A noval approach for dual probabilistic linguistic multi-criteria decision making based on prioritized Maclaurin symmetric mean operators and VIKOR

Abstract

The evolving landscape of decision-making, especially in complex scenarios, poses a challenge in accurately capturing decision-makers’ cognitive information. This challenge becomes even more intricate in group decision-making situations due to the diverse cognitive perspectives of the participants. In response to these complexities, this study delves into the realm of dual probabilistic linguistic 2-tuple sets, an extension of dual-hesitant fuzzy sets known for their effectiveness in handling perplexing data. Dual probabilistic linguistic 2-tuple sets introduces a nuanced approach, considering multiple linguistic memberships, non-membership degrees, and their associated probabilistic and deviation information. We have devised diverse mechanisms aimed at augmenting the comprehension and utilization of dual probabilistic linguistic 2-tuple sets. These mechanisms encompass transformation functions, scoring methodologies and accuracy metrics. In this study, we introduce two aggregation operators for dual probabilistic linguistic 2-tuple sets based on prioritized averaging and Maclaurin symmetric mean principles. These operators serve as the cornerstone for a pioneering hybrid approach meticulously crafted for multiple criteria decision making within the framework of dual probabilistic linguistic 2-tuple sets. To demonstrate the practicality and efficacy of our approach, we present a real-world case study. The subsequent analytical comparison and discussion underscore the effectiveness of our methodology. The outcomes of this study not only showcase the excellent performance of our proposed method but also offer valuable practical implications for decision-makers navigating complex and diverse decision environments. Our work stands as a significant stride toward enhancing decision-making processes amid contemporary challenges.

Keywords

Decision making type-2 fuzzy set prioritized averaging operator MSM operator VIKOR

1 Introduction

In the intricate landscape of multiple criteria decision making (MCDM), individuals grapple with the complex task of evaluating alternatives against predetermined criteria to draw definitive conclusions. The initial phase of this process, where decision-makers articulate their thoughts, stands as the foundation of effective decision-making. However, this phase is riddled with challenges stemming from the inherent fuzziness, unpredictability, and complexity of the human mind. The inability of experts to precisely convey their perspectives has prompted extensive exploration within the realm of linguistic sets-a domain acknowledged and examined by numerous scholars.

Zadeh’s semnal introduction of fuzzy sets (FS) [1] in 1965 marked a pivotal moment, offering a framework to model uncertainty and fuzzy information. Despite their utility, traditional FS encountered limitations in representing qualitative information, leading to explorations of extensions such as intuitionistic fuzzy sets (IFSs) [2], interval-valued intuitionistic fuzzy sets (IVIFSs) [3], type-2 fuzzy sets [4], and hesitation fuzzy sets (HFSs) [5 –7]. While these extensions adeptly handled quantitative data, integrating qualitative dimensions into decision-making models remained a challenge.

To bridge this gap, scholars delved into semantic expansions aimed at capturing the nuanced preferences of decision-makers. Two prominent trajectories emerged: the first focused on overcoming individual and inter-individual uncertainties, resulting in models like the 2-tuple fuzzy linguistic representation model [8] and the linguistic model based on type-2 fuzzy sets [9 –11]. The second trajectory addressed proportional relationships among individuals, leading to pioneering work in the form of probabilistic linguistic term sets (PLTSs) [12]. PLTSs, offering a comprehensive approach to probability distribution and partial value scenarios, laid the foundation for subsequent research in comparisons, operations [13 –15], preference relations [16, 17], and decision-making methodologies.

Within probabilistic linguistic environments, efforts focused on enhancing conventional methods such as probabilistic linguistic VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [21], Technique for Order Preference by Similarity to Ideal Solution(TOPSIS) [22], and Preference Ranking Organization Method for Enrichment Evaluations(PROMETHEE) [23]. Despite this progress, existing aggregation operators like probabilistic linguistic weighted average (PLWA) [24] and probabilistic linguistic weighted geometric (PLWG) [25] fell short in handling the interrelated nature of real-world MCDM problems. To address this complexity, integrating PLTs with advanced aggregation operators like Bonferroni mean (BM) [26], Heronian mean (HM) [27], and Maclaurin symmetric mean (MSM) [28] became crucial. In the intricate realm of decision-making, understanding the relationships among diverse input arguments is pivotal. Traditional methods, such as BM and HM operators, often falter when faced with intricate interconnections among data points. Addressing this limitation, the MSM emerges as a potent solution, defining its prowess through its unique characteristics. The fundamental distinction between MSM and its counterparts, like BM, lies in MSM’s ability to represent interactions among a multitude of input data arguments-an achievement unattainable by BM. In our current study, we harness the capabilities of MSM aggregation operators to adeptly consolidate probabilistic linguistic information. Through this strategic utilization, we aim to pave the way for more nuanced and comprehensive decision-making processes.

In the domain of decision-making methodologies, a profound analysis of existing literature reveals critical gaps in risk assessment methods that demand attention. While conventional representation models using 2-tuple language offer continuous domain representation, enabling comprehensive information expression, their limitations become apparent when experts require nuanced or intricate language terms. Similarly, probabilistic linguistic terms, while enhancing flexibility in expert judgments, fall short in capturing the non-membership degree essential for policy-making decisions. Both 2-tuple language-based and probabilistic linguistic-based models tend to overlook deviations from linguistic terms, highlighting the need for a more refined approach.

The challenge posed by the absence of sophisticated operators capable of merging dual probability language preferences, while considering their generalization and adaptability, has been a persistent concern. In response to this gap, this study delves into the realm of relative theory in new fuzzy representations, aiming to provide a more nuanced understanding of human subjective cognitions. Within this endeavor, the Maclaurin symmetric mean and the Dual Maclaurin symmetric mean (DMSM) operators emerge as pivotal tools. These operators not only offer a flexible mechanism for fusing evaluation values but also demonstrate remarkable effectiveness in resolving complex decision-making problems.

The incorporation of transformation functions, representing diverse semantic values, further amplifies the utility of these operators. Consequently, the integration of dual probabilistic linguistic representation with these Maclaurin symmetric mean aggregation operators, along with the prioritized averaging(PA) operator, specifically denoted as dual probabilistic linguistic prioritized Maclaurin symmetric mean(DPLPMSM) and dual probabilistic linguistic prioritized dual Maclaurin symmetric mean(DPLPDMSM) operators, forms the foundational cornerstone of our methodology. This innovative approach not only addresses the existing challenges but also significantly advances the field by providing a comprehensive framework for handling intricate decision-making scenarios.

This paper presents several significant contributions:

(1) Distinctive Representation Model: We introduce the dual probabilistic linguistic 2-tuple sets (DPLTSs), a unique representation model effectively capturing Decision Makers’ (DMs) assessments. DPLTS excels in expressing uncertainty, closely mirroring human cognition. This model incorporates the proportion of elements in the set, differences in representation, and deviations from linguistic terms concurrently. We define the rules, operations, and functions governing DPLTSs, enhancing their practical applicability.

(2) Innovative Aggregation Operators: Our study proposes two novel aggregation operators, DPLPMSM and DPLPDMSM, operating within the realm of DPLTSs. These operators-Prioritized Maclaurin Symmetric Mean and Dual Maclaurin Symmetric Mean, respectively-facilitate in-depth discussions on interdependencies among criteria and their priority relationships. This nuanced analysis enhances the granularity of decision-making processes.

(3) Comprehensive Decision-Making Approach: We present a holistic decision-making approach, integrating MSM or DMSM operators within the unique framework of DPLTSs. To accommodate DPLTSs’ distinctive requirements, we adopt the VIKOR technique. This strategic integration resolves conflicts between attributes, offering a balanced and optimal solution.

The paper’s structure unfolds as follows: Section 2 meticulously defines DPLTSs, elucidating their corresponding operations, laws, and properties. Section 3 delves into the development of two innovative aggregation operators, namely DPLPMSM and DPLPDMSM, while exploring their essential properties. Section 4 outlines our carefully designed decision-making methodology, integrating these aggregation operators and the VIKOR technique. To illustrate the effectiveness of our approach, Section 5 presents an insightful example, highlighting the disparities in results obtained through different methods. Finally, Section 6 concludes our study, offering a reflective analysis of the findings, and outlines promising avenues for future research endeavors.

2 Dual probabilistic linguistic 2-tuple set

In this section, we delve into the theory surrounding linguistic term sets and provide a clear definition of DPLTSs.

2.1 Concepts of dual probabilistic linguistic 2-tuple set

In the section, we present a significant extension to the DPLTSs framework, introducing crucial theoretical advancements.

We define a linguistic term set as T = {s_α : α = - τ, . . . -1, 0, 1 . . . , τ}. The 2-tuple linguistic representation model conveys linguistic information through pairs of values, denoted as linguistic 2-tuple (s_α, δ), where s_α ∈ T signifies the central value of the linguistic term, and δ indicates the deviation from this central value. The deviation δ adheres to specific conditions: $δ \in (- \frac{1}{4 τ}, \frac{1}{4 τ})$ for α ≠ - τ, τ, $δ \in (0, \frac{1}{4 τ})$ for α = - τ, and $δ \in (- \frac{1}{4 τ}, 0)$ for α = τ. Let the linguistic 2-tuple set be denoted as LT. Expanding on the concept introduced by Xie et al. [31], we employ the following equation to represent the DPLTSs:

Definition 1. Let X = {x₁, x₂, ⋯ , x_N} be fixed set and T = {s_α : α = - τ, . . . -1, 0, 1 . . . , τ} be a linguistic term set (LTS). Then a DPLTS Y_s (P) on X is defined as $Y_{s} (P) = {< x_{i}, U_{s} (P), V_{s} (P) > : x_{i} \in X},$ (1) where x_i ∈ X and $U_{s} (P) = {u_{s}^{(t)} (p_{u}^{(t)}) | u_{s}^{(t)} \in LT, t = 1, 2, . . ., ♯ U_{s} (P), p_{u}^{(t)} \geq 0, \sum_{t = 0}^{♯ U_{s} (P)} p_{u}^{(t)} \leq 1},$ $V_{s} (P) = {v_{s}^{(k)} (p_{v}^{(k)}) | v_{s}^{(k)} \in LT, k = 1, 2, . . ., ♯ V_{s} (P),$

$p_{v}^{(k)} \geq 0, \sum_{k = 0}^{♯ V_{s} (P)} p_{v}^{(k)} \leq 1},$ U_s (P) , V_s (P) stand for the possible membership and non-membership degrees to the element x_i ∈ X satisfy the conditions that -τ ≤ U⁺ + V⁺ ≤ τ and -τ ≤ U^- + V^- ≤ τ; $u_{s}^{(k)} = (s_{u^{(t)}}, δ_{u}^{(t)})$ , $v_{s}^{(k)} = (s_{v^{(k)}}, δ_{v}^{(k)})$ are linguistic 2-tuple, s_{u
^(t)}, s_{v
^(k)} represent the central value of $u_{s}^{(t)}, v_{s}^{(k)}$ , $δ_{u}^{(t)}, δ_{v}^{(k)}$ represent the deviation to the central value of $u_{s}^{(t)}, v_{s}^{(k)}$ ; $p_{u}^{(t)}, p_{v}^{(k)}$ are associate probabilities, u^(t) and v^(k) are subscripts of LTS s_{u
^(t)} and s_{v
^(k)}, respectively.

$U^{+} = max_{t = 1, 2, . . ., ♯ U_{s} (P)} (u^{(t)} + 2 τ \cdot δ_{u}^{(t)})$ ,

$U^{-} = min_{t = 1, 2, . . ., ♯ U_{s} (P)} (u^{(t)} + 2 τ \cdot δ_{u}^{(t)})$ ,

$V^{+} = max_{k = 1, 2, . . ., ♯ V_{s} (P)} (v^{(k)} + 2 τ \cdot δ_{v}^{(k)})$ ,

$V^{-} = min_{k = 1, 2, . . ., ♯ V_{s} (P)} (v^{(k)} + 2 τ \cdot δ_{v}^{(k)})$ . If the Y_s (P) contains only one element, then it becomes to a DPLTS element $\tilde{Y_{s} (P)} = < \tilde{U_{s} (P)}, \tilde{V_{s} (P)} >$ where $\tilde{U_{s} (P)} = {u_{s} (p_{u}) | u_{s} \in linguistic 2 - tuple, t = 1, 2, . . ., ♯ U_{s} (P), p_{u} \geq 0, \sum_{t = 0}^{♯ U_{s} (P)} p_{u} \leq 1},$ $\tilde{V_{s} (P)} = {v_{s} (p_{v} | v_{s} \in linguistic 2 - tuple, k = 1, 2, . . ., ♯ V_{s} (P), p_{v} \geq 0, \sum_{k = 0}^{♯ V_{s} (P)} p_{v} \leq 1} .$

Building upon the foundational theory of LTS, we enhance the theory by introducing improvements to the reversible transformation functions of linguistic 2-tuples.

Definition 2. Consider a linguistic 2-tuple (s_h, δ), where s_h ∈ T = {s_α : α = - τ, . . . -1, 0, 1 . . . , τ}, we define the transformation functions as follows:

Forward transformation function: $\tilde{g} (s_{h}, δ) = \frac{h}{2 τ} + \frac{1}{2} + δ = β .$ (2)

Reverse transformation function: ${\tilde{g}}^{- 1} (β) = (s_{h}, δ),$ (3) where $h = [(β - \frac{1}{2}) 2 τ], δ = β - (\frac{h}{2 τ} + \frac{1}{2}),$ $δ \in (- \frac{1}{4 τ}, \frac{1}{4 τ}) (δ \neq - τ, τ)$ , $δ \in (0, \frac{1}{4 τ}) (δ = - τ)$ , $δ \in (- \frac{1}{4 τ}, 0) (δ = τ)$ . On the other hand, suppose that ${\tilde{g}}_{1}^{- 1} (β) = h$ , ${\tilde{g}}_{2}^{- 1} (β, h) = δ$ .

In the groundbreaking research by Xie et al. [31], the score function for dual probabilistic linguistic elements (DPLE) was initially defined as follows: $E (D) = s_{\bar{α} - \bar{β}} .$ Additionally, the accuracy function of DPLE is formulated as:

$σ (D) = \frac{(\sum_{i = 1}^{♯ L (p)} (p^{i} (r^{i} - \bar{α}))^{2})^{\frac{1}{2}}}{\sum_{i = 1}^{♯ L (p)} p^{i}} + \frac{(\sum_{j = 1}^{♯ U (p)} (p^{j} (r^{j} - \bar{β}))^{2})^{\frac{1}{2}}}{\sum_{j = 1}^{♯ U (p)} p^{j}} .$

In this context, $D = 〈 L (p), U (p) 〉 = 〈 s_{u^{(t)} + δ_{u}^{(t)} \cdot 2 τ} (p^{i}), s_{v^{(k)} + δ_{v}^{(k)} \cdot 2 τ} (p^{j}) 〉,$ where $\bar{α} = \frac{\sum_{i = 1}^{♯ L (p)} (u^{(t)} + δ_{u}^{(t)} \cdot 2 τ) p^{i}}{\sum_{i = 1}^{♯ L (p)} p^{i}}$ and

$\bar{β} = \frac{\sum_{j = 1}^{♯ U (p)} (v^{(k)} + δ_{v}^{(k)} \cdot 2 τ) p^{j}}{\sum_{j = 1}^{♯ U (p)} p^{j}}$ .

Expanding upon these foundations and the concept of DPLTSs, our study proposes a refinement to both the score function and the accuracy function. Specifically, we advocate substituting rⁱ and r^j with $u^{(t)} + δ_{u}^{(t)} \cdot 2 τ$ and $v^{(k)} + δ_{v}^{(k)} \cdot 2 τ$ respectively. This subtle yet pivotal modification enables a nuanced comparison. By evaluating the subscripts of s, we inherently assess the score and accuracy functions, providing a sophisticated perspective within our model.

In DPLTSs, the parameters adhere to specific constraints: $u^{(t)} - 0.5 \leq u^{(t)} + δ_{u}^{(t)} \cdot 2 τ \leq u^{(t)} + 0.5$ , $v^{(k)} - 0.5 \leq v^{(k)} + δ_{u}^{(t)} \cdot 2 τ \leq v^{(k)} + 0.5$ Here, u^(t), v^(k) are discrete variables within the set {- τ, . . . , -1, 0, 0, 1, . . . , τ}. The range of $s_{u^{(t)} + δ_{u}^{(t)} \cdot 2 τ}$ , $s_{v^{(k)} + δ_{u}^{(t)} \cdot 2 τ}$ is limited to [s_-τ, s_τ]. Based on the above theory, the score value of DPLTSs is defined as follows:

Definition 3. Let $Y_{s} (P) = 〈 {U_{s}^{(t)} (p_{u}^{(t)})}, {V_{s}^{(k)} (p_{v}^{(k)})} 〉$

$= 〈 {(s_{u^{(t)}}, δ_{u}^{(t)}) (p_{u}^{(t)})}, {(s_{v^{(k)}}, δ_{v}^{(k)}) (p_{v}^{(k)})} 〉$

be a DPLTS, where t = 1, 2, . . . , ♯ U_s (P), k = 1, 2, . . . , ♯ V_s (P). The score value and accuracy value function of Y_s (P) are defined as follows: $ℓ (Y_{s} (P)) = ξ_{1} - ξ_{2},$ (4) $\begin{matrix} A (Y_{s} (P)) & = \frac{\sqrt{\sum_{t = 1}^{U_{s} (P)} {(p_{u}^{(t)} ((u^{(t)} + δ_{u}^{(t)} \cdot 2 τ) - ξ_{1}))}^{2}}}{\sum_{t = 1}^{U_{s} (P)} p_{u}^{(t)}} \\ + \frac{\sqrt{\sum_{k = 1}^{V_{s} (P)} {(p_{v}^{(k)} ((v^{(k)} + δ_{v}^{(k)} \cdot 2 τ) - ξ_{2}))}^{2}}}{\sum_{k = 1}^{V_{s} (P)} p_{v}^{(k)}} . \end{matrix}$ (5) where $ξ_{1} = \frac{\sum_{t = 1}^{U_{s} (P)} p_{u}^{(t)} (u^{(t)} + δ_{u}^{(t)} \cdot 2 τ)}{\sum_{t = 1}^{U_{s} (P)} p_{u}^{(t)}}$ ,

$ξ_{2} = \frac{\sum_{k = 1}^{V_{s} (P)} p_{v}^{(k)} (v^{(k)} + δ_{v}^{(k)} \cdot 2 τ)}{\sum_{k = 1}^{V_{s} (P)} p_{v}^{(k)}}$ .

The comparison of DPLTSs is analogous to other sets of linguistic terms.

In preparation for addressing decision-making problems, we define the deviation degree between DPLTSs as follows.

Definition 4. Two adjusted versions of DPLTSs are presented as follows:

$Y_{s_{1}} (P) = 〈 {U_{s_{1}}^{(t_{1})} (p_{u}^{(t_{1})})}, {V_{s_{1}}^{(k_{1})} (p_{v}^{(k_{1})})} 〉$

$= 〈 {(s_{u_{s_{1}}^{(t_{1})}}, δ_{u_{s_{1}}^{(t_{1})}}) (p_{u}^{(t_{1})})}, {(s_{v_{s_{1}}^{(t_{1})}}, δ_{v_{s_{1}}^{(t_{1})}}) (p_{v}^{(k_{1})})} 〉$ ,

$Y_{s_{2}} (P) = 〈 {U_{s_{2}}^{(t_{2})} (p_{u}^{(t_{2})})}, {V_{s_{2}}^{(k_{2})} (p_{v}^{(k_{2})})} 〉$

$= 〈 {(s_{u_{s_{2}}^{(t_{2})}}, δ_{u_{s_{2}}^{(t_{2})}}) (p_{u}^{(t_{2})})}, {(s_{v_{s_{2}}^{(t_{2})}}, δ_{v_{s_{2}}^{(t_{2})}}) (p_{v}^{(k_{2})})} 〉$ ,

where t₁ = 1, 2, . . . , ♯ U_{s
₁}, t₂ = 1, 2, . . . , ♯ U_{s
₂}, k₁ = 1, 2, . . . , ♯ V_{s
₁}, k₂ = 1, 2, . . . , ♯ V_{s
₂}. Suppose that U_n = ♯ U_{s
₁} = ♯ U_{s
₂} = t₁ = t₂ = t and V_n = ♯ V_{s
₁} = ♯ V_{s
₂} = k₁ = k₂ = k.

The deviation degree measure between Y_{s
₁} (P) and Y_{s
₂} (P) is defined as $\begin{matrix} d (Y_{s_{1}} (P), Y_{s_{2}} (P)) \\ = & (\frac{1}{2} (\frac{1}{U_{n}} (\sum_{t = 1}^{U_{n}} ((p_{u}^{(t_{1})} (u_{s_{1}}^{(t_{1})} + δ_{u_{s_{1}}^{(t_{1})}} \cdot 2 τ)) \\ - (p_{u}^{(t_{2})} (u_{s_{2}}^{(t_{2})} + δ_{u_{s_{2}}^{(t_{2})}} \cdot 2 τ))^{2}) \\ + \frac{1}{V_{n}} (\sum_{k = 1}^{V_{n}} ((p_{v}^{(k_{1})} (v_{s_{1}}^{(k_{1})} + δ_{v_{s_{1}}^{(k_{1})}} \cdot 2 τ)) \\ - (p_{v}^{(k_{2})} (v_{s_{2}}^{(k_{2})} + δ_{v_{s_{2}}^{(k_{2})} \cdot 2 τ)))^{2}})))^{\frac{1}{2}} . \end{matrix}$ (6)

3 Dual probabilistic linguistic prioritized Maclaurin symmetric mean operators

In this section, we introduce novel prioritized MSM operators and DMSM operators, namely DPLPMSM and DPLPDMSM, under the framework of DPLTSs, utilizing PA and MSM operators.

3.1 DPLPMSM

Definition 5.

$Y_{s_{i}} (P_{i}) = 〈 {U_{s_{i}}^{(t_{i})} (p_{u}^{(t_{i})})}, {V_{s_{i}}^{(k_{i})} (p_{v}^{(k_{i})})} 〉$

$= 〈 {(s_{u^{(t_{i})}}, δ_{u}^{(t_{i})}) (p_{u}^{(t_{i})})}, {(s_{v^{(k_{i})}}, δ_{v}^{(k_{i})}) (p_{v}^{(k_{i})})}$ (i = 1, 2, . . . , n) be n DPLTSs, then we define DPLPMSM as follows:

DPLPMSM^d (Y_{s
₁} (P₁) , Y_{s
₂} (P₂) , . . . , Y_{s
_n} (P_n))

$= (\frac{\underset{1 \leq i_{1} \leq . . . i_{d} \leq n}{\oplus} (\otimes_{j = 1}^{d} w_{i_{j}} Y_{i_{j}} (P_{i_{j}}))}{C_{n}^{d}})^{\frac{1}{d}}$

$= ⋃_{{HPE}_{i_{j}}} 〈 {{\tilde{g}}^{- 1} ((1 - (\underset{1 \leq i_{1} \leq . . . \leq i_{d} \leq n}{Π} (1 - Π_{j = 1}^{d} (1 - (1 - \tilde{g} (H_{i_{j}}))^{w_{i_{j}}})))^{\frac{1}{C_{n}^{d}}})^{\frac{1}{d}}) (\underset{1 \leq i_{1} \leq . . . \leq i_{d} \leq n}{Π} Π_{j = 1}^{d} P_{H_{i_{j}}})}, {{\tilde{g}}^{- 1} (1 - (1 - ((\underset{1 \leq i_{1} \leq . . . \leq i_{d} \leq n}{Π} (1 - Π_{j = 1}^{d} (1 - \tilde{g} (E_{i_{j}})^{w_{i_{j}}})))^{\frac{1}{C_{n}^{d}}}))^{\frac{1}{d}})$ $(\underset{1 \leq i_{1} \leq . . . \leq i_{d} \leq n}{Π} Π_{j = 1}^{d} P_{E_{i_{j}}}))} 〉,$ (7) where $w_{i_{j}} = \frac{T_{i_{j}}}{\sum_{j = 1}^{n} T_{i_{j}}}$ , $T_{i_{j}} = Π_{t = 1}^{i - 1} ℓ^{'} (Y_{s_{t}} (P)) (i = 2, . . ., n)$ and T_{1_j} = 1,

${HPE}_{i_{j}} = {(H_{i_{j}}, P_{H_{i_{j}}}, E_{i_{j}}, P_{E_{i_{j}}}) | H_{i_{j}} \in U_{s_{i_{j}}}^{(t_{i_{j}})},$ $P_{H_{i_{j}}} \in p_{u}^{(t_{i_{j}})}, E_{i_{j}} \in V_{s_{i_{j}}}^{(k_{i_{j}})}, P_{E_{i_{j}}} \in p_{v}^{(k_{i_{j}})}}$ . The modified score function value, denoted as ℓ′ (Y_{s
_t} (P)), is formulated by integrating transformation functions into the original score function ℓ (Y_{s
_t} (P)). Mathematically, it is defined as:

$- \frac{\sum_{k = 1}^{V_{s} (P)} p_{v}^{(k)} (\frac{v^{(k)}}{2 τ} + \frac{1}{2} + δ_{v}^{(k)})}{\sum_{k = 1}^{V_{s} (P)} p_{v}^{(k)}} |$ .

In alignment with the score function, the terms $u^{(t)} + δ_{u}^{(t)} \cdot 2 τ$ and $v^{(k)} + δ_{v}^{(k)} \cdot 2 τ$ are adjusted to one 2τ-th of their original values and then increased by 0.5. This adjustment ensures that the resulting values surpass 0, as mandated. The tuple (i₁, i₂, . . . , i_d) encompasses all d-tuple combinations of (1, 2, . . . , n), with $C_{n}^{d}$ representing the binomial coefficient.

Remark 6. If d = 1, then DPLPMSM operator is transformed into the form of dual probabilistic linguistic prioritized averaging (DPLPA) operators, shown below: $\begin{matrix} {DPLPMSM}^{1} (Y_{s_{1}} (P_{1}), Y_{s_{2}} (P_{2}), . . ., Y_{s_{n}} (P_{n})) \\ = ⋃_{{HPE}_{i}} 〈 {{\tilde{g}}^{- 1} (1 - (Π_{j = 1}^{n} (1 - \tilde{g} (H_{i}))^{w_{i_{j}}})^{\frac{1}{n}}) (P_{H_{i}})}, \\ {{\tilde{g}}^{- 1} ((Π_{j = 1}^{n} (\tilde{g} (E_{i})^{w_{i_{j}}})))^{\frac{1}{n}}) (P_{E_{i}})} 〉, \end{matrix}$ (8) where ${HPE}_{i} = {(H_{i}, P_{H_{i}}, E_{i}, P_{E_{i}}) | H_{i} \in U_{s_{i}}^{(t_{i})}, P_{H_{i}} \in p_{u}^{(t_{i})}, E_{i} \in V_{s_{i}}^{(k_{i})}, P_{E_{i}} \in p_{v}^{(k_{i})}}, i = 1, 2, . . ., n .$

Remark 7. If d = 2, in the view of (7), we have

$D P L P M S M^{2} (Y_{s_{1}} (P_{1}), Y_{s_{2}} (P_{2}), ., Y_{s_{n}} (P_{n})) = ⋃_{{HPE}_{i_{1}} \cup {HPE}_{i_{2}}} 〈 {{\tilde{g}}^{- 1} ((1 - (\prod_{i_{1}, i_{2} = 1, i_{1} \neq i_{2}}^{n} (1 - ((1 - (1 - \tilde{g} (H_{i_{1}}))^{w_{i_{j}}})) ((1 - (1 - \tilde{g} (H_{i_{2}}))^{w_{i_{j}}}))))^{\frac{2}{n (n - 1)}})^{\frac{1}{2}}) (P_{H_{i_{1}}} P_{H_{i_{2}}})}, {{\tilde{g}}^{- 1} (1 - (1 - ((\prod_{i_{1}, i_{2} = 1, i_{1} \neq i_{2}}^{n} (1 - (1 - \tilde{g} (E_{i_{1}})^{w_{i_{j}}}) (1 - \tilde{g} (E_{i_{2}})^{w_{i_{j}}})))^{\frac{2}{n (n - 1)}}))^{\frac{1}{2}}) (P_{E_{i_{1}}} P_{E_{i_{2}}})} 〉,$ (9) where ${HPE}_{i_{1}} = {(H_{i_{1}}, P_{H_{i_{1}}}, E_{i_{1}}, P_{E_{i_{1}}}) | H_{i_{1}} \in U_{s_{i_{1}}}^{(t_{i_{1}})}, P_{H_{i_{1}}} \in p_{u}^{(t_{i_{1}})}, E_{i_{1}} \in V_{s_{i_{1}}}^{(k_{i_{1}})}, P_{E_{i_{1}}} \in p_{v}^{(k_{i_{1}})}},$ ${HPE}_{i_{2}} = {(H_{i_{2}}, P_{H_{i_{2}}}, E_{i_{2}}, P_{E_{i_{2}}}) | H_{i_{2}} \in U_{s_{i_{2}}}^{(t_{i_{2}})}, P_{H_{i_{2}}} \in p_{u}^{(t_{i_{2}})}, E_{i_{2}} \in V_{s_{i_{2}}}^{(k_{i_{2}})}, P_{E_{i_{2}}} \in p_{v}^{(k_{i_{2}})}},$ i₁ ≠ i₂, 1 ≤ i₁, i₂ ≤ n .

Example 8. In this study, we examine three DPLTSs denoted as follows:Y_{s
₁} (P) = 〈 {(s₃, - 0.01) (0.4) , (s₂, 0.02) (0.4) ,

(s₁, 0.01) (0.2)} , {(s_-3, 0.01) (0.8) , (s_-2, - 0.01) (0.2)} 〉,

Y_{s
₂} (P) = 〈 {(s₃, - 0.01) (1)} , {(s_-2, 0.01) (1)} 〉,

Y_{s
₃} (P) = 〈 {(s₃, - 0.01) (1)} , {(s_-2, 0.02) (1)} 〉 .}

Utilizing equation (7), the priority degrees T¹,T²,T³ are calculated as 1, 0.8333, and 0.6778, respectively. Additionally, employing the modified version of equation (9), the aggregated result is determined as follows: $\begin{matrix} {DPLPMSM}^{2} (Y_{s_{1}} (P_{1}), Y_{s_{2}} (P_{2}), Y_{s_{3}} (P_{3})) \\ = 〈 {(s_{2}, - 0.0545) (0.16), (s_{1}, 0.0098) (0.16), \\ (s_{1}, - 0.0482) (0.04)}, {(s_{0}, - 0.0318) (0.32), \\ (s_{0}, 0.0605) (0.04)} 〉 \end{matrix}$ (10)

Remark 9. If d = n, then based on (7), we have $\begin{matrix} {DPLPMSM}^{n} (Y_{s_{1}} (P_{1}), Y_{s_{2}} (P_{2}), . . ., Y_{s_{n}} (P_{n})) \\ = ⋃_{{HPE}_{i}} 〈 {{\tilde{g}}^{- 1} (1 - (\prod_{i = 1}^{n} (1 - (1 - \tilde{g} (H_{i}))^{w_{i}}))^{\frac{1}{n}}) (\prod_{i = 1}^{n} P_{H_{i}})}, \\ {{\tilde{g}}^{- 1} ((\prod_{i = 1}^{n} (1 - g (E_{i}))^{w_{i}})^{\frac{1}{n}}) (\prod_{i = 1}^{n} P_{E_{i}})} 〉, \end{matrix}$ (11) where ${HPE}_{i} = {(H_{i}, P_{H_{i}}, E_{i}, P_{E_{i}}) | H_{i} \in U_{s_{i}}^{(t_{i})},$ $P_{H_{i}} \in p_{u}^{(t_{i})}, E_{i} \in V_{s_{i}}^{(k_{i})}, P_{E_{i}} \in p_{v}^{(k_{i})}} .$

3.2 DPLPDMSM

We introduce a novel operator, DPLPDMSM, which combines the PA and DMSM operators to synthesize dual probabilistic linguistic information effectively.

Definition 10. $Y_{s_{i}} (P_{i}) = 〈 {U_{s_{i}}^{(t_{i})} (p_{u}^{(t_{i})})}, {V_{s_{i}}^{(k_{i})} (p_{v}^{(k_{i})})} 〉 = 〈 {(s_{u^{(t_{i})}}, δ_{u}^{(t_{i})}) (p_{u}^{(t_{i})})}, {(s_{v^{(k_{i})}}, δ_{v}^{(k_{i})}) (p_{v}^{(k_{i})})}$ (i = 1, 2, . . . , n) be n DPLTSs, then we define DPLPDMSM operator as follows:

DPLPDMSM^d (Y_{s
₁} (P₁) , Y_{s
₂} (P₂) , . . . , Y_{s
_n} (P_n))

$= \frac{1}{d} (\underset{1 \leq i_{1} \leq . . . i_{d} \leq n}{\otimes} (\oplus_{j = 1}^{d} Y_{i_{j}} (P_{i_{j}})^{w_{i_{j}}})^{\frac{1}{C_{n}^{d}}}) = ⋃_{{HPE}_{i_{j}}} 〈 {{\tilde{g}}^{- 1} (1 - (1 - (\underset{1 \leq i_{1} \leq . . . \leq i_{d} \leq n}{Π} (1 - Π_{j = 1}^{d} (1 - \tilde{g} (H_{i_{j}})^{w_{i_{j}}}))^{\frac{1}{C_{n}^{d}}}))^{\frac{1}{d}}) (\underset{1 \leq i_{1} \leq . . . \leq i_{d} \leq n}{Π} Π_{j = 1}^{d} P_{H_{i_{j}}})}, {{\tilde{g}}^{- 1} ((1 - \underset{1 \leq i_{1} \leq . . . \leq i_{d} \leq n}{Π} (1 - Π_{j = 1}^{d} (1 - (1 - \tilde{g} (E_{i_{j}}))^{w_{i_{j}}}))^{\frac{1}{C_{n}^{d}}})^{\frac{1}{d}})$ $\begin{matrix} (\underset{1 \leq i_{1} \leq . . . \leq i_{d} \leq n}{Π} Π_{j = 1}^{d} P_{E_{i_{j}}})} 〉, \end{matrix}$ (12) where $w_{i_{j}} = \frac{T_{i_{j}}}{\sum_{j = 1}^{n} T_{i_{j}}}$ , $T_{i_{j}} = Π_{t = 1}^{i - 1} ℓ^{'} (Y_{s_{t}} (P)) (i = 2, . . ., n)$ and T_{1_j} = 1, is the modified score function value of Y_{s
_t} (P), (i₁, i₂, . . . i_d) traversal all the d-tuple combination of (1, 2, . . , n) and $C_{n}^{d}$ represents the binomial coefficient, ${HPE}_{i_{j}} = {(H_{i_{j}}, P_{H_{i_{j}}}, E_{i_{j}}, P_{E_{i_{j}}}) | H_{i_{j}} \in U_{s_{i_{j}}}^{(t_{i_{j}})}, P_{H_{i_{j}}} \in p_{u}^{(t_{i_{j}})},$ $E_{i_{j}} \in V_{s_{i_{j}}}^{(k_{i_{j}})}, P_{E_{i_{j}}} \in p_{v}^{(k_{i_{j}})}} .$

Examining the result of (12), we can draw the following remarks from the varied values of the parameter d = 1:

Remark 11. If d = 1, then DPLPDMSM operator is transformed into the form of dual probabilistic linguistic prioritized geometric (DPLPG) operators, shown below: $\begin{matrix} {DPLPDMSM}^{1} (Y_{s_{1}} (P_{1}), Y_{s_{2}} (P_{2}), . . ., Y_{s_{n}} (P_{n})) \\ = ⋃_{{HPE}_{i}} 〈 {{\tilde{g}}^{- 1} (Π_{i = 1}^{n} \tilde{g} (H_{i})^{\frac{w_{i_{j}}}{n}}) (P_{H_{i}})}, \\ {{\tilde{g}}^{- 1} (1 - Π_{i = 1}^{n} ((1 - \tilde{g} (E_{i}))^{\frac{w_{i_{j}}}{n}})) (P_{E_{i}})} 〉, \end{matrix}$ (13) where ${HPE}_{i} = {(H_{i}, P_{H_{i}}, E_{i}, P_{E_{i}}) | H_{i} \in U_{s_{i}}^{(t_{i})},$ $P_{H_{i}} \in p_{u}^{(t_{i})}, E_{i} \in V_{s_{i}}^{(k_{i})}, P_{E_{i}} \in p_{v}^{(k_{i})}}, i = 1, 2, . . ., n .$

Remark 12. If d = 2, then based on (12), we have

DPLPDMSM² (Y_{s
₁} (P₁) , Y_{s
₂} (P₂) , . . . , Y_{s
_n} (P_n))

$D P L P D M S M^{2} (Y_{s_{1}} (P_{1}), Y_{s_{2}} (P_{2}), ., Y_{s_{n}} (P_{n})) = ⋃_{{HPE}_{i_{1}} \cup {HPE}_{i_{2}}} 〈 {{\tilde{g}}^{- 1} (1 - (1 - (\prod_{i_{1}, i_{2} = 1, i_{1} \neq i_{2}}^{n} (1 - (1 - \tilde{g} (E_{i_{1}})^{w_{i_{j}}}) (1 - \tilde{g} (E_{i_{2}})^{w_{i_{j}}}))^{\frac{2}{n (n - 1)}})^{\frac{1}{2}}) (P_{H_{i_{1}}} P_{H_{i_{2}}})}, {{\tilde{g}}^{- 1} ((1 - (\prod_{i_{1}, i_{2} = 1, i_{1} \neq i_{2}}^{n} (1 - (1 - (1 - \tilde{g} (E_{i_{1}}))^{w_{i_{j}}}) (1 - (1 - \tilde{g} (E_{i_{2}}))^{w_{i_{j}}}))^{\frac{2}{n (n - 1)}}))^{\frac{1}{2}}) (P_{E_{i_{1}}} P_{E_{i_{2}}})} 〉,$ (14) where ${HPE}_{i_{1}} = {(H_{i_{1}}, P_{H_{i_{1}}}, E_{i_{1}}, P_{E_{i_{1}}}) | H_{i_{1}} \in U_{s_{i_{1}}}^{(t_{i_{1}})}, P_{H_{i_{1}}} \in p_{u}^{(t_{i_{1}})}, E_{i_{1}} \in V_{s_{i_{1}}}^{(k_{i_{1}})}, P_{E_{i_{1}}} \in p_{v}^{(k_{i_{1}})}},$ ${HPE}_{i_{2}} = {(H_{i_{2}}, P_{H_{i_{2}}}, E_{i_{2}}, P_{E_{i_{2}}}) | H_{i_{2}} \in U_{s_{i_{2}}}^{(t_{i_{2}})},$ $P_{H_{i_{2}}} \in p_{u}^{(t_{i_{2}})}, E_{i_{2}} \in V_{s_{i_{2}}}^{(k_{i_{2}})}, P_{E_{i_{2}}} \in p_{v}^{(k_{i_{2}})}},$

1 ≤ i₁, i₂ ≤ n, i₁ ≠ i₂ .

Example 13. In this study, we examine three DPLTSs denoted as follows:

Y_{s
₁} (P) = 〈 {(s₃, - 0.01) (0.4) , (s₂, 0.01) (0.4) ,

(s₁, 0.01) (0.2)} , {(s_-3, 0.01) (1)} 〉,

Y_{s
₂} (P) = 〈 {(s₂, 0.02) (0.4) , (s₁, 0.01) (0.4) ,

(s₁, - 0.01) (0.2)} , {(s_-3, 0.01) (0.4) ,

(s_-2, - 0.01) (0.4) , (s_-1, - 0.01) (0.2)} 〉,

Y_{s
₃} (P) = 〈 {(s₀, - 0.01) (1)} , {(s_-2, - 0.01) (1)} 〉 . Utilizing equation (12), the priority degrees T¹,T²,T³ are calculated as 1, 0.8587, and 0.5324, respectively. Additionally, employing the modified version of equation (14), the aggregated result is determined as follows: $\begin{matrix} squo & {DPLPDMSM}^{2} (Y_{s_{1}} (P), Y_{s_{2}} (P), Y_{s_{3}} (P)) \\ = 〈 {(s_{3}, - 0.0546) (0.16), (s_{2}, 0.0538) (0.16), \\ (s_{2}, 0.0210) (0.04)}, {(s_{- 3}, 0.0101) (0.16), \\ (s_{- 3}, - 0.0294) (0.16), (s_{- 2}, - 0.0430) (0.04)} 〉 . \end{matrix}$

Remark 14. If d = n, then based on (12), we have $\begin{matrix} {DPLPDMSM}^{n} (Y_{s_{1}} (P_{1}), Y_{s_{2}} (P_{2}), . . ., Y_{s_{n}} (P_{n})) \\ = ⋃_{{HPE}_{i}} 〈 {{\tilde{g}}^{- 1} (1 - (\prod_{i = 1}^{n} (1 - g (H_{i}))^{w_{i}})^{\frac{1}{n}}) (\prod_{i = 1}^{n} P_{H_{i}})}, \\ {{\tilde{g}}^{- 1} ((\prod_{i = 1}^{n} (1 - (1 - \tilde{g} (E_{i}))^{w_{i}}))^{\frac{1}{n}}) (\prod_{i = 1}^{n} P_{E_{i}})} 〉, \end{matrix}$ (15) where ${HPE}_{i} = {(H_{i}, P_{H_{i}}, E_{i}, P_{E_{i}}) | H_{i} \in U_{s_{i}}^{(t_{i})}, P_{H_{i}} \in p_{u}^{(t_{i})}, E_{i} \in V_{s_{i}}^{(k_{i})}, P_{E_{i}} \in p_{v}^{(k_{i})}} .$

4 Decision-making approach for DPLTSs based on VIKOR

In this section, we introduce a decision-making approach tailored for DPLTSs employing the VIKOR method. Furthermore, we refine the VIKOR method as well as the PA and MSM operators to accommodate the distinct characteristics of DPLTS representations.

4.1 Problem description

In this subsection, we delineate the fundamental characteristics of MCDM problems within the framework of DPLTSs as follows:

Consider $A = {U_{s_{1}}^{(1)}, U_{s_{1}}^{(2)}, . . ., A_{m}}$ as a discrete set of m alternatives, C = {C₁, C₂, . . . , C_n} as a finite collection of n criteria, and DM = {dm₁, dm₂, . . . , dm_f, . . . , dm_q} as a group of DMs. Let S = {S_t|t = - τ, . . . , -1, 0, 1, . . . , τ} be a linguistic scale. Each DM provides their evaluations for each alternative with respect to each criterion. Suppose that $D^{f} = (Y_{ij}^{f} (P_{ij}^{f}))_{m \times n}$ represents the decision matrix presented by dm_f. The matrix D^f can be expressed as follows: $D^{f} = \begin{matrix} U_{s_{1}}^{(1)} \\ U_{s_{1}}^{(2)} \\ ⋮ \\ A_{m} \end{matrix} (\begin{matrix} C_{1} & C_{2} & \dots & C_{n} \\ Y_{11}^{f} (P_{11}^{f}) & Y_{12}^{f} (P_{12}^{f}) & \dots & Y_{1 n}^{f} (P_{1 n}^{f}) \\ Y_{21}^{f} (P_{21}^{f}) & Y_{22}^{f} (P_{22}^{f}) & \dots & Y_{2 n}^{f} (P_{2 n}^{f}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ Y_{m 1}^{f} (P_{m 1}^{f}) & Y_{m 2}^{f} (P_{m 2}^{f}) & \dots & Y_{m n}^{f} (P_{m n}^{f}) \end{matrix})$ where the element $Y_{ij}^{f} (P_{ij}^{f})$ represents evaluation over alternative A_i with respect to the criterion C_j by DM dm_f ((i = 1, 2, . . . , m ; j = 1, 2, . . . , n)).

4.2 Decision procedures for DPLTSs based on VIKOR

Standardization of the decision matrix is essential, especially when it encompasses both benefit and cost criteria. This standardization process transforms the decision matrix into a uniform format, accomplished through the following equation: ${\hat{Y}}_{ij}^{f} ({\hat{P}}_{ij}^{f}) = {\begin{matrix} Y_{ij}^{f} (P_{ij}^{f}), & for benefit criterion C_{j}, \\ Neg (Y_{ij}^{f} (P_{ij}^{f})), & for cost criterion C_{j}, \end{matrix}$ (16) }where i = 1, 2, . . . , m ; j = 1, 2, . . . , n .

The individual decision matrix ${\hat{D}}^{f} = {\hat{Y}}_{ij}^{f} ({\hat{P}}_{ij}^{f}) = {〈 {\hat{U}}_{f_{ij}} (P_{ij}), {\hat{V}}_{f_{ij}} (P_{ij}) 〉}$ can be normalized as shown below: $\begin{matrix} {\hat{D_{1}}}^{f} = & {\hat{Y_{1}}}_{ij}^{f} ({\hat{P}}_{ij}^{f}) = 〈 {\hat{U_{1}}}_{f_{ij}} (P_{ij}), {\hat{V_{1}}}_{f_{ij}} (P_{ij}) 〉 \\ = & 〈 {({\hat{U_{1}}}_{f_{ij}}^{(k_{ij})}, {\hat{δ}}_{f_{ij}}^{u (k_{ij})}) ({\hat{δ}}_{p_{ij}}^{u (k_{ij})})}, \\ {({\hat{V_{1}}}_{f_{ij}}^{(t_{ij})}, {\hat{δ}}_{f_{ij}}^{v (t_{ij})}) ({\hat{δ}}_{p_{ij}}^{v (t_{ij})})} 〉 . \end{matrix}$ Based on the adjustment of DPLTS, the number of elements in ${\hat{U_{1}}}_{f_{ij}} (P_{ij})$ and ${\hat{V_{1}}}_{f_{ij}} (P_{ij})$ are equal. Let $O = max {♯ {\hat{U}}_{f_{ij}} (P_{ij}), ♯ {\hat{V}}_{f_{ij}} (P_{ij})}$ denote the number of elements in ${\hat{U_{1}}}_{f_{ij}} (P_{ij})$ and ${\hat{V_{1}}}_{f_{ij}} (P_{ij})$ .

We determine the priority degrees among the DMs through the following calculation: ${\hat{T_{ij}}}^{f} ({\hat{P}}_{ij}^{f}) = {\begin{matrix} 1, & f = 1, \\ \prod_{b = 1}^{f - 1} ℓ ({\hat{Y_{1}}}_{ij}^{b} ({\hat{P_{1}}}_{ij}^{b})), & f = 2, . . ., q . \end{matrix}$ (17) In the subsequent stage, we must consolidate the individual matrices into a collective evaluation matrix using either the DPLPMSM or DPLPDMSM operator.

The group evaluation matrix DA = (YR_ij (P_ij)) _m×n is derived from equations (12) or (7) as outlined below: $D A = \begin{matrix} U_{s_{1}}^{(1)} \\ U_{s_{1}}^{(2)} \\ ⋮ \\ A_{m} \end{matrix} (\begin{matrix} C_{1} & C_{2} & \dots & C_{n} \\ Y R_{11} (P_{11}) & Y R_{12} (P_{12}) & \dots & Y R_{1 n} (P_{1 n}) \\ Y R_{21} (P_{21}) & Y R_{22} (P_{22}) & \dots & Y R_{2 n} (P_{2 n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ Y R_{m 1}^{f} (P_{m 1}) & Y R_{m 2} (P_{m 2}) & \dots & Y R_{m n} (P_{m n}) \end{matrix}),$ where the element YR_ij (P_ij) represents evaluation over alternative A_i with respect to the criterion C_j. ((i = 1, 2, . . . , m ; j = 1, 2, . . . , n)).

Subsequently, we employ the VIKOR method to address multiple attribute decision-making. Here, we compute the positive ideal solutions and negative ideal solutions based on the matrix DA = (YR_ij (P_ij)) _m×n, where ${YR}_{ij} (P_{ij}) = 〈 {(s_{u_{ij}}^{(k)}, δ_{ij}^{u (k)}) (p_{ij}^{u (k)})},$

${(s_{v_{ij}}^{(k)}, δ_{ij}^{v (k)}) (p_{ij}^{v (k)})} 〉$ (k = 1, 2, . . . , O).

Prior to calculating the ideal solutions, we evaluate the following equations: $J_{_{i}^{"} j} (k) = \max_{1 \leq i \leq m} {p_{i j}^{u (k)} (u_{i j}^{(k)} + δ_{i j}^{u (k)} \cdot 2 τ) - p_{i j}^{v (k)} (v_{i j}^{(k)} + δ_{i j}^{v (k)} \cdot 2 τ)}$ (18) $\begin{matrix} J_{i^{"} j} (k) = & min_{1 \leq i \leq m} {p_{ij}^{u (k)} (u_{ij}^{(k)} + δ_{ij}^{u (k)} \cdot 2 τ) \\ - p_{ij}^{v (k)} (v_{ij}^{(k)} + δ_{ij}^{v (k)} \cdot 2 τ)} . \end{matrix}$ (19)

The positive ideal solutions and negative ideal solutions are denoted as $Y^{+} = (Y_{1}^{+}, . . ., Y_{j}^{+}, . . ., Y_{n}^{+})$ and $Y^{-} = (Y_{1}^{-}, . . ., Y_{j}^{-}, . . ., Y_{n}^{-})$ . Here, $Y_{j}^{+}$ represents the best solution for criterion C_j, and $Y_{j}^{-}$ signifies the worst solution for criterion C_j, as illustrated below: $Y_{j}^{+} = {{YD}_{j}^{+} (1), {YD}_{j}^{+} (2), . . ., {YD}_{j}^{+} (k), . . ., {YD}_{j}^{+} (O)},$ (20) where

${YD}_{j}^{+} (k) = 〈 (s_{u_{j}^{+}}^{(k)}, δ_{j^{+}}^{u (k)}) (p_{j^{+}}^{u (k)}), (s_{v_{j}^{+}}^{(k)}, δ_{j^{+}}^{v (k)}) (p_{j^{+}}^{v (k)}) 〉$

Similarly, the negative ideal solutions are shown as $Y_{j}^{-} = {{YD}_{j}^{-} (1), {YD}_{j}^{-} (2), . . ., {YD}_{j}^{-} (k), . . ., {YD}_{j}^{-} (O)},$ (21) where

${YD}_{j}^{-} (k) = 〈 (s_{u_{j}^{-}}^{(k)}, δ_{j^{-}}^{u (k)}) (p_{j^{-}}^{u (k)}), (s_{v_{j}^{-}}^{(k)}, δ_{j^{-}}^{v (k)}) (p_{j^{-}}^{v (k)}) 〉$

$= 〈 (s_{u_{i^{"} j}}^{(k)}, δ_{i^{"} j}^{u (k)}) (p_{i^{"} j}^{u (k)}), (s_{v_{i^{"} j}}^{(k)}, δ_{i^{"} j}^{v (k)}) (p_{i^{"} j}^{v (k)}) 〉$

= YR_{i
^"
j} (P_{i
^"
j}) . We proceed to calculate the group utility S_i and individual regret R_i based on the positive ideal solutions and negative ideal solutions. Each decision maker assigns subjective weights to each criterion, and the aggregated assessments for these criteria are calculated using equations (12) or (7). Subsequently, we determine the combined weight for each criterion using equation in [32].

According to (6), the group utility S_i is represented as follows: $S_{i} = \sum_{j = 1}^{m} w_{j} \frac{d (Y_{j}^{+}, {YR}_{ij} ({\tilde{P}}_{ij}))}{d (Y_{j}^{+}, Y_{j}^{-})},$ (22) The individual regret R_i can be computed by the following equation: $R_{i} = max_{1 \leq j \leq n} w_{j} \frac{d (Y_{j}^{+}, {YR}_{ij} ({\tilde{P}}_{ij}))}{d (Y_{j}^{+}, Y_{j}^{-})} .$ (23) The calculation of combined weight w_j is seen in the [32]. We can calculate the total utility Q_i associated with alternative A_i using the following equation: $Q_{i} = ɛ \frac{S_{i} - S^{+}}{S^{-} - S^{+}} + (1 - ɛ) \frac{R_{i} - R^{+}}{R^{-} - R^{+}},$ (24) where $S^{+} = min_{i} S_{i}$ , $S^{-} = max_{i} S_{i}$ , $R^{+} = min_{i} R_{i}$ , $R^{-} = max_{i} R_{i}$ , ɛ signifies the weight assigned to group utility, 1 - ɛ represents the weight allocated to individual regret. Usually, the value of ɛ is 0.5.

The VIKOR method proves invaluable in multicriteria decision-making, particularly in scenarios where decision makers may lack the expertise or initial knowledge to articulate their preferences during system design. The resulting compromise solution is conducive to acceptance by decision makers, as it maximizes group utility for the ’majority’ (as indicated by min S in equation (22)), while simultaneously minimizing individual regret for the ’opponent’ (as indicated by min R in equation (23)). These compromise solutions form a robust foundation for negotiation, allowing decision makers to incorporate their preferences through criteria weights.

Subsequently, we determine the ranking of alternatives based on S_i, R_i, and Q_i. Alternatives, denoted as A_i, are sorted based on the values of S_i, R_i and Q_i, with lower values indicating higher rankings. The final step involves proposing a promising and compromised solution. The promising solution is to check alternative ( $U_{s_{1}}^{(1)}$ ) according to the conditions [33]. If any of these conditions are not met, further evaluation is conducted based on Q_i to arrive at a compromised solution.

5 Numerical example

In this section, we employ the proposed method to evaluate personalized healthcare systems implemented in various Chinese hospitals. Furthermore, we provide a comprehensive comparative analysis to showcase the applicability and reliability of our suggested approach.

5.1 Background

This situation poses a formidable challenge: how to effectively manage limited healthcare resources in China amidst an aging population and rising healthcare costs. Despite these obstacles, it is imperative to find efficient ways to allocate resources and maximize their utility.To address this challenge, China must transition from its outdated healthcare system to a personalized one. This innovative approach should recognize the crucial link between the environment and public health, emphasizing intelligent resource management. While some progress has been made in Chinese hospitals, further efforts are essential to safeguard public health and optimize resource utilization in the face of environmental challenges. Given the environmental degradation and scarcity of medical resources, optimizing resource allocation and improving efficiency is paramount. Several Chinese hospitals have initiated research on resource management within intelligent medical and health systems. Evaluating numerous domestic hospitals is crucial, focusing particularly on their individualized healthcare systems. This evaluation will be guided by three professional decision-makers (DM = dm₁, dm₂, dm₃). The ideal hospital will be determined by assessing a_i (i = 1, 2, 3, 4) against three primary criteria:

Environmental Impact of Healthcare Services (c₁): Evaluating the environmental impact of healthcare services.

Effectiveness of Personalized Healthcare Solutions (c₂): Assessing the effectiveness of personalized healthcare solutions.

Allocation of Social Resources within Intelligent Healthcare Systems (c₃): Examining the allocation of social resources within intelligent healthcare systems.

Decision-makers communicate their evaluations using the DPLTSs with a linguistic term set S = {s_t|t = -3, - 2, - 1, 0, 1, 2, 3}. The decision matrices are presented in Tables 1, 2 and 3. Notably, as the qualities being assessed are benefit criterion, normalization is not required in our analysis. We adjust the dual probabilistic linguistic 2-tuple set for consistency in the decision matrix.

To facilitate accurate calculations, it is crucial to maintain consistency in both the number of elements and their associated probabilities within the decision matrix.

Table 1
DM₁’s assessment matrix

C ₁ 〈 {(s₁, 0.02) (0.6) , (s₀, 0.01) (0.2) , (s₀, 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-2, 0.01) (0.4) , (s_-2, 0.01) (0.2)} 〉

a ₁ C ₂ 〈 {(s₂, 0.01) (0.6) , (s₂, 0.01) (0.2) , (s₂, 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉

C ₃ 〈 {(s₀, 0.01) (0.6) , (s₀, 0.01) (0.2) , (s₀, 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉

C ₁ 〈 {(s₃, - 0.01) (0.6) , (s₃, - 0.01) (0.2) , (s₃, - 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉

a ₂ C ₂ 〈 {(s₀, 0.01) (0.6) , (s₀, 0.01) (0.2) , (s₀, 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉

C ₃ 〈 {(s₃, - 0.01) (0.4) , (s₂, 0.02) (0.4) , (s₁, 0.01) (0.2)} , {(s_-2, - 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉

C ₁ 〈 {(s₁, 0.02) (0.6) , (s₁, 0.02) (0.2) , (s₁, 0.02) (0.2)} , {(s₀, - 0.01) (0.4) , (s₀, - 0.01) (0.4) , (s₀, - 0.01) (0.2)} 〉

a ₃ C ₂ 〈 {(s₂, 0.05) (0.6) , (s₁, 0.05) (0.2) , (s₁, 0.05) (0.2)} , {(s₀, 0.01) (0.4) , (s₁, - 0.01) (0.4) , (s₁, - 0.01) (0.2)} 〉

C ₃ 〈 {(s₃, - 0.01) (0.4) , (s₂, 0.01) (0.4) , (s₂, 0.01) (0.2)} , {(s₁, 0.01) (0.4) , (s₂, - 0.02) (0.4) , (s₂, - 0.02) (0.2)} 〉

C ₁ 〈 {(s₃, - 0.02) (0.4) , (s₂, 0.01) (0.2) , (s₂, 0.01) (0.4)} , {(s_-3, - 0.01) (0.4) , (s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.2)} 〉

a ₄ C ₂ 〈 {(s₁, - 0.01) (0.6) , (s₀, 0.02) (0.2) , (s₀, 0.02) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.2)} 〉

C ₃ 〈 {(s₁, 0.01) (0.6) , (s₁, 0.01) (0.2) , (s₁, 0.01) (0.2)} , {(s_-2, 0.02) (0.4) , (s_-2, - 0.02) (0.4) , (s_-2, - 0.02) (0.2)} 〉

Table 2

DM₂’s assessment matrix

	C ₁	〈 {(s₁, 0.01) (0.6) , (s₀, - 0.01) (0.2) , (s₀, - 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.2)} 〉
a ₁	C ₂	〈 {(s₂, - 0.01) (0.6) , (s₁, 0.01) (0.2) , (s₁, 0.01) (0.2)} , {(s_-2, - 0.01) (0.6) , (s_-1, 0.01) (0.2) , (s_-1, 0.01) (0.2)} 〉
	C ₃	〈 {(s₃, - 0.01) (0.6) , (s₂, - 0.01) (0.2) , (s₁, - 0.01) (0.2)} , {(s_-2, - 0.01) (0.6) , (s_-1, 0.01) (0.2) , (s_-1, 0.01) (0.2)} 〉
	C ₁	〈 {(s₃, - 0.01) (0.4) , (s₂, 0.01) (0.2) , (s₁, 0.01) (0.4)} , {(s_-3, 0.02) (0.4) , (s_-3, 0.02) (0.4) , (s_-3, 0.02) (0.2)} 〉
a ₂	C ₂	〈 {(s₀, - 0.01) (0.4) , (s₀, - 0.01) (0.2) , (s₀, - 0.01) (0.4)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉
	C ₃	〈 {(s₃, - 0.01) (0.4) , (s₃, - 0.01) (0.4) , (s₃, - 0.01) (0.2)} , {(s_-2, - 0.01) (0.4) , (s_-2, 0.01) (0.4) , (s_-2, 0.01) (0.2)} 〉
	C ₁	〈 {(s₁, 0.01) (0.6) , (s₀, - 0.01) (0.2) , (s₀, - 0.01) (0.2)} , {(s_-2, 0.01) (0.4) , (s_-1, - 0.01) (0.4) , (s_-1, - 0.01) (0.2)} 〉
a ₃	C ₂	〈 {(s₁, 0.02) (0.6) , (s₁, 0.02) (0.2) , (s₁, 0.02) (0.2)} , {(s₀, 0.01) (0.4) , (s₀, 0.01) (0.4) , (s₀, 0.01) (0.2)} 〉
	C ₃	〈 {(s₃, - 0.01) (0.6) , (s₃, - 0.01) (0.2) , (s₃, - 0.01) (0.2)} , {(s₂, - 0.01) (0.4) , (s₂, - 0.01) (0.4) , (s₂, - 0.01) (0.2)} 〉
	C ₁	〈 {(s₃, - 0.01) (0.8) , (s₁, 0.02) (0.1) , (s₁, 0.02) (0.1)} , {(s_-3, 0.01) (0.2) , (s_-2, 0.01) (0.4) , (s_-2, 0.01) (0.4)} 〉
a ₄	C ₂	〈 {(s₁, 0.01) (0.8) , (s₁, 0.01) (0.1) , (s₁, 0.01) (0.1)} , {(s_-1, 0.01) (0.4) , (s_-1, 0.01) (0.2) , (s_-1, 0.01) (0.4)} 〉
	C ₃	〈 {(s₁, 0.02) (0.8) , (s₀, - 0.02) (0.1) , (s₀, - 0.02) (0.1)} , {(s_-3, 0.01) (0.2) , (s_-2, 0.02) (0.6) , (s_-2, - 0.02) (0.2)} 〉

Table 3

DM₃’s assessment matrix

	C ₁	〈 {(s₁, 0.01) (0.6) , (s₁, 0.01) (0.2) , (s₁, 0.01) (0.2)} , {(s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.2)} 〉
a ₁	C ₂	〈 {(s₂, 0.01) (0.6) , (s₂, 0.01) (0.2) , (s₂, 0.01) (0.2)} , {(s_-1, 0.01) (0.4) , (s_-1, 0.01) (0.4) , (s_-1, 0.01) (0.2)} 〉
	C ₃	〈 {(s₀, - 0.01) (0.6) , (s₀, - 0.01) (0.2) , (s₀, - 0.01) (0.2)} , {(s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.2)} 〉
	C ₁	〈 {(s₃, - 0.02) (0.6) , (s₃, - 0.02) (0.2) , (s₃, - 0.02) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉
a ₂	C ₂	〈 {(s₀, - 0.01) (0.6) , (s₀, - 0.01) (0.2) , (s₀, - 0.01) (0.2)} , {(s_-2, 0.01) (0.4) , (s_-2, 0.01) (0.4) , (s_-2, 0.01) (0.2)} 〉
	C ₃	〈 {(s₃, - 0.01) (0.6) , (s₃, - 0.01) (0.2) , (s₃, - 0.01) (0.2)} , {(s_-2, 0.02) (0.4) , (s_-2, 0.02) (0.4) , (s_-2, 0.02) (0.2)} 〉
	C ₁	〈 {(s₂, 0.01) (0.6) , (s₁, - 0.01) (0.2) , (s₁, - 0.01) (0.2)} , {(s₀, - 0.01) (0.4) , (s₁, 0.01) (0.4) , (s₂, 0.01) (0.2)} 〉
a ₃	C ₂	〈 {(s₂, - 0.01) (0.2) , (s₁, 0.02) (0.6) , (s₁, 0.02) (0.2)} , {(s₀, 0.02) (0.2) , (s₁, 0.03) (0.4) , (s₁, 0.03) (0.4)} 〉
	C ₃	〈 {(s₂, - 0.01) (0.8) , (s₁, - 0.01) (0.1) , (s₁, - 0.01) (0.1)} , {(s₁, 0.01) (0.2) , (s₂, - 0.02) (0.2) , (s₂, - 0.01) (0.6)} 〉
	C ₁	〈 {(s₃, - 0.02) (0.6) , (s₃, - 0.02) (0.2) , (s₃, - 0.02) (0.2)} , {(s_-3, 0.01) (0.6) , (s_-3, 0.01) (0.2) , (s_-3, 0.01) (0.2)} 〉
a ₄	C ₂	〈 {(s₁, 0.01) (0.6) , (s₁, 0.01) (0.2) , (s₁, 0.01) (0.2)} , {(s_-1, 0.01) (0.4) , (s_-1, 0.01) (0.4) , (s_-1, 0.01) (0.2)} 〉
	C ₃	〈 {(s₁, - 0.02) (0.6) , (s₁, - 0.02) (0.2) , (s₁, - 0.02) (0.2)} , {(s_-1, 0.01) (0.4) , (s_-1, 0.01) (0.4) , (s_-1, 0.01) (0.2)} 〉

5.2 The proposed decision-making method for the case

Using the equation (17) as the foundation, we determine the priority degrees among the decision makers (DMs) in the following manner: $P_{ij}^{1} = (\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}), T 1$

$P_{ij}^{2} = (\begin{matrix} 0.5060 & 0.8333 & 0.5000 \\ 0.9800 & 0.5000 & 0.8040 \\ 0.1967 & 0.2267 & 0.1433 \\ 0.8000 & 0.5040 & 0.5300 \end{matrix}), T 2$ $P_{ij}^{3} = (\begin{matrix} 0.2550 & 0.4444 & 0.3293 \\ 0.7990 & 0.2400 & 0.6604 \\ 0.0663 & 0.0400 & 0.0239 \\ 0.6288 & 0.1720 & 0.2661 \end{matrix}) . T 3$

The aggregate result for each alternative a_i concerning can be obtained using the DPLPMSM operator (9) when d = 2. The subjective weights assigned by experts are [0.2, 0.1, 0.7].

Building upon equations [32], we derive the objective weights for the criteria as follows: OW = {0.3388, 0.3193, 0.3419} .

The overall weights for the criteria are determined through the integration of objective and subjective weights, as illustrated below: $CW = {0.3021, 0.1898, 0.5081} .$

Setting ɛ = 0.5, we proceed to calculate the group utility S_i, individual regret R_i, and total utility Q_i associated with a_i using the provided equations. The computations and rankings for all alternatives are consolidated and presented in Table 4.

Table 4
The results for all alternatives

Alternative S _i Ranking result R _i Ranking result Q _i Ranking result

a ₁ 0.6547 3 0.3685 2 0.6446 2

a ₂ 0.0929 1 0.0929 1 0 1

a ₃ 1.0000 4 0.5043 4 1.0000 4

a ₄ 0.5917 2 0.4066 3 0.6563 3

Alternative	S _i	Ranking result	R _i	Ranking result	Q _i	Ranking result
a ₁	0.6547	3	0.3685	2	0.6446	2
a ₂	0.0929	1	0.0929	1	0	1
a ₃	1.0000	4	0.5043	4	1.0000	4
a ₄	0.5917	2	0.4066	3	0.6563	3

5.3 Comparison analysis and discussion

5.3.1 Parameter analysis

In the following section, we illustrate the process outlined in the previous example using the same assessment matrices. We employ varying values of d in both the DPLPMSM and DPLPDMSM operators. The corresponding outcomes are presented in Tables 5–9.

Table 5
The ranking results of example with d = 1 in DPLPMSM operator

Alternative S _i Ranking result R _i Ranking result Q _i Ranking result

DPLPMSM d = 1

a ₁ 0.6303 3 0.4053 2 0.6474 2

a ₂ 0.1171 1 0.1171 1 0 1

a ₃ 1.0000 4 0.5182 4 1.0000 4

a ₄ 0.6298 2 0.4616 3 0.7173 3

Alternative	S _i	Ranking result	R _i	Ranking result	Q _i	Ranking result
DPLPMSM d = 1
a ₁	0.6303	3	0.4053	2	0.6474	2
a ₂	0.1171	1	0.1171	1	0	1
a ₃	1.0000	4	0.5182	4	1.0000	4
a ₄	0.6298	2	0.4616	3	0.7173	3

Table 6

The ranking results of example with d = n in DPLPMSM operator

Alternative	S _i	Ranking result	R _i	Ranking result	Q _i	Ranking result
DPLPMSM d = n
a ₁	0.5324	3	0.3341	2	0.5069	2
a ₂	0.1236	1	0.1236	1	0	1
a ₃	1.0000	4	0.5081	4	1.0000	4
a ₄	0.5292	2	0.3567	3	0.5345	3

Table 7

The ranking results of example with d = 1 in DPLPDMSM operator

Alternative	S _i	Ranking result	R _i	Ranking result	Q _i	Ranking result
DPLPDMSM d = 1
a ₁	0.4047	3	0.2171	3	0.2668	3
a ₂	0.1452	1	0.1452	1	0	1
a ₃	1.0000	4	0.4580	4	1.0000	4
a ₄	0.2795	2	0.1452	1	0.0786	2

Table 8

The ranking results of example with d = 2 in DPLPDMSM operator

Alternative	S _i	Ranking result	R _i	Ranking result	Q _i	Ranking result
DPLPDMSM d = 2
a ₁	0.6003	3	0.3213	3	0.4444	3
a ₂	0.2061	1	0.2061	1	0	1
a ₃	0.9897	4	0.5049	4	1.0000	4
a ₄	0.4588	2	0.2972	2	0.3138	2

Table 9

The ranking results of example with d = n in DPLPDMSM operator

Alternative	S _i	Ranking result	R _i	Ranking result	Q _i	Ranking result
DPLPDMSM d = n
a ₁	0.6894	3	0.3864	2	0.6095	2
a ₂	0.1945	1	0.1945	1	0	1
a ₃	0.9732	4	0.5234	4	1.0000	4
a ₄	0.5953	2	0.4610	3	0.6624	3

When examining the DPLPMSM operators across a range of d values, subtle variations in the outcomes become apparent. However, discrepancies emerge when d = n in the case of the DPLPDMSM operator, specifically concerning R_i and Q_i. a₁ and a₄ exhibit contrasting rankings, yet their data are remarkably similar, with R_i values of 0.3864 and 0.4610, and Q_i values of 0.6095 and 0.6624, respectively. This indicates that there are slight differences in the performance of individual regret and total utility. Individual regret minimizes regret for the opponent, while total utility integrates the results of both group utility and individual regret, and compares the proportion of differences directly with the extremes.

Variations in results between DPLPMSM and DPLPDMSM operators arise from their distinct operator structures. For instance, when d = 1, the DPLPMSM operator’s membership is akin to an averaging operator, while non-membership takes on the form of a geometric operator. Conversely, the behavior of the DPLPDMSM operator is the reverse. When d = n, the outcomes align between the two operators.

5.3.2 Comparison with other methods

We compare other methods with our proposed method. The decision outcomes are detailed in Table 10.

Table 10
The decision results of different methods

Method Ranking result

The extended TOPSIS with PLTSs [34] a₂ > a₃ > a₄ > a₁

PLMCGDM method [35] a₂ > a₁ > a₃ > a₄

HFLBM method [30] a₃ > a₄ > a₂ > a₁

HFLWA method [30] a₃ > a₂ > a₄ > a₁

Modified PHFS-based MCDM method [36] a₂ > a₃ > a₁ > a₄

VEHFS-based score method [37] a₂ > a₃ > a₄ > a₁

Method	Ranking result
The extended TOPSIS with PLTSs [34]	a₂ > a₃ > a₄ > a₁
PLMCGDM method [35]	a₂ > a₁ > a₃ > a₄
HFLBM method [30]	a₃ > a₄ > a₂ > a₁
HFLWA method [30]	a₃ > a₂ > a₄ > a₁
Modified PHFS-based MCDM method [36]	a₂ > a₃ > a₁ > a₄
VEHFS-based score method [37]	a₂ > a₃ > a₄ > a₁

In our comprehensive study, we delved into the intricacies of two distinct groups of decision-making methods: the first group, encompassing the extended TOPSIS method utilizing PLTSs and the probabilistic linguistic multi-criteria group decision-making (PLMCGDM) method, and the second group, focusing on the hesitant fuzzy linguistic Bonferroni mean (HFLBM) and the hesitant fuzzy linguistic weighted averaging (HFLWA) methods. The differentiation between these groups lies in their fundamental approaches: the first group leaning on probabilistic linguistic information and the second on fuzzy linguistic operators. The third group is base on probabilistic hesitant fuzzy set (PHFS).

First Group: Extended TOPSIS with PLTSs and PLMCGDM Method

Upon meticulous examination of Table 10, a distinct variance in the outcomes of the extended TOPSIS method employing PLTSs and the PLMCGDM method, as opposed to our approach, becomes apparent. In comparison to the extended TOPSIS method with PLTSs, notable distinctions arise. Specifically, when employing the DPLPMSM operator, discrepancies in rankings are evident, except for alternatives a₂ and a₄. Interestingly, the ranking of alternative a₄ remains consistent for the DPLPDMSM operator when d = 1 and d = 2.

Furthermore, when contrasting with the PLMCGDM method, similar discrepancies in rankings emerge. Notably, for the DPLPMSM operator, differences in rankings persist, except for alternative a₄. Similarly, for the DPLPDMSM operator, disparities in rankings are evident, except for alternative a₁. These comparative analyses shed light on the nuanced variations among the methods under consideration.

The divergence emanates from their disparate representation methods:

Representation Discrepancies: Both the extended TOPSIS method with PLTSs and the PLMCGDM method incorporate probabilistic information but lack the capability to handle hesitant information effectively. The methods do not take into account the offset values associated with the linguistic terms. Their inability to differentiate between actual linguistic values and discrete linguistic terms hampers their evaluation accuracy, leading to divergent results.

Methodological Limitations: The extended TOPSIS method utilizing PLTSs overlooks the intricate interconnections between individual evaluations, dismissing the importance of subscript-symmetric additive linguistic terms essential for meticulous calculations. In contrast, the PLMCGDM method, which relies on the use of BM and HM operators, fails to adequately address the complex interrelationships among data points. As a consequence, both methods exhibit inherent limitations in their ability to handle the multifaceted nature of decision-making scenarios.

Second Group: HFLBM and HFLWA Methods

Upon scrutinizing Table 10, substantial disparities in the ranking of alternatives emerge between the HFLBM, HFLWA methods, and our methodology. In contrast to the HFLBM method, the ranking of alternative a₃ remains consistent when utilizing the DPLPDMSM operator for both d = 1 and d = 2. However, disparities in rankings are observed for other alternatives. Similarly, in comparison to the HFLWA method, the ranking of alternative a₄ aligns with the DPLPMSM operator, yet discrepancies are evident for other alternatives. Furthermore, the rankings of alternatives a₃ and a₄ coincide with those from the DPLPMSM operator, while disparities persist for the remaining alternatives. These variations underscore the nuanced differences in outcomes among the methods analyzed.

Impact of Representation Methods: The HFLBM and HFLWA methods employ a linguistic term set where all elements carry equal weight. In contrast, our approach amalgamates probabilistic information and offset data from s_i, resulting in nuanced representation. These discrepancies in representation methods lead to variations in aggregation operators and sorting techniques, thereby influencing the final rankings.

Methodological Differences: HFLBM overlooks the individual criterion weights, posing a significant limitation to its effectiveness. In contrast, the HFLWA method considers the importance degree of each criterion, employing appropriate weights. However, it relies solely on a simple weighted average operator, neglecting the intricate interrelationships among criteria. This oversight limits its holistic evaluation capacity.

Significance of Linguistic Term Sets: The selection of linguistic term sets plays a pivotal role in decision-making processes. HFLWA’s utilization of linguistic terms denoted by S = {s₀, s₁, . . . , s_2τ} lacks the logic of subscript-symmetric additive linguistic terms, leading to limitations in additive operations among different linguistic terms. Our approach, integrating probabilistic information and offset data, overcomes this limitation, facilitating a more comprehensive and accurate evaluation of linguistic terms.

Third group: Modified PHFS-based MCDM method and VEHFS-based score method

In the Modified PHFS-based MCDM method, rankings vary among options, except for a₂, which maintains a consistent position. However, when applying the DPLPMSM operator with both d = 1 and d = 2 ranking values, discrepancies arise in the rankings. In the VEHFS-based score method, a₂ and a₄ share a ranking, while other alternatives display varying rankings under the DPLPMSM operator. Notably, a₄’s ranking remains unchanged under these conditions, with DPLPDMSM set at both d = 1 and d = 2, whereas other rankings differ.

Impact of Representation Methods: PHFS utilizes precise numerical values and associated probabilities, enabling direct scoring and continuous value assignment within a specified range. In contrast, employing linguistic terms unifies evaluation criteria and uses exact numerical values for scoring. Discrepancies arise due to diverse interpretations of numerical values; what one expert might rate as excellent at 80 points, another might consider excellent at 70 points. To address this, our proposed method considers deviations from linguistic terms to resolve range-related issues. Additionally, it incorporates non-membership considerations, enhancing its adaptability and accuracy in evaluation.

Methodological Differences: For the Modified PHFS-based MCDM method, relying solely on PHFS representation for aggregated operators fails to overcome inherent limitations. The weighted average operator lacks flexibility in accounting for varying criteria priorities, treating all criteria equally and disregarding their diverse importance levels. Moreover, it struggles with non-linear correlations, leading to oversimplified evaluations in complex scenarios and being highly sensitive to outliers, potentially biasing decisions. In contrast, Maclaurin Symmetric Mean operators, especially the dual variant, offer improved robustness against outliers, providing a more reliable evaluation method. The VEHFS-based score method depends solely on the scoring function for ranking, overlooking a comprehensive decision-making process. Furthermore, neither method considers the combined weight of criteria.

Comparative Advantages of Our Proposed Method

In the realm of decision-making, subtle nuances in evaluation expressions play a pivotal role. Conventional methods often focus solely on either probabilistic information or membership and non-membership degrees. Our innovative approach integrates both probabilistic data and linguistic variances by introducing a dual probabilistic linguistic 2-tuple set. This integration enables us to comprehensively capture the intricacies of decision-making scenarios, including subtle linguistic nuances and probabilistic data intricacies.

To accommodate this intricate expression mechanism, we meticulously refined two MSM-based operators. These operators form the cornerstone of our methodology, enabling the capture of intricate interplays between criteria interdependence and priority relationships. Importantly, these innovative operators adeptly encapsulate diverse priorities and interdependencies discerned among decision-makers.

This unique integration of probabilistic data, linguistic subtleties, and advanced operator designs sets our approach apart. It provides a sophisticated, comprehensive, and nuanced perspective on decision-making processes, fostering a deeper understanding of the intricate dynamics at play in complex decision-making scenario.

6 Conclusions

In conclusion, this study has tackled the intricate challenges prevalent in real-world decision-making scenarios marked by diverse attributes, ambiguity, limited data, and the involvement of multiple decision-makers. The innovative MCDM technique proposed here for handling probabilistic hesitant fuzzy language data represents a significant advancement in this domain, offering profound implications for practical applications.

6.1 Methodological implications

A key contribution of this research lies in the introduction of DPLTSs as a pioneering evaluation framework. DPLTSs, an extension of conventional fuzzy linguistic sets, provides decision-makers with a versatile platform to express probabilistic hesitant information. Unlike conventional linguistic terms, DPLTSs employs linguistic 2-tuples with possibilities in membership and non-membership sets, ensuring precision even when decision-makers’ cognitive information does not align seamlessly with initial linguistic terms. This nuanced approach preserves the integrity of information, facilitating the accurate capture of preference relationships between criteria.

Building upon the foundational concept of DPLTSs, this paper introduces transformative functions, score functions, and pioneering aggregation operators—the DPLPMSM operator and the DPLPDMSM operator. These operators not only enhance the decision-making process but also pave the way for a novel extension of the VIKOR method and an innovative matching decision-making strategy. The efficacy of these methodologies has been rigorously validated through a comprehensive case study, bolstered by meticulous multiple comparisons that illuminate subtle differences in results.

6.2 Limitations

However, it is essential to acknowledge the limitations of the proposed technique. While DPLTSs provides a flexible framework, its seamless application across diverse real-world contexts necessitates further refinement and adaptation. The inherent complexity of the method might pose implementation and computational challenges, particularly in expansive decision-making scenarios. Addressing these limitations will be pivotal to ensuring the practical viability and broad applicability of the proposed approach.

6.3 Future directions

In our forthcoming research, we aim to enhance our proposed methods by developing more suitable aggregation operators and decision-making approaches tailored for MCDM problems involving DPLTSs. Additionally, we plan to explore the integration of DPLTSs with interval-valued fuzzy information.

This trajectory of our research will involve refining these methods further, enabling their integration with machine learning techniques. This integration will pave the way for solving decision-making problems in various scientific fields more effectively. The application of these advanced methods is crucial in addressing practical challenges such as supplier selection, investment appraisal, risk assessment, and other decision-making scenarios. In terms of future research directions, updating the aggregated operator to enhance its ability to summarize the opinions expressed within the single-valued neutrosophic set could be a valuable avenue for exploration. This development holds the potential to refine the precision and effectiveness of the summarization process, further advancing the field of neutrosophic theory and its applications.

7 Funding

This research was financially supported by Natural Science Foundation of Jiangsu Higher Education Institutions of China (No.:22KJD520002), Research on Modern Educational Technology in Jiangsu Province (2022-R-101356), Universities Philosophy and Social Science Researches in Jiangsu Province (No.: 2020S-JA0534) and Research Initiation Fund for High-level Talents of Jinling Institute Technology (No.: jit-b-201817, No.: jit-b-201906), China Postdoctoral Science Foundation (No.: 2020T130129ZX).

References

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–356.

Atanassov

K.T.

, Intuitionistic fuzzy set, Fuzzy Sets and Systems 20 (1986), 87–96.

Atanassov

K.T.

and Gargov

, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343–349.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoningpart I, Information Science 8(3) (1975), 199–249.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25(6) (2010), 529–539.

Kaur

and Garg

, A novel algorithm for autonomous parking vehicles using adjustable probabilistic neutrosophic hesitant fuzzy set features, Expert Systems with Applications 226 (2023), 120101.

Liao

, Cai

, Garg

, et al., Novel gained and lost dominance score method based on cumulative prospect theory for group decision-making problems in probabilistic hesitant fuzzy environment, International Journal of Fuzzy Systems 25 (2023), 1414–1428.

Herrera

, Herrera-Viedma

, Garg

, et al., Linguistic decision analysis: steps for solving decision problems under linguistic information, Fuzzy Sets and Systems 115 (2000), 67–82.

Tërksen

I.B.

, Type 2 representation and reasoning for CWW, Fuzzy Sets and Systems 127 (2002), 17–36.

10.

, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Information Science 166 (2002), 19–31.

11.

Wang

, Extended hesitant fuzzy linguistic term sets and their aggregation in group decision making, International Journal Of Computational Intelligence Systems 8 (2015), 14–33.

12.

Pang

, Wang

and Xu

, Probabilistic linguistic term sets in multi-attribute group decision making, Information Sciences 369 (2016), 128–143.

13.

, Decision-making based on probabilistic linguistic term sets without loss of information, Complex & Intelligent Systems 8 (2022), 2435–2449.

14.

Zhang

W.Y.

, Luo

, Gao

, Zhang

C.Q.

, Zhao

S.Y.

, A multi-attribute decision-making method for probabilistic language VIKOR based on correlation coefficient and entropy, in: Advances in Natural Computation, Fuzzy Systems and Knowledge Discovery, ed., Springer International Publish, Berlin, 2023, pp. 789–797.

15.

Fei

and Feng

, Modeling interactive multiattribute decision-making via probabilistic linguistic term set extended by dempstershafer theory, International Journal of Fuzzy Systems 23 (2021), 599–613.

16.

, Xu

Z.W.

, Liu

and Wei

C.P.

, Social network group decision-making for probabilistic linguistic information based on GRA, Computers & Industrial Engineering 175 (2023), 108861.

17.

Hua

and Xue

, A maximum consensus improvement method for group decision making under social network with probabilistic linguistic information, Neural Process Letters 54 (2022), 437–465.

18.

Zhang

, Li

and Lu

, Asymmetric normalized probabilistic linguistic term set based on prospect theory and its application to multi-attribute decision-making, Soft Computing 27 (2023), 10427–10445.

19.

Liu

, Fei

L.G.

and Mi

J.N.

, A multi-attribute decision-making method using belief-based probabilistic linguistic term sets and its application in emergency decision-making, Computer Modeling in Engineering & Sciences 136(2) (2023), 2039–2067.

20.

Jin

, Zhang

, Garg

, et al., Evaluation of small and medium-sized enterprises sustainable development with hesitant fuzzy linguistic group decision-making method, Applied Intelligence 52 (2022), 4940–4960.

21.

Opricovic

, Multicriteria optimization of civil engineering systems, Faculty of Civil Engineering 2(1) (1998), 5–21.

22.

Zhang

, Yu

, Jiang

, Zeng

and Wang

, Common prosperity level evaluation: A comprehensive method based on probabilistic linguistic ordered weighted distance measure, prospect theory, and TOPSIS, Frontiers in Psychology 14 (2023), 1152333.

23.

Chen

, Xu

and Ke

G.Y.

, A promethee II approach based on probabilistic hesitant fuzzy linguistic information with applications to multi-criteria group decision-making, prospect theory, and TOPSIS, International Journal of Fuzzy Systems 23 (2021), 1556–1580.

24.

Han

, Tao

, Zhou

, Chen

and Liu

, A new computational model based on archimedean copula for probabilistic unbalanced linguistic term set and its application to multiple attribute group decision making, Computers & Industrial Engineering 140 (2020), 106264.

25.

Feng

, Pang

and Zhang

, On consistency and priority weights for interval probabilistic linguistic preference relations, Fuzzy Optimization and Decision Making 19 (2020), 529–560.

26.

Liang

, Dai

, Wang

and Zhang

, Web celebrity shop assessment and improvement based on online review with probabilistic linguistic term sets by using sentiment analysis and fuzzy cognitive MAP, Fuzzy Optimization and Decision Making 19 (2020), 561–586.

27.

Naz

, Akram

, Al-Shamiri

M.M.A.

, Khalaf

M.M.

and Yousaf

, A new magdm method with 2-tuple linguistic bipolar fuzzy heronian mean operators, Mathematical Biosciences and Engineering 19(4) (2022), 3843–3878.

28.

Qin

, Hashim

S.R.M.

and Sulaiman

, Probabilistic linguistic multi-attribute decision making approach based upon novel gmsm operators, AIMS Mathematics 8(5) (2023), 11727–11751.

29.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning-III, Information Sciences 9(1) (1975), 43–80.

30.

Gou

, Xu

and Liao

, Multiple criteria decision making based on bonferroni means with hesitant fuzzy linguistic information, Soft Computing 21 (2017), 6515–6529.

31.

Xie

, Xu

, Ren

. Multiple criteria decision making based on bonferroni means with hesitant fuzzy linguis tic information, In 2017 IEEE International Conference on Industrial Engineering and Engineering Management, 2017.

32.

, et al., A novel risk assessment model based on failure mode and effect analysis and probabilistic linguistic electre II method, Computers Industrial Engineering 38(4) (2020), 4675–4691.

33.

Awasthi

and Kannan

, Green supplier development program selection using NGT and VIKOR under fuzzy environment, Computers Industrial Engineering 19 (2016), 100–108.

34.

Pang

, Wang

and Xu

, Probabilistic linguistic term sets in multi-attribute group decision making, Information Science 369 (2016), 128–143.

35.

Liang

, Kobina

and Quan

, Grey relational analysis method for probabilistic linguistic multi-criteria group decision-making based on geometric bonferroni mean, International Journal of Fuzzy Systems 20(7) (2018), 2234–2244.

36.

Farhadinial

and Herrera-Viedma

, A modification of probabilistic hesitant fuzzy sets and its application to multiple criteria decision making, Iranian Journal of Fuzzy Systems 17(4) (2020), 151–166.

37.

Farhadinial

and Taghavi

, A single-valued extended hesitant fuzzy score-based technique for probabilistic hesitant fuzzy multiple criteria decision-making, Complexity 2021 (2021), 1–19.

	C ₁	〈 {(s₁, 0.02) (0.6) , (s₀, 0.01) (0.2) , (s₀, 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-2, 0.01) (0.4) , (s_-2, 0.01) (0.2)} 〉
a ₁	C ₂	〈 {(s₂, 0.01) (0.6) , (s₂, 0.01) (0.2) , (s₂, 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉
	C ₃	〈 {(s₀, 0.01) (0.6) , (s₀, 0.01) (0.2) , (s₀, 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉
	C ₁	〈 {(s₃, - 0.01) (0.6) , (s₃, - 0.01) (0.2) , (s₃, - 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉
a ₂	C ₂	〈 {(s₀, 0.01) (0.6) , (s₀, 0.01) (0.2) , (s₀, 0.01) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉
	C ₃	〈 {(s₃, - 0.01) (0.4) , (s₂, 0.02) (0.4) , (s₁, 0.01) (0.2)} , {(s_-2, - 0.01) (0.4) , (s_-3, 0.01) (0.4) , (s_-3, 0.01) (0.2)} 〉
	C ₁	〈 {(s₁, 0.02) (0.6) , (s₁, 0.02) (0.2) , (s₁, 0.02) (0.2)} , {(s₀, - 0.01) (0.4) , (s₀, - 0.01) (0.4) , (s₀, - 0.01) (0.2)} 〉
a ₃	C ₂	〈 {(s₂, 0.05) (0.6) , (s₁, 0.05) (0.2) , (s₁, 0.05) (0.2)} , {(s₀, 0.01) (0.4) , (s₁, - 0.01) (0.4) , (s₁, - 0.01) (0.2)} 〉
	C ₃	〈 {(s₃, - 0.01) (0.4) , (s₂, 0.01) (0.4) , (s₂, 0.01) (0.2)} , {(s₁, 0.01) (0.4) , (s₂, - 0.02) (0.4) , (s₂, - 0.02) (0.2)} 〉
	C ₁	〈 {(s₃, - 0.02) (0.4) , (s₂, 0.01) (0.2) , (s₂, 0.01) (0.4)} , {(s_-3, - 0.01) (0.4) , (s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.2)} 〉
a ₄	C ₂	〈 {(s₁, - 0.01) (0.6) , (s₀, 0.02) (0.2) , (s₀, 0.02) (0.2)} , {(s_-3, 0.01) (0.4) , (s_-2, - 0.01) (0.4) , (s_-2, - 0.01) (0.2)} 〉
	C ₃	〈 {(s₁, 0.01) (0.6) , (s₁, 0.01) (0.2) , (s₁, 0.01) (0.2)} , {(s_-2, 0.02) (0.4) , (s_-2, - 0.02) (0.4) , (s_-2, - 0.02) (0.2)} 〉

A noval approach for dual probabilistic linguistic multi-criteria decision making based on prioritized Maclaurin symmetric mean operators and VIKOR

Abstract

Keywords

1 Introduction

2 Dual probabilistic linguistic 2-tuple set

2.1 Concepts of dual probabilistic linguistic 2-tuple set

3.1 DPLPMSM

4.1 Problem description

4.2 Decision procedures for DPLTSs based on VIKOR

5.1 Background

Table 4 The results for all alternatives Alternative S i Ranking result R i Ranking result Q i Ranking result a 1 0.6547 3 0.3685 2 0.6446 2 a 2 0.0929 1 0.0929 1 0 1 a 3 1.0000 4 0.5043 4 1.0000 4 a 4 0.5917 2 0.4066 3 0.6563 3

5.3.1 Parameter analysis

Table 5 The ranking results of example with d = 1 in DPLPMSM operator Alternative S i Ranking result R i Ranking result Q i Ranking result DPLPMSM d = 1 a 1 0.6303 3 0.4053 2 0.6474 2 a 2 0.1171 1 0.1171 1 0 1 a 3 1.0000 4 0.5182 4 1.0000 4 a 4 0.6298 2 0.4616 3 0.7173 3

6.1 Methodological implications

6.2 Limitations

6.3 Future directions

7 Funding

References

Table 4
The results for all alternatives

Alternative S _i Ranking result R _i Ranking result Q _i Ranking result

a ₁ 0.6547 3 0.3685 2 0.6446 2

a ₂ 0.0929 1 0.0929 1 0 1

a ₃ 1.0000 4 0.5043 4 1.0000 4

a ₄ 0.5917 2 0.4066 3 0.6563 3

Table 5
The ranking results of example with d = 1 in DPLPMSM operator

Alternative S _i Ranking result R _i Ranking result Q _i Ranking result

DPLPMSM d = 1

a ₁ 0.6303 3 0.4053 2 0.6474 2

a ₂ 0.1171 1 0.1171 1 0 1

a ₃ 1.0000 4 0.5182 4 1.0000 4

a ₄ 0.6298 2 0.4616 3 0.7173 3