Dozens of scientific studies illustrate the evolution and integration of MCDM methodologies across diverse applications, reinforcing their significance in decision-making under complex and uncertain conditions when stakeholders and policymakers select personnel. Demirel and Çubukçu (2021) proposed a decision-making system using the fuzzy logic method, one of the AI approaches. The process is related to the performance assessment of employment seekers. Performance measurement includes all applications and develops a rule based on academic qualifications and experience. Esangbedo et al. (2021) analysed some vendors’ human resource information systems through two novel hybrid MCDM methods that take ordinal data as input. They introduced a Grey-Point-Allocation Full-Consistency (Grey-PA-FUCOM) weighting approach. This approach extends the FUCOM (Pamučar et al., 2018) method with Grey-Point-Allocation. The Grey-PA-FUCOM integrates the straightforward point-allocation technique commonly used by practitioners in human resources with the sophisticated FUCOM method familiar to experts in grey system theory. Ozgormus et al. (2021) presented a systematic approach to solving the Turkish textile industry’s Personnel Selection Problem (PSP), considering multiple performance objectives and factors. The authors used a fuzzy Decision-Making Trial and Evaluation Laboratory (DEMATEL) method to assign weights to social criteria. The origins of the DEMATEL technique (Gabus and Fontela, 1972; Fontela and Gabus, 1974) trace back to Leontief’s input-output model (Leontief, 1949), a widely recognised framework in economics. A key advantage of the DEMATEL method lies in its ability to construct a structural model of a systematic problem by analysing the strength of binary relationships (pairwise comparisons) between elements. Finally, a ranking among the alternatives is derived using the GRA technique regarding the scores related to each criterion in the previous step. Nong and Ha (2021) proposed an integrated MCDM approach to help select qualified workers in distribution science. They used an integrated methodology consisting of AHP (Saaty, 1977) and TOPSIS (Hwang and Yoon, 1981) to solve the problem of staff selection. AHP was used to obtain the weights for selection criteria, and TOPSIS was applied to rank the available options. Popović (2021) researched applying MCDM approaches in personnel selection. They used the CoCoSo method (Yazdani et al., 2019) to rank possibilities and choose the best candidate. The fuzzy Analytic Hierarchy Process (FAHP) is an extension of the AHP that integrates fuzzy set theory. This integration, which allows decision-makers to use fuzzy membership functions and linguistic variables to address uncertainty, was pioneered by Bellman and Zadeh (1970). They and Zimmermann’s (1978) significant advancements laid the foundation for applying fuzzy set theory in decision-making frameworks, marking a crucial evolution in the field. Bellman and Zadeh’s (1970) framework utilised the maximin principle, a significant development in the field. This principle focuses on the worst-case scenario and carries a profound weight in decision-making. It influenced seminal works by Yager and Basson (1975) and Baas and Kwakernaak (1977) in fuzzy MADM that proposed additive weighting models. De Graan (1980) and Lootsma (1980) extended Saaty’s theory for using fuzzy sets. Van Laarhoven and Pedrycz (1983) introduced the first FAHP method, which used triangular fuzzy numbers (TFNs) in pairwise comparison, significantly contributing to the field. Chen and Hwang (1992) categorised fuzzy methods into two groups: ranking methods, such as Hamming distance and linguistic ranking, and MADM methods like fuzzy simple additive weighting, fuzzy outranking, and FAHP. Uslu et al. (2021) aimed to measure the exact criteria for selecting qualified management at a healthcare facility. They employed Fuzzy AHP and MULTIMOORA (Brauers and Zavadskas, 2010) methodologies in choosing a health manager, considering the evaluation of 8 candidates in 12 personnel selection criteria. Several advanced MCDM approaches have been proposed to improve decision-making processes. The study by Zavadskas et al. (2010) highlighted the peculiarities of determining attribute weights in MCDM methods, emphasising the variability in expert knowledge across different fields. Keršulienė and Turskis (2011) integrated the ARAS-F and SWARA techniques for architect selection, demonstrating the effectiveness of combining fuzzy and ratio-based methods. Zavadskas et al. (2011) proposed a methodology incorporating SWOT analysis, AHP, expert judgment, and QUALIFLEX to determine management strategies for construction enterprises. Further contributions to personnel selection include the hybrid fuzzy MCDM approaches of Keršulienė and Turskis (2014a, 2014b), who integrated ARAS-F, the fuzzy weighted-product model, and AHP to enhance chief accountant selection. Turskis and Keršulienė (2024) introduced the SHARDA-ARAS methodology, effectively prioritising project managers for sustainable development. Ghorabaee et al. (2017) extended the EDAS method using interval type-2 fuzzy sets, providing a robust framework for multi-criteria group decision-making under uncertainty. Zavadskas et al. (2018) applied the TOPSIS-F method to assess air pollution, demonstrating the applicability of fuzzy MCDM techniques to environmental problems. Hashemi et al. (2018) combined grey-intuitionistic fuzzy ELECTRE and VIKOR for contractor assessment, enhancing decision-making in construction management. Erdogan et al. (2019) proposed a comprehensive MCDM model for sustainable construction management, integrating AHP and expert judgment to improve project selection processes. Gigović et al. (2016) introduced a new technique for multi-criteria decision-making – Multi-Attributive Ideal-Real Comparative Analysis method (MAIRCA). Boral et al. (2020) proposed a novel integrated MCDM approach by combining the FAHP with the modified Fuzzy MAIRCA (FMAIRCA). Liou and Wang (1992) suggested modifying the fuzzy weighted average (FWA) method developed by Dong and Wong (1987). Xu and Yager (2006) introduced some geometric aggregation operators based on intuitionistic fuzzy sets. Atanassov (1986, 1989) and Atanassov and Gargov (1989) introduced the concept of an intuitionistic fuzzy set, which is a generalisation of the fuzzy set (Zadeh, 1965). Kara et al. (2022) defined the choice of human resources managers in logistics organisations using the Intuitionistic Fuzzy Weighted Averaging (IFWA) technique to assign the weights to the criteria and the FMAIRCA technique to rank candidates for managers. The key determinants influencing the implementation of green human resource management into petrochemical firms by Bushehr City are the integrated approach of fuzzy hierarchical analysis and type-2 DEMATEL by Rajabpour et al. (2022). Keršulienė et al. (2010) introduced the Stepwise Weight Assessment Ratio Analysis (SWARA) method. Yager (2013a, 2013b) introduced the concept of the Pythagorean fuzzy set, a generalisation of the intuitionistic fuzzy set, offering enhanced capabilities for addressing uncertainty. Following its development, the introduction of Pythagorean fuzzy aggregation operators, such as the Pythagorean fuzzy weighted averaging and Pythagorean fuzzy ordered weighted averaging operators, proposed by Yager and Abbasov (2013), has sparked significant interest and engagement in the research community. Saeidi et al. (2022) modified the SWARA method by integrating it with the TOPSIS using Pythagorean fuzzy sets (PFSs) and named it the PF-SWARA-TOPSIS method. Sarucan et al. (2022) applied Hesitant Fuzzy AHP, Fuzzy-COPRAS, and Fuzzy-TOPSIS methods for job evaluation research in a food enterprise. This approach helps form an equal compensation policy for increasing employee satisfaction with job evaluation analysis of various positions. Qi (2023) constructed an advanced model of the TOPSIS model to present new approaches to assessing quality performance in public charging service quality. The research ranked alternative options using the TOPSIS combined with the FUCOM method with probabilistic, hesitant and fuzzy concepts. Yiğit (2023) proposed an integrated DSS approach using MCDM to evaluate trainers within organisations and select the best candidate(s) to participate in the training program. The proposed model also considers the training budget and the limitation on the number of assignments. This proposed model comprises three phases: Delphi, the Interval-Valued Neutrosophic AHP (IVN-AHP), and Fuzzy C-Means (FCM). Alrashedi (2024) mentioned that optimising the Human Resources Management Process or HRMP is possible with Markov chain and fuzzy MCDM methodologies. Dhruva et al. (2024a) introduced a decision framework for selecting CVs in healthcare sectors. The solution resolves the issue of excessive value hesitation in criteria using the LOPCOW (logarithmic percentage change-driven objective weighting) method, similar to the logarithmic normalisation method (Zavadskas and Turskis, 2008). The ranking system acts as the measure for individual CV appraisal using the CoCoSo methodology. Taylan et al. (2024) developed a new set of criteria and sub-criteria to assess twelve candidate pilots. The research explored numerically unmeasurable, imprecise, and non-linear continuous fuzzy linguistic characteristics that made the work distinct and challenging because of differing preferences and variations among the decision-makers (DMs). The application of three different procedures of fuzzy MCDM methods–fuzzy TOPSIS, fuzzy VIKOR, and fuzzy PROMETHEE–has been evaluated through the trapezoidal fuzzy number for ranking different positions in potential pilots.

Saaty (1977) introduced the Analytic Hierarchy Process (AHP), a systematic and comprehensive approach for addressing complex decision-making problems (Saaty and Bennett, 1977; Saaty and Vargas, 1979; Saaty, 1980). While the AHP effectively structures decision-making processes and prioritises alternatives, it is important to acknowledge that it does not account for the ambiguity or indecisiveness that decision-makers may exhibit in real-world scenarios. A significant limitation should be considered when applying the AHP (Wind and Saaty, 1980; Saaty, 1982, 1996). To address this, researchers employed fuzzy AHP, a technique utilising comparison matrices that incorporate fuzzy integers. Zadeh introduced his concept of fuzzy sets in 1965. Fuzzy set theory solely considers the degree of approval, neglecting the certainty associated with decision-making. Zadeh (1965) founded the basic principles of the intuitionistic fuzzy sets (IFS) notion. Atanassov (1994) developed the IFS, a robust tool for managing ambiguity and uncertainty. The distinguishing characteristic of IFS is its ability to ascertain membership, non-membership, and indeterminacy levels for each element. Peng and Wang (2011) employed the GRA variant of the TOPSIS decision-making approach to address the optimal supplier selection problem utilising interval numbers. Hladík (2012) employed interval numbers to resolve the Linear Programming problem. Liao and Xu (2014) presented a study on Intuitionistic Fuzzy Systems. Classical AHP and Fuzzy AHP terminate at IFS values for comparison matrices, although the AHP extends well beyond this point. A triangular collection of conceptually fuzzy integers was employed for vendor selection using AHP (Kaur, 2014). Pal and Mahapatra (2017) devised a parametric representation of interval numbers in functional form, incorporating arithmetic operations in symmetric, asymmetric, and convex combinations. Wang et al. (2015) employed TOPSIS and the Response Surface Method in MCDM scenarios utilising interval data. Nirmala and Uthra (2019) used the AHP with the TIFN methodology of MCDM to solve the supplier selection problem. In recent years, numerous scholars have concentrated on evaluating the efficacy of university instructors. Nikoomaram et al. (2009) assessed the efficacy of administration sciences educators by utilising fuzzy set theory, AHP, and TOPSIS. Jana et al. (2024) employed fuzzy AHP for the priority-based selection of financial indices for investors. Huang and Feng (2015) advocate for using RAHPTOPSIS, an augmented variant of AHP and TOPSIS, to evaluate the reliability of college physical education courses. Mazumdar et al. (2010) assessed the efficacy of individual educators by utilising grey relational analysis (GRA) and the COPRAS technique. Wu et al. (2012) employed AHP and VIKOR MCDM to evaluate the 12 private universities designated as a case study by the Ministry of Education. Mondal and Pramanik (2014) employed a neutrosophic methodology and MCDM techniques for faculty recruitment problems. Modification of the extent analysis fuzzy AHP and the proposed fuzzy comprehensive evaluation method for educational performance, as outlined by Chen et al. (2015), regarding factor and sub-factor weight estimations. Daniawan (2018) integrated the AHP and SAW methodologies to determine the criterion weights and rank alternatives for evaluating lecturer performance. Alaa et al. (2019) developed a Fuzzy Delphi technique and multi-criteria research to assess and rank the English competency of prospective educators. Nandi et al. (2023) assessed the therapeutic alternatives for COVID-19 patients employing generalised hesitant fuzzy MCDM methodologies. Hussain et al. (2019) employed MCDM methods to evaluate the quality of services rendered by the telecommunications sector. Employing statistical techniques, the writers successfully identified the specific parameters affecting the quality of the service rendered. Ultimately, the Fuzzy RASCH method was employed to compute the weights. Furthermore, the TOPSIS method was employed to ascertain the best minimal quantity of lubrication (MQL), while the fuzzy MCDM approach was applied to evaluate service provider selection. Following the establishment of a correlation between machine output and input through response surface methodology (RSM), Sen et al. (2019a) utilised the non-dominated sorting genetic algorithm-II (NSGA-II) to explore potential solutions. Sen et al. (2019b) employed the MCDM method NTOPSIS in conjunction with GEP theory and the non-dominated sorting genetic algorithm-II (NSGA-II) to determine the optimal synergy between MQL and vegetable oil. To achieve optimal site selection, Hussain et al. (2018) utilised a methodology known as Parametric Interval Valued Intuitionistic Fuzzy Number (PIVIFN) to evaluate the alternatives based on the criteria. Ghorui et al. (2021) utilised MCDM methods with fuzzy interval data to assess school teachers’ effectiveness. The authors identified the ideal choice using the fuzzy TOPSIS method. Garg (2016) and Kumar and Garg (2018) enhance a unique generalised scoring function for interval-valued intuitionistic fuzzy sets. The revised scoring system was evaluated and featured four different situations. Ghosh (2021) examined the efficiency of Indian life insurance businesses utilising Data Envelopment Analysis (DEA) and Structural Equation Modelling (SEM). Sarkar (2012) and Sarkar et al. (2011) utilised the Euler-Lagrange theory to ascertain the optimal levels of product reliability and production rate inside a defective manufacturing system, focusing on the problem of economic manufacturing quantity (EMQ). Michaelowa (2007) advocates for applying bivariate and multivariate analysis, supplemented by visual aids, to examine the qualitative impacts of lower- and upper-level education on university students. Chien (2007) applied Kano’s approach to decision-making to enhance student learning satisfaction. The proposed principles and decision-making approach may assist students, instructors, and administrators. Melón et al. (2008) employed the AHP technique, whereby a panel of decision-makers assigns weights to several aspects to identify the most effective educational project. Liao et al. (2023) enhanced the EDAS approach by utilising cumulative prospect theory for multi-attribute group decision-making by incorporating probabilistic hesitant fuzzy information Dhruva et al. (2024b). It is advisable to elucidate the stability and performance characteristics of the CoCoSo ranking technique in the context of unknown preferences. Yan et al. (2024) determined the placement of electric vehicle charging stations using the spherical fuzzy CoCoSo and the CRITIC (Diakoulaki et al., 1995) methodologies.

Many MCDM methods are used for personnel selection and similar selection problems using fuzzy AHP, Fuzzy TOPSIS, IVN-AHP, Pythagorean/intuitionistic fuzzy methods, DEMATEL, FUCOM, CoCoSo, MAIRCA, etc. However, no focused studies are known to have been published that (I) customise the PIVN-AHP+PIVN-TOPSIS pipeline specifically to Finance Officer recruitment and (II) test in practice the PIVN rankings against real post-hire performance in FO roles.

a.

How well do PIVN-AHP and PIVN-TOPSIS discriminate among candidates for a Financial Officer role compared with classical AHP–TOPSIS?

The objectives of this study are: I.

To identify the best financial officer and

Scholars have proposed various MCDM methods to assess, rank, and determine the most viable options. For example, Nikoomaram et al. (2009) applied a fuzzy MCDM approach to evaluate administrative sciences instructors, while Mazumdar et al. (2010) developed MCDM models for appraising teacher performance. Wu et al. (2012) employed a hybrid MCDM model for ranking universities based on performance evaluation. Other studies have focused on specific tools and applications in educational contexts. Huang and Feng (2015) utilised an AHP-TOPSIS approach to evaluate teaching quality in physical education. Mondal and Pramanik (2014) proposed a group decision-making model under a neutrosophic environment for teacher recruitment. Karmaker and Saha (2015) explored recruitment processes for teachers in Bangladesh using MCDM methods.

Additionally, Chen et al. (2015) evaluated teaching performance using a combination of fuzzy AHP and a comprehensive evaluation approach. Daniawan (2018) investigated lecturer performance through AHP and SAW methods, and Alaa et al. (2019) developed a framework combining fuzzy Delphi and TOPSIS to assess English skills in pre-service teachers. The AHP and TOPSIS methods described in this article rely on interval numbers to aid decision-making by identifying optimal solutions. Typically, interval numbers define the fuzziness of parameters, with triangular fuzzy numbers (TFNs) serving as an extension. When evaluating qualitative characteristics, decision-makers benefit from considering ranges rather than precise numerical values. For instance, the PIVN-based method estimates the highest degree of membership as a range rather than a single point. This approach simplifies the evaluation process, saving time and effort by reducing the need for exact numerical guesses.

In contrast to fuzzy numbers, which decision-makers divide into three intervals (low, medium, and high), interval numbers may be divided into three intervals (low, medium and high). PIVN is a more accurate depiction of reality in certain cases than TFN. For instance, in the Triangular Fuzzy Numbers (TFN) (a, b, c), membership begins at “a”, peaks at “b” with 1 and gradually decreases to 0 at “c”. An expert may be unable to determine the value “b” for an attribute when the maximum membership exists, a difficulty that arises in the actual world while hiring an FO. The notion of interval numbers offers a more confident approach to the expert that can end this hesitation. Considering the range [a, c], it becomes clear that the data is consistent throughout the interval. This research aims to recruit FOs based on an exhaustive set of attributes. The PIVN-AHP methodology helped decision-makers calculate the criteria and sub-criteria weights. After that, the authors used the TOPSIS approach with PIVN to rank the FOs. The parametric form of interval numbers (PIVN) gives decision-makers a great deal of leeway to lessen the degree of ambiguity and indeterminacy in their choices. Due to the paucity of literature on PIVN-AHP and PIVN-TOPSIS, we offer a novel idea to aid decision-makers in rapidly identifying viable options with little effort and time expenditure.

Candidate selection for a Financial Officer is a multi-criteria decision-making problem involving criteria that sometimes oppose one another: financial expertise, regulatory compliance, ethics, analytical prowess, and leadership. The traditional AHP and TOPSIS methods strive to structure criteria and rank alternatives; however, they assume crisp or vague judgments, which do not allow for hesitation, ambiguity, or partial knowledge associated with human decisions.

This research uses the Parametric Interval Number Analytic Hierarchy Process (PIVN-AHP) to form the relative weights of the selection criteria. PIVN-AHP allows experts to express their opinions via interval judgments rather than fixed values, thus preserving the uncertainty of the decision maker while reducing subjectivity. The parametric representation of interval numbers allows the performance of systematic sensitivity analysis in order to verify the robustness of the weight estimates under changes in expert perception.

Once the hierarchy has been found, Parametric Interval Number TOPSIS will rank the Financial Officer candidates. The classical TOPSIS does not allow for interval-valued evaluations; hence, the PIVN extension has also been done to model the closeness of each candidate to both ideal and anti-ideal solutions under uncertainty. The hybrid PIVN-AHP and PIVN-TOPSIS application is particularly appropriate for recruitment settings because it (i) integrates both structured weighting of hierarchical criteria and compensatory ranking of alternatives; (ii) tackles vagueness and hesitation in expert judgment explicitly; and (iii) improves the decision robustness by employing parametric sensitivity analysis.

Thus, the methodology chosen herein allows for performance measurement in Financial Officer recruitment using a rigorous, transparent, and uncertainty-resilient, high-stakes, multi-dimensional framework, and subject to incomplete information.

This study contributes to the literature by theoretically fleshing out Multi-Criteria Decision-Making (MCDM) in two important directions. Firstly, it starts with the incorporation of Parametric Interval Number AHP (PIVN-AHP) and Parametric Interval Number TOPSIS (PIVN-TOPSIS) into high-impact, high-stakes, and commercially more remunerative backgrounds of Financial Officer recruitment, where there stands decision uncertainty, evaluation ambiguity, and accountability for performance. While fuzzy, intuitionistic fuzzy, or Pythagorean fuzzy approaches capture half of the uncertainty only, parametric interval number realms parametrically membership, non-membership, and hesitation degrees, thus furnishing a much richer perspective in the guise of vagueness and judgmental inconsistency in expert evaluations. On the second count, the study advances personnel selection research theoretically by showing that the PIVN-based methods provide more robust and discriminative candidate rankings, in addition to being relatively immune to rank reversal and to sensitivity with changes in parameters, traits long perceived to be theoretical downfalls of classical AHP and TOPSIS. Further reinforcing this study is the fact that, by applying the method to Financial Officer recruitment and associating the rankings obtained with post-hire performance indicators, it establishes a new theoretical linkage bridging interval-based neutrosophic decision modelling with empirical validation in strategic human resource management.

This article describes the parametric interval number (PIVN) theory, and AHP-linked TOPSIS with interval uncertainty was used to select the optimal option. a)

The concept of the parametric representation of interval numbers helps to handle the uncertainties of real-life MCDM problems conveniently.

In particular, this study will aid company boards, HR managers, and executive recruitment panels by putting forward a systematic framework for the selection of financial officers, who may face uncertainty; policy makers and regulators, in promoting transparency in the recruitment process; academics and MCDM researchers, in extending parametric interval number methods to the problem under study; and consultants and recruitment agencies, in providing support for real-world decision-making.

We worked in a practice environment. Seven FOs participated in a simulated job interview. Five decision-makers were present: subject matter specialists, finance officials, and upper management. Figure 1 represents the study’s flow chart.

This article continues with the following structure: Mathematical preliminaries with the parametric form of interval numbers are introduced in Section 2, along with some introductory arithmetic operations. The PIVN-AHP and PIVN-TOPSIS methods are outlined in Section 3, along with a basic comparison of Classical-AHP and Classical-TOPSIS. Section 4 shows how PIVN-AHP and PIVN-TOPSIS are used to hire financial officers with numerical representation of calculations. Section 5 examines how various MCDM tools might be compared and contrasted to determine a ranked list. Tables and graphs show the various rankings since Section 6 contains the sensitivity analysis. Sections 7 and 8 present the findings of this research with limitations and suggestions for further research, respectively. Lastly, Section 9 presents conclusions.

To achieve its objectives, a corporation must hire a capable FO. The government and private sector can gain from the proposed study in the following ways: (a)

It improves the current quantitative approach to hiring FOs by using an extensive list of criteria, sub-criteria, and various decision-makers.

An interval number is a subset of the set of real numbers containing a closed interval of real numbers. It is symbolised as: (1) X = [ m l , m r ] = { x : m l ⩽ x ⩽ m r ; x ∈ R } , where m l and m r denote the lowermost value and uppermost value of the interval, respectively, and R is the set of real numbers.

Parametric interval-valued Function: Let us assume an interval [ m l , m r ], where m l , m r > 0.

The parametric interval-valued function for the interval [ m l , m r ] is defined as (2) a ( τ ) = m l 1 − τ m r τ , where τ ∈ [ 0 , 1 ] . Cases: a)

if τ = 0, we get the lower value of the interval;

Let A = [ m l , m r ] and B = [ n l , n r ] be two interval numbers with m l , m r > 0, n l , n r > 0. The Sum of two PIVN is denoted by S = A + B and defined by (3) S ( τ ) = m l 1 − τ m r τ + n l 1 − τ n r τ .

Let A = [ m l , m r ] and B = [ n l , n r ] be two interval numbers with m l , m r > 0 and n l , n r > 0. The Difference between the two PIVN is denoted by D = A − B and defined by (4) D ( τ ) = m l 1 − τ m r τ − n l 1 − τ n r τ .

Let A = [ m l , m r ] and B = [ n l , n r ] be two interval numbers with m l , m r > 0 and n l , n r > 0. The Multiplication of two PIVN is denoted by M = A × B and defined by (5) M ( τ ) = m l 1 − τ m r τ n l 1 − τ n r τ = ( m l n l ) 1 − τ ( m r n r ) τ .

Let A = [ m l , m r ] and B = [ n l , n r ] be two interval numbers with m l , m r > 0 and n l , n r > 0. The Division of two PIVN is denoted by V = A B , B ≠ 0 and defined by (6) V ( τ ) = m l 1 − τ m r τ ÷ n l 1 − τ n r τ = ( m l n l ) 1 − τ ( m r n r ) τ .

Let A ˜ = [ m l 1 − τ m r τ ] and B ˜ = [ n l 1 − τ n r τ ] be two PIVN with weight factors w ˜ 1 and w ˜ 2 respectively such that w ˜ 1 + w ˜ 2 = 1. Then PIVNWAO is (7) PIVNWAO ( A ˜ , B ˜ ) = w ˜ 1 [ m l 1 − τ m r τ ] + w ˜ 2 [ n l 1 − τ n r τ ] , where τ ∈ [ 0 , 1 ] .

Many MCDM practitioners rely on the AHP for weighing criteria. Saaty (1979) introduced this methodology. Both qualitative and quantitative information can benefit from it. AHP is a widely used decision-making tool in various domains, including engineering, economics, sustainability, and management. It provides a systematic, transparent, and mathematically rigorous approach to evaluating and prioritising alternatives based on expert judgment and pairwise comparisons. Ensuring consistency in judgments enhances the reliability and robustness of the decision-making process. Priority weights are obtained from pairwise comparisons using AHP to define the importance of each criterion. The decision maker is guided to the optimal action by the weights collected. In contrast to the possible lack of precision, the conventional AHP employs discrete values for the criteria preference and offers discrete weights. This essay develops the notion of interval numbers and applies it to the issue of hiring finance officers to address the imprecision inherent in qualitative evaluation. An interval represents a range or the extent to which a choice can vary within that range.

The fuzzy AHP is a popular method. It is widely used in decision-making, and numerous published papers discuss its applications. However, we believe this approach’s logic is flawed (Kèyù Zhü, 2014). a)

Specifically, the definition and operational rules of fuzzy numbers in fuzzy AHP contradict the core principles of fuzzy set theory and deviate from the AHP’s basic tenets.

Step 1: Formation of Comparison Matrix

A generalised representation of a comparison matrix in terms of a decision expert’s Parametric form of an interval number. Whereas interval numbers include infinite values rather than a crisp or accurate value, general methods cannot assess them. The interval numbers are chosen here as the numbers within the two values are also included, providing much scope to the decision-makers. Let the decision-makers express their judgment in the form A n × n where C k t = [ m l k t 1 − τ m r k t τ ], where k = 1 , 2 , 3 , … , n; t = 1 , 2 , 3 , … , n and τ ∈ [ 0 , 1 ] denotes the comparative preference of criteria. In the matrix C i i = 1 when k = t.

Comparison matrix of criteria in terms of PIVN for different values of τ [ m l 11 1 − τ m r 11 τ ] [ m l 12 1 − τ m r 12 τ ] ⋯ [ m l 1 n 1 − τ m r 1 n τ ] [ m l 21 1 − τ m r 21 τ ] [ m l 22 1 − τ m r 22 τ ] ⋯ [ m l 2 n 1 − τ m r 2 n τ ] ⋮ ⋮ ⋱ ⋮ [ m l n 1 1 − τ m r n 1 τ ] [ m l n 2 1 − τ m r n 2 τ ] ⋯ [ m l n n 1 − τ m r n n τ ] .

Step 2: Crispification of PIVN for all τ ∈ [ 0 , 1 ]

It is a set of numbers, one for each τ.

Step 3: Normalisation of Crisp Matrix (8) N k t = h k t ∑ k = 1 n h k t , where k = 1 , 2 , 3 , … , n ; t = 1 , 2 , 3 , … , n .

Step 4: Estimation of Criteria Weights (9) w t = ∏ t = 1 n N k t n ∑ k = 1 n ( ∏ t = 1 n N k t n ) .

Step 5: Assess the consistency of pairwise comparisons using the Consistency Index ( C I) and Consistency Ratio ( C R). Eq. (10) helps to calculate the C I. (10) C I = λ max − n n − 1 . The λ max is the largest eigenvalue of the comparison matrix, and n is the matrix size.

Saaty introduced the Random Consistency Index ( R I) of the comparison matrix to validate the consistency of the comparisons. It varies based on the matrix size. There n is the matrix size.

Eq. (11) helps to calculate C R: (11) C R = C I R I . If C R < 0.10, the consistency is acceptable.

If C R ⩾ 0.10, the judgments may be inconsistent, and the comparisons should be revised.

Step 6: Sub-Criteria and Global Weights

A similar method follows for the sub-criteria, and finally, the global weights are obtained by multiplying the criterion weight by the respective sub-criterion weight.

When using MCDM to prioritise options, the TOPSIS technique is commonly employed. The TOPSIS algorithm is based on the idea that the optimal choice is the one that is the most like the Positive Ideal Solution (PIS) and the most unlike the Negative Ideal Solution (NIS). To aid decision-makers in picking the best option, the authors (Kelemenis and Askounis, 2010) created a novel TOPSIS-based multi-criterion technique. Chen et al. (2009) explained the benefits of TOPSIS as follows: a)

It makes sense and is easy to understand.

Step 1: Formation of the Decision Matrix

Step 2: Calculation of Normalised Decision Matrix (12) x ‾ i j = x i j ∑ i = 1 n x i j 2 . Step 3: Calculation of Weighted Normalised Decision Matrix (13) x ˆ i j = x ‾ i j w j . Step 4: Determination of the Positive Ideal Solution and Negative Ideal Solution (14) A + = ( x ˆ 1 + , x ˆ 2 + , … , x ˆ n + ) = { ( max i x ˆ i j | j ∈ P ) , ( min i x ˆ i j | j ∈ C ) } , (15) A − = ( x ˆ 1 − , x ˆ 2 − , … , x ˆ n − ) = { ( min i x ˆ i j | j ∈ P ) , ( max i x ˆ i j | j ∈ C ) } , where B represents Profit Type Criteria, and NB denotes Cost Type Criteria.

Step 5: Calculation of the Euclidean Distance from the Ideal Best and Ideal Worst value: (16) D i + = ∑ j = 1 n ( x ˆ i j − x j + ) 2 ; i = 1 , 2 , … , m ; j = 1 , 2 , … , n , (17) D i − = ∑ j = 1 n ( x ˆ i j − x j − ) 2 ; i = 1 , 2 , … , m ; j = 1 , 2 , … , n .

Step 6: Calculation of Performance Score and Ranks (18) K i = S i − S i + + S i − . A higher value of K i represents better alternatives.

Step 1: Formation of a comparison matrix

The decision maker uses different linguistic terms for different criteria and sub-criteria, and Table 4 denotes the comparison of different criteria in linguistic terms. The pairwise comparison matrix is prepared based on the perception of four authors who contributed to this paper.

Step 2: Crispification of PIVN for all τ ∈ [ 0 , 1 ] (see Table 5)

Conversion of linguistic terms to a ’τ’ value decided by a decision expert. The arithmetic mean method aggregates different decision-makers assigned ’τ’ values.

Step 3: Normalisation of the crisp matrix (see Table 6)

Step 4: Estimation of Criteria Weights

Table 7 represents the criteria weights obtained by the PIVN-AHP methodology.

Table 7 shows that criterion C₅ scores the maximum weight of 0.331 (rounded off), followed by C₂, C₆, C₁, C₄, C₃ and C₇.

Step 5: Consistency Check

For this step, we must take the same crisped pairwise comparison matrix, which is not normalised. Table 8 shows values in each matrix junction after being multiplied by the criteria weights, weighted sum value (WSV), and the ratio of WSV and CW.

Now, λ max = 8.906 + 8.725 + 7.659 + 7.486 + 7.193 + 7.628 + 7.021 7 = 54.418 7 = 7.774.

Consistency Index (CI) C I = λ m a x − n n − 1 = 7.774 − 7 7 − 1 = 0.774 6 = 0.129; in this case, n = 7 as we have 7 criteria (see Table 9).

Consistency Ratio (CR) C R = C I R I = 0.129 1.32 = 0.098 < 0.10.

0.10 is the standard value, as AHP allows up to 10% inconsistency. If the inconsistency exceeds 10%, the pairwise comparison matrix must be reconsidered.

The pairwise comparison matrix is reasonably consistent. Now, we can continue the decision-making process for further calculations using the criteria weights 0.139, 0.149, 0.079, 0.111, 0.331, 0.147, and 0.047, respectively, for criteria C₁, C₂, C₃, C₄, C₅, C₆, and C₇.

Step 6: Sub-Criteria and Global Weights Calculations

In the same way, sub-criteria matrices are built, and respective weights are gained. Eventually, the global weights are acquired by combining the criteria weight with their respective sub-criteria weight.

Table 10 refers to the criteria, sub-criteria, and global weights necessary to rank the FOs in this study.

Proposed Methodology: In this approach, PIVN is utilised to rank the options and illustrate the numerical grading of the alternatives concerning the criteria. The decision-makers considered in this study are a group of 3 administrative officers o = { 1 , 2 , … , γ } and 2 external experts v = { 1 , 2 , … , ρ }.

Each voter uses a numeric scale to rate their preferences, and the Arithmetic Mean Method is used to get the aggregated results. This approach is better because it gives decision-makers greater leeway in assigning weights to the various choices based on the criteria. The options are ranked using the weight assigned to each sub-criterion.

The following Table 11 represents the linguistic preference of decision-makers in terms of PIVN.

Note 1. The different PIVN values for the TOPSIS methodology are obtained for different ‘τ’. Decision-makers rate the alternatives concerning the sub-criteria. Their aggregation is used to determine the final decision matrix. The ‘τ’ chosen by the decision-makers is wholly based on their expertise.

Step 1: Formation of the Decision Matrix (see Table 12)

Step 2: Calculation of Normalised Decision Matrix (see Table 13)

Step 3: Calculation of the Weighted Normalised Decision Matrix (see Table 14)

Step 4: Determination of the Ideal Best and Ideal Worst Value (see Table 15)

Step 5: Calculation of the Euclidean Distance from the Ideal Positive and Ideal Negative Solutions (see Table 16)

Step 6: Calculation of the Performance Score & Ranks (see Table 17)

The relative closeness value of FO F 2 is maximum at 0.774. Thus, F 2 ranks first followed by the F 1 , F 3 , F 5 , F 6 , F 4 , F 7 . Therefore, F 2 ≻ F 1 ≻ F 3 ≻ F 5 ≻ F 6 ≻ F 4 ≻ F 7 (see Fig. 3).