Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number

__________________________________________________________________________________________________ Memet Şahin, and Kübra Doğan, Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number Neutrosophic Sets and Systems, Vol. 97, 2026 University of New Mexico Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number Memet Şahin1*, and Kübra Doğan2 1* Department of Mathematics, Gaziantep University, Gaziantep 27310, Turkey. [email protected] 2 Department of Mathematics, Gaziantep University, Gaziantep 27310, Turkey. [email protected] * Correspondence: [email protected]; Tel.:+905432182646 Abstract: In this paper, we define a generalized Euclidean distance measure based on generalized setvalued neutrosophic quintuple numbers. It is shown that the Euclidean distance measure satisfies the distance conditions. Furthermore, an algorithm based on generalized set-valued neutrosophic quintuple numbers, and the Euclidean distance measure is given. Furthermore, a multi-criteria decisionmaking application for this generalized algorithm is presented. Finally, a real example from daily life is presented to demonstrate the applicability and usability of the proposed multi-criteria decisionmaking method and compared with different methods. In this application, we have examined some plants with high economic return and ranging from the food sector to the pharmacology sector. We have examined which or which of these plants grow best under the criteria we have determined in the cities in our sample, where these plants are not known to grow. Thus, we can say that this application can be used in plant selection due to its result and structure. Keywords: Generalized Set Valued Neutrosophic Quintuple Set, Distance Measures, Generalized Euclidean Distance Measure, Decision Making Applications 1. Introduction Many uncertainties arise in everyday life. Most of the time Aristotelian logic (classical logic) fails to explain these uncertainties mathematically. In 1965, the theory of fuzzy set was proposed by Zadeh [1] as a generalization of the classical set under ambiguous or uncertain information. In the proposed theory, elements can have a precise degree of membership or membership functions that determine the membership degree of each element in a fuzzy set. Several significant contributions to the field of fuzzy sets and logic extensions are highlighted in references [2-10]. Finally, Smarandache defined neutrosophic sets [11] in 1998. A neutrosophic structure is generally represented as (T, I, F). In this notation, T is defined as the degree of membership, F as the degree of non-membership and I as the degree of indeterminacy, and these three components are given independently. This facilitates the mathematical description of uncertainties. It has found its place in the application areas in many different disciplines. Therefore, researchers have been conducting studies on decision-making structures in engineering, data mining, cyber security, social sciences, and many other fields. Especially with the definition of the neutrosophic set, decision making applications have found more applications and more precise results have been obtained [12,13]. Ghimire et al. studied a new neutrosophic model Neutrosophic Sets and Systems, Vol. 97, 2026 536 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number to minimize energy consumption in Fixed node networks [14]. Wajid et al. They studied a deep learning approach for image and text classification using Neutrosophy [15]. Narmada Devi et al. studied the solution of assignment problem with Pythagorean octagonal neutrosophic fuzzy number [16]. Smarandache defined the neutrosophic quadruple set [17] in 2015. In general, a neutrosophic quadruple number is of the form (g, kT, hI, nF) (g, k, h, n ∈ ℝ or ℂ). The components T, I and F in this notation are components in neutrosophic logic. However, unlike the neutrosophic set, neutrosophic quadruple set has a known part (g) and an unknown part ((kT, hI, nF)). Therefore, neutrosophic quadruple set is a new type of set that also includes neutrosophic set. In addition, to use neutrosophic quadruple set-in application studies, set-valued neutrosophic quadruple set [18] and generalized set-valued neutrosophic quadruple sets [19] have been defined. Thanks to this definition, neutrosophic quadruple sets can be used in application problems. Most importantly, this definition, which has a more general structure than neutrosophic sets, finds more applications, and gives more objective results to many problems with the help of known part and unknown part. Kargın et al. In 2021, studied the generalized hamming similarity measure based on neutrosophic quadruple numbers and its applications to legal sciences [20]. In 2021, Sahin et al. studied the generalized Euclidean criterion based on neutrosophic quadruple numbers and its applications in health sciences [21]. Chatterjee et al. defined quadripartitioned neutrosophic sets [22] for the first time in 2016. Unlike neutrosophic sets, quadripartitioned neutrosophic sets have contradiction and unknown functions C and U, respectively, instead of the uncertainty function I. Thus, the uncertainty state is divided into two as contradiction and unknown and a more useful definition is obtained. For this reason, it has a wide usage area especially in the application field. Kargın et al. In 2024, operators based on multiple generalized set-valued neutrosophic quadruple sets were defined [23] Sahin et al. In 2022, neutrosophic quintuple sets and numbers were defined [24]. Neutrosophic quintuple set have truth (T), uncertainty (U), contradiction (C) and falsity (F) values as in quadripartitioned neutrosophic sets, but unlike quadripartitioned neutrosophic sets, they also have known and unknown parts such that (g) and (Kt, hU, nC, rF), (g,k,h,n,r ∈ ℝ or ℂ) [24]. Thus, this set is named with neutrosophic quadruple set because of five components (g, kT, hU, nC, rF). Thus, this new structure, which is a generalization of neutrosophic quadruple sets and quadripartitioned neutrosophic sets, provides the properties of both neutrosophic quadruple sets and quadripartitioned neutrosophic sets. Sahin et al. In 2023, generalized set valued neutrosophic quintupe numbers were defined [25]. Sahin et al. In 2024, some operators for interval generalized set valued neutrosophic quintuple numbers and sets were defined [26]. We introduce the notion of Euclidean distance measure for generalized neutrosophic quintuple number and present some general distance measures to derive them. We define a Euclidean distance measure on the generalized neutrosophic quintuple. The benefit of the proposed euclidean distance criterion method on generalized set-valued neutrosophic quintuple set on generalized set-valued neutrosophic quintuple numbers has been numerically studied and the usability and effectiveness of this developed method have been demonstrated on plant selection. This study is organized as follows: Section 1 presents the literature review. Section 2 provides some definitions and basic information that form the basis of our work. In Section 3 we develop generalized Neutrosophic Sets and Systems, Vol. 97, 2026 537 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number distance measures on generalized set-valued neutrosophic quintuple sets. In the first part of Section 4 we develop an algorithm to apply it to real-world problems and in the second part we consider an example to use this algorithm. In this section, we choose some plants based on growing crops based on smart agriculture. We have chosen plants that have high economic returns and provide many benefits from food to pharmacology. We have taken a sample group where these plants are grown (Provinces in the Southeastern Anatolia Region). We examined the growing conditions of the plants we selected and presented them in tables. Immediately afterwards, we looked at the climate, altitude, etc. of the sample groups and presented them separately in tables. Finally, we took an expert opinion and determined in which cities the plants we selected can grow most efficiently. Section 5 presents comparative and sensitivity analyses to demonstrate the usefulness of the proposed method. Section 6 focuses on conclusions, limitations, and future directions. 2. Preliminaries In this section some information about neutrosophic sets, some similarity measures, generalized setvalued neutrosophic quintuple set and numbers are given. Definition 2.1: [27] Let 𝑋 be the universal set. For ∀𝑥∈𝑋, 0≤𝑇𝐾(𝑥)+𝐼𝐾(𝑥)+𝐹𝐾(𝑥)≤3 by the help of the function𝑇𝐾:𝑋→[0,1] , 𝐼𝐾:𝑋→[0,1] ve 𝐹𝐾:𝑋→[0,1]a single-valued neutrosophic set 𝒦 on 𝑋 is defined by 𝑀={〈𝑥,𝑇𝐾(𝑥),𝐼𝐾(𝑥),𝐹𝐾(𝑥)〉:𝑥∈𝑋}. Here, 𝑇𝐾(𝑥), 𝐼𝐾(𝑥) and 𝐹𝐾(𝑥) are the degrees of truth, indeterminacy and falsity of 𝑥∈𝑋; respectively. Definition 2.2: [19] Let X be a set and P(X) be the power set of X. 𝑀𝐾𝑖 generalized set-valued neutrosophic quadruple set is defined by 𝑀𝐾𝑖={(𝐺𝐾𝑖,𝐾𝐾𝑖𝑇𝐾𝑖,𝐻𝐾𝑖𝐼𝐾𝑖,𝐴𝐾𝑖𝐹𝐾𝑖): 𝐺𝐾𝑖,𝐾𝐾𝑖,𝐻𝐾𝑖,𝐴𝐾𝑖∈𝑃(𝑋);𝑖=1,2,3,…,𝑛}. Here 𝑇𝐾𝑖,𝐼𝐾𝑖 𝑣𝑒 𝐹𝐾𝑖 have their known meanings in neutrosophic logic. A generalized set-valued neutrosophic quadruple numbers are defined by 𝑀𝑁𝑖=(𝐺𝐾𝑖,𝐾𝐾𝑖𝑇𝐾𝑖,𝐻𝐾𝑖𝐼𝐾𝑖,𝐴𝐾𝑖𝐹𝐾𝑖) (1) For a generalized neutrosophic quadruple number, i.e., at (1), representing any entity that can be a number, an idea, an object, etc., as in neutrosophic quadruple numbers, 𝐺𝐾𝑖 is called the known part and (𝐾𝐾𝑖𝑇𝐾𝑖,𝐻𝐾𝑖𝐼𝐾𝑖,𝐴𝐾𝑖𝐹𝐾𝑖) is called the unknown part. Definition 2.3: [21] Let 𝑋 ≠ ∅ and (𝑋) be the power set of 𝑋. Let 𝑀𝐾𝑖1=(𝐺𝐾𝑖1,𝐾𝐾𝑖1𝑇𝐾𝑖1,𝐻𝐾𝑖1𝐼𝐾𝑖1,𝐴𝐾𝑖1𝐹𝐾𝑖1) and 𝑀𝐾𝑖2=(𝐺𝐾𝑖2,𝐾𝐾𝑖2𝑇𝐾𝑖2,𝐻𝐾𝑖2𝐼𝐾𝑖2,𝐴𝐾𝑖2𝐹𝐾𝑖2) be two generalized set -valued neutrosophic quadruple numbers. Define a function 𝑆𝐸: 𝑀𝐾𝑖1×𝑀𝐾𝑖2 →[0,1] such that 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)=12 (√(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝐼𝐾𝑖1−𝐼𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 3 + √𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)+𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1) max{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2),1} +𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)+𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1) max{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖2),1} +𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)+𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1) max {𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖2),1} +𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)+𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1) max {𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖2),1} 4) (2) Neutrosophic Sets and Systems, Vol. 97, 2026 538 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number Then, (2) is called generalized Euclid distance measure for generalized set-valued neutrosophic quadruple numbers. Here, 𝐺𝐾𝑖1\𝐺𝐾𝑖2 is the difference between the sets 𝐺𝐾𝑖1 and 𝐺𝐾𝑖2. Here 𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2) is the number of elements of the difference set between sets 𝐺𝐾𝑖1 and 𝐺𝐾𝑖2. Here 𝐺𝐾𝑖1∪𝐺𝐾𝑖2 is the union set of sets 𝐺𝐾𝑖1and 𝐺𝐾𝑖2. Here 𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2) is the number of elements of the union set of sets 𝐺𝐾𝑖1 and 𝐺𝐾𝑖2. Theorem 2.4: [21] Let 𝑀𝐾𝑖1=(𝐺𝐾𝑖1,𝐾𝐾𝑖1𝑇𝐾𝑖1,𝐻𝐾𝑖1𝐼𝐾𝑖1,𝐴𝐾𝑖1𝐹𝐾𝑖1), 𝑀𝐾𝑖2=(𝐺𝐾𝑖2,𝐾𝐾𝑖2𝑇𝐾𝑖2,𝐻𝐾𝑖2𝐼𝐾𝑖2,𝐴𝐾𝑖2𝐹𝐾𝑖2), and 𝑀𝐾𝑖3=(𝐺𝐾𝑖3,𝐾𝐾𝑖3𝑇𝐾𝑖3,𝐻𝐾𝑖3𝐼𝐾𝑖3,𝐴𝐾𝑖3𝐹𝐾𝑖3) be three generalized set-valued neutrosophic quadruple numbers. The generalized Euclid distance measure in Definition 2.3 satisfies the following conditions. i) 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2) ≥0 ii) 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)=0 ⇔𝑀𝐾𝑖1=𝑀𝐾𝑖2 iii) 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)=𝑑𝐸(𝑀𝐾𝑖2,𝑀𝐾𝑖1) iv) If 𝑀𝐾𝑖1⊂𝑀𝐾𝑖2⊂𝑀𝐾𝑖3 , then 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)≤𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖3) and 𝑑𝐸(𝑀𝐾𝑖2,𝑀𝐾𝑖3)≤𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖3). Definition 2.5: [22] Let 𝑋 be the universal set. For ∀𝑥∈𝑋, with functions 𝑇𝐾:𝑋→[0,1], 𝑈𝐾:𝑋→[0,1], 𝐶𝐾:𝑋→[0,1] ve 𝐹𝐾:𝑋→[0,1] quadripartitioned neutrosophic sets on 𝑋 is defined by 𝑋={〈𝑥,𝑇𝐾(𝑥),𝑈𝐾(𝑥),𝐶𝐾(𝑥),𝐹𝐾(𝑥)〉:𝑥∈𝑋} Here, 𝑇𝐾(𝑥), 𝑈𝐾(𝑥), 𝐶𝐾(𝑥), 𝐹𝐾(𝑥) is the degree of truth, uncertainty, contradiction and falsity of 𝑥∈𝑋, respectively. Definition 2.6: [25] Let X be a set and P(X) be the power set of X A generalized set-valued neutrosophic quintuple set is defined by 𝑀𝐾𝑖={(𝐺𝐾𝑖,𝐾𝐾𝑖𝑇𝐾𝑖,𝐻𝐾𝑖𝑈𝐾𝑖,𝐴𝐾𝑖𝐶𝐾𝑖,𝑁𝐾𝑖𝐹𝐾𝑖): 𝐺𝐾𝑖,𝐾𝐾𝑖,𝐻𝐾𝑖,𝐴𝐾𝑖,𝑁𝐾𝑖∈𝑃(𝑋);𝑖=1,2,3,…,𝑛} Here, 𝑇𝐾𝑖,𝑈𝐾𝑖,𝐶𝐾𝑖 and 𝐹𝐾𝑖 has the same meaning as in the known quadripartitioned neutrosophic sets. A generalized set-valued neutrosophic quintuple number is defined by 𝑀𝑁𝑖=(𝐺𝐾,𝐾𝐾𝑖𝑇𝐾𝑖,𝐻𝐾𝑖𝑈𝐾𝑖,𝐴𝐾𝑖𝐶𝐾𝑖,𝑁𝐾𝑖𝐹𝐾𝑖) (3) For a generalized neutrosophic quintuple number representing any entity that can be a number, an idea, an object, etc., as in the neutrosophic quintuple number, i.e. For at (3), 𝐺𝐾𝑖 is called the known part (𝐾𝐾𝑖𝑇𝐾𝑖,𝐻𝐾𝑖𝑈𝐾𝑖,𝐴𝐾𝑖𝐶𝐾𝑖,𝑁𝐾𝑖𝐹𝐾𝑖) is called the unknown part. Also, is shown 𝑀𝐾𝑖={𝑀𝑁𝑖:𝑖=1,2,3,…,𝑛}. 3. Generalized Euclid Measures Based on Generalized Set-Valued Neutrosophic Quintuple Numbers Definition 3.1: Let 𝑀𝐾𝑖=(𝐺𝐾𝑖,𝐾𝐾𝑖𝑇𝐾𝑖,𝐻𝐾𝑖𝑈𝐾𝑖,𝐴𝐾𝑖𝐶𝐾𝑖,𝑁𝐾𝑖𝐹𝐾𝑖) and Neutrosophic Sets and Systems, Vol. 97, 2026 539 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number 𝑀𝐾𝑗=(𝐺𝐾𝑗,𝐾𝐾𝑗𝑇𝐾𝑗,𝐻𝐾𝑗𝑈𝐾𝑗,𝐴𝐾𝑗𝐶𝐾𝑗,𝑁𝐾𝑗𝐹𝐾𝑗) be two generalized set-valued neutrosophic quintuple numbers.𝐺𝐾𝑖⊆𝐺𝐾𝑗,𝐾𝐾𝑖⊆𝐾𝐾𝑗,𝐻𝐾𝑖⊆𝐻𝐾𝑗,𝐴𝐾𝑖⊆𝐴𝐾𝑗,𝑁𝐾𝑖⊆𝑁𝐾𝑗 and 𝑇𝐾𝑖≤𝑇𝐾𝑗,𝑈𝐾𝑖≥𝑈𝐾𝑗, 𝐶𝐾𝑖≥ 𝐶𝐾𝑗,𝐹𝐾𝑖≥𝐹𝐾𝑗 if and only if 𝑀𝐾𝑖 is called a subset 𝑀𝐾𝑗 and is denoted by 𝑀𝐾𝑖⊆𝑀𝐾𝑗. Definition 3.2: Let 𝑀𝐾𝑖=(𝐺𝐾𝑖,𝐾𝐾𝑖𝑇𝐾𝑖,𝐻𝐾𝑖𝑈𝐾𝑖,𝐴𝐾𝑖𝐶𝐾𝑖,𝑁𝐾𝑖𝐹𝐾𝑖) and 𝑀𝐾𝑗=(𝐺𝐾𝑗,𝐾𝐾𝑗𝑇𝐾𝑗,𝐻𝐾𝑗𝑈𝐾𝑗,𝐴𝐾𝑗𝐶𝐾𝑗,𝑁𝐾𝑗𝐹𝐾𝑗) be two generalized set-valued neutrosophic quintuple numbers.𝐺𝐾𝑖=𝐺𝐾𝑗,𝐾𝐾𝑖=𝐾𝐾𝑗,𝐻𝐾𝑖=𝐻𝐾𝑗,𝐴𝐾𝑖=𝐴𝐾𝑗,𝑁𝐾𝑖=𝑁𝐾𝑗and 𝑇𝐾𝑖=𝑇𝐾𝑗,𝑈𝐾𝑖=𝑈𝐾𝑗,𝐶𝐾𝑖= 𝐶𝐾𝑗,𝐹𝐾𝑖=𝐹𝐾𝑗 if and only if 𝑀𝐾𝑖 equals𝑀𝐾𝑗 and is denoted by 𝑀𝐾𝑖=𝑀𝐾𝑗. Definition 3.3: Let 𝑋 ≠ ∅ and P(𝑋) be the power set of 𝑋. Let 𝑀𝐾𝑖1=(𝐺𝐾𝑖1,𝐾𝐾𝑖1𝑇𝐾𝑖1,𝐻𝐾𝑖1𝑈𝐾𝑖1,𝐴𝐾𝑖1𝐶𝐾𝑖1,𝑁𝐾𝑖1𝐹𝐾𝑖1) and 𝑀𝐾𝑖2= (𝐺𝐾𝑖2,𝐾𝐾𝑖2𝑇𝐾𝑖2,𝐻𝐾𝑖2𝑈𝐾𝑖2,𝐴𝐾𝑖2𝐶𝐾𝑖2,𝑁𝐾𝑖2𝐹𝐾𝑖2) be two generalized set-valued neutrosophic quintuple numbers. Defined a function 𝑑𝐸: 𝑀𝐾𝑖1×𝑀𝐾𝑖2 →[0,1] such that 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)= 12 (√(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 4 + √𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)+𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1) max{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2),1} +𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)+𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1) max{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖2),1} +𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)+𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1) max{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖2),1} +𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)+𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1) max {𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖2),1} +𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2)+𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1) max {𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖2),1} 5) (4) Then, at (4) is called the generalized Euclidean distance measure for generalized set-valued neutrosophic quintuple numbers. Here, 𝐺𝐾𝑖1\𝐺𝐾𝑖2 is the difference between the sets 𝐺𝐾𝑖1 and 𝐺𝐾𝑖2. Here 𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2) is the number of elements of the difference set between sets 𝐺𝐾𝑖1 and 𝐺𝐾𝑖2. Here 𝐺𝐾𝑖1∪𝐺𝐾𝑖2 is the union set of sets 𝐺𝐾𝑖1and 𝐺𝐾𝑖2. Here 𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2) is the number of elements of the union set of sets 𝐺𝐾𝑖1 and 𝐺𝐾𝑖2. NOTE 3.4: While defining the Euclidean distance measure for generalized set-valued neutrosophic quintuple number, the difference from the Euclidean distance measure previously defined by Şahin et al. for generalized set-valued neutrosophic quadruple number [21] is that instead of I, i.e. uncertainty, U and C are taken as uncertainty and contradiction respectively. Therefore, it is different from previous studies. Example3.5:𝛾1=({℘1,℘2,℘3,℘4},{℘5,℘6}(0,2),{℘7,℘8,℘9}(0,4),∅(0),{℘10,℘11,℘12}(0,4)) and 𝛾2=({℘5,℘6,℘7,℘8},{℘14}(0),∅(1),{℘4,℘13}(0),{℘1,℘2,℘3}(0)) be two generalized set-valued neutrosophic quintuple numbers. Then, =12[((0,2)+(0,6)+(0)+(0,4) 4)+(√4+4 8+2+1 3+3+0 3+0+2 2+3+3 6 5)] =12[1,2 4+√1+1+1+1+1 5] Neutrosophic Sets and Systems, Vol. 97, 2026 540 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number =12[1,2 4+1] =0,65 Theorem 3.6: Let 𝑀𝐾𝑖1=(𝐺𝐾𝑖1,𝐾𝐾𝑖1𝑇𝐾𝑖1,𝐻𝐾𝑖1𝑈𝐾𝑖1,𝐴𝐾𝑖1𝐶𝐾𝑖1,𝑁𝐾𝑖1𝐹𝐾𝑖1), 𝑀𝐾𝑖2=(𝐺𝐾𝑖2,𝐾𝐾𝑖2𝑇𝐾𝑖2,𝐻𝐾𝑖2𝑈𝐾𝑖2,𝐴𝐾𝑖2𝐶𝐾𝑖2,𝑁𝐾𝑖2𝐹𝐾𝑖2), and 𝑀𝐾𝑖3=(𝐺𝐾𝑖3,𝐾𝐾𝑖3𝑇𝐾𝑖3,𝐻𝐾𝑖3𝑈𝐾𝑖3,𝐴𝐾𝑖3𝐶𝐾𝑖3,𝑁𝐾𝑖3𝐹𝐾𝑖3) be three generalized set-valued neutrosophic quintuple numbers. The generalized Euclidean distance measure 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2) in Definition 3.3 satisfies the following conditions i) 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)≥0 ii) 𝑑𝐸(𝑀𝐾1,𝑀𝐾𝑖2)=0 ⇔𝑀𝐾𝑖1=𝑀𝐾𝑖2 iii) 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)=𝑑𝐸(𝑀𝐾𝑖2,𝑀𝐾𝑖1) iv) If 𝑀𝐾𝑖1⊂𝑀𝐾𝑖2⊂𝑀𝐾𝑖3, then 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)≤𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖3) 𝑎𝑛𝑑 𝑑𝐸(𝑀𝐾𝑖2,𝑀𝐾𝑖3)≤𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖3). Proof: i) We assume that 𝑀𝐾𝑖1=𝑀𝐾𝑖2. From Definition 3.1, we obtain that 𝐺𝐾𝑖1=𝐺𝐾𝑖2,𝐾𝐾𝑖1=𝐾𝐾𝑖2,𝐻𝐾𝑖1=𝐻𝐾𝑖2,𝐴𝐾𝑖1=𝐴𝐾𝑖2,𝑁𝐾𝑖1=𝑁𝐾𝑖2,𝑇𝐾𝑖1=𝑇𝐾𝑖2,𝑈𝐾𝑖1=𝑈𝐾𝑖2,𝐶𝐾𝑖1=𝐶𝐾𝑖2,𝐹𝐾𝑖1 =𝐹𝐾𝑖2. Thus, 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)=12 (√(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 4+ √𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)+𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1) max{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2),1} +𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)+𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1) max{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖2),1} +𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)+𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1) max{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖2),1} +𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)+𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1) max{𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖2),1} +𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2)+𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1) max{𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖2),1} 5) =12 (0+0+0+0 4+√0+0+0+0+0 5=12.0=0. In this case, ii)⇒: We assume that 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)=0. Then, √(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 4=0 (5) and √𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)+𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1) max{𝑠(𝐺𝐾1𝑖∪𝐺𝐾2𝑖),1} +𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)+𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1) max{𝑠(𝐾𝐾1𝑖∪𝐾𝐾2𝑖),1} +𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)+𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1) max{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖2),1} +𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)+𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1) max{𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖2),1} +𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2)+𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1) max{𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖2),1} 5 =0 (6) Neutrosophic Sets and Systems, Vol. 97, 2026 541 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number From (5), we obtain that √(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2=0 and 𝑇𝐾𝑖1=𝑇𝐾𝑖2 √(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2=0 and 𝑈𝐾𝑖1=𝑈𝐾𝑖2 √(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2=0 and 𝐶𝐾𝑖1=𝐶𝐾𝑖2 √(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2=0 and 𝐹𝐾𝑖1=𝐹𝐾𝑖2 (7) Also, from (6), we obtain 𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)+𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1) max{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2),1} +𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)+𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1) max{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖2),1} +𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)+𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1) max{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖2),1} +𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)+𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1) max {𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖2),1} + 𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2)+𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1) max {𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖2),1} =0. Hence, 𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)+𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1)=0 𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)+𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1)=0 𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)+𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1)=0 𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)+𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1)=0 𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2)+𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1)=0 (8) From (8), we obtain 𝐺𝐾𝑖1=𝐺𝐾𝑖2,𝐾𝐾𝑖1=𝐾𝐾𝑖2,𝐻𝐾𝑖1=𝐻𝐾𝑖2,𝐴𝐾𝑖1=𝐴𝐾𝑖2 and 𝑁𝐾𝑖1=𝑁𝐾𝑖2 (9) Therefore, from (7), (9) and Definition 3.1, we obtain 𝑀𝐾𝑖1=𝑀𝐾𝑖2. iii) 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)= 12 (√(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 4 + √𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)+𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1) max{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2),1} +𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)+𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1) max{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖2),1} +𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)+𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1) max{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖2),1} +𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)+𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1) max {𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖2),1} +𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2)+𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1) max {𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖2),1} 5) =12 (√(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 4 + √𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1)+𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2) max{𝑠(𝐺𝐾𝑖2∪𝐺𝐾𝑖1),1} +𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1)+𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2) max{𝑠(𝐾𝐾𝑖2∪𝐾𝐾𝑖1),1} +𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1)+𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2) max{𝑠(𝐻𝐾𝑖2∪𝐻𝐾𝑖1),1} +𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1)+𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2) max{𝑠(𝐴𝐾𝑖2∪𝐴𝐾𝑖1),1} +𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1)+𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2) max{𝑠(𝑁𝐾𝑖2∪𝑁𝐾𝑖1),1} 5) =𝑑𝐸(𝑀𝐾𝑖2,𝑀𝐾𝑖1). iv) We assume that 𝑀𝐾𝑖1⊆𝑀𝐾𝑖2⊆𝑀𝐾𝑖3. From Definition 3.1, we obtain Neutrosophic Sets and Systems, Vol. 97, 2026 542 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number 𝑠(𝐺𝐾𝑖1)≤𝑠(𝐺𝐾𝑖2)≤𝑠(𝐺𝐾𝑖3), 𝑠(𝐾𝐾𝑖1)≤𝑠(𝐾𝐾𝑖2)≤𝑠(𝐾𝐾𝑖3), 𝑠(𝐻𝐾𝑖1)≤𝑠(𝐻𝐾𝑖2)≤𝑠(𝐻𝐾𝑖3), 𝑠(𝐴𝐾𝑖1)≤𝑠(𝐴𝐾𝑖2)≤𝑠(𝐴𝐾𝑖3) and 𝑠(𝑁𝐾𝑖1)≤𝑠(𝑁𝐾𝑖2)≤𝑠(𝑁𝐾𝑖3) (10) Thus, we obtain 𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)=𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖3)=𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖3)=0, 𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)=𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖3)=𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖3)=0, 𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)=𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖3)=𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖3)=0, 𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)=𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖3)=𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖3)=0, and 𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2)=𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖3)=𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖3)=0 (11) Also, we got 𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1)≤𝑠(𝐺𝐾𝑖3\𝐺𝐾𝑖1), 𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1)≤𝑠(𝐾𝐾𝑖3\𝐾𝐾𝑖1), 𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1)≤𝑠(𝐻𝐾𝑖3\𝐻𝐾𝑖1), 𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1)≤𝑠(𝐴𝐾𝑖3\𝐴𝐾𝑖1), 𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1)≤𝑠(𝑁𝐾𝑖3\𝑁𝐾𝑖1), 𝑠(𝐺𝐾𝑖3\𝐺𝐾𝑖2)≤𝑠(𝐺𝐾𝑖3\𝐺𝐾𝑖1), 𝑠(𝐾𝐾𝑖3\𝐾𝐾𝑖2)≤𝑠(𝐾𝐾𝑖3\𝐾𝐾𝑖1), 𝑠(𝐻𝐾𝑖3\𝐻𝐾𝑖2)≤𝑠(𝐻𝐾𝑖3\𝐻𝐾𝑖1), 𝑠(𝐴𝐾𝑖3\𝐴𝐾𝑖2)≤𝑠(𝐴𝐾𝑖3\𝐴𝐾𝑖1), and 𝑠(𝑁𝐾𝑖3\𝑁𝐾𝑖2)≤𝑠(𝑁𝐾𝑖3\𝑁𝐾𝑖1), (12) Also, from (10) ; 𝑚𝑎𝑥{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2),1}=𝑚𝑎𝑥{𝑠(𝐺𝐾𝑖2),1}, 𝑚𝑎𝑥{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖2),1}=𝑚𝑎𝑥{𝑠(𝐾𝐾𝑖2),1}, 𝑚𝑎𝑥{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖2),1}=𝑚𝑎𝑥{𝑠(𝐻𝐾𝑖2),1}, 𝑚𝑎𝑥{𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖2),1}=𝑚𝑎𝑥{𝑠(𝐴𝐾𝑖2),1}, 𝑚𝑎𝑥{𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖2),1}=𝑚𝑎𝑥{𝑠(𝑁𝐾𝑖2),1}, 𝑚𝑎𝑥{𝑠(𝐺𝐾𝑖2∪𝐺𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝐺𝐾𝑖3),1}, 𝑚𝑎𝑥{𝑠(𝐾𝐾𝑖2∪𝐾𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝐾𝐾𝑖3),1}, 𝑚𝑎𝑥{𝑠(𝐻𝐾𝑖2∪𝐻𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝐻𝐾𝑖3),1}, 𝑚𝑎𝑥{𝑠(𝐴𝐾𝑖2∪𝐴𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝐴𝐾𝑖3),1}, 𝑚𝑎𝑥{𝑠(𝑁𝐾𝑖2∪𝑁𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝑁𝐾𝑖3),1}, 𝑚𝑎𝑥{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝐺𝐾𝑖3),1}, 𝑚𝑎𝑥{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝐾𝐾𝑖3),1}, Neutrosophic Sets and Systems, Vol. 97, 2026 543 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number 𝑚𝑎𝑥{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝐻𝐾𝑖3),1}, 𝑚𝑎𝑥{𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝐴𝐾𝑖3),1}, and 𝑚𝑎𝑥{𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖3),1}=𝑚𝑎𝑥{𝑠(𝑁𝐾𝑖3),1}, (13) Also, since 𝑀𝐾𝑖1⊆𝑀𝐾𝑖2⊆𝑀𝐾𝑖3, From Definition 3.1, we obtain 𝑇𝐾𝑖1≤𝑇𝐾𝑖2≤𝑇𝐾𝑖3 𝑈𝐾𝑖1≤𝑈𝐾𝑖2≤𝑈𝐾𝑖3 𝐶𝐾𝑖1≤𝐶𝐾𝑖2≤𝐶𝐾𝑖3 𝐹𝐾𝑖1≤𝐹𝐾𝑖2≤𝐹𝐾𝑖3. Hence, we get √(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 ≤√(𝑇𝐾𝑖1−𝑇𝐾𝑖3)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖3)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖3)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖3)2 and √(𝑇𝐾𝑖2−𝑇𝐾𝑖3)2+√(𝑈𝐾𝑖2−𝑈𝐾𝑖3)2+√(𝐶𝐾𝑖2−𝐶𝐾𝑖3)2+√(𝐹𝐾𝑖2−𝐹𝐾𝑖3)2 ≤√(𝑇𝐾𝑖1−𝑇𝐾𝑖3)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖3)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖3)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖3)2 (14) Hence, from (10), (11), (12), (13), (14); we obtain that 𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖2)= 12 (√(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 4 + √𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖2)+𝑠(𝐺𝐾𝑖2\𝐺𝐾𝑖1) max{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖2),1} +𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖2)+𝑠(𝐾𝐾𝑖2\𝐾𝐾𝑖1) max{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖2),1} +𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖2)+𝑠(𝐻𝐾𝑖2\𝐻𝐾𝑖1) max{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖2),1} +𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖2)+𝑠(𝐴𝐾𝑖2\𝐴𝐾𝑖1) max{𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖2),1} +𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖2)+𝑠(𝑁𝐾𝑖2\𝑁𝐾𝑖1) max{𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖2),1} 5) ≤12 (√(𝑇𝐾𝑖1−𝑇𝐾𝑖2)2+√(𝑈𝐾𝑖1−𝑈𝐾𝑖2)2+√(𝐶𝐾𝑖1−𝐶𝐾𝑖2)2+√(𝐹𝐾𝑖1−𝐹𝐾𝑖2)2 4 + √𝑠(𝐺𝐾𝑖1\𝐺𝐾𝑖3)+𝑠(𝐺𝐾𝑖3\𝐺𝐾𝑖1) max{𝑠(𝐺𝐾𝑖1∪𝐺𝐾𝑖3),1} +𝑠(𝐾𝐾𝑖1\𝐾𝐾𝑖3)+𝑠(𝐾𝐾𝑖3\𝐾𝐾𝑖1) max{𝑠(𝐾𝐾𝑖1∪𝐾𝐾𝑖3),1} +𝑠(𝐻𝐾𝑖1\𝐻𝐾𝑖3)+𝑠(𝐻𝐾𝑖3\𝐻𝐾𝑖1) max{𝑠(𝐻𝐾𝑖1∪𝐻𝐾𝑖3),1} +𝑠(𝐴𝐾𝑖1\𝐴𝐾𝑖3)+𝑠(𝐴𝐾𝑖3\𝐴𝐾𝑖1) max{𝑠(𝐴𝐾𝑖1∪𝐴𝐾𝑖3),1} +𝑠(𝑁𝐾𝑖1\𝑁𝐾𝑖3)+𝑠(𝑁𝐾𝑖3\𝑁𝐾𝑖1) max{𝑠(𝑁𝐾𝑖1∪𝑁𝐾𝑖3),1} 5) =𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖3). Similarly, for (10), (11), (12), (13), (14); we obtain 𝑑𝐸(𝑀𝐾𝑖2,𝑀𝐾𝑖3)≤𝑑𝐸(𝑀𝐾𝑖1,𝑀𝐾𝑖3). 4.1 Multi-Criteria Decision-Making Algorithm with Generalized Set-Valued Neutrosophic Quintuple set and Generalized Euclidean Distance Measure Neutrosophic Sets and Systems, Vol. 97, 2026 550 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number Sunshine AprilAprilAprilAprilAprilAprilAprilAprilAprilduration May: May: May: May: May: May: May: May: May: during 7,4 clock7,4 clock7,4 clock6,8 clock7,8 clock7,3 clock6,5 clock6,5 clock7,2 clockthe 9,4 clock 9,2 clock 9,2 clock 8,4 clock 9,3 clock 9,7 clock 8,9 clock 8,9 clock 8,5 clock fruiting October – October – October – October – October – October – October – October – October period Novem Novem Novem Novem Novem Novem Novem Novem Novem [53] ber: ber: ber: ber: ber: ber: ber: ber: ber: 7,3 clock7,1 clock7,1 clock6,9 clock7,2 clock7,7 clock7,2 clock7,2 clock7,2 clock5,4 clock 5,2 clock 5,2 clock 5,3 clock 5,3 clock 5,9 clock 5,2 clock 5,2 clock 5,3 clock Septem Septem Septem Septem Septem Septem Septem Septem Septem berberberberberberberberberOctober: October: October: October: October: October: October: October: October: 9,7 clock9,3 clock9,9 clock 8,7 clock9,8 clock10,3 clock9,9 clock9,9 clock10, clock7,3 clock 7,1 clock 7,1 clock 6,9 clock 7,2 clock 7,7 clock 7,2 clock 7,2 clock 7,2 clock June – June – June – June – June – June – June – June – June – July: July: July: July: July: July: July: July: July: 11,6 clock11,6 clock11,6 clock10,3 clock11,1 clock12,4 clock11,6 clock11,6 clock10,6 clock12,1 clock 12,0 clock 12,0 clock 10,6 clock 11,6 clock 12,4 clock 11,6 clock 12,1 clock 11,2 clock Table 8. shows the average annual sunshine duration and the average sunshine duration during the growing months for the selected provinces in the sample. It is important that the plant produces its own nutrients and does not resort to additional fertilisers. Table 9. Altitude Table for the Cities in the Sample Adıyaman Batman Diyarbakır Gaziantep Kilis Mardin Siirt Şanlıurfa Şırnak Altitude 699m 550m 675m 850m 663m 1083m 986m 518m 1400m Table 9. gives the altitude of the selected sample provinces. The plant needs a suitable temperature to grow and develop, we prefer it to grow at natural temperature, minimizing the greenhouse effect. We can start implementing the algorithm given in the previous section (4.1). In the previous pages we have mentioned all the necessary information in paragraphs and tables. First, we identify the plants we have chosen Neutrosophic Sets and Systems, Vol. 97, 2026 551 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number to grow. Secondly, together with an expert, we give the criteria by which the plants should grow best. We identify an ideal city where the plants can grow best. In the third step, we again determine the weight values of the criteria with the help of an expert. In the fourth step, we obtain a generalized cluster-valued neutrosophic quintile number for each city in the sample, taking into account the expert opinion. In the fifth step, we clearly express each cluster in the table. In the sixth step, we make calculations using the Euclidean distance measure we defined in the third section. In step seven, we multiply the results by the weight values and create a new table. Finally, we compare and analyze the proposed method with method 1 and method 2. NOTE 4.3: The temperature of the months in the ‘Temperature during the growing period of the fruit’ section are the periods during the growth of Aronia, Saffron, Gilaburu, Japanese raisin, respectively. Step 1: Let 𝒫 = {𝓅1 ,𝓅2 ,𝓅3 ,𝓅4 } be a set of plants. 𝓅1= Aronia, 𝓅2= Crocus sativus, 𝓅3= Viburnum opulus, and 𝓅4= Hovenia Dulcis. Step 2: Let us write down the criteria necessary for the most suitable plant to grow and express an ideal plant S as generalized set-valued neutrosophic quintuple set for comparison. Note that when writing the ideal plant, it must fulfill all the criteria. 𝓀𝒾1= temperatures, 𝓀𝒾2= soil structure, 𝓀𝒾3= precipitation, 𝓀𝒾4= sun, 𝓀𝒾5= altitude, 𝒾=1,…, n, n is the number of provinces we have selected. 𝒮={𝓅1:({𝓀11 ,𝓀12 ,𝓀13 ,𝓀14 ,𝓀15},{𝓀11 ,𝓀12 ,𝓀13 ,𝓀14 ,𝓀15 }1,∅0,∅0,∅0),𝓅2:({𝓀21,𝓀22 ,𝓀23 ,𝓀24 ,𝓀25}, {𝓀21,𝓀22 ,𝓀23 ,𝓀24 ,𝓀25 }1,∅0,∅0,∅0),𝓅3:({𝓀31 ,𝓀32 ,𝓀33 ,𝓀34 ,𝓀35},{𝓀31 ,𝓀32 ,𝓀33 ,𝓀34 ,𝓀35 }1,∅0,∅0,∅0), 𝓅4:({𝓀41 ,𝓀42 ,𝓀43 ,𝓀44 ,𝓀45},{𝓀41 ,𝓀42 ,𝓀43 ,𝓀44 ,𝓀45 }1,∅0,∅0,∅0)} Since 𝒮 is an ideal plant, the truth set of the known part and the unknown part of the criteria should be equal and the truth value of the unknown part should be 1. Also, the other sets should be empty, and the other values should be 0 for the unknown, contradiction and inaccuracy values respectively. For example, in an ideal plant 𝒮; for 𝓅1, Set {𝓀11 ,𝓀12 ,𝓀13 ,𝓀14 ,𝓀15} is considered as the set of factors influencing the good growth of plant 𝓅1 and ({𝓀11 ,𝓀12 ,𝓀13 ,𝓀14 ,𝓀15 },1,∅0,∅0) is considered as the set of factors influencing the good growth of unknown plants. This applies to 𝓅1 and other plants. Step 3: Let W = {0.3, 0.3, 0.1, 0.2, 0.1} be the set of weighted value criteria. It is such that, 0.3 is weighted value of 𝓀𝒾1, 0.2 is weighted value of 𝓀𝒾2, Neutrosophic Sets and Systems, Vol. 97, 2026 552 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number 0.3 is weighted value of 𝓀𝒾3, and 0.1 is weighted value of 𝓀𝒾4. Step 4: Let Δ={Δ𝒜,Δℬ,Δ𝒟,Δ𝒮𝒾,Δ𝒮𝒶,Δ𝒮𝒾𝓇,Δ𝒢,Δℳ} be the set of cities in Güneydoğu Anadolu where the plants we have chosen to grow. Where Δ𝒜,Δℬ,Δ𝒟,Δ𝒮𝒾,Δ𝒮𝒶,Δ𝒮𝒾𝓇,Δ𝒢,Δℳrepresent the provinces of Adıyaman, Batman, Diyarbakır, Gaziantep, Kilis, Mardin, Siirt, Şanlıurfa, Şırnak respectively, such that Δ𝒜= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀12,𝓀13,𝓀14 ,𝓀15}(0,7),∅0,{𝓀11}(0,1),{𝓀11,𝓀13}(0,2)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀23,𝓀24}(0,4),{𝓀25}(0,1),{𝓀21}(0,2),{𝓀21,𝓀22}(0,2)) 𝓅3: ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31,𝓀32,𝓀33,𝓀34 ,𝓀35}(0,5),{𝓀34} (0,1),{𝓀33,𝓀34}(0,2),{𝓀33,𝓀34}(0,2)) 𝓅4: ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41,𝓀42,𝓀44 ,𝓀43,𝓀45}(0,7),{𝓀43}(0,1),{𝓀43}(0,1),{𝓀43}(0,1)) } Δℬ= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀12,𝓀13,𝓀14 ,𝓀15}(0,6),∅0,{𝓀11,𝓀13}(0,2),{𝓀11,𝓀13}(0,2)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀22,𝓀23,𝓀24}(0,5),{𝓀25}(0,1),{𝓀21,𝓀23}(0,2),{𝓀11,𝓀23}(0,2)) 𝓅3:({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31 ,𝓀32 ,𝓀34 ,𝓀35}(0.7),∅0,{𝓀33}(0.1),{𝓀33}(0.1)) 𝓅4:({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41,𝓀42 ,𝓀44 ,𝓀45}(0,7),∅0,{𝓀41,𝓀43}(0,2),{𝓀41,𝓀43}(0,1)) } Δ𝒟= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀12,𝓀13,𝓀14 ,𝓀15}(0,6),∅0,{𝓀11,𝓀13}(0,3),{𝓀11,𝓀13}(0,2)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀22,𝓀24}(0,7),{𝓀25}(0,1),{𝓀21}(0,1),{𝓀21}(0,1)) 𝓅3:({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31,𝓀32,𝓀34 ,𝓀35}(0,7),∅0,{𝓀31}(0,1),{𝓀11,𝓀33}(0,2)) 𝓅4:({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41,𝓀42,𝓀44 ,𝓀45}(0,7),∅0,{𝓀41}(0,2),{𝓀41,𝓀43}(0,1)) } Δ𝒢= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}(0,8),∅0,{𝓀13}(0.1),{𝓀13}(0.1)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀22,𝓀23 ,𝓀24}(0,9),{𝓀25}(0,1),∅0,∅0) 𝓅3:({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}(0,7),∅0,{𝓀31,𝓀33}(0,2),{𝓀31,𝓀33}(0,1)) 𝓅4:({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41,𝓀42,𝓀43 ,𝓀44 ,𝓀45}(0,8),∅0,{𝓀43}(0,1),{𝓀43}(0,1)) } Δ𝒦= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀14 ,𝓀15}(0,5),∅0,{𝓀11}(0,2),{𝓀11,𝓀12,𝓀13}(0,3)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀23,𝓀24}(0,4),{𝓀25}(0,1),{𝓀21,𝓀23}(0,2),{𝓀21,𝓀22,𝓀23}(0,3)) 𝓅3:({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31,𝓀32,𝓀34 ,𝓀35}(0,7),∅0,{𝓀31}(0,1),{𝓀31,𝓀33}(0,2)) 𝓅4:({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41 ,𝓀42,𝓀44 ,𝓀45}(0,9),∅0,∅0,{𝓀43}(0,1)) } Δℳ= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀14 ,𝓀15}(0,5),∅0,{𝓀11}(0,1),{𝓀111,𝓀12,𝓀13}(0,4)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀23 ,𝓀24}(0,6),{𝓀25}(0,1),{𝓀21}(0,1),{𝓀21,𝓀22}(0,2)) 𝓅3:({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31,𝓀32,𝓀34 ,𝓀33,𝓀35}(0,8),∅0,{𝓀31,𝓀33}(0,1),{𝓀31,𝓀33 }(0,1)) 𝓅4:({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41,𝓀42,𝓀43,𝓀44}(0,7),∅0,{𝓀43}(0,1),{𝓀43,𝓀45}(0,2)) } Δ𝒮𝒾= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀12,𝓀14 ,𝓀15}(0,7),∅0,{𝓀11}(0,1),{𝓀11,𝓀13}(0,2)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀22,𝓀24 }(0,6),{𝓀25}(0,1),{𝓀21}(0,1),{𝓀21,𝓀23}(0,2)) 𝓅3:({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31,𝓀32,𝓀34 ,𝓀33,𝓀35}(0,6),∅0,{𝓀31,𝓀33}(0,2),{𝓀31,𝓀33}(0,2)) 𝓅4:({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41,𝓀42,𝓀43,𝓀44}(0,5),∅0,{𝓀41,𝓀43}(0,2),{𝓀41,𝓀43,𝓀45}(0,3)) } Neutrosophic Sets and Systems, Vol. 97, 2026 553 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number Δ𝒮𝒶= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀12,𝓀14 ,𝓀15}(0,7),∅0,{𝓀11}(0,1),{𝓀11,𝓀13}(0,2)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀22 ,𝓀24}(0,5),{𝓀25}(0,1),{𝓀21,𝓀24}(0,2),{𝓀21,𝓀23}(0,2)) 𝓅3:({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31,𝓀32,𝓀34 ,𝓀35}(0,7),∅0,{𝓀31}(0,1),{𝓀31,𝓀33}(0,2)) 𝓅4:({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41 ,𝓀42,𝓀44 ,𝓀45}(0,9),∅0,∅0,{𝓀43}(0,1)) } Δ𝒮𝒾𝓇= { 𝓅1:({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15},{𝓀11,𝓀12,𝓀14 ,𝓀15}(0,7),∅0,{𝓀11}(0,1),{𝓀11,𝓀13}(0,2)) 𝓅2:({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25},{𝓀21,𝓀22,𝓀24}(0,6),{𝓀25}(0,1),{𝓀21}(0,1),{𝓀21,𝓀23}(0,2)) 𝓅3:({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35},{𝓀31,𝓀32,𝓀33,𝓀34 ,𝓀35}(0,8),∅0,{𝓀31}(0,1),{𝓀31}(0,1)) 𝓅4:({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45},{𝓀41,𝓀44}(0,5),∅0,∅0,{𝓀42,𝓀43,𝓀45}(0,5)) } Step 5: Considering the growing conditions of the plant we defined in Section 4.2 and the criteria given in the tables (Tables 5, Tables 6, Tables 7, Tables 8, Tables 9), let's show in Table 2 the set of criteria for each city we defined in Step 4. Table 10. Table of the Alternatives for Each City in Our Sample. 𝓅1 𝓅2 𝓅3 𝓅4 Δ𝒜 ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45}, {𝓀11,𝓀12,𝓀13,𝓀14 ,𝓀15}(0,7),∅0, {𝓀21,𝓀23,𝓀24}(0,4),{𝓀25}(0,1), {𝓀31,𝓀32,𝓀33,𝓀34 ,𝓀35}(0,5), {𝓀41,𝓀42,𝓀44 ,𝓀43,𝓀45}(0,7), {𝓀11}(0,1),{𝓀11,𝓀13}(0,2)) {𝓀21}(0,2),{𝓀21,𝓀22}(0,2)) {𝓀34} (0,1),{𝓀33,𝓀34}(0,2), {𝓀43}(0,1),{𝓀43}(0,1), {𝓀33,𝓀34}(0,2)) {𝓀43}(0,1)) Δℬ ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45} {𝓀11,𝓀12 ,𝓀13,𝓀14 ,𝓀15}(0,6), {𝓀21,𝓀22,𝓀23,𝓀24 }(0,5), {𝓀31 ,𝓀32 ,𝓀34 ,𝓀35}(0.7), {𝓀41,𝓀42 ,𝓀44 ,𝓀45}(0,7), ∅0,{𝓀11,𝓀13}(0,2), {𝓀25}(0,1),{𝓀21,𝓀23}(0,2), ∅0,{𝓀33}(0.1),{𝓀33}(0.1)) ∅0,{𝓀41,𝓀43}(0,2), {𝓀11,𝓀13}(0,2)) {𝓀11,𝓀23}(0,2)) {𝓀41,𝓀43}(0,1)) Δ𝒟 ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45}, {𝓀11,𝓀12,𝓀13,𝓀14 ,𝓀15}(0,6), {𝓀21,𝓀22,𝓀24}(0,7), {𝓀31,𝓀32,𝓀34 ,𝓀35}(0,7), {𝓀41,𝓀42,𝓀44 ,𝓀45}(0,7), ∅0, {𝓀11,𝓀13}(0,3), {𝓀25}(0,1),{𝓀21}(0,1), ∅0,{𝓀31}(0,1), ∅0,{𝓀41}(0,2), {𝓀11,𝓀13}(0,2)) {𝓀21}(0,1)) {𝓀11,𝓀33}(0,2)) {𝓀41,𝓀43}(0,1)) Δ𝒢 ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45}, {𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}(0,8), {𝓀21,𝓀22,𝓀23 ,𝓀24}(0,9), {𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}(0,7), {𝓀41,𝓀42,𝓀43 ,𝓀44 ,𝓀45}(0,8), ∅0,{𝓀13}(0.1),{𝓀13}(0.1)) {𝓀25}(0,1),∅0,∅0) ∅0,{𝓀31,𝓀33}(0,2), ∅0,{𝓀43}(0,1),{𝓀43}(0,1)) {𝓀31,𝓀33}(0,1)) Δ𝒦 ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45}, Neutrosophic Sets and Systems, Vol. 97, 2026 554 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45}, {𝓀11,𝓀14 ,𝓀15}(0,5),∅0, {𝓀21,𝓀23,𝓀24}(0,4), {𝓀31,𝓀32,𝓀34 ,𝓀35}(0,7),∅0, {𝓀41,𝓀42,𝓀44 ,𝓀45}(0,9), {𝓀11}(0,2), {𝓀25}(0,1), {𝓀21,𝓀23}(0,2) {𝓀31}(0,1),{𝓀31,𝓀33}(0,2)) ∅0,∅0,{𝓀43}(0,1)) {𝓀11,𝓀12,𝓀13}(0,3)) {𝓀21,𝓀22,𝓀23}(0,3)) Δℳ ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45}, {𝓀11,𝓀14 ,𝓀15}(0,5),∅0, {𝓀21,𝓀23 ,𝓀24}(0,6), {𝓀31,𝓀32,𝓀34 ,𝓀33,𝓀35}(0,8), {𝓀41,𝓀42,𝓀43,𝓀44}(0,7), {𝓀11}(0,1), {𝓀25}(0,1),{𝓀21}(0,1), ∅0,{𝓀31,𝓀33}(0,1), ∅0,{𝓀43}(0,1), {𝓀111,𝓀12,𝓀13}(0,4)) {𝓀21,𝓀22}(0,2)) {𝓀31,𝓀33 }(0,1)) {𝓀43,𝓀45}(0,2)) Δ𝒮𝒾 ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45}, {𝓀11,𝓀12,𝓀14 ,𝓀15}(0,7), {𝓀21,𝓀22,𝓀24 }(0,6), {𝓀31,𝓀32,𝓀34 ,𝓀33,𝓀35}(0,6), {𝓀41,𝓀42,𝓀43,𝓀44}(0,5), ∅0,{𝓀11}(0,1), {𝓀25}(0,1),{𝓀21}(0,1), ∅0,{𝓀31,𝓀33}(0,2), ∅0,{𝓀41,𝓀43}(0,2), {𝓀11,𝓀13}(0,2)) {𝓀21,𝓀23}(0,2)) {𝓀31,𝓀33}(0,2)) {𝓀41,𝓀43,𝓀45}(0,3)) Δ𝒮𝒶 ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15} ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45}, {𝓀11,𝓀12,𝓀14 ,𝓀15}(0,7), {𝓀21,𝓀22 ,𝓀24}(0,5), {𝓀31,𝓀32,𝓀34 ,𝓀35}(0,7), {𝓀41 ,𝓀42,𝓀44 ,𝓀45}(0,9), ∅0,{𝓀11}(0,1),{𝓀11,𝓀13}(0,2)) {𝓀25}(0,1),{𝓀21,𝓀24}(0,2), ∅0,{𝓀31}(0,1), ∅0,∅0,{𝓀43}(0,1)) {𝓀21,𝓀23}(0,2)) {𝓀31,𝓀33}(0,2)) Δ𝒮𝒾𝓇 ({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}, ({𝓀21 ,𝓀22,𝓀23 ,𝓀24 ,𝓀25}, ({𝓀31 ,𝓀32,𝓀33 ,𝓀34 ,𝓀35}, ({𝓀41 ,𝓀42,𝓀43 ,𝓀44 ,𝓀45} {𝓀11,𝓀12,𝓀14 ,𝓀15}(0,7) {𝓀21,𝓀22,𝓀24}(0,6),{𝓀25}(0,1), {𝓀31,𝓀32,𝓀33,𝓀34 ,𝓀35}(0,8), {𝓀41,𝓀44}(0,5),∅0,∅0, ∅0,{𝓀11}(0,1), {𝓀11,𝓀13}(0,2)) {𝓀21}(0,1),{𝓀21,𝓀23}(0,2)) ∅0,{𝓀31}(0,1),{𝓀31}(0,1)) {𝓀42,𝓀43,𝓀45}(0,5)) Step 6: In Table 11, we calculate the distances of the cities from the ideal set (𝒮) we defined using the Euclidean distance measure described in the third section, and present the results in a table. 𝒮 the ideal city where plants can grow most efficiently. Δ𝒜(𝓅1) is alternative criteria provided by Adıyaman province for Aronia. 𝑑𝐸(𝒮,Δ𝒜(𝓅1)) the distance between the ideal city where the plants can grow most efficiently and the alternative of the criteria provided by Adıyaman province for Aronia. 𝑑𝐸(𝒮,Δ𝒜(𝓅1))=12(√(1−0,7)2+√(0−0)2+√(0−0,1)2+√(0−0,2)2 4 + √ 𝑠({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}\{𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15})+𝑠({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}\{𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}) 𝑚𝑎𝑥{𝑠({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}∪{𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}),1} +𝑠({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}\{𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15})+𝑠({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}\{𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}) 𝑚𝑎𝑥{𝑠({𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}∪{𝓀11,𝓀12,𝓀13 ,𝓀14 ,𝓀15}),1} +𝑠(∅\∅)+𝑠(∅\∅) 𝑚𝑎𝑥{𝑠(∅∪∅),1}+𝑠(∅\{𝓀11})+𝑠({𝓀11}\∅) 𝑚𝑎𝑥{𝑠(∅∪{𝓀11}),1} +𝑠(∅\ {𝓀11,𝓀13})+𝑠({𝓀11,𝓀13}\∅) 𝑚𝑎𝑥{𝑠(∅∪{𝓀11,𝓀13}),1} 5 =12[((0,3)+(0)+(0,1)+(0,2) 4)+(√0+0 5+0+0 5+0+0 0+0+1 1+0+2 2 5)] Neutrosophic Sets and Systems, Vol. 97, 2026 555 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number =12[0,6 4+√0+0+0+1+1 5] =0,3912278 Table 11. Distance Table of Cities 𝓅1 𝓅2 𝓅3 𝓅4 Δ𝒜 0.3912278 0.6416667 0.5122983 0.4622984 Δℬ 0.4162278 0.5250000 0.3941625 0.4066625 Δ𝒟 0.4287278 0.4873106 0.4066625 0.4066625 Δ𝒢 0.3662278 0.4280311 0.3912278 0.3662278 Δ𝒦 0.4714102 0.5623106 0.4066625 0.2949490 Δℳ 0.4714102 0.5123106 0.3662278 0.4066625 Δ𝒮𝒾 0.4066625 0.5123106 0.41622778 0.4566625 Δ𝒮𝒶 0.4066625 0.5373106 0.4066625 0.2699490 Δ𝒮𝒾𝓇 0.4066625 0.5123106 0.03662278 0.4078427 Step 7: In Table 12, we obtain the weighted distances of the selected cities to the ideal city. 𝑑𝐸(𝒮,Δ𝑘(𝓅𝑐)) our distance measure results for the cities in our sample. Where, Δ𝑘= {Δ𝒜,Δℬ,Δ𝒟,Δ𝒢,Δ𝒦,Δℳ,Δ𝒮𝒾,Δ𝒮𝒶,Δ𝒮𝒾𝓇}. Table 12. Weighted Distance Table of Citys with most convenient City (0.3).𝓅1 (0.3).𝓅2 (0.2).𝓅3 (0.2).𝓅4 ∑𝑤𝑐. 4𝑐=1 𝑑𝐸(𝒮,Δ𝑘(𝓅𝑐)) Δ𝒜 0.1173683 0.1925000 0.1024597 0.09245597 0.5047877 Δℬ 0.1248683 0.1575000 0.0788325 0.0813325 0.4425333 Δ𝒟 0.1286183 0.0974621 0.0813325 0.0813325 0.3887454 Δ𝒢 0.1098683 0.1284093 0.0782456 0.0732456 0.3897688 Δ𝒦 0.1414231 0.1686932 0.0813325 0.0589898 0.4504465 Δℳ 0.1414231 0.1536932 0.0732456 0.0813325 0.4496944 Δ𝒮𝒾 0.1219988 0.1536932 0.0832456 0.0913325 0.4582593 Δ𝒮𝒶 0.1219988 0.1611932 0.0813325 0.0539898 0.4185143 Δ𝒮𝒾𝓇 0.1219988 0.1536932 0.0732456 0.0815685 0.4305061 Table 12. Let's create Table 13 to see the result part of the data we obtained more clearly. Neutrosophic Sets and Systems, Vol. 97, 2026 556 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number Table 13. Weighted Distance Table Results of Cities with Ideal City ∑𝑤𝑐. 4 𝑐=1 𝑑𝐸(𝒮,Δ𝑘(𝓅𝑐)) 0.5047877 0.4425333 0.3887454 0.3897688 0.4504465 0.4496944 0.4582593 0.4985143 0.4305061 5. Comparison Analysis In this section, we will examine the similarity measure used for neutrosophic quadruple numbers. Additionally, we is compare it with the similarity measure used in the study euclidean measures based on generalized set-valued neutrosophic quartic numbers and an application in healthcare by Şahin et al.[21].Furthermore, we will compare the distance measure used for generalized quintuple numbers (T, U, C, F) with the distance measure used for quadruple numbers (T, I, C, F). However, instead of directly adapting the distance measure for quintuple numbers to quadruple numbers, only U will be considered, without distinguishing between (U, C) in the quadruple numbers.Finally, the results obtained from the distance measure for quintuple numbers will be compared with the results when this measure is adapted to quadruple numbers.In Method 1: The calculation was made using the Euclidean similarity measure defined for generalized set-valued neutrosophic quadruple numbers [21]. It should be noted that when converting generalized neutrosophic quintuple set into generalized quadruple set, we obtained the C component, i.e. the contradiction, by discarding it. Table 14. Weighted Distance Table Results of Cities with Ideal City ∑𝑤𝑐. 4𝑐=1 𝑑𝐸(𝒮,Δ𝑘(𝓅𝑐)) Δ𝒜 0.4542775 Δℬ 0.3924542 Δ𝒟 0.4168000 Δ𝒢 0.3354917 Δ𝒦 0.4278085 Δℳ 0.4163696 Δ𝒮𝒾 0.4097867 Δ𝒮𝒶 0.4319452 Δ𝒮𝒾𝓇 0.4093523 Neutrosophic Sets and Systems, Vol. 97, 2026 557 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number According to Table 14, Gaziantep is the closest to our ideal city and the province we should prefer. It is followed by Batman, Şırnak, Siirt, Mardin, Diyarbakır, Kilis, Şanlıurfa, Adıyaman. Method 2: Let us use the definition of distance in Definition 2.3. [21] We should note here that we have included the contradiction part (C) in generalized neutrosophic quintuple numbers in the unknown part (U) in quadruple numbers, because the concept of contradiction was not considered when defining quadruple numbers, so we included the contradiction parts in the unknown. Table 15. Weighted Distance Table Results of Cities with Ideal City ∑𝑤𝑐. 4𝑐=1 𝑑𝐸(𝒮,Δ𝑘(𝓅𝑐)) Δ𝒜 0.5773655 Δℬ 0.4922996 Δ𝒟 0.7114680 Δ𝒢 0.5222461 Δ𝒦 0.5279799 Δℳ 0.5439183 Δ𝒮𝒾 0.5439183 Δ𝒮𝒶 0.6042011 Δ𝒮𝒾𝓇 0.5330019 According to Table 15, the closest province to our most convenient city and the one we should prefer is Batman. It is followed by Batman, Gaziantep, Kilis, Sırnak, Siirt, Mardin,Adıyaman, Şanlıurfa, Diyarbakır. Table 16. Comparison of distance measures Distance Measure Result Method used in chapter four Δ𝒟, Δ𝒢, Δ𝒮𝒶, Δ𝒮𝒾𝓇,Δℬ,Δℳ,Δ𝒦,Δ𝒮𝒾,Δ𝒜 Method 1 [21] Δ𝒢, Δ𝒮𝒾, Δ𝒟, Δℬ, Δ𝒮𝒾𝓇, Δℳ, Δ𝒦, Δ𝒮𝒶, Δ𝒜 Method 2 [21] Δℬ, Δ𝒢, Δ𝒦,Δ𝒮𝒾𝓇,Δ𝒮𝒾, Δℳ, Δ𝒜,Δ𝒮𝒶, Δ𝒟 Table 16 shows that different results are obtained for the three distance measures shown, and in our study, we have used the Euclidean distance measure for generalized set-valued neutrosophic quintuple numbers and this is the recommended distance measure. 6 Discussion Neutrosophic Sets and Systems, Vol. 97, 2026 558 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number In this paper, we define a Euclidean distance measure based on generalized set-valued neutrosophic quintuple numbers and show that this distance measure satisfies the distance measure conditions. Furthermore, by extending the generalized Euclidean distance measure based on generalized set-valued neutrosophic quadruple numbers, we have developed an algorithm based on generalized set-valued neutrosophic quintuple numbers. Using this algorithm, we have presented an application in the southeastern Anatolian region to determine in which cities in unknown cities, where plants with high economic return can be grown most efficiently, for which we know the growing conditions. We compared the results obtained in this application with the results obtained from the Euclidean distance measure based on generalized set-valued neutrosophic quadruple numbers in two different formats in Table 14 and Table 15 (the first format is C (contradiction) in quintuple numbers, the second format is the addition of C (contradiction) in neutrosophic quintuple numbers to I (unknown) in neutrosophic quadruple numbers) and we showed that we obtained different results. In Table 15, we have given the table comparing the results. In solving such problems, it is clear that a structure with known part (plants and growing conditions), unknown part (whether it grows in the cities we choose) and known neutrosophic membership functions T,U,C,F (expert opinion) is needed. Since it will take time to test whether a plant grows in a city we have chosen and since the temperature and climatic conditions of the year we try may mislead us, a structure such as (𝐺𝐾 , 𝐾𝐾 T, 𝐻𝐾 U, 𝐴𝐾 C, 𝑁𝐾 F ) containing both set and T, U, C, F will be needed. Therefore, the Euclidean distance measure based on generalized set-valued neutrosophic quintuple numbers can give better and more accurate results in solving such problems. In addition, by using the distance measure in this paper, it can be used as a solution for such problems not only in the southeastern Anatolia region but also in every region of the world. In addition, decision making applications can be obtained by using this distance measure and algorithm in other disciplines where we can make wise and correct choices. 7. Conclusion If we evaluate the result of the algorithm according to Table 13, we should make an evaluation by considering the following point, since we evaluate according to the Euclidean distance measure, the similarity rate decreases as the distance increases because we are looking at the distance using our ideal province. As we move away from our ideal province, our efficiency decreases. If we evaluate in the light of this information, according to this table calculated by considering the ideal growing conditions of the plant, our first preference is Diyarbakır. It is followed by Gaziantep, Şanlıurfa, Şırnak, Batman, Mardin, Kilis, Siirt, and Adıyaman. Researchers can benefit from this study and apply it not only in the provinces of the southeastern Anatolia region but also in all regions of the world, especially in Turkey, and if we know which plant to plant whereby making smart agriculture, plants can be grown with the highest yield without damaging natural resources. Researchers can extend the operators in set-valued neutrosophic quintuple sets and similar studies can be conducted with more than two expert opinions and real experts. In addition, different studies and results can be obtained by using other operators defined for generalized set-valued neutrosophic quintuple set. References [1] Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [2] Broumi, S., krishna Prabha, S., & Uluçay, V. (2023). Interval-Valued Fermatean Neutrosophic Shortest Path Problem via Score Function. Neutrosophic Systems With Applications, 11, 1–10. [3] Uluçay, V., Sahin, M., Olgun, N., & Kılıçman, A. (2016). On soft expert metric spaces. Malaysian Journal of Mathematical Sciences, 10(2), 221-231. [4] Başer, Z.; Uluçay, V. Effective Q–Fuzzy Soft Expert Sets and Its Some Properties. Uncertain. Discour. Appl. 2024, in press. Neutrosophic Sets and Systems, Vol. 97, 2026 559 __________________________________________________ _________________________________________________ Şahin and Doğan,Agricultural and Sustainable Resource Management with Euclidean Distance Measure on Generalized Set Valued Neutrosophic Quintuple Number [5] Kocasakal, E.G.; Uluçay, V. CoCoSo method based on generalized distance measure with trapezoidal fuzzy multi-numbers for solving multi-criteria decision-making method problems. Int. J. Inf. Technol. 2025, 1–13. [6] Deli, I., Uluçay, V., & Baser, Z. (2025). Neutrosophic Inference Systems Using Takagi-Sugeno-Kang Model and Its Application. Neutrosophic Sets and Systems, 88(1), 68. [7] Uluçay, V. (2026). A novel hybrid distance-based similarity measures for n-valued neutrosophic trapezoidal numbers and its applications. Neutrosophic Sets and Systems, 95, 62-74. [8] Uluçay, V., Şahin, N. M., Toz, N. İ., & Bozkurt, E. (2023). VIKOR Method for Decision-Making Problems Based on Q-Single-Valued Neutrosophic Sets: Law Application. Journal of fuzzy extension & applications (JFEA), 4(4). [9] Uluçay, V., Deli, İ., & Başer, Z. (2025). Intuitionistic fuzzy Takagi–Sugeno–Kang non-linear inference system and its applications to greenhouse control system in agriculture. International Journal of Information Technology, 1-10. [10] Uluçay, V., Kocasakal, E.G., & Başer, Z. (2025). A hybrid decision-making technique of CoCoSo and entropy methods on trapezoidal fuzzy multi numbers and its application. Journal of Fuzzy Extension and Applications, e227153. [11] Smarandache, F. A unifying field in logics. Neutrosophy: neutrosophic probability, set and logic; American Research Press: Rehoboth, USA, 1998; pp. 1–204. [12] Başer, Z.; Uluçay, V. Energy of a neutrosophic soft set and its applications to multicriteria decisionmaking problems. Neutrosophic Sets Syst. 2025, 79, 28. [13] Başer, Z.; Uluçay, V. Effective Q-Neutrosophic Soft Expert Sets and Its Application in Decision Making. In Algebraic Structures In the Universe of Neutrosophic: Analysis with Innovative Algorithmic Approaches; 2024; pp. 147. [14] Ghimire, S.; Bhurtel, N.; Jha, S.; Ahmad, S.; Abdeljaber, H.A.; Nazeer, J. Minimizing energy consumption in fixed node networks using a novel neutrosophic model: comparative analysis with standard and existing algorithms. Int. J. Inf. Technol. 2025, 1–15. [15] Wajid, M.A.; Zafar, A.; Wajid, M.S. A deep learning approach for image and text classification using neutrosophy. Int. J. Inf. Technol. 2024, 16, 853–859. [16] Narmada Devi, R.; Sowmiya, S.; Parthiban, Y. Solving of assignment problem by Pythagorean octagonal neutrosophic fuzzy number. Int. J. Inf. Technol. 2025, 17, 1079–1085. [17] Smarandache, F. Neutrosophic quadruple numbers, refined neutrosophic quadruple numbers, absorbance law, and the multiplication of neutrosophic quadruple numbers. Neutrosophic Set Syst. 2015, 10, 96–98. [18] Şahin, M.; Kargın, A. Neutrosophic triplet groups based on set-valued neutrosophic quadruple numbers. Neutrosophic Set Syst. 2019, 30, 122–131. [19] Şahin, M.; Kargın, A.; Kılıç, A. Generalized set–valued neutrosophic quadruple sets and numbers. In Quadruple Neutrosophic Theory and Applications; Pons Editions: Brussels, Belgium, 2020; Vol. 2, pp. 23–40. [20] Kargın, A.; Dayan, A.; Şahin, M. Generalized Hamming Similarity Measure Based on Neutrosophic Quadruple Numbers and Its Applications to Law Sciences. Neutrosophic Sets Syst. 2021, 40, 4. [21] Şahin, M.; Kargın, A.; Uz, M.S. Generalized Euclid measures based on generalized set valued neutrosophic quadruple numbers and multi criteria decision making applications. Neutrosophic Sets Syst. 2021, 47, 573–600. [22] Chatterjee, R.; Majumdar, P.; Samanta, S.K. On some similarity measures and entropy on quadri partitioned single valued neutrosophic sets. J. Intell. Fuzzy Syst. 2016, 30, 2475–2485. [23] Kargın, A.; Şahin, M.; Şiğva, K.A. Operators Based On Multiple Generalized Set-Valued Neutrosophic Quadruple Sets. Neutrosophic Sets Syst. 2024, 70, 107–136. [24] Şahin, M.; Kargın, A.; Doğan, K. Neutrosophic Quintuple Numbers And Set Valued Neutrosophic Quintuple Numbers. In Proceedings of the III. Başkent Int. Conf. on Multidisciplinary Studies, Ankara, Turkey, 23 September 2022. [25] Şahin, M.; Kargın, A.; Doğan, K. Generalized Set Valued Neutrosophic Quintupe Numbers. In Proceedings of the Ankara Int. Congr. On Scientific Research-IX, Ankara, Turkey, 26–29 December 2023; pp. 55–62.