Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2678 https://internationalpubls.com A Smart Technique Approach in Picture Fuzzy Schweizer-Sklar Power Geometric Aggregation Operators for Enhancing Road Construction Companies through Group Decision-Making T. Yogashanthi1 and K. Deva2* 1Department of Mathematics, SRM Institute of Science and Technology, Ramauram, Chennai, Tamilnadu, India. 2* Department of Mathematics, Vel Tech High Tech Dr.Rangarajan Dr.Sakunthala Engineering College, Avadi-600062, Tamilnadu, India. *Corresponding author: devakrishnan92@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: The objective of this paper is to develop picture fuzzy aggregation operators by utilizing the concept of power aggregation operators through Schweizer-Sklar operations. The Schweizer-Sklar t-norm and t-conorm enhance the flexibility of the data integration process and the power aggregation operator, by capturing the interrelationships between various criteria during decision-making. Motivated by Schweizer-Sklar t-norm and t- conorm, this paper aims to develop the theory of the picture fuzzy Schweizer-Sklar power weighted geometric operator and the picture fuzzy Schweizer-Sklar power ordered weighted geometric operator. The paper also explores the properties and characteristics of these proposed operators. Criteria weights play a crucial role in aggregating different criteria in multiple criteria decision-making processes. This work adopts the simple multi- attribute rating technique to compute criteria weights for solving multi-criteria group decision-making problems in a picture fuzzy environment. Finally, an illustrative example of the road construction company is provided to demonstrate the applicability of the proposed operators. A comparison with existing operators validates the effectiveness of the proposed operators. Keywords: Multi-criteria group decision-making, Picture fuzzy set, Schweizer-Sklar t- norms and t-conorms, Power aggregation operator, Road construction companies. 1. Introduction Multi-criteria group decision-making (MCGDM) is an important area of decision theory with widespread applications across various fields. In many cases, decision-makers encounter difficulties in reaching reasonable conclusions, as the process involves identifying multiple criteria and evaluating several alternatives. Moreover, in real-world decision-making, the ambiguity and subjectivity inherent in human qualitative judgments often result in expert opinions that encompass a range of responses such as yes, abstain, no, or refusal-which cannot be accurately represented by crisp values. In 1965, Zadeh (1965) introduced the foundational concept of fuzzy set (FS) theory, which incorporated the degree of membership μ. This concept has since been applied to numerous real-world problems. Later, Atanassov (1986) expanded on FS theory by developing the intuitionistic fuzzy set (IFS) theory, which adds the degree of non-membership ν alongside the degree of membership μ under the condition that 0≤μ+ν≤1. mailto:devakrishnan9@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2679 https://internationalpubls.com Figure 1: Extensions of PFS The IFS, while demonstrating strong capabilities in multi-criteria decision-making (MADM), cannot adequately express more complex fuzzy information. One limitation is its inability to capture ambiguous information, such as the degree of neutrality (η). To address this issue, Cuong et al. (2013), introduced the concept of the picture fuzzy set (PFS), which extends IFS by adding the degree of neutrality (η) alongside the degrees of membership (μ) and non-membership (ν), under the condition 0≤μ+η+ν≤1. PFS models are particularly useful in situations where human opinions involve more nuanced responses such as "yes," "no," "abstain," and "refusal." Figure 1 illustrates the extensions of a PFS. Additionally, a few students may express refusal to visit either destination (π). Cuong et al. (2016) investigated the classification of representable picture t-norms and picture t-conorms along with their properties. Yager (1988) introduced the ordered weighted aggregation (OWA) operator and explored its properties. Wei (2017) described various PF arithmetic and geometric operators, as well as PF hybrid aggregation operators (AOs) and their applications. Garg (2017) presented a series of AOs for PFSs that were applied to solve multi- criteria decision-making (MCDM) problems. Wei (2018) studied the MADM problem using arithmetic and geometric AOs based on Hamacher operations in the PF context. Khan et al. (2019) proposed AOs using PF Einstein operations and discussed their application in PF-based decision- making. Zhang et al. (2018) introduced PF Dombi Heronian mean operators for the MADM problem. Jana et al. (2019) developed PF Dombi arithmetic and geometric AOs to solve decision- making problems. Additionally, Jana et al. (2019) employed PF Hamacher AOs to construct MADM approaches for enterprise performance evaluation within the PF context. Ates et al. (2020) proposed novel PF aggregation operators and extended the Bonferroni mean operator to MCDM applications. Wang et al. (2018) proposed PF Muirhead mean operators and demonstrated their application for financial investment risk evaluation. Qiyas et al. (2020) introduced several AOs using Yager operators in the PF context. Several other MADM approaches based on AOs for PFNs have been proposed, including those by Lin et al. (2021), Qin et al. (2021), Seikh et al. (2021), Ullah (2021), and Senapati (2022). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2680 https://internationalpubls.com Table 1: Comparison of different approaches for picture fuzzy AOs Authors Geometric operators Status of weight DM method DM solution Garg, et al. (2023) IF Schweizer-Sklar power IF Schweizer-Sklar weighted Known MADM Score function Khan et al. (2022) IF Schweizer-Sklar generalized power weighted IF Schweizer-Sklar generalized power ordered weighted Known MAGDM Score function Biswas, A & Deb, N. (2021) PyF Schweizer-Sklar power weighted Known MADM Score function Wei, et al. (2022) FF Schweizer-Sklar weighted Unknown MAGDM Best- and - worst Ma et al. (2024) q-ROF Schweizer-Sklar power and q-ROF Schweizer-Sklar power weighted Known MADM Score function Hussain, et al (2022) PF Schweizer-Sklar prioritized and PF Schweizer-Sklar prioritized weighted Known MADM Score function Proposed work PF Schweizer-Sklar power weighted and PF Schweizer- Sklar power ordered weighted Unknown MAGDM Score function Furthermore, various AOs based on the Schweizer-Sklar (SS) operational law within different FS frameworks are summarized in Table 1 Harish Garg, et al. (2023) introduced the IFSS prioritized AO for solving MCDM problems. Khan et al. (2022) explored the concept of the IFSS generalized power AO. Biswas, A. & Deb, N. (2021) developed the SS power AO in a pythagorean fuzzy ( PyF ) environment. Wei, D et al. (2022) established the fermatean fuzzy (FF) SS in green supplier selection. Ma et al. (2024) proposed the q-rung orthopair fuzzy (q-ROF) SS prioritized operator for MADM problems. Hussain et al (2022) developed the PFSS prioritized operator for solving MADM problems. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2681 https://internationalpubls.com This article introduces a series of efficient AO by combining the power aggregation (PA) operator with SS operations to enhance the information aggregation process in the PF environment. A set of PF AO is proposed, including the PFSS power geometric (PFSSPG) operator, PFSS power weighted geometric PFSSPWG operator, and PFSS power ordered weighted geometric (PFSSPOWG) operator. The article also discusses special cases and desirable properties of these operators. A simple multi-criteria rating technique (SMART) is presented to solve MCGDM problems in the PF environment using the proposed operators. To demonstrate the practicality and effectiveness of the approach, an illustrative example, is solved and compared with several existing approaches. Additionally, Table 1 provides a summary of various AOs based on the SS operational law. Our study aims to address this specific research gap. The main motivations of this article are as follows: • The previous research focuses on the development of AO, with particular emphasis on the importance of SS operations, which serve as a generalization of algebraic operations. • To combine decision information from multiple experts using an PG operator that accounts for the interrelationships between criteria. • To develop an MCGDM method incorporating the proposed operators for efficiently managing conflicting criteria. • To showcase the practicality of the proposed operators through solving a numerical MCGDM application. • To validate the feasibility of the method by examining the parameters of the defined operators across different values and comparing them with existing operators. The main contributions of this article are as follows: • This article introduces innovative SS operation laws and PA operator, namely PFSSPG, PFSSPWG and PFSSPOWG operators under PF environment. • These operators are utilized to develop a novel PFMCGDM method with SMART. • The effectiveness and reliability of the proposed PFMCGDM method with SMART approach are demonstrated by applying it to solve a practical road construction company. • Furthermore, the feasibility of the developed method is illustrated through a comparative evaluation against existing operators. • The contributions of this research are illustrated in Figure 2 to provide readers with a clear visual understanding. This research work is organized into eight detailed sections as outlined below. Section 2 introduces the fundamental concept of PFSs and their related operations. In Section 3, the proposed picture fuzzy Schweizer-Sklar power geometric aggregation operators, along with their properties, are discussed. Section 4 continues this discussion by presenting the properties of these aggregation operators. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2682 https://internationalpubls.com Figure 2: Contribution of the proposed work To illustrate the practical application of this approach, a numerical example in the picture fuzzy context is provided in Section 6. Section 7 offers a comparative analysis to demonstrate the advantages of the proposed method over existing aggregation operators. Finally, Section 8presents the conclusions drawn from the study. 2. Preliminaries In this section, some basic concepts have been reviewed related to PFS. 2.1 Picture Fuzzy Set The PFS (Cuong et al. (2013,2014) ) is an extension of IFS. The mathematical form of PFS is expressed as follows: Definition 1. A picture fuzzy set 𝐴 on universal set 𝑋 is defined by, 𝐴 = {⟨𝑥, 𝜇𝐴(𝑥), 𝜂𝐴(𝑥), 𝜈𝐴(𝑥)⟩/𝑥 ∈ 𝑋} Where, 𝜇A(𝑥), 𝜂A(𝑥) and 𝜈A(𝑥) ∈ [0,1] are the degree of membership, the degree of neutral membership and the degree of non-membership of 𝑥 ∈ A respectively, with the following condition: 0 ≤ 𝜇A(𝑥) + 𝜂A(𝑥) + 𝜈A(𝑥) ≤ 1∀𝑥 ∈ 𝑋. Then, for 𝑥 ∈ 𝑋, 𝜋A(𝑥) = 1 − 𝜇A(𝑥) − 𝜂A(𝑥) − 𝜈A(𝑥) could be called the degree of refusal membership of 𝑥 in A. Geometrical representation of PFS is shown in Figure 3. For convenience, 𝛼 = (𝜇𝛼, 𝜂𝛼 , 𝜈𝛼) is called a picture fuzzy number (PFN). Figure 3: Geometrical representation of picture fuzzy set Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2683 https://internationalpubls.com 2.2 Comparison for PFNs According to Garg (2017) the score and accuracy functions of PFNs are as follows: Definition 2. Let 𝛼 = (𝜇𝛼, 𝜂𝛼, 𝜈𝛼) be a PFN, its score function 𝑆(𝛼) and its accuracy function 𝐴(𝛼) is defined by 𝑆(𝛼) = 𝜇𝛼 − 𝜂𝛼 − 𝜈𝛼; 𝑆(𝛼) ∈ [−1,1], 𝐴(𝛼) = 𝜇𝛼 + 𝜂𝛼 + 𝜈𝛼; 𝐴(𝛼) ∈ [0,1]. Based on the 𝑆(𝛼) and𝐴(𝛼) an order relationship between two PFNs is defined as follows. Definition 3. Let 𝛼1 = (𝜇𝛼1 , 𝜂𝛼1 , 𝜈𝛼1 ) and 𝛼2 = (𝜇𝛼2 , 𝜂𝛼2 , 𝜈𝛼2 ) be two PFNs. Then the following comparison rules can be used: (i) If 𝑆(𝛼1) < 𝑆(𝛼2) then 𝛼1 ≺ 𝛼2 (ii) If 𝑆(𝛼1) = 𝑆(𝛼2) then (a) If 𝐴(𝛼1) < 𝐴(𝛼2) then 𝛼1 ≺ 𝛼2 (b) If 𝐴(𝛼1) = 𝐴(𝛼2)then 𝛼1 ≈ 𝛼2. 2.3 Operation laws of Picture fuzzy numbers Schweizer-Sklar (Hussain, A. et al. (2022)) operations laws and power geometric aggregation operator (Ullah, K. et all (2023)) are defined for PFNs as follows. Definition 4. The 𝑃𝐺 operator are defined as: 𝑃𝐺(�̆�1, �̆�2, … , �̆�𝑟) = ∏   𝑟 𝑘=1   �̆�𝑘 (1+�̈�(�̆�𝑘)) ∑   𝑟++�̈�(�̆�𝑘) 𝑘=1 (1) where �̈�(�̆�𝑘) = ∑  𝑟 𝑘=1,𝑘≠ℎ sup(�̆�ℎ, �̆�𝑘), sup(�̆�ℎ, �̆�𝑘) = 1 − 𝔇(�̆�ℎ, �̆�𝑘) and the weight (1+�̈�(�̆�𝑘)) ∑  𝑟 𝑘=1  1+�̈�(�̆�𝑘) of the argument �̆�ℎ depends on all the input arguments �̆�𝑘(𝑘 = 1,2, … , 𝑟), which enables the argument values to support each other in the geometric aggregation process. Definition 5. Let 𝛼1 = ⟨𝜇𝛼1 , 𝜂𝛼1 , 𝜈𝛼1 ⟩ and 𝛼2 = ⟨𝜇𝛼2 , 𝜂𝛼2 , 𝜈𝛼2 ⟩ be any two PFNs, then the Euclidean distance between them is defined as follows: 𝔇(𝛼1, 𝛼2) = 1 3 {|𝜇𝛼1 − 𝜇𝛼2 | + |𝜂𝛼1 − 𝜂𝛼2 | + |𝜈𝛼1 − 𝜈𝛼2 |} (2) Definition 6. Let 𝛼1 = ⟨𝜇𝛼1 , 𝜂𝛼1 , 𝜈𝛼1 ⟩ and 𝛼2 = ⟨𝜇𝛼2 , 𝜂𝛼2 , 𝜈𝛼2 ⟩ be any two PFN. Then the generalized union and intersection are defined as follows: 𝛼1 ∪ 𝛼2 = {⟨𝑥, {𝔗∗{𝜇𝛼1 (𝑥), 𝜇𝛼2 (𝑥)}} , {𝔗{𝜂𝛼1 (𝑥), 𝜂𝛼2 (𝑥)}} , {𝔗{𝜈𝛼1 (𝑥), 𝜈𝛼2 }(𝑥)} ∣ 𝑥 ∈ 𝑋⟩} ; 𝛼1 ∩ 𝛼2 = {⟨𝑥, {𝔗{𝜇𝛼1 (𝑥), 𝜇𝛼2 (𝑥)}} , {𝔗∗{𝜂𝛼1 (𝑥), 𝜂𝛼2 (𝑥)}} , {𝔗∗{𝜈𝛼1 (𝑥), 𝜈𝛼2 (𝑥)}}| 𝑥 ∈ 𝑋}⟩} ; where𝔗 and 𝔗∗ respectively, express TN and TCN. The SSTN and SSTCN are defined as follow. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2684 https://internationalpubls.com 𝔗(𝑥1, 𝑥2) = (𝑥1 𝛽 + 𝑥2 𝛽 − 1) 1 𝛽 𝔗∗(𝑥1, 𝑥2) = 1 − (1 − 𝑥1 𝛽 + (1 − 𝑥2 𝛽 − 1) 1 𝛽 ) Additionally, when 𝛽 = 0, we have 𝔗(𝑥1, 𝑥2) = 𝑥1𝑥2 and 𝔗∗(𝑥1, 𝑥2) = 𝑥1 + 𝑥2 − 𝑥1𝑥2. That is, SSTN and SSTCN reduce to algebraic TN and TCN. Definition 7. Let 𝛼1 = ⟨𝜇𝛼1 , 𝜂𝛼1 , 𝜈𝛼1 ⟩ and 𝛼2 = ⟨𝜇𝛼2 , 𝜂𝛼2 , 𝜈𝛼2 ⟩ be any two PFN. Then based on SS operations, the generalized union and intersection are introduced as follows: 𝛼1 ⊕ 𝛼2 = ⟨𝔗∗(𝜇𝛼1 , 𝜇𝛼2 ), 𝔗(𝜂𝛼1 , 𝜂𝛼2 ), 𝔗(𝜈𝛼1 , 𝜈𝛼2 )⟩(3) 𝛼1 ⊗ 𝛼2 = ⟨𝔗(𝜇𝛼1 , 𝜇𝛼2 ), 𝔗∗(𝜂𝛼1 , 𝜂𝛼2 ), 𝔗∗(𝜈𝛼1 , 𝜈𝛼2 )⟩ (4) Definition 8. Let 𝛼 = (𝜇𝛼, 𝜂𝛼, 𝜈𝛼), 𝛼1 = (𝜇𝛼1 , 𝜂𝛼1 , 𝜈𝛼1 ) and 𝛼2 = (𝜇𝛼2 , 𝜂𝛼2 , 𝜈𝛼2 ) be three PFNs and some basic algebraic operations of PFNs are defined as follow: 𝛼1 ⊕ 𝛼2 = ⟨𝜇𝛼1 + 𝜇𝛼2 − 𝜇𝛼1 𝜇𝛼2 , 𝜂𝛼1 𝜂𝛼2 , 𝜈𝛼1 𝜈𝛼2 ⟩ 𝛼1 ⊗ 𝛼2 = ⟨ 𝜇𝛼1 𝜇𝛼2 , 𝜂𝛼1 + 𝜂𝛼2 − 𝜂𝛼1 𝜂𝛼2 𝜈𝛼1 + 𝜈𝛼2 − 𝜈𝛼1 𝜈𝛼2 ⟩ 𝜆𝛼 = ⟨1 − (1 − 𝜇𝛼)𝜆, (𝜂𝛼)𝜆, (𝜈𝛼)𝜆⟩ 𝛼𝜆 = ⟨(𝜇𝛼)𝜆, 1 − (1 − 𝜂𝛼)𝜆, 1 − (1 − 𝜈𝛼)𝜆⟩: 𝜆 > 0. Definition 9. Let 𝛼 = (𝜇𝛼, 𝜂𝛼, 𝜈𝛼), 𝛼1 = (𝜇𝛼1 , 𝜂𝛼1 , 𝜈𝛼1 ) and 𝛼2 = (𝜇𝛼2 , 𝜂𝛼2 , 𝜈𝛼2 ) be three PFNs and some basic 𝑆𝑆 operations of PFNs are defined as follows (𝛽 < 0) : 𝛼1 ⊕ 𝛼2 = ⟨(1 − 𝜇𝛼1 ) 𝛽 + (1 − 𝜇𝛼2 ) 𝛽 − 1) 1 𝛽 , (𝜂 𝛼1 𝛽 + 𝜂 𝛼2 𝛽 − 1) 1 𝛽 , (𝜈 𝛼1 𝛽 + 𝜈 𝛼2 𝛽 − 1) 1 𝛽 )⟩ 𝛼1 ⊗ 𝛼2 = ⟨(𝜇 𝛼1 𝛽 + 𝜇 𝛼2 𝛽 − 1) 1 𝛽 , 1 − ((1 − 𝜂𝛼1 ) 𝛽 + (1 − 𝜂𝛼2 ) 𝛽 − 1) 1 𝛽 , 1 − ((1 − 𝜈𝛼1 ) 𝛽 + (1 − 𝜈𝛼2 ) 𝛽 − 1) 1 𝛽 ⟩ 𝛼1 𝜆 = ⟨(𝜆𝜇 𝛼1 𝛽 − (𝜆 − 1)) 1 𝛽 , 1 − (𝜆(1 − 𝜂𝛼1 ) 𝜆 − (𝜆 − 1)) 1 𝛽 , 1 − (𝜆(1 − 𝜈𝛼1 ) 𝜆 − (𝜆 − 1)) 1 𝛽 ⟩ , 𝜆 > 0 𝜆𝛼1 = ⟨1 − (𝜆(1 − 𝜇𝛼1 ) 𝜆 − (𝜆 − 1)) 1 𝛽 , (𝜆𝜂 𝛼1 𝛽 − (𝜆 − 1)) 1 𝛽 , (𝜆𝜈 𝛼1 𝛽 − (𝜆 − 1)) 1 𝛽 ⟩ , 𝜆 > 0. 3. Picture fuzzy Schweizer-Sklar power geometric aggregation operators In this section, we presented the Schweizer-Sklar power geometric AOs and discussed the fundamental properties of these proposed operators within the framework of PFNs. 3.1 Picture fuzzy Schweizer-Sklar power weighted geometric aggregation operators In the next section, we introduce the confidence picture fuzzy Schweizer-Sklar power weighted geometric operator within the context of PF, and we examine the desirable properties of this proposed operator. Additionally, we applied a weighted support degree throughout our article, using the following equation: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2685 https://internationalpubls.com �̆�𝑗(𝑗 = 1,2, … , 𝑛) is a set of integrated weights, �̆�𝑗 = 𝑤𝑗(1+𝔑𝔶(𝛼𝑗)) ∑  𝑛 𝑗=1  𝑤𝑗(1+𝔑𝔶(𝛼𝑗)) ⋅ 𝔑𝔶(𝛼𝑗) = ∑  𝑛 ℎ=1,ℎ≠𝑗 Sup(𝛼𝑗, 𝛼ℎ) and 𝑤 = (𝑤1, 𝑤2, … , 𝑤𝑛)𝑇 is the weight vector of 𝛼𝑗(𝑗 = 1,2, … , 𝑛)𝑤 ∈ [0,1], ∑  𝑛 𝑗=1 𝑤𝑗 = 1. Definition 10. Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ (𝑗 = 1,2, … , 𝑛) be a set of PFN. Then the PFSSPWG operator of dimension 𝑛 is a mapping PFSSPWG: 𝛼𝑛 → 𝛼 such that PFSSPWG(𝛼1, 𝛼2, … , 𝛼𝑛) =⊕𝑗=1 𝑛 𝛼 𝑗 ( 𝑤𝑗(1+𝔫𝔶(𝛼𝑗)) ∑  𝑛 𝑗=1  𝑤𝑗(1+𝔫𝔶(𝛼𝑗)) ) (5) where𝛼 is the set of all PFN and 𝔑𝔶(𝛼𝑗) = ∑  𝑛 ℎ=1,ℎ≠𝑗 Sup(𝛼𝑗, 𝛼ℎ) and 𝑤 = (𝑤1, 𝑤2, … , 𝑤𝑛)𝑇 is the weight vector of 𝛼𝑗(𝑗 = 1,2, … , 𝑛)𝑤 ∈ [0,1], ∑  𝑛 𝑗=1 𝑤𝑗 = 1. Theorem 1. Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ (𝑗 = 1,2, … , 𝑛) be a set of PFN and 𝛽 < 0, then the aggregated value obtained using PFSSPWG operator is also a PFN and can be expressed as follows: PFSSPWG(𝛼1, 𝛼2, … , 𝛼𝑛) = ⟨(∑   𝑛 𝑗=1   �̆�𝑗𝜇𝑗 𝛽 ) 1 𝛽 , 1 − (∑   𝑛 𝑗=1   �̆�𝑗(1 − 𝜂𝑗) 𝛽 ) 1 𝛽 , 1 − (∑   𝑛 𝑗=1   �̆�𝑗(1 − 𝜈𝑗) 𝛽 ) 1 𝛽 ⟩ (6) where�̆�𝑗(𝑗 = 1,2, … , 𝑛) is a set of integrated weights, �̆�𝑗 = 1+𝔑𝔶(𝛼𝑗) ∑  𝑛 𝑗=1  (1+𝔑𝔶(𝛼𝑗)) . In order to prove Equation 6, we first prove that when �̆� = (�̆�1, �̆�2, … , �̆�𝑛)𝑇 is any value, i.e. without any constraint for �̆�, the following equation is right: PFSSPWG(𝛼1, 𝛼2, … , 𝛼𝑛) = ⟨1 − (∑   𝑛 𝑗=1   �̆�𝑗(1 − 𝜇𝑗) 𝛽 − ∑   𝑛 𝑗=1   �̆�𝑗 + 1) 1 𝛽 , (∑   𝑛 𝑗=1   �̆�𝑗𝜂𝑗 𝛽 − ∑   𝑛 𝑗=1   �̆�𝑗 + 1) 1 𝛽 , (∑   𝑛 𝑗=1   �̆�𝑗𝜈𝑗 𝛽 − ∑   𝑛 𝑗=1   �̆�𝑗 + 1) 1 𝛽 ⟩ (7) By the mathematical induction method, the Equation 7 can be proved as follows: Proof. According to the operational rules of PFs based on SS operations, we have �̆�𝑗𝛼𝑗 = ⟨(�̆�𝑗𝜇𝑗 𝛽 − �̆�𝑗 + 1) 1 𝛽 1 − (�̆�𝑗(1 − 𝜂𝑗) 𝛽 − �̆�𝑗 + 1) 1 𝛽 , 1 − (�̆�𝑗(1 − 𝜈𝑗) 𝛽 − �̆�𝑗 + 1) 1 𝛽 ⟩ 1. When n = 2, we have �̆�1𝛼1 = ⟨(�̆�1𝜇1 𝛽 − �̆�1 + 1) 1 𝛽 1 − (�̆�1(1 − 𝜂1)𝛽 − �̆�1 + 1) 1 𝛽, 1 − (�̆�1(1 − 𝜈1)𝛽 − �̆�1 + 1) 1 𝛽⟩; �̆�2𝛼2 = ⟨(�̆�2𝜇2 𝛽 − �̆�2 + 1) 1 𝛽 2 − (�̆�2(1 − 𝜂2)𝛽 − �̆�2 + 1) 1 𝛽, 1 − (�̆�2(1 − 𝜈2)𝛽 − �̆�2 + 1) 1 𝛽⟩ then PFSSPWG(𝛼1, 𝛼2) = �̆�1𝛼1 + �̆�2𝛼2 = ⟨(�̆�1𝜇1 𝛽 − �̆�1 + 1) 1 𝛽 1 − (�̆�1(1 − 𝜂1)𝛽 − �̆�1 + 1) 1 𝛽, 1 − (�̆�1(1 − 𝜈1)𝛽 − �̆�1 + 1) 1 𝛽 + (�̆�2𝜇2 𝛽 − �̆�2 + 1) 1 𝛽 2 − (�̆�2(1 − 𝜂2)𝛽 − �̆�2 + 1) 1 𝛽, 1 − (�̆�2(1 − 𝜈2)𝛽 − �̆�2 + 1) 1 𝛽⟩ = ⟨(∑𝑗=1 2  �̆�𝑗𝜇𝑗 𝛽 − ∑𝑗=1 2  �̆�𝑗 + 1) 1 𝛽 , 1 − (∑𝑗=1 2  �̆�𝑗(1 − Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2686 https://internationalpubls.com 𝜂𝑗) 𝛽 − ∑𝑗=1 2  �̆�𝑗 + 1) 1 𝛽, 1 − (∑𝑗=1 2  �̆�𝑗(1 − 𝜈𝑗) 𝛽 − ∑𝑗=1 2  �̆�𝑗 + 1) 1 𝛽 ⟩ i.e. when n = 2, Equation 7 is right. 2. Suppose n = m, Equation 7 is right, i.e. PFSSPWG(𝛼1, 𝛼2, … , 𝛼𝑚) = ⟨1 − (∑𝑗=1 𝑚  �̆�𝑗(1 − 𝜇𝑗) 𝛽 − ∑𝑗=1 𝑚  �̆�𝑗 + 1) 1 𝛽, (∑𝑗=1 𝑚  �̆�𝑗𝜂𝑗 𝛽 − ∑𝑗=1 𝑚  �̆�𝑗 + 1) 1 𝛽 , (∑𝑗=1 𝑚  �̆�𝑗𝜈𝑗 𝛽 − ∑𝑗=1 𝑚  �̆�𝑗 + 1) 1 𝛽 ⟩ then when n = m + 1, according to the operational rules of PFNs, we have �̆�𝑚+1𝛼𝑚+1 = ⟨(�̆�𝑚+1𝜇𝑚+1 𝛽 − �̆�𝑚+1 + 1) 1 𝛽 , 1 − (�̆�𝑚+1(1 − 𝜂𝑚+1)𝛽 − �̆�𝑚+1 + 1) 1 𝛽, 1 − (�̆�𝑚+1(1 − 𝜈𝑚+1)𝛽 − �̆�𝑚+1 + 1) 1 𝛽⟩ and PFSSPWG(𝛼1, 𝛼2, … , 𝛼𝑚, 𝛼𝑚+1) = PFSSPWG(𝛼1, 𝛼2, … , 𝛼𝑚) ⊕ (�̆�𝑚+1𝛼𝑚+1) = ⟨1 − (∑𝑗=1 𝑚  �̆�𝑗(1 − 𝜇𝑗) 𝛽 − ∑𝑗=1 𝑚  �̆�𝑗 + 1) 1 𝛽 , (∑𝑗=1 𝑚  �̆�𝑗𝜂𝑗 𝛽 − ∑𝑗=1 𝑚  �̆�𝑗 + 1) 1 𝛽 , (∑𝑗=1 𝑚  �̆�𝑗𝜈𝑗 𝛽 − ∑𝑗=1 𝑚  �̆�𝑗 + 1) 1 𝛽 + (�̆�𝑚+1𝜇𝑚+1 𝛽 − �̆�𝑚+1 + 1) 1 𝛽, 1 − (�̆�𝑚+1(1 − 𝜂𝑚+1)𝛽 − �̆�𝑚+1 + 1) 1 𝛽, 1 − (�̆�𝑚+1(1 − 𝜈𝑚+1)𝛽 − �̆�𝑚+1 + 1) 1 𝛽⟩ = ⟨(∑𝑗=1 𝑚+1  �̆�𝑗𝜇𝑗 𝛽 − ∑𝑗=1 𝑚+1  �̆�𝑗 + 1) 1 𝛽1 − (∑𝑗=1 𝑚+1  �̆�𝑗(1 − 𝜂𝑗) 𝛽 − ∑𝑗=1 𝑚+1  �̆�𝑗 + 1) 1 𝛽 , 1 − (∑𝑗=1 𝑚+1  �̆�𝑗(1 − 𝜈𝑗) 𝛽 − ∑𝑗=1 𝑚+1  �̆�𝑗 + 1) 1 𝛽 ⟩ So, when n = m + 1, Equation 7 is right. 3. According to steps (1) and (2), Equation 7 is established for any j. Because Equation 7 is right without any constraint for �̆�, Equation 6 is also right when �̆�𝑗 ≥ 0 and ∑𝑗=1 𝑛  �̆�𝑗 = 1. Example 1. Suppose 𝛼1 = ⟨0.8,0.1,0.05⟩, 𝛼2 = ⟨0.52,0.3,0.1⟩, 𝛼3 = ⟨0.45,0.4,0.01⟩ and 𝛼4 = ⟨0.79,0.156,0.05⟩ are four PFNs, 𝑤 = (𝑤1, 𝑤2, 𝑤3, 𝑤4) = ⟨0.2,0.1,0.3,0.4⟩𝑇 be the weight vector of 𝛼𝑗(𝑗 = 1,2,3,4) then we adopt the PFSSPWG to aggregate the four PFNs and generate a comprehensive value. The steps are shown as follows: (𝛽 = −2). Sup(𝛼_1, 𝛼_2 ) = 0.823, Sup(𝛼_1, 𝛼_3 ) = 0.77, Sup(𝛼_1, 𝛼_4 ) = 0.978, Sup(𝛼_2, 𝛼_3 ) = 0.913, Sup(𝛼_2, 𝛼_4 ) = 0.845 and Sup(𝛼_3, 𝛼_4 ) = 0.792. 𝔑𝔶(𝛼_1 ) = Sup(𝛼_1, 𝛼_2 ) + Sup(𝛼_3, 𝛼_1 ) = 2.571, 𝔑𝔶(𝛼_2 ) = 2.582, 𝔑𝔶(𝛼_3 ) = 2.475 and 𝔑𝔶(𝛼4) = 2.615. �̆�1 = 0.201, �̆�2 = 0.101, �̆�3 = 0.293and�̆�4 = 0.406. PFSSPWG(𝛼1, 𝛼2, … , 𝛼4) = ⟨1 − (∑j=1 4  �̆�j(1 − 𝜇j) 𝛽) 1 𝛽, (∑j=1 4  �̆�j𝜂j 𝛽 ) 1 𝛽, (∑j=1 4  �̆�j𝜈, ⟨0.253,0.262,0.04⟩. For a collection of PFN, we can easily prove the proposed PFSSPWG operator satisfying the idempotency, monotonicity and boundedness properties as follows: property 1. (Idempotency) Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ (𝑗 = 1,2, … , 𝑛) be a set of PFN, if 𝛼𝑗 = 𝛼 = (𝜇, 𝜂, 𝜈) then PFSSPWG(𝛼1, 𝛼2, … , 𝛼𝑛) = 𝛼 property 2. (Monotonicity) Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ and 𝛼𝑗 ′ = ⟨𝜇𝛼𝑗 ′ , 𝜂𝛼𝑗 ′ , 𝜈𝛼𝑗 ′ ⟩ 𝛼𝑗 ′(𝑗 = 1,2, … , 𝑛) as two sets of PFN; and 𝜇𝑗 ≤ 𝜇𝑗 ′ , 𝜂𝑗 ′ ≤ 𝜂𝑗 and 𝜈𝑗 ′ ≤ 𝜈𝑗 then PFSSPWG(𝛼1, 𝛼2, … , 𝛼𝑛) ≤ PFSSPWG(𝛼1 ′ , 𝛼2 ′ , … , 𝛼𝑛 ′ ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2687 https://internationalpubls.com property 3. (Boundedness) Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ (𝑗 = 1,2, … , 𝑛) be a set of PFN, and 𝛼+ = (max 𝑛 𝑗𝑗=1  , min 𝑛 𝑗𝑗=1  , min𝜈𝑗𝑗=1 𝑛 ) , 𝛼− = (min 𝑛 𝑗𝑗=1  , max 𝑛 𝑗𝑗=1  , max𝜈𝑗𝑗=1 𝑛 ) then 𝛼− ≤ 𝑃𝐹𝑆𝑆𝑃𝑊𝐺(𝛼1, 𝛼2, … , 𝛼𝑛) ≤ 𝛼+ 3.2 Picture fuzzy Schweizer-Sklar power ordered weighted geometric aggregation operators In the following section, we propose the confidence picture fuzzy Schweizer-Sklar power ordered weighted geometric operator with the PF context and the desirable properties of the proposed operator are investigated. Definition 11. Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ (𝑗 = 1,2, … , 𝑛) be a set of SVNN. Then the PFSSPOWG operator of dimension 𝑛 is a mapping PFSSPOWG: 𝛼𝑛 → 𝛼 such that PFSSPOWG(𝛼1, 𝛼2, … , 𝛼𝑛) =⊕𝑗=1 𝑛 𝛼 𝜎(𝑗) ( 𝑤𝑗(1+𝔑𝔶(𝛼𝜎(𝑗))) ∑  𝑛 𝑗=1  𝑤𝑗(1+𝔫𝔶(𝛼𝜎(𝑗))) ) (8) where𝜎(𝑗) is the permutation such that 𝛼𝜎(𝑗−1) ≥ 𝛼𝜎(𝑗) for any 𝑗 = 1,2, … 𝑛 and 𝔑𝔶(𝛼𝜎(𝑗)) = ∑  𝑛 ℎ=1,ℎ≠𝑗 Sup(𝛼𝜎(𝑗), 𝛼𝜎(ℎ)) and 𝑤 = (𝑤1, 𝑤2, … , 𝑤𝑛)𝑇 is the weight vector of 𝛼𝜎(𝑗)(𝑗 = 1,2, … , 𝑛), 𝑤 ∈ [0,1], ∑  𝑛 𝑗=1 𝑤𝑗 = 1. Theorem 2. Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ (𝑗 = 1,2, … , 𝑛) be a set of PFN and 𝛽 < 0, then the aggregated value obtained using PFSSPOWG operator is also a PFN and can be expressed as follows: PFSSPOWG(𝛼1, 𝛼2, … , 𝛼𝑛) = ⟨(∑   𝑛 𝑗=1   �̆�𝑗𝜇𝑗 𝛽 ) 1 𝛽 , 1 − (∑   𝑛 𝑗=1   �̆�𝑗(1 − 𝜂𝑗) 𝛽 ) 1 𝛽 , 1 − (∑   𝑛 𝑗=1   �̆�𝑗(1 − 𝜈𝑗) 𝛽 ) 1 𝛽 ⟩ (9) Proof. Proof of this theorem is similar to theorem 1 so we omit here. Example 2. Suppose 𝛼1 = ⟨0.8,0.1,0.05⟩, 𝛼2 = ⟨0.79,0.159,0.05⟩, 𝛼3 = ⟨0.52,0.23,0.1⟩ and 𝛼4 = ⟨0.45,0.4,0.01⟩ are four PFNs, 𝑤 = (𝑤1, 𝑤2, 𝑤3, 𝑤4) = ⟨0.2,0.1,0.3,0.4⟩𝑇 be the weight vector of 𝛼𝑗(𝑗 = 1,2,3,4) then we need to permute the PFs in order to aggregate then using Equation 9, we first calculate the score value of 𝛼𝑗 for 𝑗 = 1,2,3,4. Sup(𝛼_1, 𝛼_2 ) = 1 − 𝔇(𝛼_1, 𝛼_2 ) = 0.978, Sup(𝛼_1, 𝛼_3 ) = 0.823, Sup(𝛼_1, 𝛼_4 ) = 0.77, Sup(𝛼_2, 𝛼_3 ) = 0.845, Sup(𝛼_2, 𝛼_4 ) = 0.792andSup(𝛼3, 𝛼4) = 0.913 ⋅ 𝔑𝔶(𝛼1) = Sup(𝛼1, 𝛼2) + Sup(𝛼3, 𝛼1) = 2.571, 𝔑𝔶(𝛼2) = 2.615, 𝔑𝔶(𝛼3) = 2.582 and 𝔑𝔶(𝛼4) = 2.475. �̆�1 = 0.201, �̆�2 = 0.102, �̆�3 = 0.304and �̆�4 = 0.393. The PFSSPOWG to aggregate the four PFNs and generate a comprehensive value. The steps are shown as follows: (𝛽 = −2)PFSSPOWG(𝛼1, 𝛼2, … , 𝛼4) = ⟨1 − (∑  4 𝑗=1   �̆�𝑗(1 − 𝜇𝑗) 𝛽 ) 1 𝛽 , (∑  4 𝑗=1   �̆�𝑗𝜂𝑗 𝛽 ) 1 𝛽 , (∑  4 𝑗=1   �̆�𝑗𝜈𝑗 𝛽 ) 1 𝛽 ⟩ = ⟨0.317,0.310,0.052⟩. For a collection of PFN, we can easily prove the proposed PFSSPOWG operator satisfying the idempotency, monotonicity and boundedness properties as follows: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2688 https://internationalpubls.com property 4. (Idempotency) Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ (𝑗 = 1,2, … , 𝑛) be a set of PFN, if 𝛼𝑗 = 𝛼 = (𝜇, 𝜂, 𝜈) then PFSSPOWG(𝛼1, 𝛼2, … , 𝛼𝑛) = 𝛼 property 5. (Monotonicity) Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ and 𝛼𝑗 ′ = ⟨𝜇𝛼𝑗 ′ , 𝜂𝛼𝑗 ′ , 𝜈𝛼𝑗 ′ ⟩ 𝛼𝑗 ′(𝑗 = 1,2, … , 𝑛) as two sets of PFN; and 𝜇𝑗 ≤ 𝜇𝑗 ′ , 𝜂𝑗 ′ ≤ 𝜂𝑗 and 𝜈𝑗 ′ ≤ 𝜈𝑗 then PFSSPOWG(𝛼1, 𝛼2, … , 𝛼𝑛) ≤ PFSSPOWG(𝛼1 ′ , 𝛼2 ′ , … , 𝛼𝑛 ′ ) property 6. (Boundedness) Let 𝛼𝑗 = ⟨𝜇𝛼𝑗 , 𝜂𝛼𝑗 , 𝜈𝛼𝑗 ⟩ (𝑗 = 1,2, … , 𝑛) be a set of PFN, and 𝛼+ = (max𝜇𝑗𝑗=1 𝑛 , min𝜂𝑗𝑗=1 𝑛 , min𝜈𝑗𝑗=1 𝑛 ) , 𝛼− = (min𝜇𝑗𝑗=1 𝑛 , max𝜂𝑗𝑗=1 𝑛 , max𝜈𝑗𝑗=1 𝑛 ) then 𝛼− ≤ 𝑃𝐹𝑆𝑆𝑃𝑂𝑊𝐺(𝛼1, 𝛼2, … , 𝛼𝑛) ≤ 𝛼+ 4. An approach to solving a multi criteria group decision-making problem with the proposed operators Let 𝐴 = {𝐴𝑖}; 𝑖 = 1,2, ⋯ , 𝑚 be a finite set of alternatives and 𝐵 = {𝐵𝑗}; 𝑗 = 1,2, ⋯ , 𝑛 be a finite set of criteria which are represented by PFNs. Let 𝑤 = (𝑤1, 𝑤2, ⋯ , 𝑤𝑛)𝑇 be the weighting vector of the criteria 𝐵 = {𝐵𝑗}; 𝑗 = 1,2, … , 𝑛 with 𝑤 ∈ [0,1]and ∑  𝑛 𝑗=1 𝑤𝑗 = 1. Let 𝜑 = (𝜑1, 𝜑2, ⋯ , 𝜑𝑟)𝑇 be the weighting vector of the decision-makers 𝐶 = {𝐶𝑠}; 𝑠 = 1,2, … , 𝑟 with 𝜑 ∈ [0,1] and ∑  𝑛 𝑠=1 𝜑𝑠 = 1. Now, the procedure for selecting the best alternative using the proposed operators is explained step by step. Decision matrix 𝐶𝑠 = [𝛼𝑖𝑗] 𝑚×𝑛 𝐶𝑠 = [ ⟨𝛼11⟩ ⟨𝛼12⟩ ⋯ ⟨𝛼1𝑛⟩ ⟨𝛼21⟩ ⟨𝛼22⟩ ⋯ ⟨𝛼2𝑛⟩ ⋮ ⋮ ⋱ ⋮ ⟨𝛼𝑚1⟩ ⟨𝛼𝑚2⟩ ⋯ ⟨𝛼𝑚𝑛⟩ ] Step 1: Usually, every decision matrix contained two types of data, like benefit 𝐵1 and cost 𝐵2 type criterias, if the data is cost type criteria, then normalize the SVN decision matrix using the following Equation 10. 𝑟𝑖𝑗 = { 𝛼𝑖𝑗; 𝑗 ∈ 𝐵1 𝛼𝑖𝑗 𝑐 ; 𝑗 ∈ 𝐵2 (10) Where 𝛼𝑖𝑗 𝑐 is the complement 𝛼𝑖𝑗. Step 2: Aggregate all 𝑠 decision matrixs 𝐶𝑠, 𝑠 = 1,2, ⋯ , 𝑟 as provided by 𝑟 experts into a collective decision matrix by employing proposed operators. Step 3: Determine the weight 𝑤𝑗 for each criteria 𝐶𝑗. In this step, the SMART is used to establish the criteria weights, relying on the subjective judgment of decision experts. The decision maker ranks the criteria according to their importance, from least to most significant. The criteria deemed least important is given 10 points, while the most important criteria are assigned 100 points. The Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2689 https://internationalpubls.com remaining criteria receive points in ascending order based on their relative importance. Finally, the weight of each criteria is calculated by normalizing the total points so that they sum to one. Step 4: Aggregate PFNs by using proposed operators. Step 5: Calculate the score value for each {𝐴𝑖}. If 𝑆(𝐴𝑖) = 𝑆(𝐴𝑗) then calculate the accuracy value. Step 6: The best alternative is selected by ranking the alternatives based on their score and accuracy values of 𝐴𝑖. The proposed framework's workflow is illustrated in Figure 4 5 Numerical example The proposed operators are examined through a numerical example from the decision-making field, as outlined below: This section illustrates the selection of a road construction companies to show the applicability of MAGDM based on the proposed operators. Selecting the right road construction company is a crucial step for governments and enterprises to ensure infrastructure projects are completed efficiently and to high standards. Suppose a government agency is responsible for developing new roads in an urban area. These projects often require careful consideration of multiple factors, including Cost 𝐵1, Quality 𝐵2, Time 𝐵3, Flexibility 𝐵4, and Environmental Impact 𝐵5. To evaluate the five companies, 𝐴2, 𝐴2, 𝐴3, 𝐴4, and 𝐴5. By applying the developed algorithm for the best road construction companies, the group decision makers process ensures that all necessary steps are followed to choose the most companies. The weights assigned by the decision makers are specified as 𝜑 = (0.25,0.45,0.3). Gather the decision makers' opinions under PF environment are listed in Table 2,3 and 4 shows the decision matrices. Figure 4: Flowchart for PFMCGDM Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2690 https://internationalpubls.com Table 2: Decision matrix derived from the decision maker 𝑪𝟏 𝐵1 𝐵2 𝐵3 𝐵4 𝐵5 𝐴1 ⟨0.10,0.20,0.40⟩ ⟨0.60,0.05,0.10⟩ ⟨0.50,0.10,0.33⟩ ⟨0.54,0.23,0.14⟩ ⟨0.60,0.20,0.10⟩ 𝐴2 ⟨0.10,0.20,0.40⟩ ⟨0.50,0.10,0.10⟩ ⟨0.80,0.04,0.10⟩ ⟨0.55,0.20,0.15⟩ ⟨0.70,0.10,0.05⟩ 𝐴3 ⟨040,0.10,0.10⟩ ⟨0.10,0.20,0.40⟩ ⟨0.70,0.10,0.02⟩ ⟨0.50,0.10,0.30⟩ ⟨0.30,0.40,0.20⟩ 𝐴4 ⟨0.40,0.20,0.30⟩ ⟨0.50,0.10,0.15⟩ ⟨0.10,0.20,0.40⟩ ⟨0.10,0.20,0.40⟩ ⟨0.50,0.20,0.05⟩ 𝐴5 ⟨0.10,0.20,0.40⟩ ⟨0.20,0.04,0.06⟩ ⟨0.60,0.04,0.05⟩ ⟨0.80,0.03,0.06⟩ ⟨0.25,0.04,0.50⟩ Table 3: Decision matrix derived from the decision maker 𝑪𝟐 𝐵1 𝐵2 𝐵3 𝐵4 𝐵5 𝐴1 ⟨0.10,0.20,0.40⟩ ⟨0.20,0.04,0.06⟩ ⟨0.60,0.04,0.05⟩ ⟨0.80,0.03,0.06⟩ ⟨0.25,0.04,0.50⟩ 𝐴2 ⟨0.40,0.20,0.30⟩ ⟨0.50,0.10,0.15⟩ ⟨0.10,0.20,0.40⟩ ⟨0.10,0.20,0.40⟩ ⟨0.50,0.20,0.05⟩ 𝐴3 ⟨040,0.10,0.10⟩ ⟨0.10,0.20,0.40⟩ ⟨0.70,0.10,0.02⟩ ⟨0.50,0.10,0.30⟩ ⟨0.30,0.40,0.20⟩ 𝐴4 ⟨0.10,0.20,0.40⟩ ⟨0.50,0.10,0.10⟩ ⟨0.80,0.04,0.10⟩ ⟨0.55,0.20,0.15⟩ ⟨0.70,0.10,0.05⟩ 𝐴5 ⟨0.10,0.20,0.40⟩ ⟨0.60,0.05,0.10⟩ ⟨0.50,0.10,0.33⟩ ⟨0.54,0.23,0.14⟩ ⟨0.60,0.20,0.10⟩ Table 4: Decision matrix derived from the decision maker 𝑪𝟑 𝐵1 𝐵2 𝐵3 𝐵4 𝐵5 𝐴1 ⟨0.40,0.20,0.30⟩ ⟨0.50,0.10,0.15⟩ ⟨0.10,0.20,0.40⟩ ⟨0.10,0.20,0.40⟩ ⟨0.50,0.20,0.05⟩ 𝐴2 ⟨0.10,0.20,0.40⟩ ⟨0.20,0.04,0.06⟩ ⟨0.60,0.04,0.05⟩ ⟨0.80,0.03,0.06⟩ ⟨0.25,0.04,0.50⟩ 𝐴3 ⟨040,0.10,0.10⟩ ⟨0.10,0.20,0.40⟩ ⟨0.70,0.10,0.02⟩ ⟨0.50,0.10,0.30⟩ ⟨0.30,0.40,0.20⟩ 𝐴4 ⟨0.10,0.20,0.40⟩ ⟨0.50,0.10,0.10⟩ ⟨0.80,0.04,0.10⟩ ⟨0.55,0.20,0.15⟩ ⟨0.70,0.10,0.05⟩ 𝐴5 ⟨0.10,0.20,0.40⟩ ⟨0.60,0.05,0.10⟩ ⟨0.50,0.10,0.33⟩ ⟨0.54,0.23,0.14⟩ ⟨0.60,0.20,0.10⟩ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2691 https://internationalpubls.com 5.1 Procedure steps for group decision-making Step 1: Given that the criteria are all benefit criteria, there is no need to normalise them. Step 2: To combine decision matrices 𝐶𝑠, 𝑠 = 1,2,3 into a collective decision matrix, the PFSSPWG operator is applied, as shown in Table 5, with the decision maker's weighting vector, 𝜑 = (0.25,0.45,0.43). Table 5: Comprehensive decision matrix 𝑪 using PFSSPWG operator 𝐵1 𝐵2 𝐵3 𝐵4 𝐵5 𝐴1 ⟨0.117,0.200,0.375⟩ ⟨0.130,0.200,0.362⟩ ⟨0.400, .100,0.100⟩ ⟨0.114,0.200,0.380⟩ ⟨0.100,0.200,0.400⟩ 𝐴2 ⟨0.278,0.062,0.099⟩ ⟨0.315,0.084,0.114⟩ ⟨0.100,0.200,0.400⟩ ⟨0.500,0.100,0.113⟩ ⟨0.351,0.048,0.091⟩ 𝐴3 ⟨0.178,0.110,0.272⟩ ⟨0.152,0.118,0.264⟩ ⟨0.700,0.100,0.020⟩ ⟨0.207,0.084,0.203⟩ ⟨0.519,0.087,0.285⟩ 𝐴4 ⟨0.181,0.146,0.225⟩ ⟨0.149,0.159,0.278⟩ ⟨0.500,0.100,0.300⟩ ⟨0.196,0.200,0.235⟩ ⟨0.579,0.194,0.123⟩ 𝐴5 ⟨0.337,0.142,0.351⟩ ⟨0.381,0.137,0.279⟩ ⟨0.300,0.400,0.200⟩ ⟨0.630,0.128,0.050⟩ ⟨0.415,0.171,0.268⟩ Step 3: This step, we calculate the criteria weights using SMART. Experts assign scores ranging from 10 to 100 to each criteria. The normalized criteria weights are then determined by dividing the points assigned to each criteria by the total sum of points. Table 6 displays the points allocated by each expert and the corresponding normalized criteria weights. Table 6: Criteria weights determined by the expert sussing SMART technique. Criteria Points assigned by Sum of Normalized 𝐶1 𝐶2 𝐶3 points weights 𝑤𝑗 𝐵1 80 80 80 240 0.255319 𝐵2 90 90 90 270 0.287234 𝐵3 70 70 70 210 0.223404 𝐵4 40 30 40 110 0.117021 𝐵4 40 40 30 110 0.117021 Total 940 Step 4: Apply the PFSSPWG operator to aggregate all preference values in Table 7 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2692 https://internationalpubls.com Table 7: The overall preference value is calculated using the PFSSPWG operator PFEWG 𝐴1⟨0.131,0.234,0.234⟩ 𝐴2⟨0.184,0.233,0.233⟩ 𝐴3⟨0.204,0.228,0.228⟩ 𝐴4⟨0.199,0.234,0.235⟩ 𝐴5⟨0.360,0.235,0.235⟩ Step 5: Since 𝑆(𝐴1) = −0.336, 𝑆(𝐴2) = −0.282, 𝑆(𝐴3) = −0.252, 𝑆(𝐴4) = −0.269and 𝑆(𝐴5) = −0.110. Step 6: The alternatives are ranked according to preference. The ranking of the alternatives is as follows: 𝐴5 > 𝐴3 > 𝐴4 > 𝐴2 > 𝐴1. 6 Comparative analysis To illustrate the significance and applicability of the results, we compared the proposed MCGDM approach with existing AOs. 6.1 Comparison with existing approaches Table 8 below compares the proposed operators with various existing AOs. Table 8: A comparison of the proposed operators with several existing approaches Operators Parameter Ranking Order PFWG (Wei (2017)) Not considered 𝐴3 > 𝐴5 > 𝐴4 > 𝐴2 > 𝐴1 PFOWG (Wei (2017)) Not considered 𝐴1 > 𝐴2 > 𝐴3 > 𝐴4 > 𝐴5 The Proposed AOs PFSSPWG Considered (𝛽 = −2) 𝐴5 > 𝐴3 > 𝐴4 > 𝐴2 > 𝐴1 PFSSPOWG Considered (𝛽 = −2) 𝐴1 > 𝐴2 > 𝐴3 > 𝐴4 > 𝐴5 Table 8 compares the proposed operators with existing aggregation operators. The suggested operators yield different ranking outcomes compared to the existing ones. Therefore, the proposed operators can serve as a more effective alternative to the existing operators.Based on the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2693 https://internationalpubls.com comparative study above, the proposed approach for tackling decision-making challenges offers several advantages over the existing operators. • As observed in Table 8 the results obtained from various existing approaches are derived in an environment that does not take into account the SS power of the criteria during evaluation. In other words, these approaches assume that decision-makers consider the alternatives being assessed. However, in real-life situations, such conditions are only partially met. • Additionally, some existing operators for PFS are specific cases of the proposed operators. Consequently, it can be concluded that the proposed aggregation operators are more suitable for accurately addressing real-life problems compared to the existing ones. 7. Conclusions This study presents the SMART for addressing MAGDM problems involving PF decision values. Aggregating decision values is a critical initial step in PFMCGDM. We introduce a series of PFSSPG operators, starting with the PFSSPG operator and its desirable properties. We also develop weighted variants, including PFSSPWG and PFSSPOWG operators and examine their fundamental properties. The SMART is employed to determine criteria weights based on decision experts' subjective knowledge. A real-life supplier selection application in the chemical industry demonstrates the practical applicability of the proposed work. A comparative study with existing operators highlights the feasibility and flexibility of our solution methodology. 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