Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis
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Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis T. Mythili1, V. Jeyanthi2, D. Maheswari3, W.F.Al Omeri4 1Research Scholar, 2,3,4Assistant Professor 1,2,3Department of Mathematics, Sri Krishna Arts and Science College, Coimbatore, Tamil Nadu, India. 4Mathematics Department, Faculty of Science, Jadara University, Irbid 21110, Jordan. E-mail: [email protected] ;[email protected] [email protected] ;[email protected] Abstract: In real-life situations, we often encounter challenges involving uncertainty, vagueness, complexity, and unpredictability. Heptapartitioned Neutrosophic Sets serve as a mathematical tool to address issues that existing methods cannot effectively resolve. Heptapartitioned Neutrosophic Soft Matrices play a vital role in managing indeterminate and inconsistent information during the decision-making process. This paper focuses on exploring the concepts of Heptapartitioned Neutrosophic Sets, Heptapartitioned Neutrosophic Soft Sets, and Heptapartitioned Neutrosophic Soft Matrices, which are highly effective in handling situations involving uncertainty and imprecision. It further proposes a novel method for constructing decision matrices using these soft matrices as a practical application of the theory. The study introduces an algorithm based on Heptapartitioned Neutrosophic Soft Matrices to address problems in disease diagnosis by analyzing patient symptoms. Specifically, it examines the relationship between patients and symptoms, as well as between symptoms and diseases, using these matrices. To support decision-making, a score matrix is defined by applying max-min operations and complementation of Heptapartitioned Neutrosophic Soft Matrices. Keywords: Fuzzy sets, Soft sets, Soft matrix, Heptapartitioned Neutrosophic soft sets, Heptapartitioned Neutrosophic soft matrix. AMS Subject Classification: 54A99, 54B99. ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 University of New Mexico T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical DiagnosisT. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
1 Introduction The concept of fuzzy sets, introduced by Zadeh [16], has been applied in many research areas. Fuzzy sets are useful because they handle uncertainty and vagueness that cannot be addressed with traditional crisp sets. In fuzzy set theory, a membership function assigns each element in the universe of discourse a value between 0 and 1, representing the degree to which the element belongs to the set. Normally, the membership degree is a single value between 0 and 1. However, in real-life situations, the degree of non-membership is not always simply ”1 minus the membership degree” due to the presence of hesitation or uncertainty. To address this, Atanassov [1] introduced a generalization of fuzzy sets called intuitionistic fuzzy sets, which considers non-membership. Intuitionistic fuzzy sets, as an extension of fuzzy sets, are particularly helpful in situations involving linguistic variables. They are useful in decision-making problems, especially in domains such as medical diagnosis, sales analysis, product marketing, and financial services. In such cases, there is often some level of hesitation or uncertainty during the evaluation of an unknown project. In real-life situations, many problems in areas like economics, social sciences, and the environment involve various uncertainties. However, most existing mathematical tools used for modeling, reasoning, and computation are precise, deterministic, and rigid. While there are theories such as probability, evidence theory, fuzzy sets, intuitionistic fuzzy sets and rough sets to handle uncertainties, they each have their limitations, as noted by Molodsov [12]. To address these challenges, the concept of soft set theory was introduced. Soft set theory has great potential for solving practical problems in fields like economics, social sciences, and medical sciences. Maji et al. ([7]-[9]) explored the fuzzy soft set theory, and later, it was extended to intuitionistic fuzzy soft sets in work. Smarandache [13] further generalized soft sets into hypersoft sets, applying them in decision-making processes. Additionally, Vellapandi and Gunasekaran [15] proposed a new decision-making method using multi-soft set logic. Although intuitionistic fuzzy sets can handle incomplete information by considering both truth membership and falsity membership values. However, they cannot deal with inconsistent or indeterminate information found in belief systems. To address this, Smarandache [14] introduced the concept of neutrosophic sets as a mathematical tool for handling situations that involve uncertainty, inconsistencies, and indeterminacy. Neutrosophic sets are expected to pro2 ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 457 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
vide more accurate results compared to fuzzy sets or intuitionistic fuzzy sets. Many researchers ([2],[3],[6],[10],[11]) have studied and applied neutrosophic sets in various decision-making processes. Quadripartitioned Neutrosophic Topological Spaces were introduced by Das et.al by applying topology to quadripartitioned neutrosophic sets. In 2020, Rama Malik introduced the concept of pentapartitioned sets. Later, in 2021, R. Radha and A. Stanis Arul Mary expanded the ideas of neutrosophic sets (N-S) and quadripartitioned N-S to develop heptapartitioned neutrosophic sets. In 2023, V. Jeyanthi and T. Mythili [5] introduced heptapartitioned neutrosophic topological spaces. These refine the components U, C, and G, while T is further split into absolute truth (TA)and relative truth (TR), and F into absolute falsity (FA)and relative falsity (FR). The foundation for these ideas traces back to 1995, when F. Smarandache introduced Seven-Symbol-Valued Neutrosophic Logic. A heptapartitioned neutrosophic set provides a more detailed classification of uncertainty by dividing it into seven distinct components, allowing for finer analysis compared to the standard neutrosophic set. This makes it especially useful in complex decision-making scenarios where deeper insights are required. 2 Preliminaries This section provides fundamental definitions that will be useful in the later parts of the article. Definition 2.0.1. Soft Set. Let Ube the initial universe of discourse, and Ebe a set of parameters. Let P(U)denote the power set of U. A soft set over Uis defined as a pair (E, F), where Fis a mapping defined as:F:E→P(U).In other words, a soft set is a mapping that assigns to each parameter in Ea subset of the universe U. Example 2.0.2. Let U={u1, u2, u3, u4}represent a set of four types of ornaments, and let E={costly(e1),medium(e2),cheap(e3)}be the set of parameters. Assume A={e1, e3} ⊆ E. Define the mappings as follows: F(e1) = {u1, u4}, F(e3) = {u2, u3}.The soft set can then be expressed as:(F, E) = {(e1,{u1, u4}),(e3,{u2, u3})}.This soft set represents the “quality ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 458 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
of furniture” that Mr. X plans to buy. It can also be represented in tabular form as follows: UCostly (e1)Medium (e2)Cheap (e3) u11 0 0 u20 0 1 u30 0 1 u41 0 0 This table clearly shows which ornaments are considered costly, medium, or cheap based on Mr. X’s preferences. Definition 2.0.3. Fuzzy Soft Set. Let Ube the universe of discourse, and Ebe a set of parameters. Let P(U)denote the collection of all fuzzy subsets of U. For A⊆E, a pair (FA, E)is called a fuzzy soft set over U, where FAis a mapping defined as: FA:E→P(U).In other words, a fuzzy soft set is a mapping that assigns to each parameter in Aa fuzzy subset of U. Example 2.0.4. Let U={u1, u2, u3, u4}represent a set of four types of ornaments, and let E={costly(e1),medium(e2),cheap(e3),very cheap(e4)}be the set of parameters. Consider the following fuzzy mappings: FA(e1) = {(u1,0.7),(u2,0.8),(u3,0.0),(u4,0.5)}, FA(e3) = {(u1,0.3),(u2,0.4),(u3,0.6),(u4,0.5)}.This fuzzy soft set can be represented in tabular form as follows: UCostly (e1)Medium (e2)Cheap (e3) u10.7 0.0 0.3 u20.8 0.0 0.4 u30.0 0.0 0.6 u40.5 0.0 0.5 This table illustrates the degrees of membership of each ornament in the fuzzy subsets associated with the parameters ”costly,” ”medium,” and ”cheap.” Definition 2.0.5. Intuitionistic Fuzzy Set. Let Ube the universe of discourse. An intuitionistic fuzzy set Ais defined as an object of the form:A={hx, µA(x), νA(x)i | x∈U},where µA(x) : U→[0,1] and νA(x) : U→[0,1] represent the degree of membership and the degree of non-membership of the element x∈Xin the set A, respectively. These functions satisfy the condition: 0≤µA(x) + νA(x)≤1. ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 459 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
Definition 2.0.6. Intuitionistic Fuzzy Soft Set. Let Ube the universe of discourse, and let E be a set of parameters. Let P(U)represent the collection of all intuitionistic fuzzy subsets of U. For A⊆E, a pair (FA, E)is called an intuitionistic fuzzy soft set over U, where FAis a mapping defined as: FA:E→P(U). Definition 2.0.7. Neutrosophic Set. Let Ube the universe of discourse. A neutrosophic set Aon Uis defined as: A={hx, TA(x), IA(x), FA(x)i:x∈U},where the functions TA(x), IA(x), FA(x) : U→[0,1] are called the characteristic functions. TA(x)represents the degree of membership of xin A,IA(x)represents the degree of indeterminacy of xin A, and FA(x)represents the degree of non-membership of xin A. These functions satisfy the condition: 0≤TA(x) + IA(x) + FA(x)≤3. Definition 2.0.8. Neutrosophic Soft Set. Let Ube the universe of discourse, and let Ebe a set of parameters. Let P(U)denote the collection of all neutrosophic subsets of U. For A⊆E, a pair (FA, E)is called a neutrosophic soft set over U, where FAis a mapping defined as FA:E→P(U). Definition 2.0.9. Neutrosophic soft matrix[4] Let (FA, E)be a neutrosophic soft set over U, where FAis a mapping given by FA:E→P(U),and P(U)is the collection of all neutrosophic subsets of U. The subsets of U×Eare uniquely defined by RA={(u, e) : e∈ A, u ∈FA(e)},which is called the relation form of (FA, E). The relation RAis characterized by: The truth membership function: TA:U×E→[0,1], The indeterminacy membership function: IA:U×E→[0,1], The falsity membership function: FA:U×E→[0,1], where TA(u, e),IA(u, e), and FA(u, e)represent the truth membership value, indeterminacy membership value, and falsity membership value of the object uassociated with the parameter e, respectively. Let U={u1, u2, u3, . . . , um}be the universe of discourse and E={e1, e2, e3, . . . , en}be the set of parameters. The relation RAcan be represented in tabular form as: RAe1e2· · · en u1(TA11 , IA11 , FA11 ) (TA12 , IA12 , FA12 )· · · (TA1n, IA1n, FA1n) u2(TA21 , IA21 , FA21 ) (TA22 , IA22 , FA22 )· · · (TA2n, IA2n, FA2n) · · · · · · · · · · · · · · · um(TAm1, IAm1, FAm1) (TAm2, IAm2, FAm2)· · · (TAmn , IAmn , FAmn ) ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 460 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
If aij = (TA(ui, ej), IA(ui, ej), FA(ui, ej)), the neutrosophic soft matrix can be written as: A= a11 a12 · · · a1n a21 a22 · · · a2n . . .. . ..... . . am1am2· · · amn , where (TAmn , IAmn , FAmn )=(TA(um, en), IA(um, en), FA(um, en)) This is called the neutrosophic soft matrix corresponding to the neutrosophic soft set (FA, E) over U. 3 Heptapartitioned Neutrosophic Soft Matrices Definition 3.0.1. Heptapartitioned Neutrosophic Sets. Let Ube a non-empty universe. A heptapartitioned neutrosophic set Aon Uis an object of the form: A={hx, TA(x), MA(x), CA(x), UA(x), IA(x), KA(x), FA(x)i:x∈U},where we assume that TA(x), MA(x), CA(x), UA(x), IA(x), KA(x), FA(x) : U→[0,1],and TA(x) + MA(x) + CA(x) + UA(x) + IA(x) + KA(x) + FA(x)≤7.Here TA(x)is truth membership, MA(x)is relative truth, CA(x)is contradiction, UA(x)is unknown membership, IA(x)is ignorance, KA(x)is relative falsity, FA(x)is absolute falsity. Definition 3.0.2. Heptapartitioned Neutrosophic Soft Sets. Let Ube a non-empty universe and Ebe the set of parameters. Consider a non-empty set A⊆E. Let PHN (U)denote the set of all heptapartitioned neutrosophic sets of U. The collection (FA, E)is termed to be a heptapartitioned neutrosophic soft set over U, where FAis a mapping given by: FA:E→ PHN (U).The following example will illustrate the concept. Let U be the set of houses under consideration and E be the set of parameters where each parameter includes heptapartitioned neutrosophic words. Example 3.0.3. Let U={u1, u2, u3, u4, u5, u6},be the set of six types of ornaments, E= {expensive (e1),moderate (e2),cheap (e3)},be the set of three parameters. Let us consider the following case FA(e1): (u1,0.3,0.4,0.2,0.1,0.2,0.3,0.1),(u2,0.5,0.3,0.2,0.2,0.3,0.1,0.2), (u3,0.4,0.4,0.1,0.2,0.2,0.3,0.1),(u4,0.5,0.2,0.1,0.3,0.2,0.1,0.3), (u5,0.6,0.2,0.1,0.2,0.1,0.3,0.2),(u6,0.4,0.3,0.2,0.2,0.3,0.2,0.1). ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 461 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
FA(e2): (u1,0.4,0.5,0.1,0.2,0.3,0.1,0.1),(u2,0.3,0.4,0.2,0.1,0.2,0.3,0.2), (u3,0.2,0.6,0.1,0.2,0.2,0.1,0.2),(u4,0.6,0.2,0.2,0.1,0.2,0.3,0.2), (u5,0.5,0.3,0.1,0.2,0.1,0.2,0.2),(u6,0.4,0.2,0.2,0.3,0.1,0.1,0.3). FA(e3): (u1,0.5,0.2,0.1,0.3,0.2,0.3,0.1),(u2,0.4,0.5,0.1,0.2,0.2,0.1,0.2), (u3,0.5,0.4,0.1,0.1,0.3,0.1,0.3),(u4,0.4,0.2,0.3,0.2,0.2,0.3,0.2), (u5,0.4,0.3,0.2,0.1,0.2,0.1,0.3),(u6,0.5,0.3,0.2,0.3,0.1,0.3,0.2). The tabular representation of the HNSS is U Expensive (e1)Moderate (e2)Cheap (e3) u1(0.3,0.4,0.2,0.1,0.2,0.3,0.1) (0.4,0.5,0.1,0.2,0.3,0.1,0.1) (0.5,0.2,0.1,0.3,0.2,0.3,0.1) u2(0.5,0.3,0.2,0.2,0.3,0.1,0.2) (0.3,0.4,0.2,0.1,0.2,0.3,0.2) (0.4,0.5,0.1,0.2,0.2,0.1,0.2) u3(0.4,0.4,0.1,0.2,0.2,0.3,0.1) (0.2,0.6,0.1,0.2,0.2,0.1,0.2) (0.5,0.4,0.1,0.1,0.3,0.1,0.3) u4(0.5,0.2,0.1,0.3,0.2,0.1,0.3) (0.6,0.2,0.2,0.1,0.2,0.3,0.2) (0.4,0.2,0.3,0.2,0.2,0.3,0.2) u5(0.6,0.2,0.1,0.2,0.1,0.3,0.2) (0.5,0.3,0.1,0.2,0.1,0.2,0.2) (0.4,0.3,0.2,0.1,0.2,0.1,0.3) u6(0.4,0.3,0.2,0.2,0.3,0.2,0.1) (0.4,0.2,0.2,0.3,0.1,0.1,0.3) (0.5,0.3,0.2,0.3,0.1,0.3,0.2) Definition 3.0.4. Heptapartitioned Neutrosophic Soft matrix. Let (FA, E)be a heptapartitioned neutrosophic soft set over a universe U={u1, u2, . . . , un}, where FA:E→PHN (U), and PHN (U)is the collection of all neutrosophic subsets of U. Then the relationship between the objects in Uand parameters in Ecan be represented using the heptapartitioned neutrosophic soft matrix. The subsets of U×Eare uniquely defined by RA={(u, e) : e∈A, u ∈ FA(e)},and RAis called the relation form of (FA, E). In the heptapartitioned neutrosophic soft matrix, the membership values are characterized by seven functions for each pair (u, e)∈U×Ewhere TA(u, e)is the Truth membership, MA(u, e) is the Relative truth membership, CA(u, e)is the Contradiction membership, UA(u, e)is the Unknown membership, IA(u, e)is the Ignorance membership, KA(u, e)is the Relative falsity membership, and FA(u, e)is the Absolute falsity membership. Let U={u1, u2, u3, . . . , um}be the universe of discourse and E={e1, e2, e3, . . . , en}be the ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 462 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
set of parameters. The relation RAcan be represented in tabular form as: RAe1e2· · · en u1 (TA11 , MA11 , CA11 , UA11 , IA11 , KA11 , FA11 ) (TA12 , MA12 , CA12 , UA12 , IA12 , KA12 , FA12 ) · · · (TA1n, MA1n, CA1n, UA1n, IA1n, KA1n, FA1n) u2 (TA21 , MA21 , CA21 , UA21 , IA21 , KA21 , FA21 ) (TA22 , MA22 , CA22 , UA22 , IA22 , KA22 , FA22 ) · · · (TA2n, MA2n, CA2n, UA2n, IA2n, KA2n, FA2n) · · · · · · · · · · · · · · · um (TAm1, MAm1, CAm1, UAm1, IAm1, KAm1, FAm1) (TAm2, MAm2, CAm2, UAm2, IAm2, KAm2, FAm2) · · · (TAmn , MAmn , CAmn , UAmn , IAmn , KAmn , FAmn ) If aij = (TA(ui, ej), MA(ui, ej), CA(ui, ej), UA(ui, ej), IA(ui, ej), KA(ui, ej), FA(ui, ej)), the Heptapartitioned neutrosophic soft matrix can be written as: A= a11 a12 · · · a1n a21 a22 · · · a2n . . .. . ..... . . am1am2· · · amn , where (TAmn , MAmn , CAmn , UAmn , IAmn , KAmn , FAmn )=(TA(um, en), MA(um, en), CA(um, en), UA(um, en), IA(um, en), KA(um, en), FA(um, en)) This is called the heptapartitioned neutrosophic soft matrix corresponding to the heptapartitioned neutrosophic soft set (FA, E)over U. Example 3.0.5. Let U={u1, u2, u3, u4, u5, u6}be the universal set and E={e1, e2, e3, e4} be the set of parameters, A={e1, e2, e3} Let us consider the following case FA(e1): (u1,0.3,0.2,0.4,0.5,0.1,0.6,0.2),(u2,0.5,0.4,0.3,0.6,0.2,0.1,0.3), (u3,0.6,0.3,0.2,0.4,0.5,0.1,0.3),(u4,0.2,0.3,0.4,0.5,0.6,0.2,0.1), (u5,0.4,0.5,0.3,0.2,0.1,0.3,0.2),(u6,0.3,0.2,0.5,0.1,0.4,0.6,0.3). FA(e2): ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 463 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
(u1,0.4,0.5,0.2,0.3,0.6,0.1,0.2),(u2,0.3,0.6,0.4,0.2,0.1,0.5,0.3), (u3,0.5,0.3,0.6,0.2,0.4,0.1,0.2),(u4,0.6,0.1,0.5,0.3,0.2,0.4,0.1), (u5,0.2,0.4,0.1,0.6,0.5,0.3,0.2),(u6,0.3,0.5,0.2,0.1,0.6,0.4,0.3). FA(e3): (u1,0.5,0.3,0.4,0.2,0.1,0.6,0.2),(u2,0.4,0.2,0.5,0.3,0.6,0.1,0.2), (u3,0.6,0.5,0.3,0.1,0.2,0.4,0.3),(u4,0.3,0.6,0.1,0.5,0.2,0.4,0.3), (u5,0.2,0.4,0.6,0.3,0.1,0.5,0.2),(u6,0.4,0.3,0.2,0.6,0.5,0.1,0.3). Then the HNSS(FA, E)is a parameterized family {FA(e1), FA(e2), FA(e3)}of all HNSS over Uand gives an approximate description of the object. Hence, the heptapartitioned neutrosophic soft matrix can be represented by: A= (0.3,0.2,0.4,0.5,0.1,0.6,0.2) (0.4,0.5,0.2,0.3,0.6,0.1,0.2) (0.5,0.3,0.4,0.2,0.1,0.6,0.2) (0.5,0.4,0.3,0.6,0.2,0.1,0.3) (0.3,0.6,0.4,0.2,0.1,0.5,0.3) (0.4,0.2,0.5,0.3,0.6,0.1,0.2) (0.6,0.3,0.2,0.4,0.5,0.1,0.3) (0.5,0.3,0.6,0.2,0.4,0.1,0.2) (0.6,0.5,0.3,0.1,0.2,0.4,0.3) (0.2,0.3,0.4,0.5,0.6,0.2,0.1) (0.6,0.1,0.5,0.3,0.2,0.4,0.1) (0.3,0.6,0.1,0.5,0.2,0.4,0.3) (0.4,0.5,0.3,0.2,0.1,0.3,0.2) (0.2,0.4,0.1,0.6,0.5,0.3,0.2) (0.2,0.4,0.6,0.3,0.1,0.5,0.2) (0.3,0.2,0.5,0.1,0.4,0.6,0.3) (0.3,0.5,0.2,0.1,0.6,0.4,0.3) (0.4,0.3,0.2,0.6,0.5,0.1,0.3) Definition 3.0.6. Complement of Heptapartitioned Neutrosophic Soft matrices. Let A= [(Tij, Mij, Cij, Uij, Iij, Kij, Fij)]m×n∈HNSM then the complement of the heptapartitioned neutrosophic soft matrix Ais denoted by Acand is defined as: Ac= [(Fij, Kij, Iij,1−Uij, Cij, Mij, Tij)]m×n∈HNSM for all iand j. The complement of the matrix will be Ac= (0.2,0.6,0.1,0.5,0.4,0.2,0.3) (0.2,0.1,0.6,0.7,0.2,0.5,0.4) (0.2,0.6,0.1,0.8,0.4,0.3,0.5) (0.3,0.1,0.2,0.4,0.3,0.4,0.5) (0.3,0.5,0.1,0.8,0.4,0.6,0.3) (0.2,0.1,0.6,0.7,0.5,0.2,0.4) (0.3,0.1,0.5,0.6,0.2,0.3,0.6) (0.2,0.1,0.4,0.8,0.6,0.3,0.5) (0.3,0.4,0.2,0.9,0.3,0.5,0.6) (0.1,0.2,0.6,0.5,0.4,0.3,0.2) (0.1,0.4,0.2,0.7,0.5,0.1,0.6) (0.3,0.4,0.2,0.5,0.1,0.6,0.3) (0.2,0.3,0.1,0.8,0.3,0.5,0.4) (0.2,0.3,0.5,0.4,0.1,0.4,0.2) (0.2,0.5,0.1,0.7,0.6,0.4,0.2) (0.3,0.6,0.4,0.9,0.5,0.2,0.3) (0.3,0.4,0.6,0.9,0.2,0.5,0.3) (0.3,0.1,0.5,0.4,0.2,0.3,0.4) ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 464 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
A= s1s2s3s4s5 P1(0.8,0.1,0.1,0.7,0.2,0.9,0.6) (0.7,0.1,0.2,0.8,0.1,0.9,0.6) (0.6,0.2,0.2,0.7,0.1,0.9,0.5) (0.4,0.4,0.2,0.5,0.3,0.7,0.5) (0.5,0.3,0.2,0.6,0.2,0.8,0.4) P2(0.6,0.2,0.2,0.8,0.3,0.7,0.5) (0.5,0.3,0.2,0.7,0.4,0.7,0.5) (0.4,0.3,0.3,0.6,0.4,0.8,0.6) (0.5,0.3,0.2,0.7,0.4,0.8,0.4) (0.7,0.1,0.2,0.9,0.1,0.9,0.6) P3(0.7,0.2,0.1,0.6,0.2,0.8,0.4) (0.6,0.2,0.2,0.5,0.2,0.6,0.4) (0.8,0.1,0.1,0.8,0.2,0.9,0.7) (0.6,0.2,0.2,0.5,0.3,0.6,0.3) (0.6,0.2,0.2,0.7,0.3,0.8,0.5) P4(0.5,0.3,0.2,0.5,0.4,0.6,0.3) (0.4,0.4,0.2,0.6,0.3,0.6,0.3) (0.5,0.3,0.2,0.6,0.3,0.6,0.4) (0.7,0.1,0.2,0.8,0.2,0.9,0.7) (0.4,0.4,0.2,0.5,0.4,0.7,0.3) ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 471 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
Again let the set S={S1, S2, S3, S4, S5}be a Universal set where S1, S2, S3, S4and S5represents symptoms temperature, headaches, coughs, stomach pain, and body pain, respectively. Let the possible diseases relating to the above symptoms D={D1, D2, D3}be viral fever, typhoid, and malaria. Suppose that HNSS(G,D) over S, where G is a mapping G:D→FS gives a collection of an approximate description of medical knowledge of the three diseases and their symptoms. Let (G, D) = G(D1) = {(s1,0.6,0.2,0.3,0.5,0.1,0.7,0.2),(s2,0.3,0.5,0.4,0.6,0.2,0.8,0.1), (s3,0.1,0.8,0.1,0.2,0.3,0.6,0.4),(s4,0.4,0.5,0.3,0.5,0.2,0.7,0.3), (s5,0.7,0.4,0.2,0.6,0.3,0.8,0.5)} G(D2) = {(s1,0.6,0.2,0.2,0.4,0.3,0.7,0.1),(s2,0.2,0.6,0.3,0.5,0.3,0.6,0.2), (s3,0.5,0.4,0.3,0.6,0.1,0.7,0.2),(s4,0.7,0.2,0.1,0.5,0.3,0.6,0.4), (s5,0.1,0.8,0.2,0.3,0.5,0.7,0.2)} G(D3) = {(s1,0.3,0.4,0.3,0.5,0.2,0.6,0.3),(s2,0.7,0.2,0.4,0.6,0.3,0.5,0.2), (s3,0.7,0.2,0.3,0.5,0.3,0.6,0.2),(s4,0.3,0.4,0.4,0.5,0.3,0.6,0.4), (s5,0.2,0.7,0.3,0.6,0.2,0.7,0.3)} Heptapartitioned Neutrosophic soft set can be represented by the following neutrosophic soft matrix. B= D1D2D3 s1(0.6,0.2,0.3,0.5,0.1,0.7,0.2) (0.6,0.2,0.2,0.4,0.3,0.7,0.1) (0.3,0.4,0.3,0.5,0.2,0.6,0.3) s2(0.3,0.5,0.4,0.6,0.2,0.8,0.1) (0.2,0.6,0.3,0.5,0.3,0.6,0.2) (0.7,0.2,0.4,0.6,0.3,0.5,0.2) s3(0.1,0.8,0.1,0.2,0.3,0.6,0.4) (0.5,0.4,0.3,0.6,0.1,0.7,0.2) (0.7,0.2,0.3,0.5,0.3,0.6,0.2) s4(0.4,0.5,0.3,0.5,0.2,0.7,0.3) (0.7,0.2,0.1,0.5,0.3,0.6,0.4) (0.3,0.4,0.4,0.5,0.3,0.6,0.4) s5(0.7,0.4,0.2,0.6,0.3,0.8,0.5) (0.1,0.8,0.2,0.3,0.5,0.7,0.2) (0.2,0.7,0.3,0.6,0.2,0.7,0.3) ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 472 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
Heptapartitioned Neutrosophic soft complement matrix Bc= D1D2D3 s1(0.2,0.7,0.1,0.5,0.3,0.2,0.6) (0.1,0.7,0.3,0.6,0.2,0.2,0.6) (0.3,0.6,0.2,0.5,0.3,0.4,0.3) s2(0.1,0.8,0.2,0.4,0.4,0.5,0.3) (0.2,0.6,0.3,0.5,0.3,0.6,0.2) (0.2,0.5,0.3,0.4,0.4,0.2,0.7) s3(0.4,0.6,0.3,0.8,0.1,0.8,0.1) (0.2,0.7,0.1,0.4,0.3,0.4,0.5) (0.2,0.6,0.3,0.5,0.3,0.2,0.7) s4(0.3,0.7,0.2,0.5,0.3,0.5,0.4) (0.4,0.6,0.3,0.5,0.1,0.2,0.7) (0.4,0.6,0.3,0.5,0.4,0.4,0.3) s5(0.5,0.8,0.3,0.4,0.2,0.4,0.7) (0.2,0.7,0.5,0.7,0.2,0.8,0.1) (0.3,0.7,0.2,0.4,0.3,0.7,0.2) Max-min compositions of two heptapartitioned neutrosophic soft matrices will produce the following results: Let us suppose A∗B= [Cij]mX n where C11 =max(0.6,0.3,0.1,0.4,0.5),max(0.1,0.1,0.2,0.4,0.3),max(0.1,0.2,0.1,0.2,0.2) min(0.7,0.8,0.7,0.5,0.6),min(0.2,0.2,0.3,0.3,0.3),min(0.9,0.9,0.9,0.7,0.8), min(0.6,0.6,0.5,0.5,0.5)= (0.6,0.4,0.2,0.5,0.2,0.7,0.5) C12 =max(0.6,0.2,0.5,0.4,0.1),max(0.1,0.1,0.2,0.2,0.3),max(0.1,0.2,0.2,0.1,0.2) min(0.7,0.8,0.7,0.5,0.6),min(0.3,0.3,0.1,0.3,0.5),min(0.9,0.9,0.9,0.7,0.8), min(0.6,0.6,0.5,0.5,0.4)= (0.6,0.3,0.2,0.5,0.1,0.7,0.4) C13 =max(0.3,0.7,0.6,0.3,0.2),max(0.1,0.1,0.2,0.4,0.3),max(0.1,0.2,0.2,0.2,0.2) min(0.7,0.8,0.7,0.5,0.6),min(0.2,0.3,0.3,0.3,0.2),min(0.9,0.9,0.9,0.7,0.8), min(0.6,0.6,0.5,0.5,0.4)= (0.7,0.4,0.2,0.5,0.2,0.7,0.4) C21 =max(0.6,0.3,0.1,0.4,0.7),max(0.2,0.3,0.3,0.3,0.1),max(0.2,0.2,0.1,0.2,0.2) min(0.8,0.7,0.6,0.7,0.9),min(0.3,0.4,0.4,0.4,0.3),min(0.7,0.8,0.8,0.8,0.9), min(0.5,0.5,0.6,0.4,0.6)= (0.7,0.3,0.2,0.6,0.3,0.7,0.4) C22 =max(0.6,0.2,0.4,0.5,0.1),max(0.2,0.3,0.3,0.2,0.1),max(0.2,0.2,0.3,0.1,0.2) min(0.8,0.7,0.6,0.7,0.9),min(0.3,0.4,0.4,0.4,0.5),min(0.7,0.7,0.8,0.8,0.9), min(0.5,0.5,0.6,0.4,0.6)= (0.6,0.3,0.3,0.6,0.3,0.7,0.4) ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 473 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
C23 =max(0.3,0.5,0.4,0.3,0.2),max(0.2,0.2,0.2,0.3,0.1),max(0.2,0.2,0.3,0.2,0.2) min(0.8,0.7,0.6,0.7,0.9),min(0.3,0.4,0.4,0.4,0.2),min(0.7,0.7,0.8,0.8,0.9), min(0.5,0.5,0.6,0.4,0.6)= (0.5,0.3,0.3,0.6,0.2,0.7,0.4) C31 =max(0.6,0.3,0.1,0.4,0.6),max(0.2,0.2,0.1,0.2,0.2),max(0.1,0.2,0.1,0.2,0.2) min(0.6,0.6,0.8,0.5,0.7),min(0.2,0.2,0.3,0.3,0.3),min(0.8,0.8,0.9,0.7,0.8), min(0.4,0.4,0.7,0.3,0.5)= (0.6,0.2,0.2,0.5,0.2,0.7,0.4) C32 =max(0.6,0.2,0.5,0.6,0.1),max(0.2,0.2,0.1,0.2,0.2),max(0.1,0.2,0.1,0.1,0.2) min(0.6,0.5,0.8,0.5,0.7),min(0.3,0.3,0.2,0.3,0.5),min(0.8,0.6,0.9,0.6,0.8), min(0.4,0.4,0.7,0.4,0.5)= (0.6,0.2,0.2,0.5,0.2,0.6,0.4) C33 =max(0.3,0.6,0.7,0.3,0.2),max(0.2,0.2,0.1,0.2,0.2),max(0.1,0.2,0.1,0.2,0.2) min(0.6,0.6,0.8,0.5,0.7),min(0.2,0.3,0.3,0.3,0.3),min(0.8,0.6,0.9,0.6,0.8), min(0.4,0.4,0.7,0.4,0.5)= (0.7,0.2,0.2,0.5,0.2,0.6,0.4) C41 =max(0.5,0.3,0.1,0.4,0.4),max(0.2,0.4,0.3,0.1,0.4),max(0.2,0.2,0.1,0.2,0.2) min(0.5,0.6,0.6,0.8,0.6),min(0.4,0.3,0.3,0.2,0.4),min(0.7,0.8,0.6,0.9,0.7), min(0.3,0.3,0.4,0.7,0.5)= (0.5,0.4,0.2,0.5,0.2,0.6,0.3) C42 =max(0.5,0.2,0.5,0.7,0.1),max(0.2,0.4,0.3,0.1,0.4),max(0.2,0.2,0.1,0.2,0.2) min(0.5,0.6,0.6,0.8,0.5),min(0.4,0.3,0.3,0.3,0.5),min(0.7,0.6,0.7,0.9,0.7), min(0.3,0.3,0.4,0.7,0.3)= (0.7,0.4,0.2,0.5,0.3,0.6,0.3) C43 =max(0.3,0.4,0.5,0.3,0.2),max(0.3,0.2,0.2,0.1,0.4),max(0.2,0.2,0.2,0.2,0.2) min(0.5,0.6,0.6,0.8,0.6),min(0.4,0.3,0.3,0.3,0.4),min(0.6,0.6,0.6,0.9,0.7), min(0.3,0.3,0.4,0.7,0.3)= (0.5,0.4,0.2,0.5,0.3,0.6,0.3) ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 474 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
Then A*B = D1D2D3 P1(0.6,0.4,0.2,0.5,0.2,0.7,0.5) (0.6,0.3,0.2,0.5,0.1,0.7,0.4) (0.7,0.4,0.2,0.5,0.2,0.7,0.4) P2(0.7,0.3,0.2,0.6,0.3,0.7,0.4) (0.6,0.3,0.3,0.6,0.3,0.7,0.4) (0.5,0.3,0.3,0.6,0.2,0.7,0.4) P3(0.6,0.2,0.2,0.5,0.2,0.7,0.4) (0.6,0.2,0.2,0.5,0.2,0.6,0.4) (0.7,0.2,0.2,0.5,0.2,0.6,0.4) P4(0.5,0.4,0.2,0.5,0.2,0.6,0.3) (0.7,0.4,0.2,0.5,0.3,0.6,0.3) (0.5,0.4,0.2,0.5,0.3,0.6,0.3) V= D1D2D3 P1(−0.7) (−0.6) (−0.5) P2(−0.8) (−0.8) (−0.8) P3(−0.8) (−0.7) (−0.6) P4(−0.5) (−0.4) (−0.6) Similarly if A∗Bc, is calculated then it is obtained that A∗Bc= D1D2D3 P1(0.5,0.4,0.2,0.5,0.1,0.7,0.5) (0.4,0.4,0.2,0.5,0.2,0.7,0.4) (0.4,0.4,0.2,0.5,0.3,0.7,0.4) P2(0.5,0.3,0.3,0.7,0.2,0.7,0.4) (0.4,0.3,0.2,0.6,0.2,0.7,0.5) (0.4,0.3,0.3,0.6,0.3,0.7,0.4) P3(0.5,0.2,0.2,0.5,0.2,0.6,0.4) (0.4,0.2,0.2,0.5,0.2,0.6,0.4) (0.4,0.2,0.2,0.5,0.3,0.6,0.3) P4(0.4,0.4,0.2,0.5,0.3,0.6,0.3) (0.4,0.4,0.2,0.6,0.2,0.6,0.3) (0.4,0.4,0.2,0.5,0.3,0.6,0.3) Hence W= D1D2D3 P1(−0.7) (−0.8) (−0.9) P2(−0.9) (−1.1) (−1.1) P3(−0.8) (−0.9) (−0.9) P4(−0.7) (−0.7) (−0.7) and finally it is observed that ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 475 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
V−W= D1D2D3 P1(0.0) (0.2) (0.4) P2(0.1) (0.3) (0.3) P3(0.0) (0.2) (0.3) P4(0.2) (0.3) (0.1) From the score matrix V−W, we observe the highest score values for each patient across the diseases, indicating the most probable diagnosis: Patient P2has equal and relatively high scores for D2and D3(both 0.3), suggesting potential comorbidity. Patient P3has the highest score for D3(0.3), implying a likely diagnosis of D3. Patient P4has the highest score for D2 (0.3), indicating a strong likelihood of D2. Therefore, it can be concluded that Patients {P2, P3}are likely suffering from disease {D3}. Patients {P2, P4}are likely suffering from disease {D2}. These findings can assist medical professionals in making informed diagnostic decisions based on symptom-based scoring. 6 Conclusion In this study, we have introduced a novel approach to medical diagnosis using Heptapartitioned Neutrosophic Soft Sets (HNSS). By modeling patient symptoms and disease characteris______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 476 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
tics within a heptapartitioned neutrosophic framework, we successfully handled multiple forms of uncertainty. The decision-making matrix, say score matrix (V−W) enabled the identification of likely diseases revealing from some symptoms are identified and the patient is may convey to the doctors. The results show that this model is effective in diagnosing complex cases where uncertainty is inherent, offering a nuanced and flexible tool for decision support in healthcare. Future research can focus on extending this framework by integrating HNSMs with other intelligent systems such as fuzzy logic, rough sets, and machine learning algorithms to enhance its decision-making capabilities. Moreover, developing dynamic or time-sensitive versions of HNSMs could provide real-time solutions in healthcare monitoring, financial forecasting, and industrial fault diagnosis. The theoretical foundation can also be strengthened by exploring algebraic properties, optimization techniques, and computational algorithms. Beyond healthcare, HNSMs hold promise in diverse fields such as environmental monitoring, risk assessment, engineering systems, and social sciences, where uncertainty plays a critical role. This opens up a wide avenue for interdisciplinary research and practical implementations, encouraging deeper exploration and development of this novel mathematical tool. Conflict of Interest The authors of this paper state that they have stated that there are no conflicts of interest. Acknowledgements The authors thank the reviewers for their useful suggestions that improved the quality of this paper. References [1] Atanassov, K. (2016). Intuitionistic fuzzy sets. International Journal Bioautomation, 20(1), 87-96. [2] Das, S., Kumar, S., Kar, S., & Pal, T. (2019). Group decision making using neutrosophic soft matrix: An algorithmic approach. Journal of King Saud University-Computer and Information Sciences, 31(4), 459-468. ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 477 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
[3] Deli, I., & Broumi, S. (2015). Neutrosophic soft matrices and NSM-decision making. Journal of Intelligent & Fuzzy Systems, 28(5), 2233-2241. [4] Dhar, M. (2021). Neutrosophic soft matrices and its application in medical diagnosis. Infinite Study. [5] Jeyanthi, V., & Mythili, T. (2023). Heptapartitioned neutrosophic topological spaces. Indian Journal of Natural Sciences, 14, 0976-0997. [6] Kumar Das, S. (2020). Application of transportation problem under pentagonal neutrosophic environment. Journal of Fuzzy Extension and Applications, 1(1), 27-41. [7] Maji, P. K. (2009, December). More on intuitionistic fuzzy soft sets. International Workshop on Rough Sets, Fuzzy Sets, Data Mining, and Granular-Soft Computing (pp. 231240). Berlin, Heidelberg: Springer. [8] Maji, P. K. (2012). A neutrosophic soft set approach to a decision making problem. Annals of Fuzzy Mathematics and Informatics, 3(2), 313-319. [9] Maji, P. K. (2013). Neutrosophic soft sets. Annals of Fuzzy Mathematics and Informatics, 5(1), 157-168. [10] Maji, P. K., Biswas, R., & Roy, A. R. (2001). Fuzzy soft sets. Journal of Fuzzy Mathematics, 9(3), 589-602. [11] Maji, P. K., Biswas, R., & Roy, A. R. (2003). Soft set theory. Computers & Mathematics with Applications, 45(4-5), 555-562. [12] Molodtsov, D. (1999). Soft set theory—first results. Computers & Mathematics with Applications, 37(4-5), 19-31. [13] Smarandache, F. (2018). Extension of soft set to hypersoft set, and then to plithogenic hypersoft set. Neutrosophic Sets and Systems, 22, 168-170. [14] Smarandache, F. (2019). Neutrosophic set is a generalization of intuitionistic fuzzy set, inconsistent intuitionistic fuzzy set (picture fuzzy set, ternary fuzzy set), Pythagorean fuzzy set, spherical fuzzy set, and q-rung orthopair fuzzy set, while neutrosophication is a generalization of regret theory, grey system theory, and three-way decision (revisited). Journal of New Theory, (29), 1-31. ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 478 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________
[15] Vellapandi, R., & Gunasekaran, S. (2020). A new decision-making approach for winning strategy based on multi-soft set logic. Journal of Fuzzy Extension and Applications, 1(2), 119-129. [16] Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338-353. ______________________________________________________________________________________________ Neutrosophic Sets and Systems, Vol. 97, 2026 479 Received: May 11, 2025. Accepted: Oct 6, 2025 T. Mythili, V. Jeyanthi, D. Maheswari, W.F.Al Omeri, Heptapartitioned Neutrosophic Soft Matrices and its Application in Medical Diagnosis ______________________________________________________________________________________________