Decision Support Systems are advanced technologies that facilitate decision-making in complex environments by offering structured analysis and data-driven choices (Mahmood, 2016) [1]. As a result, these systems are becoming crucial in various disciplines, including corporate strategy, health, and well-being. In an attempt to achieve a resolution, several tools and decision-making systems were integrated, but none of them was able to adequately illustrate the complexity of problem. New theories are now being used by researchers as mathematical instruments for decision assistance and decision-making systems. Mathematical and analytical models are used in model management system (MMS) to interpret data and make choices. Mathematical models are instruments for improving decision-making systems because they contain specialized information or principles to help decision-making, such as fuzzy set theory [2] (Zadeh 1965), rough set theory [3] (Pawlak 1982), and soft set theory [4] (Molodtsov 1999). Analytical models (AM) are mathematical representations employed to analyze the data, interpret it, and predict the behavior of the process, or decision-making system. AM uses mathematical formulas, statistical techniques, or algorithms to examine input data and provide output insights, assisting in decision-making, aiding in understanding correlations between variables, and forecasting outcomes using various analytical tools like EXCEL, SPSS, MATLAB, R, and Python.

The process of decision-making (DM) entails selecting the optimal choice among feasible options, preferences and available data. Many fields, including business statistics, health, applied mathematics, and economics, employ MCDM. Zadeh (1965) [2] pioneered the concept of fuzzy set theory to deal with vague data. In 1970, Bellman and Zadeh [5] initially introduced fuzzy set theory, which has applications in decision-making. Many researchers have been attempting to solve decision-making issues by utilizing fuzzy set theory. ’Decision making and problem solving are distinct processes, where problem-solving requiring finding a solution while decision making requires making a choice’ says Michael J. Marx.

In order to solve a decision making problem Hwang and Yoon (1981) [6] constructed Technique for order preference by similarity to an ideal solution (TOPSIS), a multi-criteria decision-making (MCDM) technique that seeks to identify the optimal option among acceptable possibilities. According to the TOPSIS principle, the alternative should be positioned as close to the positive ideal solution, and as far away from the negative ideal solution as possible [11]. The TOPSIS technique has been expanded by many researchers to work in fuzzy contexts, leading to applications in multi-criteria fuzzy decision-making. Triantaphyllou and Lin (1996)[7] used fuzzy arithmetic operations to create a fuzzy version of the TOPSIS technique, which produced fuzzy relative closeness for every decision. By defining, Euclidean distance between fuzzy numbers Chen(2000) [8] widened the TOPSIS technique to fuzzy group decision-making scenarios substantially with the rating value of each alternative and weight of each criterion. Tsaur et al. (2002)[9] used the centroid method which is one of the defuzzification methods to transform a fuzzified MCDM problem into a crisp one, and solved the non fuzzified multi criteria issue by using the TOPSIS technique. Many researchers applied the TOPSIS technique in various fields (Hung, 2009) [40], (Zhi-yong Bai, 2013)[11], (Akram, 2020) [10]. Further, the TOPSIS method was extended by Jahanshahloo et al. (2006) [12],(Li, 2011) [42], (Joshi, 2014) [41], Dincer (2015) [35], Han (2018) [36], Akram (2019) [34], Sindhu (2021) [33], to handle fuzzy data, achieving a high accuracy rate when dealing with numerical values.

On the other hand, a score-valued function is frequently used in fuzzy set theory to measure a fuzzy set’s total value or membership degree. When working with ambiguous or imprecise data, a score function in fuzzy sets is a useful tool for aggregating fuzzy numbers into a single indicative value for ranking or assessment. The score functions are often used in fuzzy multi-criteria decision-making (FMCDM), which uses fuzzy criteria. The selection of scoring function might have an impact on the result, various score functions may provide different ranks for the same fuzzy data.

However, decision-making data often involves both linguistic variables and numerical data, and some information exhibits bipolarity, with both positive and negative values that must be considered [14] (Nathinee Deetae, 2021). In mathematics, bipolar values refer to quantities that can take two distinct forms, one positive and one negative; these values may also represent conflicting or opposing ideologies, often positioned at opposite ends of a spectrum. Bipolar values are crucial in decision-making, especially when addressing contrasting or opposing perspectives. By recognizing opposing values, decision-makers can weigh the pros and cons, leading to more informed and equitable decisions. Bipolar values dynamically foster creativity and innovation, ensuring the decision-making accounts for diverse factors and conflicting interests, enabling more holistic and responsible outcomes.

In 1994, Zhang [16] introduced the bipolar fuzzy set, often known as bipolar fuzzy logic, which has been extensively used to address pragmatic problems. In 1998, the concepts of bipolar fuzziness and interval-based bipolar fuzzy logic were established by Zhang [13]. Numerous researchers developed the concepts of Bipolar fuzzy set theory and applied it in various domains [15], [10] [27] [31], [38] [32],[22], [37], [21]. [0, 1] and [-1, 0] are two extremes that represent the membership degree range of bipolar fuzzy sets: 0 denotes irrelevant elements and (0, 1], [-1, 0] denotes partial fulfillment. In 2008, The significance of bipolarity in human cognition and emotional selection was addressed by Da Silva Neves et al. [?]. In 2018, MCDM techniques in a bipolar fuzzy environment were presented by Alghamdi M. A. et al. [15]. Bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE-I algorithms were offered by Muhammad Akram et al. in 2020 [10]. Many researchers developed different scoring value for bipolar function [1], [37]. Additionally, various researchers applied these scoring functions in MCDM problems [1], [18], [19], [11]. The BFS-TOPSIS approach to MCDM problem-solving was put into practice by Nathinee Deetae [14] in 2022.

By incorporating BF concepts into set theory and bipolar logical reasoning in 2004, Zhang and Zhang [20] strengthened the FS. In 2015, the notion of bipolar fuzzy aggregation operators was first presented by Gul [21]. The bipolar fuzzy weighted arithmetic (BFWA) operator, and the bipolar fuzzy weighted geometric (BFWG) operator are two key operators designed within this paradigm [22]. The hesitant bipolar fuzzy weighted averaging and geometric operators were initially proposed by Wei et al. [23] in 2017. Hamacher aggregation operator was described by Xu and Wei [37]. Dombi aggregation operator was introduced by Jana [22] in 2019 and are intended for BFS. These operators are employed to create solutions for problems involving multi-attribute group decision-making. Following their research, Jan et al. [24] created a hybrid decision-making framework in bipolar complex pictures fuzzy soft sets. Mani et al. (2023) [25] suggested the concept of intuitionistic fuzzy bipolar metric spaces. In 2024, Dilshad Alghazzawi et al.,[26] introduced the dynamic aggregator operators in bipolar fuzzy environment.

Definition 1. [Bipolar fuzzy sets [13] Zhang, 2016] Let X be the non-empty fuzzy set. The bipolar fuzzy set

− + + −

B = {B , B } is defined in X. B ∈ [ 0, 1] is the satisfaction degree of x in B, B ∈ [ −1, 0] is the satisfaction degree of the implicit counter property of x in B.

Definition 2. [Operations on Bipolar fuzzy sets:][29] Consider B1, B2 ∈ BFS. Zhang defines the following

operations.

− − + +

i) Disjunction : B1 + B2 = (−max[b1 , b2 ], max[b1 , b2 ])

− − + +

= (−[b1 ∨ b2 ], [b1 ∨ b2 ])

− − + +

ii) Parallel conjuction: B1.B2 = (−min[b1 , b2 ], min[b1 , b2 ])

− − + +

= (−[b1 ∧ b2 ], [b1 ∧ b2 ])

− + + − − − + +

iii) Serial conjuction: B1 × B2 = (−{[b1 ∧ b2 ] ∨ [b1 ∧ b2 ]}, {[b1 ∧ b2 ] ∨ [b1 ∧ b2 ]}) iv) Negation: −B1 = [−b1 , b1 ] v) Complement: ¬B1 = ¬[−1 + b1 , 1 − b1 ] Definition 3. [Score function] Wei et al. [37]defined the score function S(B) of B = (b , b ) ∈ BFS as:

1 + b + b S(B) =

S(B) ∈ [0, 1].

′ − +

Definition 4. (Score Function) T. Mahmood et al. [1] defined the score function as S (B) of B = (b , b )

∈ BFS as:

′ + −

S (B) = b + b

′ + − ′

where S (B) ∈ [−1, 1], and if b = b =⇒ S (B) = 0

Definition 5. (Measures of distance)[28] Consider f1 and f2 are two fuzzy sets. The distance measure

between f1 and f2 sets are defined as:

pP i) Euclidean Distance: de(f1, f2) = (µf (xk) − µf (xk))

k=1 1 2

Pn ii) Normalized Euclidean Distance: dne(f1, f2) = (µf (xk) − µf (xk))

n k=1 1 2

(BWAO) for the alternative βj(j = 1, 2, ..., n) is described as

+ wj − wj

BW AO(β1, β2, ..., βn) = (1 − (1 − B )(x)) , − (B (x) ),

Pn where wj ∈ [0, 1] is the weight of βj and wj = 1.

Definition 7. [Bipolar weighted geometric operator (BWGO)] [22] The bipolar weighted geometric operator

(BWGO) for the alternative βj(j = 1, 2, ..., n) is described as

+ wj − wj

BW GO(β1, β2, ..., βn) = ((1 − B )(x)) , −1 + (1 − (B (x) )),

Pn where wj ∈ [0, 1] is the weight of βj and wj = 1.

+ + + + + + T

BFPIS = [(α1 , β1 ), (α2 , β2 ), · · · , [(αm, βm)]

− − − − − − T

BFNIS == [(α1 , β1 ), (α2 , β2 ), · · · , [(αm, βm)] + − + −

where αq = maxp[apq], αj = minp[apq], βq = maxp[bpq], αj = minp[bpq].

Step 4: Now, calculate the ED between each alternative from BFPIS and BFNIS [15]

u X m + 2 + 2

ED(ti, BF P IS) = t (apq − αq ) + (bpq − βq ) 2 q=1

u X m − 2 − 2

ED(ti, BF N IS) = t (apq − αq ) + (bpq − βq ) 2 q=1

cri =

Step 6: Arrange and rank the ti where the greatest value of cri gives the most desirable solution.

Step 1: The decision making matrix in the bipolar environment is given in Table 1.

Step 2: The weighted decision matrix is presented in Table 2.

Step 3: The value of BFPIS and BFNIS are shown in Table 3.

Step 4 : Calculate the distance values of each alternative from the BFPIS and BFNIS:

Step 5: The computed value of relative closeness are ES1 = 0.48806, ES2 = 0.25174,

Step 6: Arrange and select the best alternative: ES3 > ES1 > ES4 > ES2. Among the ERP sytems ES3 is the best choice.

Now, we utilize BWAO operator in the TOPSIS method to find the most suitable ERP system.

Step WA2: Table 4 represents the value of BWAO decision matrix.

Step WA3: Table 5 provides the BFPIS and BFNIS values which is computed with the help of score function.

Step WA4: The distance values of each alternative from from the BFPIS and BFNIS are,

Step 5: The value of relative closeness: ES1 = 0.4951, ES2 = 0.4407, ES3 = 0.6323, and

Step 6: Select the best alternative using the closeness ratio: ES3 > ES1 > ES4 > ES2.

Now, we utilize BWGO operator in the TOPSIS method to find the most suitable choice.

Step WG2: Table 6 represents the BWGO decision matrix.

Step WG3: Table 7 gives the BFPIS and BFNIS values.

Step WG4: The distance values of each alternative from from the BFPIS and BFNIS are:

Step WG5: The value of relative closeness are, ES1 = 0.4393, ES2 = 0.5275, ES3 = 0.6013, and

Step WG6: Arrange and rank the best alternative using the closeness value, ES3 > ES2 > ES1 > ES4. In all the extended TOPSIS method, ES3 is the best choice.

We follow Step 1, Step 2 as it is in TOPSIS method.

Step P3: Compute the value (Sfi)using the score function.

Pn

Step P5: Arrange and rank the alternatives.

Now, applying the suggested method in the above mentioned case. Also, considering Step 1, and Step 2 as it in the TOPSIS method.

Step P3: Computed (Sfi), using the Definition [] are presented in the Table 8.

Step P4: The calculated E(Sfi) s values are, E(Sf1) = 0.228, E(Sf2) = 0.1405,E(Sf3) =

0.4085, and E(Sf4) = 0.0565 Step P5: By arranging the alternatives, ES3 > ES1 > ES2 > ES4.

By using the score function, easily we can achieve the desired outcome.

In similar way, we can apply the score function for Step WA2. Step P3 is as follows,

Step P3: The computed (Sfi), values are presented in the Table 9.

Step P4: The calculated ES(Sfi) s values are, ES(Sf1) = −0.208, ES(Sf2) =

−2.346,ES(Sf3) = −1.859, and ES(Sf4) = −2.438

In similar way, we can apply the score function for Step WG2. By applying Step P3 and Step P4, we get

ES(Sf1) = 2.992, ES(Sf2) = 2.888, ES(Sf3) = 3.19, and ES(Sf4) = 2.741. By arranging it, ES3 > ES1 > ES2 > ES4.

To corroborate our methodology, we examine the robustness and effectiveness of our suggested method to MCDM approaches, and results are shown in Table 11.

After the COVID-19 epidemic, mathematics has emerged as an important scientific tool, notably in decision-making, predictive modeling, and statistical analysis. Mathematical modeling helps immensely in forecasting epidemics of diseases, tracking infection rates, and monitoring the effectiveness of measures such as vaccination and isolation period. Significant data sets are examined using machine learning models that rely on mathematical techniques. Experts use AI and machine learning to ensure effective health care, effective therapies, and predicting medical results. The field of mathematics has progressed in generating predictive algorithms, and optimization algorithms are used to allocate constrained substances like booster shots, and hospital beds.

After COVID-19, the study of mathematics has evolved and enriched with various scientific methodologies. Even Nevertheless, there is a drop in students’ decision to pursue a career in mathematics, and curiosity in Mathematics has declined. It is influenced by several variables, notably the difficulty of learning fundamental ideas via the internet and the absence of live instruction, peer problem solving, and discussion that sustains interest in subjects. This exacerbates the situation and makes the trend towards choosing multidisciplinary, technical disciplines over pure mathematics. Meantime, there is a growing need for medical research and other related fields. The changes in mathematics syllabus with less emphasis on particular topics, make students demotivated and lose interest in continuing mathematics at a higher level, their attention is directed towards skill-based or pragmatic subjects. A survey is conducted among the college students in the Vellore district to investigate their opinions about Mathe-matics and other practical disciplines. Figures 1 and 2 depict the algorithm and the conceptual model of the study.

the college students in Vellore district to investigate reasons behind the decline in Mathematics for higher studies. Data is collected via a Google Form survey that focuses on information regarding the interest, course structure, challenges, and career prospects in Mathematics. Further, information on students’ interest in other courses, course structures, and career prospects has been collected.

In this study, we identified four variables: job opportunities in Mathematics (Job), Mathematics course

structure (Course_Struct), other courses (Other_Courses), and decline in Mathematics choice (M aths_decline). The dependent variable was the decline in Mathematics choice, while the independent

variables were job opportunities in Mathematics, Mathematics course structure, and other courses.

• To measure the extent to which job opportunities in Mathematics dissuade students from choosing

Mathematics for higher studies.

• To evaluate how other applied courses in higher education influence the decision to choose Mathematics.

• To ascertain which aspects of the Mathematics course structure deter students from selecting Mathematics

for higher studies.

Research Hypothesis:

The hypothesis are formulated based on the objective:

H01: There is no significant effect of job opportunities in Mathematics that leads to the decline in Mathematics choice.

H02: There is no significant impact of other applied courses that influence the decrease in Mathematics choice. H03: There is no significant impact of Mathematics course structure that reduces the Mathematics choice. 2. Transformed crisp data into bipolar fuzzy data.

3. Represented the data into 5-point bipolar linguistic value.

4. The score value is computed for all the linguistic values using the definition [4].

5. Observations: A multiple regression test is employed to analyze the study. A multiple correlation co-

efficient between the predictors and the dependent variable is 0.795, which gives a strong and positive correlation. With an R-Square value of 0.632, the model explains 63.2% of the variation in Decline in Mathematics. The model represents a significant amount of variance (63.2%) in the dependent variable, as depicted in Figure 3. The F-value = 45.739 in Table 12, indicates that the model is statistically significant. The statistical significance

of the model (p < 0.001), implies a strong connection between the predictors and the dependent variable.

• Job (Beta = 0.408): Among all the factors, Job has the greatest positive effect on M aths_decline.,

as depicted in Figure 4.

• Other_Courses (Beta = -0.355): This indicates that Other_Courses has a relatively substantial negative influence on M aths_decline, as shown in Figure 5.

• Course_Struct (Beta = 0.221): Course_Struct significantly and positively affects M aths_decline but less strongly.

• The study found that the t-value of 3.413 is a significant predictor, while other courses (t= -4.306) have

a strong negative effect on the independent variable. Job opportunities( (t = 4.930) have the strongest positive effect on the independent variable, while course structure (t = 2.888) has a positive effect on the dependent variable. These findings suggest that job opportunities play a crucial role in predicting outcomes.

6.Results: The regression model is statistically significant, with a large F-statistic (45.739) and a p-value (0.000) confirming its significance as shown in Figure 6. Job (Beta= 0.408) is the most important factor influencing the dependent variable, followed by other courses (Beta= -0.355), which have opposite effects on the dependent variable. All predictors are statistically significant in explaining the decline in Mathematics choice in higher studies.

In this study, we developed a novel method that yields the bipolar fuzzy TOPSIS method result in a bipolar environment. The score function is applied in the suggested technique to compute the scores and ascertain the sum of the scores for each possibility. The ranking is given according to the values. The benefits of the

recommended approach are outlined and a comparison study of the various extended fuzzy T OP SIS methods

is offered to support the proposed approach. Eventually, a case study demonstrated the potential uses and effectiveness of the recommended methodology. Additionally, we employed the bipolar scoring function to illustrate the regression results and examine the effect of other parameters in mathematics. In the forthcoming work, we will work on implementing the proposed approach in many fuzzy domains.

The authors do thankful to the editor for giving an opportunity to submit our research article in this esteemed journal.

There is no funding support for this work.

The authors declared that they have no conflict of interest regarding the publication of the research article.

Enquiries about data availability should be directed to the authors.

Not applicable

GA analyzed and developed the idea of PrIFN and JR applied the technique in the field of decision making. GA and JR verified and approved the final paper.

Not applicable.