Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment

__________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment Neutrosophic Sets and Systems, Vol. 98, 2026 University of New Mexico Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment B. Sathiyapriya1* and Dr. V. Pankajam2 1*Research Scholar, Department of Mathematics, Sri G.V.G Visalakshi College for Women, Udumalpet, Tamilnadu, India, [email protected]. 2Assistant Professor, Department of Mathematics, Sri G.V.G Visalakshi College for Women, Udumalpet, Tamilnadu, India, [email protected]. Abstract This study presents Linguistic Dual Hesitant Hypersoft Set (LDHHS) for Analysing Medical Diagnose. In LDHHS, linguistic terms are combined with Dual Hesitant logic to handle uncertainty and vagueness in Decision-Making. The Hypersoft Set extend traditional soft sets by handling multi-attributes and interdependent parameters in Decision-Making. In the context of LDHHS can be used to categorize and assess different aspects of Aggregation and using Decision-Making in Medical Diagnose. In this study the development of this framework, its application and its potential impact using a Gastroesophageal reflux disease (GERD) case study to illustrate its effectiveness. Keywords: Linguistic set, Dual Hesitant set, soft set, hypersoft set and multi-criteria decisionmaking (MCDM). 1.Introduction Language, with its inherent ambiguity and subjectivity, often complicates medical diagnosis and treatment by introducing uncertainty and vagueness. Linguistic Dual Hesitant Hypersoft Sets (LDHHS) address these challenges by effectively managing linguistic uncertainty and modeling the complexities of medical data. By assigning Dual Hesitant values to descriptive terms like "minor" or "critical" for symptoms and "dissatisfied" or "very satisfied" for treatment effectiveness, LDHHS provides a precise framework for decisionmaking. This innovative approach has the potential to enhance diagnostic accuracy, optimize treatment strategies and improve healthcare outcomes, offering significant benefits for medical practitioners. Motivation and Research Gap: In this study, medical diagnosis systems have increasingly required decision-making frameworks capable of processing vague linguistic information and multidimensional uncertainty. Traditional fuzzy and soft set-based models struggle to fully manage the nested and hierarchical nature of medical attributes such as symptom severity, disease progression, and treatment effectiveness. This creates a pressing need for an advanced model that can simultaneously incorporate • linguistic subjectivity of medical descriptions, Neutrosophic Sets and Systems, Vol. 98, 2026 2 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment • membership and non-membership in diagnostic evaluation, • multi-level and multi-attribute clinical data, • integration of decision-makers preferences. A standardized framework for handling linguistic variables with further sub-attributes under Dual Hesitant environments. Aggregate operators, distance and similarity measures that support linguistic hypersoft information. Application of such a framework to medical diagnosis, where ambiguity and complexity are dominant. Thus, no comprehensive decisionmaking model currently exists that integrates linguistic uncertainty, attribute and Dual Hesitant non-membership simultaneously. Research Questions 1. How can aggregation, distance, and similarity measures be constructed underthe LDHHS environment to support multi-criteria decision-making? 2. Does the proposed LDHHS based MCDM model provide more reliable and consistent diagnostic outcomes compared with existing approaches? Novelty and Contributions: This study presents a novel decision-making framework based on Linguistic Dual Hesitant Hypersoft Sets (LDHHS) to address the persistent issues of linguistic ambiguity and uncertainty in medical diagnosis and treatment planning. The primary contribution lies in the ability of the proposed LDHHS model to incorporate multidimensional medical factors while simultaneously capturing the degrees of membership and nonmembership associated with linguistically expressed symptoms and treatment effectiveness. In this study also contributes a practical application by demonstrating the effectiveness of NLDHHS in a real-world medical decision-making scenario, providing new insights for healthcare practitioners and policymakers. Literature review: Bin Zhu and Meimei Xia [10] introduced the concept of Dual Hesitant Fuzzy Sets (DHFSs), explores their properties and operations, and demonstrates their application in group forecasting. Dejian Yu, et al. [1] proposed new aggregation operators for dual hesitant fuzzy sets to better handle uncertainty. Their effectiveness is shown through a numerical example and applied to selecting HR outsourcing suppliers. Zhiliang Ren1 and Cuiping Wei [2] presents a prioritized multi-attribute decision-making method for dual hesitant fuzzy environments. A correctional score function and Dice similarity measure are introduced to better handle hesitant degrees and attribute priorities. The approach is demonstrated through a practical example. Baoquan Ning, et al. [3] proposed a MADM method using probabilistic dual hesitant fuzzy sets, introducing new distance and entropy measures and applying them to credit risk evaluation. Muhammad Saqlain, et al. [4] introduced NHSS-TOPSIS, a decision-making method using neutrosophic hypersoft sets with new distance and similarity measures, applied to medical diagnosis and green security system selection. Hongjum Wang, et al. [5] proposed several aggregation operators for dual hesitant fuzzy MADM problems and demonstrates their effectiveness through a technology commercialization evaluation example. Jawad Ali and Muhammad Naeem [6] introduced new distance and similarity measures for normal wiggly dual hesitant fuzzy sets to better support decision-making, demonstrated through a disease Neutrosophic Sets and Systems, Vol. 98, 2026 3 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment detection example. Babitha and Sunil Jacob John [7] introduced a hybrid hesitant fuzzy soft set, combining soft sets and hesitant fuzzy sets, and explores its basic operations and application in decision-making. Sreelekshmi et al. [8] Proposed the Hesitant Fuzzy Hyper Soft Set to enhance decision-making accuracy, with defined operations and an application in robotics. Glad Deschrijyer et al. [9] introduced aggregation operators on the lattice L∗, analyzes their properties via t-norms and implicators, and examines them under Smets–Magrez axioms. Ubaid Ur Rehman et al. [11] proposed complex dual hesitant fuzzy sets and their similarity measures, applying them to pattern recognition and medical diagnosis to demonstrate their effectiveness. Rana Muhammad Zulqarnain, et al. [12] developed algebraic operations and aggregation operators for interval-valued intuitionistic fuzzy hypersoft sets, applying them to material selection in cryogenic storage systems for improved decision-making. Sathiyapriya Bangarusamy, et al. [13] proposed approach enhances diagnostic accuracy and supports more tailored and reliable treatment strategies for improved patient care. Baoquan Ning et al. [14] proposed a novel correlation coefficient in the probabilistic dual hesitant fuzzy setting and applies it to a MADM method for evaluating project manager candidates. Florentin Smarandache [15] Introduced IndetermSoft and IndetermHyperSoft Sets to handle indeterminate data, extending soft set theory with applications in fuzzy and neutrosophic environments. Takkai Fujita and Shinjuku [16] explores advanced extensions of soft and rough sets, introducing Superhypersoft Hyperrough and Superhypersoft Superhyperrough sets to better handle uncertainty in decision-making. Saqlain, et. al [17] introduced Neutrosophiclinguistic valued hypersoft sets (N-LVHS) help manage linguistic uncertainty in medical diagnosis by assigning neutrosophic values to vague terms, improving accuracy in treatment and decision-making. In this research paper explores Linguistic Dual Hesitant Hypersoft sets (LDHHS), starting with their fundamental principles and properties. It introduces operational laws and two mathematical Aggregated operators, LDHHSOWGAO and LDHHSWGAO, explaining their significance. A framework for Multi-Criteria Decision-Making (MCDM) is presented using an LDHHS algorithm, demonstrated through a case study. The paper concludes with findings and future research directions. Acronyms DHFS - Dual Hesitant Fuzzy Sets LDHHS - Linguistic Dual Hesitant Hypersoft sets ELDHHS - Empty Linguistic Dual Hesitant Hypersoft sets LDHHSWGAO -Linguistic Dual Hesitant Hypersoft set Weighted geometric averaging Operator LDHHSOWGAO -Linguistic Dual Hesitant Hypersoft Set Ordered Weighted Geometric Averaging Operator GERD -Gastroesophageal Reflux Disease 2. Preliminary Neutrosophic Sets and Systems, Vol. 98, 2026 4 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment 2.1 Linguistic Set Let w = {w1, w2, w3, . . . wn} where n = 2p + 1: p ≥ 1 and p ∈ Ꞧ+ (finite and real valued), be a finite strictly increasing set. For example, if p = 1 then, w = {w1, w2, w3} = {dissatisfied, neutral, satisfied} For Linguistic set, which is under consideration, the relationship to its elements wn and the subscript n will be strictly increasing. To define the continuity this set is extended to w = {wω: ω∈ Ꞧ} where ω is also strictly increasing. Definition 2.1 Dual Hesitant Fuzzy Set Let Ṷ be a fixed set, then a Dual Hesitant Fuzzy Set (DHFS) D  on Ṷ is described as D  = {< e, α(e), β(e) >| e ∈ Ṷ}, in which α(e) and β(e) are two sets of some values in [0, 1], denoting the possible membership degrees and non-membership degrees of the element e ∈ Ṷ to the set D  respectively, with the conditions 0 ≤ γ, η ≤ 1, 0 ≤ γ+ + η+≤ 1, Where γ ∈ α(e), η ∈ β(e), γ+∈ α+(e) = ⋃max{γ} γ∈α and η+∈ β+(e) = ⋃max{η} η∈β for all e ∈ Ṷ. For convenience, the pair D (e) = (α(e), β(e)) is called a dual hesitant fuzzy element denoted by D  = (α, β), with the conditions: γ ∈ α, η ∈ β, γ+∈ α+ = ⋃max{γ} γ∈α and η+∈ β+ = ⋃max{η} η∈β , 0 ≤ γ, η ≤ 1 and 0 ≤ γ+ + η+≤ 1. Definition 2.2 Soft Set Let Ṷ be a universe set and let H = {h1, h2, h3, . . . . hn} be a finite set of Parameters or Attributes. Let Ṕ (Ṷ) denote the collection of all subsets of Ṷ. For any E ⸦ H, a pair (D, H) is called soft Set over Ṷ, where the mapping D is given by D: H → Ṕ (Ṷ). Definition 2.4 Hypersoft Set Let Ṷ be a universe of discourse, Ṕ (Ṷ) be a power set of Ṷ, Let H = {H1,H2,H3, ....Hn} for n ≥ 1 be n distinct attributes whose corresponding attributes values are respectively the sets a1,a2,a3,....an with ap ∩ aq = ∅, for p ≠ q and p, q ∈ {1, 2, 3, . . . n}. Then the pair (D, I) where I = {a1×a2× a3× ....×an: n is finite and real valued} is known as Hypersoft set over Ṷ with mapping D: a1 ×a2× a3× ....×an = I → Ṕ (Ṷ). Definition 2.5 Linguistic Hypersoft Set Let λ = (λ1,λ2,λ3,....λn) for n ≥1 be n distinct attributes, whose corresponding attribute values are respectively the sets u1,u2,u3,....un with up ∩ uq = ∅, where p ≠ q for each n ≥ 1 and p, q ∈ {1, 2, 3, . . . n}. Then the pair (θ,∝) where ∝ = {u1×u2× u3× ....×un : t is finite Neutrosophic Sets and Systems, Vol. 98, 2026 5 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment and real valued} is known as hypersoft set over Ṷ with mapping θ : (u1×u2× u3× ....×un) → Ꝑ (Ṷ). Then the linguistic hypersoft set will be, θ({β(Ṷ)(t)}): β ⊆ λ & t ∈ wω = {w1, w2, w3, . . . wn} where n = 2p + 1: p ≥ 1 and m ∈ Ɽ+} 3. Linguistic Dual Hesitant Hypersoft Set (LDHHS) In this section, we propose LDHHS with its set structure properties. Definition 3.1: Linguistic Dual Hesitant Hypersoft Set (LDHHS) Let Ṷ be a universe of discourse Ꝑ (Ṷ) be a power set of Ṷ. Take λ = (λ1,λ2,λ3,....λn) for n ≥1. where λ1,λ2,λ3,....λn are attributes, whose corresponding sub-attribute values are respectively the sets u1,u2,u3,....un with up ∩ uq = ∅, where p ≠ q for each n ≥ 1 and p, q ∈ {1, 2, 3, . . . n}. Let ∝ = {u1×u2× u3× ....×un: where n is finite and real valued} and θ : ∝ = (u1×u2× u3× ....×un) → Ꝑ (Ṷ). Now the pair (θ,∝) is known as the Linguistic Dual Hesitant Hypersoft Set (LDHHS) can be defined as θ(λ (w)) = {β(λ (γ+, η+)) | β ⊆ λ & γ+, η+ ∈ w = {w1, w2, w3, . . . wn}} Where w is the set of Linguistic Parameters and γ+, η+ represent the Dual Hesitant maximum membership and maximum non-membership values in Linguistic Parameters with the condition 0 ≤ γ, η ≤ 1, 0 ≤ γ+ + η+≤ 1. Numerical Example 3.1.1 Let Ṷ = {s1,s2,s3} be a universe of discourse, consisting of a set of three woods, describes the strength of wood θ(λ(w)) = {s1,s2}. consider the attributes be λ1 = Softwoods, λ2 = Hardwoods and their corresponding sub-attributes values are Softwoods = u1 = {Cedar, Pine} Hardwoods = u2 = {Teak, Beech} and set θ(λ (w)) = {s1, s2} ⊂ U. Then the function θ:∝ = u1 ×u2 → Ṕ (Ṷ) and we have five Linguistic Parameters w = {w1, w2, w3, w4, w5} = {very dissatisfied, dissatisfied, neutral, satisfied, very satisfied}, each linguistic Parameter corresponds a dual Hesitant value: w1 = 0.01 for very dissatisfied, w2 = 0.05 for dissatisfied, w3 = 0.25 for neutral, w4 = 0.35 for satisfied and w5 = 0.52 for very satisfied. Define the strength of woods in Linguistic Dual Hesitant Hypersoft Set (LDHHS) (θ,∝) = θ(λ (w)) = {β(λ (γ+, η+)) | β ⊆ λ & γ+, η+ ∈ w = {w1, w2, w3, . . . wn}} θ({Cedar, Beech}) = {s1, s2} = {(s1((dissatisfied, neutral), (very dissatisfied, dissatisfied))), (s2 (neutral, satisfied), (very dissatisfied, dissatisfied))} = G Similarly, Neutrosophic Sets and Systems, Vol. 98, 2026 6 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment θ1({Pine, Teak}) = {s1, s2} = {(s1((neutral, satisfied), (very dissatisfied, dissatisfied))), (s2(dissatisfied, neutral), (very dissatisfied, neutral))} = G1 θ2({Cedar, Teak}) = {s2, s3} = {(s2((very dissatisfied, neutral), (very dissatisfied, dissatisfied)), s3((neutral, satisfied), (very dissatisfied, dissatisfied))} = G2. Definition 3.2: Let (θ1,∝1)= G1 be a LDHHS, then the subset Gb can be defined as. θ(λ (w)) = {β(λ (γ+, η+)) | β ⊆ λ & γ+, η+ ∈ w = {w1, w2, w3, . . . wn}} 1. Gb ⸦ G1 2. ∀ w∈ Gb, θ2(w) ⸦ θ1(w). This holds only when linguistic variables wω satisfy the property i.e., each wω of (θb,∝b)≤ wω of (θ1,∝1). Where wω represents Linguistic variables associated with Dual Hesitant evaluation. Example 3.2.1 Recall Example 3.1.1. The function θ1: ∝b = u1 ×u2 → Ṕ (Ṷ) and assume the hypersoft set, θ1({Pine, Teak}) = {(s1(neutral, satisfied), (very dissatisfied, dissatisfied), (s2 (dissatisfied, neutral), (very dissatisfied, neutral)} = Gb. Where ∝b ⸦ ∝ and Gb ⸦ G1. Where Gb is a subset of the original set G1. Linguistic variables are associated with Dual Hesitant evaluation (maximum of membership and non-membership). Definition 3.3 Empty Linguistic Dual Hesitant Hypersoft Set (ELDHHS) can be defined as. θ1: ∝E = u1× u2×.....×un → Ṕ(Ṷ) Such that each up(p≤ n) is empty. θ1({GE(Ṷ)}) 1. (θ1,∝E)(∅) = GE if ∀θ1(w) = ∅: ∀ w ∈ ∝E. Example 3.3.1 Recall Example 3.1.1. The function θ1: ∝E = u1× u2 → Ṕ(Ṷ), where u1 and u2 are all empty sets (u1 = u2 = ∅) and assume the Hypersoft set, θ1(∅)= ∅ = HE, where ∝E ⸦ ∝. Definition 3.4 The AND operation on two (θ1,∝1) = G1 and (θ2,∝2) = G2 Linguistic Dual Hesitant hypersoft set (LDHHS) can be defined by 1. G1˄ G2 = (θ1˄2,∝1˄2) = G1˄2 2. (wp, wq) = wω = G1˅2, where wp∈ G1 and wq∈ G2 with p≠q 3. θ1∪2(wp, wq) = θ1(wp) ∪ θ2 (wq). Definition 3.5 Neutrosophic Sets and Systems, Vol. 98, 2026 7 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment The OR operation on two (θ1,∝1) = G1 and (θ2,∝2) = G2 Linguistic Dual Hesitant hypersoft set (LDHHS) can be defined by 1. G1˅ G2 = (θ1˅2,∝1˅2) = G1˅2 2. (wp, wq) = wω= G1˅2, where wp∈ G1 and wq∈ G2 with p≠q 3. θ1∩2(wp, wq) = θ1(wp) ∩ θ2 (wq) Definition 3.6 The NOT operation on (θ, ∝) Linguistic Dual Hesitant hypersoft set (LDHHS) can be defined by. 1. ~ G = ~(θ, ∝) = ~u1× ~u2× ....×~un 2. ~ G = ~Πwp : p = 1,2,3,…….n Definition 3.7 The Complement on (θ, ∝) = G Linguistic Dual Hesitant hypersoft set (LDHHS) can be defined by 1. (θ, ∝)~ = (θ~, ~∝), θ~: ~∝ → Ṕ(Ṷ). 2. θ~(~w) = Ṷ \ θ(w); ∀w ∈ G. Proposition 3.8: Let (θ, ∝) = G, (θ1,∝1) = G1 and (θ2,∝2) = G2 be Linguistic Dual Hesitant hypersoft set (LDHHS) then following holds. 1. (θ1,∝1) ⸦ (θ1,∝1) 2. (θ2,∝E)(∅) ⸦ (θ2,∝2) 3. ~ (~ G) = G 4. ~ (θ2,∝E) (∅) = Ṷ 5. If (θ1,∝1) ⸦ (θ2,∝2) and (θ2,∝2) ⸦ (θ1,∝1) then (θ1,∝1) = (θ2,∝2) iff each wω of (θ1,∝1) = wω of (θ2,∝2). This property holds only when Dual Hesitant variables satisfy the property i.e., each wω of (θ1,∝1) = wω of (θ2,∝2). Proof: Recall G, G1 and G2 from example 3.1.1. 1. θ1 contains Dual Hesitant Variables G1, G2, G3, . . . Gn For each wω ∈ ∝1, we have a mapping θ1(wω) ⸦ θ1(wω) each Linguistic Dual Hesitant variable map to itself. Thus, the set (θ1,∝1) is a subset of itself by the Definition of 3.2, as the mappings are trivially reflexive. ∴(θ1,∝1) ⸦ (θ1,∝1). Neutrosophic Sets and Systems, Vol. 98, 2026 8 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment 2. ∝E refers to an empty domain, (i.e., no Linguistic Dual Hesitant variable) the complement operation on LDHHS with empty domain means there are no variables to map to Linguistic Dual Hesitant values. θ2(wω) = ∅ for all wω ∈ ∝E the function maps nothing to the power set Ṷ. Since the empty set is a subset of any set, it follows that (θ2,∝E)(∅) ⸦ (θ2,∝2) because the empty set maps to no Linguistic Dual Hesitant variables, making it trivially a subset of any non-empty LDHHS. 3. Let’s assume ∼(θ1,∝1) represents the complement of the LDHHS, which is (θ1,∝1) with each Linguistic Dual Hesitant variables wω replaced by its complement. Now, apply the complement again would return the original Dual Hesitant values, as: ∼ (∼ (wω)) = wω for each wω thus ~ (~ G) = G, Confirming the double complementation property. 4. When the LDHHS is complemented and the domain is the empty set, the result is the complement of the empty set, which is the entire universal set Ṷ. Therefore, ∼ (θ2,∝E) (∅) = Ṷ. 5. Let’s assume that (θ1,∝1) ⊆ (θ2,∝2) and (θ2,∝2) ⊆ (θ1,∝1). This means that for all, wω ∈ θ1, there exists a corresponding wω ∈ θ2 such that θ1(wω) ⊆ θ2(wω) Similarly, (θ2,∝2) ⊆ (θ1,∝1) implies: θ2(wω) ⊆ θ1(wω) Since θ1(wω) ⊆ θ2(wω) and θ2(wω) ⊆ θ1(wω), we conclude: θ1(wω) = θ2(wω) Thus, (θ1,∝1) = (θ2,∝2), provided that the Linguistic Dual Hesitant variables wω from both sets match exactly. 4. Operational Laws on LDHHSS In this section, we discuss the importance of operational laws, theorems and propose for LDHHS. Let (θ1,∝1) = G1 and (θ2,∝2) = G2 be two LDHHS, where ∝1 = {u1 ×u2× u3× ....×up: p is finite and real valued} over Ṷ with mapping θ:∝1 = u1 ×u2× u3× ....×up → Ṕ(Ṷ) and ∝2= u1 ×u2× u3× ....×uq: y is finite and real valued} over Ᾰ with mapping θ:∝2 = u1 ×u2× u3× ....×uq → Ṕ(Ṷ). Such that (θ,∝) = θ(λ(w)) = {β(λ (γ+, η+)) | β ⊆ λ & γ+, η+ ∈ w = {w1, w2, w3, . . . wn}} Neutrosophic Sets and Systems, Vol. 98, 2026 9 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment where w is the set of Linguistic terms and γ+, η+ represents the Dual Hesitant Maximum of Membership and non-membership values in Linguistic terms in ascending order i. e. very dissatisfied to very satisfied. Then the operational laws on LDHHS can be defined with some necessary conditions. Definition 4.1 Union of LDHHS The union of two LDHHS, (θ1,∝1) = G1 and (θ2,∝2) = G2, can be represented as G1∪G2. Depending on the relationship between their Linguistic Dual Hesitant variables and their domains, the union is defined in two cases. Case 1: Let (θ1,∝1) = G1 and (θ2,∝2) = G2 be two LDHHS, then the union can be defined as G1∪G2 = {Π λp(wp) × Π λq(wq) ∈ ∏ up n p=1 × ∏ uq n q=1 } where, λp(wp) ∈ ∏ up n p=1 and λq(wq) ∈ ∏ uq n q=1 should be distinct with up∪uq = ∅, for p ≠ q and p, q ∈ {1, 2, …. n}and w = {w1, w2, w3, . . . wn}. Case 2: G1∪G2 = {λp(wp) ∈ ∏ up n p=1 × ∏ uq n q=1 } With p = q and Linguistic Dual Hesitant variable wp of up should be same. Example: Consider 3.1.1 Case 1: θ1({Pine, Teak})= {s1, s2} = {s1(satisfied, neutral), (very dissatisfied, dissatisfied), s2(very dissatisfied, dissatisfied), (neutral, very dissatisfied)} = G1 θ2({Cedar,Beech})= {s1, s2} = {s1(very dissatisfied, dissatisfied), (satisfied, neutral), s2(neutral, dissatisfied), (very dissatisfied, very dissatisfied)} = G2. ∴ up∪ uq= ∅ with p ≠ q G1∪G2= {s1(satisfied,dissatisfied), s2(dissatisfied,neutral),s1(disatisfied,satisfied), s2(neutral,very dissatisfied)}. Case 2: θ1({Pine,Beach}) = {s1,s2} = {s1(satisfied, neutral), (dissatisfied, very dissatisfied), s2(neutral, dissatisfied), (very dissatisfied, dissatisfied)} = G1 θ2({Pine,Teak})= {s1,s2} = {s1(satisfied, neutral), (dissatisfied, very dissatisfied), s2(neutral, dissatisfied), (very dissatisfied, dissatisfied)} = G2. ∴ up∪ uq ≠∅ with p = q G1∪G2= {s1(satisfied,dissatisfied),s2(neutral,dissatisfied)}. Case 3: (counter example) θ1({Pine,Beach})= {s1,s2} = {s1(satisfied, neutral), (dissatisfied, very dissatisfied), s2(neutral, dissatisfied), (very dissatisfied, dissatisfied)} = G1 Neutrosophic Sets and Systems, Vol. 98, 2026 16 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment very dissatisfied) (neutral, dissatisfied) very dissatisfied) P4 (neutral, dissatisfied), (neutral, very dissatisfied) (neutral, very dissatisfied), (very dissatisfied, dissatisfied) (very dissatisfied, dissatisfied), (dissatisfied, neutral) (satisfied, very dissatisfied), (dissatisfied, very dissatisfied) (very dissatisfied, dissatisfied), (very dissatisfied, dissatisfied) P5 (dissatisfied, very dissatisfied), (very dissatisfied, neutral) (neutral, very dissatisfied), (dissatisfied, very dissatisfied) (very dissatisfied, dissatisfied), (dissatisfied, very dissatisfied) (very dissatisfied, neutral), (neutral, dissatisfied) (satisfied, very dissatisfied), (very dissatisfied, dissatisfied) P6 (satisfied, very dissatisfied), (very satisfied, very dissatisfied) (dissatisfied, neutral), (very dissatisfied, neutral) (satisfied, dissatisfied), (very dissatisfied, dissatisfied) (neutral, very dissatisfied), (satisfied, very dissatisfied) (dissatisfied, satisfied), (neutral, dissatisfied) P7 (neutral, dissatisfied), (very dissatisfied, dissatisfied) (very dissatisfied, dissatisfied), (neutral, very dissatisfied) (satisfied, neutral), (dissatisfied, very dissatisfied) (neutral, dissatisfied), (neural, very dissatisfied) (satisfied, very dissatisfied), (dissatisfied, very dissatisfied) P8 (dissatisfied, satisfied), (very dissatisfied, dissatisfied) (neutral, very dissatisfied), (neutral, dissatisfied) (satisfied, dissatisfied), (dissatisfied, very dissatisfied) (very dissatisfied, neutral), (neutral, dissatisfied) (neutral, very dissatisfied), (dissatisfied, very dissatisfied) P9 (satisfied, neutral), (very dissatisfied, neutral) (satisfied, dissatisfied), (very dissatisfied, dissatisfied) (dissatisfied, neutral), (very dissatisfied, neutral) (dissatisfied, very dissatisfied), (neutral, dissatisfied) (satisfied, dissatisfied), (neutral, dissatisfied) P10 (satisfied, very dissatisfied), (neutral, dissatisfied) (satisfied, very dissatisfied), (neutral, dissatisfied) (neutral, dissatisfied), (neutral, very dissatisfied) (satisfied, neutral), (very dissatisfied, dissatisfied) (very satisfied, dissatisfied), (very dissatisfied, dissatisfied) Step 2: The LDHHSWGAO is designed to aggregate values (weights: for Dry cough = 0.2, for Bitter taste = 0.3, for Upper Abdominal Discomfort = 0.2, for Epigastric = 0.1, for Belching = 0.2). across different symptoms (attributes) for each and every single patient. Patients LDHHSWGAO Values Neutrosophic Sets and Systems, Vol. 98, 2026 17 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 = [ (dissatisfied,neutral) (satisfied,dissatisfied) (very satisfied,very dissatisfied) (very dissatisfied,dissatisfied) (very dissatisfied,dissatisfied) (satisied,dissatisfied) (satisfied,dissatisied) (neutral,dissatisfied) (satisfied,neutral) (satisfied,neutral) ] Step 3 Next, the doctor uses this operator to aggregate the Linguistic Dual Hesitant variables for each Patients. This aggregation takes into account maximum of Membership and Non membership of all symptoms to calculate an overall score for each Patients. Patients Aggregated Values P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 = [ (dissatisfied) (satisfied) (very satisfied) (very dissatisfied) (very dissatisfied) (satisfied) (satisfied) (neutral) (satisfied) (satisfied) ] Step 4: Finally, list the alternatives with maximum Membership (γ+) Values. The maximum (γ+), will represent the positive ideal alternative. Alternative Result P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 = [ (Negative) (Positive) (Positive) (Negative) (Negative) (Positive) (Positive) (Negative) (Positive) (Positive) ] Neutrosophic Sets and Systems, Vol. 98, 2026 18 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment In this case study highlights how the LDHHS algorithm helped doctors overcome diagnostic challenges with patients showing common symptoms like Dry cough, Bitter taste, Upper Abdominal Discomfort, Epigastric pain and belching which overlap across multiple illness, including Gastroesophageal Reflux Disease (GERD). By leveraging advanced language models and analysing a wide range of medical data, the algorithm provided accurate, data-driven diagnoses, reducing uncertainty. Figure: 2 visually represents the relationship between Patients and GERD. Figure 2: Negative and Positive Patients in GERD Represent Negative in GERD Represent Positive in GERD 6.3 Result Discussion Comparison and Future Direction The comparison between proposed LDHHS algorithm and traditional diagnostic methods underscores its potential to revolutionize healthcare. Unlike conventional techniques reliant on clinical judgement, LDHHS leverages advanced language models and real-time medical data to deliver more accurate and adaptive diagnoses. Table 2 illustrates its comparable performance with existing methods, while its ability to incorporate new research and manage risk, particularly crises for GERD, highlights its superiority. By complementing, rather than replacing, the expertise of healthcare professionals LDHHS paves the way for more precise, efficient and patient-centered diagnostic solutions. Table 2: Comparing Research Result with Existing Studies. METHOD POSITIVE NEGATIVE FDHS P1, P3, P4, P6, P7, P10 P2, P5, P8, P9 LDHHS P2, P5, P6, P8, P10 P1, P3, P4, P7, P9 7.Conclusion In conclusion, this study underscores the critical role of language and the challenge it poses in medical diagnosis and treatment. By introducing Linguistic Dual Hesitant Hypersoft Sets (LDHHS), a robust framework for managing Linguistic Dual Hesitant uncertainty, the study presents a promising approach to enhancing healthcare decision-making. The proposed definition, concepts, aggregation operators and algorithms demonstrate the utility of LDHHS 0 0.2 0.4 0.6 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Neutrosophic Sets and Systems, Vol. 98, 2026 19 ____________________________________________________________________________________________________ __________________________________________________________________________________ B. Sathiyapriya and Dr. V. Pankajam, Linguistic Dual Hesitant Hypersoft Set and their Application with Decision Making in Medical Diagnosis and Treatment in addressing the complexity of modern medical practice, fostering more effective and patientcentered care. Looking ahead, expanding LDHHS to encompass a broader spectrum of medical conditions and fostering collaboration between data scientists and healthcare professionals will be key to advancing its potential. Beyond healthcare, the versatile framework of LDHHS holds promise for applications in diverse fields such as market research, environmental assessments and disaster preparedness, offering a powerful tool for navigating complex and uncertain environments. References [1] Dejian Yu, Wenyu Zhang and George Huang, “Dual Hesitant Fuzzy Aggregation Operators”, Technological and Economic Development of Economy, ISSN 2029-4913 / eISSN 2029-4921 2016. [2] Zhiliang Ren1 and Cuiping Wei, “A multi-attribute Decision-making method with Prioritization relationship and Dual Hesitant Fuzzy decision information”, Original Article, DOI 10.1007/s13042-015-0356-3, 2015. 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Received: July 5, 2025. Accepted: Dec 14, 2025