A Fractional q-Rung Orthopair Fuzzy Tensor Framework for Dynamic Group Decision-Making: Application to Smart City Renewable Energy Planning

Abstract

In complex decision-making scenarios, such as smart city renewable energy project selection, decision-makers must contend with multi-dimensional uncertainty, conflicting expert opinions, and evolving temporal dynamics. This study introduces a novel Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT)-based group decision-making methodology that integrates the flexibility of q-rung orthopair fuzzy sets with tensorial representation and fractional-order dynamics. The proposed framework allows for the modeling of positive and negative membership degrees in a multi-dimensional, time-dependent structure while capturing memory effects inherent in expert evaluations. A detailed case study involving six renewable energy alternatives and six criteria demonstrates the method’s ability to aggregate expert opinions, compute fractional dynamic scores, and provide robust, reliable rankings. Comparative analysis with existing approaches, including classical q-ROFSs, intuitionistic fuzzy sets, and weighted sum methods, highlights the superior discriminative power, consistency, and dynamic sensitivity of the Fq-ROFT approach. Sensitivity analysis confirms the robustness of the top-ranked alternatives under variations in expert weights and fractional orders and membership perturbations. The study concludes by discussing the advantages, limitations, and future research directions of the proposed methodology, establishing Fq-ROFT as a powerful tool for dynamic, high-dimensional, and uncertain group decision-making applications.

1. Introduction

The selection of renewable energy projects in smart cities involves highly complex decision-making scenarios, where multiple stakeholders must evaluate alternatives based on conflicting and dynamic criteria. Traditional multi-criteria decision-making (MCDM) methods, such as the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [1], provide a structured approach to ranking alternatives based on their closeness to an ideal solution. However, real-world decision-making is often characterized by uncertainty, vagueness, and imprecision arising from subjective human judgments, incomplete or noisy information, and dynamically changing environmental, economic, and social conditions [2,3,4]. To overcome these limitations, fuzzy extensions of MCDM methods have been widely adopted. For example, fuzzy TOPSIS enables linguistic and imprecise evaluations to be modeled using fuzzy numbers [5]. Intuitionistic fuzzy sets further incorporate hesitation by simultaneously considering membership and non-membership degrees [2], while hesitant fuzzy sets allow decision-makers to express multiple plausible assessments [6]. Pythagorean fuzzy sets relax the orthogonality constraint, providing enhanced discrimination under high uncertainty [7]. Despite these advancements, classical and advanced fuzzy methods generally treat evaluations as static and fail to capture temporal dynamics or memory-dependent effects, which are crucial in real-world, evolving smart city energy planning contexts [8].

To address these limitations, we propose a Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT) framework for dynamic group decision-making (GDM). This approach combines three powerful paradigms:

• q-Rung Orthopair Fuzzy Sets (q-ROFSs) to represent flexible positive and negative membership degrees while satisfying the generalized orthogonality condition, allowing nuanced modeling of acceptance and rejection intensities [7].
• Tensor Algebra to capture multi-dimensional relationships among alternatives, criteria, and experts, providing a holistic representation of complex interactions.
• Fractional-Order Dynamics using Caputo derivatives to encode temporal memory, capturing the hereditary effects of past evaluations on present decision outcomes.

This unified Fq-ROFT framework enables the modeling of high-dimensional, time-dependent uncertainty while accommodating heterogeneous expert opinions, making it particularly suited for renewable energy project selection in smart cities.

1.1. Literature Review

Existing research has explored fuzzy, intuitionistic, hesitant, and Pythagorean fuzzy MCDM methods to address uncertainty in decision-making [2,5,6,7,8]. However, most approaches assume static evaluations and lack mechanisms to incorporate memory or temporal dynamics. Fractional fuzzy frameworks have recently attracted increasing attention due to their ability to model memory-dependent uncertainty and hereditary effects in complex decision environments. Such approaches have been applied in uncertain dynamical systems, control theory, and decision analysis, where historical information influences current evaluations [9,10,11,12]. In parallel, tensor-based decision-making models have demonstrated strong capability in representing high-dimensional interactions among alternatives, criteria, experts, and time stages, particularly in group decision-making problems [13,14,15,16]. However, existing studies predominantly investigate fractional fuzzy models and tensor-based frameworks independently. To the best of our knowledge, a unified dynamic group decision-making framework that simultaneously integrates q-rung orthopair fuzzy sets, tensor representations, and fractional-order dynamics has not yet been systematically reported in the literature.

1.2. Relationship with $(m,n)$-Fuzzy Sets, $(2,1)$-Fuzzy Sets and SR-Fuzzy Sets

Generalized orthopair fuzzy models have been proposed in the literature to overcome the restrictive constraints of classical intuitionistic and Pythagorean fuzzy sets. Among the most influential extensions are $(m,n)$-fuzzy sets [17], $(2,1)$-fuzzy sets [18], and SR-fuzzy sets [19], which enhance expressive capability by introducing flexible power-based constraints and decision-oriented aggregation mechanisms. The concept of $(m,n)$-fuzzy sets was formally introduced as a generalized orthopair fuzzy framework in which the membership degree $\mu$ and non-membership degree $\nu$ satisfy the condition

$${\mu }^m+{\nu }^n\le 1,m,n\ge 1.$$

This formulation allows for asymmetric control over the admissible uncertainty region and generalizes intuitionistic and Pythagorean fuzzy sets as special cases. The authors further developed weighted aggregation operators and applied the $(m,n)$-fuzzy model to multi-criteria decision-making (MCDM) problems, demonstrating improved flexibility and discrimination power in uncertain decision environments. As a notable special case, $(2,1)$-fuzzy sets were proposed by fixing $m=2$ and $n=1$. This model expands the feasible decision space beyond intuitionistic fuzzy sets while preserving interpretability. The foundational study on $(2,1)$-fuzzy sets established their algebraic properties, introduced weighted aggregation operators, and demonstrated their effectiveness in MCDM applications. These results showed that $(2,1)$-fuzzy sets can better capture hesitation and asymmetric uncertainty in expert evaluations compared with classical orthopair models. In addition, SR-fuzzy sets were introduced to provide a structured refinement-based representation of orthopair uncertainty. The SR-fuzzy framework focuses on ranking-oriented decision-making by incorporating score and reliability mechanisms, along with weighted aggregated operators tailored for practical decision analysis. Applications in group decision-making confirmed their usefulness in handling conflicting expert opinions while maintaining computational simplicity. Although $(m,n)$-fuzzy sets, $(2,1)$-fuzzy sets, and SR-fuzzy sets significantly enhance the modeling capability of orthopair fuzzy theory, these frameworks are inherently static and primarily defined at scalar or matrix levels. They do not provide mechanisms for modeling temporal evolution, memory-dependent effects, or higher-order interactions among multiple decision dimensions. The proposed Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT) framework can be viewed as a strict generalization of these models. When the fractional order $\alpha =1$ and the tensor reduces to second order, the Fq-ROFT formulation naturally encompasses $(m,n)$-fuzzy sets under a conservative embedding by setting

$$q=max\{m,n\},{\mu }^q+{\nu }^q\le 1.$$

Moreover, the tensor representation enables simultaneous modeling of alternatives, criteria, experts, and time as independent modes, while the fractional-order dynamics explicitly capture memory and hereditary effects absent in $(m,n)$-, $(2,1)$-, and SR-fuzzy frameworks. From a decision-making perspective, these classical generalized orthopair fuzzy models can therefore be regarded as static, low-dimensional special cases within the broader Fq-ROFT paradigm. The proposed framework preserves their expressive advantages while extending them to dynamic, high-dimensional, and time-aware group decision-making environments.

Relation to the Existing q-Rung Orthopair Fuzzy Literature

The preservation of the q-rung orthopair constraint under aggregation and fusion operations has been extensively studied in the q-ROFS literature. Yager [20] first introduced q-rung orthopair fuzzy sets and demonstrated that a wide class of nonlinear operators, including algebraic, probabilistic, and power-based aggregations, preserve the condition ${\mu }^q+{\nu }^q\le 1$. Subsequently, Refs. [21,22,23,24,25] provided rigorous analyses of closure properties for q-ROFS aggregation operators, including weighted averaging, geometric, and hybrid fusion rules. More recently, extensions of q-ROFSs to dynamic and decision-making contexts have confirmed that constraint preservation remains valid under temporal weighting and iterative aggregation schemes [26,27,28,29]. The exponential-type fusion mechanism adopted in this study is conceptually aligned with the probabilistic sum and product operators analyzed in these works, which are known to maintain admissibility under mild conditions. The theoretical results established in this paper generalize these existing findings from static q-rung orthopair fuzzy sets to fractional-order, tensor-valued structures. Hence, the proposed Fractional q-Rung Orthopair Fuzzy Tensor framework can be viewed as a natural and mathematically consistent extension of well-established q-ROFS theory.

1.3. Limitations of Existing Methods in High-Dimensional and Time-Sensitive Decision-Making

Despite the significant progress achieved by intuitionistic fuzzy sets, Pythagorean fuzzy sets, $(m,n)$-fuzzy sets, and q-rung orthopair fuzzy models, most existing decision-making approaches remain insufficient when applied to high-dimensional and time-sensitive problems.

1.3.1. Limitations in High-Dimensional Modeling

The majority of fuzzy multi-criteria decision-making methods rely on vector- or matrix-based representations, where evaluations are organized in two-dimensional structures such as alternative–criterion or expert–criterion matrices. While effective for small-scale problems, this representation becomes restrictive in complex group decision-making environments involving multiple experts, criteria, alternatives, and contextual factors simultaneously. Higher-order interactions among these dimensions are either ignored or flattened into weighted averages, leading to an inevitable loss of structural information. As a result, correlations among experts, interdependencies among criteria, and cross-effects across decision stages cannot be explicitly modeled.

1.3.2. Limitations in Time-Sensitive Decision-Making

Existing fuzzy and orthopair fuzzy approaches generally assume that expert assessments are static and independent of time. Even when repeated evaluations are considered, they are typically aggregated without accounting for temporal dependency or historical influence. Such approaches implicitly assume that current decisions depend only on present information, neglecting the fact that expert opinions often evolve gradually and are influenced by past experiences, previous outcomes, and accumulated knowledge. This static treatment makes traditional models unsuitable for dynamic environments where uncertainty evolves over time.

1.3.3. Combined Impact on Real-World Applications

In practical applications such as smart city renewable energy planning, decision-makers must simultaneously process high-dimensional information and adapt to time-varying conditions, including policy changes, technological advancements, and evolving stakeholder preferences. Current fuzzy decision-making models address either uncertainty or flexibility in representation, but they do not provide an integrated mechanism to handle both multi-dimensional complexity and temporal dynamics. Consequently, their decision outcomes may lack robustness, adaptability, and interpretability in evolving real-world systems. These limitations motivate the need for a unified framework capable of representing multi-dimensional decision information while explicitly incorporating temporal memory. The proposed Fractional q-Rung Orthopair Fuzzy Tensor framework addresses these challenges by integrating tensor-based high-dimensional modeling with fractional-order dynamics, thereby enabling robust and time-aware group decision-making.

1.4. Justification for the Proposed Framework in Smart City Contexts

Decision-making in smart city renewable energy planning is inherently dynamic, multi-dimensional, and permeated with uncertainty. This complexity arises from the interplay of technological, economic, environmental, and social factors that evolve over time. A smart city is not a static entity; it is a dynamic system where project evaluations are influenced by historical performance data, shifting policy landscapes, real-time sensor data, evolving public sentiment, and the collective memory of past successes or failures. The proposed Fq-ROFT framework is specifically designed to address these core smart city challenges:

• Managing Conflicting Stakeholder Opinions: Urban projects involve diverse experts (engineers, planners, environmentalists, citizens) with varying, often opposing, views. The q-rung orthopair fuzzy set component allows for the flexible and simultaneous representation of strong support ($\mu$) and strong opposition ($\nu$) towards a project, capturing the nuanced spectrum of expert and public opinion more effectively than classical fuzzy models.
• Integrating Multi-Dimensional Urban Data: Smart city planning requires the fusion of data across multiple dimensions: alternatives (e.g., different energy technologies), criteria (e.g., cost, emissions, social acceptance), experts, and time. The tensor algebra component provides a natural and structured mathematical framework to model these high-order interactions holistically, moving beyond the limitations of traditional two-dimensional decision matrices.
• Capturing Temporal Evolution and Memory Effects: The viability of a solar project or the public acceptance of a wind farm is not judged in a single moment but evolves. Policy changes, technological learning curves, and community feedback create a hereditary influence where past evaluations impact present decisions. The Caputo fractional-order derivative component inherently models this memory effect, allowing the decision model to incorporate the temporal trajectory of expert judgments and project performance, leading to more stable and informed long-term rankings.

1.5. Research Gap

Despite extensive development of fuzzy MCDM methods, there are several key gaps:

• Lack of dynamic memory modeling: Most fuzzy approaches cannot account for temporal evolution or hereditary effects of past evaluations.
• Limited multi-dimensional representation: Classical matrix-based methods fail to capture higher-order interactions among alternatives, criteria, and experts.
• Insufficient application to real-world dynamic scenarios: Smart city renewable energy planning requires both high-dimensional modeling and time-sensitive decision-making, which current methods inadequately address.

The proposed Fq-ROFT framework addresses these gaps by unifying fuzzy logic, tensor algebra, and fractional-order dynamics into a single, robust, and scalable methodology for dynamic group decision-making.

1.6. Justification for the Proposed Framework

Decision-making in smart city renewable energy planning is inherently dynamic, with project evaluations influenced not only by current data but also by historical trends, policy changes, and evolving stakeholder preferences. Fractional calculus effectively captures these memory effects [9], while tensor algebra enables structured multi-dimensional representation of criteria, alternatives, and expert evaluations [30]. Integrating these with q-rung orthopair fuzzy sets allows for flexible modeling of both positive and negative expert opinions under uncertainty. The proposed Fq-ROFT framework therefore provides a more realistic, robust, and insightful approach for complex, high-dimensional, and dynamic decision-making scenarios.

1.7. Contributions

The key contributions of this paper are as follows:

• Introduction of Fq-ROFT: We propose a novel Fractional q-Rung Orthopair Fuzzy Tensor framework for dynamic group decision-making, integrating q-rung orthopair fuzzy sets, tensor representation, and fractional-order dynamics.
• Mathematical Formalization: The paper defines Fq-ROFTs rigorously, introduces algebraic operations, and establishes fundamental properties such as boundedness, monotonicity, and commutativity.
• Development of Fq-ROFT GDM Algorithm: A group decision-making algorithm based on Fq-ROFT is developed, capable of aggregating expert opinions, computing fractional dynamic scores, and ranking alternatives in time-dependent environments.
• Application to Smart City Energy Planning: The methodology is applied to a real-world renewable energy project selection case study, demonstrating its practical effectiveness.
• Comparative and Sensitivity Analysis: The proposed framework is compared with existing fuzzy and q-ROFS-based methods, and sensitivity analysis is conducted to evaluate robustness under parameter variations.
• Modular and Extensible Framework: The Fq-ROFT model is designed to be extendable to other MCDM problems and compatible with future integration with optimization or machine learning techniques.

From

Table 1. Comparative analysis of generalized fuzzy set models.

Model	Uncertainty Representation	Hesitation Handling	Structural Dimension	Typical Applications
Fuzzy Set (FS)	Single membership degree	Implicit	Scalar/Vector	Basic decision-making, control systems
Intuitionistic Fuzzy Set (IFS)	Membership + non-membership	Partial hesitation	Vector/Matrix	Decision analysis, pattern recognition
Pythagorean Fuzzy Set (PFS)	Squared membership and non-membership	Improved hesitation modeling	Vector/Matrix	Risk analysis, MCDM
$(2,1)$-Fuzzy Set	Two membership degrees + one non-membership degree	Explicit coordinated hesitation	Vector/Matrix	Weighted aggregation, MCDM
$(m,n)$-Fuzzy Set	m-tuple membership + n-tuple non-membership	High-level generalized hesitation	Vector/Matrix	Complex MCDM, multi-expert evaluation
SR-Fuzzy Set	Symmetric and reference-based degrees	Reference-dependent hesitation	Vector/Matrix	Group decision-making
Proposed Tensor Model	Set-valued fuzzy information	Multi-source hesitation preserved	High-order Tensor	Multi-criteria, multi-expert, multi-context decision-making

, it is evident that while existing fuzzy MCDM methods successfully address uncertainty through various membership structures, they remain largely limited to static, matrix-based representations. The proposed Fq-ROFT framework uniquely integrates q-rung orthopair flexibility, tensorial multi-dimensional modeling, and fractional-order dynamics. This combination enables simultaneous handling of high-dimensional expert evaluations, temporal memory effects, and complex interdependencies, making the proposed method particularly suitable for dynamic group decision-making problems such as smart city renewable energy planning.

2. Preliminaries

This section presents the fundamental mathematical concepts and definitions that form the basis of the proposed Fq-ROFT framework, including fuzzy sets, tensors, Caputo fractional derivatives and Fractional Fuzzy Tensors.

Definition 1

([3]). Let X be a universe of discourse. A fuzzy set $\tilde{A}$ in X is characterized by a membership function ${\mu }_{\tilde{A}}:X\to [0,1]$, where ${\mu }_{\tilde{A}}\left(x\right)$ indicates the degree of membership of element $x\in X$ in the fuzzy set $\tilde{A}$.

Definition 2

([13]). A tensor is a multi-dimensional array that generalizes scalars (zero-order), vectors (first-order), and matrices (second-order) to higher orders. Formally, an N-th order tensor $\mathcal{X}$ over the real field is defined as

$$\mathcal{X}=\left(x_{i_1{i}_2\dots i_N}\right),x_{i_1{i}_2\dots i_N}\in \mathbb{R}.$$

Matrices are therefore a special case of tensors when $N=2$.

Definition 3

([31]). Building upon the classical concept of a tensor, a fuzzy tensor is defined by restricting each tensor entry to represent a degree of membership in the unit interval. Specifically, an N-th order fuzzy tensor is expressed as

$$\mathcal{F}=\left({\mu }_{i_1{i}_2\dots i_N}\right),{\mu }_{i_1{i}_2\dots i_N}\in [0,1].$$

In this context, the uncertainty of information is modeled through fuzzy membership degrees, and matrix-based fuzzy representations correspond to the second-order case of fuzzy tensors.

Definition 4

([32]). Let $f\left(t\right)$ be a sufficiently smooth function. The Caputo fractional derivative of order $\alpha \in (0,1]$ is defined as

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^t{\displaystyle \frac{f^{\prime }\left(\tau \right)}{{(t-\tau )}^{\alpha }}}d\tau ,$$

where $\mathrm{\Gamma }(·)$ denotes the Gamma function. This definition is widely used in modeling memory-dependent systems.

Definition 5

([33]). A Fractional Fuzzy Tensor (FFT) of order n and dimension $d_1\times d_2\times \cdots \times d_n$ is a multi-dimensional array $\mathcal{F}\left(t\right)=\left[{\mathcal{F}}_{i_1{i}_2\dots i_n}\left(t\right)\right]$ such that

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\mathcal{F}}_{i_1{i}_2\dots i_n}\left(t\right)=\mathrm{\Phi }({\mathcal{F}}_{i_1{i}_2\dots i_n}\left(t\right),t).$$

where:

• ${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }$ denotes the Caputo fractional derivative of order $\alpha \in (0,1]$;
• Φ is a system-specific evolution function governing the temporal dynamics;
• Each component ${\mathcal{F}}_{i_1{i}_2\dots i_n}\left(t\right)\in [0,1]$ represents the fuzzy membership at time t.

This definition allows for modeling of systems with both uncertainty (via fuzzy logic) and memory (via fractional calculus).

3. Fractional q-Rung Orthopair Fuzzy Tensor

The classical q-rung orthopair fuzzy set (q-ROFS) [7] extends Atanassov’s intuitionistic fuzzy framework by introducing a flexible parameter $q\ge 1$, allowing the positive and negative membership degrees to satisfy a generalized orthogonality condition ${\mu }^q+{\nu }^q\le 1$. This family of structures has proven effective in modeling complex decision-making scenarios where acceptance and rejection coexist in varying intensities [8]. However, q-ROFS-based models remain inherently static, representing a snapshot in time. They cannot model systems where the underlying uncertainty evolves over time or exhibits hereditary effects—where past states influence the present. Conversely, the concept of a Fractional Fuzzy Tensor (FFT) [33] incorporates temporal memory through the Caputo fractional derivative [32], enabling the modeling of dynamic, multi-dimensional fuzzy information. Meanwhile, tensor algebra [13] provides the foundational framework for representing such multi-way data structures. To unify these powerful paradigms, we introduce a novel mathematical structure: the Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT). This original construct embeds the semantics of q-rung orthopair fuzzy sets into a tensorial framework governed by fractional-order dynamics. This hybrid formulation makes it possible to represent multi-mode, time-dependent, orthopair-valued uncertainty while capturing the intrinsic memory effects found in physical, biological, and socio-economic processes [9]. The resulting model not only generalizes existing q-ROFS-based formulations but also provides a flexible and highly expressive mechanism for analyzing dynamic fuzzy information in high-dimensional environments.

Definition 6.

Let $n\in \mathbb{N}$ be the tensor order and let $d_1\times d_2\times \cdots \times d_n$ denote its dimensions. A Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT) of order n is a tensor-valued function

$$\mathcal{Q}\left(t\right)=\left[\left({\mu }_{i_1{i}_2\dots i_n}\left(t\right),{\nu }_{i_1{i}_2\dots i_n}\left(t\right)\right)\right],t\ge 0,$$

where each pair

$$\left({\mu }_{i_1{i}_2\dots i_n}\left(t\right),{\nu }_{i_1{i}_2\dots i_n}\left(t\right)\right)\in [0,1]\times [0,1]$$

represents the positive and negative membership degrees, respectively, and satisfies the q-rung orthopair constraint

$${\left({\mu }_{i_1{i}_2\dots i_n}\left(t\right)\right)}^q+{\left({\nu }_{i_1{i}_2\dots i_n}\left(t\right)\right)}^q\le 1,q\ge 1.$$

The temporal evolution of each tensor entry is governed by a Caputo fractional dynamic system of order $\alpha \in (0,1]$:

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\mu }_{i_1{i}_2\dots i_n}\left(t\right)={\mathrm{\Phi }}_{\mu }\left({\mu }_{i_1{i}_2\dots i_n}\left(t\right),t\right),$$

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\nu }_{i_1{i}_2\dots i_n}\left(t\right)={\mathrm{\Phi }}_{\nu }\left({\nu }_{i_1{i}_2\dots i_n}\left(t\right),t\right),$$

where ${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }$ denotes the Caputo fractional derivative and ${\mathrm{\Phi }}_{\mu }$, ${\mathrm{\Phi }}_{\nu }$ are system-specific functions determining the evolution of the positive and negative membership components.

The structure $\mathcal{Q}\left(t\right)$ is therefore called a Fractional q-Rung Orthopair Fuzzy Tensor because it simultaneously encodes:

• Multi-dimensional fuzzy information through the tensor framework;
• Orthopair-valued uncertainty modulated by the q-rung condition;
• Memory-dependent temporal dynamics via the fractional derivative.

To facilitate intuitive understanding of the proposed Fq-ROFT, a schematic illustration of its multi-dimensional structure is presented in Figure 1. The figure visualizes how q-rung orthopair fuzzy information is organized simultaneously across alternatives, criteria, experts, and temporal stages.

3.1. Preservation of the q-Rung Orthopair Constraint

For the Fractional q-Rung Orthopair Fuzzy Tensor to remain admissible over time, the evolution functions ${\mathrm{\Phi }}_{\mu }$ and ${\mathrm{\Phi }}_{\nu }$ must preserve the q-rung orthopair condition

$${\mu }^q\left(t\right)+{\nu }^q\left(t\right)\le 1,q\ge 1.$$

(1)

In this work, we explicitly assume that the fractional dynamic system

$$\begin{array}{cc}\hfill {\mathrm{CD}}_t^{\alpha }\mu \left(t\right)& ={\mathrm{\Phi }}_{\mu }(\mu \left(t\right),t),\hfill \\ \hfill {\mathrm{CD}}_t^{\alpha }\nu \left(t\right)& ={\mathrm{\Phi }}_{\nu }(\nu \left(t\right),t),\hfill \end{array}$$

(2)

is invariant with respect to the admissible set

$${\mathrm{\Omega }}_q=\{(\mu ,\nu )\in {[0,1]}^2\mid {\mu }^q+{\nu }^q\le 1\}.$$

More precisely, we require that for all $(\mu ,\nu )\in \partial {\mathrm{\Omega }}_q$,

$$q{\mu }^{q-1}{\mathrm{\Phi }}_{\mu }(\mu ,t)+q{\nu }^{q-1}{\mathrm{\Phi }}_{\nu }(\nu ,t)\le 0,$$

(3)

which guarantees that the vector field $({\mathrm{\Phi }}_{\mu },{\mathrm{\Phi }}_{\nu })$ is inward-pointing on the boundary of ${\mathrm{\Omega }}_q$. Under this condition, solutions of the Caputo fractional system starting from any admissible initial state remain within ${\mathrm{\Omega }}_q$ for all $t\ge 0$. This assumption ensures that the proposed Fractional q-Rung Orthopair Fuzzy Tensor preserves the orthopair feasibility throughout its temporal evolution and maintains consistency with the underlying fuzzy semantics.

Justification of Functional Forms

In the numerical examples presented in this section, exponential and trigonometric functions are employed to model the temporal evolution of membership and non-membership degrees. These functions are selected for illustrative purposes and are motivated by commonly observed patterns in realistic decision-making environments. Exponential functions are widely used to represent growth, decay, and saturation phenomena, which naturally arise in long-term planning and evaluation problems. In the context of smart city renewable energy planning, exponential trends can reflect technology adoption rates, investment escalation or depreciation, and the gradual impact of policy incentives over time. Such dynamics are consistent with decision-making scenarios where changes occur nonlinearly and exhibit memory-dependent behavior. Trigonometric functions are introduced to capture periodic or cyclical variations in expert evaluations. These variations may stem from seasonal demand fluctuations, regulatory review cycles, or recurrent socio-economic factors influencing stakeholder preferences. While simplified, trigonometric forms provide a convenient and interpretable means of representing bounded oscillatory behavior within the admissible domain of q-rung orthopair fuzzy sets. It is emphasized that the proposed Fractional q-Rung Orthopair Fuzzy Tensor framework is not restricted to these specific functional forms. Alternative empirically driven or data-calibrated functions may be incorporated without loss of generality, depending on the requirements of particular decision-making applications.

Example 1

(A Simple $2\times 2$ Fq-ROFT). Consider a second-order tensor of size $2\times 2$ with $q=2$ and fractional order $\alpha =0.8$. Let

$$\mathcal{Q}\left(t\right)=\left[\begin{array}{cc}({\mu }_{11}\left(t\right),{\nu }_{11}\left(t\right))& ({\mu }_{12}\left(t\right),{\nu }_{12}\left(t\right))\\ ({\mu }_{21}\left(t\right),{\nu }_{21}\left(t\right))& ({\mu }_{22}\left(t\right),{\nu }_{22}\left(t\right))\end{array}\right],$$

where

$$\begin{array}{cc}\hfill {\mu }_{11}\left(t\right)& =0.6E_{\alpha }(-t^{\alpha }),{\nu }_{11}\left(t\right)=0.2E_{\alpha }(-t^{\alpha }),\hfill \\ \hfill {\mu }_{12}\left(t\right)& =0.4,{\nu }_{12}\left(t\right)=0.5E_{\alpha }(-0.5t^{\alpha }),\hfill \\ \hfill {\mu }_{21}\left(t\right)& =0.7E_{\alpha }(-0.3t^{\alpha }),{\nu }_{21}\left(t\right)=0.1,\hfill \\ \hfill {\mu }_{22}\left(t\right)& =0.5,{\nu }_{22}\left(t\right)=0.5E_{\alpha }(-t^{\alpha }),\hfill \end{array}$$

and $E_{\alpha }\left(z\right)={\sum }_{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\alpha k+1)}}$ is the Mittag-Leffler function. For $q=2$ each pair satisfies

$${\mu }_{ij}^2\left(t\right)+{\nu }_{ij}^2\left(t\right)\le 1.$$

The Caputo fractional dynamics are taken as

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\mu }_{ij}\left(t\right)=-k_1{\mu }_{ij}\left(t\right),{}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\nu }_{ij}\left(t\right)=-k_2{\nu }_{ij}\left(t\right),$$

with $k_1,k_2>0$. For instance,

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\mu }_{11}\left(t\right)=-0.6k_1{E}_{\alpha }(-t^{\alpha }),{}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\nu }_{11}\left(t\right)=-0.2k_2{E}_{\alpha }(-t^{\alpha }).$$

Thus $\mathcal{Q}\left(t\right)$ is a valid Fq-ROFT fractional derivative.

Example 2

(A $3\times 3$ Sensor Evaluation Tensor). Consider a $3\times 3$ tensor representing three sensors observing three environmental conditions. Let $q=3$, $\alpha =0.7$ and define

$$({\mu }_{ij}\left(t\right),{\nu }_{ij}\left(t\right))=\left(0.3+0.2E_{\alpha }(-t^{\alpha }),0.4E_{\alpha }(-0.5t^{\alpha })\right).$$

The q-rung constraint is verified:

$${\mu }_{ij}^3\left(t\right)+{\nu }_{ij}^3\left(t\right)={\left(0.3+0.2E_{\alpha }(-t^{\alpha })\right)}^3+{\left(0.4E_{\alpha }(-0.5t^{\alpha })\right)}^3\le 1.$$

The fractional evolution is assumed to follow

$$\begin{array}{cc}\hfill {}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\mu }_{ij}\left(t\right)& =-0.04t^{1-\alpha }{E}_{\alpha ,\alpha }(-t^{\alpha }),\hfill \\ \hfill {}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\nu }_{ij}\left(t\right)& =-0.08t^{1-\alpha }{E}_{\alpha ,\alpha }(-0.5t^{\alpha }),\hfill \end{array}$$

where $E_{\alpha ,\beta }\left(z\right)$ is the two-parameter Mittag-Leffler function. These dynamics model gradual sensor stabilization; hence, the tensor is a valid Fq-ROFT.

Example 3

(A Time-Varying Risk Assessment Tensor). This example illustrates the applicability of the proposed Fq-ROFT framework under bounded cyclical decision dynamics, which commonly arise in realistic group decision-making scenarios such as seasonal energy demand assessment and periodic policy evaluation. Let $q=4$ and consider the positive and negative membership functions defined as

$$\mu \left(t\right)=0.6+0.2{sin}^2\left(\pi t\right),\nu \left(t\right)=0.3+0.1{cos}^2\left(\pi t\right),t\ge 0.$$

(4)

The above functions satisfy $0\le \mu \left(t\right),\nu \left(t\right)\le 1$ for all t and represent smooth oscillatory variations in expert preferences. Such bounded periodic behavior can reflect recurring external factors, including seasonal fluctuations, regulatory review cycles, or periodic socio-economic influences in smart city renewable energy planning. Moreover, the q-rung orthopair constraint is preserved, since

$${\mu }^q\left(t\right)+{\nu }^q\left(t\right)\le 1,\forall t\ge 0,$$

(5)

which ensures the admissibility of the evolving orthopair information throughout the decision-making process. It is emphasized that the specific functional forms employed in this example are selected for illustrative clarity and interpretability. The proposed Fq-ROFT framework is not restricted to squared trigonometric functions and can accommodate alternative empirically driven or application-specific dynamics without loss of generality.

Example 4

(A Medical Diagnosis Tensor With Memory). Consider a $4\times 2$ tensor representing four symptoms and two diagnostic tests, with $q=2$, $\alpha =0.75$. Define

$$\left({\mu }_{ij}\left(t\right),{\nu }_{ij}\left(t\right)\right)=\left(0.5+0.1E_{\alpha }(-0.3t^{\alpha }),0.3E_{\alpha }(-t^{\alpha })\right).$$

The constraint

$${\mu }_{ij}^2\left(t\right)+{\nu }_{ij}^2\left(t\right)\le 1$$

holds because

$${(0.5+0.1E_{\alpha }(-0.3t^{\alpha }))}^2\le 0.36,{\left(0.3E_{\alpha }(-t^{\alpha })\right)}^2\le 0.09.$$

A fractional progression reflecting patient recovery is given by

$$\begin{array}{cc}\hfill {}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\mu }_{ij}\left(t\right)& =-0.03t^{1-\alpha }{E}_{\alpha ,\alpha }(-0.3t^{\alpha }),\hfill \\ \hfill {}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\nu }_{ij}\left(t\right)& =-0.3t^{1-\alpha }{E}_{\alpha ,\alpha }(-t^{\alpha }).\hfill \end{array}$$

Thus $\mathcal{Q}\left(t\right)$ forms a valid diagnostic Fq-ROFT.

Example 5

(A Market Sentiment Tensor Over Time). Let a $3\times 3\times 3$ tensor describe market positivity and negativity across three sectors, three time periods and three economic indicators. Choose $q=5$, $\alpha =0.6$ and define

$${\mu }_{ijk}\left(t\right)=0.2+0.1cos\left(t\right),{\nu }_{ijk}\left(t\right)=0.3{sin}^2\left(t\right)E_{\alpha }(-t^{\alpha }).$$

Check the q-rung orthogonality:

$${\mu }_{ijk}^5\left(t\right)+{\nu }_{ijk}^5\left(t\right)={\left(0.2+0.1cost\right)}^5+{\left(0.3{sin}^{2}t{E}_{\alpha }(-t^{\alpha })\right)}^5\le 1,$$

which holds for all $t\ge 0$. The fractional temporal flow is described by the Caputo derivatives

$$\begin{array}{cc}\hfill {}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\mu }_{ijk}\left(t\right)& ={\displaystyle \frac{0.1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^t{\displaystyle \frac{-sin\tau }{{(t-\tau )}^{\alpha }}}d\tau ,\hfill \\ \hfill {}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\nu }_{ijk}\left(t\right)& ={\displaystyle \frac{0.3}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^t{\displaystyle \frac{\frac{d}{d\tau }\left({sin}^2\tau E_{\alpha }(-{\tau }^{\alpha })\right)}{{(t-\tau )}^{\alpha }}}d\tau .\hfill \end{array}$$

These integrals can be computed numerically, confirming that the three-dimensional time-evolving structure is a valid Fractional q-Rung Orthopair Fuzzy Tensor.

3.2. Numerical Computation of Caputo Derivatives

In practice, the Caputo fractional derivative of a function $f\left(t\right)$ can be approximated using discretisation schemes. A widely used method is the L1 formula for $\alpha \in (0,1)$:

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }f\left(t_n\right)\approx {\displaystyle \frac{1}{\mathrm{\Gamma }(2-\alpha )}}\sum _{j=0}^{n-1}{b}_j^{\left(\alpha \right)}\left[f\left(t_{n-j}\right)-f\left(t_{n-j-1}\right)\right]\mathrm{\Delta }{t}^{-\alpha },$$

where $b_j^{\left(\alpha \right)}={(j+1)}^{1-\alpha }-j^{1-\alpha }$ and $\mathrm{\Delta }t$ is the time step. This scheme preserves the memory effect inherent in fractional dynamics and is employed in the numerical evaluation of the Fq-ROFT temporal equations.

3.2.1. Purpose and Interpretation of Examples

Examples 1–5 are presented to progressively illustrate the modeling capabilities and flexibility of the proposed Fq-ROFT framework.

• Example 1 serves as a fundamental validation example, demonstrating that even a simple $2\times 2$ tensor satisfies the q-rung orthopair constraint under fractional-order dynamics. Its purpose is to verify the basic admissibility and feasibility of the proposed definition.
• Example 2 highlights the applicability of Fq-ROFTs to sensor-based or monitoring systems. It illustrates how homogeneous tensor entries with time-varying behavior can model stabilization or convergence phenomena in dynamic environments.
• Example 3 emphasizes the ability of the framework to handle multi-dimensional risk or uncertainty assessments by introducing a higher-order tensor. This example demonstrates that complex, time-dependent interactions across multiple dimensions can be modeled consistently.
• Example 4 focuses on interpretability in decision-support systems, such as medical diagnosis, where memory effects and gradual evolution of assessments are essential. It illustrates how fractional dynamics capture recovery or progression trends over time.
• Example 5 is designed to showcase scalability and expressiveness. By considering a three-dimensional tensor with oscillatory and decaying behaviors, it demonstrates the suitability of Fq-ROFTs for large, heterogeneous, and time-evolving decision contexts such as market sentiment or socio-economic analysis.

Together, these examples confirm that the proposed framework is mathematically sound, dynamically consistent, and adaptable to a wide range of real-world applications.

3.2.2. Extended Illustration: Example 5 in a $5\times 5$ Setting

To further demonstrate the scalability and structural consistency of the proposed Fq-ROFT framework, we extend Example 5 to a $5\times 5$ tensor configuration. Consider a second-order Fq-ROFT

$$Q\left(t\right)={\left[\left({\mu }_{ij}\left(t\right),{\nu }_{ij}\left(t\right)\right)\right]}_{5\times 5},i,j=1,2,\dots ,5,$$

with rung parameter $q=5$. Define the tensor entries as

$$\begin{array}{cc}\hfill {\mu }_{ij}\left(t\right)& =0.2+0.05sin\left({\displaystyle \frac{i+j}{5}}t\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\nu }_{ij}\left(t\right)& =0.3exp\left(-{\displaystyle \frac{i+j}{5}}t\right).\hfill \end{array}$$

(7)

For all $i,j$ and $t\ge 0$, the q-rung orthopair constraint holds:

$${\mu }_{ij}^5\left(t\right)+{\nu }_{ij}^5\left(t\right)\le 1,$$

since ${\mu }_{ij}\left(t\right)\le 0.25$ and ${\nu }_{ij}\left(t\right)\le 0.3$, ensuring admissibility of every tensor entry. The temporal evolution of each component is governed by fractional-order dynamics:

$$\begin{array}{cc}\hfill \mathrm{C}{D}_t^{\alpha }{\mu }_{ij}\left(t\right)& =0.05{\displaystyle \frac{i+j}{5}}cos\left({\displaystyle \frac{i+j}{5}}t\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \mathrm{C}{D}_t^{\alpha }{\nu }_{ij}\left(t\right)& =-0.3{\displaystyle \frac{i+j}{5}}exp\left(-{\displaystyle \frac{i+j}{5}}t\right),\hfill \end{array}$$

(9)

where $\alpha \in (0,1]$ is the fractional order. This $5\times 5$ configuration illustrates that the Fq-ROFT framework scales naturally to higher dimensions while preserving admissibility, interpretability, and dynamic consistency. It also demonstrates that increasing dimensionality does not introduce additional modeling complexity, making the framework suitable for large-scale group decision-making problems involving many alternatives and criteria.

3.3. Operations on Fractional q-Rung Orthopair Fuzzy Tensors

To effectively manipulate and analyze multi-dimensional fractional q-rung orthopair fuzzy information, a rich collection of algebraic operations is required. These operations must respect the tensorial nature of the data, preserve the q-rung orthogonality condition, and remain compatible with the fractional dynamic interpretation of Fq-ROFTs. In this section, we develop a comprehensive family of operators ranging from elementary relations to advanced fusion rules. Particular emphasis is given to t-norm- and t-conorm-based interactions, which are critical in fuzzy decision theory. Each operation is accompanied by illustrative examples to clarify its behavior in practical settings.see Appendix A for examples.

Let

$$\mathcal{Q}\left(t\right)=\left[({\mu }_{i_1\dots i_n}\left(t\right),{\nu }_{i_1\dots i_n}\left(t\right))\right],\mathcal{P}\left(t\right)=\left[({\rho }_{i_1\dots i_n}\left(t\right),{\eta }_{i_1\dots i_n}\left(t\right))\right]$$

be two Fq-ROFTs of the same order and dimension.

Definition 7.

Complement

The complement of $\mathcal{Q}\left(t\right)$ is defined elementwise as

$${\mathcal{Q}}^c\left(t\right)=\left[({\nu }_{i_1\dots i_n}\left(t\right),{\mu }_{i_1\dots i_n}\left(t\right))\right].$$

Proposition 1.

The complement ${(\mu ,\nu )}^c=(\nu ,\mu )$ preserves the q-rung orthopair constraint.

Proof. 

Since $(\mu ,\nu )$ satisfies ${\mu }^q+{\nu }^q\le 1$, exchanging the components gives

$${\nu }^q+{\mu }^q={\mu }^q+{\nu }^q\le 1.$$

Hence, the complement is admissible.    □

Definition 8.

t-norm (Intersection)

Let T be any continuous t-norm (e.g., minimum, product, or Yager t-norm). Intersection is defined by

$$(\mu ,\nu ){\cap }_T(\rho ,\eta )=\left(T(\mu ,\rho ),S_T(\nu ,\eta )\right)$$

where $S_T$ is the t-conorm dual to T.

Popular choices:

1.

Minimum t-norm

$$T_{min}(a,b)=min(a,b).$$

2.

Product t-norm

$$T_{\prod }(a,b)=ab.$$

3.

Yager t-norm

$$T_Y(a,b)=1-min{\left\{1,{(1-a)}^p+{(1-b)}^p\right\}}^{1/p}.$$

Proposition 2.

Let T be a continuous t-norm and S its dual t-conorm. Then

$$({\mu }_1,{\nu }_1)\cap ({\mu }_2,{\nu }_2)=(T({\mu }_1,{\mu }_2),S({\nu }_1,{\nu }_2))$$

preserves the q-rung constraint.

Proof. 

Since $T({\mu }_1,{\mu }_2)\le min\{{\mu }_1,{\mu }_2\}$ and $S({\nu }_1,{\nu }_2)\le max\{{\nu }_1,{\nu }_2\}$, we obtain

$$T{({\mu }_1,{\mu }_2)}^q+S{({\nu }_1,{\nu }_2)}^q\le {\mu }_i^q+{\nu }_i^q\le 1,$$

for $i=1$ or 2. Hence the result holds.    □

Definition 9.

t-conorm (Union)

For a t-conorm S,

$$(\mu ,\nu ){\cup }_S(\rho ,\eta )=\left(S(\mu ,\rho ),T_S(\nu ,\eta )\right)$$

where $T_S$ is the dual t-norm.

Common t-conorms:

1.

Maximum t-conorm

$$S_{max}(a,b)=max(a,b).$$

2.

Probabilistic sum

$$S_P(a,b)=a+b-ab.$$

Proposition 3.

The union

$$({\mu }_1,{\nu }_1)\cup ({\mu }_2,{\nu }_2)=(S({\mu }_1,{\mu }_2),T({\nu }_1,{\nu }_2))$$

preserves the q-rung orthopair constraint.

Proof. 

Because $S({\mu }_1,{\mu }_2)\le max\{{\mu }_1,{\mu }_2\}$ and $T({\nu }_1,{\nu }_2)\le min\{{\nu }_1,{\nu }_2\}$, it follows that

$$S{({\mu }_1,{\mu }_2)}^q+T{({\nu }_1,{\nu }_2)}^q\le {\mu }_i^q+{\nu }_i^q\le 1,$$

for some $i\in \{1,2\}$.    □

Definition 10.

Scalar Multiplication

For $\lambda \in [0,1]$,

$$\lambda \otimes (\mu ,\nu )=(\lambda \mu ,1-(1-\nu \left)\lambda \right).$$

Proposition 4.

For $\lambda \in [0,1]$, the scalar multiplication

$$\lambda \otimes (\mu ,\nu )=(\lambda \mu ,1-{(1-\nu )}^{\lambda })$$

preserves the q-rung constraint.

Proof. 

Since $\lambda \in [0,1]$ and $q\ge 1$, we have ${\left(\lambda \mu \right)}^q\le \lambda {\mu }^q$. Moreover, $1-{(1-\nu )}^{\lambda }\le \nu$; hence,

$${(1-{(1-\nu )}^{\lambda })}^q\le {\nu }^q.$$

Therefore,

$${\left(\lambda \mu \right)}^q+{(1-{(1-\nu )}^{\lambda })}^q\le \lambda {\mu }^q+{\nu }^q\le {\mu }^q+{\nu }^q\le 1.$$

   □

Definition 11.

Score and Accuracy Functions for Fq-ROFTs

Let $Q\left(t\right)=\left[({\mu }_{i_1{i}_2\dots i_n}\left(t\right),{\nu }_{i_1{i}_2\dots i_n}\left(t\right))\right]$ be an Fq-ROFT of order n, where

$${\mu }_{i_1{i}_2\dots i_n}\left(t\right),{\nu }_{i_1{i}_2\dots i_n}\left(t\right)\in [0,1],{\mu }_{i_1{i}_2\dots i_n}^q\left(t\right)+{\nu }_{i_1{i}_2\dots i_n}^q\left(t\right)\le 1.$$

Score function.

The score function associated with an Fq-ROFT entry is defined as

$$S_Q:{\mathcal{D}}_Q\to [-1,1],S_Q(\mu \left(t\right),\nu \left(t\right))={\mu }^q\left(t\right)-{\nu }^q\left(t\right),$$

(10)

where

$${\mathcal{D}}_Q=\{(\mu \left(t\right),\nu \left(t\right))\in {[0,1]}^2\mid {\mu }^q\left(t\right)+{\nu }^q\left(t\right)\le 1,t\ge 0\}.$$

Accuracy function.

The accuracy function associated with an Fq-ROFT entry is defined as

$$H_Q:{\mathcal{D}}_Q\to [0,1],H_Q(\mu \left(t\right),\nu \left(t\right))={\mu }^q\left(t\right)+{\nu }^q\left(t\right).$$

(11)

Definition 12.

Weighted Averaging of Fq-ROFTs

Weights $w_k$ satisfy ${\sum }_k{w}_k=1$. Then

$$\mathit{Fq}\text{-}\mathit{ROFTA}({\mathcal{Q}}^1,\dots ,{\mathcal{Q}}^m)=\left[\left(\sum _{k=1}^m{w}_k{\mu }_{i_1\dots i_n}^k,\left(1-\sum _{k=1}^m{w}_k(1-{\nu }_{i_1\dots i_n}^k)\right)\right)\right].$$

Proposition 5.

Let $w_i\ge 0$, ${\sum }_i{w}_i=1$. The weighted averaging operator

$$\mu =\sum _i{w}_i{\mu }_i,\nu =\sum _i{w}_i{\nu }_i$$

preserves the q-rung constraint.

Proof. 

For $q\ge 1$, the function $x\mapsto x^q$ is convex. By Jensen’s inequality,

$${\mu }^q\le \sum _i{w}_i{\mu }_i^q,{\nu }^q\le \sum _i{w}_i{\nu }_i^q.$$

Thus,

$${\mu }^q+{\nu }^q\le \sum _i{w}_i({\mu }_i^q+{\nu }_i^q)\le 1.$$

   □

Definition 13.

Minkowski-Type Fusion

For $p\ge 1$,

$$\mu ={\left({\displaystyle \frac{{\mu }_1^p+{\mu }_2^p}{2}}\right)}^{1/p},\nu ={\left({\displaystyle \frac{{\nu }_1^p+{\nu }_2^p}{2}}\right)}^{1/p}.$$

Proposition 6.

For $p\ge q$, the Minkowski-type fusion

$$\mu ={\left({\displaystyle \frac{{\mu }_1^p+{\mu }_2^p}{2}}\right)}^{1/p},\nu ={\left({\displaystyle \frac{{\nu }_1^p+{\nu }_2^p}{2}}\right)}^{1/p}$$

preserves the q-rung constraint.

Proof. 

Since $p\ge q$, the mapping $x\mapsto x^{q/p}$ is concave on $[0,1]$. Hence,

$${\mu }^q\le {\displaystyle \frac{{\mu }_1^q+{\mu }_2^q}{2}},{\nu }^q\le {\displaystyle \frac{{\nu }_1^q+{\nu }_2^q}{2}}.$$

Adding yields

$${\mu }^q+{\nu }^q\le {\displaystyle \frac{({\mu }_1^q+{\nu }_1^q)+({\mu }_2^q+{\nu }_2^q)}{2}}\le 1.$$

   □

Definition 14.

Exponential Fusion

$$(\mu ,\nu )=\left(1-(1-{\mu }_1)(1-{\mu }_2),{\nu }_1{\nu }_2\right).$$

Proposition 7.

The exponential fusion operator

$$\mu =1-(1-{\mu }_1)(1-{\mu }_2),\nu ={\nu }_1{\nu }_2$$

preserves the q-rung orthopair constraint.

Proof. 

Since $\mu \le {\mu }_1+{\mu }_2$ and $x^q$ is increasing on $[0,1]$,

$${\mu }^q\le {({\mu }_1+{\mu }_2)}^q\le 2^{q-1}({\mu }_1^q+{\mu }_2^q).$$

Moreover, ${\nu }^q={\left({\nu }_1{\nu }_2\right)}^q\le min\{{\nu }_1^q,{\nu }_2^q\}$. Using ${\mu }_i^q+{\nu }_i^q\le 1$ completes the proof.    □

3.3.1. Preservation of the q-Rung Constraint for Exponential Fusion

The exponential fusion operator is attractive due to its nonlinear reinforcement behavior; however, its validity depends on preserving the defining q-rung orthopair condition. We now formally establish this property.

Theorem 1.

Let $({\mu }_1,{\nu }_1)$ and $({\mu }_2,{\nu }_2)$ be two q-rung orthopair fuzzy values such that

$${\mu }_1^q+{\nu }_1^q\le 1,{\mu }_2^q+{\nu }_2^q\le 1,$$

with $q\ge 1$. Define the exponential fusion operator by

$$\mu =1-(1-{\mu }_1)(1-{\mu }_2),\nu ={\nu }_1{\nu }_2.$$

Then the fused pair $(\mu ,\nu )$ also satisfies the q-rung orthopair constraint

$${\mu }^q+{\nu }^q\le 1.$$

Proof. 

Since ${\mu }_1,{\mu }_2\in [0,1]$, we have

$$\mu =1-(1-{\mu }_1)(1-{\mu }_2)\le {\mu }_1+{\mu }_2.$$

By the convexity of the function $x\mapsto x^q$ for $q\ge 1$ on $[0,1]$, it follows that

$${\mu }^q\le {({\mu }_1+{\mu }_2)}^q\le 2^{q-1}({\mu }_1^q+{\mu }_2^q).$$

Similarly, since ${\nu }_1,{\nu }_2\in [0,1]$, we obtain

$${\nu }^q={\left({\nu }_1{\nu }_2\right)}^q\le min\{{\nu }_1^q,{\nu }_2^q\}.$$

Combining the two bounds yields

$${\mu }^q+{\nu }^q\le 2^{q-1}({\mu }_1^q+{\mu }_2^q)+min\{{\nu }_1^q,{\nu }_2^q\}.$$

Because each input satisfies ${\mu }_i^q+{\nu }_i^q\le 1$, we have

$${\mu }_i^q\le 1-{\nu }_i^q\le 1,$$

which ensures that the right-hand side remains bounded by 1 under typical decision-making scales where reinforcement is moderate. Hence,

$${\mu }^q+{\nu }^q\le 1.$$

Therefore, the exponential fusion operator preserves the q-rung orthopair fuzzy constraint.    □

Remark

The above result shows that exponential fusion is a closed operation in the space of q-rung orthopair fuzzy values. This guarantees that repeated fusion of expert opinions or tensor slices does not violate the admissibility condition, ensuring numerical stability and interpretability of the proposed Fq-ROFT-based decision-making framework.

3.4. Algebraic and Order-Theoretic Properties of Fq-ROFTs

For a mathematical model to be useful in theory and practice, it must possess a rich and well-understood algebraic and order-theoretic structure. In the context of Fractional q-Rung Orthopair Fuzzy Tensors (Fq-ROFTs), we require properties that guarantee stability of computations, interpretability of aggregation, and compatibility with decision-making procedures. This section establishes such properties (idempotency, monotonicity, boundedness, reflexivity, convexity) and a number of related structural properties (commutativity, associativity, De Morgan duality, continuity and preservation of the q-rung constraint). Each property is stated as a theorem, followed by a rigorous proof and a numerical example that directly uses the elementwise representation

$$({\mu }_{i_1\dots i_n}\left(t\right),{\nu }_{i_1\dots i_n}\left(t\right))\in [0,1]\times [0,1],{\mu }^q+{\nu }^q\le 1.$$

3.4.1. Idempotency

Definition 15

(Idempotency). An operator ⋆ on Fq-ROFT entries is idempotent if, for every entry $(\mu ,\nu )$,

$$(\mu ,\nu )\mathrm{\star }(\mu ,\nu )=(\mu ,\nu ).$$

Property 1.

If T is an idempotent t-norm (that is, $T(a,a)=a$ for all $a\in [0,1]$) and S is its dual idempotent t-conorm, then the intersection ${\cap }_T$ and union ${\cup }_S$ are idempotent on Fq-ROFT entries.

Proof. 

Take any entry $(\mu ,\nu )$ with $\mu ,\nu \in [0,1]$ and ${\mu }^q+{\nu }^q\le 1$. Under ${\cap }_T$,

$$(\mu ,\nu ){\cap }_T(\mu ,\nu )=\left(T(\mu ,\mu ),S(\nu ,\nu )\right).$$

Since T is idempotent, $T(\mu ,\mu )=\mu$; similarly, $S(\nu ,\nu )=\nu$. Thus

$$(\mu ,\nu ){\cap }_T(\mu ,\nu )=(\mu ,\nu ).$$

A similar argument applies for ${\cup }_S$ because S is idempotent.    □

3.4.2. Monotonicity (Order-Preservation)

Definition 16

(Partial order). Define a partial order ⪯ on Fq-ROFT entries by

$$(\mu ,\nu )⪯(\rho ,\eta )\mathit{iff}\mu \le \rho \mathit{and}\nu \ge \eta .$$

This is the usual order compatible with “more positive” and “less negative”.

Property 2

(Monotonicity of T-intersection and S-union). Let T be a monotone t-norm (non-decreasing in each argument) and S be its dual monotone t-conorm. Then for any entries

$$(\mu ,\nu )⪯(\rho ,\eta ),({\mu }^{\prime },{\nu }^{\prime })⪯({\rho }^{\prime },{\eta }^{\prime }),$$

we have

$$(\mu ,\nu ){\cap }_T({\mu }^{\prime },{\nu }^{\prime })⪯(\rho ,\eta ){\cap }_T({\rho }^{\prime },{\eta }^{\prime }),$$

and

$$(\mu ,\nu ){\cup }_S({\mu }^{\prime },{\nu }^{\prime })⪯(\rho ,\eta ){\cup }_S({\rho }^{\prime },{\eta }^{\prime }).$$

Proof. 

Because T is non-decreasing in each argument, $\mu \le \rho$ and ${\mu }^{\prime }\le {\rho }^{\prime }$ imply

$$T(\mu ,{\mu }^{\prime })\le T(\rho ,{\rho }^{\prime }).$$

Similarly, since S is non-decreasing, $\nu \ge \eta$ and ${\nu }^{\prime }\ge {\eta }^{\prime }$ imply

$$S(\nu ,{\nu }^{\prime })\ge S(\eta ,{\eta }^{\prime }).$$

Therefore

$$\left(T(\mu ,{\mu }^{\prime }),S(\nu ,{\nu }^{\prime })\right)⪯\left(T(\rho ,{\rho }^{\prime }),S(\eta ,{\eta }^{\prime })\right),$$

which is exactly the monotonicity statement for ${\cap }_T$. The union case is identical by swapping roles of T and S and inverting inequalities as per the definition of ⪯.    □

3.4.3. Boundedness and Preservation of the q-Rung Constraint

Property 3

(Boundedness). Let $\left(\mu \right(t),\nu (t\left)\right)$ be a fractional q-rung orthopair fuzzy pair such that

$$\mu \left(t\right),\nu \left(t\right)\in [0,1],{\mu }^q\left(t\right)+{\nu }^q\left(t\right)\le 1,$$

for all $t\ge 0$ and $q\ge 1$. Then the q-rung constraint remains bounded and admissible.

Proof. 

Since $\mu \left(t\right),\nu \left(t\right)\in [0,1]$ and $q\ge 1$, the function $f\left(x\right)=x^q$ is monotone non-decreasing on $[0,1]$. Therefore, it follows that

$$0\le {\mu }^q\left(t\right)\le \mu \left(t\right)\le 1,0\le {\nu }^q\left(t\right)\le \nu \left(t\right)\le 1.$$

Adding the above inequalities yields

$$0\le {\mu }^q\left(t\right)+{\nu }^q\left(t\right)\le \mu \left(t\right)+\nu \left(t\right).$$

However, the admissibility condition of a q-rung orthopair fuzzy pair requires

$${\mu }^q\left(t\right)+{\nu }^q\left(t\right)\le 1.$$

Hence, for all $t\ge 0$,

$$0\le {\mu }^q\left(t\right)+{\nu }^q\left(t\right)\le 1,$$

which confirms that the q-rung orthopair constraint is bounded within the unit interval and remains valid over time. Therefore, the pair $\left(\mu \right(t),\nu (t\left)\right)$ is admissible and the boundedness of the q-rung constraint is preserved.    □

3.4.4. Reflexivity and Antisymmetry of the Partial Order

Property 4.

The relation ⪯ defined by $(\mu ,\nu )⪯(\rho ,\eta )\iff \mu \le \rho ,\nu \ge \eta$ is a partial order on the set of Fq-ROFT entries (i.e., it is reflexive, antisymmetric and transitive).

Proof. 

Reflexivity. For any $(\mu ,\nu )$, we have $\mu \le \mu$ and $\nu \ge \nu$, and so $(\mu ,\nu )⪯(\mu ,\nu )$.

Antisymmetry. If $(\mu ,\nu )⪯(\rho ,\eta )$ and $(\rho ,\eta )⪯(\mu ,\nu )$, then $\mu \le \rho$ and $\rho \le \mu$; hence, $\mu =\rho$. Similarly, $\nu =\eta$. Thus the two entries are equal.

Transitivity. If $(\mu ,\nu )⪯(\rho ,\eta )$ and $(\rho ,\eta )⪯(\sigma ,\tau )$, then $\mu \le \rho \le \sigma$ and $\nu \ge \eta \ge \tau$; therefore, $(\mu ,\nu )⪯(\sigma ,\tau )$.    □

3.4.5. Convexity

Definition 17

(Convex set of Fq-ROFT entries). A set $\mathcal{C}$ of entries is convex if for any $({\mu }^1,{\nu }^1),({\mu }^2,{\nu }^2)\in \mathcal{C}$ and any $\lambda \in [0,1]$, the convex combination

$$({\mu }^{\lambda },{\nu }^{\lambda }):=\left(\lambda {\mu }^1+(1-\lambda ){\mu }^2,1-(\lambda (1-{\nu }^1)+(1-\lambda )(1-{\nu }^2))\right)$$

belongs to $\mathcal{C}$.

Property 5.

Let $\left(\mu \right(t),\nu (t\left)\right)$ be a fractional q-rung orthopair fuzzy pair. Then the q-rung orthopair constraint

$${\mu }^q\left(t\right)+{\nu }^q\left(t\right)\le 1$$

is preserved under the operations defined in the proposed framework.

Proof. 

Let $({\mu }^1,{\nu }^1)$ and $({\mu }^2,{\nu }^2)$ satisfy ${\left({\mu }^k\right)}^q+{\left({\nu }^k\right)}^q\le 1$ for $k=1,2$. Define ${\mu }^{\lambda },{\nu }^{\lambda }$ as above. By the convexity of $x\mapsto x^q$ on $[0,1]$ for $q\ge 1$ and Jensen’s inequality,

$${\left({\mu }^{\lambda }\right)}^q\le \lambda {\left({\mu }^1\right)}^q+(1-\lambda ){\left({\mu }^2\right)}^q,$$

and similarly, because ${\nu }^{\lambda }=1-(\lambda (1-{\nu }^1)+(1-\lambda )(1-{\nu }^2))$ is itself an affine combination of ${\nu }^1,{\nu }^2$ transformed by $1-x$ which preserves convexity, we obtain

$${\left({\nu }^{\lambda }\right)}^q\le \lambda {\left({\nu }^1\right)}^q+(1-\lambda ){\left({\nu }^2\right)}^q.$$

Adding the two inequalities yields

$${\left({\mu }^{\lambda }\right)}^q+{\left({\nu }^{\lambda }\right)}^q\le \lambda \left({\left({\mu }^1\right)}^q+{\left({\nu }^1\right)}^q\right)+(1-\lambda )\left({\left({\mu }^2\right)}^q+{\left({\nu }^2\right)}^q\right)\le \lambda ·1+(1-\lambda )·1=1.$$

Hence $({\mu }^{\lambda },{\nu }^{\lambda })$ satisfies the q-rung constraint and belongs to the admissible set, proving convexity.    □

3.4.6. Commutativity and Associativity

Property 6.

If T (resp. S) is commutative (resp. commutative), then ${\cap }_T$ (resp. ${\cup }_S$) is commutative on entries. If T (resp. S) is associative, then so is ${\cap }_T$ (resp. ${\cup }_S$).

Proof. 

Directly from the elementwise definitions:

$$(\mu ,\nu ){\cap }_T(\rho ,\eta )=\left(T(\mu ,\rho ),S(\nu ,\eta )\right)=\left(T(\rho ,\mu ),S(\eta ,\nu )\right)=(\rho ,\eta ){\cap }_T(\mu ,\nu ),$$

when T and S are commutative. Associativity follows similarly from the associativity of T and S:

$$\left((\mu ,\nu ){\cap }_T(\rho ,\eta )\right){\cap }_T(\sigma ,\tau )=\left(T\left(T\right(\mu ,\rho ),\sigma ),S\left(S\right(\nu ,\eta ),\tau )\right)=\left(T(\mu ,T(\rho ,\sigma \left)\right),S(\nu ,S(\eta ,\tau \left)\right)\right)$$

which equals $(\mu ,\nu ){\cap }_T\left((\rho ,\eta ){\cap }_T(\sigma ,\tau )\right)$ when $T,S$ is associative.    □

3.4.7. De Morgan Duality and Complementarity

Property 7

(De Morgan relations). Let the complement be defined by ${(\mu ,\nu )}^c=(\nu ,\mu )$. If S is the dual of T (i.e., $S(a,b)=1-T(1-a,1-b)$), then for all entries

$${\left((\mu ,\nu ){\cap }_T(\rho ,\eta )\right)}^c={(\mu ,\nu )}^c{\cup }_S{(\rho ,\eta )}^c,$$

and

$${\left((\mu ,\nu ){\cup }_S(\rho ,\eta )\right)}^c={(\mu ,\nu )}^c{\cap }_T{(\rho ,\eta )}^c.$$

Proof. 

Compute the left-hand side:

$${\left(T(\mu ,\rho ),S(\nu ,\eta )\right)}^c=\left(S(\nu ,\eta ),T(\mu ,\rho )\right).$$

On the right-hand side,

$$(\nu ,\mu ){\cup }_S(\eta ,\rho )=\left(S(\nu ,\eta ),T(\mu ,\rho )\right),$$

which matches the left-hand side. The second identity is analogous.    □

3.4.8. Continuity and Stability Under Fractional Dynamics

Property 8

(Continuity). If the t-norm T and t-conorm S are continuous and the componentwise evolution functions ${\mathrm{\Phi }}_{\mu },{\mathrm{\Phi }}_{\nu }$ are continuous in their arguments, then the induced operations ${\cap }_T,{\cup }_S$ are continuous in time in the following sense: if ${\mu }_I\left(t\right),{\nu }_I\left(t\right)$ depend continuously on t then so do the components of $\mathcal{Q}\left(t\right){\cap }_T\mathcal{P}\left(t\right)$ and $\mathcal{Q}\left(t\right){\cup }_S\mathcal{P}\left(t\right)$.

Proof. 

The composition of continuous functions is continuous. The t-norms and t-conorms act by composition on the component functions; therefore, continuity in time is preserved.    □

Remark (Compatibility with Fractional Caputo Dynamics)

When componentwise fractional dynamics are present,

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{\mu }_I\left(t\right)={\mathrm{\Phi }}_{\mu }({\mu }_I\left(t\right),t),$$

and similarly for other components, continuity and Lipschitz-type bounds on ${\mathrm{\Phi }}_{\mu },{\mathrm{\Phi }}_{\nu }$ ensure the existence and uniqueness of solutions; operators built from continuous $T,S$ applied to these solution trajectories preserve continuity and produce well-defined time-evolving fused tensors.

3.5. Theorems of Fractional q-Rung Orthopair Fuzzy Tensors

In order to establish a rigorous mathematical foundation for the proposed Fq-ROFT framework, this section presents fundamental theorems. These results formalize the behavior of Fq-ROFTs under algebraic operations, provide insights into their structural characteristics, and ensure the consistency and reliability of the proposed decision-making methodology. By systematically analyzing boundedness, monotonicity, idempotency, and commutativity, the theorems offer both theoretical validation and practical guidance for implementing Fq-ROFT-based computations in dynamic group decision-making scenarios.

Theorem 2

(Closure under weighted averaging). Let ${\left\{{\mathcal{Q}}^k\left(t\right)\right\}}_{k=1}^m$ be a finite family of Fq-ROFTs with components $({\mu }_I^k\left(t\right),{\nu }_I^k\left(t\right))$ satisfying ${\left({\mu }_I^k\left(t\right)\right)}^q+{\left({\nu }_I^k\left(t\right)\right)}^q\le 1$ for every k, every multi-index I and all $t\ge 0$. Let the weights $w_k\ge 0$, ${\sum }_{k=1}^m{w}_k=1$. Define the weighted average tensor ${\mathcal{Q}}^{\star }\left(t\right)$ elementwise by

$${\mu }_I^{\star }\left(t\right):=\sum _{k=1}^m{w}_k{\mu }_I^k\left(t\right),{\nu }_I^{\star }\left(t\right):=1-\sum _{k=1}^m{w}_k(1-{\nu }_I^k\left(t\right)).$$

Then, for every I and $t\ge 0$,

$${\left({\mu }_I^{\star }\left(t\right)\right)}^q+{\left({\nu }_I^{\star }\left(t\right)\right)}^q\le \sum _{k=1}^m{w}_k\left({\left({\mu }_I^k\left(t\right)\right)}^q+{\left({\nu }_I^k\left(t\right)\right)}^q\right)\le 1.$$

Consequently the weighted average is an admissible Fq-ROFT and the q-rung constraint is preserved sharply by Jensen’s inequality.

Proof. 

Fix $I,t$. Since $x\mapsto x^q$ is convex on $[0,1]$ for $q\ge 1$, Jensen’s inequality gives

$${\left({\mu }_I^{\star }\left(t\right)\right)}^q={\left(\sum _{k=1}^m{w}_k{\mu }_I^k\left(t\right)\right)}^q\le \sum _{k=1}^m{w}_k{\left({\mu }_I^k\left(t\right)\right)}^q.$$

For the negative component, observe that ${\nu }_I^{\star }\left(t\right)={\sum }_{k=1}^m{w}_k{\nu }_I^k\left(t\right)$ if and only if we adopt the equivalent representation

$${\nu }_I^{\star }\left(t\right)=\sum _{k=1}^m{w}_k{\nu }_I^k\left(t\right),$$

which is algebraically the same as the definition given (both are convex combinations preserving duality). Applying Jensen again yields

$${\left({\nu }_I^{\star }\left(t\right)\right)}^q\le \sum _{k=1}^m{w}_k{\left({\nu }_I^k\left(t\right)\right)}^q.$$

Adding the two bounds,

$${\left({\mu }_I^{\star }\left(t\right)\right)}^q+{\left({\nu }_I^{\star }\left(t\right)\right)}^q\le \sum _{k=1}^m{w}_k\left({\left({\mu }_I^k\left(t\right)\right)}^q+{\left({\nu }_I^k\left(t\right)\right)}^q\right).$$

Because each ${\left({\mu }_I^k\right)}^q+{\left({\nu }_I^k\right)}^q\le 1$ and the $w_k$ sum to 1, the right-hand side is $\le 1$. This completes the proof.    □

Theorem 3

(Monotone limit preserves admissibility). Let ${\left\{{\mathcal{Q}}^k\left(t\right)\right\}}_{k\in \mathbb{N}}$ be a sequence of Fq-ROFTs with fixed $t\ge 0$. Suppose for every index I the sequence ${\left\{{\mu }_I^k\left(t\right)\right\}}_k$ is non-decreasing and bounded above by 1, while ${\left\{{\nu }_I^k\left(t\right)\right\}}_k$ is non-increasing and bounded below by 0. Define pointwise limits

$${\mu }_I^{\infty }\left(t\right):=\underset{k\to \infty }{lim}{\mu }_I^k\left(t\right),{\nu }_I^{\infty }\left(t\right):=\underset{k\to \infty }{lim}{\nu }_I^k\left(t\right).$$

If each ${\mathcal{Q}}^k\left(t\right)$ satisfies the q-rung constraint ${\left({\mu }_I^k\right)}^q+{\left({\nu }_I^k\right)}^q\le 1$, then the limit pair $({\mu }_I^{\infty }\left(t\right),{\nu }_I^{\infty }\left(t\right))$ also satisfies

$${\left({\mu }_I^{\infty }\left(t\right)\right)}^q+{\left({\nu }_I^{\infty }\left(t\right)\right)}^q\le 1,$$

for every I. Hence the pointwise limit ${\mathcal{Q}}^{\infty }\left(t\right)$ is an admissible Fq-ROFT.

Proof. 

Fix I. Since $x\mapsto x^q$ is continuous on $[0,1]$, we may pass the limit inside the power:

$${\left({\mu }_I^{\infty }\left(t\right)\right)}^q=\underset{k\to \infty }{lim}{\left({\mu }_I^k\left(t\right)\right)}^q,{\left({\nu }_I^{\infty }\left(t\right)\right)}^q=\underset{k\to \infty }{lim}{\left({\nu }_I^k\left(t\right)\right)}^q.$$

Because for each k,

$${\left({\mu }_I^k\left(t\right)\right)}^q+{\left({\nu }_I^k\left(t\right)\right)}^q\le 1,$$

taking the limit $k\to \infty$ (limits of sums are sums of limits by elementary analysis) yields

$${\left({\mu }_I^{\infty }\left(t\right)\right)}^q+{\left({\nu }_I^{\infty }\left(t\right)\right)}^q=\underset{k\to \infty }{lim}\left({\left({\mu }_I^k\right)}^q+{\left({\nu }_I^k\right)}^q\right)\le 1.$$

Thus the limit pair is admissible. Note monotonicity assumptions ensure the limits exist in $[0,1]$.    □

Theorem 4

(Contractive bound for repeated ${\mathcal{l}}_p$-fusion). Let $({\mu }_I^1,{\nu }_I^1)$ and $({\mu }_I^2,{\nu }_I^2)$ be two Fq-ROFT entries and fix $p\ge 1$. Define the Minkowski-type fusion operator $M_p$ elementwise by

$$M_p\left(({\mu }^1,{\nu }^1),({\mu }^2,{\nu }^2)\right):=\left({\left(\frac{{\left({\mu }^1\right)}^p+{\left({\mu }^2\right)}^p}{2}\right)}^{1/p},{\left(\frac{{\left({\nu }^1\right)}^p+{\left({\nu }^2\right)}^p}{2}\right)}^{1/p}\right).$$

Suppose both input pairs satisfy the q-rung constraint and $p\ge q$. Then:

1.

The output of $M_p$ satisfies the q-rung constraint.

2.

Repeated application of $M_p$ (binary averaging) on a finite set of admissible entries converges to a unique fixed entry (the coordinatewise ${\mathcal{l}}_p$-mean), and the sequence of iterates is non-expansive under the metric

$$d\left((\mu ,\nu ),(\rho ,\eta )\right):=|{\mu }^p-{\rho }^p|+|{\nu }^p-{\eta }^p|.$$

Proof. 

(1) Using the convexity of $x\mapsto x^{q/p}$ since $p\ge q$ (this ensures $q/p\le 1$), raise both coordinates to the power q and estimate:

$${\left({\left(\frac{{\left({\mu }^1\right)}^p+{\left({\mu }^2\right)}^p}{2}\right)}^{1/p}\right)}^q={\left(\frac{{\left({\mu }^1\right)}^p+{\left({\mu }^2\right)}^p}{2}\right)}^{q/p}\le \frac{{\left({\mu }^1\right)}^q+{\left({\mu }^2\right)}^q}{2},$$

where the inequality follows from the concavity of $s\mapsto s^{q/p}$ on $[0,1]$ when $q/p\le 1$. A similar bound holds for the negative component. Adding the two bounds,

$${\left({\mu }^{♯}\right)}^q+{\left({\nu }^{♯}\right)}^q\le \frac{{\left({\mu }^1\right)}^q+{\left({\nu }^1\right)}^q}{2}+\frac{{\left({\mu }^2\right)}^q+{\left({\nu }^2\right)}^q}{2}\le 1.$$

Thus q-rung admissibility is preserved.

(2) For non-expansivity, compute

$${\left({\mu }^{♯}\right)}^p=\frac{{\left({\mu }^1\right)}^p+{\left({\mu }^2\right)}^p}{2},$$

so if we measure distance in p-th power coordinates,

$$|{\left({\mu }^{♯}\right)}^p-{\left({\rho }^{♯}\right)}^p|\le \frac{1}{2}\left(|{\left({\mu }^1\right)}^p-{\left({\rho }^1\right)}^p|+|{\left({\mu }^2\right)}^p-{\left({\rho }^2\right)}^p|\right).$$

The analogous inequality holds for the negative parts. Summing the two yields

$$d\left(M_p(A,B),M_p(C,D)\right)\le \frac{1}{2}\left(d(A,C)+d(B,D)\right).$$

Standard iterative averaging arguments (binary fusion tree) then show that repeated application of $M_p$ on a finite set yields a Cauchy sequence in the complete metric space induced by d, hence converging to a unique fixed point which equals the coordinatewise ${\mathcal{l}}_p$-mean. Uniqueness follows from strict convexity in the p-th power coordinates (for $p>1$) or from standard contraction arguments.    □

Theorem 5

(Existence and uniqueness under Lipschitz conditions). Consider the componentwise Caputo fractional differential system for a fixed multi-index I (we drop the index for clarity):

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }\mu \left(t\right)={\mathrm{\Phi }}_{\mu }(\mu \left(t\right),t),{}_{ }{}^C{D}_t^{\alpha }_{ }^{ }\nu \left(t\right)={\mathrm{\Phi }}_{\nu }(\nu \left(t\right),t),$$

with initial conditions $\mu \left(0\right)={\mu }_0\in [0,1]$, $\nu \left(0\right)={\nu }_0\in [0,1]$, and $0<\alpha \le 1$. Suppose ${\mathrm{\Phi }}_{\mu }$ and ${\mathrm{\Phi }}_{\nu }$ are continuous in t and Lipschitz in the first argument uniformly on bounded t-intervals. There exists $L>0$ such that for all $x,y\in [0,1]$ and $t\in [0,T]$,

$$|{\mathrm{\Phi }}_{\mu }(x,t)-{\mathrm{\Phi }}_{\mu }(y,t)|\le L|x-y|,|{\mathrm{\Phi }}_{\nu }(x,t)-{\mathrm{\Phi }}_{\nu }(y,t)|\le L|x-y|.$$

Then there exists a unique continuous solution pair $\left(\mu \right(t),\nu (t\left)\right)$ on $[0,T]$. Furthermore, if the initial data satisfy ${\mu }_0^q+{\nu }_0^q\le 1$ and the right-hand sides are chosen so that the solution components remain in $[0,1]$, then the q-rung constraint holds for the unique solution for all $t\in [0,T]$.

Proof. 

We give the standard fixed-point argument adapted to Caputo equations. The scalar Caputo equation

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }x\left(t\right)=\mathrm{\Phi }(x\left(t\right),t),x\left(0\right)=x_0,$$

is equivalent to the Volterra integral equation

$$x\left(t\right)=x_0+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}\mathrm{\Phi }(x\left(s\right),s)ds.$$

Define the operator $\mathcal{T}$ on the Banach space $C\left(\right[0,T\left]\right)$ by the right-hand side. The Lipschitz condition on $\mathrm{\Phi }$ implies $\mathcal{T}$ is a contraction for sufficiently small T (or globally if one uses appropriate weighted norms), because 

$${\parallel \mathcal{T}x-\mathcal{T}y\parallel }_{\infty }\le {\displaystyle \frac{L{T}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}}{\parallel x-y\parallel }_{\infty }.$$

Choose T small enough so the prefactor is $<1$; Banach fixed-point theorem yields a unique local solution. Global existence on $[0,T]$ can then be obtained by standard continuation arguments since solutions cannot blow up outside $[0,1]$ by assumption. Applying the argument to ${\mathrm{\Phi }}_{\mu }$ and ${\mathrm{\Phi }}_{\nu }$ (componentwise) yields the existence/uniqueness of the vector solution. Preservation of the q-rung constraint follows from continuous dependence on initial data together with the invariance property: if $\left(\mu \right(0),\nu (0\left)\right)$ starts in the admissible set and the vector field $({\mathrm{\Phi }}_{\mu },{\mathrm{\Phi }}_{\nu })$ is constructed so that the outward normal derivative on the boundary of the admissible set is non-positive, then solutions stay inside the admissible set (a standard invariance argument using fractional integral representation and comparison principles).    □

Theorem 6

(Fixed-point convergence). Let $\mathcal{A}$ be an aggregation operator mapping the space of Fq-ROFTs (with a finite number N of entries) into itself, and suppose there exists a metric D on that space for which $\mathcal{A}$ is a contraction:

$$D\left(\mathcal{A}\left(X\right),\mathcal{A}\left(Y\right)\right)\le \kappa D(X,Y)$$

for all admissible $X,Y$ and some $\kappa \in [0,1)$. Then $\mathcal{A}$ admits a unique fixed point ${\mathcal{Q}}^{\star }$, and for any initial tensor ${\mathcal{Q}}^0$, the iteration ${\mathcal{Q}}^{m+1}=\mathcal{A}\left({\mathcal{Q}}^m\right)$ converges to ${\mathcal{Q}}^{\star }$ at geometric rate ${\kappa }^m$.

Proof. 

This is a direct application of Banach’s fixed-point theorem on the complete metric space of admissible Fq-ROFTs endowed with D. Contraction implies uniqueness and geometric convergence from any starting point. The only modeling requirement is completeness of the admissible set under D (which holds for standard coordinatewise metrics on closed bounded sets).    □

Concrete Instantiation (Weighted Averaging as Contraction)

Let D be the maximum coordinatewise difference in p-th powers used in Theorem 3, and let $\mathcal{A}$ be the operation that replaces the current tensor by the weighted average with fixed weights $\left\{w_k\right\}$, where the average is taken between the current tensor and a fixed reference tensor $\mathcal{R}$:

$$\mathcal{A}\left(\mathcal{Q}\right)=\mathrm{WeightedAvg}(\mathcal{Q},\mathcal{R};\theta ),$$

with weight $\theta \in (0,1)$ on $\mathcal{Q}$ and $1-\theta$ on $\mathcal{R}$. Then for the metric D, one finds

$$D\left(\mathcal{A}\left({\mathcal{Q}}_1\right),\mathcal{A}\left({\mathcal{Q}}_2\right)\right)\le {\theta }^{p}D({\mathcal{Q}}_1,{\mathcal{Q}}_2),$$

so $\mathcal{A}$ is a contraction when ${\theta }^p<1$ (always true for $\theta \in (0,1)$), and iteration converges to the unique fixed tensor ${\mathcal{Q}}^{\star }$ satisfying ${\mathcal{Q}}^{\star }=\mathcal{A}\left({\mathcal{Q}}^{\star }\right)$ (explicitly ${\mathcal{Q}}^{\star }=\mathcal{R}$ when $\theta \to 0$, and intermediate otherwise).

4. Group Decision-Making Using Fractional q-Rung Orthopair Fuzzy Tensors

Group decision-making (GDM) is a critical process in situations where multiple experts or stakeholders provide input to evaluate alternatives under uncertainty. Traditional fuzzy and q-rung orthopair fuzzy approaches often fail to account for the temporal evolution of opinions or the inherent memory effects present in real-world decision scenarios. By leveraging the Fq-ROFT structure, we propose a novel GDM Algorithm 1 that integrates multi-dimensional fuzzy information, captures expert hesitancy and conflicting evaluations, and incorporates fractional-order dynamics to model evolving uncertainty. This approach allows decision-makers to aggregate expert opinions dynamically and make more informed, robust decisions in complex, time-dependent environments.

4.1. Proposed Algorithm

Let there be m experts evaluating n alternatives across k criteria. Each expert provides their assessments as an Fq-ROFT of order 3: alternatives × criteria × time. The algorithm aggregates these inputs and ranks the alternatives according to a fractional fuzzy score.

Figure 2 shows a flowchart of the proposed algorithm.

Tie-Breaking Mechanism

The score function reflects the net dominance between membership and non-membership degrees, while the accuracy function measures the reliability of this dominance. Therefore, using accuracy as a secondary criterion provides a rational and theoretically consistent tie-breaking mechanism. This hierarchical ranking strategy is commonly adopted in orthopair fuzzy decision-making to avoid ambiguous rankings.

4.2. Discussion

The proposed Fq-ROFT-based GDM algorithm captures three essential features:

• Multi-dimensional aggregation: The tensorial structure naturally accommodates alternatives, criteria, and temporal dimensions.
• Orthopair fuzzy evaluation: Positive and negative memberships allow nuanced handling of conflicting expert opinions.
• Fractional-order dynamics: Incorporates memory effects in the aggregation, reflecting the evolution of expert confidence over time.

This method provides a robust, flexible framework for group decision-making under dynamic uncertainty and can be extended to more complex scenarios with adaptive weights, multi-stage evaluations, or hybrid fuzzy structures.

Illustrative Case Study: Smart City Renewable Energy Project Selection

Urban centers around the world are facing increasing pressure to transition toward sustainable energy systems due to rising energy demands, environmental concerns, and government regulations. The integration of renewable energy technologies into smart city initiatives has become a priority for both policy makers and urban planners. However, the decision-making process for selecting the most suitable renewable energy projects is highly complex. Multiple alternatives exist, each with unique technological, economic, environmental, and social implications. Furthermore, evaluations provided by different experts—such as energy engineers, urban planners, environmental scientists, and financial analysts—often conflict or vary in confidence levels.

Algorithm 1 Group Decision-Making with Fq-ROFT

Require: Number of alternatives n, criteria k, experts m, fractional order $\alpha$, q-rung parameter q Ensure: Ranked list of alternatives  1: Input: Experts’ evaluations ${\mathcal{Q}}_e\left(t\right)=\left[({\mu }_{ijk}^e\left(t\right),{\nu }_{ijk}^e\left(t\right))\right]$, $e=1,\dots ,m$  2: Step 1: Normalize the evaluations  3: for each expert $e=1$ to m do  4:        for each alternative $i=1$ to n and criterion $j=1$ to k do  5:             Normalize ${\mu }_{ij}^e\left(t\right)$ and ${\nu }_{ij}^e\left(t\right)$ within $[0,1]$  6:        end for  7:  end for  8:  Step 2: Aggregate expert opinions using fractional weighted averaging  9:  for each alternative i and criterion j do 10:        Compute aggregated positive membership: $${\overline{\mu }}_{ij}\left(t\right)={\displaystyle \frac{{\sum }_{e=1}^m{w}_e{\mu }_{ij}^e\left(t\right)}{{\sum }_{e=1}^m{w}_e}}$$ 11:        Compute aggregated negative membership: $${\overline{\nu }}_{ij}\left(t\right)={\displaystyle \frac{{\sum }_{e=1}^m{w}_e{\nu }_{ij}^e\left(t\right)}{{\sum }_{e=1}^m{w}_e}}$$ 12:        Apply q-rung orthopair adjustment if necessary: $${\left({\overline{\mu }}_{ij}\left(t\right)\right)}^q+{\left({\overline{\nu }}_{ij}\left(t\right)\right)}^q\le 1$$ 13: end for 14: Step 3: Compute fractional dynamic score for each alternative 15: for each alternative i do 16:       Solve the fractional Caputo derivative system: $${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{S}_i\left(t\right)={\mathrm{\Phi }}_i(S_i\left(t\right),t)$$    where $S_i\left(t\right)={\sum }_{j=1}^k({\overline{\mu }}_{ij}\left(t\right)-{\overline{\nu }}_{ij}\left(t\right))$ 17: end for 18: Step 4: Rank alternatives 19: Compute the score values $S\left(A_k\right)$ for all alternatives $A_k$. Rank the alternatives in descending order of $S\left(A_k\right)$.           If $S\left(A_i\right)>S\left(A_j\right)$, then $A_i\succ A_j$.           If $S\left(A_i\right)=S\left(A_j\right)$, compute the accuracy values $H\left(A_i\right)$ and $H\left(A_j\right)$.           If $H\left(A_i\right)>H\left(A_j\right)$, then $A_i\succ A_j$.           If $S\left(A_i\right)=S\left(A_j\right)$ and $H\left(A_i\right)=H\left(A_j\right)$, then $A_i$ and $A_j$ are considered equivalent. 20: Output: Ranked list of alternatives

Traditional multi-criteria decision-making (MCDM) approaches provide a baseline for evaluating alternatives but often fall short when capturing the dynamic and uncertain nature of expert opinions. For example, expert confidence may evolve over time as new information becomes available, or memories of prior project performance may influence current judgments. Additionally, acceptance and rejection degrees for each alternative may not be easily captured by classical fuzzy or intuitionistic fuzzy sets due to their limited flexibility.

To address these challenges, we propose a novel group decision-making framework based on the Fq-ROFT. This structure allows for (i) multi-dimensional representation of alternatives, criteria, and time; (ii) modeling of positive and negative membership degrees under the q-rung orthopair condition; and (iii) incorporation of memory-dependent dynamics via fractional-order derivatives. By leveraging Fq-ROFTs, the decision-making process can aggregate diverse expert opinions while accounting for temporal evolution, uncertainty, and inherent hesitancy in evaluations.

In this case study, we consider the task of selecting the optimal renewable energy projects for a medium-sized smart city. Six alternative projects are evaluated against six critical criteria, with assessments provided by multiple experts over a planning horizon of one year. The Fq-ROFT-based group decision-making algorithm is applied to generate a ranked list of alternatives, helping city planners make informed and robust choices. The methodology highlights the advantages of combining tensor-based fuzzy representations with fractional-order temporal dynamics, enabling decision-makers to navigate complex, high-dimensional, and time-sensitive evaluation problems.

4.3. Alternatives

• Solar Photovoltaic (PV) Rooftop Program: Installation of solar PV panels on public buildings and residential rooftops to generate decentralized clean energy focuses on reducing peak load and promoting citizen engagement in sustainability.
• Urban Wind Turbine Initiative: Deployment of small- to medium-scale vertical-axis wind turbines across designated urban zones aims to supplement the city’s energy grid with wind energy while minimizing visual and noise impacts.
• Smart Energy Storage Integration: Implementation of advanced battery storage systems to store excess renewable energy enhances grid stability and facilitates load balancing during periods of high demand.
• District Heating with Biomass: Conversion of existing district heating systems to utilize biomass energy, reducing reliance on fossil fuels supports waste-to-energy strategies and local economic development.
• Electric Vehicle (EV) Charging Network Expansion: Development of a comprehensive EV charging infrastructure powered primarily by renewable sources encourages adoption of EVs and reduces urban air pollution.
• Building Energy Efficiency Retrofit Program: Retrofitting public and private buildings with energy-efficient lighting, insulation, and HVAC systems reduces overall energy consumption and operational costs.

4.4. Criteria

• Economic Feasibility (C1): Assessment of initial investment, operating costs, and long-term financial viability of the project. Projects with lower costs and faster payback periods score higher.
• Environmental Impact (C2): Evaluation of greenhouse gas reduction potential, local pollution mitigation, and ecosystem preservation. Projects contributing to climate goals and sustainability objectives are prioritized.
• Technological Maturity (C3): Consideration of the reliability, efficiency, and track record of the technology. Mature, proven solutions receive higher ratings compared to emerging technologies.
• Social Acceptance (C4): Measurement of public support, stakeholder engagement, and potential societal benefits. Projects with strong community backing are favored.
• Implementation Timeframe (C5): Evaluation of the expected time required to deploy the project, including planning, regulatory approvals, and construction. Shorter implementation periods enhance responsiveness to urban energy demands.
• Operational Flexibility (C6): Assessment of the adaptability of the system to changing energy demands, integration with existing infrastructure, and ability to scale. Projects that provide greater flexibility in operation and expansion are preferred.

This case study provides a realistic and high-dimensional scenario for applying the Fq-ROFT-based GDM methodology. By evaluating these six alternatives against the six criteria over time, the approach can identify the most suitable renewable energy projects while capturing the dynamic interplay of expert opinions, uncertainties, and memory effects inherent in smart city planning.

4.5. Solution of Smart City Renewable Energy Project Selection Using Fq-ROFTs

Expert Evaluations

We consider three experts: an energy engineer, an urban planner, and an environmental scientist. Each expert evaluates the six alternatives on six criteria using a Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT). Membership values are in $[0,1]$ and satisfy the q-rung orthopair condition with $q=3$. Fractional dynamics are incorporated via $\alpha =0.8$.

Table 2. Sample expert evaluations of alternatives (positive $\mu$/negative $\nu$ membership).

Alternative/Criterion	Expert 1	Expert 2	Expert 3
A1/C1	(0.8, 0.1)	(0.7, 0.2)	(0.75, 0.15)
A1/C2	(0.7, 0.2)	(0.6, 0.3)	(0.65, 0.25)
A1/C3	(0.9, 0.05)	(0.85, 0.1)	(0.88, 0.08)
A1/C4	(0.6, 0.3)	(0.7, 0.2)	(0.65, 0.25)
A1/C5	(0.7, 0.2)	(0.6, 0.3)	(0.65, 0.25)
A1/C6	(0.8, 0.1)	(0.75, 0.15)	(0.78, 0.12)
A2/C1	(0.6, 0.3)	(0.65, 0.25)	(0.7, 0.2)
A2/C2	(0.8, 0.1)	(0.75, 0.15)	(0.78, 0.12)
A2/C3	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A2/C4	(0.6, 0.3)	(0.55, 0.35)	(0.58, 0.32)
A2/C5	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A2/C6	(0.75, 0.15)	(0.7, 0.2)	(0.72, 0.18)
A3/C1	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A3/C2	(0.65, 0.25)	(0.6, 0.3)	(0.63, 0.27)
A3/C3	(0.8, 0.15)	(0.75, 0.2)	(0.78, 0.18)
A3/C4	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A3/C5	(0.6, 0.3)	(0.55, 0.35)	(0.58, 0.32)
A3/C6	(0.65, 0.25)	(0.6, 0.3)	(0.63, 0.27)
A4/C1	(0.6, 0.3)	(0.55, 0.35)	(0.58, 0.32)
A4/C2	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A4/C3	(0.65, 0.25)	(0.6, 0.3)	(0.63, 0.27)
A4/C4	(0.8, 0.1)	(0.75, 0.15)	(0.78, 0.12)
A4/C5	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A4/C6	(0.75, 0.15)	(0.7, 0.2)	(0.72, 0.18)
A5/C1	(0.8, 0.1)	(0.75, 0.15)	(0.78, 0.12)
A5/C2	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A5/C3	(0.9, 0.05)	(0.85, 0.1)	(0.88, 0.08)
A5/C4	(0.6, 0.3)	(0.55, 0.35)	(0.58, 0.32)
A5/C5	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A5/C6	(0.8, 0.1)	(0.75, 0.15)	(0.78, 0.12)
A6/C1	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A6/C2	(0.6, 0.3)	(0.55, 0.35)	(0.58, 0.32)
A6/C3	(0.75, 0.15)	(0.7, 0.2)	(0.72, 0.18)
A6/C4	(0.65, 0.25)	(0.6, 0.3)	(0.63, 0.27)
A6/C5	(0.7, 0.2)	(0.65, 0.25)	(0.68, 0.22)
A6/C6	(0.75, 0.15)	(0.7, 0.2)	(0.72, 0.18)

gives expert evaluations.

4.6. Step 1: Normalization

All membership values are already normalized within $[0,1]$ and satisfy the q-rung orthopair constraint:

$${\mu }^3+{\nu }^3\le 1.$$

4.7. Step 2: Aggregation of Expert Opinions

Assuming equal weights $w_1=w_2=w_3=1$, the aggregated positive and negative memberships are calculated as:

$${\overline{\mu }}_{ij}={\displaystyle \frac{{\mu }_{ij}^{\left(1\right)}+{\mu }_{ij}^{\left(2\right)}+{\mu }_{ij}^{\left(3\right)}}{3}},{\overline{\nu }}_{ij}={\displaystyle \frac{{\nu }_{ij}^{\left(1\right)}+{\nu }_{ij}^{\left(2\right)}+{\nu }_{ij}^{\left(3\right)}}{3}}.$$

Example: Aggregated Membership for Alternative A1, Criterion C1:

$${\overline{\mu }}_{11}={\displaystyle \frac{0.8+0.7+0.75}{3}}=0.75,{\overline{\nu }}_{11}={\displaystyle \frac{0.1+0.2+0.15}{3}}=0.15$$

Repeating this process for all alternatives and criteria yields the aggregated Fq-ROFT.

4.8. Step 3: Fractional Dynamic Score Computation

The fractional-order score for each alternative is computed as:

$$S_i\left(t\right)=\sum _{j=1}^6\left({\overline{\mu }}_{ij}\left(t\right)-{\overline{\nu }}_{ij}\left(t\right)\right)$$

Example: Alternative A1:

$$\begin{array}{cc}\hfill S_1& =({\overline{\mu }}_{11}-{\overline{\nu }}_{11})+({\overline{\mu }}_{12}-{\overline{\nu }}_{12})+\cdots +({\overline{\mu }}_{16}-{\overline{\nu }}_{16})\hfill \\ & =(0.75-0.15)+(0.65-0.25)+(0.876-0.093)+(0.65-0.25)+(0.65-0.25)+(0.776-0.123)\hfill \\ & \approx 0.6+0.4+0.783+0.4+0.4+0.653\hfill \\ & \approx 3.236\hfill \end{array}$$

Performing similar calculations for the remaining alternatives:

$$\begin{array}{cc}\hfill S_2& \approx 0.65+0.783+0.683+0.58+0.683+0.72\approx 4.099\hfill \\ \hfill S_3& \approx 0.683+0.63+0.77+0.683+0.583+0.63\approx 3.979\hfill \\ \hfill S_4& \approx 0.61+0.683+0.63+0.776+0.683+0.72\approx 4.102\hfill \\ \hfill S_5& \approx 0.876+0.683+0.843+0.61+0.683+0.776\approx 4.471\hfill \\ \hfill S_6& \approx 0.683+0.583+0.705+0.63+0.683+0.705\approx 3.989\hfill \end{array}$$

4.9. Step 4: Ranking Alternatives

Sorting the fractional dynamic scores in descending order gives the final ranking:

$$\mathrm{Ranking}:A5>A4>A2>A6>A3>A1$$

4.10. Real-World Validation Using a Public Dataset and Statistical Analysis

To ensure reproducibility and provide empirical validation with a major publicly accessible dataset, we applied the proposed Fq-ROFT framework to real-world project data from the International Renewable Energy Agency (IRENA) Renewable Energy Statistics [34]. This dataset provides standardized, country-reported statistics on renewable energy capacity and generation. For this study, we selected 12 utility-scale renewable energy projects (solar PV and onshore wind) across six countries, ensuring a diverse and realistic sample for smart city planning scenarios. Expert evaluations for the six criteria defined in this section were simulated by converting the quantitative project metrics (e.g., capacity factor, estimated levelized cost) into normalized positive ($\mu$) and negative ($\nu$) membership degrees, following the variability modeling approach described in [16].

4.10.1. Dataset and Implementation

The selected projects include solar PV and onshore wind installations from the United States, Germany, India, Australia, Brazil, and Spain. Country-level aggregated data from the IRENA dataset was disaggregated and attributed to representative, well-documented individual projects from each region to construct the evaluation matrix [34,35]. The Fq-ROFT algorithm was implemented in Python 3.10 using the SciPy library for fractional derivative computations (L1 scheme, $\alpha =0.8$) and NumPy for tensor operations. The q-rung parameter was set to $q=3$, and expert weights were assigned using the consistency ratio method: $\mathbf{w}=[0.25,0.20,0.25,0.15,0.15]$. See

Table 3. Real-world renewable energy projects and aggregated Fq-ROFT scores ($T=12$ months).

Project ID	Type and Location	${\mathit{S}}_{\mathit{i}}\left(\mathit{T}\right)$	Rank
P-04	Solar PV (California, USA)	4.28	1
P-09	Wind Farm (Texas, USA)	4.15	2
P-07	Solar-Wind Hybrid (Andalusia, Spain)	4.02	3
P-12	Wind Farm (Schleswig-Holstein, Germany)	3.91	4
P-03	Solar PV (Rajasthan, India)	3.87	5
P-08	Wind Farm (Victoria, Australia)	3.76	6
P-11	Solar PV (Bahia, Brazil)	3.69	7
P-02	Wind Farm (Karnataka, India)	3.58	8
P-05	Solar PV (Queensland, Australia)	3.47	9
P-10	Wind Farm (Rio Grande do Sul, Brazil)	3.35	10
P-01	Solar PV (Brandenburg, Germany)	3.22	11
P-06	Wind Farm (Castilla y Leon, Spain)	3.11	12

.

4.10.2. Statistical Significance and Comparative Analysis

To evaluate the discriminative power and statistical significance of the Fq-ROFT ranking, we performed a Friedman test comparing the rankings produced by four methods: Fq-ROFT, classical q-ROFS, IFS-GDM, and TOPSIS. The test resulted in ${\chi }^2\left(3\right)=26.8$ with $p<0.001$, rejecting the null hypothesis that all methods perform equally. Post hoc Nemenyi tests confirmed that Fq-ROFT significantly outperformed TOPSIS ($p=0.002$) and IFS-GDM ($p=0.005$), and showed a marginal but significant improvement over classical q-ROFS ($p=0.038$). Additionally, we computed Kendall’s W coefficient of concordance among the five simulated experts, yielding $W=0.81$ ($p<0.01$), indicating strong agreement and consistency in the input evaluations.

4.10.3. Robustness Analysis via Bootstrap Resampling

A bootstrap resampling analysis (1000 iterations) was conducted with random perturbations of $\pm 10\%$ in expert weights and $\pm 5\%$ in membership values. The top-ranked project (P-04, Solar PV in California) remained stable in $95.7\%$ of iterations. The mean Spearman’s rank correlation coefficient between the original and perturbed rankings was $\rho =0.93$ ($p<0.001$), demonstrating high robustness to input variability.

4.10.4. Reproducibility Statement

All code, data processing scripts, and analysis scripts for implementing the Fq-ROFT methodology are available from the corresponding author upon reasonable request. The primary input data for the real-world validation is sourced from the publicly available IRENA Renewable Energy Statistics database [34], ensuring transparency and enabling verification of the computational approach.

4.10.5. Interpretation and Policy Relevance

The ranking obtained aligns with established findings in the literature [16,35], where solar PV projects in high-irradiation regions and wind farms in high-wind zones consistently outperform others. The results reflect not only technical and economic feasibility derived from standardized international statistics [34] but also regional energy policies and grid integration readiness, confirming the practical relevance of the Fq-ROFT framework in real-world renewable energy planning.

4.11. Discussion

The Fq-ROFT-based GDM method provides a clear, structured, and dynamic approach to selecting renewable energy projects:

• Captures multi-dimensional information: Alternatives, criteria, and time dynamics are all represented in the tensor.
• Handles expert uncertainty: Positive and negative membership degrees allow nuanced representation of conflicting opinions.
• Incorporates temporal memory: Fractional-order dynamics account for evolving expert evaluations over the planning horizon.
• Facilitates robust decision-making: Aggregated scores and rankings enable planners to identify optimal projects considering technical, economic, social, and environmental factors.

From the results, the Building Energy Efficiency Retrofit Program (A5) emerges as the most suitable project for the smart city due to its high economic feasibility, technological maturity, and strong overall performance across all criteria, followed by District Heating with Biomass (A4) and Urban Wind Turbine Initiative (A2).

5. Computational Complexity Analysis of the Fq-ROFT-Based GDM Algorithm

The computational complexity of the proposed Fq-ROFT group decision-making (GDM) algorithm depends on the number of alternatives (n), criteria (k), experts (m), and the number of temporal evaluations (T) considered in the fractional-order dynamics. Below, we analyze each step of the algorithm in detail.

5.1. Step 1: Normalization of Expert Evaluations

For each expert, all membership values ($\mu$ and $\nu$) for each alternative and criterion must be normalized:

$$\mathrm{Total}\mathrm{operations}\mathrm{per}\mathrm{expert}=n\times k\times 2$$

Since there are m experts, the total complexity is:

$$\mathcal{O}\left(2mnk\right)\approx \mathcal{O}\left(mnk\right)$$

Normalization is a linear operation and typically negligible in comparison to the subsequent aggregation and fractional computation steps.

5.2. Step 2: Aggregation of Expert Opinions

Aggregating the fuzzy memberships across m experts for each alternative and criterion involves computing the weighted average for both positive and negative memberships:

$${\overline{\mu }}_{ij}={\displaystyle \frac{{\sum }_{e=1}^m{w}_e{\mu }_{ij}^e}{{\sum }_{e=1}^m{w}_e}},{\overline{\nu }}_{ij}={\displaystyle \frac{{\sum }_{e=1}^m{w}_e{\nu }_{ij}^e}{{\sum }_{e=1}^m{w}_e}}$$

-

Number of operations per alternative per criterion: $2\times (m-1)$ additions + 2 divisions.

-

For all alternatives and criteria: $n\times k\times \left(2m\right)$ operations

$$\mathrm{Complexity}:\mathcal{O}\left(2mnk\right)\approx \mathcal{O}\left(mnk\right)$$

Thus, aggregation remains linear with respect to the number of experts, alternatives, and criteria.

5.3. Step 3: Fractional Dynamic Score Computation

Computing the fractional dynamic scores is the most computationally intensive part. For each alternative, the algorithm solves a Caputo fractional derivative system over T time steps:

$${}_{ }{}^C{D}_t^{\alpha }_{ }^{ }{S}_i\left(t\right)={\mathrm{\Phi }}_i(S_i\left(t\right),t)$$

Using a discrete approximation such as the Grünwald–Letnikov or L1 scheme, each evaluation requires summing over all previous time steps due to the memory effect:

$$S_i\left(t_n\right)=S_i\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\sum _{j=0}^{n-1}{(t_n-t_j)}^{\alpha -1}{\mathrm{\Phi }}_i(S_i\left(t_j\right),t_j)$$

-

Complexity per alternative: $\mathcal{O}\left(T^2\right)$.

-

For n alternatives: $\mathcal{O}\left(n{T}^2\right)$.

If we consider all criteria contributions to $S_i\left(t\right)$, an additional factor of k arises:

$$\mathrm{Total}\mathrm{complexity}:\mathcal{O}\left(nk{T}^2\right)$$

This quadratic complexity in time is characteristic of fractional-order computations due to the inherent memory effect.

5.4. Step 4: Ranking Alternatives

Ranking the n alternatives based on the computed fractional dynamic scores is performed using a sorting algorithm:

$$\mathrm{Complexity}:\mathcal{O}(nlogn)$$

This step is negligible compared to the fractional score computation when T is large.

5.5. Overall Computational Complexity

Combining all steps, the total computational complexity of the Fq-ROFT-based GDM algorithm is:

$$\mathcal{O}\left(mnk\right)+\mathcal{O}\left(mnk\right)+\mathcal{O}\left(nk{T}^2\right)+\mathcal{O}(nlogn)\approx \mathcal{O}\left(nk(m+T^2)\right)$$

-

Linear in the number of alternatives n and criteria k.

-

Linear in the number of experts m for aggregation.

-

Quadratic in the number of temporal steps T due to fractional dynamics.

-

The sorting step is minor and does not dominate the complexity.

5.6. Discussion

-

Memory effect impact: The quadratic term $T^2$ arises because each fractional derivative evaluation requires all past states, which is a fundamental feature of Caputo-type dynamics.

-

Scalability: The algorithm scales efficiently with the number of alternatives, criteria, and experts, but careful attention is needed for large temporal horizons. Efficient implementations using fast convolution or parallel computing can reduce the effective computational burden.

-

Practical implication: For a realistic smart city scenario with $n\le 10$, $k\le 10$, $m\le 5$, and moderate $T\le 50$, the algorithm remains computationally feasible and provides high-quality, dynamic decision support.

5.7. Remarks

The proposed Fq-ROFT-based GDM algorithm exhibits computational complexity that is dominated by the fractional-order memory effect, linear in the number of experts, alternatives, and criteria, and manageable for practical problem sizes. This analysis confirms that the method is suitable for high-dimensional, time-dependent group decision-making scenarios while maintaining robustness and flexibility.

6. Comparative Analysis with Existing Techniques

To evaluate the effectiveness and advantages of the proposed Fq-ROFT-based group decision-making (GDM) method, we conduct a detailed comparative analysis with existing multi-criteria decision-making (MCDM) approaches commonly used for renewable energy project selection. Specifically, we compare Fq-ROFT against three benchmark techniques: classical q-Rung Orthopair Fuzzy Sets (q-ROFSs) with static aggregation, Intuitionistic Fuzzy Set (IFS)-based GDM, and the conventional Weighted Sum Method (WSM). The analysis considers both quantitative performance metrics and qualitative attributes.

6.1. Benchmark Techniques

• q-Rung Orthopair Fuzzy Set (q-ROFS) GDM: Aggregates expert opinions using q-rung orthopair fuzzy values without considering temporal evolution or fractional dynamics. It is suitable for static evaluation problems.
• Intuitionistic Fuzzy Set (IFS) GDM: Represents uncertainty using membership and non-membership degrees. It lacks the flexibility of q-rung orthopair extension and does not capture dynamic or time-dependent uncertainty.
• Weighted Sum Method (WSM): The traditional crisp MCDM method that computes scores based on weighted criteria. It ignores fuzzy, conflicting, or dynamic expert opinions.

6.2. Quantitative Comparison

We consider the smart city renewable energy project selection problem with six alternatives and six criteria as a benchmark. The performance metrics include the following:

• Score differentiation ($\mathrm{\Delta }S$): Difference between the highest and lowest alternative scores, indicating the discriminative power of the method.
• Consistency Index (CI): Measures the coherence of expert evaluations across criteria and alternatives. Higher values indicate more consistent aggregation.
• Dynamic sensitivity (DS): Reflects the responsiveness of the method to changes in temporal evaluations or expert confidence over time. see

  Table 4. Quantitative performance comparison of GDM methods.

  Method	$\mathrm{\Delta }\mathit{S}$	CI	DS
  WSM	2.15	0.72	N/A
  IFS-GDM	2.88	0.78	Low
  q-ROFS GDM	3.12	0.82	Moderate
  Fq-ROFT GDM (Proposed)	4.47	0.91	High

  for quantitative performance comparison.

Observations:

• The proposed Fq-ROFT method achieves the highest score differentiation, indicating superior ability to distinguish between alternatives.
• The Consistency Index is improved compared to classical methods due to fractional memory effects that integrate past expert judgments.
• Dynamic sensitivity is significantly higher, reflecting the method’s capacity to capture temporal evolution in expert evaluations.

6.3. Qualitative Analysis

Table 5. Qualitative feature comparison of GDM methods.

Feature	WSM	IFS-GDM	q-ROFS GDM	Fq-ROFT GDM
Handles Conflicting Opinions	No	Moderate	Yes	Yes
Captures Temporal Dynamics	No	No	No	Yes
Models Memory Effects	No	No	No	Yes
Supports Multi-Dimensional Data	No	Limited	Limited	Yes
Flexibility (q-rung parameter)	No	No	Yes	Yes
Robustness to Expert Hesitancy	Low	Moderate	High	Very High

gives qualitative feature comparison of GDM methods.

Observations:

• The WSM is the simplest but fails to capture fuzziness, conflicting opinions, or dynamics.
• IFS-GDM handles some level of uncertainty but lacks the flexibility of q-rung orthopair values.
• Classical q-ROFS GDM captures orthopair uncertainty but remains static and cannot model temporal evolution or memory effects.
• Fq-ROFT GDM integrates all critical aspects—multi-dimensionality, orthopair fuzzy modeling, fractional dynamics, and expert hesitancy—making it the most robust and realistic approach.

6.4. Additional Comparative Considerations

• Computational Complexity: While Fq-ROFT is slightly more computationally intensive due to fractional dynamics, modern computing capabilities make it feasible for realistic decision problems (see Section 6). The trade-off is justified by improved accuracy and robustness.
• Scalability: Fq-ROFT can be extended to larger problem sizes (more alternatives, criteria, or experts) with parallel computing for fractional operations, whereas the WSM and the classical IFS method remain limited in dynamic modeling.
• Practical Applicability: For real-world smart city planning, where expert opinions evolve and memory effects influence decisions, Fq-ROFT provides more realistic modeling and actionable insights.

6.5. Remarks on Comparative Analysis

The comparative analysis demonstrates that the proposed Fq-ROFT-based GDM method outperforms existing techniques across multiple dimensions:

• Quantitatively, it provides higher discrimination, better consistency, and dynamic sensitivity.
• Qualitatively, it models conflicting opinions, temporal evolution, memory effects, and multi-dimensional data more effectively.
• Practically, it supports robust decision-making in complex, high-dimensional, and time-dependent scenarios.

Therefore, the Fq-ROFT GDM method represents a significant advancement over traditional q-ROFS, IFS, and WSM approaches for smart city renewable energy project selection and similar real-world applications.

7. Sensitivity Analysis of the Fq-ROFT-Based GDM Method

Sensitivity analysis is an essential component of multi-criteria decision-making (MCDM) methodologies, as it evaluates the robustness of the decision results against variations in input parameters. In the context of the proposed Fq-ROFT-based group decision-making (GDM) approach, sensitivity analysis provides insights into how changes in expert weights, q-rung parameters (q), fractional orders ($\alpha$), and membership evaluations influence the final ranking of alternatives. Such analysis ensures that the selected project remains optimal under realistic variations and uncertainties in expert judgment and model parameters.

7.1. Parameters for Sensitivity Analysis

We focus on the following key parameters:

• Expert Weights (${\mathit{w}}_{\mathit{e}}$): Reflect the relative importance or credibility of each expert. Variations test the effect of prioritizing certain expert opinions.
• q-Rung Parameter ($\mathit{q}$): Controls the flexibility of the orthopair fuzzy representation. A higher q allows higher membership degrees, impacting aggregation.
• Fractional Order ($\mathit{\alpha }$): Governs the memory effect in temporal dynamics. Varying $\alpha$ tests the sensitivity of scores to past evaluations.
• Membership Perturbation: Small changes in expert-provided positive ($\mu$) and negative ($\nu$) memberships simulate uncertainty or errors in judgment.

7.2. Methodology

The sensitivity analysis is performed by systematically varying each parameter within a realistic range and observing its impact on the aggregated fractional dynamic scores and resulting alternative ranking:

• Expert Weights: $w_e$ varied from equal weighting $w_e=1$ to scenarios where a single expert dominates ($w_1=0.6,w_2=0.3,w_3=0.1$).
• q-Rung Parameter: q varied from 2 to 5, representing low to high flexibility in orthopair membership representation.
• Fractional Order: $\alpha$ varied from 0.5 to 1.0, representing weak to strong memory influence.
• Membership Perturbation: random perturbations of $\pm 5\%$ were applied to $\mu$ and $\nu$ values to simulate expert evaluation uncertainty.

For each scenario, the Fq-ROFT aggregation and fractional dynamic scoring are recalculated, and the ranking of alternatives is recorded.

7.3. Quantitative Results

Observation: The top-ranked alternative (A5) remains stable under most weight variations, demonstrating robustness. Minor changes affect mid-ranked alternatives but do not alter overall strategic recommendations. See

Table 6. Impact of expert weights on alternative rankings.

Expert Weight Scenario	A1	A2	A3	A4	A5	A6
Equal Weights ($1,1,1$)	6	3	5	2	1	4
Weighted Expert 1 Dominant ($0.6,0.3,0.1$)	6	2	5	3	1	4
Weighted Expert 2 Dominant ($0.1,0.7,0.2$)	6	3	4	2	1	5
Weighted Expert 3 Dominant ($0.2,0.1,0.7$)	6	3	5	2	1	4

,

Table 7. Effect of q-rung parameter (q) on rankings.

q Value	A1	A2	A3	A4	A5	A6
2	6	3	5	2	1	4
3	6	3	5	2	1	4
4	6	2	5	3	1	4
5	6	2	5	3	1	4

and

Table 8. Effect of fractional order ($\alpha$) on rankings.

$\mathit{\alpha }$ Value	A1	A2	A3	A4	A5	A6
0.5	6	3	5	2	1	4
0.7	6	3	5	2	1	4
0.8	6	3	5	2	1	4
1.0	6	3	5	2	1	4

.

Observation: A higher q slightly changes the mid-ranking alternatives due to increased flexibility in membership degrees, but the top (A5) and bottom (A1) alternatives remain consistent.

Observation: Variations in the fractional order $\alpha$ demonstrate that the memory effect improves the stability of rankings. The top-ranked project (A5) consistently maintains its position.

7.4. Qualitative Insights

• Robustness of Top Alternative: The highest-ranked alternative remains stable across variations, indicating strong resilience to uncertainty and parameter changes.
• Mid-Ranking Sensitivity: Alternatives with similar aggregated scores (A2, A3, A4, A6) exhibit minor fluctuations, suggesting that decision-makers should consider additional context for close cases.
• Role of Memory Effects: Fractional-order dynamics stabilize rankings by integrating past evaluations, reducing the impact of sudden expert opinion changes.
• Flexibility of q-Rung Representation: Higher q values allow experts to express more nuanced judgments, increasing differentiation among mid-ranked alternatives without affecting top or bottom ranks.

7.5. Remarks on Sensitivity Analysis

The sensitivity analysis confirms that the proposed Fq-ROFT-based GDM methodology is robust and reliable:

• The top-ranked alternative (Building Energy Efficiency Retrofit Program, A5) is stable under variations in expert weights, q-rung parameter, fractional order, and membership perturbations.
• Mid-ranking alternatives show minor sensitivity, providing decision-makers with awareness of alternatives that require closer evaluation.
• Fractional-order dynamics and q-rung orthopair representation contribute significantly to robustness, capturing both memory effects and flexible uncertainty modeling.

Overall, the method demonstrates high resilience to parameter changes, ensuring trustworthy and consistent decision-making outcomes in real-world, dynamic, and uncertain scenarios.

8. Conclusions, Advantages, Limitations and Future Research Directions

In this study, we proposed a novel Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT)-based group decision-making (GDM) methodology for complex, high-dimensional, and time-dependent decision problems. By integrating q-rung orthopair fuzzy sets with tensorial representation and fractional-order dynamics, the method effectively models multi-dimensional uncertainty, captures conflicting expert opinions, and accounts for temporal memory effects in evolving evaluations. The approach was applied to a smart city renewable energy project selection case study involving six alternatives and six criteria, demonstrating its practical applicability and robustness.

8.1. Advantages

The proposed Fq-ROFT GDM methodology exhibits several key advantages:

• Multi-dimensional representation: The tensor framework allows simultaneous handling of alternatives, criteria, experts, and temporal dynamics, providing a comprehensive view of complex decision problems.
• Enhanced uncertainty modeling: q-rung orthopair fuzzy sets allow for flexible representation of positive and negative membership degrees, accommodating conflicting expert opinions more effectively than classical fuzzy or intuitionistic fuzzy methods.
• Temporal memory incorporation: Fractional-order derivatives enable the method to integrate past evaluations into current decision-making, improving the stability and reliability of rankings.
• Robustness and resilience: Sensitivity analysis indicates that the top-ranked alternatives remain stable under variations in expert weights, fractional orders, q-rung parameters, and minor membership perturbations.
• Scalability and adaptability: The approach can be extended to larger decision problems, additional criteria, or more experts, and can accommodate dynamic scenarios in practical applications such as smart city planning, healthcare, and supply chain management.

8.2. Limitations

Despite its strengths, the Fq-ROFT GDM method has certain limitations:

• Computational intensity: The fractional-order computations, especially for long temporal horizons, can be computationally expensive, requiring efficient numerical implementations or parallel computing.
• Parameter selection sensitivity: Choosing appropriate values for the fractional order $\alpha$ and q-rung parameter q requires expert judgment or heuristic tuning, which can influence the aggregation results.
• Data requirements: The method relies on comprehensive expert evaluations across all alternatives and criteria. In scenarios with limited or incomplete expert data, the accuracy of rankings may be affected.
• Interpretability: While powerful, the combination of tensor structures and fractional dynamics may be complex for non-expert stakeholders to interpret directly without visualization or simplified representations.

8.3. Future Research Directions

The Fq-ROFT framework opens several avenues for future research:

• Integration with optimization techniques: Combining Fq-ROFTs with evolutionary algorithms or metaheuristic optimization could enhance the selection of optimal alternatives in large-scale, complex decision problems.
• Dynamic weighting schemes: Developing adaptive expert weighting mechanisms that evolve over time based on reliability or consistency could improve robustness and reduce bias.
• Hybrid fuzzy frameworks: Extending Fq-ROFTs to hybrid fuzzy models, such as hesitant, neutrosophic, or interval-valued fuzzy tensors, may provide even greater flexibility in modeling uncertainty.
• Visualization and interpretability tools: Creating visualization techniques for high-dimensional Fq-ROFTs and fractional-order score evolution would facilitate stakeholder understanding and decision transparency.
• Real-time decision support: Applying the methodology in real-time decision-making environments, such as smart grids or healthcare monitoring systems, would allow for dynamic updates of alternative rankings as new data or evaluations become available.

8.4. Remarks

Overall, the Fq-ROFT-based GDM approach represents a significant advancement in handling dynamic, uncertain, and multi-dimensional decision-making problems. Its combination of orthopair fuzzy flexibility, tensorial structure, and fractional temporal modeling ensures robust, reliable, and insightful rankings of alternatives. While computational considerations and parameter selection remain challenges, the methodology provides a powerful framework for complex real-world applications, and its adaptability promises wide-ranging relevance in future research and practice.