(0.7,0.3)	(0.4,0.0)	(0.0,0.2)	(0.8,0.1)	(0.6,0.1)
	E ₅	(0.3,0.3)	(0.2,0.4)	(0.5,0.5)	(0.2,0.7)	(0.4,0.6)	(0.2,0.8)	(0.6,0.3)	(0.4,0.4)
S ₅	E ₁	(0.8,0.2)	(0.5,0.1)	(0.4,0.2)	(0.3,0.3)	(0.4,0.0)	(0.0,0.4)	(0.3,0.3)	(0.5,0.5)
	E ₂	(0.5,0.4)	(0.2,0.4)	(0.6,0.1)	(0.4,0.4)	(0.2,0.2)	(0.5,0.3)	(0.2,0.5)	(0.1,0.4)
	E ₃	(0.5,0.3)	(0.2,0.0)	(0.4,0.3)	(0.6,0.2)	(0.2,0.4)	(0.7,0.3)	(0.5,0.5)	(0.1,0.3)
	E ₄	(0.0,0.3)	(0.2,0.4)	(0.6,0.4)	(0.5,0.5)	(0.6,0.4)	(0.2,0.5)	(0.1,0.4)	(0.2,0.6)
	E ₅	(0.2,0.4)	(0.3,0.3)	(0.5,0.2)	(0.6,0.3)	(0.1,0.7)	(0.5,0.4)	(0.4,0.0)	(0.5,0.4)

5.1 Example illustration and matching process

Jiangsu Proprietary Technology Exchange Center (JTEC) is an online service platform that integrates technology suppliers and demanders as well as professional service institutions. It is positioned as a “nationally leading and internationally influential” provincial-level technology trading market. JTEC can provide services such as supply and demand information release, information query, value evaluation of scientific and technological achievements, and coordinating trading between both sides. The center is dedicated to addressing the prevalent challenges in the transformation of scientific and technological achievements, particularly issues related to discovery (“unable to find”), negotiation (“can’t agree on”), and execution (“difficult to implement”).

In a certain period, eight demanders {D₁, D₂, ⋯ , D₈} submitted technical requirement information to JTEC. The platform screened out five qualified technologies {T₁, T₂, ⋯ , T₅} owned by suppliers {S₁, S₂, ⋯ , S₅} through a preliminary search of the technical achievement database. Based on data analysis of successful technology trading in the past, JTEC has found that the criteria considered by suppliers when selecting the demanders for trading are usually expected returns C₁, enterprise technology application capabilities C₂, market development and marketing capabilities C₃, and expected technical goals of the enterprise C₄. When selecting suitable scientific and technological achievements, demanders typically prioritize several attributes. These include the maturity of the scientific and technological achievements E₁, the technical level E₂, production and usage conditions E₃, market application prospects E₄ and the expected trading methods of the seller E₅. To achieve the optimal matching of technology supply and demand, JTEC implements the subsequent decision-making steps.

Table 3
Weight of evaluation attributes

C ₁ C ₂ C ₃ C ₄ E ₁ E ₂ E ₃ E ₄ E ₅

S ₁ 0.2 0.3 0.4 0.1 — — — — —

S ₂ 0.5 0.2 0.2 0.1 — — — — —

S ₃ 0.3 0.3 0.2 0.2 — — — — —

S ₄ 0.4 0.2 0.2 0.2 — — — — —

S ₅ 0.1 0.2 0.5 0.2 — — — — —

D ₁ — — — — 0.2 0.3 0.1 0.2 0.2

D ₂ — — — — 0.1 0.2 0.2 0.4 0.1

D ₃ — — — — 0.5 0.1 0.1 0.2 0.1

D ₄ — — — — 0.2 0.1 0.2 0.2 0.3

D ₅ — — — — 0.2 0.3 0.1 0.1 0.3

D ₆ — — — — 0.1 0.2 0.1 0.3 0.3

D ₇ — — — — 0.1 0.4 0.3 0.1 0.1

D ₈ — — — — 0.2 0.2 0.2 0.2 0.2

	C ₁	C ₂	C ₃	C ₄	E ₁	E ₂	E ₃	E ₄	E ₅
S ₁	0.2	0.3	0.4	0.1	—	—	—	—	—
S ₂	0.5	0.2	0.2	0.1	—	—	—	—	—
S ₃	0.3	0.3	0.2	0.2	—	—	—	—	—
S ₄	0.4	0.2	0.2	0.2	—	—	—	—	—
S ₅	0.1	0.2	0.5	0.2	—	—	—	—	—
D ₁	—	—	—	—	0.2	0.3	0.1	0.2	0.2
D ₂	—	—	—	—	0.1	0.2	0.2	0.4	0.1
D ₃	—	—	—	—	0.5	0.1	0.1	0.2	0.1
D ₄	—	—	—	—	0.2	0.1	0.2	0.2	0.3
D ₅	—	—	—	—	0.2	0.3	0.1	0.1	0.3
D ₆	—	—	—	—	0.1	0.2	0.1	0.3	0.3
D ₇	—	—	—	—	0.1	0.4	0.3	0.1	0.1
D ₈	—	—	—	—	0.2	0.2	0.2	0.2	0.2

Step 1: Obtain preference information from both technology suppliers and demanders.

JTEC disseminates information regarding potential suppliers and demanders to their respective counterparts, subsequently soliciting their feedback on various evaluation attributes. Given the inherent challenges in acquiring comprehensive and precise trading details from the opposing party, both sets of agents typically offer evaluation information that is inherently fuzzy with respect to the criteria. The intuitionistic fuzzy satisfaction evaluation data furnished by the technology trading suppliers and demanders are delineated in Tables 1 and 2, respectively. Concurrently, the weight assigned to each evaluation criterion, as provided by both parties, is encapsulated in Table 3.

Step 2: Get overall satisfaction from both supply and demand sides. Firstly, the weighted intuitionistic fuzzy satisfaction matrixs $R_{i} = [r_{ij}^{p}]_{n \times f}$ and $T_{j} = [t_{ij}^{q}]_{m \times h}$ are calculated by Equations (4) and (5). Then, the positive and negative ideal solutions of suppliers and demanders are determined by Equations

(6)–(9) are shown in Tables 4 and 5. The Euclidean distance can be derived based on Equations (10)–(13). On this basis, the relative closeness (overall satisfaction) between two-sided agents and the positive ideal solution can be generated, as shown in Tables 6 and 7.

Step 3: The optimization model (16a)–(16f) of supply and demand matching for five technology suppliers and eight demanders is built based on the overall satisfaction matrix ∂ = [α_ij] _m×n and ℘ = [β_ij] _m×n.

Step 4: To ensure the matching fairness of suppliers and demanders in technology trading, the weight coefficient w₁ and w₂ are set to w₁ = w₂ = 0.5. The model (16a)- (16f) is transformed into the single-objective linear optimization model (17a)–(17f).

Step 5: Using Lingo software to solve the single objective linear programming model, the optimal solution is x₁₄ = x₂₅ = x₃₇ = x₄₆ = x₅₈ = 1 and other decision variables are x_ij = 0. It can be seen that the optimal stable matching scheme of technology trading is $μ_{1}^{*} = {(S_{1}, D_{4}), (S_{2}, D_{5}), (S_{3}, D_{7}), (S_{4}, D_{6}),$ (S₅, D₈)}. This means the demander D₄ and supplier S₁ form a potential cooperation intention on the technology T₁, other matching pairs, and so on.

5.2 Sensitivity analysis

We conduct sensitivity analysis of the weight coefficient on the matching results of technology trading. For the optimization model (17a)- (17f), the weight coefficients w₁ and w₂ reflect the intermediary’s different preferences on technology trading demanders and suppliers. Concretely, w₁ > w₂ and w₁ < w₂ respectively indicate that intermediaries prioritize the satisfaction of technology demanders and suppliers. When (w₁, w₂) takes (0.0,1.0), (0.1,0.9), (0.2,0.8), (0.3,0.7), (0.4,0.6), (0.5,0.5), (0.6,0.4), (0.7,0.3), (0.8,0.2), (0.9,0.1), (1.0,0.0) in turn, the optimal technology trading matching scheme obtained by solving the optimization model (17a)- (17f) is all $μ_{S}^{*} = {(S_{1}, D_{4}), (S_{2}, D_{5}), (S_{3}, D_{7}), (S_{4}, D_{6}), (S_{5}, D_{8})}$ . This shows that different preferences of intermediaries for technology demanders and suppliers will not affect the matching scheme of technology trading, and further reflects the stability of the matching scheme obtained by the decision-making method proposed in this paper.

5.3 Comparison analysis

The models constructed in the existing technology trading matching research [10 –14] have mainly focused on optimizing the satisfaction of suppliers and demanders, without considering the impact of stability on trading matching. In what follows, we compare the stable matching method proposed in this paper with the unstable matching method. From sensitivity analysis, it can be seen that the optimal trading matching scheme obtained using the stable matching decision method in this paper is the same for different weight coefficient combinations of w₁ and w₂. The overall satisfaction of the technology suppliers and demanders for $μ_{S}^{*}$ is Z₁ = 0.9007and Z₂ = 1.0847, respectively. The optimal matching scheme obtained using the unstable matching method of existing research is shown in Table 8.

Upon comparative analysis, it becomes evident that the technology trading matching scheme derived from the decision-making method introduced in this paper boasts pronounced stability. In contrast, the unstable matching method yields multiple matching schemes contingent on varying weight coefficients.

Table 4
Positive and negative ideal solutions of suppliers

$N_{S_{i}}^{+}$ $N_{S_{i}}^{-}$

C ₁ C ₂ C ₃ C ₄ C ₁ C ₂ C ₃ C ₄

S ₁ (0.2752, 0.6310) (0.3830, 0.0000) (0.6019, 0.3981) (0.1487, 0.7943) (0.0436, 0.9029) (0.0648, 0.8579) (0.0000, 0.8152) (0.0350, 0.9502)

S ₂ (0.2929, 0.0000) (0.2140, 0.0000) (0.2752, 0.0000) (0.1487, 0.0000) (0.1633, 0.8367) (0.0436, 0.8706) (0.0689, 0.83269) (0.0350, 0.9502)

S ₃ (0.3830, 0.0000) (0.4988, 0.0000) (0.2752, 0.0000) (0.2752, 0.0000) (0.1015, 0.8579) (0.0000, 0.8123) (0.0436, 0.9029) (0.0689, 0.8706)

S ₄ (0.3822, 0.3981) (0.2140, 0.6310) (0.3960, 0.6310) (0.2752, 0.0000) (0.1848, 0.75759) (0.0000, 0.8326) (0.0689, 0.9029) (0.0209, 0.8706)

S ₅ (0.1134, 0.7943) (0.2140, 0.6310) (0.5528, 0.0000) (0.3690, 0.0000) (0.0221, 0.9502) (0.0221, 0.9502) (0.0689, 0.8326) (0.0209, 0.8706)

	$N_{S_{i}}^{+}$				$N_{S_{i}}^{-}$
S ₁	(0.2752, 0.6310)	(0.3830, 0.0000)	(0.6019, 0.3981)	(0.1487, 0.7943)	(0.0436, 0.9029)	(0.0648, 0.8579)	(0.0000, 0.8152)	(0.0350, 0.9502)
S ₂	(0.2929, 0.0000)	(0.2140, 0.0000)	(0.2752, 0.0000)	(0.1487, 0.0000)	(0.1633, 0.8367)	(0.0436, 0.8706)	(0.0689, 0.83269)	(0.0350, 0.9502)
S ₃	(0.3830, 0.0000)	(0.4988, 0.0000)	(0.2752, 0.0000)	(0.2752, 0.0000)	(0.1015, 0.8579)	(0.0000, 0.8123)	(0.0436, 0.9029)	(0.0689, 0.8706)
S ₄	(0.3822, 0.3981)	(0.2140, 0.6310)	(0.3960, 0.6310)	(0.2752, 0.0000)	(0.1848, 0.75759)	(0.0000, 0.8326)	(0.0689, 0.9029)	(0.0209, 0.8706)
S ₅	(0.1134, 0.7943)	(0.2140, 0.6310)	(0.5528, 0.0000)	(0.3690, 0.0000)	(0.0221, 0.9502)	(0.0221, 0.9502)	(0.0689, 0.8326)	(0.0209, 0.8706)

Table 5

Positive and negative ideal solutions of demanders

	$N_{D_{j}}^{+}$					$N_{D_{j}}^{-}$
	E ₁	E ₂	E ₃	E ₄	E ₅	E ₁	E ₂	E ₃	E ₄	E ₅
D ₁	(0.2752, 0.7248)	(0.3032, 0.0000)	(0.1487, 0.7943)	(0.2752, 0.0000)	(0.1674, 0.7860)	(0.1294, 0.8326,	(0.1421, 0.7597)	(0.0670, 0.9330)	(0.0000, 0.7860)	(0.0436, 0.8326)
D ₂	(0.0876, 0.0000)	(0.0971, 0.7248)	(0.2752, 0.0000)	(0.3822, 0.5253)	(0.0670, 0.0000)	(0.0350, 0.9502)	(0.0436, 0.8326)	(0.0436, 0.9564)	(0.0413, 0.6931)	(0.0221, 0.9502)
D ₃	(0.4623, 0.3162)	(0.1134, 0.7943)	(0.1134, 0.0000)	(0.3690, 0.6310)	(0.1134, 0.8513)	(0.0000, 0.8367)	(0.0350, 0.9502)	(0.0498, 0.8866)	(0.0436, 0.9029)	(0.0221. 0.9330)
D ₄	(0.2752, 0.7248)	(0.0876, 0.8866)	(0.1674, 0.0000)	(0.2140, 0.0000)	(0.2403, 0.5012)	(0.0000, 0.9029)	(0.0498, 0.9124)	(0.0209, 0.8326)	(0.0971, 0.8706)	(0.0648, 0.8985)
D ₅	(0.1674, 0.0000)	(0.3032, 0.6170)	(0.0498, 0.9124)	(0.1487, 0.0000)	(0.2403, 0.0000)	(0.0436, 0.8706)	(0.0648, 0.6968)	(0.0221, 0.9650)	(0.0350, 0.9330)	(0.0311. 0.8985)
D ₆	(0.1134, 0.8866)	(0.1294, 0.0000)	(0.1134, 0.8513)	(0.1877, 0.0000)	(0.1877, 0.6170)	(0.0000, 0.9502)	(0.0209, 0.8326)	(0.0221, 0.9330)	(0.0000, 0.8123)	(0.0311, 0.9352)
D ₇	(0.1487, 0.0000)	(0.2421, 0.6178)	(0.2403, 0.6170)	(0.1487, 0.7943)	(0.0876, 0.0000)	(0.0350, 0.9650)	(0.0854, 0.7579)	(0.0311, 0.8579)	(0.0105, 0.9502)	(0.0221, 0.9650)
D ₈	(0.1674, 0.7860)	(0.2140, 0.6310)	(0.1674, 0.0000)	(0.2140, 0.6310)	(0.2752, 0.7248)	(0.0436, 0.9564)	(0.0209, 0.8706)	(0.0209, 0.8326)	(0.0000, 0.9029)	(0.0436, 0.8326)

Table 6

Relative closeness coefficient of suppliers

	D ₁	D ₂	D ₃	D ₄	D ₅	D ₆	D ₇	D ₈
S ₁	0.3199	0.3668	0.3639	0.2269	0.3185	0.4167	0.5804	0.2758
S ₂	0.0853	0.3901	0.3587	0.3990	0.0532	0.2927	0.1606	0.4439
S ₃	0.4317	0.3923	0.2561	0.0811	0.4714	0.1877	0.1504	0.4032
S ₄	0.7522	0.2815	0.2689	0.2689	0.3821	0.2686	0.3255	0.1922
S ₅	0.3096	0.3251	0.6698	0.2235	0.1463	0.4953	0.2040	0.2016

Table 7

Relative closeness coefficient of demanders

	D ₁	D ₂	D ₃	D ₄	D ₅	D ₆	D ₇	D ₈
S ₁	0.6871	0.2235	0.4690	0.2197	0.2035	0.2119	0.4931	0.3254
S ₂	0.5097	0.4622	0.7862	0.3135	0.4041	0.8734	0.1405	0.1550
S ₃	0.5051	0.5896	0.1293	0.8088	0.1844	0.2588	0.1335	0.3211
S ₄	0.2409	0.1311	0.2854	0.0955	0.4475	0.1766	0.5294	0.7734
S ₅	0.1277	0.4417	0.3002	0.1831	0.4097	0.1662	0.5240	0.1508

Such an observation underscores that stability amplifies the constraints of the optimization model, thereby curtailing the number of optimal matching schemes. Furthermore, this suggests that variances in intermediary preferences towards the involved parties in technology trading do not influence the trading matching outcomes. Consequently, the stable trading matching model presented in this study demonstrates enhanced robustness.

Fig. 2

Satisfaction comparison of matching results.

Table 8

Unstable matching results

(w₁, w₂)	Matching scheme	Matching pairs	Z ₁	Z ₂
(0.0,1.0)	$μ_{1}^{*}$	S₁ ↔ D₁, S₂ ↔ D₆, S₃ ↔ D₄, S₄ ↔ D₈, S₅ ↔ D₇	1.0899	3.6667
(0.1,0.9)
(0.2,0.8)
(0.3,0.7)
(0.4,0.6)	$μ_{2}^{*}$	S₁ ↔ D₇, S₂ ↔ D₆, S₃ ↔ D₄, S₄ ↔ D₈, S₅ ↔ D₃	1.8162	3.2489
(0.5,0.5)
(0.6,0.4)
(0.7,0.3)
(0.8,0.2)	$μ_{3}^{*}$	S₁ ↔ D₇, S₂ ↔ D₆, S₃ ↔ D₂, S₄ ↔ D₁, S₅ ↔ D₃	2.6874	2.4972
(0.9,0.1)	$μ_{4}^{*}$	S₁ ↔ D₇, S₂ ↔ D₈, S₃ ↔ D₅, S₄ ↔ D₁, S₅ ↔ D₃	2.9177	1.3736
(1.0,0.0)

From a satisfaction standpoint, the cumulative satisfaction of agents in a stable matching scheme is somewhat diminished compared to that in an unstable matching scheme. This suggests that achieving stable trading matching comes at a cost, potentially necessitating a compromise in the satisfaction levels of both trading parties. Figure 2 offers a lucid and direct visualization of the satisfaction disparities among each matched agent. Consequently, it can be inferred that the pursuit of stability in technology trading matching may invariably lead to a marginal reduction in mutual satisfaction. However, if we do not care about matching stability, there will be unstable matching pairs in the optimal trading matching scheme. Taking the unstable matching scheme $μ_{3}^{*}$ as an example, (S₃, D₁) is an unstable matching pair for $μ_{3}^{*}$ . In this case, to obtain higher satisfaction, both S₃ and D₁ will have an incentive to violate the original trading matching scheme and form a new trading cooperation relationship between them. As a result, the potential trading formed by the original matching scheme is invalid. Therefore, in the process of technology trading matching, the stability between suppliers and demanders cannot be ignored.

6 Conclusions

This research introduces a stable matching decision-making method tailored to the technology trading challenge, characterized by multi-attribute and intuitionistic fuzzy information from both suppliers and demanders. By integrating the TOPSIS approach with intuitionistic fuzzy set theory, we effectively aggregate the satisfaction of technology traders. Furthermore, we construct an innovative multi-objective optimization model that emphasizes both the stability and satisfaction of technology traders. The linear weighted method is employed to pinpoint optimal technology trading pairs. To validate the practicality of our approach, we apply the two-sided matching method, as delineated in this study, to a real-world scenario involving the JTEC platform. Sensitivity analysis with weights for different objective functions and the comparative analysis with the unstable matching method is conducted.

Although the TOPSIS method is widely used in MADM problems, its deployment in measuring the satisfaction of technology traders— where two-sided agents employ multi-attribute intuitionistic fuzzy information to evaluate their counterparts— remains underexplored. This research stands as a pioneering effort in melding intuitive fuzzy set theory with TOPSIS for the realm of technology trade matching. Empirical evidence underscores the effectiveness of the intuitionistic fuzzy TOPSIS method in addressing the satisfaction evaluation intricacies inherent to technology demanders and suppliers.

The paramount advantage of the method proposed herein lies in its holistic optimization model, which concurrently addresses both the satisfaction and stability of technology traders. This approach effectively rectifies the limitations of prior research, which predominantly focused on the satisfaction of two-sided agents. Both sensitivity and comparative analyses indicate that while a stable matching scheme might marginally compromise the satisfaction levels of technology traders, it fortifies the commitment of one-sided agents. This reduces their propensity to breach contracts in favor of private dealings with alternative partners. Consequently, technology traders can foster enduring collaborative ties, bolstering the model’s robustness.

It is imperative to recognize that stakeholders in technology trading encompass not only the supply and demand sides but also intermediaries pivotal to the trading process. Incorporating the benefit of these intermediaries into the decision-making model of technology trading matching, and discerning the ramifications of varied intermediary fee structures on matching outcomes— particularly concerning stability and satisfaction— merits scholarly attention. A notable constraint of this study is its singular focus on one-to-one technology trades, rendering the proposed decision-making framework ill-suited for scenarios where a single technology might be licensed to multiple demanders concurrently. As such, theirs is pressing need for academia to devise decision-making paradigms tailored to many-to-one technology trading matching.

Footnotes

Acknowledgment

This work was partly supported by the National Social Science Fund of China (NSSFC) under Grant 20BGL006, the National Natural Science Foundation of China (NSFC) under Grant 71874067, and sponsored by Qing Lan Project of Jiangsu Province (2022).

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