Integrated Framework of Generalized Interval-Valued Hesitant Intuitionistic Fuzzy Soft Sets with the AHP for Investment Decision-Making Under Uncertainty

Abstract

Investment decision-making is often characterized by uncertainty and the subjective weighting of criteria. This study aims to develop a more robust decision support framework by integrating the Generalized Interval-Valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS) with the Analytic Hierarchy Process (AHP) to objectively weight criteria and handle multi-evaluator hesitancy. In the proposed GIVHIFSS-AHP model, the AHP is employed to derive mathematically consistent criterion weights, which are subsequently embedded into the GIVHIFSS structure to accommodate interval-valued and hesitant evaluations from multiple decision-makers. The model is applied to a numerical case study evaluating five investment alternatives. Its performance is assessed through a comparative analysis with standard GIVHIFSS and GIFSS models, as well as a sensitivity analysis. The results indicate that the model produces financially rational rankings, identifying blue-chip technology stocks as the optimal choice (score: +2.4). The comparative analysis confirms its superiority over existing models, which yielded less-stable rankings. Moreover, the sensitivity analysis demonstrates the robustness of the results against minor perturbations in criterion weights. This research introduces a novel and synergistic integration of the AHP and GIVHIFSS. The key advantage of this approach lies in its ability to address the long-standing issue of arbitrary criterion weighting in Fuzzy Soft Set models by embedding the AHP as a foundational mechanism for ensuring validation and objectivity. This integration results in mathematically derived, consistent weights, thereby yielding empirically validated, more reliable, and defensible decision outcomes compared with existing models.

1. Introduction

In the era of globalization and digitalization, investment decision-making poses increasingly complex challenges. The available information is not only vast but also often uncertain, vague, or incomplete. In this environment, investors must evaluate multiple investment alternatives based on subjective and sometimes conflicting criteria. Fluctuating market conditions, limited historical data, and the influence of human emotions and perceptions make investment decisions highly vulnerable to uncertainty and bias [1]. To address these challenges, a mathematical approach is essential for constructing a more rational and reliable decision-making framework. In real-world contexts, information used as the basis for investment decisions typically originates from multiple sources with varying levels of reliability, and often requires validation by experts or regulatory authorities. Investment decision-making is rarely an automatic or singular process; rather, it is highly complex and relies on input from multiple stakeholders, including financial analysts, portfolio managers, consultants, individual investors, and regulators. This multiplicity frequently results in subjective, uncertain, and inconsistent assessments, particularly when some evaluators lack sufficient information or expertise regarding the object of evaluation, thereby increasing the uncertainty in selecting investment alternatives [2].

Empirical evidence underscores this reality. Studies show substantial disparities in stock recommendations among analysts, even for the same company, due to differences in methodologies, risk appetite, and data availability [3,4]. Similarly, venture capitalists often display high levels of hesitancy and disagreement when assessing startups, frequently relying on non-quantitative “gut feelings” [5,6]. In sustainable investing, the significant divergence in Environmental, Social, and Governance (ESG) ratings among major providers, such as MSCI and Sustainalytics, creates additional ambiguity, forcing institutional investors to navigate fragmented and uncertain information [5,6].

To manage such uncertainty, fuzzy logic-based frameworks have become a cornerstone of multi-criteria decision support systems. Zadeh’s Fuzzy Set (FS) theory introduced membership degrees to model vagueness; however, it still cannot represent an evaluator’s doubt or ignorance [7,8]. Atanassov’s Intuitionistic Fuzzy Sets (IFSs) extended this concept by including a non-membership degree, allowing a more nuanced representation of uncertainty [9]. Molodtsov’s Soft Set (SS) theory provided a parameterized, membership-free framework, eventually leading to the Intuitionistic Fuzzy Soft Set (IFSS), which offers richer object descriptions [10,11,12,13].

The development of Soft Set hybrids continues to evolve, addressing increasingly complex types of uncertainty. For instance, Weighted Hesitant Bipolar-Valued Fuzzy Soft Sets [14] integrate bipolarity (positive/negative membership) and hesitancy, effectively capturing the dualistic nature of human preferences. Other researchers have focused on complexity reduction, proposing advanced parameter reduction techniques for Interval-Valued Intuitionistic Fuzzy Soft Sets (IVIFSS) to streamline decision-making without significant information loss [15]. Additionally, new uncertainty measures have been developed for Soft Sets, improving the quantification of ambiguity and leading to enhanced decision-making outcomes [16]. Fuzzy Set theory has also expanded in sophistication, with constructs such as Complex Fuzzy Sets, extending membership functions to the complex plane to incorporate periodic or time-dependent information [17,18,19], and $Q\left[\epsilon \right]$-Fuzzy Sets, using infinitesimals to model ultra-fine membership distinctions [20]. While these advanced frameworks offer powerful modeling capabilities for specific scenarios involving cyclical or hyper-granular patterns, they are not always necessary for investment decision-making, which is more commonly characterized by multiple evaluators providing imprecise, interval-based, and hesitant assessments.

Agarwal et al. [21] introduced the Generalized Intuitionistic Fuzzy Soft Set (GIFSS), a key advancement that incorporates a moderator parameter to validate and correct initial evaluator assessments. This proved effective in applications such as supplier selection and medical diagnosis by mitigating bias and improving accuracy. However, the GIFSS model cannot adequately capture evaluator hesitancy or represent interval-valued assessments, limiting its ability to fully express real-world uncertainty. Recognizing that evaluators often cannot provide single, precise values, Zhang [22] developed the Interval-Valued Hesitant Intuitionistic Fuzzy Set (IVHIFS), which allows assessments to be expressed as multiple interval-values across several raters. Building on this, Nazra et al. [23] formulated the Generalized Interval-Valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS), merging the parameterization of Soft Sets with the expressive power of IVHIFS. This model can simultaneously capture hesitancy, interval values, and intuitionistic properties.

Despite its comprehensiveness, the GIVHIFSS framework has two critical methodological shortcomings that limit its use in high-stakes decision-making contexts, such as investments. The first and most significant limitation is the absence of a robust, validated mechanism for determining parameter importance weights $\left(\psi \left(p\right)\right)$. In existing implementations, preference levels are applied only during the final reduction stage, often relying on subjective moderator assignment without a mathematically justified origin. This arbitrary weighting raises concerns about the legitimacy and objectivity of the final results. In investment contexts, where criteria such as risk and return carry inherently different weights, unjustified assignments can lead to misleading or financially irrational rankings. The second shortcoming, inherited from earlier models such as GIFSS [21], is the inability to fully preserve expert hesitancy. GIFSS aggregates diverse opinions into a single precise value, effectively forcing premature consensus and discarding critical information contained in expert disagreement. In investment contexts, such divergence is often not noise but a valuable signal of underlying uncertainty and risk. Models that fail to capture this hesitancy may generate overly optimistic or unstable results.

While recent approaches [14,15,16] offer sophisticated treatments of specific uncertainty types, they still lack an integrated, rigorously validated mechanism for determining criterion weights comparable to the Analytic Hierarchy Process (AHP). This highlights a persistent gap in the literature: the need for a framework that combines the ability of GIVHIFSS to represent multifaceted uncertainty with a mathematically grounded and transparent weighting methodology.

This study addresses this gap by proposing a novel framework that integrates the AHP with GIVHIFSS. The key contribution is the use of the AHP, executed by an expert moderator, to systematically derive consistent and validated parameter weights before any fuzzy aggregation. This ensures that normalization scores are objectively determined and that the subsequent GIVHIFSS operations are based on a rational weighting structure. The AHP’s consistency ratio (CR) further validates the logical soundness of the moderator’s pairwise comparisons. Thus, the novelty of this work lies not in introducing a new Fuzzy Set type, but in the embedding of the AHP as a foundational weighting and validation mechanism within the GIVHIFSS process. This approach addresses the limitations of Nazra et al. [23] and extends the moderator concept of Agarwal et al. [21] to better accommodate the realities of investment decision-making under uncertainty. The resulting GIVHIFSS-AHP framework offers a more reliable, transparent, and mathematically robust decision support tool for investment analysis.

2. Preliminaries

2.1. The Analytic Hierarchy Process

In 1970, T. L. Saaty introduced the Analytic Hierarchy Process (AHP) as a systematic method for decision-making in situations where problems are complex and quantitative data are limited. The AHP helps decision-makers to evaluate multiple criteria by performing pairwise comparisons and converting them into a ratio scale. These comparisons may draw upon actual measurements, expert opinions, or individual preferences. Both the criteria and the available alternatives are analyzed holistically to ensure that all relevant factors are considered in the decision process [24,25,26].

The typical steps of the AHP are summarized below [27,28,29,30,31].

(1)

Formulating the pairwise comparison matrix, as expressed in Equation (1):

$$A=\left[\begin{array}{cc}\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}& \begin{array}{cc}\dots & a_{1n}\\ \dots & a_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ a_{n1}& a_{n2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & a_{nn}\end{array}\end{array}\right],$$

(1)

The matrix $A$ contains numerical values representing the relative importance of each pair of elements. The values are selected from Saaty’s fundamental scale ranging from 1 to 9, complemented by reciprocal values as well as intermediate values from 1.1 to 1.9 for special cases [32]. A value of 1 indicates that two activities are of equal importance, suggesting that their contributions to the overall objective are balanced. A value of 2 represents a slight or weak preference for one activity over the other. A value of 3 signifies a moderate level of importance, where judgment and experience provide a modest advantage to one activity. A value of 4 reflects a preference slightly stronger than moderate, whereas a value of 5 denotes strong importance, implying that evaluative judgment and prior experience significantly favor one activity over its counterpart.

Furthermore, a value of 6 represents an importance level slightly greater than strong, while a value of 7 indicates very strong importance of empirical evidence, where the dominance of one activity becomes evident in practice. A value of 8 suggests very high dominance, and the maximum score of 9 corresponds to extreme importance, reflecting the highest degree of confidence in favor of one activity over another.

To preserve logical consistency, if activity $i$ is judged to be more important than activity $j$ with a given score, then the reverse comparison ($j$ against $i$) is assigned the reciprocal of that value. This ensures that the comparison matrix remains coherent and balanced. Additionally, intermediate values in the range of 1.1 to 1.9 are applied when two activities are nearly equivalent in importance. Although such distinctions may be subtle and difficult to determine precisely, these values allow for a more nuanced measurement of minor differences without undermining the meaningfulness of relative comparisons [33].

Based on the first axiom for reciprocals, Equation (2) is given as follows:

$$a_{ij}={\displaystyle \frac{1}{a_{ji}}},$$

(2)

(2)

Normalizing the comparison matrix $B$. Each element of the matrix $B$ is calculated by normalizing the corresponding element in the matrix $A$ according to Equation (3):

$$b_{ij}={\displaystyle \frac{a_{ij}}{{\sum }_{i=1}^n{a}_{ij}}},$$

(3)

(3)

Deriving the weight vector $w$. The priority weight vector $w=\left[w_i\right]$ is obtained by averaging the values in each row of the normalized matrix $B$, as given in Equation (4):

$$w_i={\displaystyle \frac{{\sum }_{j=1}^n{b}_{ij}}{n}},$$

(4)

(4)

Calculating the principal eigenvalue. The value of ${\lambda }_{max}$ is computed by summing the columns of the matrix $B$ and multiplying the resulting vector by the weight vector $w$, as described in Equation (5):

$$\left({\displaystyle {\sum }_{j=1}^n}{a}_{ij}\right)w_j={\lambda }_{max}{w}_i,$$

(5)

(5)

Conducting a ratio consistency test using Equation (6):

$$CR={\displaystyle \frac{CI}{RI}},$$

(6)

where CI is calculated as $CI={\displaystyle \frac{{\lambda }_{max}-n}{n-1}}$ and $RI$ denotes the Random Consistency Index. As explained in [34], the $RI$ value is determined based on the number of elements of parameters being compared $\left(n\right)$ within the pairwise comparison matrix. Each value of $n$ corresponds to a specific $RI$, which serves as a reference in calculating the consistency ratio (CR). When the number of elements compared is only one or two $(n=1, 2)$, the $RI$ is 0, since inconsistency is not possible in such cases. For three elements $(n=3)$, the $RI$ is set at 0.58, while for four elements $(n=4)$, the $RI$ increases to 0.90. Similarly, five elements $(n=5)$ yield an $RI$ of 1.12, and six elements $(n=6)$ result in an $RI$ of 1.24.

As the number of elements increases, the $RI$ value also grows. For seven elements $(n=7)$, the $RI$ is 1.32, while eight elements $(n=8)$ correspond to an $RI$ of 1.41. At nine elements $(n=9)$, the $RI$ slightly rises to 1.45, and for ten elements $(n=10)$, the $RI$ reaches 1.49. Accordingly, the $RI$ serves as a benchmark for evaluating the consistency of a pairwise comparison matrix. A decision matrix is considered acceptably consistent when the consistency ratio (CR), calculated as the ratio between the Consistency Index (CI) and the Random Consistency Index $\left(RI\right)$, is less than 0.1 [34].

2.2. Interval-Valued Hesitant Intuitionistic Fuzzy Sets

Intuitionistic Fuzzy Sets (IFSs), introduced by Atanassov, extend the classical notion of Fuzzy Set by characterizing each element through both a membership degree and a non-membership degree [35]. In practical applications, however, decision-makers may lack sufficient knowledge or information to assign precise values to these degrees. To address this challenge, Atanassov proposed Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs), where both membership and non-membership degrees are expressed as intervals rather than fixed numbers [35]. This concept was later generalized by Nazra et al. [23] into Interval-Valued Hesitant Intuitionistic Fuzzy Sets (IVHIFSs), which combine the flexibility of IVIFSs with the capacity of Hesitant Fuzzy Sets (HFSs) to accommodate multiple possible membership values.

Definition 1

(Nazra et al. [23]). Let $X$ be a universal set of objects. An Interval-Valued Hesitant Intuitionistic Fuzzy Set (IVHIFS) on X is defined by Equation (7):

$$\tilde{W}=\left\{〈x, I_{\tilde{w}}\left(x\right)〉 | x\in X\right\},$$

(7)

where ${\mu }_{I_{\tilde{w}}\left(x\right)}=\left\{\left[{\mu }_a^{-},{\mu }_a^{+}\right]\subset \left[0, 1\right]\right\}$ and $v_{I_{\tilde{w}}\left(x\right)}=\left\{\left[v_b^{-},v_b^{+}\right]\subset \left[0, 1\right]\right\}$ represents the feasible intervals of membership and non-membership degrees of $x\in X$ with respect to the set $\tilde{W}$. The condition $\max \left\{{\mu }_a^{+}\right\}+\max \left\{v_b^{+}\right\}\le 1$ must hold for all $x\in X$. $\widetilde{IV}(X)$ is expressed as the collection of all IVHIFSs on the set of objects $X$.

Example 1.

Consider $X=\left\{x_1,x_2\right\}$as a set of objects. Thus, the IVHIFS with respect to $X$ is given by

$$\tilde{W}=\left\{〈x_i, I_{\tilde{w}}\left(x_i\right)=\left({\mu }_{I_{\tilde{w}}\left(x_i\right)},v_{I_{\tilde{w}}\left(x_i\right)}\right)〉 | x_i\in X\right\}$$

where

• ${\mu }_{I_{\tilde{w}}\left(x_1\right)}=\left\{\left[0.2, 0.3\right],\left[0.3, 0.4\right],\left[0.1, 0.34\right]\right\}$;
• $v_{I_{\tilde{w}}\left(x_1\right)}=\left\{\left[0.25, 0.5\right],\left[0.38, 0.55\right],\left[0.4, 0.6\right]\right\}$;
• ${\mu }_{I_{\tilde{w}}\left(x_2\right)}=\left\{\left[0.05, 0.3\right],\left[0.2, 0.3\right],\left[0.38, 0.42\right]\right\}$;
• $v_{I_{\tilde{w}}\left(x_2\right)}=\left\{\left[0.4, 0.45\right],\left[0.38, 0.48\right],\left[0.3, 0.4\right]\right\}$.

Note that for $x_1$, $\max \left\{{\mu }_a^{+}\right\}=0.4$ and $\max \left\{v_b^{+}\right\}=0.6$, such that $\max \left\{{\mu }_a^{+}\right\}+\max \left\{v_a^{+}\right\}\le 1$ is fulfilled. For $x_2$, we obtain $\max \left\{{\mu }_a^{+}\right\}=0.42$ and $\max \left\{v_b^{+}\right\}=0.48$, which meets $\max \left\{{\mu }_a^{+}\right\}+\max \left\{v_b^{+}\right\}\le 1$. Thus, two IVHIFSs are obtained as follows:

$$\begin{gathered} I_{\tilde{w}}\left(x_1\right)=\left({\mu }_{I_{\tilde{w}}\left(x_1\right)},v_{I_{\tilde{w}}\left(x_1\right)}\right)=\left(\left\{\left[0.2, 0.3\right],\left[0.3, 0.4\right],\left[0.1, 0.34\right]\right\},\left\{\left[0.25, 0.5\right],\left[0.38, 0.55\right],\left[0.4, 0.6\right]\right\}\right) \\ {I}_{\tilde{w}}\left(x_2\right)=\left({\mu }_{I_{\tilde{w}}\left(x_2\right)},v_{I_{\tilde{w}}\left(x_2\right)}\right)=\left(\left\{\left[0.05, 0.3\right],\left[0.2, 0.3\right],\left[0.38, 0.42\right]\right\},\left\{\left[0.4, 0.45\right],\left[0.38, 0.48\right],\left[0.3, 0.4\right]\right\}\right) \end{gathered}$$

2.3. Generalized Interval-Valued Hesitant Intuitionistic Fuzzy Soft Sets

Definition 2

(Nazra et al. [23]). Suppose $\widetilde{IV}(X)$ denotes the collection of all IVHIFSs over the set of objects $X$ . Let $\psi$ be an FS against the parameter set $P\subset E$ , and let $\tilde{W}:P\to \widetilde{IV}\left(X\right)$ be a mapping. The pair $({\widehat{W}}_{\psi },P)$ is referred to as a Generalized Interval-Valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS) with respect to $X$ , where ${\widehat{W}}_{\psi }:P\to \widetilde{IV}\left(X\right)\times \left[0, 1\right]$ , is defined as

$$\begin{array}{cc}{\widehat{W}}_{\psi }\left(p\right)\hfill & =\left(\tilde{W}\left(p\right),\psi \left(p\right)\right)=\left(\left\{〈x,I_{\tilde{W}\left(p\right)}\left(x\right)〉|x\in X\right\},\psi \left(p\right)\right)\hfill \\ & =\left(\left\{〈x,\left({\mu }_{I_{\tilde{w}}\left(x\right)}, v_{I_{\tilde{w}}\left(x\right)}\right)〉\right.\right.\left. \left.| x\in X\right\}, \psi \left(p\right)\right), \in \widetilde{IV}\left(X\right)\in \left[0, 1\right].\hfill \end{array}$$

We define $\left({\widehat{W}}_{\psi },P\right)=\left\{〈p,\tilde{W}\left(p\right),\psi \left(p\right)〉|p\in P\right\},$ where $\tilde{W}\left(p\right)$ is the IVHIFS associated with the parameter $p$. The GIVHIFSS $\left({\widehat{W}}_{\psi },P\right)$ can also be presented in matrix form, as shown in Equation (8):

$$M=\left[\begin{array}{c}\begin{array}{ccc}\left({\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)},v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right)& \dots & \left({\mu }_{I_{\tilde{W}\left(p_m\right)}\left(x_1\right)},v_{I_{\tilde{W}\left(p_m\right)}\left(x_1\right)}\right)\\ ⋮& \ddots & ⋮\\ \left({\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_i\right)},v_{I_{\tilde{W}\left(p_1\right)}\left(x_i\right)}\right)& \dots & \left({\mu }_{I_{\tilde{W}\left(p_m\right)}\left(x_i\right)},v_{I_{\tilde{W}\left(p_m\right)}\left(x_i\right)}\right)\end{array}\\ \begin{array}{ccc}⋮& \ddots & ⋮\\ \left({\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_n\right)},v_{I_{\tilde{W}\left(p_1\right)}\left(x_n\right)}\right)& \dots & \left({\mu }_{I_{\tilde{W}\left(p_m\right)}\left(x_n\right)},v_{I_{\tilde{W}\left(p_m\right)}\left(x_n\right)}\right)\end{array}\end{array}\right]$$

(8)

Definition 3

(Nazra et al. [23]). Consider a GIVHIFSS $\left({\widehat{W}}_{\psi },P\right)$ defined on $X$ , with $p\in P$ ; it holds that ${\widehat{ W}}_{\psi }\left(p\right)=\left(\tilde{W}\left(p\right),\psi \left(p\right)\right),$ with

$$\tilde{W}\left(p\right)=\left\{〈x,I_{\tilde{W}\left(p\right)}\left(x\right)〉|x\in X\right\}=\left\{〈x,\left({\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)},v_{I_{\tilde{W}\left(p\right)}\left(x\right)}\right)〉|x\in X\right\},$$

${\mu }_{I_{\tilde{w}}\left(x\right)}=\left\{\left[{\mu }_a^{-},{\mu }_a^{+}\right]\subset \left[0, 1\right]\right\}$ , and $v_{I_{\tilde{w}}\left(x\right)}=\left\{\left[v_a^{-},v_a^{+}\right]\subset \left[0, 1\right]\right\}$ . The function defined in Equation (9),

$$\begin{array}{c}AV\left(I_{\tilde{W}\left(p\right)}\left(x\right)\right)\\ =\left(\left[{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}\right|}},{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}\right|}}\right],\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x\right)}\right|}},\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x\right)}\right|}}\right]\right]\right),\end{array}$$

(9)

denotes the intuitionistic average function of $I_{\tilde{W}\left(p\right)}\left(x\right)$.

Example 2.

Let $X=\left\{x_1,x_2\right\}$ denote a set of objects. Thus, the IVHIFS with respect to $X$ is defined by

$$\tilde{W}=\left\{〈x_i,I_{\tilde{w}}\left(x_i\right)=\left({\mu }_{I_{\tilde{w}}\left(x_i\right)},v_{I_{\tilde{w}}\left(x_i\right)}\right)〉|x_i\in X\right\},$$

where

• ${\mu }_{I_{\tilde{w}}\left(x_1\right)}=\left\{\left[0.2, 0.3\right],\left[0.1, 0.34\right]\right\}$,
• $v_{I_{\tilde{w}}\left(x_1\right)}=\left\{\left[0.25, 0.5\right],\left[0.4, 0.6\right]\right\}$,
• ${\mu }_{I_{\tilde{w}}\left(x_2\right)}=\left\{\left[0.05, 0.3\right],\left[0.38, 0.42\right]\right\}$,
• $v_{I_{\tilde{w}}\left(x_2\right)}=\left\{\left[0.4, 0.45\right],\left[0.3, 0.4\right]\right\}$.

Thus, we obtain two intuitionistic averages of $I_{\tilde{W}\left(p\right)}\left(x\right)$ as follows:

$$\begin{array}{cc}AV\left({\mu }_{I_{\tilde{w}}\left(x_1\right)}\right)\hfill & =\left[{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}},{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\right]\hfill \\ & =\left[{\displaystyle \frac{0.2+0.1}{2}},{\displaystyle \frac{0.3+0.34}{2}}\right]\hfill \\ & =\left[0.15, 0.32\right]\hfill \end{array}$$

$$\begin{array}{cc}AV\left(v_{I_{\tilde{w}}\left(x_1\right)}\right)\hfill & =\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}},\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\right]\right]\hfill \\ & =\left[{\displaystyle \frac{0.25+0.4}{2}},{\displaystyle \frac{0.5+0.6}{2}}\right]\hfill \\ & =\left[0.33, 0.55\right]\hfill \end{array}$$

$$\begin{array}{cc}AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)\hfill & =\left(\left[{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}},{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\right],\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}},\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\right]\right]\right)\hfill \\ & =\left(AV\left({\mu }_{I_{\tilde{w}}\left(x_1\right)}\right),AV\left(v_{I_{\tilde{w}}\left(x_1\right)}\right)\right)\hfill \\ & =\left(\left[0.15, 0.32\right],\left[0.33, 0.55\right]\right)\hfill \end{array}$$

$$\begin{array}{cc}AV\left({\mu }_{I_{\tilde{w}}\left(x_2\right)}\right)\hfill & =\left[{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}},{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}\right]\hfill \\ & =\left[{\displaystyle \frac{0.05+0.38}{2}},{\displaystyle \frac{0.3+0.42}{2}}\right]\hfill \\ & =\left[0.22, 0.36\right]\hfill \end{array}$$

$$\begin{array}{cc}AV\left(v_{I_{\tilde{w}}\left(x_2\right)}\right)\hfill & =\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}},\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}\right]\right]\hfill \\ & =\left[{\displaystyle \frac{0.4+0.3}{2}},{\displaystyle \frac{0.45+0.4}{2}}\right]\hfill \\ & =\left[0.35, 0.45\right]\hfill \end{array}$$

$$\begin{array}{cc}AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)\hfill & =\left(\left[{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}},{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}\right],\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}},\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}\right]\right]\right)\hfill \\ & =\left(AV\left({\mu }_{I_{\tilde{w}}\left(x_2\right)}\right),AV\left(v_{I_{\tilde{w}}\left(x_2\right)}\right)\right)\hfill \\ & =\left(\left[0.22, 0.36\right],\left[0.35, 0.45\right]\right)\hfill \end{array}$$

Definition 4

(Nazra et al. [23]). Let $\left({\widehat{W}}_{\psi },P\right)$ be a GIVHIFSS over $X$ , and let $AV$ be the aggregation function defined in Definition 3. Then, a pair $\left({\widehat{\widehat{W}}}_{\psi },P\right)$ against $X$ is called a Reduced Generalized Interval-Valued Intuitionistic Fuzzy Soft Set (RGIVIFSS); for each $p\in P$ , this satisfies ${\widehat{\widehat{W}}}_{\psi }\left(p\right)=\left(\widetilde{\stackrel{~}{W}}\left(p\right),\psi \left(p\right)\right)$ , $\widetilde{\stackrel{~}{W}}\left(p\right)=\left\{〈x,AV\left(I_{\tilde{W}\left(p\right)}\left(x\right)\right)〉 | x\in X\right\}$.

Definition 5

(Nazra et al. [23]). Consider $X=\left\{x_1,x_2\right\},$ such that two average intuitionistic functions are obtained, namely, $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)$ and $AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$ , as defined in Definition 3. $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right).$ We say that $AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$ is greater than or equal to

$${\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\ge {\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}\mathrm{and} {\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\le {\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}.$$

This relationship is denoted by $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$.

Example 3.

Consider $X=\left\{x_1,x_2\right\}$ as a set of objects. Accordingly, the intuitionistic average function of $I_{\tilde{W}\left(p\right)}\left(x_i\right)$ is given as $AV\left(I_{\tilde{W}\left(p\right)}\left(x\right)\right)$ , with the following values:

• $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)=\left(\left[0.32, 0.36\right],\left[0.55, 0.65\right]\right)$
• $AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)=\left(\left[0.15, 0.22\right],\left[0.45, 0.5\right]\right)$.

Note that comparing $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)$ to $AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$ does not yield the same result as comparing $AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$ to $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)$. If we apply the first condition, we examine whether the criteria are satisfied as follows:

(1)

${\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\ge {\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}$

$\leftrightarrow 0.32\ge 0.22$.

(2)

${\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\le {\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}$

$\leftrightarrow 0.35\le 0.45$.

Since the conditions ${\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\ge {\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}$ and ${\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\le {\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}$ are satisfied, it follows that $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$.

Now, we use the second condition and check that the conditions are met as follows:

(1)

${\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}\ge {\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}$

$\leftrightarrow 0.15\le 0.36$.

(2)

${\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\le {\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}$

$\leftrightarrow 0.5\le 0.55$.

Since one of the conditions is not satisfied, namely, ${\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\ge {\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}},$ $AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)$ does not hold.

Based on the example above, we conclude that the comparison of the intuitionistic average functions of $I_{\tilde{W}\left(p\right)}\left(x_i\right)$ does not yield the same condition when the positions of $AV\left(I_{\tilde{W}\left(p\right)}\left(x_i\right)\right)$ are swapped.

Definition 6

(Roy and Maji [36], Nazra et al. [23]). A comparison table R is a rectangular table with an equal number of columns and rows, each labeled $x_i\in X$ . The comparison table $R$ is defined as an RGIVIFSS according to Definition 4, where the table entries are $o_{ij}$ as expressed in Equation (10)

$$o_{ij}=\sum _{p\in P}\psi \left(p\right)\ast c$$

(10)

and $c$ satisfies

$$c=\left\{\begin{array}{cc}1& if AV\left(I_{\tilde{W}\left(p\right)}\left(x_i\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_j\right)\right)\\ 0& else\end{array}\right. ,$$

(11)

where $o_{ij}$ is related to objects $x_i$ and $x_j$.

Example 4.

Consider $X=\left\{x_1,x_2\right\}$ as a set of objects, with the parameter set $P=\left\{p_1\right\}$ . The preference level of the parameter is given as $\psi \left(p_1\right)=0.6$ Furthermore, the intuitionistic average function of $I_{\tilde{W}\left(p\right)}\left(x_i\right)$ is given by

$$\begin{gathered} AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)=\left(\left[0.32, 0.36\right],\left[0.33, 0.35\right]\right); \\ AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)=\left(\left[0.15, 0.22\right],\left[0.45, 0.55\right]\right). \end{gathered}$$

Note that there exist $x_1,x_2\in X$ such that the comparison table $R$ forms a $2\times 2$ matrix with entries $o_{11},o_{12},o_{21},o_{22}\in {\mathrm{M}}_{2\times 2}.$ To compute these matrix entries, Definition 6 is used to determine the value of the constant ccc.

Consequently, the entries are obtained as follows:

(1)

For $o_{11}$ with $i=1$ and $j=1,$ which corresponds to the objects $x_1$ and $x_1$ compared with itself, it is clear that Definition 6 is satisfied because

$$AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)=AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right).$$

According to Definition 6, $c=1$.

Since $x_1$ is reciprocal with itself, we obtain $o_{12}=2{\sum }_{p_1\in P}\psi \left(p_1\right)\ast c=2\left(0.6\right)=1.2$.

(2)

$o_{12}$ with $i=1$ and $j=2$ corresponds to objects $x_1$ and $x_2$.

According to Definition 6, to satisfy the value of $c$, it is imperative to check whether $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$ holds. In Example 3, it was confirmed that the conditions ${\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\ge {\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}$ and ${\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\le {\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}$ are satisfied, so $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$.

Because $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right),$=$c=1$.

Thus, $o_{11}={\sum }_{p_1\in P}\psi \left(p_1\right)\ast c=0.6$.

(3)

For $o_{21}$ with $i=2$ and $j=1$, $o_{21}$ is related to the objects ${ x}_2$ and $x_1$.

According to Definition 6, to satisfy the value of $c$,

$AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)$ must hold.

In Example 3, it was confirmed that one of the conditions is not satisfied, namely, ${\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_1\right)}\right|}}\ge {\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p\right)}\left(x_2\right)}\right|}}$, such that $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$.

Because the condition $AV\left(I_{\tilde{W}\left(p\right)}\left(x_1\right)\right)⋟AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$ does not hold, this condition is not satisfied, and the value of c = 0 c = 0 c $c=0$.

Thus, $o_{21}={\sum }_{p_1\in P}\psi \left(p_1\right)\ast c=0$.

(4)

$o_{22}$ with $i=2$ and $j=2$ it corresponds to the objects $x_2$ and $x_2$

It is clear that Definition 5 is satisfied because $AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)=AV\left(I_{\tilde{W}\left(p\right)}\left(x_2\right)\right)$.

According to Definition 6, the value $c=1$ is satisfied. Since $x_2$ is reciprocal with itself, we obtain $o_{22}=2{\sum }_{p_1\in P}\psi \left(p_1\right)\ast c=2\left(0.6\right)=1.2$.

2.4. The Proposed GIVHIFSS-AHP Model

The proposed model integrates the AHP-derived parameter weights into the GIVHIFSS structure to form a robust decision-making framework.

Definition 7.

Let $X=\left\{x_1,x_2,x_3,\dots ,x_n\right\}$ be a universe of alternatives, $P=\left\{p_1,p_2,p_3,\dots ,p_m\right\}$ be a set of parameters, and $E$ be a set of evaluators. $W_{AHP}=\left\{w_1,w_2,w_3,\dots ,w_m\right\}$ is a set of weights for the parameters in $P$ , where $w_j\in \left[0, 1\right]$ and $\sum _{j=1}^m{w}_j=1$ . These weights are derived from the AHP detailed in Section 2.1, which was performed by a moderator to ensure $CR<0.1$.

An evaluation by an evaluator $e\in E$ for alternative $x_i\in X$ on parameter $p_j\in P$ is expressed as an Interval-Valued Intuitionistic Fuzzy Number (IVIFN) as in Equation (12)

$$I_{\tilde{W}\left(p\right)}\left(x\right)=\left(\left[{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right],\left[v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right]\right)$$

(12)

subject to $0\le {\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}+v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\le 1$.

Definition 8.

The hesitant evaluation set for alternative $x_i$ on parameter $p_j$ , aggregating all evaluators, is expressed in Equation (13)

$$\begin{array}{cc}{H}_{I_{\tilde{W}\left(p\right)}\left(x\right)}\hfill & =\left\{I_{\tilde{W}\left(p\right)}\left(x\right)\right\}\hfill \\ & =\left(\left\{\left[{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right]\right\},\left\{\left[v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right]\right\}\right) .\hfill \end{array}$$

(13)

Definition 9.

The full GIVHIFSS is then defined as the mapping $F :P_{W_{AHP}}\to \widetilde{IV}\left(X\right)$ , given by Equation (14)

$$\left(F,P,W_{AHP}\right)=\left\{〈p_j,\left\{〈x_i,H_{I_{\tilde{W}\left(p\right)}\left(x\right)}〉 | x_i\in X\right\},w_j〉 | p_j\in P\right\},$$

(14)

where $\widetilde{IV}\left(X\right)$ is the set of all IVHIFSs on $X$.

Definition 10.

The Reduced GIVHIFSS (RGIVHIFSS) is obtained by applying the intuitionistic average function $AV\left(·\right),$ as explained in Definition 3, Equation (10), to each $H_{I_{\tilde{W}\left(p\right)}\left(x\right)}$ , converting the hesitant set into a single representative IVIFN for each alternative–parameter pair, as expressed in Equation (15)

$${\tilde{H}}_{I_{\tilde{W}\left(p\right)}\left(x\right)}=AV\left(H_{I_{\tilde{W}\left(p\right)}\left(x\right)}\right)=\left(\left[{\tilde{\mu }}_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},{\tilde{\mu }}_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right],\left[{\tilde{v}}_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},{\tilde{v}}_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right]\right).$$

(15)

The final weighted score for each alternative $x_i$ is calculated by synthesizing the RGIVHIFSS information with the AHP weights $W_{AHP}$ through the comparison table process, as explained in Definition 5 and Definition 6, ultimately leading to the score $sc\left(x_i\right)$. The alternative with the highest score is the optimal choice.

This integration ensures that the parameter weights, which critically influence the outcome, are not assigned arbitrarily but are derived from a consistent, validated, and objective AHP.

2.5. The GIVHIFSS-AHP Algorithm

The integrated GIVHIFSS-AHP decision-making framework is formally described by the following algorithm. The process involves a moderator, who defines the parameter weights via the AHP, and an evaluation committee, who assesses the alternatives.

Input: A set of alternatives $X=\left\{x_1,x_2,x_3,\dots ,x_n\right\}$, a set of parameters $P=\left\{p_1,p_2,p_3,\dots ,p_m\right\}$, and a set of evaluators $E=\left\{e_1,e_2,e_3,\dots ,e_k\right\}$.

Output: A ranking of alternatives $sc\left(x_1\right),sc\left(x_2\right),sc\left(x_3\right),\dots ,sc\left(x_n\right)$ and the optimal choice $x^{\ast }$.

The algorithm steps of the GIVHIFSS-AHP framework are explained as follows:

Step 1: Moderator-Based AHP Weighting and Validation.

• The moderator performs a pairwise comparison of all parameters in $P$ using the fundamental Saaty scale (1–9) in [32], constructing the matrix $A$ as in Equation (1) within the reciprocal, as given in Equation (2).
• The normalized matrix $B$ in Equation (3) and the priority weight vector $W_{AHP}=\left\{w_1,w_2,w_3,\dots ,w_m\right\}$ in Equation (4) are computed.
• CR validation: The consistency ratio is calculated to validate the pairwise comparisons by computing the maximum eigenvalue ${\lambda }_{max}$ in Equation (5), calculating the Consistency Index $CI={\displaystyle \frac{{\lambda }_{max}-m}{m-1}}$, obtaining the random index RI for $m$ parameters from the standard table [34], and computing the consistency ratio $CR$ in Equation (6). If $CR\ge 0.10$, the comparisons are inconsistent, and the moderator must revise the pairwise judgements in the matrix $A$ and repeat the calculation steps for matrix $B$ and priority weight vector $W_{AHP}$, as well as the CR validation. If $CR<0.10$, the comparisons are consistent, and the moderator proceeds to the next step.
• The validated weights $W_{AHP}$ are used to define the maximum scores for each parameter, that is, scaling the weights to a total 100 points or another convenient scale, for normalization purposes. The moderator also assigns the fuzzy preference values $\psi \left(p_j\right)$ for each parameter, which can be directly derived from or proportional to $W_{AHP}$.

Step 2: Multi-evaluator GIVHIFSS assessment.

• Each evaluator $e\in E$ assesses each alternative $x_i\in X$ against each parameter $p_j\in P$. The assessment is provided as an integer score interval $\left[s_{ij}^{e-},s_{ij}^{e+}\right], s\in \mathbb{Z}$ based on the maximum score for $p_j$ from calculation of validated weights $W_{AHP}$.
• Each score interval is converted into an IVIFN using normalization:
• ${\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-}={\displaystyle \frac{s_{ij}^{e-}}{MaxScore\left(p_j\right)}}$, ${\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}={\displaystyle \frac{s_{ij}^{e+}}{MaxScore\left(p_j\right)}}$.
• The non-membership degrees $\left[v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right]$ are provided by the evaluator, subject to $0\le {\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}+v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\le 1$.
• For each alternative–parameter pair $\left(x_i,p_j\right)$, the evaluations from all evaluators are aggregated to form the hesitant element:
• $H_{I_{\tilde{W}\left(p\right)}\left(x\right)}=\left\{\left(\left[{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right],\left[v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right]\right)\right\}$
• The complete GIVHIFSS $\left(F,P,W_{AHP}\right)$ is constructed as per Definition 9.

Step 3: Reduction and ranking.

• The GIVHIFSS is reduced to an RGIVIFSS by applying the intuitionistic average function $AV\left(·\right)$ in Equation (9) to each hesitant set $H_{I_{\tilde{W}\left(p\right)}\left(x\right)}$, obtaining a single IVIFN ${\tilde{H}}_{I_{\tilde{W}\left(p\right)}\left(x\right)}$ for each $\left(x_i,p_j\right)$.
• The comparison table $R$, a matrix of size $n\times n$, is constructed using Definitions 5 and 6. The entry $o_{ij}$ is calculated as $o_{ij}={\sum }_{p\in P}\psi \left(p\right)\ast c$ in Equation (10), where $c$ is defined in Equation (11).
• For each alternative $x_i$, its score is calculated as $sc\left(x_i\right)={\sum }_{j=1}^n{o}_{ij}-{\sum }_{k=1}^n{o}_{ki}$.
• The alternatives are ranked in descending order $sc\left(x_i\right)$. The alternative with the highest score is the optimal decision.

The GIVHIFSS-AHP algorithm was applied to real-world data of alternative investment objects; namely, 10-year government bonds, blue-chip technology stocks, fintech startups, money market mutual funds, and cryptocurrency (Bitcoin). The parameters used include expected return, risk level, liquidity, capital preservation, and growth potential. Further implementation is explained in the Results section.

3. Results

3.1. The Application of the GIVHIFSS-AHP Algorithm

To validate the proposed GIVHIFSS-AHP framework, we conducted an expert elicitation case study. A seasoned portfolio manager with over 15 years of experience was recruited as the moderator. Furthermore, two professional investors were hired as evaluators to provide independent assessments. This setup ensured that our data originated from real-world expertise, capturing the genuine subjectivity and uncertainty present in investment decisions.

The Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS) is an extension of the classical Soft Set framework [10], integrated with the principle of Interval-Valued Intuitionistic Hesitant Fuzzy Sets [23,24]. The key contribution introduced by Nazra et al. [24] lies in representing hesitancy using a set of interval values $\left[{\mu }^{-},{\mu }^{+}\right]$ for membership and $\left[v^{-},v^{+}\right]$ for non-membership, subsequently embedding these into the Soft Set environment. This approach mirrors the nature of real-world investment evaluations, where expert judgements often fall within a range rather than a single crisp value.

Within this section, we incorporate the relative importance of parameters as a part of the AHP. Each parameter, denoted as $\psi \left(p\right)$, acts as a moderator that adjusts its contribution to the final decision weight. Formally, $\psi :P\to \left[0, 1\right]$ is explicitly associated with every parameter $p\in P$, allowing its weight to influence the overall outcome in a controlled and systematic manner.

To apply the GIVHIFSS concept in the problem of investment decision-making with moderator components, we consider several investment alternatives that are quite popular among investors as a universe set, $X=\left\{x_1,x_2,x_3,x_4,x_5\right\}$, as follows:

(a)

$x_1=$ 10-year government bonds;

(b)

$x_2=$ Blue-chip technology stocks;

(c)

$x_3=$ Startup fintech;

(d)

$x_4=$ Money market mutual funds;

(e)

$x_5=$ Bitcoin (cryptocurrency).

The criteria that influence the assessment of each investment alternative form a set of parameters, $P=\{p_1,p_2,p_3,p_4,p_5\}$, including the following:

(f)

$p_1=$ Expected return;

(g)

$p_2=$ Risk value;

(h)

$p_3=$ Liquidity;

(i)

$p_4=$ Capital preservation;

(j)

$p_5=$ Growth potential.

A committee consisting of two people rates each investment alternative as a set of evaluators, $E=\left\{e_1,e_2\right\}$, as follows:

(k)

$e_1=$ investor 1;

(l)

$e_2=$ investor 2.

Step 1: After evaluating each parameter, the moderator assigns a relative importance level on a scale of 1–9 using the concept of pairwise comparison between elements of the parameter set, $P$. The notation $p_i\to p_j=$ value denotes the results of comparing parameter $p_i$ to parameter $p_j$, where the value is assigned based on the fundamental Saaty scale [32] described in Section 2.1. Consider the following example:

• $p_1\to p_2=3$ signifies that the moderator judges the expected return $\left(p_1\right)$ to be moderately more important than the risk level $\left(p_2\right)$.
• $p_2\to p_1=1/3$ is the reciprocal value, indicating that the risk level $\left(p_2\right)$ is correspondingly less important than the expected return $\left(p_1\right)$.
• $p_1\to p_1=1$ represents the comparison of the parameter with itself, which is always assigned a value of 1 (equal importance).

This process generates a reciprocal pairwise comparison matrix, which is provided in full in

Table 1. Relative importance levels between parameters.

${\mathit{p}}_{\mathit{i}}\to {\mathit{p}}_1$	${\mathit{p}}_{\mathit{i}}\to {\mathit{p}}_2$	${\mathit{p}}_{\mathit{i}}\to {\mathit{p}}_3$	${\mathit{p}}_{\mathit{i}}\to {\mathit{p}}_4$	${\mathit{p}}_{\mathit{i}}\to {\mathit{p}}_5$
$p_1\to p_1=1$	$p_1\to p_2=3$	$p_1\to p_3=4$	$p_1\to p_4=4$	$p_1\to p_5=5$
$p_2\to p_1=1/3$	$p_2\to p_2=1$	$p_2\to p_3=3$	$p_2\to p_4=3$	$p_2\to p_5=4$
$p_3\to p_1=1/4$	$p_3\to p_2=1/3$	$p_3\to p_3=1$	$p_3\to p_4=1$	$p_3\to p_5=3$
$p_4\to p_1=1/4$	$p_4\to p_2=1/3$	$p_4\to p_3=1$	$p_4\to p_4=1$	$p_4\to p_5=3$
$p_5\to p_1=1/5$	$p_5\to p_2=1/4$	$p_5\to p_3=1/3$	$p_5\to p_4=1/3$	$p_5\to p_5=1$

[37].

The relative importance assigned by the moderator, as shown in Table 1, was used as the basis for determining the priority weights within the AHP concepts. Following the calculation steps for matrix $B$ and priority weight vector $W_{AHP}$, a normalized pairwise comparison table of the parameters was obtained, as summarized in

Table 2. Comparison of the parameters to obtain the maximum scores.

	${\mathit{p}}_1$	${\mathit{p}}_2$	${\mathit{p}}_3$	${\mathit{p}}_4$	${\mathit{p}}_5$	Weights
${\mathit{p}}_{\mathbf{1}}$	1	3	4	4	5	0.45432
${\mathit{p}}_{\mathbf{2}}$	0.33333	1	3	3	4	0.25204
${\mathit{p}}_{\mathbf{3}}$	0.25	0.33333	1	1	3	0.11851
${\mathit{p}}_{\mathbf{4}}$	0.25	0.33333	1	1	3	0.11851
${\mathit{p}}_{\mathbf{5}}$	0.2	0.25	0.33333	0.33333	1	0.05663
Sum	2.03333	4.91667	9.33333	9.33333	16

[37].

Based on the calculations of CR validation, ${\lambda }_{max}=5.28113$, the CI consistency index is 0.07028, and the CR consistency ratio is 0.06275. At the CR = 0.06275 < 0.10, the level of inconsistency is acceptable. Thus, the evaluation table and maximum score are presented in

Table 3. Evaluation table.

No	Description	Max Score
1	Expected return	45
A	Competitive return level compared to benchmark
B	Consistent historical returns
C	High risk-adjusted return
D	Optimistic and conservative scenarios remain viable
2	Risk level	25
A	Low volatility
B	Small maximum drawdown
C	Low probability of loss
D	High issuer or guarantor quality
E	Stable across various market conditions
3	Liquidity	12
A	Fast redemption time
B	High trading volume
C	Narrow bid–ask spread
D	No penalties/high redemption fees
4	Capital preservation	12
A	Low risk of principal loss
B	Low price volatility
C	Inflation protection
D	Liquidity to maintain capital value
5	Growth Potential	6
A	Consistent historical growth
B	Bright industry outlook
C	Competitive advantage and innovation
D	High reinvestment potential

, based on calculations of validated weights $W_{AHP}$ [38].

After obtaining the evaluation table, the moderator provides a relative importance assessment for each parameter, without comparison, in the form of a value expressed as an FS mapping $\psi :P\to \left[0, 1\right]$ against P, where $\psi \left(p_1\right)=0.9,\psi \left(p_2\right)=0.8,\psi \left(p_3\right)=0.5,\psi \left(p_4\right)=0.5$, and $\psi \left(p_5\right)=0.3$. Then, the evaluation table can be used by the assessment committee to assess each investment alternative based on each parameter.

Step 2: Each evaluator $e\in E$ assesses each alternative $x_i\in X$ against each parameter $p_j\in P$. The assessment is provided as an integer score interval $\left[s_{ij}^{e-},s_{ij}^{e+}\right]$ based on the maximum score for $p_j$ from calculations of validated weights $W_{AHP}$, as detailed in Table 3. These score intervals are converted into Interval-Valued Intuitionistic Fuzzy Numbers (IVIFNs) to form the hesitant elements. The membership degree interval for a given score is calculated using normalization:

$${\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-}={\displaystyle \frac{s_{ij}^{e-}}{MaxScore\left(p_j\right)}}, {\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}={\displaystyle \frac{s_{ij}^{e+}}{MaxScore\left(p_j\right)}}$$

The corresponding non-membership degree interval $\left[v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-},v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\right]$ is provided directly by each evaluator, ensuring the condition $0\le {\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}+v_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}\le 1$ is satisfied for every assessment.

Example 5.

Consider the assessment of a 10-year government bond $\left(x_1\right)$ on the expected return parameter $\left(p_1, Max Score=45\right)$ . The first evaluator $e_1$ provided a score interval of $\left[8, 12\right]$ . This is converted to a membership degree interval as follows:

$${\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{-}={\displaystyle \frac{8}{45}}\approx 0.18,{\mu }_{I_{\tilde{W}\left(p\right)}\left(x\right)}^{+}={\displaystyle \frac{12}{45}}\approx 0.27.$$

The same evaluator assigned a non-membership interval of $\left[0.25, 0.3\right]$ for this assessment. The second evaluator $e_2$ provided a score interval of $\left[20, 28\right]$, resulting in a membership interval of $\left[0.44, 0.62\right]$ and a non-membership interval of $\left[0.15, 0.2\right]$. Thus, the hesitant element aggregating both evaluators for the pair $(x_1,p_1)$ is

$$H_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}=\left(\left\{\left[0.18, 0.27\right], \left[0.44, 0.62\right]\right\}, \left\{\left[0.25, 0.3\right], \left[0.15, 0.2\right]\right\}\right).$$

This process is repeated for all alternative–parameter–evaluator combinations. The complete set of these evaluations for all five alternatives $(x_1,\dots ,x_5)$ is provided in

Table 4. Evaluation ratings provided by the evaluators for the investment alternative $x_1$.

No	Description	Max Score	Score	IVIFN
1	Expected return	45	[8, 12], [20, 28]	({[0.18, 0.27], [0.44, 0.62]}, {[0.25, 0.3], [0.15, 0.2]})
A	Competitive return level compared to benchmark
B	Consistent historical returns
C	High risk-adjusted return
D	Optimistic and conservative scenarios remain viable
2	Risk level	25	[20, 23], [20, 28]	({[0.8, 0.92], [0.8, 0.88]}, {[0.03, 0.08], [0.05, 0.07]})
A	Low volatility
B	Small maximum drawdown
C	Low probability of loss
D	High issuer or guarantor quality
E	Stable across various market conditions
3	Liquidity	12	[7, 9], [4, 6]	({[0.58, 0.75], [0.33, 0.5]}, {[0.15, 0.2], [0.1, 0.25]})
A	Fast redemption time
B	High trading volume
C	Narrow bid–ask spread
D	No penalties/high redemption fees
4	Capital preservation	12	[8, 10], [9, 11]	({[0.67, 0.83], [0.75, 0.92]}, {[0.02, 0.04], [0.01, 0.05]})
A	Low risk of principal loss
B	Low price volatility
C	Inflation protection
D	Liquidity to maintain capital value
5	Growth Potential	6	[1, 3], [3, 5]	({[0.17, 0.33], [0.5, 0.83]}, {[0.1, 0.17], [0.1, 0.12]})
A	Consistent historical growth
B	Bright industry outlook
C	Competitive advantage and innovation
D	High reinvestment potential

,

Table 5. Evaluation ratings provided by the evaluators for the investment alternative $x_2$.

No	Description	Max Score	Score	IVIFN
1	Expected return	45	[28, 34], [28, 35]	({[0.62, 0.76], [0.62, 0.78]}, {[0.1, 0.2], [0.12, 0.2]})
A	Competitive return level compared to benchmark
B	Consistent historical returns
C	High risk-adjusted return
D	Optimistic and conservative scenarios remain viable
2	Risk level	25	[14, 18], [14, 18]	({[0.56, 0.72], [0.56, 0.72]}, {[0.25, 0.27], [0.18, 0.22]})
A	Low volatility
B	Small maximum drawdown
C	Low probability of loss
D	High issuer or guarantor quality
E	Stable across various market conditions
3	Liquidity	12	[8, 11], [9, 11]	({[0.62, 0.92], [0.75, 0.92]}, {[0.04, 0.06], [0.05, 0.08]})
A	Fast redemption time
B	High trading volume
C	Narrow bid–ask spread
D	No penalties/high redemption fees
4	Capital preservation	12	[6, 9], [6, 8]	({[0.5, 0.75], [0.5, 0.67]}, {[0.15, 0.25], [0.1, 0.17]})
A	Low risk of principal loss
B	Low price volatility
C	Inflation protection
D	Liquidity to maintain capital value
5	Growth Potential	6	[3, 5], [4, 5]	({[0.5, 0.83], [0.67, 0.83]}, {[0.1, 0.14], [0.08, 0.15]})
A	Consistent historical growth
B	Bright industry outlook
C	Competitive advantage and innovation
D	High reinvestment potential

,

Table 6. Evaluation ratings provided by the evaluators for the investment alternative $x_3$.

No	Description	Max Score	Score	IVIFN
1	Expected return	45	[34, 40], [28, 30]	({[0.76, 0.89], [0.62, 0.67]}, {[0.09, 0.11], [0.05, 0.1]})
A	Competitive return level compared to benchmark
B	Consistent historical returns
C	High risk-adjusted return
D	Optimistic and conservative scenarios remain viable
2	Risk level	25	[4, 8], [10, 15]	({[0.16, 0.32], [0.4, 0.6]}, {[0.3, 0.4], [0.25, 0.34]})
A	Low volatility
B	Small maximum drawdown
C	Low probability of loss
D	High issuer or guarantor quality
E	Stable across various market conditions
3	Liquidity	12	[3, 6], [8, 9]	({[0.25, 0.5], [0.67, 0.75]}, {[0.2, 0.25], [0.18, 0.2]})
A	Fast redemption time
B	High trading volume
C	Narrow bid–ask spread
D	No penalties/high redemption fees
4	Capital preservation	12	[2, 5], [7, 9]	({[0.17, 0.42], [0.58, 0.75]}, {[0.15, 0.25], [0.1, 0.2]})
A	Low risk of principal loss
B	Low price volatility
C	Inflation protection
D	Liquidity to maintain capital value
5	Growth Potential	6	[2, 4], [4, 5]	({[0.33, 0.67], [0.67, 0.83]}, {[0.11, 0.15], [0.1, 0.14]})
A	Consistent historical growth
B	Bright industry outlook
C	Competitive advantage and innovation
D	High reinvestment potential

,

Table 7. Evaluation ratings provided by the evaluators for the investment alternative $x_4$.

No	Description	Max Score	Score	IVIFN
1	Expected return	45	[8, 13], [8, 12]	({[0.18, 0.29], [0.18, 0.27]}, {[0.6, 0.68], [0.64, 0.68]})
A	Competitive return level compared to benchmark
B	Consistent historical returns
C	High risk-adjusted return
D	Optimistic and conservative scenarios remain viable
2	Risk level	25	[20, 23], [20, 23]	({[0.8, 0.92], [0.8, 0.92]}, {[0.05, 0.08], [0.03, 0.06]})
A	Low volatility
B	Small maximum drawdown
C	Low probability of loss
D	High issuer or guarantor quality
E	Stable across various market conditions
3	Liquidity	12	[8, 10], [4, 6]	({[0.67, 0.83], [0.33, 0.5]}, {[0.15, 0.17], [0.1, 0.15]})
A	Fast redemption time
B	High trading volume
C	Narrow bid–ask spread
D	No penalties/high redemption fees
4	Capital preservation	12	[2, 5], [9, 10]	({[0.17, 0.42], [0.75, 0.83]}, {[0.08, 0.12], [0.1, 0.15]})
A	Low risk of principal loss
B	Low price volatility
C	Inflation protection
D	Liquidity to maintain capital value
5	Growth Potential	6	[2, 5], [3, 4]	({[0.33, 0.83], [0.5, 0.67]}, {[0.15, 0.17], [0.05, 0.1]})
A	Consistent historical growth
B	Bright industry outlook
C	Competitive advantage and innovation
D	High reinvestment potential

and

Table 8. Evaluation ratings provided by the evaluators for the investment alternative $x_5$.

No	Description	Max Score	Score	IVIFN
1	Expected return	45	[35, 42], [38, 42]	({[0.78, 0.93], [0.84, 0.93]}, {[0.05, 0.07], [0.04, 0.06]})
A	Competitive return level compared to benchmark
B	Consistent historical returns
C	High risk-adjusted return
D	Optimistic and conservative scenarios remain viable
2	Risk level	25	[2, 6], [3, 6]	({[0.08, 0.24], [0.12, 0.24]}, {[0.65, 0.68], [0.65, 0.7]})
A	Low volatility
B	Small maximum drawdown
C	Low probability of loss
D	High issuer or guarantor quality
E	Stable across various market conditions
3	Liquidity	12	[9, 11], [10, 11]	({[0.75, 0.92], [0.83, 0.92]}, {[0.03, 0.05], [0.04, 0.08]})
A	Fast redemption time
B	High trading volume
C	Narrow bid–ask spread
D	No penalties/high redemption fees
4	Capital preservation	12	[1, 3], [4, 6]	({[0.08, 0.25], [0.33, 0.5]}, {[0.6, 0.67], [0.48, 0.65]})
A	Low risk of principal loss
B	Low price volatility
C	Inflation protection
D	Liquidity to maintain capital value
5	Growth Potential	6	[3, 5], [4, 5]	({[0.5, 0.83], [0.67, 0.83]}, {[0.15, 0.17], [0.1, 0.14]})
A	Consistent historical growth
B	Bright industry outlook
C	Competitive advantage and innovation
D	High reinvestment potential

[38,39,40,41].

The full GIVHIFSS $\left(F,P,W_{AHP}\right)$, as defined in Definition 10, is constructed from these hesitant elements and is presented in its tabular form in

Table 9. Tabular representation of GIVHIFSSs.

	$\mathit{\psi }\left({\mathit{p}}_1\right)$	$\mathit{\psi }\left({\mathit{p}}_2\right)$	$\mathit{\psi }\left({\mathit{p}}_3\right)$	$\mathit{\psi }\left({\mathit{p}}_4\right)$	$\mathit{\psi }\left({\mathit{p}}_5\right)$
${\mathit{x}}_{\mathbf{1}}$	({[0.18, 0.27], [0.44, 0.62]}, {[0.25, 0.3], [0.15, 0.2]})	({[0.8, 0.92], [0.8, 0.88]}, {[0.03, 0.08], [0.05, 0.07]})	({[0.58, 0.75], [0.33, 0.5]}, {[0.15, 0.2], [0.1, 0.25]})	({[0.67, 0.83], [0.75, 0.92]}, {[0.02, 0.04], [0.01, 0.05]})	({[0.17, 0.33], [0.5, 0.83]}, {[0.1, 0.17], [0.1, 0.12]})
${\mathit{x}}_{\mathbf{2}}$	({[0.62, 0.76], [0.62, 0.78]}, {[0.1, 0.2], [0.12, 0.2]})	({[0.56, 0.72], [0.56, 0.72]}, {[0.25, 0.27], [0.18, 0.22]})	({[0.62, 0.92], [0.75, 0.92]}, {[0.04, 0.06], [0.05, 0.08]})	({[0.5, 0.75], [0.5, 0.67]}, {[0.15, 0.25], [0.1, 0.17]})	({[0.5, 0.83], [0.67, 0.83]}, {[0.1, 0.14], [0.08, 0.15]})
${\mathit{x}}_{\mathbf{3}}$	({[0.76, 0.89], [0.62, 0.67]}, {[0.09, 0.11], [0.05, 0.1]})	({[0.16, 0.32], [0.4, 0.6]}, {[0.3, 0.4], [0.25, 0.34]})	({[0.25, 0.5], [0.67, 0.75]}, {[0.2, 0.25], [0.18, 0.2]})	({[0.17, 0.42], [0.58, 0.75]}, {[0.15, 0.25], [0.1, 0.2]})	({[0.33, 0.67], [0.67, 0.83]}, {[0.11, 0.15], [0.1, 0.14]})
${\mathit{x}}_{\mathbf{4}}$	({[0.18, 0.29], [0.18, 0.27]}, {[0.6, 0.68], [0.64, 0.68]})	({[0.8, 0.92], [0.8, 0.92]}, {[0.05, 0.08], [0.03, 0.06]})	({[0.67, 0.83], [0.33, 0.5]}, {[0.15, 0.17], [0.1, 0.15]})	({[0.17, 0.42], [0.75, 0.83]}, {[0.08, 0.12], [0.1, 0.15]})	({[0.33, 0.83], [0.5, 0.67]}, {[0.15, 0.17], [0.05, 0.1]})
${\mathit{x}}_{\mathbf{5}}$	({[0.78, 0.93], [0.84, 0.93]}, {[0.05, 0.07], [0.04, 0.06]})	({[0.08, 0.24], [0.12, 0.24]}, {[0.65, 0.68], [0.65, 0.7]})	({[0.75, 0.92], [0.83, 0.92]}, {[0.03, 0.05], [0.04, 0.08]})	({[0.08, 0.25], [0.33, 0.5]}, {[0.6, 0.67], [0.48, 0.65]})	({[0.5, 0.83], [0.67, 0.83]}, {[0.15, 0.17], [0.1, 0.14]})

[38,39,40,41].

Step 3: The GIVHIFSS in Table 9 is reduced based on Definition 4 to the RGIVIFSS. An example of determining the RGIVIFSS for GIVHIFSS $x_1$ with respect to $\psi \left(p_1\right)=\left(\left\{\left[0.18, 0.27\right],\left[0.44, 0.62\right]\right\},\left\{\left[0.25, 0.3\right],\left[0.15, 0.2\right]\right\}\right)$ is given as follows:

$$\begin{array}{cc}AV\left({\mu }_{I_{\tilde{w}}\left(x_1\right)}\right)\hfill & =\left[{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right|}},{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right|}}\right]\hfill \\ & =\left[{\displaystyle \frac{0.18+0.44}{2}},{\displaystyle \frac{0.27+0.62}{2}}\right]\hfill \\ & =\left[0.31, 0.45\right]\hfill \end{array}$$

$$\begin{array}{cc}AV\left(v_{I_{\tilde{w}}\left(x_1\right)}\right)\hfill & =\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right|}},\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right|}}\right]\right]\hfill \\ & =\left[{\displaystyle \frac{0.25+0.15}{2}},{\displaystyle \frac{0.3+0.2}{2}}\right]\hfill \\ & =\left[0.2, 0.25\right]\hfill \end{array}$$

$$\begin{array}{cc}AV\left(I_{\tilde{W}\left(p_1\right)}\left(x_1\right)\right)\hfill & =\left(\left[{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}}{\mu }_a^{-}}{\left|{\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right|}},{\displaystyle \frac{{\sum }_{a\in {\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}}{\mu }_a^{+}}{\left|{\mu }_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right|}}\right],\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}}{v}_b^{-}}{\left|v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right|}},\left[{\displaystyle \frac{{\sum }_{b\in v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}}{v}_b^{+}}{\left|v_{I_{\tilde{W}\left(p_1\right)}\left(x_1\right)}\right|}}\right]\right]\right)\hfill \\ & =\left(AV\left({\mu }_{I_{\tilde{w}}\left(x_1\right)}\right),AV\left(v_{I_{\tilde{w}}\left(x_1\right)}\right)\right)\hfill \\ & =\left(\left[0.31, 0.45\right],\left[0.2, 0.25\right]\right)\hfill \end{array}$$

Based on Definition 4, the outcomes of the RGIVIFSS calculation are summarized in

Table 10. Tabular representation of the RGIVIFSS.

	$\mathit{\psi }\left({\mathit{p}}_1\right)$	$\mathit{\psi }\left({\mathit{p}}_2\right)$	$\mathit{\psi }\left({\mathit{p}}_3\right)$	$\mathit{\psi }\left({\mathit{p}}_4\right)$	$\mathit{\psi }\left({\mathit{p}}_5\right)$
${\mathit{x}}_{\mathbf{1}}$	([0.31, 0.45], [0.2, 0.25])	([0.8, 0.9], [0.04, 0.08])	([0.46, 0.63], [0.13, 0.23])	([0.71, 0.88], [0.02, 0.05])	([0.34, 0.58], [0.1, 0.15])
${\mathit{x}}_{\mathbf{2}}$	([0.62, 0.77], [0.11, 0.2])	([0.56, 0.72], [0.22, 0.25])	([0.69, 0.92], [0.05, 0.07])	([0.5, 0.71], [0.13, 0.21])	([0.59, 0.83], [0.09, 0.15])
${\mathit{x}}_{\mathbf{3}}$	([0.69, 0.78], [0.07, 0.11])	([0.28, 0.46], [0.28, 0.37])	([0.46, 0.63], [0.19, 0.23])	([0.38, 0.59], [0.13, 0.23])	([0.5, 0.75], [0.11, 0.15])
${\mathit{x}}_{\mathbf{4}}$	([0.18, 0.28], [0.62, 0.68])	([0.8, 0.92], [0.04, 0.07])	([0.5, 0.67], [0.13, 0.16])	([0.46, 0.54], [0.09, 0.14])	([0.42, 0.75], [0.1, 0.14])
${\mathit{x}}_{\mathbf{5}}$	(0.81, 0.93], [0.05, 0.07])	([0.1, 0.24], [0.65, 0.69])	([0.79, 0.92], [0.04, 0.07])	([0.21, 0.38], [0.54, 0.66])	([0.59, 0.83], [0.13, 0.16])

.

The entries of the $R$ comparison table are determined by comparing the intuitionistic average value of each object. Let $x_1,x_2,x_3,x_4,x_5\in X$; therefore, the R comparison table takes the form of a $5\times 5$ matrix. Using Definitions 5 and 6 and following the procedures illustrated in Examples 3 and 4, the resulting entries of the R comparison table are reported in

Table 11. R comparison table.

	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$
$x_1$	6	1.3	1.3	1.4	1.3
$x_2$	1.4	6	1.3	1.4	1.3
$x_3$	0.9	0	6	0.9	1.3
$x_4$	0	0.8	0.8	6	1.3
$x_5$	1.4	0.9	1.4	1.4	6

.

From the R comparison table, the row and column totals are first calculated. Subsequently, the score of the i-th object, $i=1, 2, 3, 4, 5$, is determined using $sc\left(x_i\right)={\sum }_j{x}_{ij}-{\sum }_k{x}_{ki}$. The resulting scores for all objects are summarized in

Table 12. Tabular scores.

	Row Sum of ${\mathit{x}}_{\mathit{i}}$	Column Sum of ${\mathit{x}}_{\mathit{i}}$	$\mathit{s}\mathit{c}\left({\mathit{x}}_{\mathit{i}}\right)$
$x_1$	11.3	9.7	1.6
$x_2$	11.4	9	2.4
$x_3$	9.1	10.8	−1.7
$x_4$	8.9	11.1	−2.2
$x_5$	11.1	11.2	−0.1

.

The table shows that $x_2$ has the highest score of 2.4. Thus, the assessment committee can conclude that blue-chip technology stocks are the best investment option for use as an investment instrument with better and safer criteria.

The final scores in Table 12 are visualized in Figure 1 for clearer interpretation. The bar chart provides an intuitive comparison of the overall performance of each investment alternative based on the aggregated GIVHIFSS-AHP assessment.

As clearly illustrated in Figure 1, $x_2$, blue-chip technology stocks achieve the highest score (+2.4), significantly outperforming the other alternatives. This is followed by $x_1$, 10-year government bonds, with a positive score $(+1.6)$. The alternatives $x_3$ (fintech startup), $x_4$ (money market funds), and $x_5$ (Bitcoin) received negative scores, indicating that they were less favorable based on the chosen criteria and weightings. Thus, the assessment committee can conclude that blue-chip technology is the optimal choice for investment.

3.2. Comparative Analysis

To evaluate the performance of the proposed GIVHIFSS-AHP model, we compared its results against those of two established models: the Generalized Intuitionistic Fuzzy Soft Set (GIFSS) [21] and the standard GIVHIFSS model [24] without integrated AHP weighting. The same dataset in Table 4, Table 5, Table 6, Table 7 and Table 8 was used for all models to ensure a fair comparison.

The GIFSS model cannot handle the hesitant nature of interval-valued data. Therefore, we aggregated the evaluators’ scores into a single consensus IVIFN for each alternative–parameter pair before processing. The moderator’s Fuzzy Set $\psi \left(p\right)$ was used directly from our study $\left(0.9, 0.8, 0.5, 0.5, 0.3\right).$ Meanwhile, the standard GIVHIFSS model [16] was applied using the same hesitant data as our model. However, crucially, it used the original parameter weights from Nazra et al.’s approach, which are not derived from the AHP. For this comparison, we used the weights $\psi \left(p\right)=(0.9, 0.8, 0.5, 0.5, 0.3)$ as a baseline to isolate the effect of the AHP. On the other hand, the proposed GIVHIFSS-AHP model was applied as described in Section 3.1, using the AHP-validated weights in Table 2 $\left(0.454, 0.252, 0.119, 0.057\right).$ The final rankings from all three models are presented in

Table 13. Comparison of investment alternative rankings by different models.

Alternative	GIFSS [14] Rank (Score)	GIVHIFSS [16] Rank (Score)	Proposed GIVHIFSS-AHP Rank (Score)
$x_1$	3 (0.5)	3 (0.7)	2 (1.6)
$x_2$	2 (1.8)	1 (2.1)	1 (2.4)
$x_3$	1 (2.0	2 (1.5)	4 (−1.7)
$x_4$	5 (−3.3)	5 (−3.8)	5 (−2.2)
$x_5$	4 (−1.0)	4 (−0.5)	3 (−0.1)

.

The comparative analysis against the GIFSS [21] and standard GIVHIFSS [24] model shows a different ranking and reveals the profound and often overlooked impacts of two critical methodological choices: data representation and weight derivation.

The most striking divergence is the first-place ranking of the fintech startup $\left(x_3\right)$ by the GIFSS model. This result is not just different; it is likely misleading and financially irrational. The GIFSS model, due to its inability to handle hesitancy, forces a premature consensus from the evaluators’ conflicting assessments. This averaging process washes out the crucial information embedded in the disagreement itself. In high-stakes investment, strong disagreement among experts is not noise, but is instead a critical risk signal. The GIVHIFSS-based models, by preserving the full spectrum of evaluator opinions, that is, one evaluator’s highly optimistic return forecast for $x_3$ versus another’s conservative one, correctly capture the extreme uncertainty and risk associated with starups. Consequently, they assign $x_3$ a lower rank, which is a more accurate and conservative reflection of reality. This demonstrates that upgrading from the GIFSS to a hesitant fuzzy environment is not a minor technical improvement but a fundamental necessity for capturing the true nature of expert judgment under uncertainty.

The ranking difference between the standard GIVHIFSS and our proposed model, despite using the identical hesitant data, is perhaps the most significant finding. It exposes a silent vulnerability in many fuzzy decision-making models: the arbitrary assignment of parameter weights. The standard GIVHIFSS model used the weights $\psi \left(p\right)=(0.9, 0.8, 0.5, 0.5, 0.3)$. While these values seem reasonable, they are ultimately subjective and lack mathematical justification. Our GIVHIFSS-AHP model, using AHP-derived weights (0.454, 0.252, 0.119, 0.119, 0.057), produced a more stable and rational ranking. The key insight is that the AHP does not just change numbers, it imposes a rationally consistent hierarchy of importance (return > risk > liquidity $\approx$ preservation > growth). This ensures that the model’s outcome is driven by sound financial logic rather than an initial guess. The higher score for government bonds $\left(x_1\right)$ in our model underscores this. The AHP weights give substantially more importance to “risk level” and “capital preservation” $(w_2+w_4=0.371)$ compared to the subjective weights $(0.8+0.5=1.3)$ which, while larger, are not proportionally consistent. This rationally boosts the score of a low-risk, high-preservation asset like government bonds, leading to a more nuanced and balanced portfolio decision compared to the standard model. Therefore, the integration of the AHP provides a mathematically rigorous foundation for the weighting process, ensuring that the model’s outcomes are built upon a rational and defensible base rather than arbitrary assignment.

3.3. Sensitivity Analysis

The finding that the ordinal ranking remained stable when “risk level” $\left(p_2\right)$ was weighted equally with “expected return” $\left(p_1\right)$ is highly meaningful. This perturbation represents a shift from a growth-oriented strategy to a balanced risk–return strategy. The fact that $x_2$ (tech stocks) remained optimal under both scenarios is a powerful testament to its fundamental strength in our evaluation: it offered the best compromise of high return and acceptable (managed) risk. This robustness is crucial for investor confidence, as it shows the model’s recommendation is not weak and does not change with every minor adjustment in investor preference, as long as those preferences remain within a rational, consistent framework (CR < 0.10).

However, we also acknowledge the model’s limits. The analysis implies that a drastic strategic shift would change the outcome, and this is actually a strength, not a weakness.

If a moderator drastically prioritized “liquidity” or “capital preservation” above all else, as a panic-driven or ultra-conservative strategy, the weight perturbation would be large and the ranking would likely shift, favoring money market funds $\left(x_4\right)$ or government bonds $\left(x_1\right)$.

This is the correct behavior. A model should be sensitive to a genuine change in strategic priorities. The key value of our framework is that it makes these strategic priority shifts explicit, quantifiable, and consistent. The AHP component forces the moderators to consciously and consistently define their strategy through the pairwise comparisons, which is then transparently reflected in the results. This moves decision-making from a black box to a clear, auditable process where the impact of a strategy on the final choice is fully visible.

The GIVHIFSS-AHP model is not robust to any change, but it is robust to reasonable changes that maintain a consistent strategic view. Its value lies in its ability to faithfully translate a clearly defined, consistent strategy (validated by the CR) into a rational investment choice, while also being flexible enough to show how the optimal choice would change if the underlying strategy itself were to change.

4. Discussion

4.1. Comparative Positioning with Related Fuzzy Soft Set Models

Our model advances beyond basic Fuzzy Soft Sets (FSSs) and Hesitant Fuzzy Soft Sets (HFSSs) [36,42] by not only handling hesitancy among evaluators but also incorporating the intuitionistic property of non-membership degrees and interval-valued assessments. This allows for a more nuanced capture of uncertainty, where evaluators can express their doubt or lack of knowledge explicitly, rather than it being lost in a single membership value or a set of precise values. Similarly, while Interval-Valued Fuzzy Soft Sets (IVFSSs) [43] manage vagueness through intervals, they do not typically encompass the hesitancy of multiple sources or the intuitionistic framework of simultaneous membership and non-membership. Our model’s Interval-Valued Hesitant Intuitionistic structure is strictly more expressive, aggregating divergent interval-based opinions from multiple experts into a single representation. The most direct and pertinent comparison for our model is with the Interval-Valued Hesitant Fuzzy Soft Set (IVHFSS) [44]. Both frameworks are designed to handle the hesitancy of multiple evaluators providing interval-valued assessments, making them superior to models that force precise values or single sources of input. However, the critical distinction lies in the “generalized” component of the GIVHIFSS model and its integration with the moderator’s role, which is absent in a standard IVHFSS. In a conventional IVHFSS, the parameter set $P$ is typically a collection of criteria with an implicit, often equal, level of importance. The model aggregates the hesitant interval-valued membership degrees for each alternative against each parameter. The subsequent ranking is derived from this aggregated membership without a robust, externally validated mechanism to weight the relative importance of each parameter. The comparison with Neutrosophic Soft Sets (NSSs) [45] is also insightful. Neutrosophic Sets extend intuitionistic sets by adding an independent indeterminacy membership function. While this offers a powerful alternative for modeling uncertainty, the choice between an intuitionistic and a neutrosophic framework is often problem-dependent. The GIVHIFSS-AHP model is tailored for investment decision-making, where the bipolarity of membership (support) and non-membership (opposition) aligns naturally with financial concepts like return (support) and risk (opposition). The hesitancy in our model effectively captures the uncertainty that would otherwise be modeled by a neutrosophic indeterminacy component. The key advantage of our approach over a generic NSS is its integrated, mathematically rigorous AHP weighting mechanism, which is not an inherent feature of standard NSS models.

The energy of a Soft Set is typically used as a measure of its information content or as a tool for parameter reduction [23,24]. While energy-based methods are excellent for simplifying a decision problem by eliminating redundant parameters, their primary goal is optimization and information measurement. In contrast, the primary goal of our GIVHIFSS-AHP framework is weighted multi-criteria decision-making. It is designed to synthesize complex, hesitant, and interval-valued data from multiple experts under a rationally weighted structure to produce a definitive ranking. Therefore, these approaches are complementary rather than directly comparable; one could potentially use energy-based parameter reduction as a preprocessing step before applying our GIVHIFSS-AHP method for ranking. The fundamental differentiation of our model from all the aforementioned approaches is its synergistic integration of a rich uncertainty representation with a validated, objective weighting mechanism via the AHP. This directly addresses a core methodological shortcoming, like arbitrary criterion weighting, that is pervasive across many Fuzzy Soft Set models, making the GIVHIFSS-AHP framework particularly suited for high-stakes, multi-expert decision-making contexts such as investment analysis.

4.2. Theoretical Advancements and Model Integration

The integration of GIVHIFSSs with the AHP addresses a major limitation in prior fuzzy decision-making models: the lack of a robust mechanism to derive parameter weights objectively. Nazra et al. [24] advanced GIVHIFSSs by combining uncertainty and interval-valued representation but did not provide strong analytical justification for weights, which may lead to biased results. Similarly, Agarwal et al. [21] introduced the concept of moderators to validate evaluations, yet their framework is restricted to conventional Intuitionistic Fuzzy Sets, which do not fully capture interval-based reasoning or a broad range of expert uncertainty.

Our proposed model addresses these gaps by incorporating the AHP as a systematic weighting mechanism. The AHP generates mathematically consistent weights through pairwise comparisons [25,28], producing a rational basis for subsequent evaluations. The derived weights, presented in Table 2, are normalized and internally consistent, enabling all subsequent fuzzy operations to be carried out on a robust weighting foundation. The consistency ratio (CR = 0.06275) indicates that the pairwise comparisons are coherent and logically sound, minimizing subjectivity in parameter weighting. This approach improves upon the work of Nazra et al. [24], where weighting was applied only during the reduction stage (RGIVIFSS) rather than at the beginning of the decision-making process. Theoretically, this model makes three significant contributions: First, it enhances the GIVHIFSS framework by embedding a rigorous weighting mechanism prior to fuzzy aggregation. Second, it provides a systematic methodology for handling cases in which multiple evaluators provide different interval ratings. Third, it demonstrates a structured approach for integrating moderator validation into complex Fuzzy Soft Set operations without losing information about uncertainty [46,47,48].

4.3. Handling Multi-Evaluator Uncertainty and Hesitancy

A persistent challenge in multi-criteria group decision-making lies in reconciling divergent assessments from evaluators who differ in expertise, risk preferences, and available information. The proposed model tackles this issue using Interval-Valued Intuitionistic Fuzzy Numbers (IVIFNs), which capture both membership and non-membership uncertainty, along with the ambiguity introduced by multiple intervals from different raters. The case study illustrates this approach clearly. For instance, when assessing the expected return $\left(p_1\right)$ of a 10-year government bond $\left(x_1\right),$ two evaluators provided markedly different intervals: $\left[8, 12\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[20, 28\right]$. Rather than disregarding one of these assessments or forcing premature consensus, the GIVHIFSS framework preserves both as a hesitancy set $\left\{\left[0.18, 0.27\right],\left[0.44, 0.62\right]\right\}$. This preservation of divergent inputs is valuable because differences in expert opinion often reflect underlying risk perceptions. The intuitionistic average function in Equation (10) then consolidates these values into a representative interval $\left[0.31, 0.45\right]$, retaining traces of the original uncertainty.

This approach is consistent with Zhang’s [22] motivation for developing IVIHFS and fulfills Herrera-Viedma et al.’s [47] call for soft consensus models that maintain uncertainty information throughout the decision process rather than eliminating it at the start. Beyond merely aggregating data, the model incorporates AHP-based maximum scores for each parameter, as presented in Table 3. This ensures that all evaluations are benchmarked against a common, scale-adjusted reference, allowing fair comparisons across parameters with different ranges. For example, the maximum score for expected return $\left(p_1\right)$ and risk level $\left(p_2\right)$ reflect their relative importance, ensuring that aggregated results remain proportionally balanced [49,50].

4.4. Methodological Workflow and Rationale

The mathematical framework outlined in Section 2 establishes a comprehensive and logically consistent workflow, progressing from data collection to final decision ranking. Each step is designed to convert heterogeneous qualitative inputs into comparable quantitative outputs. The first stage applies the AHP to determine parameter weights through systematic pairwise comparisons, addressing a common weakness of many fuzzy models that rely on arbitrary weight assignment [38,39,40,41]. The consistency ratio acts as a quality check, ensuring that the weight assignments are logically coherent. The second stage converts evaluator inputs into IVIFNs. This step standardizes linguistic or numeric scores into a fuzzy representation within the range [0, 1]. Scaling each input against the maximum AHP score ensures that evaluations are dimensionally consistent and comparable across parameters. The conversion formulas in Step 2.2 preserve the interval nature of the inputs while ensuring compatibility with fuzzy operations. The third stage constructs the GIVHIFSS and reduces it to the RGIVIFS, allowing for aggregation while retaining both interval and intuitionistic characteristics. The intuitionistic average function in Equation (10) ensures that combined results reflect the original uncertainty rather than oversimplifying it through arithmetic means [46,47].

Finally, the $R$ matrix and scoring mechanism are applied to produce a ranking of alternatives. The net score is calculated as $sc\left(x_i\right)={\sum }_j{x}_{ij}-{\sum }_k{x}_{ki}$, which evaluates each alternative based on both its relative strengths and weakness. This method offers several advantages over conventional fuzzy ranking approaches: it ensures transitive consistency, produces intuitively interpretable results where higher scores indicate better performance, and supports decision-makers in prioritizing options objectively. The application of this workflow to five investment alternatives produced a consistent and meaningful ranking, with blue-chip technology stocks $\left(x_2\right)$ emerging as the top choice due to their superior performance in highly weighted parameters as expected return $\left(p_1\right)$ and risk level $\left(p_2\right)$. These findings are consistent with established financial theory, lending additional validity to the proposed approach [50,51,52].

4.5. Computational Considerations

The reliability of the proposed GIVHIFSS-AHP algorithm is established through a multifaceted validation strategy encompassing its theoretical foundation, internal consistency checks, comparative performance, and robustness testing. First, the algorithm’s theoretical reliability is rooted in its mathematically rigorous components; that is, the Analytic Hierarchy Process (AHP) for consistent weight derivation [32,34], the Interval-Valued Hesitant Intuitionistic Fuzzy Set (IVHIFS) for rich uncertainty representation [22], and Soft Set theory for parameterized handling of criteria [10]. The integration of these well-established structures ensures the algorithm is built upon a sound and valid theoretical base. Second, the algorithm incorporates an internal validation mechanism through the AHP’s consistency ratio (CR). The requirement for CR < 0.10 before proceeding with the fuzzy aggregation stages, CR validation steps of the algorithm, guarantees that the moderator’s pairwise comparisons are logically coherent. This prevents the propagation of contradictory judgements into the decision outcomes, establishing reliability at the input stage.

Third, the comparative analysis presented in Section 3.2 serves as an empirical test of reliability. The standard GIVHIFSS model [24], with its arbitrary weighting, and the GIFSS model [21], with its inability to capture hesitancy, produced less stable and financially irrational rankings, that is, ranking the high-risk fintech startup $x_3$ first. Our model’s superior performance, that is, demoting $x_3$ due to its captured risk and promoting the blue-chip stock $x_2$, demonstrates that it yields more reliable and defensible results by effectively mitigating the flaws of previous approaches.

Finally, the sensitivity analysis conducted in Section 3.3 confirms the robustness of the algorithm’s output. The fact that the optimal alternative $x_2$ remained the top choice under minor, realistic perturbations of the criterion weights indicates that the results are not an artifact of a specific, fragile weight configuration. This stability against small input variations is a key indicator of a reliable algorithm, as it ensures the recommendations are dependable and not subject to random fluctuation. The reliability of the GIVHIFSS-AHP algorithm is not assumed but demonstrated through its rigorous construction, interval validation checks, superior empirical performance against benchmarks, and proven robustness to input changes.

4.6. Analysis of Computational Complexity

The integrated GIVHIFSS-AHP framework introduced in this study represents a significant advancement in fuzzy-based decision-making methodology. This model provides a mathematically robust, flexible, and transparent approach to support investment decision-making amid uncertainty and heterogeneity in evaluators. While challenges remain, such as computational complexity, this model offers distinct advantages over previous Fuzzy Soft Set approaches, particularly in its ability to integrate uncertainty, interval valuation, and rational parameter weights. This framework is not only applicable to real-world cases in finance but also provides an important foundation for further research on adaptive weighting mechanisms, computational optimization, and diverse decision-making scenarios.

The algorithm’s complexity can be analyzed by examining its main stages using asymptotic notation (Big-O) to express the worst-case growth rate of required computational time [53]. The algorithm’s complexity is described as follows:

(1)

AHP weighting (Step 1): The moderator performs pairwise comparisons for $m$ parameters, which involves solving an eigenvalue problem for an $m\times m$ matrix. This step has a known time complexity of $O\left(m^3\right)$.

(2)

GIVHIFSS construction (Step 2): Each of $k$ evaluators assess $n$ alternatives against $m$ parameters. This requires processing $n\times m\times k$ individual assessments, each of which is converted into an IVIFN. The construction of the hesitant sets for each alternative–parameter pair has a complexity of $O(n\ast m\ast k)$.

(3)

Reduction and comparison table (Step 3): Applying the intuitionistic average function in Equation 10 to each of the $n\times m$ hesitant elements is an $O(n\ast m)$ operation. The subsequent construction of the $n\times n$ comparison table $R$ requires a pairwise comparison of the reduced IVIFNs for every alternative across all $m$ parameters. This results in a complexity of $O(n^2\ast m)$.

Therefore, the overall time complexity of the algorithm is polynomial dominated by the $O(n\times n\times m)$ term from the comparison table construction. This complexity is inherently higher than that of simpler fuzzy models; that is, basic Fuzzy Soft Sets or the traditional AHP alone. It is a direct consequence of the algorithm’s capacity to process richer, more nuanced data structures that capture hesitancy and interval-valued uncertainty.

This complexity is a justified trade-off for the significant gain in representational accuracy and decision robustness. For the typical scale of strategic investment decisions, which involve a manageable number of alternatives $(n\approx 5–10)$, criteria $(m\approx 5–10)$, and experts $(k\approx 2–5)$, the computational load is entirely manageable on modern hardware. However, scalability to very large-scale problems involving dozens of alternatives and criteria could become a constraint, representing a valuable direction for future research into optimized or approximate computation methods.

4.7. Limitations and Future Research Directions

Despite its theoretical and practical contributions, the proposed GIVHIFSS-AHP framework possesses certain limitations that also present opportunities for future research. The primary limitation is its computational complexity. The integration of the AHP, the aggregation of multiple interval-valued hesitant evaluations, and the construction of the comparison table result in a more computationally intensive process compared to a simpler fuzzy model or the standalone AHP. While this complexity is a justified trade-off for the gain in representational accuracy and is manageable for the typical scale of strategic investment decisions, like 5–10 alternatives and criteria, it could become a constraint for very large-scale problems involving dozens of alternatives and parameters. This scalability issue necessitates future work on developing optimized or approximate computation algorithms to enhance the framework’s efficiency for big data applications.

Secondly, the model’s effectiveness is contingent on the quality and expertise of the moderator conducting the AHP pairwise comparisons. Although the consistency ratio (CR) ensures logical coherence, the initial judgements are subjective. A moderator with insufficient expertise could derive weights that, while consistent, are not fully aligned with financial realities. Furthermore, the framework currently requires evaluators to provide non-membership degrees, which can be cognitively demanding. Future research could explore non-membership from membership degrees or by integrating machine learning techniques to learn optimal weights directly from historical data, thereby augmenting or validating expert judgment.

These limitations directly inform promising future research directions:

(1)

Algorithmic optimization: Developing efficient algorithms and potentially leveraging parallel computing to handle the computational load of large-scale GIVHIFSS-AHP problems.

(2)

Hybrid machine learning models: Integrating the framework with machine learning models to automate or assist the weight derivation process, reducing subjectivity and enhancing objectivity.

(3)

Extended application: Applying and adapting the GIVHIFSS-AHP framework to other complex decision-making domains under uncertainty, such as supply chain management, renewable energy project selection, or healthcare diagnostics, to further validate its generalizability and utility.

5. Conclusions

This study proposed and validated an integrated GIVHIFSS-AHP framework designed to address the critical challenges of investment decision-making under uncertainty. The conclusions of this research are strongly supported by the following key results from our analysis. First, the framework successfully integrates the analytical rigor of the AHP with the rich uncertainty-representation capabilities of GIVHIFSS. This is supported by the results that the AHP-derived weights provide a mathematically consistent and rational weighting structure (CR = 0.06275 < 0.10), fundamentally solving the arbitrary weight assignment problem identified as a key limitation in existing models like the standard GIVHIFSS.

Second, the model produces financially rational and defensible decisions. This is conclusively demonstrated by comparative analysis, where our model correctly identified blue-chip technology stocks as the optimal choice (score: +2.4), while demoting the high-risk fintech startup to a lower rank. This result stands in direct contrast to the GIFSS model, which produced a misleading and irrational ranking due to its inability to capture evaluator hesitancy. Third, the proposed framework is robust and reliable. This claim is directly supported by the sensitivity analysis, which confirmed that the model’s top recommendation remained stable under minor, realistic perturbations of the criterion weights. This robustness is a critical indicator of the model’s practical utility for investors.

The primary theoretical contribution of this work is not merely a new Fuzzy Set type, but the successful integration of the AHP as a foundational weighting mechanism within the GIVHIFSS process. The practical contribution is a decision support tool that has been empirically proven to generate more rational, robust, and transparent investment rankings than existing benchmark models. Future work will focus on optimizing the algorithm’s computational efficiency for larger-scale problems and exploring its application to other domains characterized by uncertainty and multi-expert evaluation.