Plane Thermoelastic Waves in Infinite Half-Space Caused Decision Making: Applications in Management and Engineering ISSN: 2560-6018 eISSN:2620-0104 DOI: https://doi.org/10.31181/dmame0306102022c * Corresponding author. E-mail addresses: divya.jain@juetguna.in (D. Jain), hrishi4676@gmail.com (R. Chaurasiya). HYBRID MCDM METHOD ON PYTHAGOREAN FUZZY SET AND ITS APPLICATION Rishikesh Chaurasiya1 and Divya Jain1 * 1 Department of Mathematics, Jaypee University of Engineering and Technology, India Received: 8 May 2022; Accepted: 14 July 2022; Available online: 6 October 2022. Original scientific paper Abstract: Here in this article a hybrid MCDM method on the Pythagorean fuzzy-environment is presented. This method is based on the Pythagorean Fuzzy Method based on Removal Effects of Criterion (PF-MEREC) and Stepwise Weight Assessment Ratio Analysis (SWARA) approaches. The objective and subjective weights are assessed by PF-MEREC, SWARA model and the preference order ranking of the various alternatives is done through Complex Proportional Assessment (COPRAS) framework on the PFS. The proposed method is the hybrid model of MEREC, SWARA and COPRAS methods. Further, the proposed model is used to identify the best banking management software (BMS) so that the bank can choose the robust bank management software tool to enhance its efficiency and excellence. Thereafter, sensitivity analysis and comparative discussion of the proposed model is done with the existing techniques to judge the reasonability and efficiency of the proposed model. Key words: Pythagorean fuzzy set, decision-making, MEREC, SWARA, COPRAS, banking management software. 1. Introduction There are many uncertain, fuzzy and incomplete problems in the real world. Hence, the fuzzy set theory, originated by Zadeh (1965) is a successful and vigorous tool for determining many same issues. To overcome its primary extension and shortcomings, intuitionistic fuzzy sets (IFS) has been established by Atanassov (1986), it satisfies the requirement of sum total of membership function (MF) and non-MF (N-MF) is less than or equal to one. Nevertheless, there may be difficulties in the policymaking procedure when both the FS and the IFS theories are not capable of addressing the uncertain and incompatible data. Viz., if a decision expert assigns 0.8 and 0.4 as his preference of belonging and non-belongingness of any object, then plainly, it can be easily seen that 0.8 + 0.4 > 1. Hence, this situation is not handled by IFSs. To beat these shortcomings of IFSs, initially, Yager (2013) presented the fundamental of the mailto:divya.jain@juetguna.in mailto:hrishi4676@gmail.com Chaurasiya et al./Decis. Mak. Appl. Manag. Eng. (2022) 2 PFSs. In a Pythagorean fuzzy set MF and N-MF satisfies the condition (0.8)2 + (0.4)2 ≤ 1. In the PFS is a good device for expressing uncertain information ascending in practical, complicated MCDM problems. It has the same provision as IFSs, however has a lot of flexibility and more space to express fuzzy information than IFS. In this regard, PFS has attracted the eye of many scholars and has been studied extensively in management. Some PF-aggregation operators are also presented such as PF-weighted averaging operators (Garg, 2019; Pamucar & Jankovic, 2020; Rong et al., 2020; Akram et al., 2021; Farid & Riaz, 2022) to help tackle MCDM problems. “Einstein geometric aggregation operators employing a new complex-IVPFS” (Ali et al., 2021). Some researchers presented the score functions on PFS (Zhang & Xu, 2014; Peng & Yuan, 2017) can accurately rank general choices and also has a strong sense of partiality by taking hesitant information into account. Moreover, some researchers focused on Pythagorean fuzzy objective weighting methods (Biswas & Sarkar, 2019; Ozdemir & Gul, 2019) and subjective weight (Wei, 2019; Chen, 2019; Wang et al., 2019; Zavadskas et al., 2020). The subjective weights are submitted by DMs supported in their own knowledge, whereas they neglect the primary weight info explained by the valuation data. Some novel approaches to obtaining objective weight from assessment data don’t take into account the DEs’ preferences. So, a combined weighting approach is submitted, which may amalgamate each subjective and objective weight (OW). Many multi-criteria decision-making (MCDM) approaches are dealing with a massive quantity of problems and estimating alternative and help the user in mapping the problem. Criterion weights has an important role in the decision-making (DM) procedure, as the suitable selection of criterion weights is best for ranking of alternatives. Thereafter, it’s vital to discover a method to define the weights. Some approaches have been available in the literature. As a result, many scholars have studied the OW by criteria importance through intercriteria correlation (CRITIC) and entropy measure of PFSs. Xu et al. (2020) proposed an entropy measure on PFS to solve MCDM problems. Chaurasiya and Jain (2021) proposed MARCOS method on IFSs. The authors have applied the predictable MCDM method in various fields (Rani & Jain, 2019; Petrovic et al., 2019; Eiegwa, 2020; Mishra et al., 2021; Li et al., 2022; Yildirim & Yildirim, 2022). In addition, criterion weight is very significant in solving MCDM difficulties. Therefore, the authors have moved their attention to methods related to criterion weight. Keshavarz et al. (2021) developed MEREC technique is one of the powerful approaches for defining the objective criterion weights (OCWs). Whereas, among the innovative technique to determining criteria weight (Zizovic & Pamucar, 2019). Hadi and Abdullah (2022) presented integrate MEREC-TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) method for IoT-based hospital place selection. Hezam et al. (2022) proposed an IF-MEREC-ranking sum-double normalization-based multi-aggregation method for evaluating alternative fuel vehicles concerning sustainability. Marinkovic et al. (2022) employed the MEREC- Combined Compromise Solution multi-criteria method to evaluate the application of waste and recycled materials to production. Integrated MEREC method on Fermatean fuzzy environment proposed Rani et al. (2022), MEREC-MULTIMOORA (Mishra et al., 2022), MEREC-MARCOS (Nguyen et al., 2022), Level based weight assessment-Z- MAIRCA method (Bozanic et al., 2020). Kersuliene et al. (2010) has established the SWARA approach to be an effective device for calculating the SCWs. Alipour et al. (2021) employed a combined SWARA and COPRAS technique to assess the supplier selection of fuel cell and hydrogen constituents in the PFS domain. Saraji et al. (2021) proposed the hesitant fuzzy- Hybrid MCDM method on pythagorean fuzzy set and its application 3 SWARA-MULTIMURA method for online education. Some researchers have drawn attention to integrated methods to solve the MCDM problems, such as (Rani et al., 2020) developed a new integrate SWARA-ARAS method on PFS for healthcare waste treatment problem. Since at present, many scholars have developed the following ranking methods to solve MCDM problems. For examples, Badi and Pamucar (2020) proposed integrate Grey-MARCOS methods for supplier selection. Durmic et al. (2020) proposed an integrated Full Consistency Method-Rough-Simple Additive Weighting (FUCOM-R- SAW) method has been employed to choose sustainable suppliers. Tesic et al. (2022) presented DIBR-fuzzy-MARCOS framework. Puska et al. (2020) suggested a way for measurement of alternatives and ranking according to compromise solution (MARCOS) method for project management software. Some researchers applied MCDM methods such as (Kaya, 2020; Pamucar, 2020; Keshavarz-Ghorabaee, 2021; Ashraf et al., 2022). The COPRAS method, established by Zavadskas et al. (1994) is one of the practical well-orderly approaches to solve intricate MCDM difficulties. The main objective of the COPRAS approach, includes: (i) it is an appropriate and assess method to obtain the solution to the DM issue. (ii) It considers the ratio of the worst and the best outcome; (iii) it provides results in a short-time as compared to other MCDM methods. Several researchers have used the COPRAS technique for various applications (Mishra et al., 2020). These days, various academicians have expanded the traditional COPRAS technique under a range of vague environments. Zheng et al. (2018) studied a hesitant fuzzy (HF) COPRAS approach to solving the health decision-making problem. Thereafter, Mishra et al. (2019) proposed the integrated HF-COPRAS method to solve service quality problems. The PF-COPRAS approach has been used by (Rani et al., 2020a) to appraise pharmaceutical therapy for type-2 diabetic disease. Song and Chen (2021) proposed the COPRAS method on the probabilistic HFS, which is based on the new distance measures of probabilistic HF-elements. For waste-to-energy technology selection, (Mishra et al., 2022) suggested the COPRAS approach based on the IVPF- similarity measure. Currently, Chaurasiya and Jain (2022) have submitted the COPRAS technique on PFS in the MCDM problems which, competently launch the interrelation among criteria & permits decision experts (DE’s) to catch the uncertainty elaborate in judgments of numerous incompatible criterions. The main motivation for this study is, a new hybrid PF-MEREC-SWARA-COPRAS method is established that can efficiently deal with the implicit vagueness and uncertainty concerned with DE’s judgment. Therefore, the summary of the article is as follows: 1) To develop a novel hybrid PF-MEREC-SWARA-COPRAS method under the PF- domain. 2) We calculate the decision experts’ weights in PFS based on (Boran et al., 2009) formula. 3) To calculate objective criterion weights, by new MEREC and subjective criterion weight by SWARA method. Thereafter, we calculate combined criterion weights. 4) The proposed technique is employed to solve the problem of selecting banking management software. Subsequently, its method is compared with other existing methods and sensitively analysis by taking a set of criterion weights. The paper is planned as follows: In section 2, we describe fundamental on PFSs. Section 3, presents a novel hybrid MCDM method on Pythagorean fuzzy set. Section 4, a case study of banking management software selection, which illustrates the Chaurasiya et al./Decis. Mak. Appl. Manag. Eng. (2022) 4 efficiency and applicability of the advanced method. Along with it, sensitivity analysis and the results are compared with already existing methods to validate. Finally, in 5th section, conclusion and future outlook is considered. 2. Basic concept of PFS This section delivers a brief-overview of the PFS. Definition 2.1. (Yager, 2013) A PFS 𝐴 ⊂ 𝑈 in a fixed set defined as: 𝐴 = {〈𝑢𝑖,𝜇𝐴(𝑢𝑖),𝜈𝐴(𝑢𝑖)〉| 𝑢𝑖 ∈ 𝑈} (1) where 𝜇𝐴(𝑢𝑖):𝑈 → [0,1] indicate the MF and 𝜈𝐴(𝑢𝑖):𝑈 → [0,1] indicate the N-MF that mollify the state 0 ≤ 𝜇𝐴 2(𝑢𝑖) + 𝜈𝐴 2(𝑢𝑖) ≤ 1. The hesitancy function 𝜋𝐴(𝑢𝑖) is denoted by 𝜋𝐴(𝑢𝑖) = √1 − 𝜇𝐴 2(𝑢𝑖) − 𝜈𝐴 2(𝑢𝑖), then it is Pythagorean fuzzy-index. Definition 2.2. (Peng & Li, 2019) Let 𝛽 = (μ𝛽,ν𝛽) be a PFN. The modified normalized score and accuracy function of 𝛽 is given as: 𝒮∗(𝛽) = 2(μ𝛽) 2 +(1−(ν𝛽) 2 )+((μ𝛽) 2 ) 2 4 and ℏ°(𝛽) = 1 − ℏ(𝛽), (2) where 𝒮∗(𝛽),ℏ°(𝛽) ∈ [0,1]. Definition 2.3. (Yager, 2013a, b) Assume 𝛽 = (𝜇𝛽,ν𝛽),𝛽1 = (𝜇𝛽1,ν𝛽1) and 𝛽2 = (𝜇𝛽2,ν𝛽2) be PFNs. Where the operations on the PFNs are depicted below as: (i) 𝛽𝑐 = (μ𝛽,ν𝛽); (ii) 𝛽1 ⊕ 𝛽2 = (√μ𝛽1 2 + μ𝛽2 2 − μ𝛽1 2 μ𝛽2 2 , ν𝛽1ν𝛽2 ); (iii) 𝛽1⨂ 𝛽2 = (μ𝛽1μ𝛽2 , √ν𝛽1 2 + ν𝛽2 2 − ν𝛽1 2 ν𝛽2 2 ); (iv) 𝜆𝛽 = (√1 − (1 − μ𝛽 2) 𝜆 , (ν𝛽) 𝜆 ),𝜆 > 0; (v) 𝛽𝜆 = ((μ𝛽) 𝜆 , √1 − (1 − ν𝛽 2) 𝜆 ) ,𝜆 > 0. 3. Pythagorean Fuzzy MEREC-SWARA-COPRAS Method In this section, we have developed a new decision-making scheme, as hybrid PF- MEREC-SWARA-COPRAS method, to deal with the MCDM problems on PFS domain. The present method uses the MEREC method to evaluate the OCWs. This method uses the removal effect of each criterion on the performance of the alternatives to calculate the objective criterion weights. The SWARA method is an effective tool for evaluating SCWs. Thereafter, we have calculated criteria weights by combined formula. Whereas, the COPRAS technique uses the notion of relative degree to assess the importance of the ranking of the alternatives. During this method, the relative degree that describes the complex relative proficiency best selection is directly proportional to the comparative outcome and criterion weights pondered in the decision-making issue. So, we combine these three methods on PFSs to get additional precise and suitable judgments in an ambiguous reference. It is based on MEREC-SWARA and COPRAS Hybrid MCDM method on pythagorean fuzzy set and its application 5 method under the PFS. The working procedure of the hybrid framework is as given below (see Figure 1): Step 1. For a MCDM problem under PF-domain, assume alternatives 𝑇 = {𝑇1,𝑇2,… ,𝑇𝑚} and the features/criteria 𝐹 = {𝐹1,𝐹2,… ,𝐹𝑛}. A group of decision expert’s (DE’s) 𝐸 = {𝐷𝐸1,𝐷𝐸2,… ,𝐷𝐸𝑙 } represents their ideas on each alternative 𝑇𝑖 with respect to each criterion 𝐹𝑗 in terms of linguistic values (LVs). Let 𝑋 = (𝑥𝑖𝑗 (𝑘) ) be a linguistic decision matrix recommended by the DE’s, where 𝑥𝑖𝑗 (𝑘) present to the assessment of an alternative 𝑇𝑖 regarding a criterion 𝐹𝑗 in forms of LVs for 𝑘 𝑡ℎ DE. Step 2. Calculate a primary DE’s weights (𝜆𝑘). For the judgements of the 𝑘 𝑡ℎ DE’s weight, let 𝐸𝑘 = (𝜇𝑘,𝜈𝑘,𝜋𝑘) be a PFNs, then 𝜆𝑘 = (𝜇𝑘 2 + 𝜋𝑘 2 × ( 𝜇𝑘 2 𝜇𝑘 2 + 𝜈𝑘 2)) ∑ (𝜇𝑘 2 + 𝜋𝑘 2 × ( 𝜇 𝑘 2 𝜇𝑘 2 + 𝜈𝑘 2)) ℓ 𝑘=1 (3) Here 𝜆𝑘 ≥ 0, ∑ 𝜆𝑘 = 1 ℓ 𝑘=1 . Step 3. Define the aggregated pythagorean fuzzy decision matrix (APF-DM), corresponding to expert’s weight. Let ℕ = ( 𝑖𝑗)𝑚×𝑛 be the APF-DM, where 𝑖𝑗 = (√1 − ∏ (1 − 𝜇𝑘 2)𝜆𝑘 ℓ 𝑘=1 , ∏ (𝜈𝑘) 𝜆𝑘 ℓ 𝑘=1 ) (4) Step 4. Determination of criteria weights (CWs) Step 4.1. Estimate objective criteria weights (OCWs) by MEREC technique using following steps: Step 4.1a. Evaluate the score matrix 𝒮∗( 𝑘𝑗) = ( 𝑖𝑗)𝑚×𝑛 using equation (2) of each PFN 𝑖𝑗. Step 4.1b. Normalize the APF-DM (ℕ1) = 𝑛𝑖𝑗 𝑥 . The decision-matrix components are scaled using a linear normalization. The elements of the normalized-DM are denoted by 𝑛𝑖𝑗 𝑥 . Here 𝐹𝑏 represents beneficial criteria and 𝐹𝑐 represents cost criteria. 𝑛𝑖𝑗 𝑥 = { 𝑚𝑖𝑛 𝑘 𝑥𝑘𝑗 𝑥𝑖𝑗 , 𝑖𝑓 𝑗 ∈ 𝐹𝑏 𝑥𝑖𝑗 𝑚𝑎𝑥 𝑘 𝑥𝑘𝑗 , 𝑖𝑓 𝑗 ∈ 𝐹𝑐 (5) Step 4.1c. Compute the entire performance of the alternatives (Ω𝑖). A logarithmic function with identical CWs is employed to get alternative entire performance. Ω𝑖 = ln(1 + ( 1 𝑛 ∑ |ln(𝑛𝑖𝑗 𝑥 )| 𝑛 𝑗=1 )) (6) Step 4.1d. Estimate the behavior of the alternatives by eliminating each criterion. The same logarithmic function as in step 4.1c is employed, the only difference is, the alternative appraisals are calculated on the basis of eliminating each criterion individually in this step. Hence, we have n sets of appraisals corresponding to 𝑛 criteria. Assume Ω𝑖𝑗 ′ represent the entire evaluation of 𝑖𝑡ℎ alternative for eliminating the jth criterion. The following process of appraisal using Eq. (7): Chaurasiya et al./Decis. Mak. Appl. Manag. Eng. (2022) 6 Ω𝑖𝑗 ′ = ln(1 + ( 1 𝑛 ∑ |ln(𝑛𝑖𝑘 𝑥 )| 𝑘,𝑘≠𝑗 )) (7) Step 4.1e. Calculate the summation of absolute deviations (𝐷𝑗). We use the Eqs. (6), (7) 𝐷𝑗 = ∑ |𝛺𝑖𝑗 ′ − Ω𝑖| 𝑚 𝑖=1 (8) Step 4.1f. Evaluate final OCWs. The 𝐷𝑗 is employed to compute the objective weight of each criterion in this step. The process is applied to calculate ϖj. ϖj = 𝐷𝑗 ∑ 𝐷𝑗 𝑛 𝑗=1 (9) Step 4.2. Determine the subjective criteria weights (SCWs) by SWARA technique. The procedures for assessment of the SCWs using the SWARA technique is given follow as: Step 4.2a. Analyze the conventional values. Primary, score values 𝒮∗( 𝑘𝑗) of PFNs by (2) are calculated using APF-DM. Step 4.2b. Compute the rank of criteria by the expert’s insight from the greatest significant to the smallest significant criteria. Step 4.2c. Find the relative significance (𝑠𝑗) of the mean value. Relative position is evaluated from the criteria that are placed at second location. The subsequent relative importance is obtained by comparing the criteria located at 𝐹𝑗 to 𝐹𝑗−1. Step 4.2d. Evaluate the relative coefficient (𝑐𝑗) by Eq. (10) 𝑐𝑗 = { 1 , 𝑗 = 1 𝑠𝑗 + 1, 𝑗 > 1 (10) where, 𝑠𝑗 is relative significance. Step 4.2e. Calculate the weights (𝑝𝑗), as given by Eq. (11). 𝑝𝑗 = { 1 , 𝑗 = 1 𝑐𝑗−1 𝑐𝑗 , 𝑗 > 1 (11) Step 4.2f. Compute scaled weight. In common, the criterion weights are discussed by the expression. 𝜔𝑗 = 𝑝𝑗 ∑ 𝑝𝑗 𝑛 𝑗=1 (12) Step 4.3. Evaluate the combining CWs. In the MCDM technique, all criteria have varying degrees of significance. Let 𝑤 = (𝑤1,𝑤2,…,𝑤𝑛) 𝑇 be a set of CWs with ∑ 𝑤𝑗 = 1 𝑛 𝑗=1 and 𝑤𝑗 ∈ [0,1], given as: 𝑤𝑗 = ϖj∗𝜔𝑗 ∑ ϖj∗𝜔𝑗 𝑛 𝑗=1 (13) Step 5. Ranking of the alternative by COPRAS method. The values of benefit-(𝜎𝑖) and cost-(𝜑𝑖) type criteria, 𝑖 = 1(1)𝑚 is given as: 𝜎𝑖 =⊕ 𝑗=1 𝑛 𝑤𝑗 𝑖𝑗, (For benefit-type) (14) 𝜑𝑖 = ⊕ 𝑗=𝑙+1 𝑛 𝑤𝑗 𝑖𝑗, (For cost-type) (15) Hybrid MCDM method on pythagorean fuzzy set and its application 7 Step 6. Evaluate the relative degree (𝛿𝑖) of each alternative as follows: 𝛿𝑖 = 𝒮 ∗(𝜎𝑖) + ∑ 𝒮∗(𝜑𝑖) 𝑚 𝑖=1 𝒮∗(𝜑𝑖) ∑ 𝒮 ∗(𝜑𝑖) 𝑚 𝑖=1 (16) Where 𝒮∗(𝜎𝑖) and 𝒮 ∗(𝜑𝑖) represents the score values of 𝜎𝑖 and 𝜑𝑖. Step 7. Compute the utility degree (𝛾𝑖). Using Eq. (17) 𝛾𝑖 = 𝛿𝑖 max (𝛿𝑖) × 100% (17) Step 8. Find the best ranking of alternatives. Figure 1. Representation of the PF-hybrid method for BMS selection 4. Application in Banking Management Software All over the world, the banks are being digitized with the assistance of information technology tools. It provides extraordinary speed to banking operations. Thus, to be successful in banking services, one has to offer the best banking software choosing the best banking software requirements. The opinion of banking experts needs to become technologically more innovative to meet all the requirements and expectations of the clients. Banking software is a means of communication between the bank and the user. Chaurasiya et al./Decis. Mak. Appl. Manag. Eng. (2022) 8 It serves to improve the workflow within the company and its branches, for easier investment policies, and to provide services that address the necessity of the users. Which is offers greater functionality, convenience, flexibility, reliability, security, instant transfers, mobile apps, the ability to remain adaptable and modern to meet the changing nature of market needs and competitiveness. Innovations in information communication technology (ICT) and globalization are constantly changing business processes. These alterations range from easy structural changes to paradigm shifts Laudon and Laudon (2015). The bank’s goal is to alleviate costs, increase efficiency and guarantee client holding with the use of technology. In the banking sector, the relationship among organizations and its clients is vital. Technological advancement enables closer and longer-terms affinities with clints. The CBS developed in the 1970s and has undergone important changes over time. The upgraded core banking system has capability of real-time processing and multichannel unification (Kreca & Barac, 2015). Due to the growing issues of electronic payments, some researchers and managers have turned their attention to banking software. For this MCDM methods are best suited that can based on numerous criteria. Recently, due to digitization in the banking sector, it became very important to select the best banking management software. It provides extraordinary speed to banking operations. Thus, to be successful in banking services, one has to offer the best banking software choosing the best banking software requirements. Here, a case study of BMS for a banking area in India is measured to demonstrate the applicability and practicality of the evolved PF-MEREC-SWARA-COPRAS method. In the procedure of existing method, the bank shaped a team entailing of four decision experts who are responsible for BMS. Let the various banking tools available with us are (Figure 2): Mambu (𝑇1), Temenos (𝑇2), Oracle Flexcube (𝑇3), Finastra (𝑇4) and Finacle (𝑇5). We have to identify the best software tool for any banking management based on the following important features (criteria’s): Customizable interfaces (𝐹1), Data management and history tracking (𝐹2), Documentation (𝐹3), Live customer support (𝐹4), Online payments and bills (𝐹5), Mobile version (𝐹6), Self-service options for clients (𝐹7), Transaction processing (𝐹8). Figure 2. The selection of BMS https://www.softwaresuggest.com/caresoft-his https://www.softwaresuggest.com/hospital-mgt-sy-by-genipulse Hybrid MCDM method on pythagorean fuzzy set and its application 9 Table 1 presents the LVs given in PFNs for the relative behavioral rating of weights. Table 1. Linguistic values (LVs) in terms of PFNs LVs PFNs Thrillingly Significant (THS) (0.90, 0.10) Typically Significant (TS) (0.80, 0.20) Noteworthy (N) (0.60, 0.40) Reasonable (R) (0.50, 0.50) Inconsequential (IC) (0.45, 0.55) Trivial (TR) (0.30, 0.75) Pitty (P) (0.10, 0.90) Table 2 shows the weight of each DE’s as calculated using Eq. (3). Table 2. Decision expert weights Decision Experts LVs PFNs Weights (𝝀𝒌) DE1 N (0.60,0.40) 0.2749 DE2 R (0.50,0.50) 0.2257 DE3 TS (0.80,0.20) 0.3058 DE4 IC (0.45,0.55) 0.1936 For assessing the alternatives linguistic values are transformed in terms of PFNs. Table 3. LV for assessing the alternatives LVs PFNs Extremely small (ES) (0,1) Very small (VS) (0.10, 0.90) small (S) (0.20, 0.80) Slightly small (SS) (0.30, 0.70) Below intermediate (BI) (0.40, 0.60) Intermediate (I) (0.50, 0.50) Above intermediate (AI) (0.60, 0.40) Slightly big (SB) (0.70, 0.30) Big (B) (0.80, 0.20) Very big (VB) (0.90, 0.10) Extremely big (EB) (1, 0) Here, Table 4 represents the ideas of DE’s on each of the alternative 𝑇𝑖 respect to each criterion 𝐹𝑗 in terms of LVs defined in Table 3. Table 4. The LV’s calculation of alternatives given by DE’s Alternative DEs Criteria 𝐹1 𝐹2 𝐹3 𝐹4 𝐹5 𝐹6 𝐹7 𝐹8 𝑻𝟏 DE1 VB VB B B VB B VB BI DE2 B SB SB I B VB B SS DE3 B VB VB SB VB B VB I Chaurasiya et al./Decis. Mak. Appl. Manag. Eng. (2022) 10 DE4 VB VB B B SS B B AI 𝑻𝟐 DE1 VB B SB SB VB VB VB BI DE2 B B I SB B SB B AI DE3 VB VB B AI B I B I DE4 B B VB VB AI VB SB BI 𝑻𝟑 DE1 VB B SB SB B B VB I DE2 VB B SB AI AI SB B SS DE3 B VB VB I VB SB B BI DE4 VB AI BI VB I B BI AI 𝑻𝟒 DE1 B B A I SB B B VB I DE2 B B SB AI SS SB I AI DE3 B B SB SB AI AI SB BI DE4 B VB VB SS SB S BI I 𝑻𝟓 DE1 VB B SB SB B B B I DE2 B B B BI SB BI AI S DE3 B B AI SB B B SB I DE4 VB SB VB I AI AI AI SB In Table 5, the LV’s of alternatives given by DE’s in Table 4 is converted to APF- DM using Eq. (4). Table 5. Computed APF-DM 𝑭𝟏 𝑭𝟐 𝑭𝟑 𝑭𝟒 𝑭𝟓 𝑭𝟔 𝑭𝟕 𝑭𝟖 𝑻𝟏 (0.8562 , 0.1445) (0.8733, 0.1281) (0.8244, 0.1773) (0.7262, 0.2784) (0.8383, 0.1704) (0.8297, 0.1710) (0.8669, 0.1337) (0.4649, 0.5432) 𝑻𝟐 (0.8669 , 0.1337) (0.8389, 0.1618) (0.7660, 0.2404) (0.7406, 0.2648) (0.8139, 0.1890) (0.7992, 0.2096) (0.8228, 0.1788) (0.4867, 0.5178) 𝑻𝟑 (0.8769 , 0.1236) (0.8179, 0.1850) (0.7646, 0.2452) (0.7078, 0.3025) (0.7819, 0.2259) (0.7529, 0.2481) (0.8026, 0.2045) (0.4619, 0.5463) 𝑻𝟒 (0.8266 , 0.1710) (0.8258, 0.1749) (0.7427, 0.2625) (0.6321, 0.3772) (0.6578, 0.3547) (0.6622, 0.3543) (0.7299, 0.2847) (0.5009, 0.5027) 𝑻𝟓 (0.8562 , 0.1445) (0.7841, 0.2163) (0.7633, 0.2417) (0.6204, 0.3873) (0.7514, 0.2506) (0.7161, 0.2931) (0.7002, 0.3028) (0.5139, 0.5036) 4.1. MEREC Technique This measure reflects the difference between the performance of the composite option and its performance in removing the criterion. The following steps are used to calculate the OCWs by MEREC method: we compute the score matrix using Eq. (2). As {𝑭𝟓,𝑭𝟖} a set of cost/non-benefit and others are benefit type of criteria, so, we normalized-APF-DM using Eq. (5) and shown in Table 6. Hybrid MCDM method on pythagorean fuzzy set and its application 11 Table 6. Normalized APF-DM 𝑭𝟏 𝑭𝟐 𝑭𝟑 𝑭𝟒 𝑭𝟓 𝑭𝟔 𝑭𝟕 𝑭𝟖 𝑻𝟏 0.9405 0.8286 0.8383 0.9689 0.6713 0.6888 0.6981 0.8804 𝑻𝟐 0.9197 0.8903 0.9506 1.0000 0.7064 0.7358 0.7658 0.9384 𝑻𝟑 0.9008 0.9308 0.9541 0.9286 0.7569 0.8119 0.7999 0.8729 𝑻𝟒 1.0000 0.9150 1.0000 0.7804 1.0000 1.0000 0.9386 0.9759 𝑻𝟓 0.9405 1.0000 0.9559 0.7594 0.8078 0.8823 1.0000 1.0000 To obtain the OCWs by MEREC method, we compute the overall performance of the Table 7. Calculate the performance of the alternatives by removing each criterion. 𝑭𝟏 𝑭𝟐 𝑭𝟑 𝑭𝟒 𝑭𝟓 𝑭𝟔 𝑭𝟕 𝑭𝟖 𝑻𝟏 0.1879 0.1747 0.1760 0.1910 0.1524 0.1551 0.1566 0.1811 𝑻𝟐 0.1345 0.1309 0.1381 0.1436 0.1052 0.1098 0.1143 0.1367 𝑻𝟑 0.1221 0.1257 0.1284 0.1255 0.1027 0.1105 0.1089 0.1186 𝑻𝟒 0.0517 0.0411 0.0517 0.0218 0.0517 0.0517 0.0442 0.0488 𝑻𝟓 0.0792 0.0862 0.0810 0.0541 0.0614 0.0718 0.0862 0.0862 alternative values from Eq. (6), given as (Ω1 = 0.1943, Ω2 = 0.1436, Ω3 = 0.1336, Ω4 = 0.0517, Ω5 = 0.0862). Apply the Eq. (7), we appraise alternatives overall performances (𝛺𝑖𝑗 ′ ) in removing criterion and are given in Table 7. Afterward, we compute the absolute deviation (𝐷𝑗) values from Eq. (8). Finally, we compute OCWs (𝜛𝑗) using Eq. (9) by MEREC method. Absolute deviation (𝐷𝑗) = (0.0340, 0.0508, 0.0342, 0.0734, 0.1360, 0.1105, 0.0992, 0.0380). Objective weight (ϖj) = (0.0590, 0.0882, 0.0594, 0.1274, 0.2361, 0.1918, 0.1722, 0.0659). 4.2. Subjective Weights by SWARA Technique The following steps are used to compute the SCWs by SWARA method. Table 8. Evaluate of criteria weights by DE’s Criteria DE1 DE2 DE3 DE4 Aggregated PFNs Crisp values 𝓢∗(𝜺𝒌𝒋) 𝑭𝟏 VB B SB VB (0.8385, 0.1636) 0.7184 𝑭𝟐 VB SB I B (0.7688, 0.2397) 0.6185 𝑭𝟑 B AI I SB (0.6717, 0.3348) 0.4985 𝑭𝟒 SB B BI AI (0.6528, 0.3578) 0.4765 𝑭𝟓 VB I SB SS (0.7250, 0.2933) 0.5604 𝑭𝟔 B SB B AI (0.7514, 0.2506) 0.5963 𝑭𝟕 VB AI BI I (0.6974, 0.3230) 0.5262 𝑭𝟖 BI B I SB (0.6257, 0.3872) 0.4466 Chaurasiya et al./Decis. Mak. Appl. Manag. Eng. (2022) 12 Table 9. Criteria weights evaluated by SWARA method Criteria Crisp values Relative significance (𝐬𝒋) Relative coefficient (𝒄𝒋) Recalculate d weight (𝒑𝒋) Criteria weight (𝝎𝒋) 𝑭𝟏 0.7184 - 1.0000 1.0000 0.1459 𝑭𝟐 0.6185 0.0999 1.0999 0.9092 0.1327 𝑭𝟔 0.5963 0.0222 1.0222 0.8895 0.1298 𝑭𝟓 0.5604 0.0359 1.0359 0.8587 0.1253 𝑭𝟕 0.5262 0.0342 1.0342 0.8303 0.1211 𝑭𝟑 0.4985 0.0277 1.0277 0.8079 0.1179 𝑭𝟒 0.4765 0.0220 1.0220 0.7905 0.1153 𝑭𝟖 0.4466 0.0299 1.0299 0.7676 0.1120 (𝜔𝑗) = (0.1459,0.1327,0.1179,0.1153,0.1253,0.1298,0.1211,0.1120). There after we calculated the weights of the criteria by Eqs. (13). Combined weight (𝑤𝑗 ) = (0.0690, 0.0938, 0.0562, 0.1178, 0.2372, 0.1996, 0.1672, 0.0592) T. Table 10. Calculate the values from 𝜎𝑖 and 𝜑𝑖 𝝈𝒊 𝝋𝒊 𝑺 ∗(𝝈𝒊) 𝑺 ∗(𝝋𝒊) 𝜹𝒊 𝜸𝒊 𝑻𝟏 (0.8048, 0.2161) (0.3124, 0.8296) 0.6671 0.1291 0.7758 100.0 𝑻𝟐 (0.7729, 0.2504) (0.3223, 0.8224) 0.6222 0.1356 0.7257 93.54 𝑻𝟑 (0.7475, 0.2786) (0.3027, 0.8381) 0.5880 0.1223 0.7027 90.58 𝑻𝟒 (0.6753, 0.3593) (0.2726, 0.8559) 0.4977 0.1054 0.6307 81.30 𝑻𝟓 (0.7036, 0.3231) (0.2695, 0.8587) 0.5327 0.1033 0.6685 86.17 From equations (14)-(17), the values of 𝜎𝑖,𝜑𝑖,𝑆 ∗(𝜎𝑖),𝑆 ∗(𝜎𝑖),𝛿𝑖 𝑎𝑛𝑑 𝛾𝑖 of 𝑇𝑖 are assessed with respect to criteria 𝐹𝑗, shown in Table 10. Displayed in Table 10, the rank descending sequence of the banking management software choice is 𝑇1 ≻ 𝑇2 ≻ 𝑇3 ≻ 𝑇5 ≻ 𝑇4. Thus, alternative 𝑇1 is the best selection. 4.3. Sensitivity Analysis Here sensitivity analysis is undertaken to calibrate the presented methods behavior. Eight different CW sets are taken and displayed in Table 11. The table shows for each set, one of the criteria has the highest weight, whereas the others have lesser weights. Using this procedure, a sufficient range of criterion weights has been built to examine the sensitivity of the evolved method to variants of CWs. The ranking outcomes of BMS amenity alternative and the relative degree 𝛿𝑖 of various criteria weight, according to the sensitivity analysis outcomes are displayed in Table 12 and figure 3. When the DE’s provide weighting Table 11. Diverse criteria weight sets for BMS alternative Set-I Set-II Set-III Set-IV Set-V Set-VI Set-VII Set-VIII 𝑭𝟏 0.0690 0.0938 0.0562 0.1178 0.2372 0.1996 0.1672 0.0592 𝑭𝟐 0.0938 0.0562 0.1178 0.2372 0.1996 0.1672 0.0592 0.0690 𝑭𝟑 0.0562 0.1178 0.2372 0.1996 0.1672 0.0592 0.0690 0.0938 𝑭𝟒 0.1178 0.2372 0.1996 0.1672 0.0592 0.0690 0.0938 0.0562 Hybrid MCDM method on pythagorean fuzzy set and its application 13 Set-I Set-II Set-III Set-IV Set-V Set-VI Set-VII Set-VIII 𝑭𝟓 0.2372 0.1996 0.1672 0.0592 0.0690 0.0938 0.0562 0.1178 𝑭𝟔 0.1996 0.1672 0.0592 0.0690 0.0938 0.0562 0.1178 0.2372 𝑭𝟕 0.1672 0.0592 0.0690 0.0938 0.0562 0.1178 0.2372 0.1996 𝑭𝟖 0.0592 0.0690 0.0938 0.0562 0.1178 0.2372 0.1996 0.1672 Table 12. Relative degree for BMS alternatives for different criteria weight sets Set-I Set-II Set-III Set-IV Set-V Set-VI Set-VII Set-VIII T1 0.7758 0.7838 0.7804 0.8156 0.7678 0.7481 0.7719 0.7592 T2 0.7257 0.7311 0.7106 0.6862 0.7152 0.7143 0.7198 0.6987 T3 0.7027 0.7261 0.7160 0.6996 0.7232 0.7263 0.7188 0.6812 T4 0.6307 0.6789 0.6847 0.6717 0.6724 0.6639 0.6473 0.5987 T5 0.6685 0.7277 0.7139 0.6966 0.6891 0.6767 0.6638 0.6239 sets I, VII, and VIII, the BMS ranks them in the same order, while for other sets its different. According to the description above, the BMS selection is dependent on, and sentient to, these CW sets, As the proposed method is stable with a variety of weight sets. Figure 3. Outcome of 𝛿𝑖 for each alternative with various weight sets of criteria Chaurasiya et al./Decis. Mak. Appl. Manag. Eng. (2022) 14 Table 13 The comparative study with existing techniques Methods Standar d Expert’s weight Criteria weights Ranking BMS alterna tive Zhang and Xu (2014) PF- TOPSIS method Not evaluate Assumed 𝑇1 ≻ 𝑇2 ≻ 𝑇3 ≻ 𝑇4 ≻ 𝑇5 𝑇1 Kumari and Mishra (2020) IF- COPRAS method Evaluate Completely unknown 𝑇1 ≻ 𝑇3 ≻ 𝑇2 ≻ 𝑇5 ≻ 𝑇4 𝑇1 Peng et al. (2020) PF- COCOSO method Not evaluate Assumed 𝑇1 ≻ 𝑇2 ≻ 𝑇3 ≻ 𝑇5 ≻ 𝑇4 𝑇1 proposed method PF- COPRAS Evaluate MEREC- SWARA combined method 𝑇1 ≻ 𝑇2 ≻ 𝑇3 ≻ 𝑇5 ≻ 𝑇4 𝑇1 4.5. Comparison and Discussion In this section, now, we see that the framework submitted here has a lot of similarities with the existing methods. The PF-MEREC-SWARA-COPRAS method is found to be proficient for handling qualitative and quantitative MCDM issues, especially in cases where there are many conflicting criteria. The advantages or features of the presented framework can be discussed as follows:  The method PF-TOPSIS (Zhang & Xu, 2014) and PF-COCOSO (Peng et al., 2019) and the proposed PF-hybrid method are submitted in the context of PFS, whereas (Kumari & Mishra, 2020) have described IF-COPRAS method is used.  In the developed PF-hybrid method, we have evaluated expert weights on the basis of expert opinion, leaving no space to treat vagueness, whereas PF- TOOPSIS (Zhang & Xu, 2014) and PF-COCOSO (Peng et al., 2019) the procedure does not involve expert opinion.  PF-COPRAS outperformed PF-TOPSIS and IF-COPRAS in terms of effectiveness and proficiency. In addition, the hybrid COPRAS method is more powerful and stable in terms of criterion weight disparity than PF-COCOSO (Peng et al., 2019).  The practical outcomes of the presented method provide some significant perceptions related to the evaluation criteria and the alternative for BMS in India. As may be displayed in Table 10, the most significant is the effectiveness of the BMS. We find the best alternative among the existing ones. The problem of banking management can be solved to a great extent by seeing the outcomes of this paper. We also analyzed the performance of BMS alternatives and compared the results for each criterion evaluated. According to the results, Mambu (𝑇1) first rank among all alternatives and (𝑇4) is the last in the ranking. Therefore, (𝑇1) can be chosen as the best alternative meeting all the valuation. 5. Conclusions Currently, with the speedy growth of IT, it is a composite problem to select the best software for the diverse work of bank. MCDM is the best tool to deal with it. The key Hybrid MCDM method on pythagorean fuzzy set and its application 15 purpose of the present paper is to develop an MCDM method in a pythagorean fuzzy environment. To do this, we first submitted a new MEREC method and score function on PFS. The PFSs provide a precise and practical solution of the ambiguous real-life DM difficulties; consequently, a new hybrid PF-MEREC-SWARA-COPRAS method has been developed under PFS. Finally, the PF-COPRAS methodology is proposed for ranking the alternatives. In addition, the discussion of comparative study of the presented method with the existing methods. Based on a comparison with existing method, it is worth saying that the PF-COPRAS method provide an effortless calculation with accurate and effective outcomes for the development of MCDM difficulties. The application of the proposed hybrid method on selecting the optimal banking software tool helps in finding the best BMS.  A new normalization score function for PFN is submitted, which minimizes intimation loss by taking uncertainty information into account. Compared to existing score functions, it has a more robust ability to differentiate when comparison two PFNs.  The combined weight framework has been submitted on the basis of MEREC and SWARA weighted extensive methods, which consider both objective and subjective weight.  MEREC presented a novel PF-decision-making technique basis on the COPRAS method, which can get the best alternative without any adverse events, get the outcome of the decision without segmentation, and has a robust ability and stability. Some short comings of the projected structure are significant. A practical problem is that DM necessity skilled in the flexibility and ability to properly use the preferred style of PFS. The projected structure will help as a useful device for selecting the best BMS under multiple-criteria situations and ambiguous environments. In the future, the evolved MCDM method may be further proceed to Fermatean-FSs, interval-valued PFS, and hesitant PFS. In addition, the researchers can extend our research via various MCDM platforms (for example, Mixed Aggregation by Comprehensive Normalization Technique (MACONT), Gained and Lost Dominance Score (GLDS), MAIRCA, and CoCoSo) to choose the most suitable BMS selection. The limitation of the current study is that only a small number of DE’s were included, and it does not take into account the interrelationships among the criteria, which somehow limits the scope of the application of the proposed framework. Consequently, further research is still needed, which considers huge number of decision experts. Author Contributions: Research Problem, R.K.C. and D.J.; Methodology, R.K.C.; Formal Analysis, R.K.C.; Writing-Original Draft Preparation, R.K.C; Modification, D.J.; Writing-Review & Editing, D.J Ackno wledgme nt: The authors would like to express their gratitude to the editors and anonymous referees for their informative, helpful remarks and suggestions to improve this paper as well as the important guiding significance to our researches. Conflicts of Interest: There is no conflict of interest. Funding: This research received no external funding. Data Availability Statement: Not applicable. Chaurasiya et al./Decis. Mak. Appl. Manag. Eng. (2022) 16 References Akram, M., Khan, A., & Borumand Saeid, A. (2021). Complex Pythagorean dombi fuzzy operators using aggregation operators and their decision‐making. Expert Systems, 38(2), 12626. Ali, Z., Mahmood, T., Ullah, K., & Khan, Q. (2021). 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