Introduction

In today’s world, with the accelerating development of economic globalization and urban–rural integration, many cities are enhancing their influence and competitiveness by adjusting their industrial structure and transforming their economic development models. A city’s brand positioning is the core element that guides its healthy and sustainable development, determining its development direction and model. As the level of social modernization increases, the status and importance of sports in people’s lives are becoming increasingly prominent. Sports events are the most active and influential component of this, serving as an important means of promoting and popularizing sports activities. They are also a key avenue for advancing human modernization and shaping a city’s brand. Hosting sports events to drive a city’s large-scale development has become a growing trend. Dai1 analyzed the relationship between sports event and city brand building through the Xiamen International Marathon, emphasizing the crucial role of media in enhancing city vitality and image. The study highlighted that successful sports events and their brand promotion could significantly strengthen a city’s comprehensive capabilities. Following this, Li2 critically examined the development of SEB culture in China, identifying weaknesses, particularly the lack of globally influential event brands, and proposed strategies to enhance brand culture dissemination. By 2013, using the Xiamen International Marathon as a case study, Weng, Zhang and Ju3 investigated the interaction between SEBs and city brands. They emphasized the importance of city culture and spirit in event branding and proposed a theoretical framework for the coordinated development of event and city brands. Subsequently, Ji4 also focused on international marathons, analyzing Yangzhou’s experience in using event communication to shape its city brand. The study found that effective event communication could stimulate city vitality, improve its image, and enhance overall strength. In 2015, Qu and Zu5 discussed the characteristics of branded sports events, exploring how factors like event scale, brand identity, and communication influence brand building. They proposed that a diversified branding strategy could significantly enhance the competitiveness of sports events. Using the Guangzhou International Marathon as an example, Jiao6 analyzed the role of large-scale sports events in shaping city brands, discussing Guangzhou’s advantages and shortcomings in hosting the marathon and proposing relevant development strategies. Liu7 further explored the role of the Changsha International Marathon in city branding, emphasizing the synergy between sports events and city image promotion. The study suggested that successful sports events could become powerful carriers for city brands. Xu, Li and Liu8 analyzed micro-communication strategies for grassroots sports events. The results showed that micro-communication effectively met the marketing needs of grassroots event organizers and expanded the event’s influence. Wang9 studied the current state of university SEB in China, noting that while certain university SEBs had progressed, there remained significant potential for overall development. The study suggested further strengthening branding efforts in the future. Xiong and Han10 analyzed the impact of branded sports events on enhancing Dongguan’s city image. Their research indicated that SEB significantly boosted the city’s economic, cultural, and ecological vitality. Zhang and Yu11 studied the Tour of Poyang Lake cycling race, exploring strategies for integrating landscape sports events with local cultural branding to enhance regional brand value. In 2022, Yao12 focused on the branding of university sports events, analyzing its necessity and development potential, and concluded that university SEB could improve the social image of schools. Cheng and Wen13 studied the impact of large-scale sports events on city branding in Lu’an, finding that hosting sports events significantly enhanced the city’s reputation and competitiveness. Zhang and Wu14 conducted a case study on the Xiangyang Marathon, analyzing the specific pathways through which branded sports events shape city images, and proposed a "four-wheel-drive" development model that includes city advantages, media promotion, event support, and technological innovation. Finally, Zheng and Zheng15 focused on sports event consumption behavior patterns and brand building strategies, exploring how consumer characteristics influence event consumption. They proposed branding strategies based on consumer traits, such as emotional and experiential marketing, to enhance brand influence.

Multi-attribute group decision-making (MAGDM) is an extremely important decision-making method in modern management science22-24]. In real life, we often encounter complex decision-making problems that require comprehensive consideration of multiple factors16,17,18,19,20,21. For example, when selecting a sponsor for a sports event, we need to evaluate not only the sponsorship amount but also factors such as brand alignment, market influence, and long-term partnership potential. This represents a typical multi-attribute decision-making problem. The significance of this decision-making approach lies first in its ability to systematically address complex real-world problems. Traditional single-criterion decision-making tends to oversimplify issues, making it difficult to reflect the complete picture22,23,24,25,26. Multi-attribute decision-making, by establishing a comprehensive evaluation system, incorporates all relevant influencing factors, rendering decisions more holistic and scientific27,28,29,30,31,32,33. Secondly, MAGDM effectively integrates opinions from different experts. In practice, experts from various fields often focus on different priorities34,35,36. Marketing experts may emphasize communication effectiveness, while financial experts might prioritize return on investment. Through standardized group decision-making methods, we can scientifically synthesize these diverse perspectives, avoiding biases that may arise from individual viewpoints37,38. In our study, we employed expert weight allocation and opinion aggregation methods to ensure the objectivity of decisions. From the perspective of objectives, MAGDM primarily pursues three goals: first, identifying the optimal solution; second, enhancing the transparency of the decision-making process; and third, supporting long-term dynamic optimization. To achieve these goals, we need to select appropriate decision models, such as TODIM, TOPSIS or VIKOR, and design reasonable implementation processes39,40,41,42,43,44,45. In our sports brand evaluation research, we placed particular emphasis on the traceability of the decision-making process, ensuring that every score and weight was well-documented. It is worth noting that MAGDM has unique advantages in handling uncertain information46,47. Many real-world decision-making inputs are inherently ambiguous, such as qualitative descriptions like "the brand has significant influence." By introducing fuzzy set theory, we can transform these qualitative assessments into computable numerical values, significantly improving the scientific rigor of decisions48,49,50,51. With the advancement of big data and artificial intelligence technologies, multi-attribute group decision-making is evolving toward greater intelligence52,53,54,55. In the future, we can anticipate more intelligent decision-making systems that incorporate machine learning algorithms, further enhancing decision-making efficiency and accuracy. However, regardless of how it develops, the core value of MAGDM will always remain: using systematic methods to solve complex real-world problems and making decisions more scientific and rational56,57,58,59.

The evaluation of SEB building involves addressing MAGDM challenges. To tackle these complexities, recent methods have incorporated the ExpTODIM technique60,61 and GRA technique40,62. This research presents three fundamental advantages through the novel integration of exponential TODIM (ExpTODIM) with GRA in the 2TLNN decision-making framework. First, the hybrid approach establishes a comprehensive decision-making paradigm that effectively combines behavioral psychology with relational pattern analysis. The ExpTODIM component accurately models decision-makers’ risk preferences and cognitive behaviors through its distinctive exponential gain/loss function, while the GRA component systematically examines the geometric relationships among alternatives. This dual mechanism ensures decisions are both psychologically valid and mathematically rigorous. Second, the exponential formulation in TODIM significantly enhances the method’s sensitivity and discrimination capability, particularly when processing vague linguistic assessments. The modified function demonstrates superior performance in distinguishing between closely-ranked alternatives, while GRA’s relational coefficients contribute to maintaining solution stability across varying data conditions. Third, the integrated framework provides robust uncertainty management through complementary mechanisms. ExpTODIM’s nonlinear approach to probabilistic uncertainty works in concert with GRA’s proficiency in handling incomplete information, creating a more resilient decision-making system capable of delivering reliable outcomes in complex, uncertain environments. This synergy results in more stable and confident decision outcomes compared to conventional approaches.

Furthermore, 2TLNSs63 have been introduced as an effective tool for representing uncertain information during the evaluation process. This study demonstrates three key advantages of employing 2TLNSs for SEB building evaluation. First, 2TLNSs provide a superior framework for capturing the inherent uncertainty and vagueness in brand assessment by simultaneously modeling truth, indeterminacy, and falsity degrees through linguistic terms. This tripartite structure enables more precise representation of experts’ qualitative judgments about brand perception, popularity, and reputation factors. Second, the 2-tuple linguistic model effectively eliminates information distortion during computational processes by preserving both symbolic translation and numerical values. This proves particularly valuable when aggregating diverse expert opinions about multidimensional brand elements like event identity, fan engagement, and media impact, ensuring no loss of linguistic information. Third, the proposed approach offers enhanced flexibility in handling the complex interdependencies among various brand evaluation criteria. The neutrosophic nature of 2TLNSs allows for comprehensive analysis of conflicting attributes and incomplete information commonly encountered in sports branding, while the linguistic component maintains interpretability for decision-makers. This combination facilitates more nuanced brand strength assessment compared to conventional fuzzy methods.

This paper presents the ExpTODIM-GRA technique for addressing MAGDM within the 2TLNSs framework. Initially, the paper reviews the foundational principles of 2TLNSs. Following this, the ExpTODIM-GRA method is applied to solve MAGDM issues under 2TLNSs by integrating the entropy method for attribute weight determination. A numerical study is then conducted to validate the proposed 2TLNN-Com-ExpTODIM-GRA model, demonstrating its effectiveness in the context of SEB building evaluation. The motivation for this study arises from several key factors: (1) the application of Entropy39 to calculate the weight under 2TLNSs, ensuring a reliable weighting technique; (2) the implementation of 2TLNN-Com-ExpTODIM-GRA method to efficiently handle MAGDM involving uncertain information; (3) a comprehensive numerical example showcasing the practical utility of the 2TLNN-Com-ExpTODIM-GRA approach in evaluating SEB building; and (4) a detailed comparative analysis to highlight the advantages of the proposed method over existing techniques.

Through these contributions, this study provides a structured and innovative framework for tackling MAGDM problems in SEB evaluation. By leveraging the strengths of the ExpTODIM-GRA method and the flexibility of 2TLNSs, the approach offers a robust solution for dealing with uncertainty and complexity in decision-making processes. The comparative analysis further emphasizes the superiority of the proposed method in terms of accuracy and practicality, making it a valuable tool for future applications in similar evaluation scenarios.

This paper is structured as follows: Section "Preliminaries" introduces the concept of 2TLNSs. Section "2TLNN-Com-ExpTODIM-GRA method for MAGDM with entropy model" explains the application of the 2TLNN-Com-ExpTODIM-GRA technique within the 2TLNSs framework, incorporating the entropy method. A numerical example is provided in Section "Numerical example and comparative analysis" to illustrate the evaluation process for SEB building, accompanied by a comparative analysis. Finally, Section "Conclusion" offers concluding remarks, summarizing the key findings of the study.

Preliminaries

Wang et al.63 operated the 2TLNSs.

Definition 1

28,29. Let \(\xi s_{1} ,\xi s_{2} , \ldots ,\xi s_{\phi }\) be linguistic information and \(\xi s\) is operated:

$$\xi s = \left\{ \begin{gathered} \xi s_{0} = extremely\;poor,\xi s_{1} = very\;poor,\xi s_{2} = poor,\xi s_{3} = medium, \hfill \\ \xi s_{4} = good,\xi s_{5} = very\;good,\xi s_{6} = extremely\;good. \hfill \\ \end{gathered} \right\}$$

Definition 2

6,7. The SVNSs is operated:

$$\xi \eta = \left\{ {\left. {\left( {\theta ,\phi_{\xi \eta } \left( \theta \right),\varphi_{\xi \eta } \left( \theta \right),\gamma_{\xi \eta } \left( \theta \right)} \right)} \right|\theta \in \Theta } \right\}$$
(1)

where \(\phi_{\xi \eta } \left( \theta \right),\varphi_{\xi \eta } \left( \theta \right),\gamma_{\xi \eta } \left( \theta \right) \in \left[ {0,1} \right]\) is named as the truth-membership (TM), indeterminacy-membership (IM) and falsity-membership (FM),\(0 \le \phi_{\xi \eta } \left( \theta \right) + \varphi_{\xi \eta } \left( \theta \right) + \gamma_{\xi \eta } \left( \theta \right) \le 3.\)

Definition 3

63. Let \(\xi s_{j} \left( {j = 1,2,3, \ldots ,\phi } \right)\) be 2TLSs. If \(\xi \delta = \left\langle {\left( {\xi s_{t} ,\xi \xi } \right),\left( {\xi s_{i} ,\xi \psi } \right),\left( {\xi s_{f} ,\xi \zeta } \right)} \right\rangle\) is operated for \(\xi s_{t} ,\xi s_{i} ,\xi s_{f} \in \xi s\), \(\xi \xi ,\xi \psi ,\xi \zeta \in \left[ {0,\left. {0.5} \right)} \right.\), where \(\left( {\xi s_{t} ,\xi \xi } \right),\left( {\xi s_{i} ,\xi \psi } \right),\left( {\xi s_{f} ,\xi \zeta } \right)\) is operated for TM, IM and FM based on the 2TLSs, the 2TLNSs is operated:

$$\xi \delta = \left\langle {\left( {\xi s_{t} ,\xi \xi } \right),\left( {\xi s_{i} ,\xi \psi } \right),\left( {\xi s_{f} ,\xi \zeta } \right)} \right\rangle$$
(2)

where \(0 \le \Delta^{ - 1} \left( {\xi s_{t} ,\xi \xi } \right) \le \phi ,\) \(0 \le \Delta^{ - 1} \left( {\xi s_{i} ,\xi \psi } \right) \le \phi ,\) \(0 \le \Delta^{ - 1} \left( {\xi s_{f} ,\xi \zeta } \right) \le \phi\), \(0 \le \Delta^{ - 1} \left( {\xi s_{t} ,\xi \xi } \right) + \;\)\(\Delta^{ - 1} \left( {\xi s_{i} ,\xi \psi } \right) + \;\)\(\Delta^{ - 1} \left( {\xi s_{f} ,\xi \zeta } \right) \le 3\phi .\)

Definition 3

27. Let \(\xi \delta = \left\langle {\left( {\xi s_{t} ,\xi \xi } \right),\left( {\xi s_{i} ,\xi \psi } \right),\left( {\xi s_{f} ,\xi \zeta } \right)} \right\rangle\). Uncertain score function (USF) and uncertain accuracy function (UAF) are operated:

$$USF\left( {\xi \delta } \right) = \frac{{\left( {2\phi + \Delta^{ - 1} \left( {\xi s_{t} ,\xi \xi } \right) - \Delta^{ - 1} \left( {\xi s_{i} ,\xi \psi } \right) - \Delta^{ - 1} \left( {\xi s_{f} ,\xi \zeta } \right)} \right)}}{3\phi },USF\left( {\xi \delta } \right) \in \left[ {0,1} \right]$$
(3)
$$UAF(\xi \delta ) = \frac{{\phi + \Delta^{ - 1} \left( {\xi s_{t} ,\xi \xi } \right) - \Delta^{ - 1} \left( {\xi s_{f} ,\xi \zeta } \right)}}{2\phi },UAF(\xi \delta ) \in \left[ {0,1} \right].$$
(4)

Definition 4

27. Let \(\xi \delta_{1} = \left\langle {\left( {\xi s_{{t_{1} }} ,\xi \xi_{1} } \right),} \right.\) \(\xi \delta_{1} = \left( {\xi s_{{i_{1} }} ,\xi \psi_{1} } \right),\) \(\xi \delta_{1} = \left. {\left( {\xi s_{{f_{1} }} ,\xi \zeta_{1} } \right)} \right\rangle\) and \(\xi \delta_{2} = \left\langle {\left( {\xi s_{{t_{2} }} ,\xi \xi_{2} } \right),} \right.\) \(\xi \delta_{2} = \left( {\xi s_{{i_{2} }} ,\xi \psi_{2} } \right),\) \(\xi \delta_{2} = \left. {\left( {\xi s_{{f_{2} }} ,\xi \zeta_{2} } \right)} \right\rangle\), then

  1. 1.

    \(if\;USF\left( {\xi \delta_{1} } \right) \prec USF\left( {\xi \delta_{2} } \right),\)\(\xi \delta_{1} \prec \xi \delta_{2} ;\)

  2. 2.

    \(if\;\;USF\left( {\xi \delta_{1} } \right) = USF\left( {\xi \delta_{2} } \right),UAF\left( {\xi \delta_{1} } \right) \prec UAF\left( {\xi \delta_{2} } \right)\), \(\xi \delta_{1} \prec \xi \delta_{2} ;\)

  3. 3.

    \(if\;\;USF\left( {\xi \delta_{1} } \right) = USF\left( {\xi \delta_{2} } \right),UAF\left( {\xi \delta_{1} } \right) = UAF\left( {\xi \delta_{2} } \right),\) \(\xi \delta_{1} = \xi \delta_{2} ;\)

Definition 5

27. Let \(\xi \delta_{1} = \left\langle {\left( {\xi s_{{t_{1} }} ,\xi \xi_{1} } \right),} \right.\) \(\xi \delta_{1} = \left( {\xi s_{{i_{1} }} ,\xi \psi_{1} } \right),\) \(\xi \delta_{1} = \left. {\left( {\xi s_{{f_{1} }} ,\xi \zeta_{1} } \right)} \right\rangle\) and \(\xi \delta_{2} = \left\langle {\left( {\xi s_{{t_{2} }} ,\xi \xi_{2} } \right),} \right.\) \(\xi \delta_{2} = \left( {\xi s_{{i_{2} }} ,\xi \psi_{2} } \right),\) \(\xi \delta_{2} = \left. {\left( {\xi s_{{f_{2} }} ,\xi \zeta_{2} } \right)} \right\rangle\) and \(\xi \delta = \left\langle {\left( {\xi s_{t} ,\xi \xi } \right),} \right.\) \(\xi \delta = \left( {\xi s_{i} ,\xi \psi } \right),\) \(\xi \delta = \left. {\left( {\xi s_{f} ,\xi \zeta } \right)} \right\rangle\) be three 2TLNNs, then

1.

$$\xi \delta _{1} \oplus \xi \delta _{2} = \left\{ \begin{aligned} & \Delta \left( {\phi \left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{t_{1} }} ,\xi \xi _{1} } \right)}}{\phi } + \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{t_{2} }} ,\xi \xi _{2} } \right)}}{\phi } - \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{t_{1} }} ,\xi \xi _{1} } \right)}}{\phi } \cdot \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{t_{2} }} ,\xi \xi _{2} } \right)}}{\phi }} \right)} \right), \\ & \Delta \left( {\phi \left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{i_{1} }} ,\xi \psi _{1} } \right)}}{\phi } \cdot \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{i_{2} }} ,\xi \psi _{2} } \right)}}{\phi }} \right)} \right), \\ & \Delta \left( {\phi \left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{f_{1} }} ,\xi \zeta _{1} } \right)}}{\phi } \cdot \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{f_{2} }} ,\xi \zeta _{2} } \right)}}{\phi }} \right)} \right) \\ \end{aligned} \right\};$$

2.

$$\xi \delta _{1} \otimes \xi \delta _{2} = \left\{ \begin{aligned} & \Delta \left( {\phi \left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{t_{1} }} ,\xi \xi _{1} } \right)}}{\phi } \cdot \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{t_{2} }} ,\xi \xi _{2} } \right)}}{\phi }} \right)} \right), \\ & \Delta \left( {\phi \left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{i_{1} }} ,\xi \psi _{1} } \right)}}{\phi } + \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{i_{2} }} ,\xi \psi _{2} } \right)}}{\phi } - \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{i_{1} }} ,\xi \psi _{1} } \right)}}{\phi } \cdot \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{i_{2} }} ,\xi \psi _{2} } \right)}}{\phi }} \right)} \right), \\ & \Delta \left( {\phi \left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{f_{1} }} ,\xi \zeta _{1} } \right)}}{\phi } + \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{f_{2} }} ,\xi \zeta _{2} } \right)}}{\phi } - \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{f_{1} }} ,\xi \zeta _{1} } \right)}}{\phi } \cdot \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{f_{2} }} ,\xi \zeta _{2} } \right)}}{\phi }} \right)} \right) \\ \end{aligned} \right\};$$

3.

$$o\xi \delta = \left\{ \begin{aligned} & \Delta \left( {\phi \left( {1 - \left( {1 - \frac{{\Delta ^{{ - 1}} \left( {\xi s_{t} ,\xi \xi } \right)}}{\phi }} \right)^{o} } \right)} \right), \\ & \Delta \left( {\phi \left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{i} ,\xi \psi } \right)}}{\phi }} \right)^{o} } \right),\Delta \left( {D\left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{f} ,\xi \zeta } \right)}}{\phi }} \right)^{o} } \right) \\ \end{aligned} \right\},o > 0;$$

4.

$$\xi \delta ^{o} = \left\{ \begin{aligned} & \Delta \left( {\phi \left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{t} ,\xi \xi } \right)}}{\phi }} \right)^{o} } \right),\Delta \left( {\phi \left( {1 - \left( {1 - \frac{{\Delta ^{{ - 1}} \left( {\xi s_{i} ,\xi \psi } \right)}}{\phi }} \right)^{o} } \right)} \right), \\ & \Delta \left( {\phi \left( {1 - \left( {1 - \frac{{\Delta ^{{ - 1}} \left( {\xi s_{f} ,\xi \zeta } \right)}}{\phi }} \right)^{o} } \right)} \right) \\ \end{aligned} \right\},o > 0.$$

Definition 6

64. Let \(\xi \delta_{1} = \left\langle {\left( {\xi s_{{t_{1} }} ,\xi \xi_{1} } \right),} \right.\) \(\xi \delta_{1} = \left( {\xi s_{{i_{1} }} ,\xi \psi_{1} } \right),\) \(\xi \delta_{1} = \left. {\left( {\xi s_{{f_{1} }} ,\xi \zeta_{1} } \right)} \right\rangle\) and \(\xi \delta_{2} = \left\langle {\left( {\xi s_{{t_{2} }} ,\xi \xi_{2} } \right),} \right.\) \(\xi \delta_{2} = \left( {\xi s_{{i_{2} }} ,\xi \psi_{2} } \right),\) \(\xi \delta_{2} = \left. {\left( {\xi s_{{f_{2} }} ,\xi \zeta_{2} } \right)} \right\rangle,\) 2TLNN Hamming distance (2TLNNHD) and 2TLNN Euclidean distance (2TLNNED) is operated:

$$2TLNNHD\left( {\xi \delta_{1} ,\xi \delta_{2} } \right) = \frac{1}{3}\left( \begin{gathered} \left| {\frac{{\Delta^{ - 1} \left( {\xi s_{{t_{1} }} ,\xi \xi_{1} } \right) - \Delta^{ - 1} \left( {\xi s_{{t_{2} }} ,\xi \xi_{2} } \right)}}{\phi }} \right| \hfill \\ + \left| {\frac{{\Delta^{ - 1} \left( {\xi s_{{i_{1} }} ,\xi \psi_{1} } \right) - \Delta^{ - 1} \left( {\xi s_{{i_{2} }} ,\xi \psi_{2} } \right)}}{\phi }} \right| \hfill \\ + \left| {\frac{{\Delta^{ - 1} \left( {\xi s_{{f_{1} }} ,\xi \zeta_{1} } \right) - \Delta^{ - 1} \left( {\xi s_{{f_{2} }} ,\xi \zeta_{2} } \right)}}{\phi }} \right| \hfill \\ \end{gathered} \right)$$
(5-a)
$$2TLNNED\left( {\xi \delta_{1} ,\xi \delta_{2} } \right) = \sqrt {\frac{1}{3}\left( \begin{gathered} \left| {\frac{{\Delta^{ - 1} \left( {\xi s_{{t_{1} }} ,\xi \xi_{1} } \right) - \Delta^{ - 1} \left( {\xi s_{{t_{2} }} ,\xi \xi_{2} } \right)}}{\phi }} \right|^{2} \hfill \\ + \left| {\frac{{\Delta^{ - 1} \left( {\xi s_{{i_{1} }} ,\xi \psi_{1} } \right) - \Delta^{ - 1} \left( {\xi s_{{i_{2} }} ,\xi \psi_{2} } \right)}}{\phi }} \right|^{2} \hfill \\ + \left| {\frac{{\Delta^{ - 1} \left( {\xi s_{{f_{1} }} ,\xi \zeta_{1} } \right) - \Delta^{ - 1} \left( {\xi s_{{f_{2} }} ,\xi \zeta_{2} } \right)}}{\phi }} \right|^{2} \hfill \\ \end{gathered} \right)}$$
(5-b)

The 2TLNNWA technique is operated:

Definition 7

63. Let \(\xi \delta_{j} = \left\langle {\left( {\xi s_{{t_{j} }} ,\xi \xi_{j} } \right),} \right.\) \(\xi \delta_{j} = \left( {\xi s_{{i_{j} }} ,\xi \psi_{j} } \right),\) \(\xi \delta_{j} = \left. {\left( {\xi s_{{f_{j} }} ,\xi \zeta_{j} } \right)} \right\rangle,\) the 2TLNNWA technique is operated:

$$\begin{aligned} & {\text{2TLNNWA}}\left( {\xi \delta _{1} ,\xi \delta _{2} , \ldots ,\xi \delta _{n} } \right) \\ & \quad {\text{ = }}\xi w_{1} \xi \delta _{1} \oplus \xi w_{2} \xi \delta _{2} \ldots \oplus \xi w_{n} \xi \delta _{n} = \mathop \oplus \limits_{{j = 1}}^{n} \xi w_{j} \xi \delta _{j} \\ & \quad = \left\{ \begin{aligned} & \Delta \left( {\phi \left( {1 - \prod\limits_{{j = 1}}^{n} {\left( {1 - \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{t_{j} }} ,\xi \alpha _{j} } \right)}}{\phi }} \right)^{{\xi w_{j} }} } } \right)} \right), \\ & \Delta \left( {\phi \prod\limits_{{j = 1}}^{n} {\left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{i_{j} }} ,\xi \beta _{j} } \right)}}{\phi }} \right)^{{\xi w_{j} }} } } \right), \\ & \Delta \left( {\phi \prod\limits_{{j = 1}}^{n} {\left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{f_{j} }} ,\xi \chi _{j} } \right)}}{\phi }} \right)^{{\xi w_{j} }} } } \right) \\ \end{aligned} \right\} \\ \end{aligned}$$
(6)

2TLNN-Com-ExpTODIM-GRA method for MAGDM with entropy model

2TLNN-MAGDM data

The 2TLNN-Com-ExpTODIM-GRA method is operated for MAGDM. Let \(DA = \left\{ {DA_{1} ,DA_{2} , \cdots ,DA_{m} } \right\}\) be alternatives and \(DG = \left\{ {DG_{1} ,DG_{2} , \cdots ,DG_{n} } \right\}\) be attributes with weight \(\xi w = \left( {\xi w_{1} ,\xi w_{2} , \cdots ,\xi w_{n} } \right),\) \(\xi w_{j} \in \left[ {0,1} \right],\) \(\sum\limits_{j = 1}^{n} {\xi w_{j} } = 1\) and experts \(DE\; = \;\)\(\left\{ {DE_{1} ,DE_{2} , \cdots ,DE_{q} } \right\}\) with weight numbers \(\xi \omega = \left( {\xi \omega_{1} ,\xi \omega_{2} , \cdots ,\xi \omega_{q} } \right)\) \(\xi \omega_{j} \in \left[ {0,1} \right],\) \(\sum\limits_{j = 1}^{q} {\xi \omega_{j} } = 1.\)

Then, 2TLNN-Com-ExpTODIM-GRA technique is operated for MAGDM.

Step 1 Build the 2TLNN-matrix \(DM = \left[ {DM_{ij}^{\left( t \right)} } \right]_{m \times n} = \;\)\(\left\{ {\left( {\xi s_{{t_{ij} }}^{\left( t \right)} ,\xi \alpha_{ij}^{\left( t \right)} } \right),} \right.\) \(\left( {\xi s_{{i_{ij} }}^{\left( t \right)} ,\xi \beta_{ij}^{\left( t \right)} } \right),\)\(\left. {\left( {\xi s_{{f_{ij} }}^{\left( t \right)} ,\xi \chi_{ij}^{\left( t \right)} } \right)} \right\}_{m \times n}\) and manage the average matrix \(DM = \left[ {DM_{ij} } \right]_{m \times n}:\)

$$\begin{aligned} & \;\begin{array}{*{20}c} {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;DG_{{\text{1}}} } & {\;\;\;\;\,DG_{{\text{2}}} } & {\;\, \ldots } & {\;DG_{n} } \\ \end{array} \\ DM^{{\left( t \right)}} = \left[ {DM_{{ij}}^{{\left( t \right)}} } \right]_{{m \times n}} = & \;\begin{array}{*{20}c} {DA_{1} } \\ {DA_{2} } \\ \vdots \\ {DA_{m} } \\ \end{array} \left[ {\begin{array}{*{20}c} {DM_{{11}}^{{\left( t \right)}} } & {DM_{{12}}^{{\left( t \right)}} } & \ldots & {DM_{{1n}}^{{\left( t \right)}} } \\ {DM_{{21}}^{{\left( t \right)}} } & {DM_{{22}}^{{\left( t \right)}} } & \ldots & {DM_{{2n}}^{{\left( t \right)}} } \\ \vdots & \vdots & \vdots & \vdots \\ {DM_{{m1}}^{{\left( t \right)}} } & {DM_{{m2}}^{{\left( t \right)}} } & \ldots & {DM_{{mn}}^{{\left( t \right)}} } \\ \end{array} } \right] \\ \end{aligned}$$
(7)
$$\begin{aligned} & \;\begin{array}{*{20}c} {\;\;\;\;\;\;\;\;\;\,\;\;\;\;\;\;\,DG_{{\text{1}}} } & {\;\;\;\,DG_{{\text{2}}} } & {\;\;\, \ldots } & {\;\;\;DG_{n} } \\ \end{array} \\ DM = \left[ {DM_{{ij}} } \right]_{{m \times n}} = & \;\begin{array}{*{20}c} {DA_{1} } \\ {DA_{2} } \\ \vdots \\ {DA_{m} } \\ \end{array} \left[ {\begin{array}{*{20}c} {DM_{{11}} } & {DM_{{12}} } & \ldots & {DM_{{1n}} } \\ {DM_{{21}} } & {DM_{{22}} } & \ldots & {DM_{{2n}} } \\ \vdots & \vdots & \vdots & \vdots \\ {DM_{{m1}} } & {DM_{{m2}} } & \ldots & {DM_{{mn}} } \\ \end{array} } \right] \\ \end{aligned}$$
(8)

In line with the 2TLNNWA technique, the \(DM = \left[ {DM_{ij} } \right]_{m \times n} = \;\)\(\left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),} \right.\) \(\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\) \(\left. {\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\}_{m \times n}\) is:

$$\begin{aligned} DM_{{ij}} = & \;\xi \omega _{1} DM_{{ij}}^{1} \oplus \xi \omega _{2} DM_{{ij}}^{2} \oplus \cdots \oplus \xi \omega _{t} DM_{{ij}}^{t} \\ = & \;\left\{ \begin{aligned} & \Delta \left( {\phi \left( {1 - \prod\limits_{{t = 1}}^{q} {\left( {1 - \frac{{\Delta ^{{ - 1}} \left( {\xi s_{{t_{{ij}} }}^{{\left( t \right)}} ,\xi \alpha _{{ij}}^{{\left( t \right)}} } \right)}}{\phi }} \right)^{{\xi w_{t} }} } } \right)} \right), \\ & \Delta \left( {\phi \prod\limits_{{t = 1}}^{q} {\left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{i_{{ij}} }}^{{\left( t \right)}} ,\xi \beta _{{ij}}^{{\left( t \right)}} } \right)}}{\phi }} \right)^{{\xi w_{t} }} } } \right), \\ & \Delta \left( {\phi \prod\limits_{{t = 1}}^{q} {\left( {\frac{{\Delta ^{{ - 1}} \left( {\xi s_{{f_{{ij}} }}^{{\left( t \right)}} ,\xi \chi _{{ij}}^{{\left( t \right)}} } \right)}}{\phi }} \right)^{{\xi w_{t} }} } } \right) \\ \end{aligned} \right\} \\ \end{aligned}$$
(9)

Step 2 Operate the normalized \(NDM = \left[ {NDM_{ij} } \right]_{m \times n} = \;\) \(\left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),} \right.\) \(\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\) \(\left. {\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\}_{m \times n}\) with \(DM = \left[ {DM_{ij} } \right]_{m \times n} = \;\).\(\left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),} \right.\) \(\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\) \(\left. {\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\}_{m \times n}\).

For beneficial attributes:

$$NDM_{ij} = DM_{ij} = \left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\}$$
(10)

For non-beneficial attributes:

$$\begin{aligned} NDM_{ij} = & \;\left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\} \\ = & \;\left\{ {\Delta \left( {\phi - \Delta^{ - 1} \left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right)} \right),\Delta \left( {\phi - \Delta^{ - 1} \left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right)} \right),\Delta \left( {\phi - \Delta^{ - 1} \left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right)} \right\} \\ \end{aligned}$$
(11)

Operate the weight values with entropy

This study leverages uncertain score function (USF) and uncertain accuracy function (UAF) to determine weights in the 2TLNN framework, offering two key advantages. First, USF enhances interpretability by converting complex neutrosophic information into a simplified, comparable scalar value. Since 2TLNNs simultaneously model truth, indeterminacy, and falsity degrees, direct comparisons can be challenging. The score function synthesizes these three dimensions into a single metric, facilitating clearer ranking and decision-making. This is particularly valuable in sports brand evaluation, where experts often assess multiple conflicting criteria (e.g., fan loyalty vs. sponsor satisfaction). By reducing linguistic ambiguity, SF ensures more consistent weight assignments. Second, UAF refines decision precision by distinguishing between alternatives with similar scores. In cases where two options yield comparable SF values, the accuracy function acts as a tiebreaker by evaluating the reliability of the neutrosophic information. A higher UAF indicates greater confidence in the assessment, which helps prioritize more robust data in weight calculation. This dual-layered approach—first ranking via USF, then refining via UAF—strengthens the objectivity of entropy-based weighting65 while maintaining alignment with expert intuition. Together, USF and UAF provide a balanced methodology that captures both the clarity of quantitative ranking and the nuanced reliability needed for high-stakes evaluations like sports brand building.

Step 3 Normalized hybrid decision matrix \(NHDM_{ij}^{{}}\) is operated with Entropy65:

$$NHDM_{ij}^{{}} = \frac{{\frac{{UAF\left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\} + 1}}{{USF\left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\} + 1}}}}{{\sum\limits_{i = 1}^{m} {\left( {\frac{{UAF\left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\} + 1}}{{USF\left\{ {\left( {\xi s_{{t_{ij} }}^{{}} ,\xi \alpha_{ij}^{{}} } \right),\left( {\xi s_{{i_{ij} }}^{{}} ,\xi \beta_{ij}^{{}} } \right),\left( {\xi s_{{f_{ij} }}^{{}} ,\xi \chi_{ij}^{{}} } \right)} \right\} + 1}}} \right)} }},\;\;$$
(12)

Then, the Shannon entropy (UHISE) is operated:

$$UHISE_{j} = - \frac{1}{\ln m}\sum\limits_{i = 1}^{m} {NHDM_{ij}^{{}} \ln NHDM_{ij}^{{}} }$$
(13)

and \(NHDM_{ij}^{{}} \ln NHDM_{ij}^{{}} = 0\) if \(NHDM_{ij}^{{}} = 0\).

Then, the weights values \(\xi w = \left( {\xi w_{1} ,\xi w_{2} , \cdots ,\xi w_{n} } \right)\) is managed:

$$\xi w_{j} = \frac{{1 - UHISE_{j} }}{{\sum\limits_{j = 1}^{n} {\left( {1 - UHISE_{j} } \right)} }}$$
(14)

2TLNN-Com-ExpTODIM-GRA method for MAGDM

2TLNN-Com-ExpTODIM-GRA method is operated for MAGDM.

Step 4 Operate relative weight values:

$$r\xi w_{j} = \frac{{\xi w_{j} }}{{\mathop {\max }\limits_{j} \xi w_{j} }},$$
(15)

Step 5 2TLNN dominance degree (2TLNNDD) of \(DA_{i}\) under \(DA_{t}\) for \(DG_{j}\) is operated based on the 2TLNNHD and 2TLNNED.

$$2TLNNDD_{j} \left( {DA_{i} ,DA_{t} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{2}\left( \begin{gathered} \frac{{r\xi w_{j} \times \left( {1 - 10^{{ - \rho 2TLNNHDHD\left( {NDM_{{ij}} ,NDM_{{tj}} } \right)}} } \right)}}{{\sum\nolimits_{{j = 1}}^{n} {r\xi w_{j} } }} \hfill \\ + \frac{{r\xi w_{j} \times \left( {1 - 10^{{ - \rho 2TLNNHDED\left( {NDM_{{ij}} ,NDM_{{tj}} } \right)}} } \right)}}{{\sum\nolimits_{{j = 1}}^{n} {r\xi w_{j} } }} \hfill \\ \end{gathered} \right)} \hfill & {{\text{if }}USF\left( {NDM_{{ij}} } \right) > USF\left( {NDM_{{tj}} } \right)} \hfill \\ 0 \hfill & {{\text{if }}USF\left( {NDM_{{ij}} } \right) = USF\left( {NDM_{{tj}} } \right)} \hfill \\ {\frac{1}{2}\left( \begin{gathered} - \frac{1}{\theta }\frac{{\sum\nolimits_{{j = 1}}^{n} {r\xi w_{j} \times \left( {1 - 10^{{ - \rho 2TLNNHDHD\left( {NDM_{{ij}} ,NDM_{{tj}} } \right)}} } \right)} }}{{r\xi w_{j} }} \hfill \\ - \frac{1}{\theta }\frac{{\sum\nolimits_{{j = 1}}^{n} {r\xi w_{j} \times \left( {1 - 10^{{ - \rho 2TLNNHDHD\left( {NDM_{{ij}} ,NDM_{{tj}} } \right)}} } \right)} }}{{r\xi w_{j} }} \hfill \\ \end{gathered} \right)} \hfill & {{\text{if }}USF\left( {NDM_{{ij}} } \right) < USF\left( {NDM_{{tj}} } \right)} \hfill \\ \end{array} } \right.$$
(16)

where \(\theta\) is operated66 and \(\rho \in \left[ {1,5} \right]\)60.

The 2TLNNDD for \(DG_{j}\) is operated:

$$\begin{aligned} & 2TLNNDD_{j} \left( {DA_{i} } \right) = \left[ {2TLNNDD_{j} \left( {DA_{i} ,DA_{t} } \right)} \right]_{{m \times m}} \\ & \begin{array}{*{20}c} {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;DA_{1} } & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,DA_{2} } & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \cdots } & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;DA_{m} } \\ \end{array} \\ & \quad = \begin{array}{*{20}c} {DA_{1} } \\ {DA_{2} } \\ \vdots \\ {DA_{m} } \\ \end{array} \left[ {\begin{array}{*{20}c} 0 & {2TLNNDD_{j} \left( {DA_{1} ,DA_{2} } \right)} & \cdots & {2TLNNDD_{j} \left( {DA_{1} ,DA_{m} } \right)} \\ {2TLNNDD_{j} \left( {DA_{2} ,DA_{1} } \right)} & 0 & \cdots & {2TLNNDD_{j} \left( {DA_{2} ,DA_{m} } \right)} \\ \vdots & \vdots & \cdots & \vdots \\ {2TLNNDD_{j} \left( {DA_{m} ,DA_{1} } \right)} & {2TLNNDD_{j} \left( {DA_{m} ,DA_{2} } \right)} & \cdots & 0 \\ \end{array} } \right] \\ \end{aligned}$$

(3) Operate the 2TLNNDD of \(DA_{i}\) under other decision alternatives for \(DG_{j}\):

$$2TLNNDD_{j} \left( {DA_{i} } \right) = \sum\limits_{t = 1}^{m} {2TLNNDD_{j} \left( {DA_{i} ,DA_{t} } \right)}$$
(17)

The overall 2TLNNDD information is operated:

$$\begin{aligned} & 2TLNNDD = \left( {2TLNNDD_{{ij}} } \right)_{{m \times n}} \\ & \quad = \left[ {\begin{array}{*{20}c} {} & {DG_{1} } & {DG_{2} } & \ldots & {DG_{n} } \\ {DA_{1} } & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{1} \left( {DA_{1} ,DA_{t} } \right)} } & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{2} \left( {DA_{1} ,DA_{t} } \right)} } & \ldots & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{n} \left( {DA_{1} ,DA_{t} } \right)} } \\ {DA_{2} } & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{1} \left( {DA_{2} ,DA_{t} } \right)} } & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{2} \left( {DA_{2} ,DA_{t} } \right)} } & \ldots & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{n} \left( {DA_{2} ,DA_{t} } \right)} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {DA_{m} } & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{1} \left( {DA_{m} ,DA_{t} } \right)} } & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{2} \left( {DA_{m} ,DA_{t} } \right)} } & \ldots & {\sum\limits_{{t = 1}}^{m} {2TLNNDD_{n} \left( {DA_{m} ,DA_{t} } \right)} } \\ \end{array} } \right] \\ \end{aligned}$$
(18)

Step 6 Operate the 2TLNN positive ideal decision solution (2TLNNPIDS) and 2TLNN negative ideal decision solution (2TLNNNIDS):

$$2TLNNPIDS = \left( {2TLNNPIDS_{1} ,2TLNNPIDS_{1} , \cdots ,2TLNNPIDS_{n} } \right)$$
(19)
$$2TLNNNIDS = \left( {2TLNNNIDS_{1} ,2TLNNNIDS_{1} , \cdots ,2TLNNNIDS_{n} } \right)$$
(20)
$$2TLNNPIDS_{j} = \mathop {\max }\limits_{i} \left( {2TLNNDD_{ij} } \right),2TLNNNIDS_{j} = \mathop {\min }\limits_{i} \left( {2TLNNDD_{ij} } \right)$$
(21)

Step 7 Operate the 2TLNN grey rational coefficients (2TLNNGRC) from 2TLNNPIDS and 2TLNNNIDS.

$$2TLNNGRC\left( {\zeta_{ij}^{ + } } \right) = \frac{{\left( {\mathop {\min }\limits_{1 \le i \le m} \mathop {\min }\limits_{1 \le j \le n} \left| {2TLNNDD_{ij} - 2TLNNPIDS_{j} } \right| + \rho \mathop {\max }\limits_{1 \le i \le m} \mathop {\max }\limits_{1 \le j \le n} \left| {2TLNNDD_{ij} - 2TLNNPIDS_{j} } \right|} \right)}}{{\left( {\left| {2TLNNDD_{ij} - 2TLNNPIDS_{j} } \right| + \rho \mathop {\max }\limits_{1 \le i \le m} \mathop {\max }\limits_{1 \le j \le n} \left| {2TLNNDD_{ij} - 2TLNNPIDS_{j} } \right|} \right)}}$$
(22)
$$2TLNNGRC\left( {\zeta_{ij}^{ - } } \right) = \frac{{\left( {\mathop {\min }\limits_{1 \le i \le m} \mathop {\min }\limits_{1 \le j \le n} \left| {2TLNNDD_{ij} - 2TLNNNIDS_{j} } \right| + \rho \mathop {\max }\limits_{1 \le i \le m} \mathop {\max }\limits_{1 \le j \le n} \left| {2TLNNDD_{ij} - 2TLNNNIDS_{j} } \right|} \right)}}{{\left( {\left| {2TLNNDD_{ij} - 2TLNNNIDS_{j} } \right| + \rho \mathop {\max }\limits_{1 \le i \le m} \mathop {\max }\limits_{1 \le j \le n} \left| {2TLNNDD_{ij} - 2TLNNNIDS_{j} } \right|} \right)}}$$
(23)

Step 8 Operate the 2TLNNGRD (2TLNN grey relation degree) from 2TLNNPIDS and 2TLNNNIDS.

$$2TLNNGRD\left( {\zeta_{i}^{ + } } \right) = \sum\limits_{j = 1}^{n} {\xi w_{j} 2TLNNGRC\left( {\zeta_{ij}^{ + } } \right)}$$
(24)
$$2TLNNGRD\left( {\zeta_{i}^{ - } } \right) = \sum\limits_{j = 1}^{n} {\xi w_{j} 2TLNNGRC\left( {\zeta_{ij}^{ - } } \right)}$$
(25)

Step 9 Operate the 2TLNN relative relational degree (2TLNNRRD) from 2TLNNPIDS.

$$\begin{aligned} 2TLNNRRD\left( {\zeta_{i} } \right) = & \;\frac{{2TLNNGRD\left( {\zeta_{i}^{ + } } \right)}}{{2TLNNGRD\left( {\zeta_{i}^{ - } } \right) + 2TLNNGRD\left( {\zeta_{i}^{ + } } \right)}} \\ = & \;\frac{{\sum\limits_{j = 1}^{n} {\xi w_{j} 2TLNNGRC\left( {\zeta_{ij}^{ + } } \right)} }}{{\sum\limits_{j = 1}^{n} {\xi w_{j} 2TLNNGRC\left( {\zeta_{ij}^{ + } } \right)} + \sum\limits_{j = 1}^{n} {\xi w_{j} 2TLNNGRC\left( {\zeta_{ij}^{ - } } \right)} }} \\ \end{aligned}$$
(26)

Step 10 Sort and select the optimal alternative through largest 2TLNNRRD.

Numerical example and comparative analysis

Numerical example

A branded sports event is a collective and uniquely branded sports spectacle. It goes beyond the traditional scope of sports competitions, serving as a distinct showcase for the event’s brand. In this context, “brand” refers not only to commercial logos but also to the event’s unique identity and its value at social and cultural levels. The process of SEB building blends elements of sports, culture, and commerce, making it a powerful tool for cities to present themselves on the global stage. In today’s society, branded sports events are no longer just athletic competitions; they have evolved into a bridge that connects cities with the world, offering unique opportunities to enhance and develop a city’s brand image. As a significant branch of the sports industry, branded sports events can profoundly impact a city’s development, economy, and culture. They symbolize a city’s rising economic stature and can serve as a postcard of the city’s image. A city’s brand image is a comprehensive reflection of its historical and cultural heritage, as well as its economic development. This image includes natural resources, cultural environment, historical traditions, economic progress, scientific and educational achievements, architectural landscapes, and overall style—all of which are projected into the public’s perception. A city’s brand is not static; it evolves over time, adapting to the city’s development. As the core of urban promotion, the city brand shapes how the public perceives and understands the city. There is a subtle yet profound relationship between SEB building and city brand image, with each influencing and complementing the other. Branded sports events are an integral part of a city’s brand identity, vividly reflecting its culture and sporting prowess. Through the spectacular presentation of the event, a city can showcase its rich cultural heritage, passion for sports, and openness to the world. Conversely, a well-developed city brand provides a broad platform for SEBing. The city, as the host, offers a rich backdrop for the event, with both the event and the city blending together to present the audience with a comprehensive feast for the senses and emotions.

For example, the Xiangyang Marathon serves as a deep showcase of the city, vividly interpreting its rich history and dynamic development. A successful sports event can help shape a positive brand image for a city or region, highlighting qualities of health, vitality, and unity, while also enhancing visibility and attractiveness, drawing in more tourists and investment. Hosting branded sports events has a positive impact on economic, tourism, and commercial development. Large-scale events attract significant attention from audiences and the media, driving growth in related industries such as hospitality, catering, retail, and other service sectors, creating jobs, and promoting the city’s economic development.

The interaction between SEB building and city brand image is not a one-off process but a continuous cycle. Successful events not only enhance the city’s brand but also attract more attention, resources, and investments, further driving the city’s overall development. The improvement of the city’s brand, in turn, provides strong support for the success of future events, creating a virtuous cycle. In the era of globalization and information, the integration of SEB building and city brand image is becoming increasingly important. Branded sports events are a showcase of urban culture, a vivid reflection of a city’s vitality and inclusiveness. The city’s brand, on the other hand, provides a broader stage for the event, enhancing its attractiveness and uniqueness. The SEB building evaluation is MAGDM. Five potential host cities or regions \(DA_{i} \left( {i = 1,2,3,4,5} \right)\) are to chosen through four attributes (Table 1).

Table 1 Four attributes for SEB building evaluation.
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Five host cities or regions \(DA_{i} \left( {i = 1,2,3,4,5} \right)\) are assessed using linguistic scales (refer to Table 264) under four attributes involving three experts \(DE_{t} \left( {t = 1,2,3} \right)\), each with their respective weight \(d\omega = \left( {{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3},{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3},{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}} \right)\) in the assessment process.

Table 2 Linguistic information scales and 2TLNNs.
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The SEB building evaluation is operated based on 2TLNN-Com-ExpTODIM-GRA technique.

Step 1 Construct the 2TLNN-matrix \(DM = \left[ {DM_{ij}^{\left( t \right)} } \right]_{5 \times 4} \left( {t = 1,2,3} \right)\) (Tables 3, 4, and 5).

Table 3 Decision data from \(DE_{1}\).
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Table 4 Decision data from \(DE_{2}\).
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Table 5 Decision data from \(DE_{3}\).
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Then, from 2TLNNWA technique, the \(DM = \left[ {DM_{ij} } \right]_{5 \times 4}\) is operated (Table 6).

Table 6  The \(DM = \left[ {DM_{ij} } \right]_{5 \times 4}\).
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Step 2 Normalize the \(DM = \left[ {DM_{ij} } \right]_{5 \times 4}\) into \(NDM = \left[ {NDM_{ij} } \right]_{5 \times 4}\) (See Table 7).

Table 7 The \(NDM = \left[ {NDM_{ij} } \right]_{5 \times 4}\).
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Step 3 Operate the weight (Table 8):

Table 8 Weight information.
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Step 4 Operate the relative weight (Table 9):

Table 9 Relative weight information.
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Step 5 Operate the \(2TLNNDD = \left( {2TLNNDD_{ij} } \right)_{5 \times 4}\) (Table 10):

Table 10 \(2TLNNDD = \left( {2TLNNDD_{ij} } \right)_{5 \times 4}\).
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Step 6 Operate the 2TLNNPIDS and 2TLNNNIDS (Table 11).

Table 11 2TLNNPIDS and 2TLNNNIDS.
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Step 7 Operate the \(2TLNNGRC\left( {\zeta_{ij}^{ + } } \right)\) and \(2TLNNGRC\left( {\zeta_{ij}^{ - } } \right)\) (Tables 12, 13).

Table 12 \(2TLNNGRC\left( {\zeta_{ij}^{ + } } \right)\).
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Table 13 \(2TLNNGRC\left( {\zeta_{ij}^{ - } } \right)\).
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Step 8 Operate the weighted \(2TLNNGRC\left( {\zeta_{ij}^{ + } } \right)\) and weighted \(2TLNNGRC\left( {\zeta_{ij}^{ - } } \right)\) (See Tables 14, 15).

Table 14 The weighted \(2TLNNGRC\left( {\zeta_{ij}^{ + } } \right)\).
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Table 15 The weighted \(2TLNNGRC\left( {\zeta_{ij}^{ - } } \right)\).
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Step 9 Establish the \(2TLNNGRD\left( {\zeta_{i}^{ + } } \right)\) and \(2TLNNGRD\left( {\zeta_{i}^{ - } } \right)\) (Table 16).

Table 16 The \(2TLNNGRD\left( {\zeta_{i}^{ + } } \right)\) and \(2TLNNGRD\left( {\zeta_{i}^{ - } } \right)\).
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Step 10 Construct the 2TLNNRRD (Table 17).

Table 17 2TLNNRRD and order.
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The order is operated: \(DA_{4} > DA_{3} > DA_{2} > DA_{1} > DA_{5}\) and the optimal host city is \(DA_{4}\).

Sensitivity analysis

Building upon the methodology for sensitivity investigation established in reference67, an extensive parametric evaluation is carried out to systematically examine how fluctuations in key values influence the efficacy of the 2TLNN-Com-ExpTODIM-GRA technique. The model’s robustness is tested across two fundamentally different decision-maker (DM) behavioral archetypes: one characterized by risk aversion and the other by risk-seeking tendencies. Throughout the experiment, the critical parameters are adjusted across a predefined spectrum from 1 to 5. The comprehensive outcomes derived from this detailed sensitivity appraisal are meticulously compiled in Table 18, while their dynamic variations are graphically represented in Fig. 1.

Table 18 The 2TLNNRRD for different θ information.
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Fig. 1
Fig. 1
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The 2TLNNRRD for different θ information.

This analytical process aims to quantify the degree to which modifications in input parameters alter the model’s outputs and final rankings. By adopting the foundational principles from the cited source, the study ensures methodological consistency while extending the analysis to a novel decision-making context. Each parameter set within the specified range is iteratively applied, and the corresponding performance of the 2TLNN-Com-ExpTODIM-GRA technique is recorded and compared under both behavioral scenarios. This allows for a clear understanding of the model’s stability and reactivity to changing conditions.

The use of a structured sensitivity approach not only validates the reliability of the proposed technique but also highlights its adaptability to different decision-making personalities. The results underscore how specific parameter values can drive significantly different outcomes depending on the risk propensity of the decision-maker. Such insights are critical for practical applications, as they inform users about the model’s behavior in diverse realistic settings. Furthermore, the dual presentation of results—in both tabular and graphical formats—facilitates a multi-faceted interpretation, catering to different analytical preferences and enhancing the accessibility of the findings.

Ultimately, this rigorous examination serves to reinforce the credibility and applicability of the 2TLNN-Com-ExpTODIM-GRA technique, demonstrating its operational flexibility under varying parametric and behavioral conditions.

Comparative analysis

Then, 2TLNN-Com-ExpTODIM-GRA technique is compared with 2TLNNWA63, 2TLNNWG63, 2TLNNWMM technique68, 2TLNNWDMM model68, 2TLNN-MABAC approach69, 2TLNN-CODAS technique70, 2TLNN-GRA technique71, 2TLNN-CLVA technique72 and 2TLNN-TODIM technique64. The comparative result is operated in Tables 19 and 20.

Table 19 Order for different techniques.
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Table 20 Core features and limitations for different techniques.
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The proposed 2TLNN-Com-ExpTODIM-GRA method is systematically compared with nine existing techniques, including aggregation operators (2TLNNWA, 2TLNNWG, 2TLNNWMM, 2TLNNWDMM) and MADM approaches (2TLNN-MABAC, 2TLNN-CODAS, 2TLNN-GRA, 2TLNN-TODIM, 2TLNN-CLVA). The comparative results in Tables 18 and 19 demonstrate its superior performance in handling 2TLNNs under uncertainty.

Key Advantages of the Proposed 2TLNN-Com-ExpTODIM-GRA Method are outlined:

  • Integration of Behavioral Psychology and Relational Analysis

    Unlike conventional methods (e.g., 2TLNNWA or 2TLNN-CODAS), which rely solely on mathematical aggregation or distance measures, the proposed method combines TODIM’s prospect theory (capturing risk-aversion behavior) with Grey Relational Analysis (GRA) to enhance decision stability. This hybrid approach ensures that rankings align with human cognitive preferences while maintaining robustness in ambiguous scenarios.

  • Enhanced Sensitivity via Exponential Function

    Traditional TODIM uses linear functions, which may fail to distinguish alternatives with closely ranked scores. By introducing an exponential gain/loss function, the proposed method amplifies critical differences between attributes, leading to sharper and more reliable rankings. This innovation reduces the risk of rank reversal, especially in cases with subtle criterion variations.

  • Superior Adaptability in Complex Environments

    While existing methods excel in specific contexts (e.g., 2TLNNWMM for attribute correlations or 2TLNN-MABAC for boundary analysis), they often lack versatility. The proposed framework integrates behavioral modeling, relational analysis, and nonlinear sensitivity enhancement, making it more adaptable to highly uncertain and dynamic decision-making environments.

The 2TLNN-Com-ExpTODIM-GRA method outperforms existing techniques by balancing psychological realism, discrimination power, and computational robustness. Its holistic design is particularly valuable for high-stakes applications like medical diagnosis or supply chain risk assessment, where both data uncertainty and human judgment play pivotal roles.

Managerial insights

Based on the research content and the novel 2TLNN-Com-ExpTODIM-GRA method proposed, the following three management implications can be derived, encompassing theoretical significance, practical application, and policy advice.

  1. 1.

    Theoretical Advancement This approach makes a substantial contribution to MAGDM by effectively capturing the complexity of human cognition and real-world ambiguity. Through two-type linguistic neutrosophic sets, it provides a sophisticated mathematical framework that processes qualitative, imprecise expert judgments more accurately than conventional fuzzy models. The integration of the ExpTODIM model incorporates behavioral economics principles, recognizing that decision-makers are influenced by psychological factors like loss aversion rather than being purely rational. Combined with distance measures and grey relational analysis, this creates a comprehensive theoretical model that mirrors how experts actually think and compare alternatives under uncertainty.

  2. 2.

    Practical Application The method serves as a powerful tool for sports event organizers, brand managers, and marketing agencies in strategic planning and investment decisions. It enables comprehensive brand health audits, quantifying competitive strengths and weaknesses through measurable attributes. For sponsorship directors, it provides a data-driven approach to demonstrate value to partners beyond traditional metrics like viewership, facilitating more strategic partnerships and justifying investment decisions. The method’s group decision capability allows integration of cross-functional insights, while its attribute prioritization feature helps managers allocate resources more effectively to maximize returns and enhance long-term competitiveness.

  3. 3.

    Policy Implications The methodology offers policymakers a scientific framework for evaluating and planning major sports events. Municipal and regional authorities can incorporate this approach in bid assessment processes and post-event audits, shifting focus from short-term infrastructure planning to long-term brand legacy development. The system enables measurement of crucial but often overlooked policy goals such as social cohesion, cultural promotion, and urban branding. This leads to more informed public spending on events, ensuring they serve as sustainable catalysts for regional development rather than merely spectacular occasions, ultimately creating lasting positive impacts on host communities.

Conclusion

The evaluation of SEB building is a comprehensive analysis of a sports event’s performance, influence, and long-term development in the market. It involves multiple dimensions, including brand awareness, audience engagement, economic benefits, and socio-cultural impact. First, brand awareness is key to measuring the event’s popularity with the public, assessed through media coverage, social media interaction, and advertising effects. Second, audience engagement and experience are important criteria, reflecting the emotional connection and depth of interaction between the event and its audience. Third, the event’s economic impact on the host city, such as growth in tourism, hospitality, and retail sectors, is also a critical indicator of successful brand building. Lastly, the socio-cultural influence of the brand, especially its role in enhancing the city’s image and global recognition, reflects the long-term value of the event. Through a comprehensive brand building evaluation, event organizers can optimize their branding strategies and enhance the event’s market competitiveness and social impact. The evaluation of SEB building requires addressing MAGDM challenges. Recently, the ExpTODIM method and GRA have been utilized as effective tools for managing such decision-making scenarios. To better handle the inherent uncertainty in this process, 2TLNSs have emerged as a powerful approach for representing and processing uncertain or imprecise information. This study introduces a novel method, the 2TLNN-Com-ExpTODIM-GRA technique, which integrates the strengths of 2TLNSs, ExpTODIM, and GRA to provide a comprehensive solution to MAGDM problems. By combining these components, the proposed framework is designed to evaluate and rank alternatives in a more robust and reliable manner, particularly in the context of SEB building, where uncertainty and subjective judgments often play a significant role. To demonstrate the practicality and effectiveness of this approach, a numerical example is conducted. This validation process highlights how the 2TLNN-Com-ExpTODIM-GRA method can be applied to real-world SEB building evaluations, effectively addressing the complexities of decision-making under uncertain conditions. The results confirm the utility of this integrated technique, offering a structured and efficient solution for tackling MAGDM problems in similar domains. The key contribution of this paper is the creation of the ExpTODIM-GRA method, designed to solve MAGDM problems using 2TLNSs. This approach is particularly applied to the evaluation of SEB building.

Although the 2TLNN-Com-ExpTODIM-GRA method proposed in this study provides a novel and powerful decision-making framework for evaluating sports event brand building, effectively integrating linguistic uncertainty, decision-makers’ psychological behaviors, and grey relational analysis, any research has certain limitations. These limitations precisely point the way for future exploration.

This study has some shortcomings that warrant further discussion First, the method is relatively complex in terms of model construction. Its computational process involves exponential functions, distance formulas, and correlation calculations, which require users to have a strong mathematical background. This may, to some extent, limit its popularity and application in management practice. Second, although the paper provides a numerical example to verify the feasibility of the method, it lacks a large-scale real-world case study to comprehensively test its robustness and effectiveness in complex real-world environments. Finally, key parameters in the model (such as the loss attenuation coefficient θ in ExpTODIM) mostly rely on expert subjective settings or default values, and their sensitivity and potential impact on the final ranking results have not been fully discussed. This may introduce a degree of subjectivity and uncertainty into the decision-making outcomes.

In light of this, future research can proceed in the following directions to address the existing shortcomings and advance the field First, developing intuitive software implementations with graphical user interfaces would significantly enhance accessibility for practitioners. By hiding complex computations behind user-friendly dashboards for inputting indicators and visualizing results, the method could become a practical "out-of-the-box" tool for sports brand managers and government agencies. Second, comprehensive empirical validation through real-world case studies is essential. Applying the method to evaluate major sporting events like the Olympics or World Cup using actual decision-maker data would verify its practical utility. Systematic comparisons with traditional methods like AHP or TOPSIS would objectively demonstrate its advantages and optimal use cases. Third, the model’s parameterization and structural capabilities need enhancement. Implementing machine learning algorithms such as genetic algorithms could optimize parameter determination based on historical data, reducing subjectivity. Expanding the framework to incorporate expert weight determination methods and advanced fuzzy sets like probabilistic linguistic term sets would strengthen its ability to handle complex uncertainties, making the decision-support system more robust and comprehensive.