Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators

University of New Mexico S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators S. Annadurai1, R. Sundareswaran2, M. Shanmugapriya3, M. Mohanalakshmi4 1Department of Mathematics, St. Joseph's College of Engineering, India. 2,3Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, India. 4Department of Chemical Engineering, Sri Sivasubramaniya Nadar College of Engineering, India. Abstract: The Linguistic Pythagorean Neutrosophic (LPN) set is a powerful framework for handling uncertainty in assessments by integrating linguistic variables with Pythagorean Neutrosophic numbers (PNNs). In this study, we define new fundamental operations on Linguistic Pythagorean Neutrosophic Numbers (LPNNs) based on Einstein operations and examine their interrelationships. To address the challenges of LPNN fusion, we propose several LPN aggregation operators, namely the LPN Einstein Weighted Averaging (LPNEWA), and LPN Einstein Order Weighted Averaging (LPNEOWG) operators, and investigate their key characteristics. To demonstrate the proposed methodology’s usefulness, we present an illustrative case study in sustainability agriculture. This case study highlights the practicality and effectiveness of the proposed decision-making model. Keywords: Linguistic Pythagorean Neutrosophic set; LPN Einstein Weighted Average Operator, LPN Einstein Order Weighted Average Operator, Multi-Criteria Decision Making i. Introduction In 1998, Smarandache [1] introduced the concept of Neutrosophic sets (𝑁𝑠𝑒𝑡), as an extension of intuitionistic fuzzy sets (𝐼𝐹𝑠𝑒𝑡), which provides a more comprehensive framework for handling uncertainty. Unlike 𝐼𝐹𝑠𝑒𝑡𝑠, those that are characterized by degrees of truth and falsity, 𝑁𝑠𝑒𝑡 incorporates an additional dimension of uncertainty, enabling decision-makers to evaluate problems in terms of independent truth (T), indeterminacy (I), and falsity (F) values. This independence makes 𝑁𝑠𝑒𝑡 a more powerful and generalized mathematical framework for representing and processing vague or imprecise information. Since its inception, researchers have extensively studied [2-5] both the theoretical foundations and applications of 𝑁𝑠𝑒𝑡𝑠. Linguistic variables (𝐿𝑉𝑠) are used to express qualitative evaluations in complex decision-making. Zadeh [6] concept of 𝐿𝑉𝑠 for preference information in fuzzy reasoning gained broad research interest and led to further advancements in decision-making (DM) science. Fang and Ye [7] first introduced linguistic neutrosophic numbers (𝐿𝑁𝑁𝑠), incorporating linguistic values for truth, indeterminacy, and falsity, and enabling the use of Neutrosophic Sets and Systems, Vol. 94, 2025 13 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators all three kinds of linguistic information simultaneously. They further developed score and accuracy functions, along with aggregation operators, for effective decision-making. Recently, many researchers [8-12] have been exploring the applications, enhancements, and integration of 𝐿𝑁𝑁𝑠 into various decision-making frameworks and fuzzy logic systems. Zhao [13] introduced generalized aggregation operators based on 𝐼𝐹𝑠𝑒𝑡𝑠 and showed that the arithmetic aggregation (AA) and geometry aggregation (GA) are special cases of these operators. These operators are derived using the algebraic sum and product of number sets, corresponding to the Archimedes t-conorm and t-norm for defining union and intersection operations. Wang and Liu [14] developed several 𝐼𝐹𝐸𝐴 operators and demonstrated that the Einstein aggregation operator offers better results compared to the AA operator. Zhao and Wei [15] introduced the 𝐼𝐹𝐸𝐻𝐴 and 𝐼𝐹𝐸𝐻𝐺 operators. Guo et al. [16] applied the Einstein operations to hesitant fuzzy sets. Later Li et al. [17] introduced the generalized Neutrosophic number for the Einstein aggregation operator. Recently, numerous researchers [18-20] have been exploring the Neutrosophic Einstein operator and its application in various decision-making processes. When combined with 𝐿𝑁𝑁𝑠 ,the Einstein operators enable effective aggregation of linguistic values involving truth, indeterminacy, and falsity probabilities. This integration enhances decision-making by managing uncertainty and offering smooth aggregation methods, such as weighted or geometric averages. Recently, many researchers [21-23] have focused on the use of the Einstein operator with 𝐿𝑁𝑁𝑠 to manage uncertain or vague data in real-world decision-making settings. 1.2 Motivation Aggregation operators are vital in decision support systems for consolidating information and ranking alternatives. While traditional algebraic T-norm and S-norm operators lack flexibility and robustness, Einstein T-norm and S-norm provide a superior alternative with smooth approximation properties. To enhance decision support systems, we develop Linguistic Pythagorean Neutrosophic Einstein Operators (LPNEO), enabling more effective aggregation of uncertain information. In sustainable agriculture, decision-making is often challenged by imprecise data, conflicting expert opinions, and dynamic environmental conditions. Tasks such as selecting appropriate crop varieties, optimizing resource allocation and infrastructure, or assessing the environmental impact of farming practices typically involve uncertain, incomplete, or ambiguous information. By integrating LPNEO, these challenges are effectively addressed, enabling more accurate handling of uncertainty and vagueness in agricultural decision processes. 1.3 Novelty ➢ This study extends the Einstein T-norm and T-conorm to LPNEO, improving their capability to manage uncertainty and imprecision more effectively. ➢ Establish a Multi-Attribute Group Decision-Making (MAGDM) framework based on the newly introduced Einstein operators, providing a more efficient and accurate approach for decision-making in uncertain environments. 1.4 Objective The key research objectives and contributions of this study are: Neutrosophic Sets and Systems, Vol. 94, 2025 14 ➢ Extending the Einstein T-norm and T-conorm to LPNEO to enhance flexibility and robustness. ➢ Introducing various LPNEOs, including LPN Einstein averaging operators, LPN Einstein geometric operators, and LPN Einstein hybrid operators, while exploring their fundamental properties. ➢ Developing a novel decision-making (DM) method based on the proposed operators to effectively address MAGDM problems in real-world scenarios. 2 Preliminaries In this section, some fundamental concepts related to LPNS have been presented. Definition: 1 Neutrosophic set (𝑁𝑠𝑒𝑡): [1] Let Θ be a universe set. A 𝑁𝑠𝑒𝑡, 𝐴󰆻 on Θ is defined as 𝐴󰆻= {〈𝑥,𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝐹𝐴(𝑥)〉:𝑥∈Θ}, where 𝑇𝐴(𝑥):Θ→−]0,1[+ is said to be the TMF, which represents the degree of confidence, 𝐼𝐴(𝑥):Θ→−]0,1[+is said to be the IMF, which represents the degree of uncertainty, and 𝐹𝐴(𝑥):Θ→−]0,1[+ is said to be the FMF, which represents the degree of skepticism, respectively of the element 𝑥∈Θ in 𝐴𝑁  , such that 0≤𝑇𝐴(𝑥)+𝐼𝐴(𝑥)+𝐹𝐴(𝑥)≤3. Definition: 2 Pythagorean Neutrosophic sets (𝑃𝑁𝑠𝑒𝑡): [2] Let Θ be a universe set. A 𝑃𝑁𝑠𝑒𝑡 𝐴󰆻 on Θ is defined as 𝐴󰆻={〈𝑥,𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝐹𝐴(𝑥)〉:𝑥∈Θ}, such that ( 𝑇𝐴(𝑥))2+( 𝐼𝐴(𝑥))2+( 𝐹𝐴(𝑥))2≤2, where 𝑇𝐴(𝑥):Θ→−]0,1[+ is the TMF, 𝐼𝐴(𝑥):Θ→−]0,1[+is the IMF, and 𝐹𝐴(𝑥):Θ→−]0,1[+ is the FMF. Definition: 3 Linguistic Neutrosophic Set (𝐿𝑁𝑠𝑒𝑡): [3] Let Θ be a universe set. A 𝐿𝑁𝑠𝑒𝑡 in Θ is defined as 𝐴󰆻={〈𝑥,𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝐹𝐴(𝑥)〉:𝑥∈Θ}, where 𝑇𝐴(𝑥):Θ→−]0,1[+ is the LTMF, 𝐼𝐴(𝑥):Θ→−]0,1[+is the LIMF, and 𝐹𝐴(𝑥):Θ→−]0,1[+ is the LFMF. Each membership functions 𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝑎𝑛𝑑 𝐹𝐴(𝑥) takes linguistic values from a predefined linguistic term set 𝑆. Definition: 4 Linguistic Pythagorean Neutrosophic Set (𝐿𝑃𝑁𝑠𝑒𝑡): [24] Let Θ be a universe set. A 𝐿𝑃𝑁𝑠𝑒𝑡 in Θ is defined as 𝐴󰆻={〈𝑥,𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝐹𝐴(𝑥)〉:𝑥∈Θ}, such that ( 𝑇𝐴(𝑥))2+( 𝐼𝐴(𝑥))2+( 𝐹𝐴(𝑥))2≤2, where 𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝑎𝑛𝑑 𝐹𝐴(𝑥) are represented using linguistic terms. Definition: 5 Einstein T-Norm and S-Norm [5]: For arbitrary two real numbers (𝑎,𝑏)∈[0.1], the Einstein sums and product are defined as follows: 𝑆󰆻𝐸(𝑎 ,𝑏 )=𝑎 ⊕𝜖𝑏=𝑎+𝑏 1+𝑎∙ 𝑏 , 𝑇𝐸(𝑎 ,𝑏 )=𝑎 ⊗𝜖𝑏=𝑎∙ 𝑏 1+(1−𝑎)∙(1−𝑏), ∀(𝑎 ,𝑏 )∈[0,1]2. , Garg [24] introduced new different functions for ordering the alternatives using the score function with an accuracy function to build the comparison approach of LPNNs. Definition: 6 Let 𝑣 = (𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) be a LPNN. Then the score function 𝒯 and accuracy function ℋ of 𝒜 are defined as: 𝔗(𝒜) = 𝜁√𝑘2+𝛼12−𝛽12−𝛾12 3 ℌ(𝒜) = 𝜁√𝛼12+𝛽12−𝛾12 Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 15 For comparing two LPNNs A and B, the comparison method is given as: i. if 𝒯(𝒜)>𝒯(ℬ), then 𝒜≻ℬ; ii. if 𝒯(𝒜)=𝒯(ℬ), then ➢ if ℋ(𝒜)<ℋ(ℬ), then 𝒜≺ ℬ; ➢ if ℋ(𝒜)=ℋ(ℬ), then 𝒜∼ ℬ. 3 Einstein Operation of Linguistic Pythagorean Neutrosophic Numbers (LPNNs) Linguistic Pythagorean Neutrosophic Numbers (LPNNs) offer a novel and powerful framework for handling uncertainty, overcoming the limitations of traditional linear approaches. Unlike previous works that focus on fuzzy or standard neutrosophic numbers, LPNNs combine the enhanced flexibility of Pythagorean logic with the interpretability of linguistic terms. The application of Einstein operations, known for their nonlinear, bounded, and smooth aggregation behavior, further strengthens the robustness of this approach in complex decision-making scenarios. In this section, we introduce the Einstein sum (⊕𝜖) and Einstein product (⊗𝜖) operations within the LPNN framework, along with two aggregation operators such as LPN Einstein Weighted Average (LPNEWA) operator, and LPN Einstein Ordered Weighted Average (LPNEOWA) operator. Definition: 7 Let 𝒫=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) and 𝒬=(𝜁𝛼2,𝜁𝛽2,𝜁𝛾2) be two LPNNs and 𝜆≥0, then the Einstein operation of ⊕𝜖 and ⊗𝜖 under the LPNN are defined as follows: i. 𝒫⊕𝜖𝒬=(𝜁𝑡√𝑡2(𝛼12+𝛼22) 𝑡4+𝛼12𝛼22,𝜁𝑡𝛽1𝛽2 √𝑡4+(𝑡2−𝛽12)(𝑡2−𝛽22),𝜁𝑡𝛾1𝛾2 √𝑡4+(𝑡2−𝛾12)(𝑡2−𝛾22)); ii. 𝒫⊗𝜖𝒬=(𝜁𝑡𝛼1𝛼2 √𝑡4+(𝑡2−𝛼12)(𝑡2−𝛼22),𝜁𝑡√𝑡2(𝛽12+𝛽22) 𝑡4+𝛽12𝛽22,𝜁𝑡√𝑡2(𝛾12+𝛾22) 𝑡4+𝛾12𝛾22); iii. 𝜆𝒫= ( 𝜁𝑡√(𝑡2+𝛼12)𝜆−(𝑡2−𝛼12)𝜆 (𝑡2+𝛼12)𝜆+(𝑡2−𝛼12)𝜆,𝜁𝑡√2 𝛽1𝜆 √(2𝑡2−𝛽12)𝜆+(𝛽12)𝜆,𝜁𝑡√2 𝛾1𝜆 √(2𝑡2−𝛾12)𝜆+(𝛾12)𝜆 ) ; iv. 𝒫𝜆= ( 𝜁𝑡√2 𝛼1𝜆 √(2𝑡2−𝛼12)𝜆+(𝛼12)𝜆,𝜁𝑡√(𝑡2+𝛽12)𝜆−(𝑡2−𝛽12)𝜆 (𝑡2+𝛽12)𝜆+(𝑡2−𝛽12)𝜆,𝜁𝑡√(𝑡2+𝛾12)𝜆−(𝑡2−𝛾12)𝜆 (𝑡2+𝛾12)𝜆+(𝑡2−𝛾12)𝜆 ) . Theorem: 1 Let 𝒫=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) and 𝒬=(𝜁𝛼2,𝜁𝛽2,𝜁𝛾2) be two LPNNs and 𝜆1,𝜆2,𝜆3≥0, then the Einstein operation of ⊕𝜖 and ⊗𝜖 have the following performance: i. 𝒫⊕𝜖𝒬=𝒬⊕𝜖𝒫; ii. 𝒫⊗𝜖𝒬=𝒬⊗𝜖𝒫; iii. 𝜆(𝒫⊕𝜖𝒬)=𝜆𝒫⊕𝜖𝜆𝒬; iv. (𝒫⊗𝜖𝒬)𝜆=𝒫𝜆⊗𝜖𝒬𝜆; Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 16 v. (𝜆1⊕𝜖 𝜆2)𝒫=𝜆1𝒫⊕𝜖𝜆2𝒫; vi. 𝒫𝜆1⊗𝜖𝒫𝜆2=𝒫𝜆1+𝜆2. Proof: Performance (𝑖) 𝑎𝑛𝑑 (𝑖𝑖) 𝑎𝑟𝑒 𝑒𝑎𝑠𝑦.𝑆𝑜,𝑤e proves (𝑖𝑖𝑖),𝑎𝑛𝑑 (𝑣). According to Definition 5, we can get 𝒫⊕𝜖𝒬=(𝜁𝑡√𝑡2(𝛼12+𝛼22) 𝑡4+𝛼12𝛼22,𝜁𝑡𝛽1𝛽2 √𝑡4+(𝑡2−𝛽12)(𝑡2−𝛽22),𝜁𝑡𝛾1𝛾2 √𝑡4+(𝑡2−𝛾12)(𝑡2−𝛾22)); =(𝜁𝑡√(𝑡2+𝛼12)(𝑡2+𝛼22)−(𝑡2−𝛼12)(𝑡2−𝛼22) (𝑡2+𝛼12)(𝑡2+𝛼22)+(𝑡2−𝛼12)(𝑡2−𝛼22),𝜁𝑡√2 𝛽12𝛽22 𝛽12𝛽22+(2𝑡2−𝛽12)(2𝑡2−𝛽22),𝜁𝑡√2𝛾12𝛾22 𝛾12𝛾22+(2𝑡2−𝛾12)(2𝑡2−𝛾22)); =(𝜁𝑡√𝑎−𝑏 𝑎+𝑏,𝜁𝑡√2𝑐 𝑐+𝑑,𝜁𝑡√2 𝑒 𝑒+𝑓󰆻) where 𝑎=(𝑡2+𝛼12)(𝑡2+𝛼22),𝑏=(𝑡2−𝛼12)(𝑡2−𝛼22),𝑐=𝛽12𝛽22,𝑑󰆻=(2𝑡2−𝛽12)(2𝑡2−𝛽22),𝑒= 𝛾12𝛾22,𝑓󰆻=(2𝑡2−𝛾12)(2𝑡2−𝛾22). 𝒫⊕𝜖𝒬=𝜆(𝜁𝑡√𝑎−𝑏 𝑎+𝑏,𝜁𝑡√2𝑐 𝑐+𝑑,𝜁𝑡√2 𝑒 𝑒+𝑓󰆻) =(𝜁𝑡√(𝑡2+𝛼12)𝜆(𝑡2+𝛼22)𝜆−(𝑡2−𝛼12)𝜆(𝑡2−𝛼22)𝜆 (𝑡2+𝛼12)𝜆(𝑡2+𝛼22)𝜆+(𝑡2−𝛼12)𝜆(𝑡2−𝛼22)𝜆,𝜁𝑡√2 (𝛽12)𝜆(𝛽22)𝜆 (𝛽12)𝜆(𝛽22)𝜆+(2𝑡2−𝛽12)𝜆(2𝑡2−𝛽22)𝜆,𝜁𝑡√2(𝛾12)𝜆(𝛾22)𝜆 (𝛾12)𝜆(𝛾22)𝜆+(2𝑡2−𝛾12)𝜆(2𝑡2−𝛾22)𝜆) =(𝜁𝑡√ 𝑎 𝜆−𝑏𝜆 𝑎𝜆+𝑏𝜆,𝜁𝑡√2𝑐𝜆 𝑐𝜆+𝑑 𝜆,𝜁 𝑡√2 𝑒𝜆 𝑒𝜆+𝑓󰆻𝜆). Now, 𝜆𝒫= ( 𝜁𝑡√(𝑡2+𝛼12)𝜆−(𝑡2−𝛼12)𝜆 (𝑡2+𝛼12)𝜆+(𝑡2−𝛼12)𝜆,𝜁𝑡√2 (𝛽12)𝜆 (𝛽12)𝜆+(2𝑡2−𝛽12)𝜆,𝜁𝑡√2(𝛾12)𝜆 (𝛾12)𝜆+(2𝑡2−𝛾12)𝜆 ) =(𝜁𝑡√𝑎1−𝑏1 𝑎1+𝑏1,𝜁𝑡√2𝑐1 𝑐1+𝑑1,𝜁𝑡√2 𝑒1 𝑒1+𝑓󰆻1) and 𝜆𝒬=(𝜁𝑡√(𝑡2+𝛼22)𝜆−(𝑡2−𝛼12)𝜆 (𝑡2+𝛼22)𝜆+(𝑡2−𝛼12)𝜆,𝜁𝑡√2 (𝛽22)𝜆 (𝛽22)𝜆+(2𝑡2−𝛽22)𝜆,𝜁𝑡√2(𝛾22)𝜆 (𝛾22)𝜆+(2𝑡2−𝛾22)𝜆)=(𝜁𝑡√𝑎2−𝑏2 𝑎2+𝑏2,𝜁𝑡√2𝑐2 𝑐2+𝑑2,𝜁𝑡√2 𝑒2 𝑒2+𝑓2) then 𝜆𝒫⊕𝜖𝜆𝒬=(𝜁𝑡√𝑎1−𝑏1 𝑎1+𝑏1,𝜁𝑡√2𝑐1 𝑐1+𝑑1,𝜁𝑡√2 𝑒1 𝑒1+𝑓1)⊕𝜖(𝜁𝑡√𝑎2−𝑏2 𝑎2+𝑏2,𝜁𝑡√2𝑐2 𝑐2+𝑑2,𝜁𝑡√2 𝑒2 𝑒2+𝑓2) =(𝜁𝑡√𝑎1𝑎2−𝑏1𝑏2 𝑎1𝑎2+𝑏1𝑏 2,𝜁𝑡√2𝑐1𝑐2 𝑐1𝑐 2+𝑑1𝑑2,𝜁𝑡√2 𝑒1𝑒2 𝑒1𝑒2+𝑓󰆻1𝑓 2) =(𝜁𝑡√(𝑡2+𝛼12)𝜆(𝑡2+𝛼22)𝜆−(𝑡2−𝛼12)𝜆(𝑡2−𝛼22)𝜆 (𝑡2+𝛼12)𝜆(𝑡2+𝛼22)𝜆+(𝑡2−𝛼12)𝜆(𝑡2−𝛼22)𝜆,𝜁𝑡√2 (𝛽12)𝜆(𝛽22)𝜆 (𝛽12)𝜆(𝛽22)𝜆+(2𝑡2−𝛽12)𝜆(2𝑡2−𝛽22)𝜆,𝜁𝑡√2(𝛾12)𝜆(𝛾22)𝜆 (𝛾12)𝜆(𝛾22)𝜆+(2𝑡2−𝛾12)𝜆(2𝑡2−𝛾22)𝜆) Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 17 where 𝑎1=(𝑡2+𝛼12)𝜆,𝑎2=(𝑡2+𝛼22)𝜆,𝑏1=(𝑡2−𝛼12)𝜆,𝑏2=(𝑡2−𝛼22)𝜆,𝑐1=(𝛽12)𝜆,𝑐2=(𝛽22)𝜆,𝑑1=(2𝑡2− 𝛽12)𝜆(2𝑡2−𝛽22)𝜆,𝑑2=(2𝑡2−𝛽22)𝜆,𝑒1=(𝛾12)𝜆,𝑒2=(𝛾22)𝜆 ,𝑓1=(2𝑡2−𝛾12)𝜆,𝑓2=(2𝑡2−𝛾22)𝜆. Hence, we can obtain 𝜆(𝒫⊕𝜖𝒬)=𝜆𝒫⊕𝜖𝜆𝒬. Now, we prove the performance of (𝑣): 𝜆1𝒫= ( 𝜁𝑡√(𝑡2+𝛼12)𝜆1−(𝑡2−𝛼12)𝜆1 (𝑡2+𝛼12)𝜆1+(𝑡2−𝛼12)𝜆1,𝜁𝑡√2 (𝛽12)𝜆1 (𝛽12)𝜆1+(2𝑡2−𝛽12)𝜆1,𝜁𝑡√2(𝛾12)𝜆1 (𝛾12)𝜆1+(2𝑡2−𝛾12)𝜆1 ) =(𝜁𝑡√𝑎1−𝑏1 𝑎1+𝑏1,𝜁𝑡√2𝑐1 𝑐1+𝑑1,𝜁𝑡√2 𝑒1 𝑒1+𝑓1), 𝜆2𝒫= ( 𝜁𝑡√(𝑡2+𝛼12)𝜆2−(𝑡2−𝛼12)𝜆2 (𝑡2+𝛼12)𝜆2+(𝑡2−𝛼12)𝜆2,𝜁𝑡√2 (𝛽12)𝜆2 (𝛽12)𝜆2+(2𝑡2−𝛽12)𝜆2,𝜁𝑡√2(𝛾12)𝜆2 (𝛾12)𝜆2+(2𝑡2−𝛾12)𝜆2 ) =(𝜁𝑡√𝑎1−𝑏1 𝑎1+𝑏1,𝜁𝑡√2𝑐1 𝑐1+𝑑1,𝜁𝑡√2 𝑒1 𝑒1+𝑓1), where 𝑎1=(𝑡2+𝛼12)𝜆1,𝑎2=(𝑡2+𝛼22)𝜆2,𝑏1=(𝑡2−𝛼12)𝜆1,𝑏1=(𝑡2−𝛼22)𝜆2,𝑐1=(𝛽12)𝜆1,𝑐2=(𝛽22)𝜆2,𝑑1= (2𝑡2−𝛽12)𝜆1,𝑑2=(2𝑡2−𝛽22)𝜆2,𝑒1=(𝛾12)𝜆2,𝑒2=(𝛾22)𝜆1 ,𝑓1=(2𝑡2−𝛾12)𝜆1,𝑓2=(2𝑡2−𝛾22)𝜆2. 𝜆1𝒫⊕𝜖𝜆2𝒫=(𝜁𝑡√𝑎1−𝑏1 𝑎1+𝑏1,𝜁𝑡√2𝑐1 𝑐1+𝑑1,𝜁𝑡√2 𝑒1 𝑒1+𝑓1)⊕𝜖(𝜁𝑡√𝑎1−𝑏1 𝑎1+𝑏1,𝜁𝑡√2𝑐1 𝑐1+𝑑1,𝜁𝑡√2 𝑒1 𝑒1+𝑓1) =(𝜁𝑡√𝑎1𝑎2−𝑏1𝑏2 𝑎1𝑎2+𝑏1𝑏2,𝜁𝑡√2𝑐1𝑐1 𝑐1𝑐2+𝑑1𝑑2,𝜁𝑡√2 𝑒1𝑒2 𝑒1𝑒2+𝑓1𝑓2) =(𝜁𝑡√(𝑡2+𝛼12)𝜆1+𝜆2−(𝑡2−𝛼12)𝜆1+𝜆2 (𝑡2+𝛼12)𝜆1+𝜆2+(𝑡2−𝛼12)𝜆1+𝜆2,𝜁𝑡√2 (𝛽12)𝜆1+𝜆2 (𝛽12)𝜆1+𝜆2+(2𝑡2−𝛽12)𝜆1+𝜆2,𝜁𝑡√2(𝛾12)𝜆1+𝜆2 (𝛾12)𝜆1+𝜆2+(2𝑡2−𝛾12)𝜆1+𝜆2)=(𝜆1⊕𝜖𝜆2)𝒫. Hence, 𝜆1𝒫⊕𝜖𝜆2𝒫=(𝜆1⊕𝜖𝜆2)𝒫. 4. LPN Einstein Aggregation Operators 4.1 LPN Einstein weighted average (LPNEWA) operator Definition: 8 Let LPNN 𝒫𝑖=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) in 𝜁, for 𝑖=1,2,3,…𝑛. Then the LPNEWA operator is defined as: LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=𝜔1𝒫1⊕𝜖𝜔2𝒫2⊕𝜖𝜔3𝒫3⊕𝜖…⊕𝜖𝜔𝑛𝒫𝑛, with the weight vector 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1 𝑛𝑖=1 and 𝜔𝑖∈[0,1]. Theorem: 2 Set a collection 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖) in 𝜁, for 𝑖=1,2,3,…𝑛, then the fusion value generated by LPNEWA operator is also a LPNN and LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=(𝜁𝑡√∏(𝑡2+𝛼𝑖2) 𝑛 𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2) 𝑛 𝑖=1 𝜔𝑖 ∏(𝑡2+𝛼𝑖2) 𝑛 𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2) 𝑛 𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖 𝑛 𝑖=1 ∏ (𝛽𝑖2)𝜔𝑖 𝑛 𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖 𝑛 𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖 𝑛 𝑖=1 ∏ (𝛾𝑖2)𝜔𝑖 𝑛 𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖 𝑛 𝑖=1 ) with the weight vector 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1 𝑛𝑖=1 and 𝜔𝑖∈[0,1]. Proof: When 𝑛=2,LPNEWA(𝒫1,𝒫2)=𝜔1𝒫1⊕𝜖𝜔2𝒫2. By definition 5 , we get Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 18 𝜔1𝒫1=(𝜁𝑡√(𝑡2+𝛼12)𝜔1−(𝑡2−𝛼12)𝜔1 (𝑡2+𝛼12)𝜔1+(𝑡2−𝛼12)𝜔1,𝜁𝑡√2 (𝛽12)𝜔1 (𝛽12)𝜔1+(2𝑡2−𝛽12)𝜔1,𝜁𝑡√2(𝛾12)𝜔1 (𝛾12)𝜔1+(2𝑡2−𝛾12)𝜔1), 𝜔2𝒫2=(𝜁𝑡√(𝑡2+𝛼22)𝜔2−(𝑡2−𝛼22)𝜔2 (𝑡2+𝛼22)𝜔2+(𝑡2−𝛼22)𝜔2,𝜁𝑡√2 (𝛽22)𝜔2 (𝛽22)𝜔2+(2𝑡2−𝛽22)𝜔2,𝜁𝑡√2(𝛾22)𝜔2 (𝛾22)𝜔2+(2𝑡2−𝛾22)𝜔2). 𝜔1𝒫1⊕𝜖𝜔2𝒫2 = ( 𝜁𝑡 √ (𝑡2+𝛼12)𝜔1−(𝑡2−𝛼12)𝜔1 (𝑡2+𝛼12)𝜔1+(𝑡2−𝛼12)𝜔1+(𝑡2+𝛼22)𝜔2−(𝑡2−𝛼22)𝜔2 (𝑡2+𝛼22)𝜔2+(𝑡2−𝛼22)𝜔2 1+((𝑡2+𝛼12)𝜔1−(𝑡2−𝛼12)𝜔1 (𝑡2+𝛼12)𝜔1+(𝑡2−𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2−(𝑡2−𝛼22)𝜔2 (𝑡2+𝛼22)𝜔2+(𝑡2−𝛼22)𝜔2),𝜁𝑡 √ (2 (𝛽12)𝜔1 (𝛽12)𝜔1+(2𝑡2−𝛽12)𝜔1)∙( 2 (𝛽22)𝜔2 (𝛽22)𝜔2+(2𝑡2−𝛽22)𝜔2) 1+((𝑡2−2 (𝛽12)𝜔1 (𝛽12)𝜔1+(2𝑡2−𝛽12)𝜔1)∙ 2 (𝛽22)𝜔2 (𝛽22)𝜔2+(2𝑡2−𝛽22)𝜔2),𝜁𝑡 √ (2 (𝛾12)𝜔1 (𝛾12)𝜔1+(2𝑡2−𝛾12)𝜔1)∙( 2 (𝛾22)𝜔2 (𝛾22)𝜔2+(2𝑡2−𝛾22)𝜔2) 1+((𝑡2−2 (𝛾12)𝜔1 (𝛾12)𝜔1+(2𝑡2−𝛾12)𝜔1)∙ 2 (𝛾22)𝜔2 (𝛾22)𝜔2+(2𝑡2−𝛾22)𝜔2) ) = ( 𝜁𝑡√((𝑡2+𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2)−((𝑡2+𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2) ((𝑡2+𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2)+((𝑡2+𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2),𝜁𝑡√2 (𝛽12)𝜔1∙(𝛽22)𝜔2 (𝛽12)𝜔1(𝛽22)𝜔2+(2𝑡2−𝛽12)𝜔1∙(2𝑡2−𝛽22)𝜔2,𝜁𝑡√2 (𝛾12)𝜔1∙(𝛾22)𝜔2 (𝛾12)𝜔1(𝛽𝛾22)𝜔2+(2𝑡2−𝛾12)𝜔1∙(2𝑡2−𝛾22)𝜔2 ) Hence,LPNEWA(𝒫1,𝒫2)=𝜔1𝒫1⊕𝜖𝜔2𝒫2,𝑣𝑎𝑙𝑖𝑑 𝑓𝑜𝑟 𝑛=2. When the consequence is valid for 𝑛=𝑘, we have LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑘)= ( 𝜁𝑡√∏(𝑡2+𝛼𝑖2) 𝑘𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2) 𝑘𝑖=1 𝜔𝑖 ∏(𝑡2+𝛼𝑖2) 𝑘𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2) 𝑘𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖 𝑘𝑖=1 ∏ (𝛽𝑖2)𝜔𝑖 𝑘𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖 𝑘𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖 𝑘𝑖=1 ∏ (𝛾𝑖2)𝜔𝑖 𝑘𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖 𝑘𝑖=1 ) . When 𝑛=𝑘+1, we have LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑘+1)=LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑘⊕𝜖𝜔𝑘+1𝒫𝑘+1) = ( 𝜁𝑡√∏(𝑡2+𝛼𝑖2) 𝑘𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2) 𝑘𝑖=1 𝜔𝑖 ∏(𝑡2+𝛼𝑖2) 𝑘𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2) 𝑘𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖 𝑘𝑖=1 ∏ (𝛽𝑖2)𝜔𝑖 𝑘𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖 𝑘𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖 𝑘𝑖=1 ∏ (𝛾𝑖2)𝜔𝑖 𝑘𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖 𝑘𝑖=1 ) ⊕𝜖 ( 𝜁𝑡√(𝑡2+𝛼𝑘+1 2)𝜔𝑘+1−(𝑡2−𝛼𝑘+1 2)𝜔𝑘+1 (𝑡2+𝛼𝑘+1 2)𝑘+1+(𝑡2−𝛼𝑘+1 2)𝜔𝑘+1,𝜁𝑡√2 (𝛽𝑘+1 2)𝜔𝑘+1 (𝛽𝑘+1 2)𝜔𝑘+1+(2𝑡2−𝛽𝑘+1 2)𝜔𝑘+1,𝜁𝑡√2(𝛾𝑘+1 2)𝜔𝑘+1 (𝛾𝑘+1 2)𝜔𝑘+1+(2𝑡2−𝛾𝑘+1 2)𝜔𝑘+1 ) , = ( 𝜁𝑡√∏(𝑡2+𝛼𝑖2) 𝑘+1 𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2) 𝑘+1 𝑖=1 𝜔𝑖 ∏(𝑡2+𝛼𝑖2) 𝑘+1 𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2) 𝑘+1 𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖 𝑘+1 𝑖=1 ∏ (𝛽𝑖2)𝜔𝑖 𝑘+1 𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖 𝑘+1 𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖 𝑘+1 𝑖=1 ∏ (𝛾𝑖2)𝜔𝑖 𝑘+1 𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖 𝑘+1 𝑖=1 ) . Therefore, LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛) holds for any 𝑛. Hence, Theorem 2 is proved. Theorem: 3 Set a collection 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖), 𝒬𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)in 𝜁, for 𝑖=1,2,3,…𝑛, be two LPNNs with the weight vector 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1 𝑛𝑖=1 and 𝜔𝑖∈[0,1]. We can deduce the following properties: i. 𝐼𝑑𝑒𝑚𝑝𝑜𝑡𝑒𝑛𝑐𝑦: 𝐼𝑓 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)=(𝜁𝛼,𝜁𝛽,𝜁𝛾) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖,𝑡ℎ𝑒𝑛 Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 19 LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=(𝜁𝛼,𝜁𝛽,𝜁𝛾). ii. 𝑀𝑜𝑛𝑜𝑡𝑜𝑛𝑖𝑐𝑖𝑡𝑦:𝐼𝑓 𝒫𝑖≤ 𝒬𝑖,𝑡ℎ𝑎𝑡 𝑖𝑠,𝜁𝛼𝑖≤𝜁𝛼𝑖,𝜁𝛽𝑖≥𝜁𝛽𝑖,𝑎𝑛𝑑 𝜁𝛾𝑖≥𝜁𝛾𝑖,𝑡ℎ𝑒𝑛 𝑤𝑒 ℎ𝑎𝑣𝑒 LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤LPNEWA(𝒬1,𝒬2,𝒬3,…𝒬𝑛). iii. Boundedness: Suppose 𝒫−=𝑚𝑖𝑛(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝒫+=𝑚𝑎𝑥(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝑡ℎ𝑒𝑛 𝒫−≤LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤ 𝒫+. Proof: Let 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖), 𝒬𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)in 𝜁, for 𝑖=1,2,3,…𝑛, be two collections of LPNNs. Then i. 𝑤ℎ𝑒𝑛 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)=(𝜁𝛼,𝜁𝛽,𝜁𝛾) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖,𝑜𝑛𝑒 ℎ𝑎𝑠 LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛) = ( 𝜁𝑡√∏(𝑡2+𝛼𝑖2) 𝑛 𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2) 𝑛 𝑖=1 𝜔𝑖 ∏(𝑡2+𝛼𝑖2) 𝑛 𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2) 𝑛 𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖 𝑛 𝑖=1 ∏ (𝛽𝑖2)𝜔𝑖 𝑛 𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖 𝑛 𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖 𝑛 𝑖=1 ∏ (𝛾𝑖2)𝜔𝑖 𝑛 𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖 𝑛 𝑖=1 ) , = ( 𝜁𝑡√(𝑡2+𝛼𝑖2)∏𝜔𝑖 𝑛 𝑖=1 −(𝑡2+𝛼𝑖2)∏𝜔𝑖 𝑛 𝑖=1 (𝑡2+𝛼𝑖2)∏𝜔𝑖 𝑛 𝑖=1 +(𝑡2+𝛼𝑖2)∏𝜔𝑖 𝑛 𝑖=1 ,𝜁𝑡√2(𝛽𝑖2)∏ 𝜔𝑖 𝑘𝑖=1 (𝛽𝑖2)∏ 𝜔𝑖 𝑘𝑖=1 +(2𝑡2−𝛽𝑖2)∏ 𝜔𝑖 𝑘𝑖=1 ,𝜁𝑡√2(𝛾𝑖2)∏ 𝜔𝑖 𝑘𝑖=1 (𝛾𝑖2)∏ 𝜔𝑖 𝑘𝑖=1 +(2𝑡2−𝛾𝑖2)∏ 𝜔𝑖 𝑘𝑖=1 ) , =(𝜁𝑡(𝛼𝑖 𝑡),𝜁𝑡(𝛽𝑖 𝑡),𝜁𝑡(𝛾𝑖 𝑡))=𝒫𝑖. ii. For 𝒫𝑖≤ 𝒬𝑖,𝑡ℎ𝑒𝑛 𝜔𝑖𝒫𝑖≤ 𝜔𝑖𝒬𝑖. So, we can obtain ⊕𝜖𝑖=1 𝑛𝜔𝑖𝒫𝑖≤ ⊕𝜖𝑖=1 𝑛𝜔𝑖𝒬𝑖. For LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=⊕𝜖𝑖=1 𝑛𝜔𝑖𝒫𝑖, 𝑎𝑛𝑑 LPNEWA(𝒬1,𝒬2,𝒬3,…𝒬𝑛)=⊕𝜖𝑖=1 𝑛𝜔𝑖𝒬𝑖, then we can get LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤ LPNEWA(𝒬1,𝒬2,𝒬3,…𝒬𝑛). iii. Since 𝒫−=𝑚𝑖𝑛(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝒫+=𝑚𝑎𝑥(𝒫1,𝒫2,𝒫3,…𝒫𝑛). By the previous proof (𝑖𝑖), we have LPNEWA(𝒫−,𝒫−,𝒫−,…𝒫−)≤LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤LPNEWA(𝒫+,𝒫+,𝒫+,…𝒫+). In addition, by the previous proof (𝑖), we have LPNEWA(𝒫+,𝒫+,𝒫+,…𝒫+)=𝒫+,𝑎𝑛𝑑 LPNEWA(𝒫−,𝒫−,𝒫−,…𝒫−)=𝒫−. From all the above, we can get 𝒫−≤LPNEWA(𝒫+,𝒫+,𝒫+,…𝒫+)=𝒫+. 4.2 LPN Einstein order weighted average (LPNEOWA) operator Definition: 9 Set a LPNNs 𝒫𝑖=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) in 𝜁, for 𝑖=1,2,3,…𝑛, then the LPNEOWA operator is defined as: LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=𝜔1𝒫𝜌(1)⊕𝜖𝜔2𝒫𝜌(2)⊕𝜖𝜔3𝒫𝜌(3)⊕𝜖…⊕𝜖𝜔𝑛𝒫𝜌(𝑛), where (𝜌(1),𝜌(2),𝜌(3),…𝜌(𝑛)) is a permutation of (𝑖=1,2,3,…𝑛),𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝒫𝜌(𝑖−1)≥𝒫𝜌(𝑖) for each 𝑖, with the weight vector 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1 𝑛𝑖=1 and 𝜔𝑖∈[0,1]. Theorem: 4 Set a collection 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖) in 𝜁, for 𝑖=1,2,3,…𝑛, then the fusion result by LPNEOWA operator is obtained as: Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 20 LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)= ( 𝜁𝑡√∏(𝑡2+𝛼𝜌(𝑖) 2) 𝑛 𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝜌(𝑖) 2) 𝑛 𝑖=1 𝜔𝑖 ∏(𝑡2+𝛼𝜌(𝑖) 2) 𝑛 𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝜌(𝑖) 2) 𝑛 𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝜌(𝑖) 2)𝜔𝑖 𝑛 𝑖=1 ∏ (𝛽𝜌(𝑖) 2)𝜔𝑖 𝑛 𝑖=1 +∏ (2𝑡2−𝛽𝜌(𝑖) 2)𝜔𝑖 𝑛 𝑖=1 ,𝜁𝑡√2∏ (𝛾𝜌(𝑖) 2)𝜔𝑖 𝑛 𝑖=1 ∏ (𝛾𝜌(𝑖) 2)𝜔𝑖 𝑛 𝑖=1 +∏ (2𝑡2−𝛾𝜌(𝑖) 2)𝜔𝑖 𝑛 𝑖=1 ) , where (𝜌(1),𝜌(2),𝜌(3),…𝜌(𝑛)) is a permutation of (𝑖=1,2,3,…𝑛),𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝒫𝜌(𝑖−1)≥𝒫𝜌(𝑖) for each 𝑖, with the weight vector 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1 𝑛𝑖=1 and 𝜔𝑖∈[0,1]. Evidently, if 𝜔= (1𝑛,1𝑛,1𝑛,…,1𝑛 ),𝑡ℎ𝑒 LPNEOWA operator will reduce to LPNWA operator. Theorem: 5 Set a collection 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖), 𝒬𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)in 𝜁, for 𝑖=1,2,3,…𝑛, be two LPNNs with the weight vector 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1 𝑛𝑖=1 and 𝜔𝑖∈[0,1]. We can deduce the following properties: i. 𝐼𝑑𝑒𝑚𝑝𝑜𝑡𝑒𝑛𝑐𝑦: 𝐼𝑓 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)=(𝜁𝛼,𝜁𝛽,𝜁𝛾) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖,𝑡ℎ𝑒𝑛 LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=(𝜁𝛼,𝜁𝛽,𝜁𝛾). ii. 𝑀𝑜𝑛𝑜𝑡𝑜𝑛𝑖𝑐𝑖𝑡𝑦:𝐼𝑓 𝒫𝑖≤ 𝒬𝑖,𝑡ℎ𝑎𝑡 𝑖𝑠,𝜁𝛼𝑖≤𝜁𝛼𝑖,𝜁𝛽𝑖≥𝜁𝛽𝑖,𝑎𝑛𝑑 𝜁𝛾𝑖≥𝜁𝛾𝑖,𝑡ℎ𝑒𝑛 LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤LPNEWOA(𝒬1,𝒬2,𝒬3,…𝒬𝑛). iii. Boundedness: Suppose 𝒫−=𝑚𝑖𝑛(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝒫+=𝑚𝑎𝑥(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝑡ℎ𝑒𝑛 𝒫−≤LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤ 𝒫+. iv. Commutativity: 𝒬𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖) (𝑖=1,2,3,…𝑛) is any permutation of 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖), then LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=LPNEWOA(𝒬1,𝒬2,𝒬3,…𝒬𝑛). The proof is similar to that of Theorem 3; therefore, we omit it here. 4.3 LPN Einstein weighted geometry (LPNEWG) operator Definition: 10 Let LPNNs 𝒫𝑖=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) in 𝜁, for 𝑖=1,2,3,…𝑛. Then the LPNEWG operator is defined as: LPNEWG (𝒫1,𝒫2,𝒫3,…𝒫𝑛)=𝒫1𝜔1⊗𝜖𝒫2𝜔2⊗𝜖…𝒫𝑛𝜔𝑛, with the weight vector 𝜔= (𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1 𝑛𝑖=1 and 𝜔𝑖∈[0,1]. Theorem: 6 Set a collection 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖) in 𝜁, for 𝑖=1,2,3,…𝑛, then the fusion value generated by LPNEWG operator is also a LPNN and LPNEWG(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=( 𝜁𝑡√2∏ (𝛼𝑖2)𝜔𝑖 𝑛 𝑖=1 ∏ (𝛼𝑖2)𝜔𝑖 𝑛 𝑖=1 +∏ (2𝑡2−𝛼𝑖2)𝜔𝑖 𝑛 𝑖=1 ,𝜁𝑡√∏(𝑡2+𝛽𝑖2) 𝑛 𝑖=1 𝜔𝑖−∏(𝑡2−𝛽𝑖2) 𝑛 𝑖=1 𝜔𝑖 ∏(𝑡2+𝛽𝑖2) 𝑛 𝑖=1 𝜔𝑖+∏(𝑡2−𝛽𝑖2) 𝑛 𝑖=1 𝜔𝑖,𝜁𝑡√∏(𝑡2+𝛾𝑖2) 𝑛 𝑖=1 𝜔𝑖−∏(𝑡2−𝛾𝑖2) 𝑛 𝑖=1 𝜔𝑖 ∏(𝑡2+𝛾𝑖2) 𝑛 𝑖=1 𝜔𝑖+∏(𝑡2−𝛾𝑖2) 𝑛 𝑖=1 𝜔𝑖) with the weight vector 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1 𝑛𝑖=1 and 𝜔𝑖∈[0,1]. Proof: When 𝑛=2,LPNEWA(𝒫1,𝒫2)=𝒫1𝜔1⊗𝜖𝒫2𝜔2. By definition 5, we get 𝒫1𝜔1=(𝜁𝑡√2(𝛼12)𝜔1 (𝛼12)𝜔1(2𝑡2−𝛼12)𝜔1,𝜁𝑡√(𝑡2+𝛽12)𝜔1−(𝑡2−𝛽12)𝜔1 (𝑡2+𝛽12)𝜔1+(𝑡2−𝛽12)𝜔1,𝜁𝑡√(𝑡2+𝛾12)𝜔1−(𝑡2−𝛾12)𝜔1 (𝑡2+𝛾12)𝜔1+(𝑡2−𝛾12)𝜔1), 𝒫2𝜔2=(𝜁𝑡√2(𝛼22)𝜔2 (𝛼22)𝜔2(2𝑡2−𝛼22)𝜔2,𝜁𝑡√(𝑡2+𝛽22)𝜔2−(𝑡2−𝛽22)𝜔2 (𝑡2+𝛽22)𝜔2+(𝑡2−𝛽22)𝜔2,𝜁𝑡√(𝑡2+𝛾22)𝜔2−(𝑡2−𝛾22)𝜔2 (𝑡2+𝛾22)𝜔2+(𝑡2−𝛾22)𝜔2) Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 27 SDG2 focuses on ending hunger, achieving food security and improved nutrition and promoting sustainable agriculture. In the Indian agriculture sector, companies like Godrej Agrovet, AgNext Technologies, and Coromandel International are known for their sustainability initiatives and focus on sustainable farming practices. In this work, we consider four of these companies ℭ=(ℭ1,ℭ2,ℭ3,ℭ4) in India depends on their product selling strategies for achieving sustainability in agriculture. based on these factors how the companies can maintain their sustainability in the agriculture field. There are decision makers/ experts 𝔇=(𝔇1,𝔇2,𝔇3) are invited to evaluate according to six factors based on the company’s performance with weight vector is 𝓌 = (0.37,0.33,0.3) . The evaluations based on the experts make evaluations on the four alternative factors 𝒮ℱ=(𝒮ℱ1, 𝒮ℱ2, 𝒮ℱ3, 𝒮ℱ4) with the weight vector 𝒲𝑓= (0.26,024,0.21,0.29). Now, the experts use LPNNs to make the evaluation values with a linguistic set 𝜁 = {𝜁0 = 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑝𝑜𝑜𝑟,𝜁1 = 𝑣𝑒𝑟𝑦 𝑝𝑜𝑜𝑟,𝜁2= 𝑝𝑜𝑜𝑟,𝜁3= 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑝𝑜𝑜𝑟,𝜁4= 𝑚𝑒𝑑𝑖𝑢𝑚 ,𝜁5= 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑔𝑜𝑜𝑑,𝜁6= 𝑔𝑜𝑜𝑑,𝜁7= 𝑣𝑒𝑟𝑦 𝑔𝑜𝑜𝑑,𝜁8 = 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑔𝑜𝑜𝑑}. The decision evaluation matrix are given below (tables 1– 4). Table 1: The first decision maker 𝔇1 gives the following values in the matrix form 𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒 ℭ1 (𝜁6,𝜁1,𝜁3) (𝜁7,𝜁1,𝜁3) (𝜁8,𝜁1,𝜁3) (𝜁5,𝜁2,𝜁3) ℭ2 (𝜁6,𝜁2,𝜁3) (𝜁6,𝜁7,𝜁5) (𝜁6,𝜁6,𝜁3) (𝜁5,𝜁3,𝜁3) ℭ3 (𝜁6,𝜁3,𝜁3) (𝜁6,𝜁4,𝜁3) (𝜁6,𝜁1,𝜁6) (𝜁5,𝜁3,𝜁3) ℭ4 (𝜁6,𝜁2,𝜁2) (𝜁6,𝜁1,𝜁5) (𝜁8,𝜁1,𝜁3) (𝜁6,𝜁3,𝜁3) Table 2: The second decision maker 𝔇2 gives the following values in the matrix form 𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒 ℭ1 (𝜁6,𝜁2,𝜁3) (𝜁7,𝜁4,𝜁3) (𝜁6,𝜁1,𝜁3) (𝜁6,𝜁3,𝜁3) ℭ2 (𝜁6,𝜁2,𝜁3) (𝜁7,𝜁1,𝜁4) (𝜁6,𝜁1,𝜁3) (𝜁7,𝜁2,𝜁3) ℭ3 (𝜁5,𝜁3,𝜁3) (𝜁5,𝜁2,𝜁4) (𝜁6,𝜁1,𝜁3) (𝜁8,𝜁3,𝜁3) ℭ4 (𝜁6,𝜁4,𝜁3) (𝜁5,𝜁4,𝜁3) (𝜁6,𝜁1,𝜁3) (𝜁5,𝜁2,𝜁3) Table 3: The third decision maker 𝔇3 gives the following values in the matrix form 𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒 ℭ1 (𝜁7,𝜁1,𝜁3) (𝜁6,𝜁2,𝜁5) (𝜁3,𝜁3,𝜁3) (𝜁8,𝜁1,𝜁3) ℭ2 (𝜁6,𝜁4,𝜁3) (𝜁6,𝜁2,𝜁5) (𝜁5,𝜁5,𝜁5) (𝜁6,𝜁2,𝜁2) ℭ3 (𝜁6,𝜁1,𝜁4) (𝜁6,𝜁5,𝜁3) (𝜁6,𝜁5,𝜁3) (𝜁6,𝜁2,𝜁3) ℭ4 (𝜁6,𝜁1,𝜁5) (𝜁6,𝜁5,𝜁3) (𝜁4,𝜁4,𝜁3) (𝜁6,𝜁3,𝜁3) Based on the 𝐋𝐏𝐍𝐄𝐖𝐀 and 𝐋𝐏𝐍𝐄𝐖𝐆 operators, we solve the above decision-making problem in the following manner and the obtained values are in Table 5 and Table 6. Table 5. The overall decision matrix Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 28 𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒 ℭ1 (𝜁6.0397,𝜁1.1230,𝜁3) (𝜁6.6098,𝜁1.4128,𝜁3.5219) (𝜁8,𝜁1.1862,𝜁3) (𝜁8,𝜁1.2946,𝜁3) ℭ2 (𝜁5.7547,𝜁1.17956,𝜁3) (𝜁6.2061,𝜁2.6956,𝜁4.6548) (𝜁5.5650,𝜁2.5979,𝜁3.5219) (𝜁5.3915,𝜁1.6758,𝜁2.6612) ℭ3 (𝜁5.4334,𝜁1.81.42,𝜁3.2762) (𝜁5.4334,𝜁2.4168,𝜁3.3049) (𝜁5.7547,𝜁1.3046,𝜁3.9515) (𝜁5.3915,𝜁1.6758,𝜁3) ℭ4 (𝜁5.7547,𝜁1.6443,𝜁3.0485) (𝜁5.4334,𝜁1.6460,𝜁3.6536) (𝜁8,𝜁1.2478,𝜁3) (𝜁5.4334,𝜁1.9888,𝜁3) Step 2: The total collective LPNN ℭ𝑖 (𝑖 = 1,2, …, 𝑚) can be obtained by the LPNEWA operator: ℭ1=(𝜁8,𝜁2.3655,𝜁3.1190);ℭ2=(𝜁5.0618,𝜁3.1022,𝜁3.3464); ℭ3=(𝜁4.7291,𝜁3.2301,𝜁3.3319); ℭ4= (𝜁7.6441,𝜁3.0416,𝜁3.1604) Step 3: By using definition 5, we calculate the expected values of 𝔗(ℭ𝑖) for ℭ𝑖 (𝑖 = 1,2,3,4) 𝔗(ℭ1)=6.1285; 𝔗(ℭ2)= 4.7889 ;𝔗(ℭ3)=4.6486 ; 𝔗(ℭ4)=5.8649. Based on the expected values, four alternatives can be ranked ℭ1 ≻ ℭ4≻ ℭ2≻ ℭ3,. Thus, company ℭ3 is the optimal choice. Now, we find the optimal choice using the LPNEWG operator. Table 6. The overall decision matrix 𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒 ℭ1 (𝜁3.5005,𝜁1.3752,𝜁2.848) (𝜁3.7147,𝜁2.558,𝜁3.391) (𝜁3.589,𝜁1.585,𝜁3.391) (𝜁3.392,𝜁1.418,𝜁2.848) ℭ2 (𝜁3.4122,𝜁2.4606,𝜁2.848) (𝜁3.5091,𝜁4.968,𝜁4.447) (𝜁3.327,𝜁4.471,𝜁3.391) (𝜁3.197,𝜁2.116,𝜁2.667) ℭ3 (𝜁3.3187,𝜁2.5533,𝜁3.089) (𝜁3.3187,𝜁3.534,𝜁4.447) (𝜁3.412,𝜁2.442,𝜁4.388) (𝜁3.197,𝜁2.116,𝜁2.848) ℭ4 (𝜁3.4122,𝜁2.6547,𝜁3.111) (𝜁3.3187,𝜁3.307,𝜁3.785) (𝜁3.685,𝜁1.991,𝜁2.848) (𝜁3.319,𝜁2.542,𝜁2.848) Step 2: The total collective LPNN ℭ𝑖 (𝑖 = 1,2, …, 𝑚) can be obtained by the LPNEWA operator: ℭ1=(𝜁4.8605,𝜁1.5982,𝜁2.593);ℭ2=(𝜁4.6882,𝜁3.4566,𝜁3.093); ℭ3=(𝜁4.6193,𝜁2.4321,𝜁3.054); ℭ4= (𝜁4.7284,𝜁2.2984,𝜁2.803) Step 3: By using definition 5, we calculate the expected values of 𝔗(ℭ𝑖) for ℭ𝑖 (𝑖 = 1,2,3,4) 𝔗(ℭ1)=5.1103; 𝔗(ℭ2)= 4.6356 ;𝔗(ℭ3)=4.8338 ; 𝔗(ℭ4)=4.9402. Based on the expected values, four alternatives can be ranked ℭ1 ≻ ℭ4≻ ℭ3≻ ℭ2. Thus, company ℭ2 is the optimal choice. 6.2 Comparative Analysis We compare the proposed LPNEWA and LPNEWG methods with other LIFEWA and LPFEWA approaches. The results of this comparison are presented in Figure 4. From Figure 4, it is evident that alternatives ℭ3 and ℭ2 emerge as the most optimal choices when evaluated using the LPNEWA and LPNEWG methods. The ranking orders produced by these two methods are: ℭ1 ≻ ℭ4≻ ℭ2≻ ℭ3 for LPNEWA, and ℭ1 ≻ ℭ4≻ ℭ3≻ ℭ2. for LPNEWG. To validate the effectiveness of the proposed method, a comparison is made with existing approaches, including the linguistic intuitionistic fuzzy weighted average (LIFWA) operator introduced by Chen et al. [25], the LPF weighted average (LPFWA) operator developed by Garg [26], Sine Single-Valued Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators 29 Neutrosophic Einstein Weighted Averaging (S-SvNVEWA) and Sine Single-Valued Neutrosophic Einstein Weighted Geometric (S-SvNVEWG) aggregation operators developed by Zhang et al. [27]. Unlike these earlier methods [25–27], the proposed LPNN-based approach can effectively represent and handle purely linguistic evaluation values—something that traditional MCDM methods cannot achieve. By integrating LPNS with Einstein operations, the proposed method clearly demonstrates its flexibility and effectiveness. Fig. 4. Comparative analysis of different MCDM methods 7. Conclusion This paper proposed a novel approach to solving MCDM problems. Initially, the Einstein operation was applied to Linguistic Pythagorean Neutrosophic Numbers (LPNNs), and new operational rules were established based on this operator. Subsequently, several aggregation operators were integrated with the LPNNs to define the Linguistic Pythagorean Neutrosophic Einstein Weighted Average (LPNEWA) operator and the Linguistic Pythagorean Neutrosophic Einstein Weighted Geometric (LPNEWG) operator, in accordance with the newly developed rules. Using the LPNEWA and LPNEWG operators, two methods were introduced to effectively address MCDM problems. To demonstrate the practicality and benefits of the proposed methods, they were applied to a real-world example. Acknowledgments: The authors wish to express gratitude to the Management, Principal, Sri Sivasubramaniya Nadar College of Engineering, Chennai, India. Conflicts of Interest: The authors declare no conflicts of interest. Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. 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Evaluation of Practice-Based Curriculum Objectives Achievement Degree Using Einstein Aggregation Operators of Single-Valued Neutrosophic Credibility Numbers. Neutrosophic Sets and Systems, 2025, 86, 159-170. Received: May 11, 2025. Accepted: Aug 24, 2025 Neutrosophic Sets and Systems, Vol. 94, 2025 S. Annadurai, R. Sundareswaran, M. Shanmugapriya, M. Mohanalakshmi , Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators