12096 FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 21, No 3, Special Issue, 2023, pp. 359 - 386 https://doi.org/10.22190/FUME230901034M © 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper ASSESSMENT OF SUSTAINABLE WASTEWATER TREATMENT TECHNOLOGIES USING INTERVAL-VALUED INTUITIONISTIC FUZZY DISTANCE MEASURE-BASED MAIRCA METHOD Arunodaya Raj Mishra1, Pratibha Rani2, Fausto Cavallaro3, Adel Fahad Alrasheedi4 1Department of Mathematics, Government College Raigaon, Madhya Pradesh, India 2Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, Guntur Andhra Pradesh, India 3Department of Economics, University of Molise, Via De Sanctis, Campobasso, Italy 4Statistics and Operations Research Department, College of Science, King Saud University, Riyadh, Saudi Arabia Abstract. Effective wastewater treatment has significant effects on saving water and preventing unnecessary water scarcity. An appropriate wastewater treatment technology (WWTT) brings economic benefits through reuse in different sectors and benefits the society and environment. This study aims to develop a decision-making framework for evaluating the sustainable WWTTs under interval-valued intuitionistic fuzzy set (IVIFS) environment. The proposed MCDM framework is divided into two stages. First, a new Hellinger distance measure is developed to determine the degree of difference between IVIFSs and also discussed its desirable characteristics. Second, an interval-valued intuitionistic fuzzy extension of multi-attribute ideal-real comparative analysis (MAIRCA) model is developed using the proposed Hellinger distance measure- based weighting tool. Further, the proposed model is implemented on an empirical study of sustainable WWTTs evaluation problem. Sensitivity and comparative studies are made. The results indicate that odor impacts, sludge production, maintenance and operation are the most effective sustainable factors and Microbial fuel cell (MFC) technology is the best WWTT followed by natural treatment methods. Key words: Interval-valued intuitionistic fuzzy sets, Distance measure, Sustainability, MAIRCA, Rank sum model, Wastewater treatment Received: September 01, 2023 / Accepted October 06, 2023 Corresponding author: Pratibha Rani Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, Guntur-522302, Andhra Pradesh, India E-mail: pratibha138@gmail.com mailto:pratibha138@gmail.com 360 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI 1. INTRODUCTION Most of the water that is not used as an ingredient finally appears in the wastewater stream. Wastewater treatment (WWT) is the elimination of impurities from wastewater before it reaches aquifers or natural bodies of water such as lakes, rivers and oceans. It is one of the most imperative necessities of the current scenario that can provide wider environmental and societal benefits including improved ambient water quality, protect wildlife, reduced greenhouse gas emissions and so on [1-3]. Treated wastewater also brings economic benefits through reuse in different sectors like agriculture, groundwater recharge, vehicle washing, golf course irrigation, toilet flushes, cooling purposes in thermal power plants and building construction activities. The quality of treated wastewater used in agriculture has a great influence on the operation and performance of the wastewater-soil-plant or aquaculture system. WWT solutions can recover valuable resources from wastewater such as biodiesel, electricity, recycled water and nutrients and serve as the components of fertilizer [4]. Thus, it is significant for maintaining the health of human beings and ecosystems. The process of WWT consists of using suitable technology to improve or upgrade the quality of a wastewater. Usually, WWT will involve collecting the wastewater in a central, segregated location and subjecting the wastewater to various treatment processes. WWT process needs to be a technology type that will be suitable for a particular development and not necessarily the best available technology. Selection of suitable WWT technologies that is highly correlated with sustainable development, presents a challenge to the local, regional, national and global policy makers. In developing countries, there is a need for a decision-making tool to evaluate the wastewater treatment technology (WWTT) selection. Many researchers have presented their works on the development new technologies and also reviewed the current systems for wastewater treatment. For instance, Choudhury et al. [5] highlighted the working strategies of Microbial fuel cell (MFC) technology for WWT and reuse of wastewater for power generation. In a study, Tarpani and Azapagic [6] assessed the life cycle environmental impacts of advanced WWT methods for removal of pharmaceuticals and personal care products. Arroyo and Molinos-Senante [7] presented a choosing-by-advantages approach to evaluate seven WWT alternatives. Based on different aspects of sustainability, they selected the most suitable WWT alternative. A systematic review has presented to explain the advantages and disadvantages of different membrane technologies for water treatment [8]. Munoz-Cupa et al. [9] highlighted the benefits and technical barriers of current MFC systems for WWT. Moreover, they have presented the effects of different reaction conditions on chemical oxygen demand removal and electricity generation from MFCs. Zhang et al. [10] stated the current trends of WWT plants in China. Their study provided some useful implications to the authorities and policy makers by analyzing the industries’ current status in the WWT process. Saravanan et al. [11] reviewed several WWT technologies and presented their remarkable power for toxic pollutants removal from wastewater. In addition, they discussed the difficulties related to commercial development of WWT technologies and suggested the future research directions. Saravanan et al. [12] presented sustainable strategy on MFC technology to treat the wastewater for the green energy production. For this purpose, the authors have reviewed diverse MFC technologies and their core performance in the direction of waste management and energy conversion. As per the existing studies, several technologies, Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 361 ranging from conventional to advanced treatment processes, are available to treat the wastewater. Finding the most suitable wastewater treatment technology (WWTT) is a complex issue due to involvement of multiple sustainability aspects of criteria. An alternative WWTT is considered “most suitable” to the degree that it is consistent with the economic, environment, social, technical, cultural and political aspects of the society. Thus, the multi-criteria decision-making (MCDM) approaches are more appropriate to systematically solve this problem. Ullah et al. [13] established a decision-making agenda for the selection of WWTT. For this purpose, they firstly provided the comprehensive review of the state-of-the-art in WWT. Srivastava and Singh [14] established a decision support model for the selection of most suitable WWTT from multiple criteria perspective in which the criteria weights are determined through Full consistency method. Salamirad et al. [15] proposed an integrated decision support system using the best worst method and the behavioral Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). In that study, the authors have evaluated seven WWTTs with respect to economic, environmental and social dimensions of sustainability. Pennelilini et al. [16] presented a novel utility interval-based evidential reasoning approach to evaluate and prioritize the WWTT alternatives for agricultural reuse. With the use of analytic hierarchy process, Ćetković et al. [17] reviewed the current situation and problems related to WWTTs and evaluated the optimal variant of WWTTs. In the process of multi-criteria WWTT selection, ambiguity arises in some form due to the genuine limits on the human mind and imprecise information. The notion of fuzzy set (FS) has widely been used in practice to handle such types of uncertain decision- making problems [18-20]. In order to choose the WWTT, Dursun [21] developed a hybrid method by incorporating the Decision making trial and evaluation laboratory (DEMATEL) and TOPSIS methods with 2-tuple fuzzy linguistic set and applied to evaluate the WWTT candidates. Attri et al. [22] presented the combined use of three MCDM methods such as the Multi-Objective Optimization by Ratio Analysis (MOORA), the Stepwise Weight Assessment Ratio Analysis (SWARA) and the TOPSIS under fuzzy environment. In this study, fuzzy SWARA method has applied to evaluate the significance values of the considered criteria, while the fuzzy MOORA and TOPSIS approaches have used to determine the rank of six WWTTs. A hybrid decision support system has developed based on the Pivot pairwise relative criteria importance assessment and an interactive MCDM approach with linear diophantine fuzzy information to handle the multi-criteria WWTT selection problem [23]. As the FS only contains a membership grade (MG), therefore, Atanassov [24] extended the classical FS and investigated the notion of intuitionistic fuzzy set (IFS), which assigns a MG, a non-membership grade (NG) and an indeterminacy grade (IG) to each element with sum of the MG and NG is bounded to 1. After the pioneering innovation by Atanassov [24], various significant results have been achieved based on IFS theory [25-28]. In the intuitionistic fuzzy set theory, the degrees of membership and non-membership are exact numbers, which is hard for the experts to define their exact value in several decision- making problems. To conquer this issue, Atanassov and Gargov [29] gave the idea of interval-valued intuitionistic fuzzy set (IVIFS), which deals with uncertainty in practical decision-making problems. Its basic feature is that both membership and non-membership functions of an element to a given set are considered and taken as interval values rather than exact numbers. Due to the broad range of information coverage, the theory of IVIFS is used 362 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI in our proposed work. In the context of IVIFS, some relations and basic operations score and accuracy functions have been presented [30]. As an extended version of IFS, the IVIFS theory provides a more effective and reasonable way to cope with imprecise and uncertain information. Due to its higher flexibility in dealing with uncertain data, the IVIFS doctrine has been broadly explored from different perspectives [31-33]. To deal with MADM problems, several novel methods have been developed in the recent past [34-39]. The Multi-Attribute Ideal-Real Comparative Analysis (MAIRCA) [40] method is a newly developed MCDM method, which determines the gap between the ideal and empirical ratings during the assessment of alternatives [41]. For each alternative, summation of the gaps for all the criteria determines the total gap and option with the least total distance is considered as the best choice. This ranking technique uses different linear normalization method, which is characterized by easy mathematical computations and solution stability [42]. In a study, the classical MAIRCA method has integrated with the well-known DEMATEL model and applied to evaluate the sites for multimodal logistics centre development from sustainability perspectives. Kaya [43] analyzed the consequence of COVID-19 pandemic in the countries’ sustainable development level through MAIRCA method. The classical MAIRCA method has been combined with LBWA model and interval rough numbers to develop a hybrid MCDM model [44]. In the context of uncertainty, Boral et al. [41] incorporated the standard MAIRCA approach with analytic hierarchy process, failure mode and effects analysis and fuzzy numbers, and developed a hybrid decision support system for solving complicated MCDM problem. Further, Ecer [45] proposed a hybrid intuitionistic fuzzy MAIRCA method with an application in the assessment of COVID-19 vaccines. In the recent past, Hezam et al. [46] discussed an approach based on symmetry point of criterion-based MAIRCA method and applied to select the most suitable biomass resources for biofuel production under intuitionistic fuzzy environment. The classical MAIRCA approach has been extended from interval-valued neutrosophic perspective and applied to select the multi-criteria sustainable materials [47]. Rani et al. [48] proposed a Pythagorean fuzzy information-based MAIRCA model, in which the criteria weights are determined through standard deviation-based method. In addition, they presented the drawbacks of existing studies in the context of Pythagorean fuzzy set. Based on the literature review, this paper identifies some research gaps in the existing studies, given as ▪ Existing distance measures (Zhang et al. [49], Düğenci [50], Baccour and Alimi [51], Mishra et al. [52]) present some counter-intuitive cases in order to measure the degree of difference between two IVIFSs. ▪ Few authors (Kaya [43], Božanić et al. [44], Boral et al. [42], Ecer [45], Haq et al. [47], Hezam et al. [46], Rani et al. [48]) have developed the extensions of classical MAIRCA method from fuzzy, intuitionistic fuzzy, interval-valued neutrosophic, Pythagorean fuzzy perspectives, but the these methods cannot deal with IVIF information in which the alternatives’ information is represented in terms of intervals rather than the crisp numbers. ▪ In the literature, some authors (Ullah et al. [13], Srivastava and Singh [14], Salamirad et al. [15], Pennelilini et al. [16], Ćetković et al. [17]) have proposed different MCDM methods for solving WWTTs assessment problem. As an extension of IFS, the theory of IVIFS has the advantage that both membership and non-membership degrees are interval values and can used to characterize the uncertain information more flexibly Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 363 due to its constraint condition. Unfortunately, existing decision support models are unable to deal with the interval-valued membership and non-membership degrees. Inspired by the limitations of existing studies, this study develops an interval-valued intuitionistic fuzzy MADM framework to evaluate and prioritize the WWTT alternatives. The developed methodology does not only evaluate the considered options through IVIF- distance measure-MAIRCA model, but also consider the weights of considered criteria and decision makers. The proposed model can assist the decision makers (DMEs) to get more confident for ranking the blockchain platforms. The key contributions of this paper are presented as follows: ▪ New Hellinger distance measure is proposed to overcome the limitations of existing IVIF-distance measures. Comparative study is presented to prove the effectiveness of the proposed measure. ▪ A novel extension of MAIRCA method is developed in the context of interval- valued intuitionistic fuzzy information perspective in which the information about the criteria and decision makers is completely unknown. ▪ In the proposed method, the proposed distance measure and total support degree- based model is presented to compute the criteria weights. ▪ The presented MAIRCA method is implemented on a case study of WWTTs assessment, which proves its applicability and powerfulness. The leading question of the WWTT alternative selection problem is “which one is the most appropriate WWTT alternative among a set of alternatives from sustainability perspective?” To solve this problem, the following questions need to be answered: (i) What are the main criteria to evaluate the most appropriate WWTT alternatives with interval-valued intuitionistic fuzzy information? (ii) Which is the most significant criterion for WWTTs assessment? (iii) Which is the most suitable MCDM technique to select and prioritize the WWTT alternatives based on the economic, social, environmental and technical aspects of sustainability? The key objectives of this study are as follows: (i) Identify the main criteria to choose the most suitable WWTT alternative through literature survey and DE opinions. (ii) Find the importance weights of each criterion under the context of uncertain information. Introduce a model to compute the weights of considered evaluation criteria. (iii) Determine a hybrid MCDM methodology to prioritize the WWTT alternatives under interval-valued intuitionistic fuzzy environment. The rest part of this study is organized as follows: Section 2 firstly presents the basic concepts and then proposes a new distance measure for IVIFSs. Section 3 develops a hybrid MAIRCA method for assessing the MCDM problems under IVIFS context. Section 4 implements the proposed method on a case study of WWTTs assessment. Section 5 presents the comparative analysis, discussion on the results and implications. Section 6 concludes the whole study and recommends for future researches. 2. PROPOSED DISTANCE MEASURE FOR IVIFSS This section firstly discusses the fundamental notions related to an IVIFS and then proposes a new measure to describe the degree of difference between two IVIFSs. 364 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI 2.1. Basic Concepts Definition 2.1. Consider Φ = {ϕ1, ϕ2, ..., ϕt} be a finite universal set. In the following way, Atanassov and Gargov [29] mathematically defined the IVIFS Q on Φ: { , ([ ( ), ( )],[ ( ), ( )]) : },i Q i Q i Q i Q i iQ          − + − +=   where 0 ( ) ( ) 1,Q i Q i   − +   0 ( ) ( ) 1Q i Q i   − +   and 0 ( ) ( ) 1.Q i Q i   + + +  Here, ( ) [ ( ), ( )]Q i Q i Q i     − += and ( ) [ ( ), ( )]Q i Q i Q i     − += define the degrees of interval-valued membership and non-membership of an object ϕi in Q, respectively. The function ( ) [ ( ), ( )]Q i Q i Q i     − += represents the indeterminacy degree of ϕi to Q, where ( ) 1 ( ) ( )Q i Q i Q i     − + += − − and ( ) 1 ( ) ( ).Q i Q i Q i     + − −= − − For the simplicity, the term ([ ( ), ( )],[ ( ), ( )])Q i Q i Q i Q i       − + − + is defined as the “interval-valued intuitionistic fuzzy value/number (IVIFV/IVIFN)” and denoted by ([ , ],[ , ])       − + − += which fulfills 0 ( ) ( ) 1.  + + +  Definition 2.2. Xu and Gou [30] defined some operational laws on IVIFVs 1 1 1 1 1([ , ],[ , ])    − + − += and 2 2 2 2 2([ , ],[ , ]),    − + − += presented as (a) 1 2  if and only if 1 2 1 2 1 2( ) ( ), ( ) ( ), ( ) ( )i i i i i i           − − + + − −   and 1 2( ) ( ), ,i i i    + +   (b) 1 2 = if and only if 1 2  and 1 2 ,  (c) 1 1 1 1 1{( ,[ ( ), ( )],[ ( ), ( )])| },c i i i i i i          − + − +=  Definition 2.3. For any IVIFN ([ , ],[ , ]),       − + − += Xu and Gou [30] studied the normalized score function and accuracy function, given by Eq. (1) and Eq. (2), respectively. ( ) 1 1 ( ) 1 , 2 2        − + − +  = + − − +    (1) 1 ( ) ( ). 2        − + − += + + + (2) Definition 2.4. For a set of IVIFNs ω = {ω1, ω2, ..., ωt}, Xu and Gou [30] defined the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) and interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) operators as 1 1 1 1 1 1 (1 ) , 1 (1 ) , ( ) , ( ) , k k k k t t t tt k k k k k k k k k k k         − + − + = = = = =      =  − − − −              (3) 1 1 1 1 1 ( ) , ( ) , 1 (1 ) , 1 (1 ) . k k k k t t t t k k k k k k k k k k k         − + − + = = = = =       =  − − − −              (4) Definition 2.5 [30]. Assume that Q, R, S IVIFSs (Φ), A real-valued function D: IVIFSs (Φ)×IVIFSs (Φ) → [0,1] is said to be IVIF-distance measure if it holds the following axioms: (a1). 0 ≤ D (Q, R) ≤ 1, Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 365 (a2). D (Q, R) if and only if Q = R, (a3). D (Q, R) = D (R, Q), (a4). If ,Q R S  then D (Q, R) ≤ D (Q, S) and D (R, S) ≤ D (Q, S). 2.2. Hellinger Distance Measure on IVIFSs Hellinger [54] originated the idea of Hellinger distance to quantify the degree of difference between two discrete probability distributions. Based on the concept of Hellinger distance, this section proposes a Hellinger distance measure on IVIFSs. Moreover, the important characteristics of this measure are discussed in this section. Definition 2.6 Suppose that Q, R IVIFSs(Φ), then the distance measure on IVIFSs is presented as ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) . 2 ( ) ( ) ( ) ( ) Q i R i Q i R i t Q i R i Q i R i i Q i R i Q i R i D Q R t                         − − + + − + + = − − + +  − + −   + − + −=    + − + −    (5) Remark 2.1. The bigger value of D (Q, R) signifies the larger difference between two IVIFSs Q and R. In a similar way, the lesser value of D (Q, R) signifies the smaller difference between two IVIFSs Q and R. The properties of D (Q, R) are deduced as follows: Property 2.1: For Q, R IVIFSs(Φ), 0 ≤ D (Q, R) ≤ 1. Proof: In IVIFS, we have ( ) ( ) ( ) 1,Q i Q i Q i     − − −+ + = ( ) ( ) ( ) 1,Q i Q i Q i     + + ++ + = ( ) ( ) ( ) 1R i R i R i     − − −+ + = and ( ) ( ) ( ) 1.R i R i R i     + + ++ + = Thus, we deduce that ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) 1 ( , ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) Q i Q i R i R i Q i Q i R i R i Q i Q i R i R i Q i Q i R i R i Q i Q i R i R i Q i Q i R i D Q R t                                                − − − − + + + + − − − − + + + + − − − − + + +  − + +   − + + − = + + − + + − + + − + 1 ( ) t i R i = +    ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) 1 ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) Q i Q i R i R i Q i Q i R i R i Q i Q i R i R i Q i Q i R i R i Q i Q i R i R i Q i Q i R i R i t                                                 − − − − + + + + − − − − + + + − − − − + + + +  − + +   − + + − = + + − + + − + + − + 1 t i=   366 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI ( ) )1 4 2 ( ) ( ) ( ) ( ) ( ) ( ) 1 2 ( ) ( ) ( ) ( ) ( ) ( ) t Q i R i Q i R i Q i R i i Q i R i Q i R i Q i R i t                         − − + + − − + + − − + +=  − + +  = + + +   ( ) )1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1. ( ) ( ) ( ) ( ) ( ) ( ) t Q i R i Q i R i Q i R i i Q i R i Q i R i Q i R i t z                        − − + + − − + + − − + +=  − + + =  + + +   Therefore, we can prove that 0 ≤ D (Q, R) ≤ 1. Property 2.2: D (Q, R) = 0, if and only if Q = R. Proof: For given two IVIFSs Q and R, we have ( ) ( ),Q i R i   − −= ( ) ( ),Q i R i   + += ( ) ( ),Q i R i   − −= ( ) ( ),Q i R i   + += ( ) ( )Q i R i   − −= and ( ) ( ).Q i R i   + += Then we find that ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) 0 2 ( ) ( ) ( ) ( ) Q i R i Q i R i t Q i R i Q i R i i Q i R i Q i R i D Q R t                         − − + + − + + = − − + +  − + −   + − + −= =   + − + −    For any ϕiΦ, if D (Q, R) = 0, then we have ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 0. 2 ( ) ( ) ( ) ( ) ( ) ( ) Q i R i Q i R i Q i R it i Q i R i Q i R i Q i R i t                         − − + + − − + + − − + +=  − + − + −  =  + − + − + −    It implies ( ) 2 ( ) ( ) 0,Q i R i   − −− = ( ) 2 ( ) ( ) 0,Q i R i   + +− = ( ) 2 ( ) ( ) 0,Q i R i   − −− = ( ) 2 ( ) ( ) 0,Q i R i   − − = ( ) 2 ( ) ( ) 0,Q i R i   + +− = ( ) 2 ( ) ( ) 0Q i R i   − −− = and ( ) 2 ( ) ( ) 0.Q i R i   + +− = Hence, ( ), 0D Q R = if and only if .Q R= Property 2.3: D (Q, R) = D (R, Q). Proof: For given two IVIFSs Q and R, we have Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 367 ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) 2 ( ) ( ) ( ) ( ) Q i R i Q i R i t Q i R i Q i R i i R i R i Q i R i D Q R t                         − − + + − − + + = − − + +  − + −   + − + −=    + − + −    ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( , ). 2 ( ) ( ) ( ) ( ) R i Q i R i Q i t R i Q i R i Q i i R i Q i R i Q i D R Q t                         − − + + − − + + = − − + +  − + −   + − + −= =   + − + −    Hence, D (Q, R) = D (R, Q). Property 2.4: If ,Q R S  then D (Q, R) ≤ D (Q, S) and D (R, S) ≤ D (Q, S). Proof: For given three IVIFSs Q, R and S, if ,Q R S  then ( ) ( ) ( ),Q i R i S i     − − −  ( ) ( ) ( ),Q i R i S i     + + +  ( ) ( ) ( ),Q i R i S i     − − −  ( ) ( ) ( ),Q i R i S i     + + +  ( ) ( ) ( )Q i R i S i     − − −  and ( ) ( ) ( ).Q i R i S i     + + +  ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) 2 ( ) ( ) ( ) ( ) Q i S i Q i S i t Q i s i Q i S i i Q i S i Q i S i D Q S t                         − − + + − − + + = − − + +  − + −   + − + −=    + − + −    ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( , ). 2 ( ) ( ) ( ) ( ) , Q i R i Q i R i t Q i R i Q i R i i Q i R i Q i R i D Q R t                         − − + + − + + = − − + +  − + −   + − + − =   + − + −    Similarly, we can prove that when ,Q R S  then D (Q, R) ≤ D (Q, S) and 0 ≤ D (R, S) ≤ D (Q, S). 368 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI Proposition 2.1: From the Properties 2.1-2.4, Eq. (5) holds all the necessary conditions of Definition 2.5. Hence, Eq. (5) is valid IVIF-distance measure on IVIFSs (Φ). Definition 2.7. Suppose that Q, R, S IVIFSs(Φ), then the weighted Hellinger distance measure Dα: IVIFSs(Φ)×IVIFSs(Φ) → [0,1] is given by ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) 2 ( ) ( ) ( ) ( ) , Q i R i Q i R i t Q i R i Q i R ii i Q i R i Q i R i D Q R                         − − + + − + + = − − + +  − + −   + − + −=    + − + −    (6) where αi is the weight of ϕi on Φ satisfying αi  [0, 1] and 1 1. t ii  = = 2.3. Comparative Study In this section, we firstly recall some of the previously developed distance measures in the context of IVIFS (Zhang et al. [49], Düğenci [50], Baccour and Alimi [51], Mishra et al. [52]). Further, we apply the proposed and existing IVIF-distance measures on some common data sets and obtain some useful results. 1 ( ) ( ) ( ) ( )1 ( , ) , 4 ( ) ( ) ( ) ( ) t Q i R i Q i R i H i Q i R i Q i R i D Q R t                 − − + + − − + + =  − + −  =   + − + −    (7) 2 2 2 2 1 ( ( ) ( )) ( ( ) ( ))1 ( , ) , 4 ( ( ) ( )) ( ( ) ( )) t Q i R i Q i R i E i Q i R i Q i R i D Q R t                 − − + + − − + + =  − + −  =  + − + −   (8) 1 ( ) ( ) , ( ) ( ) ,1 ( , ) max , 4 ( ) ( ) , ( ) ( ) t Q i R i Q i R i HH i Q i R i Q i R i D Q R t                 − − + + − − + + =  − −  =   − −    (9) 2 2 2 2 1 ( ( ) ( )) , ( ( ) ( )) ,1 ( , ) max , 4 ( ( ) ( )) , ( ( ) ( )) t Q i R i Q i R i HE i Q i R i Q i R i D Q R t                 − − + + − − + + =  − −  =  − −   (10) 1 ( ) ( ) ( ) ( ) , 1 2 ( , ) max , ( ) ( ) ( ) ( ) 2 Q i R i Q i R i t Z i Q i R i Q i R i D Q R t                 − − + + − − + + =  − + −     =   − + −       (11) Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 369 ( ) 1 ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( ))1 ( , ) 4 1 ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) p p Q i R i Q i R i t Q i R i Q i Q i D p p p i Q i R i Q i R i Q i R i Q i R i D Q R t                                      − − − − − − − − + + + += + + + +    − −   +   − − − −   =   +  − −  +    − − − −     1 , p       (12) where  = 2, 3, 4, . . . and p = 1, 2, 3, . . . ( ) ( ) ( ) ( ) 2 2 1 2 2 1 1 ( ) ( ) ( ) ( ) 1 1 ( , ) , 4 4 ( ) ( ) ( ) ( ) t tQ i R i Q i R i B i i Q i R i Q i R i D Q R t t                 − − + + − + += =     − −       = +     + − + −          (13) 2 2 2 1 1 ( ) ( ) ( ) ( ) 1 1 ( , ) , 8 8 ( ) ( ) ( ) ( ) t tQ i R Q i R i B i i Q i R i Q i R i D Q R t t                 − − + + − − + + = =     − −    = +       + − + −       (14) 1 1 ( ) ( ) ( ) ( ) 1 1 exp ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) , 1 exp( (1)) , Q i R i Q i R i t Q i R i Q i R i i Q i R i Q i R i M t D Q R                                − − + + − − + + = − − + +      − + −          − − + − + −         + − + −        − − =  (15) where γ > 0, γ ≠ 1. Table 1 Comparisons of diverse IVIF-distance measures (Bold shows the counter-intuitive cases) Qi ([0.25, 0.35], [0.25, 0.35]) [0.25, 0.35], [0.35, 0.45]) ([1, 1], [0, 0]) ([0.5, 0.5], [0.5, 0.5]) ([0.35, 0.45], [0.15, 0.25]) ([0.35, 0.45], [0.15, 0.25]) Ri ([0.35, 0.45], [0.35, 0.45]) [0.35, 0.45], [0.25, 0.35]) ([0, 0], [0, 0]) ([0, 0], [0,0]) ([0.45, 0.55], [0.25, 0.35]) ([0.45, 0.55], [0.15, 0.25]) DH (Qi, Ri) 0.1 0.1 0.5 0.5 0.1 0.05 DE (Qi, Ri) 0.1 0.1 0.707 0.5 0.1 0.071 DHH (Qi, Ri) 0.025 0.025 0.25 0.125 0.025 0.025 DHE (Qi, Ri) 0.05 0.05 0.5 0.25 0.05 0.05 DZ (Qi, Ri) 0.100 0.100 1.000 0.500 0.100 0.100 DD (Qi, Ri) 0.033 0.100 0.500 0.208 0.033 0.050 DB1 (Qi, Ri) 0.007 0.007 0.500 0.375 0.008 0.003 DB2 (Qi, Ri) 0.100 0.100 0.250 0.313 0.100 0.025 DM (Qi, Ri) 0.252 0.151 1.000 0.802 0.252 0.151 D (Qi, Ri) 0.165 0.086 1.000 0.804 0.167 0.082 370 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI Table 1 presents the computational results obtained by the proposed and previously developed IVIF-distance measures (Zhang et al. [49], Düğenci [50], Baccour and Alimi [51], Mishra et al. [52]). The distance measures DH, DE, DHH, DHE, DB1, DB2, DZ, DD and DM present the counter-intuitive results for some examples. For instance, DH (Q1, R1) = 0.1 and DH (Q2, R2) = 0.1, DE (Q1, R1) = 0.1 and DE (Q2, R2) = 0.1, DHH (Q1, R1) = 0.025 and DHH (Q2, R2) = 0.025, DHE (Q1, R1) = 0.05 and DHE (Q2, R2) = 0.05, DZ (Q1, R1) = 0.1 and DZ (Q2, R2) = 0.1, DB1 (Q1, R1) = 0.007 and DB1 (Q2, R2) = 0.007, DB2 (Q1, R1) = 0.1 and DB2 (Q2, R2) = 0.1, when Q1 = [(0.25, 0.35), (0.25,0.35)], R1 = [(0.35, 0.45), (0.35, 0.45)], Q2 = [(0.25, 0.35), (0.35, 0.45)] and R2 = [(0.35, 0.45), (0.25, 0.35)]. Also, DHH (Q5, R5) = 0.025 and DHH (Q6, R6) = 0.025, DHE (Q5, R5) = 0.05 and DHE (Q6, R6) = 0.05, DZ (Q5, R5) = 0.1 and DZ (Q6, R6) = 0.1, when Q5 = [(0.35, 0.45), (0.15,0.25)], R5 = [(0.45, 0.55), (0.25, 0.35)], Q6 = [(0.35, 0.45), (0.15, 0.25)] and R6 = [(0.45, 0.55), (0.15, 0.25)]. For the above discussed sets, the Hellinger distance measure proposed in this study can successfully discriminate the given IVIFSs and the results are as follows: D (Q1, R1) = 0.165 and D (Q2, R2) = 0.086, D (Q5, R5) = 0.167 and D (Q6, R6) = 0.082. For a different pair of IVIFSs Q3 = [(1, 1), (0, 0)], R3 = [(0, 0), (0, 0)] and Q4 = [(0.5, 0.5), (0.5, 0.5)], R4 = [(0, 0), (0, 0)], we get DH (Q3, R3) = DH (Q4, R4) = 0.5, while other measures provide reasonable results. In addition, the counter-intuitive cases arise for Q1 = ([0.25, 0.35], [0.25, 0.35]), R1 = ([0.35, 0.45], [0.35, 0.45]), Q5 = ([0.35, 0.45], [0.15, 0.25]), R5 = ([0.45, 0.55], [0.25, 0.35]) and R2 = [(0.25, 0.35), (0.35, 0.45)] and R2 = [(0.35, 0.45), (0.25, 0.35)], R6 = [(0.35, 0.45), (0.15, 0.25)] and R6 = [(0.45, 0.55), (0.15, 0.25)], respectively. In all the discussed cases, the present measure is free from all the counter-intuitive cases, see Table 1. 3. AN INTEGRATED IVIF-DISTANCE MEASURE-BASED MAIRCA METHOD The current part of the study develops an extended MAIRCA model for solving MCDM problems in which the information about the criteria and DMs is completely known. This model combines the proposed IVIF-distance measure and the MAIRCA method with IVIF information. The proposed model comprises the following procedure (see Fig.1): Step 1: Create the decision matrix. A group of DMEs H = {h1, h2, ..., hn} is invited for the assessment of an optimal alternative among a set of options Y = {y1, y2, ..., ys} by means of the criteria set W = {w1, w2, ..., wt} The DMs present the linguistic assessment rating of each option yi with respect to criteria wj, j = 1, 2, ..., t. Let D = (δij (k))s×t be the linguistic assessment matrix (LAM), where δij (k) denotes the linguistic variable (LV) of each candidate yi over a criterion wj presented by kth DME. Based on the given linguistic scale’s table, the LAM is switched into IVIF decision matrix (IVIFDM). Step 2: Compute the DMEs’ significance values. Assume that ([ , ],[ , ]), 1,2,...,k k k k kh k n   − + − += = be the performance of kth DME. Then the procedure for estimating the numeric significance value of kth DME is presented in the following steps: Step 2a: Determine the matrix using score function. Each IVIFN hk is normalized and computed as Mishra et al. [54]: 1 ( )(2 ) , . (( )(2 )) k k k k k n k k k k k h k         − + − + − + − + = + + + =  + + + (16) Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 371 Step 2b: Determine the rank of DMEs’ performances and compute the DME’s significance value n-ρk+1, wherein ρk denotes the priority of kth expert. The normalization process is used to normalize each significance value (Zhu et al. [55]): 1 1 , . ( 1) r k k n k k n h k n   = − + =  − + (17) Step 2c: Compute the weights. In accordance with the combination of Eq. (7) and Eq. (8), DME’s weighting formula is given as 1 (( ) ( )), , 2 r k k kh h k = +  where 0k  and 1 1. n k k  = = (18) Fig. 1 Graphical structure of the proposed IVIF-distance measure-MAIRCA model 372 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI Step 3: Aggregate the individual DME’s opinions. As each decision maker has their own opinion regarding the performance of options with respect to the criteria. To make an optimal decision, there is a need to combine the individual decision opinions and create the aggregated IVIFDM (A-IVIFDM) A = (δij)s×t, where δij denotes the aggregated IVIFN, computed based on IVIFWA operator. (1) (2) ( )([ , ],[ , ]) ( , ,..., ) .n ij ij ij ij ij ij ij ijIVIFWA       − + − += = (19) Step 4: Compute the criteria weights. Assume that the significance value of each criterion is different and independent of each other. Let X = (x1, x2, ..., xt)T be the weight vector of criteria set, satisfying xj [0, 1] and 1 1. t jj x = = Next, we present the procedure to compute the criteria weights as follows: Step 4.1: Determine the support degree sup (δij, δil) between the considered criteria wj and wl using the proposed IVIF-distance measure, which as sup( , ) 1 ( , ), 1,2,..., , , 1,2,..., , ,ij il ij ilD i s j l t j l   = − = =  (20) where D (δij, δil) denotes the IVIF-distance measure given in Eq. (5). Step 4.2: Compute the total support degree T (δij) for each criterion wj, by means of Eq. (21). 1, ( ) sup( , ). t ij ij il l l j T    =  =  (21) Step 4.3: Compute the rationality degree θj of each criterion wj, given as 1 1 1 ( ), [0,1]. ( 1) s t j ij j i j T s t    = = =  −  (22) Step 4.4: Determine the comprehensive index (weight of criteria) xj, of jth criterion wj, given as 1 , j j t jj x   = =  (23) where j = 1, 2, …, t. Step 5: Calculate the positive distance grade psij and the negative distance grade pnij between an element δij in an A-IVIFDM A = (δij)s×t and the PIS ω+ and the NIS ω-, respectively, where ([ , ],[ , ]),ij ij ij ij ij    − + − += ([ , ],[ , ]) j j j j        + + + + + − + − += and ([ , ],[ , ]), j j j j        − − − − − − + − += shown as follows: 2 2 2 2 2 2 1 2 , j j j j j j ij ij ij ij ij ij ij ps                   + + + + + + − − + + − − + + − − + +       − + − + −            =       + − + − + −              (24) Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 373 2 2 2 2 2 2 1 2 . j j j j j j ij ij ij ij ij ij ij pn                   − − − − − − − − + + − − + + − − + +       − + − + −            =       + − + − + −              (25) Here, i = 1, 2, …, s and j = 1, 2, …, t. An IVIFN has a positive ideal solution ω+ and a negative ideal solution ω-, where ω+ = ([1, 1], [0, 0]) and ω- = ([0, 0], [1, 1]) are the IVIFNs. Step 6: With the use of Eq. (24) and Eq. (25), create the relative closeness decision matrix R = (rcij)s×t, where . ij ij ij ij pn rc pn ps = + (26) Step 7: Normalize the relative closeness-decision matrix R = (rcij)s×t into the normalized form R = (nrcij)s×t, where min max min max max min ( ) , for is thebenefit criterion, ( ) ( ) ( ) , for is thecost criterion. ( ) ( ) ij ij j ij ij ij ij ij j ij ij rc rc w rc rc nrc rc rc w rc rc −  − =  −  − (27) Step 8: Make the IVIF theoretical matrix T = (εij)s×t, where , iij C jP x = (28) 1 , iCP t= (29) where xj signifies the jth criterion’s weight, where j = 1, 2, …, t. Step 9: Based on the obtained IVIF theoretical matrix T = (εij)s×t and the obtained normalized relative closeness decision matrix R = (nrcij)s×t, construct the interval-valued intuitionistic fuzzy real assessment matrix β = (βij)s×t, where . .ij ij ijnrc = (30) Step 10: Based on the obtained IVIF-TM T = (εij)s×t and the obtained real assessment matrix β = (βij)s×t, construct the interval-valued intuitionistic fuzzy gap matrix G = (gij)s×t, where .ij ij ijg  = − (31) Step 11: Calculate the utility degree Ci of alternative yi, shown as follows: 1 , 1,2,..., . t i ij j C g i s = = = (32) 374 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI Prioritize the options as per the obtained utility degrees C1, C1, ..., and Cs of the alternatives y1, y1, ..., and ys, respectively. The lesser the utility degree of an option yi, the better the ranking order of option yi, where i = 1, 2, …, s. 4. CASE STUDY: WWTT SELECTION PROBLEM Environmental challenges related to the chemical and biological pollution of water have become important for the industrial sector, society and public agencies. Most of the domestic and industrial activities generate wastewater that contains harmful and undesirable pollutants, thus, it requires a proper management and treatment. The wastewater management and treatment aim to the sustainable development of natural resources together with the protection of environment and public health. In this section, the present MAIRCA model is firstly executed on a case study of the WWTTs assessment problem with respect to several factors. Further, the sensitivity analysis and comparative study by means of existing methods are discussed under IVIFS environment, which shows the stability and robustness of the presented methodology. With the increasing complexity, time boundedness and lack of precise knowledge/ information, it is quite hard to evaluate the candidates with regard to given criteria in realistic situations. In this section, a group of three DMs is formed to identify the criteria and evaluate the WWTTs based on considered criteria. These DEs are having more than 15 years of experience in their respective fields. Two of them are from the environmental engineering department and the other one is from sustainable planning and management. Based on the literature review and online questionnaire, we have considered five WWTT alternatives and nine criteria. The presented case study is for the demonstration purposes to prove the practicality of the proposed method. Readers may reduce some criteria or add more criteria as per their requirements. Description of the alternatives is presented as follows: ▪ Microbial fuel cell (MFC) (y1): MFC is relatively a new promising technology for producing renewable energy while treating wastewater. This technology is a chemical reactor system that generates electricity from the biodegradation of organic materials with the help of suitable microbial substrate. It is used to acquire a higher energy density and pollutants removal. ▪ Membrane Filtration (y2): It is physiochemical process for the treatment of water from different wastewater streams and makes it possible to reuse. This process of treatment is defined on the size of the material that needs to be separated from the liquid. ▪ Automatic Variable Filtration (AVF) (y3): It is a simple water filtration technology used for wastewater treatment where the upward flow of influent is purified or cleaned by the downward flow of filter media. It can effectively remove the bacterial contamination and micro-organisms while treating wastewater. ▪ Natural treatment methods (y4): It is a biological treatment method to treat the wastewater naturally by removing contaminants from wastewater. These methods are eco-friendly, cost-effective, and can be jointly driven by public bodies and communities. Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 375 ▪ Advanced Photo-Oxidation Process (APOP) (y5): APOP is a type of chemical treatment that oxidizes organic molecules in wastewater that are hard to manage biologically. Table 2 Criteria used for WWTT selection extracted from the literature Dimensions Criteria Meanings References Economic (EC) Maintenance and operation cost (EC1) Repair, personnel, chemical and energy costs to manage WWT Curiel-Esparza et al. [56], Molinos-Senante et al. [57], Saghafi et al. [58], Obaideen et al. [59] Land requirement (EC2) Enough space for WWT plant/future expansion Kalbar et al. [60], Mahjouri et al. [61], Saghafi et al. [58], Obaideen et al. [59] Environmental (EN) Energy consumption (EN1) Energy consumption amount during WWT activities Molinos-Senante et al. [57], Piadeh et al. [62], Srdjevic et al. [63], Salamirad et al. [15], Narayanamoorthy et al. [23] Sludge production (EN2) Sludge generation of the system Molinos-Senante et al. [57], Saghafi et al. [58], Salamirad et al. [15] Odor impacts (EN3) Undesired smell potential of the system Plakas et al. [64], Eseoglu et al. [65], Salamirad et al. [15] Social (S) Public acceptance (S1) Public awareness Molinos-Senante et al. [57], Plakas et al. [64], Obaideen et al. [59], Salamirad et al. [15] Aesthetic (S2) Acceptability of plant conditions and appearance Eseoglu et al. [65], Salamirad et al. [15] Technical (T) COD removal capacity (T1) The removal capacity of amount of oxygen consumed to oxidize all organic material by chemical oxidants Zhang et al. [10], Eseoglu et al. [65], Srivastava and Singh [14] BOD removal capacity (T2) The removal capacity of amount of oxygen consumed by microorganisms while decomposing organic matter Zhang et al. [10], Eseoglu et al. [65], Srivastava and Singh [14] Further, an online survey has been prepared with the purpose of determining the significance of criteria to assess the WWTT alternatives. In addition, the criteria that may have an effect on the WWTT alternatives’ evaluation are assembled through literature survey. Table 2 presents the source and type of each considered criterion. Fig. 2 presents the hierarchical structure of the considered criteria and alternatives. 376 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI Fig. 2 Hierarchical structure for sustainable WWTT selection Step 1: Table 3 presents the linguistic variables and their corresponding IVIFNs (Alrasheedi et al. [32] and Mishra et al. [52]). Based on the DMs’ opinions, the assessment rating of each WWTT alternative with respect to each criterion and form a linguistic assessment matrix in Table 4. Table 3 LVs for sustainable WWTTs assessment LVs IVIFNs Absolutely significant (AS) ([0.90,0.95],[0.01,0.05]) Very significant (VS) ([0.80,0.90],[0.05,0.10]) Significant (S) ([0.70,0.80],[0.10,0.15]) Quite significant (QS) ([0.65,0.70],[0.15,0.25]) Moderate (M) ([0.55,0.65],[0.20,0.35]) Quite insignificant (QI) ([0.40,0.50],[0.40,0.45]) Insignificant (I) ([0.25,0.40],[0.45,0.50]) Very insignificant (VI) ([0.15,0.20],[0.60,0.75]) Absolutely insignificant (AI) ([0.05,0.10],[0.80,0.90]) Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 377 Table 4 The LAM for sustainable WWTT selection problem Criteria y1 y2 y3 y4 y5 w1 (VI,VI,QI) (AI,I,QI) (I,VI,VI) (VI,AI,M) (VI,QI,QI) w2 (S,S,VS) (S,VS,M) (M,S,VS) (QS,QS,S) (M,QS,S) w3 (VI,SI,M) (QI,I,M) (QI,I,I) (VI,I,VI) (VI,M,VI) w4 (M,I,VI) (I,VI,AI) (I,VI,M) (I,QI,I) (VI,QI,AI) w5 (VI,I,VI) (QI,I,VI) (VI,VI,M) (I,I,AI) (QI,I,AI) w6 (S,QS,VS) (VS,QS,S) (S,S,QS) (M,VS,S) (VS,VS,S) w7 (VS,QS,QI) (S,QS,QS) (VS,VS,QI) (S,VS,S) (AS,M,QS) w8 (QS,AS,S) (S,S,QS) (QS,M,S) (VS,S,QI) (VS,M,QI) w9 (M,M,VS) (QS,VS,S) (S,S,S) (M,VS,M) (QI,VS,M) Step 2: With the use of linguistic scales of Table 3 and Eqs. (16-18), the significance values of DMs are derived and shown in Table 5 for sustainable WWTT selection problem. Table 5 DMEs’ weights for sustainable WWTT selection DMEs h1 h2 h3 LVs H VH EH IVIFNs ([0.70,0.80],[0.10,0.15]) ([0.80,0.90],[0.05,0.10]) ([0.90,0.95],[0.01,0.05]) n-ρk+1 1 2 3 Weights 0.2382 0.3343 0.4274 Fig. 3 Representation of the criteria weights using IVIF-distance measure-based tool 378 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI Table 6 Aggregated decision matrix for sustainable WWTT selection Criteria y1 y2 y3 y4 y5 w1 ([0.268, 0.346], [0.505, 0.603]) ([0.279, 0.389], [0.491, 0.550]) ([0.175, 0.253], [0.560, 0.681]) ([0.328, 0.416], [0.413, 0.576]) ([0.348, 0.441], [0.441, 0.508]) w2 ([0.748, 0.851], [0.074, 0.126]) ([0.688, 0.798], [0.107, 0.188]) ([0.722, 0.830], [0.088, 0.154]) ([0.672, 0.748], [0.126, 0.177]) ([0.652, 0.738], [0.135, 0.202]) w3 ([0.423, 0.520], [0.328, 0.457]) ([0.428, 0.544], [0.309, 0.419]) ([0.289, 0.425], [0.438, 0.488]) ([0.185, 0.273], [0.545, 0.655]) ([0.313, 0.393], [0.416, 0.581]) w4 ([0.299, 0.403], [0.420, 0.546]) ([0.135, 0.214], [0.634, 0.736]) ([0.371, 0.475], [0.350, 0.492]) ([0.304, 0.435], [0.433, 0.483]) ([0.207, 0.281], [0.593, 0.684]) w5 ([0.185, 0.273], [0.545, 0.655]) ([0.250, 0.350], [0.495, 0.580]) ([0.352, 0.438], [0.375, 0.542]) ([0.170, 0.286], [0.575, 0.643]) ([0.229, 0.317], [0.560, 0.627]) w6 ([0.734, 0.830], [0.085, 0.139]) ([0.657, 0.770], [0.135, 0.197]) ([0.680, 0.762], [0.119, 0.170]) ([0.711, 0.819], [0.094, 0.160]) ([0.762, 0.865], [0.067, 0.119]) w7 ([0.614, 0.713], [0.176, 0.240]) ([0.663, 0.728], [0.136, 0.184]) ([0.680, 0.801], [0.122, 0.207]) ([0.738, 0.841], [0.079, 0.131]) ([0.718, 0.794], [0.087, 0.173]) w8 ([0.784, 0.861], [0.051, 0.111]) ([0.596, 0.672], [0.177, 0.276]) ([0.644, 0.734], [0.139, 0.213]) ([0.634, 0.749], [0.153, 0.218]) ([0.580, 0.698], [0.193, 0.289]) w9 ([0.682, 0.795], [0.111, 0.205]) ([0.728, 0.825], [0.087, 0.140]) ([0.700, 0.800], [0.100, 0.150]) ([0.657, 0.770], [0.126, 0.230]) ([0.632, 0.749], [0.148, 0.244]) Step 3: By means of Table 3, Table 4 and Eq. (19), the aggregated decision matrix is created to combine the individual opinions of three DMEs and presented in Table 6. Step 4: To determine criteria weights, the total support degree using the proposed IVIF-distance measure and rationality degree of the aggregated decision matrix are calculated using Eqs. (20-22) and depicted in Table 7. Based on the rationality degree of each criterion, we determine the weights of criteria for sustainable WWTT selection using Eq. (23) and portrayed in Table 7. Here, Fig. 3 presents the significance degrees or weights of considered evaluation criteria in HSWPP locations assessment. Based on the obtained results, Odor impacts (EN3) is the most important criterion among a set of twelve criteria for assessing the WWTT. Sludge production (EN2) is the second most important criterion for WWTT evaluation. Maintenance and operation cost (EC1) has third with significance value 0.123, Energy consumption (EN1) with weight value 0.11 has fourth most important criterion for WWTT evaluation and others are considered crucial sub-criteria for the taken case study. Table 7 Rationality and comprehensive degree of option for sustainable WWTT selection Criteria y1 y2 y3 y4 y5 δj xj w1 2.058 1.789 2.492 1.846 1.592 0.244 0.123 w2 1.723 1.608 1.621 1.672 1.571 0.205 0.103 w3 1.604 1.441 1.769 2.245 1.677 0.218 0.110 w4 1.848 2.660 1.517 1.777 2.170 0.249 0.126 w5 2.443 1.894 1.630 2.275 1.995 0.256 0.129 w6 1.661 1.438 1.473 1.715 2.027 0.208 0.105 w7 1.489 1.525 1.669 1.858 1.720 0.207 0.104 w8 1.896 1.335 1.363 1.513 1.409 0.188 0.095 w9 1.730 1.746 1.533 1.773 1.512 0.207 0.105 Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 379 Table 8 Normalized relative closeness-decision matrix for sustainable WWTT selection Criteria y1 y2 y3 y4 y5 w1 0.509 0.325 1.000 0.120 0.000 w2 1.000 0.403 0.738 0.157 0.000 w3 0.082 0.000 0.460 1.000 0.534 w4 0.248 1.000 0.000 0.173 0.769 w5 0.956 0.541 0.000 1.000 0.816 w6 0.676 0.000 0.128 0.497 1.000 w7 0.000 0.298 0.532 1.000 0.726 w8 1.000 0.017 0.286 0.273 0.000 w9 0.513 1.000 0.729 0.241 0.000 Steps 5-6: Based on Eqs (24-26), we obtain the positive distance grade psij and the negative distance grade pnij and relative closeness matrix, where ω+ = ([1,1], [0, 0]) and ω- = ([0,0], [1,1]) are positive and negative ideal solutions, respectively for sustainable WWTT selection as rc11 = 0.431, rc12 = 0.718, rc13 = 0.522, and others. Table 9 IVIF-Theoretical matrix for sustainable WWTT selection Criteria y1 y2 y3 y4 y5 w1 0.025 0.025 0.025 0.025 0.025 w2 0.021 0.021 0.021 0.021 0.021 w3 0.022 0.022 0.022 0.022 0.022 w4 0.025 0.025 0.025 0.025 0.025 w5 0.026 0.026 0.026 0.026 0.026 w6 0.021 0.021 0.021 0.021 0.021 w7 0.021 0.021 0.021 0.021 0.021 w8 0.019 0.019 0.019 0.019 0.019 w9 0.021 0.021 0.021 0.021 0.021 Step 7: Using Eq. (27), we obtain the normalized relative closeness decision matrix for sustainable WWTT selection and given in Table 8, where nrc11 = 0.509, nrc12 = 1.000, nrc13 = 0.083, and others. Step 8: Based on Eq. (28) and Eq. (29), we obtain the interval-valued intuitionistic fuzzy theoretical matrix for sustainable WWTT selection and discussed in Table 9, where ε11 = 0.025, ε12 = 0.021, ε13 = 0.022, and others. Step 9: Based on Eq. (30), we obtain the interval-valued intuitionistic fuzzy real assessment matrix for sustainable WWTT selection and presented in Table 10, where β11 = 0.013, β12 = 0.021, β13 = 0.002, and others. Table 10 IVIF-Real assessment matrix for sustainable WWTT selection Criteria y1 y2 y3 y4 y5 w1 0.013 0.008 0.025 0.003 0.000 w2 0.021 0.008 0.015 0.003 0.000 w3 0.002 0.000 0.010 0.022 0.012 w4 0.006 0.025 0.000 0.004 0.019 w5 0.025 0.014 0.000 0.026 0.021 w6 0.014 0.000 0.003 0.010 0.021 w7 0.000 0.006 0.011 0.021 0.015 w8 0.019 0.000 0.005 0.005 0.000 w9 0.011 0.021 0.015 0.005 0.000 380 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI Step 10: Based on Eq. (31), we obtain the interval-valued intuitionistic fuzzy gap matrix for sustainable WWTT selection and discussed in Table 11, where g11 = 0.012, g12 = 0.000, g13 = 0.020, and others. Table 11 IVIF-Gap matrix for sustainable WWTT selection Criteria y1 y2 y3 y4 y5 w1 0.012 0.017 0.000 0.022 0.025 w2 0.000 0.012 0.005 0.017 0.021 w3 0.020 0.022 0.012 0.000 0.010 w4 0.019 0.000 0.025 0.021 0.006 w5 0.001 0.012 0.026 0.000 0.005 w6 0.007 0.021 0.018 0.011 0.000 w7 0.021 0.015 0.010 0.000 0.006 w8 0.000 0.019 0.014 0.014 0.019 w9 0.010 0.000 0.006 0.016 0.021 Step 11: Based on Eq. (32), we obtain the utility score Ci of alternative yi, where i = 1, 2, 3, 4, 5, C1 = 0.0902, C2 = 0.1171, C3 = 0.1155, C4 = 0.1001 and C5 = 0.1118. Because C1 >C4 >C5>C3 >C2, therefore ranking order of the WWTT alternatives y1, y2, y3, y4 and y5 is: y1 y4 y5 y3 y2. Thus, the alternative y1 is the best alternative for sustainable WWTT selection. 5. COMPARATIVE ANALYSIS AND DISCUSSION In this part of the study, we compare the results obtained by the proposed and some of the extant MADM methods including IVIF-weighted aggregated sum product assessment (IVIF-WASPAS) [66] model, IVIF-complex proportional assessment (IVIF-COPRAS) [67] model and IVIF-combined compromise solution (IVIF-CoCoSo) [32] model and IVIF- technique for order of preference by similarity to ideal solution (IVIF-TOPSIS) [68] model. Here, Fig. 4 depicts the obtained ranking results by different MCDM approaches. From Fig. 4, we can observe that the ranking orders are different by all the methods but the optimum alternative is same in case of the obtained results by introduced method, IVIF-WASPAS, IVIF-COPRAS and IVIF-CoCoSo methods, while IVIF-TOPSIS model provides different optimal candidate. The preference orders of WWTT candidate by means of different weighting approaches are shown in Fig. 5. To accomplish better insight from the IVIF-distance measure-based MAIRCA technique in the assessment of sustainable WWTT, we calculate the utility score of each WWTT over considered sustainable aspects and factors, as given in Fig. 5. Since MFC (y1) has extensively highest score for all extant and proposed models social, economic environmental aspects of attributes’ weighting (see Fig. 5), consequently, it is chosen as the best WWTT option. In accordance with the aforementioned analysis, it can be easily noticed that observing the various weighting frameworks will enhance the utility and effectiveness of the developed IVIF-MAIRCA method. Table 12 presents the parameters to compare different approaches including proposed and existing MCDM approaches. Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 381 Table 12 Parameters to compare different approaches Standards IVIF- WASPAS IVIF-COPRAS IVIF-CoCoSo IVIF-TOPSIS Proposed method Interrelationships between the arguments Not considered Not considered Not considered Not considered Considered Criteria weights Computed Computed Computed Assumed Computed MCDM procedure Group Group Group Group Group DMEs’ weights Not considered Computed Computed Not considered Computed Does the ranking tool consider type of criteria No No Yes No Yes Preference order y1 y4 y2 y5 y3 y1 y4 y5 y2 y3 y1 y4 y2 y5 y3 Y4 y5 y1 y2 y3 y1 y4 y5 y3 y2 Optimal option y1 y1 y1 y4 y1 The main advantages of the developed IVIF-distance measure-based MAIRCA are listed as ▪ In this method, the weights of the decision makers are computed through score function-based model. Thus, the proposed approach gives a more accurate decision as compared to existing IVIF MCDM methods. ▪ The distance measure proposed in this study avoids the limitations of several existing IVIF-distance measures. Thus, the criteria weighting model based on the proposed distance measure provides more efficient result in the assessment of WWTT alternatives. ▪ The MAIRCA method determines the best solution considering the deviation between the defined theoretical and the real results. The key benefits of the MAIRCA approach are presented as follows: (i) it can solve the MCDM problems with mixed qualitative and quantitative assessment criteria; (ii) this method considers the concept of the positive and negative ideal solutions and (iii) the MAIRCA has a distinctive linear normalization algorithm which can obtain highly reliable discrepancies and generate consistent results. Fig. 4 Comparison of proposed with extant methods for sustainable WWTT selection 382 A. R. MISHRA, P. RANI, F. CAVALLARO, A. F. ALRASHEEDI 5.1. Discussion and Implications The weighting outcomes revealed that the Odor impacts (EN3), Sludge production (EN2) and Maintenance and operation cost (EC1) had become the most significant factors for sustainable WWTT assessment (see Fig. 3). As a result, these criteria should be taken sincerely, while energy consumption (EN1), public acceptance (S1), BOD removal capacity (T2), aesthetic (S2), land requirement (EC2) and COD removal capacity (T1) should be also emphasized with small weights. Moreover, assessment outcomes of sustainable aspects of WWTT assessments are prioritized as: Environmental (0.365) ≻ Economic (0.226) ≻ Social (0.209) ≻ Technical (0.2), which means Environmental dimension have highest impact on prioritization order of sustainable WWTT selection followed by economic, social, technical aspects. The utility scores of WWTT option are 0.0902, 0.1171, 0.1155, 0.1001 and 0.1118, the prioritization order of alternatives y1, y2, y3, y4 and y5 is: 1 4 5 3 2.y y y y y By means of the concept of IVIF-distance measure-based MAIRCA methodology, we have combined the weight-determining models based on IVIF-distance measure and MAIRCA tool, which reduces information loss during the process of making decision. Fig. 5 Ranking results obtained by extant methods for WWTTs assessment This study suggests the policymakers to understand the performance of WWTT alternatives using different aspects of sustainability from uncertain perspective. The proposed work has the following implications for practitioners and scholars: ▪ The most suitable WWTT alternative performs better with respect to economic, environmental social and technical dimensions of sustainability with minimum cost and higher efficiency in order to remove the effluents from the wastewater. Assessment of Sustainable Wastewater Treatment Technologies using Intervall-valued Intuitionistic... 383 ▪ Managers and policymakers can use the information presented in this study to support their decision for assessing the WWTT alternative. ▪ The proposed model not only evaluates the significant degrees of considered criteria but also tackles the ambiguity and fuzziness arisen during the process of WWTT alternatives assessment process. 6. CONCLUSIONS To select the best sustainable WWTT for agricultural purposes, we develop a hybrid MCDM methodology that incorporates the sustainability concept. The methodology presented in this study combines the IVIF-distance measure, criteria weighting tool and MAIRCA model, and also considers the uncertainty level of the decision makers. Here, we have firstly proposed a new Hellinger IVIF-distance measure and analyzed the causes of counter-intuitive results of extant distance measures. Further, we have proposed a new criteria weight determining model based on the proposed Hellinger IVIF-distance measure and total support degree-based model. Further, we have proposed an integrated MAIRCA approach for dealing with the MCDM problems in which the information about the criteria and decision makers is completely unknown. The proposed MAIRCA method integrates the normalization process and ideal solutions, and then determines the option with the smallest total distance (gap) is the best option, which is the main advantage of the proposed work. Comparative assessments have been presented to reveal the outcomes obtained by the hybrid approach. As per the comparative study, it can be observed that the proposed MAIRCA model is very robust and appropriate for the decision support problems under IVIFS environment. This study does not consider the geographical and cultural aspects of the criteria, which is one of the main limitations of this research. In addition, we consider only the independent characteristics of the criteria. In future, we can develop a model to evade the drawbacks of present work. In addition, future works should be deliberated towards utilizing a wider number of global DMs who will assess the factors affecting the healthcare blockchain platforms evaluation process. In addition, we can extend the MAIRCA model under different environments such as "interval-valued hesitant q-rung orthopair fuzzy sets (IVHq-ROFSs)", "q-rung orthopair soft rough sets (q-ROFSRSs)" and "interval-valued picture fuzzy sets (IVPiFSs)". Funding: This research was conducted under a project titled “Researchers Supporting Project”, funded by King Saud University, Riyadh, Saudi Arabia under grant number (RSP2023R323). REFERENCES 1. 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