Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 227 https://internationalpubls.com A Comprehensive study on the Women Disparities in the society using the Methodology of Multi-Criteria Decision Making S. Shafeena Miraclinl1, R. Irene Hepzibah2 1& 2 PG & Research Department of Mathematics, T.B.M.L.College, Porayar, Tamil Nadu, India. Affiliated to Annamalai University E-mail: shafeesam12@gmail.com, ireneraj74@gmail.com ORCID iD:10009-0006-0459-6181, 20000-0003-1019-573X Article History: Received: 13-11-2024 Revised:25-12-2024 Accepted:09-01-2025 Abstract: MCDMs are the unique tool for making decisions and variety of MCDM methods are used halfway around the world aiding in choosing the best option among the multiple alternatives based on various criteria. In this work we have applied the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method that represents a solution methodology for Fuzzy TOPSIS with Heptagonal Fuzzy numbers and each alternative's rating is indicated by Heptagonal Fuzzy Numbers. So, when compared to TOPSIS using triangular numbers we obtain improved and more accurate results. For application and verification, a numerical instance is provided that deals with the Disparities of Women faced in the community. Keywords: MCDM method – Positive Ideal Solution (PIS) - Negative Ideal Solution (NIS) -TOPSIS – Heptagonal Fuzzy Numbers (HFN) – Distance of two heptagonal fuzzy numbers – Disparities of Women. 1. Introduction People make decisions on a regular basis, whether they realize it or not. It's similar to speaking prose. Therefore, having a thorough grasp of the decision-making processes could aid in both preventing and encouraging the incorrect decisions. To handle such complexity, a large number of models, theories, and decision support systems are produced [8]. TOPSIS, which “Hwang and Yoon” created [7]. It is used to express the rankings of the alternatives based on the criteria [12]. Here, we have implemented a TOPSIS decision- making method by utilizing heptagonal fuzzy numbers to integrate expert statements. We use this fuzzy number to more closely evaluate the information provided by the experts, and we find that the opinions are 7-tuple fuzzy values. TOPSIS decision-making approach using hexagonal, pentagonal, trapezoidal, and triangular fuzzy numbers to be a little ambiguous due to its enhanced dimensionality [6]. A fuzzy method for assessing consumers (buyers) was presented by Lin and Chang [10]. The fuzzy TOPSIS was then applied to the assessment findings to screen orders. An integrated approach of fuzzy TOPSIS and fuzzy AHP was presented by Kamble and Naziya [9] and used to address the staff selection problem. The TOPSIS approach has been effectively utilized in a variety of domains, and numerous articles on its implementation have been completed so we have suggested TOPSIS decision-making method that integrates the opinions using heptagonal fuzzy numbers in order to address this inadequacy. Thus, we could make decision in a precise manner using this method. The framework of the study is followed. In section 2, the Fuzzy set, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 228 https://internationalpubls.com Heptagonal fuzzy number, Operations and Distance formula of the Heptagonal numbers are introduced and we have proposed the procedure for TOPSIS method. In section 3, the method and algorithm are given. In section 4 the numerical illustration is given related with disparities faced by women in the society is given. The paper is concluded by ranking the alternatives by their performance score. Acknowledgement and References are followed. 2. Methodology 2.1 Preliminaries 2.1 Heptagonal Fuzzy Number [HFN] A fuzzy number �̃� = (𝑞, 𝑟, 𝑠, 𝑡, 𝑢, 𝑣, 𝑤) is a customary Heptagonal fuzzy number where 𝑞, 𝑟, 𝑠, 𝑡, 𝑢, 𝑣, 𝑤 are real numbers and 𝑞 ≤ 𝑟 ≤ 𝑠 ≤ 𝑡 ≤ 𝑢 ≤ 𝑣 ≤ 𝑤 with membership function is given below, 𝜇�̃�(𝑥) = { 1 2 (𝑥−𝑞) (𝑟−𝑠) , 𝑞 ≤ 𝑥 ≤ 𝑟 1 2 , 𝑓𝑜𝑟 𝑟 ≤ 𝑥 ≤ 𝑠 1 2 + 1 2 (𝑥−𝑠) (𝑡−𝑠) , 𝑓𝑜𝑟 𝑠 ≤ 𝑥 ≤ 𝑡 1 2 + 1 2 (𝑢−𝑥) (𝑣−𝑢) , 𝑓𝑜𝑟 𝑡 ≤ 𝑥 ≤ 𝑢 1 2 , 𝑓𝑜𝑟 𝑢 ≤ 𝑥 ≤ 𝑣 1 2 (𝑤−𝑥) (𝑤−𝑣) , 𝑓𝑜𝑟 𝑣 ≤ 𝑥 ≤ 𝑤 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Figure 1: Graphical representation of HFN 2.2 Arithmetic Operations on Heptagonal Fuzzy Numbers Consider two positive HFN �̃� = (𝑞1, 𝑟1, 𝑠1, 𝑡1, 𝑢1, 𝑣1, 𝑤1) and �̃� = (𝑞2, 𝑟2, 𝑠2, 𝑡2, 𝑢2, 𝑣2, 𝑤2) then operations on these two positive HFNs are given below, 1. Addition: �̃� ⊕ �̃� = (𝑞1 + 𝑞2, 𝑟1 + 𝑟2, 𝑠1 + 𝑠2, 𝑡1 + 𝑡2, 𝑢1 + 𝑢2, 𝑣1 + 𝑣2, 𝑤1 + 𝑤2) 2. Subtraction: �̃� − �̃� = (𝑞1 − 𝑤2, 𝑟1 − 𝑣2, 𝑠1 − 𝑢2, 𝑡1 − 𝑡2, 𝑢1 − 𝑠2, 𝑣1 − 𝑟2, 𝑤1 − 𝑞2) 3. Multiplication: �̃� ∗ �̃� = (𝑞1 ∗ 𝑤2, 𝑟1 ∗ 𝑣2, 𝑠1 ∗ 𝑢2, 𝑡1 ∗ 𝑡2, 𝑢1 ∗ 𝑠2, 𝑣1 ∗ 𝑟2, 𝑤1 ∗ 𝑞2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 229 https://internationalpubls.com 4. Division: �̃� �̃� ⁄ = 𝑞1/𝑤2, 𝑟1/𝑣2, 𝑠1/𝑢2, 𝑡1/𝑡2, 𝑢1/𝑠2, 𝑣1/𝑟2, 𝑤1/𝑞2 2.3 The distance between two Heptagonal fuzzy numbers Let �̃� = (𝑞1, 𝑟1, 𝑠1, 𝑡1, 𝑢1, 𝑣1, 𝑤1) and �̃� = (𝑞2, 𝑟2, 𝑠2, 𝑡2, 𝑢2, 𝑣2, 𝑤2) are two HFNs, then the hamming distance of �̃� from �̃� is given by, 𝑑(�̃�,𝑀)̃ = 1 7 (|𝑞1 − 𝑞2| + |𝑟1 − 𝑟2| + |𝑠1 − 𝑠2| + |𝑡1 − 𝑡2| + |𝑢1 − 𝑢2| + |𝑣1 − 𝑣2| + |𝑤1 − 𝑤2|) 3. Methods and Algorithm • Describe the linguistic variable. HFN are used to express the variables. • Create a fuzzy decision matrix using the HFNs. Every entry in the decision matrix contains the HFN and it is given by, 𝐷𝑀 = ( 𝜁11 𝜁12 . . . 𝜁1𝑚 𝜁21 𝜁22 . . . 𝜁2𝑚 . . . . . . . . . . . . . . . . . . 𝜁𝑛1 𝜁𝑛2 . . . 𝜁𝑛𝑚) (1) Where, 𝜁𝑖𝑗 = (𝜁𝑖𝑗1, 𝜁𝑖𝑗2, 𝜁𝑖𝑗3, 𝜁𝑖𝑗4, 𝜁𝑖𝑗5, 𝜁𝑖𝑗6, 𝜁𝑖𝑗7), 𝑖 = 1,2,3, … ,𝑚 ; 𝑗 = 1,2,3, … , 𝑛 denotes the number of criteria and alternatives. • Form a normalized decision matrix denoted by 𝜑𝑖𝑗 and is calculated as, 𝜑𝑖𝑗 = 𝜁𝑖𝑗 √∑ (𝜁𝑖𝑗)̃2 𝑛 𝑗=1 , 𝑖 = 1,2,3, … ,𝑚 (2) • Formulate a weighted normalized decision matrix denoted by the value �̃�𝑖𝑗 = �̃�𝑖𝑗 × �̃�𝑖𝑗 𝑖 = 1,2,3, … ,𝑚 ; 𝑗 = 1,2,3, … , 𝑛 (3) �̃�𝑖𝑗 is the weightage of the criteria. • Applying Equations (4) and (5), determine the PIS and NIS, respectively. 𝜉𝐽 + = {�̃�1 +, �̃�2 +, �̃�3 +, �̃�4 +} (4) 𝜉𝐽 − = {�̃�1 −, �̃�2 −, �̃�3 −, �̃�4 −} (5) • Establish the measures for separation from PIS and NIS using the equations (6) and (7) respectively. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 230 https://internationalpubls.com 𝜂𝑖 + = √∑ 𝑑(�̃�𝑗 +, �̃�𝑖𝑗)2 𝑛 𝑗=1 , 𝑖 = 1,2,3, … ,𝑚. (6) 𝜂𝑖 − = √∑ 𝑑(�̃�𝑗 −, �̃�𝑖𝑗)2 𝑛 𝑗=1 , 𝑖 = 1,2,3, … ,𝑚. (7) Where, 𝑑(�̃�𝑗 +, �̃�𝑖𝑗) 𝑎𝑛𝑑 𝑑(�̃�𝑗 −, �̃�𝑖𝑗) are determined by the distance formula of Heptagonal fuzzy number. • Determine how far each option is from the ideals of the positive and negative solutions which are defined as the relative proximity. 𝑐𝑙𝑖 + = (𝜂𝑖 +) (𝜂𝑖+ + 𝜂𝑖−) ⁄ (8) 𝑐𝑙𝑖 − = (𝜂𝑖 −) (𝜂𝑖+ + 𝜂𝑖−) ⁄ (9) • Ranking the alternatives: For 𝑐𝑙𝑖 + all of the options are shown in order of decreasing proximity to the highest. Additionally, the ranking order of all possibilities for 𝑐𝑙𝑖 −is provided, ranging from the closest to the furthest. Figure 2: Proposed Framework Flowchart 2. Numerical Illustration The following characteristics are gathered for 50 women using a self-separate approach. The TOPSIS method is suggested as a tool. The criteria 𝒮1, 𝒮2, 𝒮3, 𝒮4 are related to the disparities that women confront, and the options 𝒬1, 𝒬2, 𝒬3, 𝒬4, 𝒬5, 𝒬6, 𝒬7 are related to the effects of disparities. Characteristics associated with the inequalities that women experience are used as standards. 𝒮1 − 𝐸𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛 𝒮2 − 𝐻𝑒𝑎𝑙𝑡ℎ 𝑎𝑛𝑑 𝑊𝑒𝑙𝑙 𝑏𝑒𝑖𝑛𝑔 𝒮3 − 𝐸𝑐𝑜𝑛𝑜𝑚𝑖𝑐 𝑝𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝒮4 − 𝑃𝑜𝑙𝑖𝑡𝑖𝑐𝑎𝑙 𝑃𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑡𝑖𝑜𝑛 Attributes associated with the absence of inequalities in society are considered alternatives. 𝒬1 −𝑊𝑜𝑟𝑘 − 𝑙𝑖𝑓𝑒 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝒬2 − 𝑆𝑎𝑓𝑒𝑡𝑦 𝑐𝑜𝑛𝑐𝑒𝑟𝑛𝑠 Step 1 Gathering information Step 2 Survey Data in the form of Linguistic terms Step 3 TOPSIS Decision Matrix Step 4 Calculating weightage normalized matrix Step 5 Performance score and ranking computations Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 231 https://internationalpubls.com 𝒬3 − 𝑉𝑖𝑜𝑙𝑒𝑛𝑐𝑒 𝑎𝑔𝑎𝑖𝑛𝑠𝑡 𝑤𝑜𝑚𝑒𝑛 𝒬4 − 𝐺𝑒𝑛𝑑𝑒𝑟 𝑆𝑡𝑒𝑟𝑒𝑜𝑡𝑦𝑝𝑒𝑠 𝒬5 − 𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝐼𝑚𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠 𝒬6 − 𝑃𝑜𝑣𝑒𝑟𝑡𝑦 𝒬7 − 𝐻𝑒𝑎𝑙𝑡ℎ 𝑖𝑠𝑠𝑢𝑒𝑠 The decision-maker assigns the weights based on a subjective method for the criteria W = (0.4,0.3,0.2,0.1). 1. The decision-maker uses the linguistic terms listed in Tables 1 and 2 to compare criteria or alternatives. 2. Create a decision matrix using the data gathered. The HFN, which is displayed in Table 3, provides the entries of a decision matrix. 3. Creating the normalized matrix with Eq. (2), as indicated in Table 4. 4. Constructing a weighted matrix with Eq. (3), the other values are calculated which is shown in Table 4. 5. Using Eqns. (4) and (5) compute the PIS and NIS. 𝜉𝐽 + = {�̃�1 +, �̃�2 +, �̃�3 +, �̃�4 +} 𝜉𝐽 − = {�̃�1 −, �̃�2 −, �̃�3 −, �̃�4 −} 6. Equations (6) and (7) are used to compute each alternative's distance. Table 1 Linguistic terms and their corresponding HFN S.No. Linguistic Variables Heptagonal fuzzy numbers 1 Very high (VH) (7,9,11,13,15,17,19) 2 High (H) (6,8,10,12,14,16,18) 3 Medium High (MH) (5,7,9,11,13,15,17) 4 Neutral (N) (4,6,8,10,12,14,16) 5 Medium low (ML) (3,5,7,9,11,13,15) 6 Low (L) (2,4,6,8,10,12,14) 7 Very Low (VL) (1,3,5,7,9,11,13) Table 2 Rating of alternatives by decision maker 𝒮1 𝒮2 𝒮3 𝒮4 𝒬1 MH H N ML 𝒬2 MH L H N 𝒬3 L N ML VL 𝒬4 H N VH H 𝒬5 MH L N ML Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 232 https://internationalpubls.com 𝒬6 VL MH MH ML 𝒬7 L ML N VL Table 3 The decision matrix using HFN 𝒮1 𝒮2 𝒮3 𝒮4 𝒬1 (5,7,9,11, 13,15,17) (6,8,10, 12,14,16,18) (4,6,8,1 0,12,14,16) (3,5,7, 9,11,13,15) 𝒬2 (5,7,9,11, 13,15,17) (2,4,6, 8,10,12,14) (6,8,10, 12,14,16,18) (4,6,8, 10,12,14,16) 𝒬3 (2,4,6, 8,10,12,14) (4,6,8, 10,12,14,16) (3,5,7, 9,11,13,15) (1,3,5, 7,9,11,13) 𝒬4 (6,8,10, 12,14,16,18) (4,6,8, 10,12,14,16) (7,9,11, 13,15,17,19) (6,8,10, 12,14,16,18) 𝒬5 (5,7,9, 11,13,15,17) (2,4,6, 8,10,12,14) (4,6,8, 10,12,14,16) (3,5,7, 9,11,13,15) 𝒬6 (1,3,5, 7,9,11,13) (5,7,9, 11,13,15,17) (5,7,9, 11,13,15,17) (3,5,7, 9,11,13,15) 𝒬7 (2,4,6, 8,10,12,14) (3,5,7, 9,11,13,15) (4,6,8, 10,12,14,16) (1,3,5, 7,9,11,13) Table 4 Normalized Matrix 𝒮1 𝒮2 𝒮3 𝒮4 𝒬1 (0.19,0.27,0.35,0.4 2, 0.50, 0.58,0.65) (0.23,0.30,0.39,0.4 6, 0.54, 0.62,0.69) (0.14,0.21,0.28,0.3 5, 0.42, 0.49,0.56) (0.14,0.21,0.28,0.3 5, 0.42, 0.49,0.56) 𝒬2 (0.19,0.27,0.35,0.4 2, 0.50, 0.58,0.65) (0.08,0.15,0.23,0.3 1, 0.39, 0.46,0.54) (0.21,0.28,0.35,0.4 2, 0.49, 0.56,0.63) (0.21,0.28,0.35,0.4 2, 0.49, 0.56,0.63) 𝒬3 (0.07,0.15,0.23,0.3 0, 0.39, 0.46,0.53) (0.15,0.23,0.31,0.3 9, 0.46, 0.54,0.62) (0.11,0.16,0.25,0.3 2, 0.39, 0.46,0.53) (0.11,0.16,0.25,0.3 2, 0.39, 0.46,0.53) 𝒬4 (0.23,0.30,0.39,0.4 6, 0.53, 0.61,0.69) (0.15,0.23,0.31,0.3 9, 0.46, 0.54,0.62) (0.24,0.32,0.39,0.4 6, 0.53, 0.60,0.67) (0.24,0.32,0.39,0.4 6, 0.53, 0.60,0.67) 𝒬5 (0.19,0.27,0.35,0.4 2, 0.50, 0.58,0.65) (0.08,0.15,0.23,0.3 1, 0.39, 0.46,0.54) (0.14,0.21,0.28,0.3 5, 0.42, 0.49,0.56) (0.14,0.21,0.28,0.3 5, 0.42, 0.49,0.56) 𝒬6 (0.03,0.11,0.19,0.2 7, 0.35, 0.42,0.50) (0.19,0.27,0.35,0.4 2, 0.50, 0.58,0.66) (0.18,0.25,0.31,0.3 9, 0.46, 0.53,0.60) (0.18,0.25,0.31,0.3 9, 0.46, 0.53,0.60) 𝒬7 (0.07,0.15,0.23,0.3 0, 0.39, 0.46,0.53) (0.12,0.19,0.27,0.3 5, 0.42, 0.50,0.58) (0.14,0.21,0.28,0.3 5, 0.42, 0.49,0.56) (0.14,0.21,0.28,0.3 5, 0.42, 0.49,0.56) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 233 https://internationalpubls.com Table 5 Weighted Normalized Matrix 𝒮1 𝒮2 𝒮3 𝒮4 𝒬1 (0.08,0.11,0.14,0 .17, 0.2, 0.23,0.26) (0.06,0.09,0.12, 0.14, 0.16,0.19,0.21) (0.03,0.04,0.05,0.07,0.08,0.0 9,0.11) (0.01,0.02,0.03, 0.04, 0.05,0.06,0.07) 𝒬2 (0.08,0.11,0.14,0 .17, 0.2, 0.23,0.26) (0.02,0.05,0.07, 0.09, 0.12,0.14,0.16) (0.04,0.05,0.07,0.08,0.09,0.1 1,0.13) (0.01,0.02,0.03, 0.04, 0.05,0.06,0.07) 𝒬3 (0.03,0.06,0.10, 0.12, 0.16,0.18,0.21) (0.05,0.07,0.09, 0.12, 0.14,0.16,0.19) (0.02,0.03,0.05,0.06,0.08,0.0 9,0.11) 0.00,0.01,0.02,0. 03, 0.04, 0.05,0.06) 𝒬4 (0.09,0.12,0.16, 0.18, 0.21,0.24,0.28) (0.05,0.07,0.09, 0.12, 0.14,0.16,0.19) (0.05,0.06,0.08,0.09,0.11,0.1 2,0.13) (0.02,0.03,0.04, 0.05, 0.06,0.07.0.08) 𝒬5 (0.08,0.11,0.14,0 .17, 0.2, 0.23,0.26) (0.02,0.05,0.07, 0.09, 0.12,0.14,0.16) (0.03,0.04,0.05,0.07,0.08,0.0 9,0.11) (0.01,0.02,0.03, 0.04, 0.05,0.06,0.07) 𝒬6 (0.01,0.04,0.08, 0.11, 0.14,0.17,0.2) (0.06,0.08,0.11,0 .13, 0.15,0.17,0.20) (0.04,0.05,0.06,0.08,0.09,0.1 1,0.12) (0.01,0.02,0.03, 0.04, 0.05,0.06,0.07) 𝒬7 (0.03,0.06,0.10, 0.12, 0.16,0.18,0.21) (0.04,0.06,0.08, 0.11, 0.13,0.15,0.17) (0.03,0.04,0.05,0.07,0.08,0.0 9,0.11) (0.00,0.01,0.02, 0.03, 0.04,0.05,0.06) �̃�𝑗 + = (0.1829,0.1386,0.0814,0.05) �̃�1 + =(0.09,0.12,0.16,0.18,0.21,0.24,0.28) �̃�2 + =(0.06,0.09,0.12,0.14,0.16,0.19,0.21) �̃�3 + =(0.04,0.05,0.07,0.08,0.09,0.11,0.13) �̃�4 + =(0.02,0.03,0.04,0.05,0.06,0.07.0.08) �̃�𝑗 − = (0.1229,0.0929,0.0629,0.03) �̃�1 − =(0.03,0.06,0.10,0.12,0.16,0.18,0.21) �̃�2 − =(0.02,0.05,0.07,0.09,0.12,0.14,0.16) �̃�3 − =(0.02,0.03,0.05,0.06,0.08,0.09,0.11) �̃�4 − =(0.00,0.01,0.02,0.03,0.04,0.05,0.06) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 234 https://internationalpubls.com Table 6 Performance rating for every option using the suggested approach S.No Alternatives Performance score Ranking 1 𝒬1 0.83556 2 2 𝒬2 0.85691 4 3 𝒬3 0.86516 5 4 𝒬4 0.74944 1 5 𝒬5 0.83907 3 6 𝒬6 0.8758 6 7 𝒬7 0.90344 7 Figure 3: Graphical analysis for Performance Score 3. Discussion We infer that the accuracy rises with increasing the fuzzy number. Heptagonal fuzzy numbers with subjective opinions incorporated into them actually reduce ambiguity and the resulting results provide a better option than the other way. In this proposed method alternative "𝓠𝟒", which states that gender stereotypes are more prevalent in society and that women must deal with them comes in first. 𝓠𝟏, 𝓠𝟓, 𝓠𝟐, 𝓠𝟑, 𝓠𝟔 that ranks the second, third, fourth, fifth, sixth respectively and 𝓠𝟕 ranks the last. While in comparing the other fuzzy numbers [17] we get the better results. Acknowledgement The authors would like to thank all the unknown referees for their guidance and expertise throughout this research process. References [1] B. Ashtiani, F. Haghighirad, A. Makui, G.A. Montazer: Extension of fuzzy TOPSIS method based on interval-valued fuzzy sets. Appl. Soft Comput. (2008) [2] A.K. Bhargava: Fuzzy Set Theory Fuzzy Logic and Their Applications. S. Chand and Company (2008) [3] C. 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