Unveiling Water Quality Insights by Exploring Intuitionistic Fuzzy TOPSIS in Multi-criteria Decision Analysis

Water is a priceless natural resource that is essential for societal development and human life [1] and serves as the foundation for long-term economic growth. But as the global economy has grown and the population has increased exponentially, there is a growing shortage of water sources and a decline in the quality of the water, which has led to serious problems [2]. Due to the overflow of domestic sewage from cities, agricultural effluents, and industrial wastewaters, water quality integrity has suffered significantly [3]. It is crucial to conduct thorough examinations into the water quality levels of various rivers in order to stop further deterioration in light of the escalating water resource problem [4, 5].

Numerous academics from various countries have studied methods for determining water quality [6]. Notably, Horton et al. developed the Water Quality Index (WQI) approach in the US in 1965, which is a framework that includes eight different water quality indices [7-9]. Since the 1990s, a variety of mathematical methods and models have been used to study the worldwide water quality, with encouraging results. These approaches include, among others, evaluations based on neural networks [10, 11], studies of matter elements [12], the complete approach to identifying water quality [13], the projection pursuit method [14] and the grey index method [15]. These methods have significantly accelerated the development of water quality evaluation. However, underlying flaws and deficiencies continue [16].

Enter the world of intuitionistic fuzzy set theory, a field awash in uses for decision-making, multi-criteria decision-making, networking, computing, smart systems, economics, and more. The notion of Multi-Criteria Decision-Making (MCDM), which seeks to identify the best option from a variety of options, is well suited to meet the difficulty of assessing several alternatives, which is a necessity in decision-making. MCDM, which was founded by Bellman and Zadeh, has recently experienced exceptional growth.

A new paradigm that tackles common issues in the field is presented by combining intuitionistic fuzzy set theory with the TOPSIS model. Mishra et al. [17] have observed that conventional fuzzy mathematics approaches have faced difficulties related to notable discrepancies, subjective perceptions, and complex computation. Alternatively, our novel strategy departs from the state of the art by introducing non-membership functions into intuitionistic fuzzy sets. By enabling a more nuanced depiction of fuzziness, this augmentation lessens the drawbacks of conventional methods.

The TOPSIS Model has been well recognized for its geometric clarity, careful use of original data, little information loss, and wide applicability; nevertheless, when combined with intuitionistic fuzzy set theory, it is a groundbreaking development [18-20]. Together, they provide a potent new model that outperforms existing approaches. In addition to addressing the drawbacks of current methods, the merging adds a degree of efficiency and adaptability that goes beyond the capabilities of tried-and-true multi-attribute decision-making strategies.

The introduction of CCME-WQI, pioneered by Horton in 1965, marks a paradigm-shifting moment in the evaluation of water quality. This all-encompassing gauge quickly won praise for its unmatched usefulness, crossing geographical boundaries to encroach on areas as different as Asia, Africa, and Europe [21]. The CCME-WQI's global applicability underlined its fundamental relevance and encouraged widespread adoption across continents, establishing it as a de facto standard for evaluating water quality worldwide.

The paper is structured as follows: In Section 1, the research area's engineering overview is first presented. A novel method of assessing water quality is provided in Section 2. It is predicated on the intuitionistic fuzzy set model. The IFT model is created for the Cauvery River's water quality evaluation in Section 3, and the evaluation results of the suggested model are examined. Likewise, the WQI is examined. Section 4 presents the conclusions.

The AHP technique made it easier to organise the decision problem in a hierarchical manner. AHP provides a structured method to prioritise and assess alternatives through pairwise comparisons and the creation of weights. The IFT approach was used in parallel to deal with the inherent uncertainties and ambiguity in the decision-making process. This method accommodated uncertainty and hesitancy in the evaluation of criteria and alternatives by using intuitionistic fuzzy sets. A strong basis for efficient decision-making in the setting of complex material selection considerations was created by the combination of the structural clarity of the AHP and the ability of the IFT to manage uncertainty.

3.1 Analytic Hierarchy Process (AHP)

The AHP appears as a potent instrument in the decision-making space, particularly when dealing with complicated scenarios. The late 1970s Thomas L. Saaty, a mathematician and operations researcher, develop AHP, which provides a structured method for addressing the difficulties MCDM presents [12]. AHP serves as a pillar of rational decision analysis by assisting decision-makers in making well-informed decisions through a systematic process of prioritization and comparison.

AHP proves useful when a decision-making team must make complicated decisions and develop a structured method to deal with complexity [23]. The core of AHP entails creating a hierarchical arrangement, similar to a ranking, of decision components as shown in Figure 2. Decision-makers can systematically weigh the importance of various components by using this hierarchy, which captures the links between them. Matrix comparisons are made between each potential pair inside each cluster of choice components to aid in the evaluation. The consistency ratio helps decision-makers to evaluate the internal consistency of their preferences and judgements, ensuring that the decision-making process is logical and well-founded [24]. By assigning weights to each component in a cluster based on their relative relevance, AHP increases the usability of its results. These weightings give you a methodical way to quantify the importance of various components.

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Figure 2. Structure of AHP

Table 1. Comparison scale

Step 1. Identification and selection of attributes for the decision tree in a hierarchical structure.

Every AHP analysis begins with defining the study’s hierarchy structure, which can be defined as a split of a series of levels of attributes, each of which represents a number of tiny groups of interrelated sub-attributes.

Step 2. The matrix of pair-wise comparisons has to be setup:

The matrix of pair-wise comparison is a matrix (MPC) Collects findings of expert and expert ratings. Professional judgments are stated in an MPS Analysis of the MADM problem using AHP, where a result. The producer specifies a judgment by inserting the entry $a_{i j}$ where ($a_{i j}$>0), indicating how much more important attribute i is than attribute j.

A MPC is defined as:

$\mathrm{A}=\left(a_{i j}\right)=\left[\begin{array}{cccccc}a_{11} & a_{12} & \ldots & & \ldots & a_{1 n} \\ a_{21} & a_{21} & \ldots & & \ldots & a_{2 n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &\\ a_{m 1} & a_{m 2} & \ldots & \ldots & a_{m n}\end{array}\right]$    （1）

where, $a_{i j}$ is the weighted average of the ascribed $a_i$ and $a_j$. The MPC would be a square matrix A, encompassing n qualities each with relative weights. The weights of all qualities in this matrix are measured in multiples of that unit with regard to each other.

Step 3. Weighting vectors of attributes to be calculated

Cardinal numerical values that represent the overall preference of each defined choice are taken into account via additive weighting methods. Saaty offered comparable scores ranging from 1 to 9 (Table 1) as a basis for translating language judgments into cardinal.

Step 4. Calculating the relative weights to approach principal eigen vector

The weights of characteristics are calculated during the averaging procedure over the normalized columns. A priority matrix reflecting the estimation of the matrix’s eigen values is necessary to provide the best match for attributes. so that the weights added together equal one to accomplish this, divide the relative weights of each individual characteristic by the column-sum of the acquired weights.

Step 5. Checking of the consistency of attributes

If the calculated discrepancies are greater than 10, the result of the manufacturer may have to make compensatory transactions within the characteristic values. Only if there are comparative metrics the calculated priorities may be consistent or almost consistent is valid. The equation can be used to calculate approximations stability ratio.

$C R=\frac{C I}{R I}$     (2）

where, CR represents consistence ratio, CI represents consistency index, RI represents random index for the size of the matrix ‘n’ proceeding with Eq. (2) the consistency ratio satisfies with CR<0.1

Based on the above steps the criteria weights are evaluated for the parameters DO, BOD and FC to evaluate the quality of the water as shown in Table 2.

Table 2. Criteria weight by AHP

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Figure 3. Structure of IFT

3.2 Intuitionistic Fuzzy Sets-TOPSIS (IFT)

IFT emerges as a sophisticated framework that effortlessly combines the strength of fuzzy logic with the analytical skill of decision analysis in the decision-making domain, where uncertainty, imprecision, and subjectivity abound. This approach, which was developed to address situations when choice qualities and preferences are characterized by increased ambiguity and indistinctness, is an advanced extension of the traditional TOPSIS [25-29]. The working of IFT is given in Figure 3.

Using intuitionistic fuzzy sets and the TOPSIS model, the priority ranking of MCDM items is solved [30]. Additionally, the evaluation of water quality risk can be transformed into a problem comprising different degrees of probability rankings. The procedures for using this specific approach are as follows:

Step 1. The production of the water quality evaluation indices and categorization criteria is possible following a thorough examination of the factors that affect the water quality.

Step 2. Weighting factors are calculated, and the significance of the index weights is conveyed using intuitionistic fuzzy numbers that are based on the idea of intuitionistic fuzzy sets. The weight coefficients of various indices are then solved using the AHP weight theory.

Step 3. Assuming that µ=µ₁, µ₂, .... µ_n, and γ=γ₁, γ₂, .... γ_n, respectively, are the weights of the membership function and the non-membership function, respectively, the following weights coefficients are used to represent the combination of the two functions:

$\varphi_n=\left(\mu_n, \gamma_n\right)=\left(\min \left(\mu_n, \gamma_n, 1\right)-\max \left(\mu_n, \gamma_n\right)\right)$    (3)

Step 4. At various stable levels of assessment indices, the evaluation systems calculate the corresponding solutions C⁺ and C⁻.

$C^{-}=\left[\left(\mu_1^{-}, \gamma_1^{-}\right),\left(\mu_2^{-}, \gamma_2^{-}\right), \ldots .\left(\mu_n^{-}, \gamma_n^{-}\right)\right]$   (4)

Step 5. The evaluation of the water quality to determine the degree of membership by calculating the Euclidean distance between positive and negative ideal solutions.

Step 6. The magnitudes of the pasting schedule can be used to calculate the levels of water quality. Following the determination of the degrees of membership regarding water quality, the magnitudes of each degree are rated from large to small, and the highest degree of membership is then chosen as the evaluation level of the related level of water quality.

The CCME, which consists of Canadian jurisdictions, has created a CCME WQI that may be utilised by several water authorities in numerous other countries. It came from the British Columbia WQI but was tweaked. The CCME WQI does not use typical index aggregation, subindexes, or weights for the variables [20]. According to the CCME, the CCME WQI measures three essential parts (factors) in order to produce a single unitless number that, in the end, represents the whole water quality. For each parameter that shouldn’t be exceeded, goal values (objectives or guidelines) are employed.

Step 1. The percentage of parameters (failed parameters) that at least once during the time period under consideration failed to achieve their criteria is represented by $F_1$ (Scope) when compared to the overall number of parameters analysed. The words “target values” and “objectives” are equivalent to “guidelines.”

$F_1=\left(\frac{\text { No. of Failed Parameter }}{\text { Total no. of Parameter }}\right) \times 100$    (5)

Step 2. The word $F_2$ (Frequency) denotes the frequency of individual tests that fail to meet standards. A test compares a parameter’s value from a specific sample campaign and the corresponding guideline directly.

$F_2=\left(\frac{N o . \text { of Failed tests }}{\text { Total no.of tests }}\right) \times 100$     (6)

Step 3. Calculating $F_3$ (Amplitude), which shows how far test results that failed are from the norm, takes three steps. An excursion is the number of times a certain concentration exceeds (or falls below, if the guideline is a minimum) the guideline and is shown as follows:

When the objective (guideline) of the j^th parameter cannot be exceeded by the i^th test value:

excursion $_i=\left(\frac{\text { Failed test value }_i}{\text { objective }_i}\right)-1$     (7)

For situations when the test value must not be lower than the objective (guideline):

excursion $_i=\left(\frac{\text { objective }_j}{\text { Failed test value }_i}\right)-1$     (8)

Step 4. Divide the total number of tests (both those that adhere to the standards and those that do not) by the sum of the individual tests’ deviations from the requirements to get the overall amount by which each test is out of compliance. The parameter known as the normalised sum of excursions (nse) is calculated as:

nse $=\frac{\sum_{i=1}^n \text { excursion }_i}{\text { Total no.of tests }}$    (9)

Step 5. After scaling the normalised sum of the excursions from the guidelines (nse) to produce a range between 0 and 100, an asymptotic function calculates $F_3$.

$F_3=\left(\frac{n s e}{0.01 n s e+0.01}\right)$     (10)

Step 6. After obtaining the factors, the index can be determined by adding the three factors together as follows:

$C C M E-W Q I=100-\left[\frac{\sqrt{F_1^2+F_2^2+F_3^2}}{1.732}\right]$    (11)

The resultant values are normalised to a range between 0 and 100 using the divisor 1.732, where 0 denotes the “worst” water quality and 100 denotes the “best” water quality.

The various rankings given to the nine locations scattered throughout the length of the Cauvery River are shown in Figure 6. The first rank in this graphical representation denotes regions with excellent water quality, while the ninth rank is designated for places with poor water quality. The variability of water quality along the river’s length includes a wide variety of characteristics. In this context, Vairapalayam stands out as a regrettable exception because it suffers from poor water quality. In contrast, Komarapalayam is highlighted as a significant instance of a location where the water quality reaches a level deemed acceptable. This comparison highlights the notable differences in the state of the water resources at several locations along the Cauvery River.

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Figure 6. Rank representation by CCME WQI