A novel hesitant fuzzy tensor-based group decision-making approach with application to heterogeneous wireless network evaluation

Abstract

In decision environments characterized by vagueness and uncertainty, traditional models often struggle to accommodate the inherent hesitation in expert judgments. To address this challenge, this study introduces the novel concept of the Hesitant Fuzzy Tensor (HFT), a multidimensional extension of hesitant fuzzy sets, capable of representing multiple opinions across complex criteria spaces. The proposed HFT framework captures hesitation more effectively by organizing hesitant fuzzy data within tensorial structures, enabling more comprehensive analysis in group decision-making scenarios. Theoretical foundations of HFT are rigorously developed, including formal definitions, fundamental operations, and algebraic properties. Several theorems are presented to establish the mathematical consistency and operational soundness of the structure. Furthermore, a group decision-making algorithm tailored for HFT is formulated, leveraging aggregation operators and a scoring mechanism that accounts for maximum hesitation values. To demonstrate the practical utility of the proposed framework, an application to the selection of optimal heterogeneous wireless communication networks is conducted. Multiple wireless technologies are evaluated under various performance criteria using expert assessments expressed in hesitant fuzzy form. The results highlight the robustness, interpretability, and effectiveness of the HFT-based approach in complex real-world decision problems. This research not only advances the theoretical landscape of hesitant fuzzy modeling but also provides a scalable and realistic tool for uncertainty-based decision analysis in emerging technological domains.

Introduction

In real-world decision-making scenarios, uncertainty and hesitation frequently arise due to incomplete, imprecise, or conflicting expert opinions. Traditional fuzzy models often fail to adequately capture the hesitation inherent in human judgment. Hesitant fuzzy sets (HFS), introduced to address this issue, allow decision makers to express uncertainty by providing multiple possible membership values. However, existing HFS-based models are typically limited to low-dimensional data structures and struggle with complex, multi-criteria decision problems involving diverse expert inputs.

To overcome these limitations, this study proposes the concept of the Hesitant Fuzzy Tensor (HFT), which extends hesitant fuzzy logic into a multidimensional tensor framework. The HFT model enables the representation and manipulation of hesitant information across multiple criteria, alternatives, and decision makers simultaneously. In addition to establishing its fundamental structure, we define key operations, derive algebraic properties, and prove relevant theorems that ensure mathematical soundness.

Furthermore, a group decision-making algorithm based on HFT is developed to support rational selection under uncertainty. The proposed method is applied to a practical case study involving the evaluation of heterogeneous wireless communication networks, demonstrating the model’s effectiveness in handling complex, uncertain, and high-dimensional decision environments.

In modern data analysis, the methodologies used to interpret and process collected information have progressed significantly. However, as datasets become increasingly complex–due to uncertain measurements, incomplete human expertise, or multidimensional relationships–conventional crisp set approaches often fail to capture underlying ambiguity. To address this limitation, advanced mathematical theories have been introduced to enable robust reasoning under uncertainty. Among them, Fuzzy Set (FS) theory, pioneered by Zadeh in 1965¹, provides a flexible way of expressing partial truths by assigning membership degrees between 0 and 1. This foundational framework has since proven instrumental in numerous domains where imprecision is inherent.

Building upon FS theory, researchers have proposed several enhancements to tackle more sophisticated real-world scenarios. One such development is cut set theory, crucial for analyzing fuzzy matrices (FMs), where Fan and Liu^19,20 contributed significantly by formulating decomposition theorems to improve FM applicability. Anti-fuzzy theory^21,22 later emerged, offering a complementary structure for algebraic modeling in uncertain contexts. Expanding the capabilities of fuzzy modeling to multidimensional spaces, Chen and Lu^23,24 merged fuzzy theory with tensor algebra to propose fuzzy tensors (FT) and intuitionistic fuzzy tensors (IFT). These innovations enable effective representation and analysis of complex, high-order data, making them particularly suitable for modern data science problems.

A further leap in uncertainty modeling was achieved with Atanassov’s introduction of Intuitionistic Fuzzy Sets (IFS) in 1986², which incorporate both membership and non-membership degrees with the constraint that their sum does not exceed unity. This dual representation allows a more refined modeling of hesitation or indeterminacy in data. Applications of IFS in decision analysis are numerous, particularly through Intuitionistic Fuzzy Matrices (IFMs), which have been widely adopted for practical evaluations. Several researchers have refined decomposition techniques for IFMs: Yuan et al.³ applied cut sets for structural analysis, Muthuraji and Sriram⁴ utilized cut matrices for simplification, Lee and Jeong⁵ introduced decompositions into nilpotent and symmetric parts, and Murugadas and Lalitha⁶ employed implication operators for algebraic factorization.

Recognizing the expressive limitations of IFS, Yager⁷ introduced the Pythagorean Fuzzy Set (PFS) in 2013, where the sum of squares of membership and non-membership degrees is constrained to be . This relaxed condition provides increased flexibility in representing uncertainty, especially in complex multi-criteria decision-making (MCDM) problems. The practical utility of PFS has been demonstrated through enhancements of established decision methods–for example, Zhang and Xu⁸ extended the TOPSIS method to handle PFS environments, reinforcing the applicability of fuzzy logic in real-world decisions.

Meanwhile, tensor-based methodologies have gained prominence in applied mathematics and data analysis, especially since 2005^9,10. Tensors generalize matrices to higher dimensions, allowing for richer representations of structured data. Their decomposition–analogous to matrix factorization–has become a vital technique in extracting patterns from multi-dimensional data. Prominent decomposition methods include Tucker Decomposition^13,14, which offers a PCA-like structure but becomes computationally expensive for high-order tensors; CP Decomposition^15,16, which represents tensors as sums of rank-one components, although identifying the correct rank can be difficult; and Tensor Train (TT) Decomposition^17,18, which addresses efficiency by decomposing high-order tensors into sequences of third-order cores. The origins of tensor decomposition can be traced to Hitchcock’s early work in 1927^11,12, and continued advancements have enabled wide applications in signal processing, image analysis, and machine learning.

In response to the growing need for tools that can address both human hesitation and data complexity, this study proposes the concept of the Hesitant Fuzzy Tensor (HFT). By integrating hesitant fuzzy logic into the tensor framework, HFT allows for more comprehensive modeling of multi-criteria decision problems. The proposed model defines algebraic operations, establishes key properties and theorems, and introduces a novel decision-making algorithm. Its practical effectiveness is demonstrated through an application to heterogeneous wireless network selection, highlighting the model’s potential for handling real-world uncertainties in high-dimensional environments.

Motivating real-world problem: Heterogeneous wireless network selection

In today’s digital ecosystem, users and devices rely on a variety of wireless communication technologies–such as Wi-Fi, 5G, Zigbee, and LoRa–each with different performance characteristics in terms of speed, latency, energy efficiency, coverage, and cost. Selecting the most suitable heterogeneous wireless network (HWN) for a given application is a critical multi-criteria decision-making (MCDM) problem, especially in environments where user requirements, technical conditions, and expert assessments are uncertain or inconsistent.

Experts may hesitate when assigning evaluations across multiple alternatives and criteria due to incomplete data, subjective judgments, or conflicting priorities. This inherent uncertainty and hesitation make the HWN selection problem an ideal candidate for modeling using Hesitant Fuzzy Sets (HFS) and their extensions. However, existing HFS-based models are limited in handling high-dimensional data involving multiple experts, alternatives, and criteria simultaneously.

To address this challenge, we propose a new decision-making model based on Hesitant Fuzzy Tensors (HFTs), which can capture the complex, multi-way relationships and hesitations encountered in HWN selection problems. The proposed framework is theoretically rigorous and practically applicable to real-world technology evaluation scenarios where multiple expert opinions must be fused under uncertainty.

Contribution and novelty of the proposed HFT framework

The proposed work introduces a novel Hesitant Fuzzy Tensor (HFT) model for multi-criteria group decision-making under uncertainty. While traditional hesitant fuzzy models such as hesitant fuzzy sets, hesitant fuzzy numbers, and hesitant fuzzy decision matrices have been extensively applied in decision theory, they exhibit limitations in representing multi-way data and retaining the full spectrum of expert hesitation.

The main contributions and distinguishing novelties of the proposed HFT framework are as follows:

• Tensor-Based Extension of Hesitant Fuzzy Models: The HFT model extends conventional hesitant fuzzy approaches into a higher-order tensor structure. This allows simultaneous modeling of alternatives, criteria, and decision-makers within a unified framework, enhancing structural expressiveness.

• Preservation of Multidimensional Hesitancy: Unlike prior models that defuzzify or average hesitant values early in the process, HFT retains the full set of hesitant values across all dimensions throughout computation. This preserves uncertainty information and improves interpretability.

• Unified Aggregation and Ranking Mechanism: A robust aggregation mechanism based on tensor union operations and score functions is proposed. It avoids inconsistencies introduced by separate handling of expert inputs, and allows for direct ranking of alternatives based on normalized tensor scores.

• Scalability and Real-World Applicability: The model is naturally scalable to large decision spaces with multiple experts and criteria, making it suitable for complex systems such as wireless networks, healthcare planning, and financial analysis.

• Comparative Advantage Over Traditional MCDM Methods: Through a comparative case study, we demonstrate that the HFT model provides more nuanced and reliable results than classical MCDM techniques like TOPSIS, MOORA, EDAS, WASPAS, and GRA – particularly when dealing with vague, hesitant, and group-based evaluations.

These contributions establish the proposed HFT framework as a significant extension of existing hesitant fuzzy models and a powerful tool for decision-making in uncertain and multidimensional environments.

Structure of article

This article is organized as: Section Introduction gives introduction. Section Basic Definitions gives basic definitions. Section Hesitant Fuzzy Tensor proposes novel approach, examples of novel approach, operations, properties and basic theorems. Section Group decision-making algorithm based on hesitant fuzzy tensors (HFT) presents proposed decision making algorithm, problem statement of heterogeneous wireless networks and solution of this problem by using proposed algorithm. Section Theoretical Correctness and superiority of the proposed approach gives advantages and empirical support. Section Reflexivity gives comparative analysis and limitations. And at the end, we gave conclusion and future research directions.

Basic definitions

In this section we gave basic definitions of fuzzy set, hesitant fuzzy set, tensor and fuzzy tensor. These basic definitions are helpful for next study.

Definition 1

A Fuzzy Set (FS), introduced by Zadeh in 1965¹, is a generalization of a classical set where each element has a membership degree ranging between 0 and 1. Formally, given a universe of discourse X, a fuzzy set A is defined as

where denotes the membership function that quantifies the degree to which x belongs to A.

Definition 2

A Hesitant Fuzzy Set (HFS) was introduced to handle situations where there is hesitation in assigning membership values. As defined by Torra and Narukawa²⁵, an HFS on a universe X is characterized by a function that assigns to each element a set of possible membership degrees:

where is a finite set of membership values expressing the hesitation or uncertainty of x belonging to the set.

Definition 3

A Tensor is a multidimensional array generalizing matrices to higher orders. Formally, an m-order tensor over a field is an element of the tensor product space

where each denotes the dimension in the kth mode^9,10. Tensors are widely used to represent multiway data in signal processing, machine learning, and data analysis.

Definition 4

A Fuzzy Tensor (FT) extends the concept of fuzzy sets into the tensor domain by associating a membership degree to each element of an m-order tensor. Formally, an FT is defined as

where are the tensor elements, and is the fuzzy membership function defined on each tensor entry^23,24.

Hesitant fuzzy tensor

In real-world scenarios, especially under uncertain and conflicting expert opinions, decision-makers often hesitate among several possible membership degrees when evaluating an element. This leads to the development of the hesitant fuzzy set (HFS) framework, introduced to capture such hesitation. To extend this concept for multi-dimensional, structured, and dynamic data representation, we define the Hesitant Fuzzy Tensor (HFT) – a higher-order generalization of hesitant fuzzy matrices and hesitant fuzzy vectors, suitable for applications involving complex decision systems, multidimensional clustering, and intelligent analytics.

Definition 5

Let be a universe of discourse. A hesitant fuzzy tensor of order k and dimension is a mapping:

where for , and denotes a finite set of possible membership degrees expressing the hesitation of experts in assigning a precise membership value. Each is known as a hesitant fuzzy element (HFE).

Example 1

2D Hesitant Fuzzy Tensor (Matrix Form)

Consider a hesitant fuzzy tensor of order 2 and dimension :

Each entry in the tensor denotes the set of possible membership degrees due to hesitation in evaluation.

Example 2

3D Hesitant Fuzzy Tensor (Tensor Cube)

Let be a hesitant fuzzy tensor of order 3 and dimension :

Example 3

4D Hesitant Fuzzy Tensor in Healthcare Evaluation

Suppose we are evaluating medical devices (i), criteria (j), experts (k), and scenarios (l). Let the HFT represent hesitant assessments:

This enables capturing hesitation across multiple dimensions and stakeholders.

Example 4

Application in Smart City Decision Making

In a smart city context, consider decisions influenced by location (i), infrastructure type (j), sensor type (k), and time slot (l). A sample HFT could be:

This allows urban planners to analyze the degree of support or concern for different technologies and deployment strategies over time.

Basic operations on hesitant fuzzy tensor

Hesitant Fuzzy Tensors (HFTs) extend hesitant fuzzy sets into multi-dimensional spaces, enabling rich modeling of complex, uncertain information. To support analysis and decision-making over HFTs, it is essential to define core operations. These operations are generally performed element-wise across corresponding hesitant fuzzy elements (HFEs) in tensors of identical shape. In what follows, we describe and illustrate the most fundamental operations.

1. Addition

Definition 6

Let and be two HFTs. The addition of HFEs is given by:

Example 5

Let and , then:

2. Subtraction

Definition 7

The subtraction is defined by:

Example 6

If and :

3. Union

Definition 8

Example 7

, :

4. Intersection

Definition 9

Example 8

If and :

5. Complement

Definition 10

The complement of a HFT is defined as:

Example 9

If :

6. Algebraic Sum

Definition 11

Example 10

Let , :

7. Algebraic Product

Definition 12

Example 11

, :

8. Norm (L2-norm-like for HFE)

Definition 13

For each HFE:

Example 12

:

9. T-norm (Minimum-based)

Definition 14

Example 13

, :

10. T-conorm (Maximum-based)

Definition 15

Example 14

, :

Basic properties of hesitant fuzzy tensor

Hesitant Fuzzy Tensors (HFTs) allow each element to hold a set of possible membership degrees instead of a single value. To ensure meaningful operations and analysis, it is critical to explore the mathematical properties of HFTs. The following section outlines several fundamental properties – monotonicity, idempotency, boundedness, reflexivity, and convexity – each with a formal proof and example.

1. Monotonicity

Definition 16

An operation on two HFTs and is said to be monotonic if:

for all tensors , where the ordering is defined as: iff for every , there exists such that .

Proof

Let and so that . Let . Under addition:

Since and , the property holds.

2. Idempotency

Definition 17

An operation is idempotent if:

Proof

For hesitant fuzzy intersection (minimum-based):

Example 15

Let . Then:

3. Boundedness

Definition 18

A HFT is bounded if for all entries , we have:

Proof

By definition, every hesitant fuzzy element (HFE) is a non-empty finite subset of [0, 1]. Thus:

Example 16

Let – clearly within [0, 1].

4. Reflexivity

Definition 19

A hesitant fuzzy relation (or tensor expressing similarity) is reflexive if:

Proof

In a reflexive similarity context, the degree to which an element is similar to itself is always maximum (i.e., 1). Hence, diagonal elements must be:

Example 17

For similarity across 2 items:

This tensor satisfies reflexivity.

5. Convexity

Definition 20

A HFT is convex if for any , and any , we have:

Proof

Let and consider :

If , then convexity holds. However, if is closed under convex combination, the condition is satisfied.

Example 18

Let . For , , and :

Basic theorems of hesitant fuzzy tensor

To build a rigorous mathematical framework for Hesitant Fuzzy Tensors (HFTs), it is essential to establish foundational theorems that govern their structure and operations. These theorems help verify closure, consistency, and the preservation of essential properties such as associativity and commutativity, which are vital in ensuring their applicability in real-world modeling and computational frameworks.

Theorem 1

(Closure under Addition) Let and be two hesitant fuzzy tensors of the same dimension. Then, the element-wise addition defined by:

is also a hesitant fuzzy tensor.

Proof

Since and , we have . To preserve membership within [0, 1], we redefine addition as:

Thus, every result of lies in [0, 1], and since the result is a finite set, is a valid HFT.

Example 19

Let and . Then,

Theorem 2

(Associativity of Algebraic Sum) Let , , and be hesitant fuzzy tensors. The algebraic sum is associative, i.e.,

Proof

Let us verify for a single entry . Define:

We have:

Applying truncation at 1:

which is the same as:

Hence, associativity holds.

Example 20

Let . Then:

Theorem 3

(Commutativity of Intersection) Let and be two hesitant fuzzy tensors. Then:

Proof

The hesitant fuzzy intersection is defined as:

Since for all real numbers a, b, the result remains unchanged when switching and .

Example 21

Let and . Then:

Theorem 4

(Distributivity of over ) Let be HFTs. Then:

Proof

For each position :

Since by real number lattice theory, the result follows.

Example 22

Let :

Theorem 5

(De Morgan’s Laws for HFT)

Let and be hesitant fuzzy tensors. Then, the following De Morgan’s laws hold:

Proof

Let and . Then:

On the other hand:

Thus, both sides are equal. Similarly for the second identity using .

Example 23

Let . Then:

Theorem 6

(Duality Theorem) Let and be hesitant fuzzy tensors. Then:

Proof

This directly follows from the De Morgan’s laws for hesitant fuzzy tensors, as proven above. Therefore, the operations of union and intersection are dual under the complement operation.

Example 24

Let :

Theorem 7

(Identity Element Theorem) There exists an identity hesitant fuzzy tensor such that for any hesitant fuzzy tensor :

Proof

Let for all indices. Then:

Example 25

Let . Then:

Computational complexity analysis

The computational complexity of the proposed HFT-based decision-making framework depends primarily on the following operations:

• Union of Tensors: The aggregation step requires element-wise union across m decision-makers for each position. Assuming average hesitant set length l, the union has a complexity of per element (due to duplicate removal and sorting).

• Score Computation: For each alternative , the algorithm computes the maximum hesitant value for each criterion. This requires operations.

• Normalization and Ranking: Score normalization and ranking over n alternatives is performed in time.

Hence, the overall complexity of the algorithm is approximately:

This is linear in the number of alternatives and criteria, but increases linearly with the number of decision-makers and logarithmically with the size of hesitant sets.

Memory Usage: The HFT structure stores sets of floating-point values per position, requiring space in the order of . For large-scale applications or higher-order tensors, optimized storage or sparse representations may be adopted to improve performance.

Scalability Considerations: Although the model is computationally efficient for moderate dimensions, in real-time or large-scale systems it may benefit from parallelization or distributed tensor processing methods.

Group decision-making algorithm based on hesitant fuzzy tensors (HFT)

Introduction. In many real-life scenarios, decision-makers hesitate among several membership degrees when evaluating alternatives. To accommodate such uncertainty, we propose a novel group decision-making algorithm built on the framework of Hesitant Fuzzy Tensors. This method supports multi-criteria, multi-expert evaluations and provides a structured way to fuse and analyze these hesitant inputs.

Proposed algorithm steps

Let there be decision-makers (DMs), alternatives , and criteria . Each DM provides hesitant fuzzy evaluations across criteria and alternatives.

• Step 1:
  Construct Individual Hesitant Fuzzy Tensors. Each decision-maker provides evaluations in the form of a hesitant fuzzy tensor , where:

  
• Step 2:
  Aggregate Evaluations Across Decision-Makers. Compute a collective hesitant fuzzy tensor using the union-based aggregation:

  

  Remove duplicates and sort each resulting set.
• Step 3:
  Compute Score Function for Each Alternative. Define a score function for each alternative based on average of max values:

  
• Step 4:
  Normalize Scores (Optional). Normalize the score vector if required:

  
• Step 5:

  Rank the Alternatives. Rank the alternatives based on descending order of or . The alternative with the highest score is the most preferable.

• Step 6:

  Sensitivity or Robustness Check (Optional). Analyze the impact of variations in hesitant values or weight distributions (if used later) to validate the robustness of the ranking.

Highlights of the algorithm

• Accommodates hesitation and ambiguity in expert judgments.

• Preserves multi-dimensional structure using tensor representation.

• Provides a simple yet powerful aggregation and scoring method.

• Can be extended to incorporate criteria weights, decision-maker weights, or fuzzy distance measures.

Case study: Heterogeneous wireless networks

The increasing adoption of digital technologies and the Internet of Things (IoT) has significantly intensified the demand for reliable and diverse wireless communication networks. As users and devices become more mobile and interconnected, effective selection of wireless networks is crucial to ensure optimal connectivity. Heterogeneous Wireless Networks (HWNs) encompass a variety of communication technologies, including low-power wide-area networks (LPWANs) such as LoRa and Zigbee, local area networks like Wi-Fi, and mobile cellular networks including LTE and 5G. Each technology offers distinct advantages and disadvantages in terms of coverage, capacity, energy consumption, and security. Figure 1 illustrates the various HWN technologies considered in this study.

Fig. 1.

Heterogeneous wireless network structure.

Selecting the most suitable wireless network for a specific application is a complex process influenced by multiple, often competing, criteria. Traditional methods that rely on a single factor such as cost or bandwidth are typically insufficient to capture the multifaceted nature of this decision. Consequently, fuzzy neural networks combined with Multi-Criteria Decision-Making (MCDM) techniques have proven effective in evaluating and ranking various alternatives based on several criteria.

• : A low-power wide-area network (LPWAN) technology designed for long-range communication with minimal energy consumption. It is widely used in IoT applications such as smart cities and agricultural monitoring.

• : Worldwide Interoperability for Microwave Access (WiMAX) provides long-distance, high-speed broadband wireless connectivity, serving as an alternative to wired broadband for both fixed and mobile users.

• : Also known as 802.11ax, Wi-Fi 6 is the latest generation of wireless local area network technology, designed to enhance capacity, speed, and efficiency in environments with numerous connected devices.

• (Long-Term Evolution Advanced): An enhanced version of LTE offering improved coverage, capacity, and data rates, incorporating advanced features to support multimedia and mobile broadband services.

• : A low-power, low-data-rate wireless protocol often used for short-range communication in applications such as industrial control systems and home automation.

• (New Radio): The next-generation mobile communication technology aiming to provide higher capacity, reduced latency, and faster speeds, supporting diverse applications from IoT to enhanced mobile broadband.

The selection of the most appropriate network is based on six critical criteria:

• : The delay in data transmission, crucial for real-time communication applications.

• : The data throughput capacity, essential for applications requiring high data rates.

• : Power consumption levels, important for battery-powered and IoT devices.

• : The geographic range over which the network can reliably operate.

• : Measures including encryption and data protection to safeguard communications.

• : The expenses associated with network deployment, operation, and maintenance.

Problem context and expert evaluation details

The case study focuses on selecting the most suitable heterogeneous wireless network technology from six alternatives: Zigbee (Z1), WiFi (Z2), WiMAX (Z3), Bluetooth (Z4), LTE (Z5), and 5G NR (Z6). This decision is complex due to the variability in performance, cost, energy efficiency, and user requirements across different contexts (e.g., IoT, mobile broadband, or smart infrastructure).

Expert selection process

To ensure informed and balanced evaluations, three domain experts (denoted as , , and ) were selected based on the following criteria:

• Minimum 5 years of professional experience in wireless communication technologies,

• Representation across academia, industry, and research sectors,

• Prior involvement in technology assessment or network deployment planning.

The experts were asked to evaluate the six technologies based on a common set of technical and practical criteria using hesitant fuzzy linguistic assessments.

Evaluation criteria

The following six criteria were used to assess the wireless technologies:

1. Data Rate (C1): The achievable transmission speed,

2. Coverage (C2): The effective operational range of the network,

3. Latency (C3): The responsiveness or delay in communication,

4. Deployment Cost (C4): The infrastructure and operational cost involved,

5. Energy Efficiency (C5): Power consumption characteristics of devices and network nodes,

6. Security & Reliability (C6): The resilience and protection against attacks or faults.

These criteria were chosen based on their relevance to real-world wireless deployment decisions, particularly in smart environments where trade-offs between performance and cost are significant.

The hesitant fuzzy evaluations provided by the experts reflect their uncertainty or hesitation in assigning precise membership values. Each evaluation is represented as a set of values in the [0, 1] range, capturing multiple possible ratings for each criterion-alternative pair.

Problem solution

Setup

Let:

Alternatives:

• 

• 

• 

• 

• 

• 

Criteria:

• 

• 

• 

• 

• 

• 

Decision-makers: .

Step 1: Individual Hesitant Fuzzy Tensors

Tensors for :

Step 2: Aggregation (Union + Sorting)

Step 3: Score Computation

Use:

Step 4: Normalization

Step 5: Ranking

Order by descending :

Figure 2 shows the output of proposed approach.

Fig. 2.

Output of the proposed approach.

Sensitivity and robustness analysis

To examine the reliability of the proposed Hesitant Fuzzy Tensor (HFT)-based decision-making model under minor variations in expert inputs, two controlled experiments were conducted.

Experiment 1: Perturbation of hesitant values

In this experiment, we modified certain hesitant membership values given by decision-maker D3. Specifically, small perturbations of ±0.1 were applied to evaluations across two randomly selected criteria. Following aggregation and recalculation, the resulting alternative rankings remained unchanged. This suggests that the model is stable under mild fluctuations in the input data.

Experiment 2: Inclusion of an additional decision-maker

To further evaluate the robustness of the HFT model, a fourth decision-maker (denoted as D4) was synthetically introduced. This expert’s evaluations were similar to existing ones but included slight variations to reflect a new perspective. The hesitant fuzzy evaluations of D4 across six alternatives ( to ) and six criteria ( to ) are as follows:

This tensor was aggregated with those from D1, D2, and D3 using the union operation. After aggregation, the maximum membership values across each criterion were extracted for scoring. The updated scores for all alternatives are:

Final Ranking with D4 Included:

Interpretation:

The inclusion of an additional decision-maker (D4) resulted in only slight changes in score magnitudes, while the overall ranking structure remained intact. The top three alternatives–, , and –continued to occupy the highest positions. This confirms the robustness of the HFT-based approach under dynamic group structures and uncertain expert assessments.

Incorporating weights for criteria and decision-makers

To enhance the realism of the proposed HFT-based decision-making model, we extend the original framework to incorporate weighting mechanisms for both criteria and decision-makers.

Weighted aggregation of decision-makers

Let be the weight of the p-th decision-maker, such that . The weighted union aggregation is defined as:

Where scalar multiplication of weights over hesitant fuzzy elements is applied entry-wise as:

Weighted scoring across criteria

Let be the weight assigned to criterion , with . The weighted score for each alternative is then computed as:

These extensions enable prioritization based on domain-specific knowledge or stakeholder preferences. In the absence of prior knowledge, equal weights may still be assumed as a baseline.

Computational complexity and scalability discussion

The proposed HFT-based decision-making algorithm operates on a multidimensional structure where hesitant evaluations are given across n alternatives, k criteria, and m decision-makers. We now evaluate the computational cost of each stage:

• Tensor Aggregation (Union): Each hesitant fuzzy tensor entry contains a finite set of membership values, typically of size l (average cardinality of a hesitant set). For each element , the union of m sets (each of size ) is computed and then sorted. The complexity is per element, leading to an overall aggregation complexity of .

• Score Computation: For each alternative, the algorithm computes the maximum value from each aggregated hesitant fuzzy element. This requires operations.

• Normalization and Ranking: The final scores for n alternatives are normalized and sorted for ranking, which takes time.

Total Complexity:

Scalability Considerations: While the algorithm is efficient for small-to-moderate scale decision problems, its performance may degrade when applied to high-dimensional datasets with many experts or large hesitant sets. To address this, future versions of the model may integrate the following optimizations:

• Parallelized aggregation and score computation,

• Sparse tensor representation for memory efficiency,

• Pruning techniques to discard low-influence hesitant values.

These improvements would enhance the feasibility of the HFT-based model in big-data decision environments such as smart cities, healthcare systems, and IoT networks.

Extended sensitivity and robustness analysis

To rigorously test the reliability of the HFT-based decision-making model under uncertain or evolving decision data, we performed a multi-level sensitivity analysis with controlled perturbations.

Scenario 1: Perturbation of hesitant values

We introduced perturbations to randomly selected hesitant values in the evaluations of decision-maker 2 across three criteria and all six alternatives. The resulting normalized scores and rankings were computed and compared with the baseline.

As Table 1 shows, the ranking order remains unchanged, indicating strong robustness to small fluctuations in hesitant data.

Table 1.

Effect of hesitant value perturbation on rankings.

Alternative	Original Rank	Perturbed Rank
Z1	3	3
Z2	5	5
Z3	2	2
Z4	4	4
Z5	6	6
Z6	1	1

Scenario 2: Addition of a new decision-maker

A fourth decision-maker () with a slightly different evaluation profile was added. We re-aggregated the hesitant tensors and recalculated the scores. The top-3 alternatives remained unchanged.

• Top-3 Alternatives (before):

• Top-3 Alternatives (after):

Interpretation: These experiments demonstrate that the HFT-based approach produces stable and consistent rankings under moderate changes to decision inputs. This supports its suitability for real-world decision-making where data may be uncertain, incomplete, or evolving over time.

Future work may involve Monte Carlo simulation-based sensitivity analyses to assess model behavior under stochastic fluctuations across large datasets.

Theoretical correctness and superiority of the proposed HFT approach

The proposed Hesitant Fuzzy Tensor (HFT)-based decision-making model has been established on a rigorous mathematical foundation. Several formal theorems and algebraic properties–including closure under addition, associativity, commutativity, boundedness, De Morgan’s laws, and the identity element theorem–have been thoroughly proven in Section Hesitant fuzzy tensor. These proofs confirm the internal consistency, correctness, and operational soundness of the proposed HFT structure.

To further support the practical correctness and effectiveness of the model, we applied the HFT-based group decision-making algorithm to a real-world problem involving the evaluation of heterogeneous wireless networks. This case study demonstrates the model’s ability to aggregate hesitant expert evaluations and produce interpretable and robust rankings under uncertainty.

Advantages over traditional MCDM methods

The comparative evaluation against established MCDM methods–WASPAS, MOORA, EDAS, TOPSIS, and GRA–highlights the superiority of the proposed approach. The key advantages are as follows:

• Handling Expert Hesitation: Traditional methods often require single-valued or crisp inputs, whereas the HFT model allows for sets of possible membership values, effectively capturing expert hesitation and ambiguity.

• High-Order Data Modeling: The tensor structure enables the representation of multi-dimensional relationships among alternatives, criteria, and decision-makers, which is not feasible with classical matrix-based methods.

• Improved Accuracy and Interpretability: As shown in Table 1, the proposed method consistently yields higher performance scores across the alternatives. This reflects a more precise evaluation that accommodates uncertainty and hesitancy.

• Robustness in Dynamic Environments: Unlike methods that assume static or uniform data inputs, the HFT approach is inherently adaptable to changing criteria weights or evolving expert judgments due to its flexible tensor representation.

• Computational Feasibility: Despite its expressive power, the algorithm involves simple element-wise operations and max-based scoring, making it computationally efficient for practical deployment.

Empirical support

Tables 2 and 3 demonstrate that the 5G NR technology (Z6) is consistently ranked as the best-performing alternative across all methods. However, the proposed HFT-based approach assigns the highest score and exhibits clearer separation among alternatives, thereby improving the decision-maker’s confidence. Figure 3 visually affirms the superiority of the proposed method.

Table 2.

Comparative Results of MCDM Techniques

Method	Z_1	Z_2	Z_3	Z_4	Z_5	Z_6
WASPAS	0.3041	0.2519	0.2809	0.2340	0.2359	0.3445
MOORA	0.6240	0.3903	0.5306	0.4900	0.3694	0.7555
EDAS	0.6220	0.1991	0.4524	0.1722	0.1382	0.9260
TOPSIS	0.5602	0.3653	0.4811	0.2808	0.2630	0.7084
GRA	0.5349	0.4332	0.4833	0.4454	0.4146	0.5726
Proposed	0.7333	0.5833	0.7833	0.6500	0.5166	0.9666

Table 3.

Ranking and Best Treatment Method by Different Techniques

Methods	Ranking	Best HWN
WASPAS
MOORA
EDAS
TOPSIS
GRA
Proposed

Fig. 3.

The comparative analysis.

These results confirm that the HFT-based model not only aligns with conventional MCDM outcomes but also offers enhanced granularity, interpretability, and correctness in environments where decision data is uncertain, hesitant, or multidimensional.

Comparative analysis

To validate the effectiveness and robustness of our proposed approach, we conducted a comprehensive comparative study against several well-established Multi-Criteria Decision-Making (MCDM) methods, including:

•WASPAS (Weighted Aggregated Sum Product Assessment)²⁶

•MOORA (Multi-Objective Optimization by Ratio Analysis)²⁷

•EDAS (Evaluation based on Distance from Average Solution)²⁸

•TOPSIS (Technique for Order Preference by Similarity to Ideal Solution)²⁹

•GRA (Grey Relational Analysis)³⁰

Table 2 shows the comparative results of proposed approach with existing techniques. The comparison was performed using evaluations provided by three domain experts specializing in heterogeneous wireless networks. Their assessments served as the foundation for our analysis, ensuring real-world applicability and credibility.

Key Observations from the Comparative Study

1. Handling Hesitation and Uncertainty: Unlike traditional MCDM techniques, our proposed hesitant fuzzy tensor-based approach effectively captures and processes uncertainty and expert hesitancy, which is often overlooked in conventional methods.

2. Flexibility in Data Representation: While WASPAS and TOPSIS rely on crisp or fuzzy inputs, our model incorporates higher-order tensor structures, allowing for more nuanced and multi-dimensional decision-making.

3. Robustness in Dynamic Environments: MOORA and EDAS perform well in static scenarios, but our method demonstrates superior adaptability in dynamic wireless network environments where criteria weights and alternatives frequently change.

4. Computational Efficiency: GRA and EDAS offer simplicity but may lack precision in complex scenarios. In contrast, our approach balances computational efficiency with high accuracy, making it suitable for real-time applications.

The results highlight that our proposed technique not only aligns with existing methods but also addresses their limitations, particularly in handling imprecise and incomplete data. This makes it a more reliable and advanced tool for decision-making in heterogeneous wireless networks.

According to the results of the WASPAS, MOORA, EDAS, TOPSIS, and GRA techniques, the 5G New Radio (NR) performs better than other heterogeneous wireless networks. The proposed approach is successful in improving decision support systems, demonstrating its efficacy and validity, according to the comparison analysis. Figure 3, which graphically depicts the 5G NR network’s greater efficiency in comparison to other networks, and Table 2 which offer comprehensive performance information.

In order to identify the best HWN, the final rankings were determined using the proposed approach along with several well-known MCDM techniques, including WASPAS, MOORA, EDAS, TOPSIS, and GRA. The rankings generated by each approach are compared in Table 3. The overall results consistently indicate that the 5G NR network is the best-performing HWN, with only minor differences in the rankings of individual HWNs across the methods. Despite slight variations in the rankings of the other assessed networks, this consistency reinforces the reliability of 5G NR as the top HWN selection.

Limitations of Traditional MCDM methods and the need for HFT

Although traditional MCDM methods such as TOPSIS, MOORA, EDAS, WASPAS, and GRA have been widely used across application domains, they exhibit several limitations when dealing with real-world decision problems that involve uncertainty, hesitation, and high-dimensional data. A summary of these limitations and the corresponding advantages of the proposed HFT model is provided below.

• Single-Valued Input Assumption: Most traditional methods assume that decision inputs are crisp or scalar values. They are not equipped to handle hesitation in expert evaluations, where multiple membership values may exist for a single criterion-alternative pair. In contrast, HFT directly supports hesitant fuzzy sets, allowing for richer and more realistic modeling of expert uncertainty.

• Flat Decision Matrices: Traditional MCDM models use 2D matrices to represent alternatives and criteria. This structure cannot inherently accommodate evaluations from multiple decision-makers or multidimensional relationships. HFT overcomes this by using a 3D (or higher) tensor structure, which allows simultaneous handling of alternatives, criteria, and experts in a unified framework.

• Aggregation Constraints: Methods like TOPSIS or MOORA rely on normalization or distance calculations that are sensitive to scale and may not robustly aggregate hesitant or uncertain data. The HFT approach uses union-based aggregation and score-maximization that are better suited for non-exact inputs.

• Limited Expressiveness under Ambiguity: Classical methods struggle in scenarios with subjective or conflicting evaluations from experts. HFT accommodates multiple membership degrees, retaining the inherent ambiguity in human judgments instead of forcing premature precision.

• Loss of Information during Simplification: Many methods require defuzzification or averaging of hesitant inputs, leading to potential information loss. HFT preserves all original hesitant values throughout the computation pipeline, enhancing interpretability and traceability.

These structural and theoretical distinctions underscore the value of the HFT model, especially in uncertain and multi-expert decision contexts such as network selection, medical treatment planning, or urban infrastructure evaluation. The comparative results shown in Tables 2 and 3 along with Fig. 3, quantitatively reinforce the model’s robustness and practicality.

Emphasis on results and practical insights

The ranking results derived from the proposed HFT-based approach demonstrate its robustness and effectiveness in handling multidimensional hesitant information. As shown in Table 2, Alternative (5G NR) consistently ranks first across all decision-maker inputs, affirming its superior performance in terms of data rate, latency, and reliability.

Compared to traditional MCDM methods (TOPSIS, MOORA, EDAS, etc.), the HFT model preserves hesitant information without premature averaging or defuzzification. This leads to more nuanced and reliable ranking outcomes. In addition, the HFT framework accommodates complex expert evaluations while maintaining interpretability – a key advantage for real-world deployment in domains such as telecommunications, healthcare, and finance.

The results indicate that the proposed model is not only theoretically sound but also practically applicable, especially in decision-making scenarios with uncertainty and hesitation among expert inputs.

Conclusion and future research directions

Conclusion

This article has introduced a novel decision-making framework based on the concept of the Hesitant Fuzzy Tensor (HFT). By integrating the flexibility of hesitant fuzzy sets with the multi-dimensional capacity of tensors, the proposed approach effectively addresses complex decision environments characterized by ambiguity, hesitation, and multi-source evaluations. The methodology captures expert hesitation in assigning membership degrees and enables the aggregation and analysis of multidimensional decision data in a structured and scalable format. A comprehensive group decision-making algorithm using HFT was developed and validated through an application to the selection of heterogeneous wireless communication networks. The results demonstrate that the HFT model provides a robust and adaptable solution, offering improved precision and reliability compared to traditional hesitant fuzzy models.

Future research directions

Cross-domain validation for enhanced generality

Although the wireless network selection case study demonstrates the effectiveness of the proposed HFT model, further validation using real-world data from additional domains is essential for demonstrating the full potential and robustness of the framework. Future studies will aim to apply the HFT-based decision-making algorithm in the following domains:

• Healthcare: Evaluating treatment plans or medical technologies where decisions are based on multiple expert opinions under uncertain clinical conditions.

• Finance: Portfolio management or credit scoring, where different investment options must be assessed under market uncertainty and investor hesitation.

• Urban Planning: Selecting infrastructure projects, sustainability policies, or smart city components involving multidimensional and conflicting expert criteria.

By conducting empirical studies in these domains, we aim to validate the flexibility, adaptability, and robustness of the HFT framework under diverse decision-making contexts.

• Integration with Other Fuzzy Extensions: Incorporating neutrosophic, intuitionistic, or interval-valued hesitant fuzzy components into the tensor framework could further improve modeling of more complex uncertainties.

• Optimization Algorithms: Hybridizing the HFT decision-making process with metaheuristic optimization methods (e.g., PSO, GA, ACO) may aid in solving large-scale multi-objective problems.

• Dynamic and Time-Dependent Tensors: Developing temporal hesitant fuzzy tensors could extend the method to dynamic decision environments, such as real-time systems or evolving expert opinions.

• Machine Learning Synergy: Embedding HFT into machine learning models (e.g., clustering, classification, recommendation systems) may lead to improved interpretability and performance in uncertain data-driven tasks.

• Software and Computational Tools: Creating a dedicated toolbox or package for hesitant fuzzy tensor operations would promote wider adoption and simplify implementation for researchers and practitioners.

• Open-Source Toolbox Development: A future objective is to develop a Python- or MATLAB-based open-source toolbox that supports HFT construction, operations, aggregation, and decision-making workflows. Such a toolkit would significantly enhance usability for researchers and practitioners across domains.

In conclusion, the hesitant fuzzy tensor model lays a strong foundation for advanced decision analysis under uncertainty and opens new avenues for future exploration in both theory and application.

Applicability in diverse domains

Although the primary case study in this work focuses on telecommunications, the proposed Hesitant Fuzzy Tensor framework is designed to be domain-agnostic. The following application scenarios are suggested to demonstrate its adaptability:

• Healthcare: HFT can be used to evaluate treatment strategies for chronic diseases (e.g., cancer), where multiple experts assess treatment options under uncertain outcomes and patient conditions.

• Finance: Portfolio selection or credit risk assessment often involves multiple conflicting criteria and opinions from financial analysts. HFT enables robust modeling of such hesitant evaluations.

• Urban Planning: In smart city initiatives, decisions about infrastructure, environmental sustainability, or public safety require integration of conflicting expert views across multiple stakeholders and locations. HFT supports multidimensional assessment and priority mapping in such contexts.

These domains highlight the model’s potential to serve as a unified decision support framework under uncertainty across sectors. In future work, we aim to develop full case studies to validate HFT in these domains.

Author contributions

All Authors Contributed Equally.

Data availability

All data generated or analyzed for this article are included in this article.

Declarations

Competing interests

The authors declare no competing interests.