Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets

University of New Mexico Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets B. Anitha1, M. Lavanya2 1Assistant Professor, Department of Mathematics, Annamalai University, Annamalai Nagar - 608002, Tamilnadu, India; anibhav[email protected] 2Research Scholar, Department of Mathematics, Annamalai University, Annamalai Nagar - 608002, Tamilnadu, India; lavan[email protected] ∗Correspondence: lavany[email protected] Abstract.This study presents a multi criteria decision making (MCDM) approach based on the cubic neutrosophic TOPSIS method with a score function to evaluate kidney failure across different age groups. Patients are assessed based on symptoms and these symptoms are linked to diseases to form a complete patient disease relationship. The cubic neutrosophic framework handles uncertainty and imprecision in medical data, and the score function ensures accurate ranking of alternatives. Real patient data are analyzed to determine the level of susceptibility to kidney failure among children (0 - 17), adults (18 - 49), and the elderly (50 +). The findings show that the elderly group is the most susceptible to kidney failure, emphasizing the importance of early diagnosis and preventive measures in this population. Keywords: Cubic Neutrosophic Set, Kidney Failure, Age Groups, Topsis Method, Score Function. —————————————————————————————————————————- 1. Introduction In 1965, Zadeh [25] introduced fuzzy sets (FS), assigning each element a membership value between 0 and 1 to represent uncertainty. Later, in 1975, he extended this to interval-valued fuzzy sets (IVFS), where elements are given intervals instead of single values, offering a more flexible way to capture imprecision. Jun et al. [15] proposed cubic sets (CS), which use cubic functions for membership grades, building on FS and IVFS to better model uncertainty. Smarandache [21] introduced neutrosophic sets (NS), incorporating truth, indeterminacy, and falsity. Wang et al. [23] extended this to interval-valued neutrosophic sets (IVNS), allowing interval representation for all three B. Anitha , M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 components. Y. B. Jun et al. [15] later defined neutrosophic cubic sets, with further applications researched by Ajay et al. [2] and M. Ali et al. [3]. Inspired by this, we propose cubic neutrosophic sets, which differ from neutrosophic cubic sets but share a common foundation. TOPSIS has been widely extended under fuzzy and neutrosophic environments for multicriteria decision-making (MCDM). Wang et al. [23] developed fuzzy TOPSIS with subjective and objective weights, while Nadaban et al. [18] introduced neutrosophic TOPSIS for indeterminate data. Simplifications were offered by Elhassouny et al. [10], and further extensions were made by Imtaz et al. [13], Li et al. [17], and Sindhu et al. [20]. Applications in medical diagnosis include works by Broumi et al. [8], Guleria et al. [12], and Farooq et al. [11]. This paper evaluates the risk of kidney failure among children, adults, and the elderly. The TOPSIS approach based on cubic neutrosophic sets is applied to manage uncertainty in medical data and to rank age groups according to their susceptibility. 2. Preliminaries Definition 2.1. Let Xbe a non-empty set. A Cubic Set(CS) Cin Xis a structure of the form C={`n, ˜%(`n), %(`n)/`n∈X)}and denoted by C=<˜%, % >, where ˜%= [%L, %U] is an IVFS in Xand %is a FS in X. Definition 2.2. Let Xbe a universe set. Then a Neutrosophic Set(NS) Non XN= {`n, ζ(`n), %(`n), ς(`n)/`n∈X}, with ζ, %, ς ∈[0,1]. Definition 2.3. An Interval Valued Neutrosophic Set(IVNS) of Xis denoted as ˜ N= {`n, ˜ ζ(`n),˜%(`n),˜ς(`n)/`n∈X}where ˜ ζ, ˜%, ˜ς∈[0,1]. 3. Topsis Method for cubic neutrosophic set(CNS) TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is a method used to make decisions involving multiple criteria. It evaluates alternatives by comparing their affinity with the best solution and their deviation from the least advantageous solution. Each criterion is assigned a weight, and a relative closeness score determines the final ranking. Definition 3.1. Let Xbe a nonempty set. A cubic neutrosophic set (CNS) on Xis represented as C={`n, ζF(`n), %F(`n), ςF(`n)/`n∈ X} Where, ζF=<˜ ζ, ζ >∈[0,1],˜ ζis an i-v truth membership function and ζis a truth membership funtion, %F=<˜%, % >∈[0,1],˜%is an i-v indeterminacy membership function and %is a indeterminacy membership funtion, ςF=<˜ς, ς >∈[0,1],˜ςis an i-v falsity membership function and ςis a falsity membership funtion, with the condition that 0 ≤˜ ζ+ ˜%+ ˜ς≤3 and 0 ≤ζ+%+ς≤3. B. Anitha, M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 648 Definition 3.2. Let C={(ζ−, ζ+), ζ, (%−, %+), %, (ς−, ς+), ς}be a cubic neutrosophic number. Its score function is defined as follows S(C) = (ζ−+ζ+)+ζ 3+[3−(%−+%+)+%] 3+[3−(ς−+ς+)+ς] 3 3 Definition 3.3. Let Nand Kbe two cubic neutrosophic numbers. The maxmin composition between them is defined as follows N ∗ K =                  {max(min(˜ ζN,˜ ζK)),min(max(ζN, ζK))}, {min(max(˜%N,˜%K)),max(min(%N, %K))}, {min(max(˜ςN,˜ςK)),max(min(ςN, ζK))}                  Topsis Algorithm Step:1 Create a table values The alternatives and criteria are structured in a table value to evaluate their performance under each criterion. In this table, rows represent alternatives and columns represent criteria. This table value serves as the foundation for further analysis and comparison.                  (ζ? 11, %? 11, ς? 11) (ζ? 12, %? 12, ς? 12). . . (ζ? 1n, %? 1n, ς? 1n) (ζ? 21, %? 21, ς? 21) (ζ? 22, %? 22, ς? 22). . . (ζ? 2n, %? 2n, ς? 2n) . . .. . .. . . (ζ? m1, %? m1, ς? m1) (ζ? m2, %? m2, ς? m2). . . (ζ? mn, %? mn, ς? mn)                  Step:2 Score the table values Using the definition 3.2 to create score table value. Step:3 Construct the weighted score values Each criterion is assigned a weight according to its importance. The performance of each alternative is multiplied by these weights to compute the weighted scores. Wj=1−Ej m P j=1 (1−Ej) Here, j= 1,2, ..., m and PWj= 1 Step:4 Determaine the PIS and NIS Positive Ideal Solution: N+ j=D+ 1,D+ 2,...,D+ n where D+ i={max(Dij); j∈J+,min(Dij); j∈J+} Negative Ideal Solution: N− j=D− 1,D− 2,...,D− n B. Anitha, M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 649 where D− i={max(Dij); j∈J−,min(Dij); j∈J−} Here, j= 1,2, ..., n Step:5 Compute the separation distances for the alternatives Seperation measure of each alternatives is to be measured from PIS and NIS respectively. P+ i=sm P j=1 (D+ j− Dij)2 P− i=sm P j=1 (D− j− Dij)2 Step:6 Calculate the relative similarity to the ideal solution For each alternatives, closeness coefficient is calculated by Si=P− i P+ i+P− i Step:7 Result Alternatives are ordered based on their closeness coefficient, and the one attaining the maximum value is regarded as optimal. The algorithm is represented in the form of a flow chart. It shows the step-by-step process of TOPSIS from constructing the table of values to ranking the alternatives. Create a table values Score the table values Construct the weighted score values Determaine the PIS and NIS Calculate the separation measures Calculate the relative closeness value Calculate Final Ranking of Alternatives Numerical Example Kidney failure can affect individuals of all age groups, though the underlying causes may vary. Lifestyle factors such as an unhealthy diet, physical inactivity, irregular daily routines, and insufficient hydration can gradually impair renal function. Chronic stress and prolonged B. Anitha, M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 650 exposure to harmful substances, including certain medications and environmental toxins, further increase the risk of kidney damage. In addition to these lifestyle-related factors, medical conditions such as diabetes mellitus, hypertension and recurrent or chronic infections can progressively injure the renal structures over time. Genetic predisposition and inherited disorders may also contribute, with some conditions presenting symptoms during childhood. Understanding these risk factors is essential for early detection, effective prevention, and the improvement of long-term renal health outcomes. Chronic kidney disease develops gradually, often as a consequence of diabetes, hypertension, chronic infections or prolonged use of nephrotoxic drugs. These factors damage the glomeruli and other renal filtering structures, leading to progressive loss of kidney function. Patients commonly present with fatigue, edema in the extremities or face, nausea, decreased appetite, reduced urine output and pruritus caused by accumulation of toxins in the blood. Nephrotic syndrome arises from damage to the glomeruli, resulting in significant protein loss in the urine. It may develop due to primary kidney disorders, systemic conditions such as diabetes and lupus or infections. Clinical manifestations include puffiness around the eyes, generalized edema, frothy urine, fatigue and hyperlipidemia. Glomerulonephritis is characterized by inflammation of the glomerular structures, often triggered by immune responses following infections or due to autoimmune diseases such as lupus. Patients typically present with hematuria, edema, elevated blood pressure and decreased urine output, reflecting impaired renal filtration. Acute kidney injury is a sudden decline in renal function, commonly caused by dehydration, severe infections, blood loss, exposure to nephrotoxic agents or urinary obstruction. Clinical features include rapid reduction in urine output, edema, fatigue, confusion, shortness of breath and in severe cases, cardiovascular complications due to fluid overload. Polycystic kidney disease is an inherited disorder characterized by the progressive formation of multiple fluid filled cysts within the kidneys, which gradually replace normal renal tissue. It is associated with flank or back pain, hypertension, hematuria, abdominal enlargement and frequent urination. Over time, these changes can culminate in significant renal impairment and eventual kidney failure. Let P = {P1,P2,P3}represent the age groups of patients in the hospital, where P1−Children(0 - 17), P2−Adults(18 - 49), and P3−Elderly(50+). Consider a set of symptoms S={S1,S2,S3,S4,S5}, where S1−Edema, S2−Low urine output, B. Anitha, M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 651 S3−High blood pressure, S4−Protein in urine, and S5−Flank/Back pain. Similarly, let the set of diseases be D={D1,D2,D3,D4,D5}, where D1−Chronic kidney disease, D2−Nephrotic syndrome, D3−Glomerulonephritis, D4−Acute kidney injury, and D5−Polycystic kidney disease. Figure 1. Types of Kidney Disease Table 1. Cubic Neutrosophic table value = N ∗ S1S2 P1([.3, .5].7, [.4, .6].8, [.4, .7].9) ([.7, .9].2, [.5, .7].3, [.6, .8].4) P2([.4, .7].8, [.3, .5].7, [.6, .9].2) ([.3, .5].7, [.4, .6].8, [.4, .7].9) P3([.7, .9].2, [.5, .7].3, [.6, .8].4) ([.1, .3].5, [.6, .8].7, [.3, .5].6) B. Anitha, M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 652 ∗ S3S4 P1([.3, .7].2, [.1, .4].8, [.4, .5].5) ([.4, .8].2, [.2, .4].5, [.6, .7].8) P2([.4, .7].3, [.2, .8].5, [.3, .4].6) ([.6, .7].1, [.7, .9].4, [.2, .3].6) P3([.6, .8].7, [.3,.5].2, [.4, .6].8) ([.5, .7].3, [.7, .9].2, [.3, .4].6) ∗ S5 P1([.4, .6].2, [.3, .4].5, [.1, .2].9) P2([.8, .9].1, [.7, .8].2, [.6, .8].4) P3([.3, .5].7, [.4, .6].8, [.3, .6].5) Table 2. Cubic Neutrosophic table value = K ∗ D1D2 S1([.2, .4].6, [.3, .4].5, [.6, .7].1) ([.3, .5].4, [.1, .2].1, [.2, .6].4) S2([.4, .6].2, [.3, .4].5, [.1, .2].9) ([.2, .4].6, [.3, .4].5, [.6, .7].1) S3([.4, .7].3, [.2, .8].5, [.3, .4].6) ([.6, .8].7, [.3, .5].2, [.4, .6].8) S4([.1, .2].4, [.4, .5].3, [.6, .7].2) ([.3, .5].7, [.4, .6].8, [.4, .7].9) S5([.8, .9].2, [.7, .9].1, [.6, .8].3) ([.7, .9].2, [.5, .7].3, [.6, .8].4) B. Anitha, M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 653 ∗ D3D4 S1([.3, .5].7, [.4, .6].8, [.3, .6].5) ([.7, .9].2, [.5, .7].3, [.6, .8].4) S2([.4, .7].3, [.2, .8].5, [.3, .4].6) ([.6, .7].1, [.7, .9].4, [.2, .3].6) S3([.2, .4].6, [.3, .4].5, [.6, .7].1) ([.2, .4].6, [.3, .4].5, [.6, .7].1) S4([.3, .7].2, [.1, .4].8, [.4, .5].5) ([.3, .7].2, [.1, .4].8, [.4, .5].2) S5([.4, .8].2, [.2, .4].5, [.6, .7].8) ([.8, .9].2, [.7, .9].1, [.6, .8].3) ∗ D5 S1([.6, .8].7, [.3, .5].2, [.2, .6].4) S2([.4, .7].3, [.2, .8].5, [.3, .4].6) S3([.1, .2].4, [.4, .5].3, [.6, .7].2) S4([.4, .8].6, [.2, .4].7, [.6, .7].2) S5([.3, .5].7, [.4, .6].8, [.3, .6].5) B. Anitha, M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 654 Step 1 Prepare a cubic neutrosophic table of values Table 3. N ∗ K Cubic Neutrosophic table value ∗ D1D2 P1([.4, .7].2, [.2, .5].5, [.4, .5].5) ([.4, .7].2, [.3, .5].5, [.4, .8].8) P2([.8, .9].2, [.2, .5].5, [.3, .4].9) ([.7, .9].2, [.3, .5].5, [.4, .6].6) P3([.4, .7].4, [.3, .7].5, [.3, .5].6) ([.6, .8].4, [.3, .5].5, [.4, .6].8) ∗ D3D4 P1([.4, .7].2, [.2, .4].8, [.4, .7].8) ([.6, .7].2, [0.2, .4].5, [.6, .7].4) P2([.4, .8].2, [.3, .6].7, [.4, .5].6) ([.8, .9].2, [.3, .7].5, [.4, .5].6) P3([.3, .7].3, [.3, .5].5, [.3, .5].6) ([.7, .9].2, [.3, .5].4, [.3, .5].6) ∗ D5 P1([.4, .8].3, [.2, .4].5, [.3, .6].5) P2([.4, .7].4, [.3, .5].5, [.4, .7].6) P3([.6, .8].5, [.4, .5].8, [.3, .5].6) B. Anitha, M. Lavanya, Evaluating Kidney Failure in Age Groups with Cubic Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 97, 2026 655