
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12451-6
10.1016/j.heliyon.2024.e36420
e36420
Research Article
A spherical fuzzy objective weighting framework for factors affecting bidding decisions for construction projects: A case study of Taiwan
Chen Yih-Tzoo a
Dao Thi-Hien I112109110@nkust.edu.tw
b⁎
Chiu I-Hsin a
a Department of Construction Engineering, National Kaohsiung University of Science and Technology, Kaohsiung, 807618, Taiwan
b Ph.D. Program in Engineering Science and Technology, College of Engineering, National Kaohsiung University of Science and Technology, Kaohsiung, 807618, Taiwan
⁎ Corresponding author. I112109110@nkust.edu.tw
17 8 2024
30 8 2024
17 8 2024
10 16 e3642031 12 2023
15 8 2024
15 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
In response to the intensely competitive bidding environment in Taiwan's construction industry, this study innovates with a spherical fuzzy extension of the modified CRITIC method (SFmCRITIC), a pioneering approach crafted to refine the decision-making process. The research centers on the pivotal role of accurately weighting various significant factors, such as financial stability, market insight, and specialized expertise, to optimize bid success rates. The SFmCRITIC method stands out by quantitatively incorporating the uncertainty and subjectivity typically encountered in bid evaluation. The findings from this methodological application demonstrate a clear hierarchy of influencing factors, offering a nuanced view of strategic priorities for construction firms. This hierarchical understanding not only aids firms in targeted resource distribution and strategic development but also holds promise for adaptation in diverse industry contexts beyond Taiwan, potentially revolutionizing bidding strategies at a global scale.

Keywords

Bidding
Construction industry
Fuzzy theory
Multi-criteria decision-making
Objective weighting method
Project evaluation
Spherical fuzzy set
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pmc1 Introduction

The competitive nature of construction projects underscores the crucial role that bidding decisions play in the success of construction companies. These decisions, which involve selecting the appropriate projects to bid for, are pivotal in ensuring a firm's sustainability and profitability. A strategic approach to bidding not only influences the odds of winning a project but also determines the potential for future business growth [1]. In this intricate process, the identification and consideration of various factors that affect bidding decisions are of paramount importance. In the context of construction bidding, numerous factors influence the decisions made by contractors. These factors range from economic conditions, project size, and complexity to the availability of resources and the competitive environment. A study by Binshakir et al. showed that economic indicators, such as interest rates and inflation, have a statistically significant impact on the bidding strategy of Taiwanese construction firms. The interplay between these factors determines the feasibility and attractiveness of construction projects [2]. Recognizing their influence is a critical step toward developing a nuanced bidding strategy that aligns with a company's capabilities and business goals.

The accurate weighting of these factors is indispensable for making informed bidding decisions. This is especially relevant in diverse and dynamic construction markets, such as Taiwan's, where rapid economic development and unique local conditions introduce additional complexity to the bidding process. Yu et al. highlighted that successful contractors in Taiwan's construction sector effectively prioritize and weigh critical bidding factors [3]. Weighting allows firms to prioritize the factors most critical to their unique situation, thereby refining the decision-making process and enhancing the chances of successful bids.

The application of multiple criteria decision-making (MCDM) approaches presents a structured framework for dealing with the complexities inherent in bidding decisions. MCDM methodologies enable decision-makers to evaluate various factors systematically, considering both quantitative and qualitative data [4]. Such approaches facilitate a more comprehensive analysis, leading to more robust and strategic bidding decisions [5]. Among the MCDM methodologies, the Criteria Importance Through Intercriteria Correlation (CRITIC) method stands out due to its ability to discern the objective weight of each factor based on the degree of contrast and conflict between criteria [6]. This method provides a distinct advantage, as it does not solely rely on the subjective judgments of decision-makers, thus introducing an element of objectivity into the process of factor weighting [7]. Fuzzy theory, pioneered by Lotfi A. Zadeh in the 1960s, provides a mathematical framework for addressing uncertainty, vagueness, and imprecision in decision-making [8]. At its core, fuzzy theory employs "fuzzy sets" to represent concepts that are not easily defined with crisp boundaries [9]. Unlike traditional sets, fuzzy sets allow for degrees of membership between 0 and 1, expressing the extent to which an element belongs to the set. Fuzzy numbers, a fundamental component of fuzzy theory, represent numerical values with associated degrees of fuzziness, enabling the representation of imprecise data and preferences in decision-making contexts [10]. When integrated with MCDM methods, fuzzy theory enhances the handling of complex decision problems [11]. In this integration, fuzzy theory enables the representation of uncertainty and imprecision in criteria and preferences used in MCDM. Decision matrices, a standard tool in MCDM, can incorporate fuzzy numbers to model imprecise evaluations. This integration empowers decision-makers to make more informed and robust choices when faced with intricate decision problems characterized by vague or uncertain information, bridging the gap between real-world complexity and mathematical decision-support tools [12,13].

In an industry where competitive bidding is a cornerstone of success, the construction sector in Taiwan presents a particularly complex tableau of challenges and considerations. This research aims to unveil a groundbreaking methodological approach, the spherical fuzzy extension of the modified CRITIC (SFmCRITIC) method, to finesse the art of bidding. While traditional decision-making processes in construction rely on the subjective judgment of myriad factors, from economic conditions to project specifics, our method stands apart by injecting an unparalleled level of precision and objectivity into the equation. The SFmCRITIC method, with its theoretical and practical advancements, is set to mark a paradigm shift in the decision-making landscape. It innovatively marries the robustness of the CRITIC method with the nuanced handling of uncertainty intrinsic to fuzzy theory. This hybridization is not merely a theoretical exercise; it is an actionable framework that provides decision-makers with a tool that is finely tuned to the complexities of Taiwan's vibrant construction market. Unlike its predecessors, the SFmCRITIC method transcends traditional approaches by embracing the vagueness and imprecision that characterize real-world decision-making, offering a more informed and resilient basis for bidding strategies.

The introduction of this method represents a novel stride in the construction industry's bidding processes, significantly enhancing the weighting of factors with a level of meticulousness yet to be seen. By leveraging the spherical fuzzy sets' ability to express degrees of membership, the SFmCRITIC method enriches the MCDM toolbox with a solution that is better aligned with the dynamic and often uncertain environment of construction projects. This novel approach is poised to equip industry practitioners with a powerful ally in their quest for sustainability and growth, setting a new standard for bidding strategies in complex markets.

The structure of this article unfolds as follows: After the introduction, a comprehensive literature review will delve into the factors influencing bidding decisions and the MCDM approaches. Subsequently, the methodology section will detail the SFmCRITIC method employed in this study. The case study section will apply the framework to the Taiwanese construction industry, followed by a conclusion that synthesizes our findings and proposes directions for future research.

2 Literature review

The bid/no-bid decision-making process is essential for construction companies as it affects project success and organizational sustainability. This literature review synthesizes various studies focusing on decision-making models, critical factors influencing bidding, behavioral and performance impacts, risk perception and assessment, predictive tools and approaches, and sector-specific and regional studies.

In their extensive survey, Patel et al. assessed the importance of various factors affecting bidding strategy in residential construction [14]. The study stands out for its breadth, identifying 49 factors and ranking them using the Relative Importance Index and Importance Index methods. This comprehensive identification of factors provides a foundation for contractors to develop informed bidding strategies. Yan and colleagues delved into group bidding decisions, identifying 20 factors influencing this process and ultimately determining 14 critical factors through extensive surveys and interviews [15]. Their work underscores the significance of risk perception and team preferences in collective decision-making, offering a categorization that serves as a strategic tool for construction firms. The study by Wibowo et al. is particularly informative in linking bidding strategies with project and company performance [16]. Using structural equation modeling partial least squares (SEM-PLS) to analyze questionnaire and interview data offered a sophisticated understanding of how strategic decisions at the bidding phase have far-reaching implications for overall company success, mediated through project performance. The research by Bageis et al. investigated how internal and external factors manifest in behavioral differences among contractors during the bid/no-bid decision process [17]. Utilizing quantitative methods, the study provided evidence of the significant and distinct roles these factors play, revealing insights into contractor behavior and decision-making processes in the Saudi Arabian context.

Concerning risk, the research by Laryea and Hughes offers a critical lens on the practical application of risk in the bidding process [18]. Their participant observation of United Kingdom construction firms reveals a disparity between theoretical models and actual practices, highlighting the need for models that can capture the commercial realities faced by contractors. Chen and colleagues' investigation into risk assessment practices among Chinese contractors illuminates the psychological underpinnings of risk perception [19]. Their findings point to the complex interplay between past outcomes and risk propensity, significantly shaping bid/no-bid decisions and emphasizing the subjective nature of risk evaluation in the construction industry. On the other hand, Chisala's introduction of a weighted scoring model provided a predictive tool with high accuracy for bidding decisions [20]. The validation of this model on real-life projects presents it as a viable tool for contractors looking to streamline their decision-making processes. Kalan and Ozbek's development of an AHP-based decision-making tool demonstrates the applicability of structured decision-making methodologies in the construction industry. Their illustrative examples based on hypothetical case studies offer a tangible guide for contractors in applying these tools in practice [21]. In other efforts in several specific regions, Dissanayake et al. evaluated critical factors for the bid/no-bid decision in Sri Lankan construction projects, emphasizing traditional project management factors [22]. Mahamid identified the top factors influencing bid/no-bid decisions from contractors' perspectives in the West Bank, Palestine [23]. Binshakir et al. (2023) explored factors impacting the bidding decision in sustainable construction projects in the UAE, indicating a significant difference from conventional construction bidding [2].

Recent literature on renewable energy emphasizes the importance of comprehensive approaches to address challenges such as material scarcity and waste generation. Pan and Hashemizadeh (2023) advocate for a circular economy approach to evaluate renewable energy resources, integrating methodologies like FBWM and eMEREC to assess the sustainability throughout the lifecycle of these resources [24]. They also introduce a novel compromise solution methodology to adapt the framework to different regional conditions, particularly within Belt and Road Initiative countries. On the practical side of renewable energy execution, Hashemizadeh, Ju, and Dong (2019) present a model that uses Geographic Information Systems (GIS) and MCDM)to optimize site selection for solar photovoltaic projects [25]. This model effectively prioritizes various factors to identify optimal locations, emphasizing the use of Best–Worst Method for criteria prioritization. Furthermore, Hashemizadeh et al. (2021) explore investment risks in renewable energies, identifying risk factors and using methods like TODIM and fuzzy-analytic network processes to rank different energy sources under uncertainty conditions. This approach aids policymakers and investors in assessing investment projects more reliably [26]. Additionally, Chia-Nan Wang and colleagues (2024) apply MCDM frameworks to identify optimal locations for wave energy sites in Chile, demonstrating how technical, environmental, and socio-economic assessments can guide strategic investment and policy-making [27]. Similarly, their study in 2023 applies these frameworks to South African wave energy sites, underscoring the global applicability and effectiveness of MCDM in renewable energy site selection [28].

In the realm of decision-making models and frameworks for construction bidding, various academic contributions have paved the way for more nuanced and sophisticated approaches. El-Mashaleh proposed using Data Envelopment Analysis (DEA), offering a flexible tool that relies on an organization's historical data for bid comparison, underlining the need for a robust historical database to fully leverage this method [29]. Cheng et al., on the other hand, offered a blend of fuzzy preference relations and cumulative prospect theory with multi-criteria decision analysis in their Multi-Criteria Prospect Model for Bidding Decision (BD-MCPM) model. This notable contribution brings psychological factors into the strategic decision-making arena despite the challenges in assessing competitors' utility functions [30]. Meanwhile, Borges et al. introduced a fuzzy ELECTRE TRI-C (TRI-C variant of Elimination and Choice Expressing Reality) procedure, which stands out for its innovative use of fuzzy set theory to assist contractors in gauging the attractiveness of potential projects, providing a systematic way to filter out less appealing bids [31]. On a different note, Zaqout et al. (2022) developed a Mamdani-type Fuzzy Inference System to streamline the determination of markups, although their model's validation was limited due to a small sample size, reflecting the unforeseen impact of the COVID-19 pandemic [32]. Collectively, these studies reflect a trend towards incorporating more complexity and rigor into the bid decision-making process, acknowledging the diverse factors and uncertainties that construction companies face.

The existing literature highlights several decision-making models for construction bidding, yet there appears to be a research gap in specifically addressing the weighting of critical factors using a fuzzy extension of the CRITIC method for the Taiwanese market. Previous studies have not combined fuzzy logic with the CRITIC method's objective weighting approach, which is particularly relevant given the uncertain and subjective nature of bidding environments. The proposed research seeks to address this gap by developing a FmCRITIC method. This method aims to enhance the precision of bidding decisions by incorporating the inherent ambiguity in human judgment, a feature not yet tailored to the Taiwanese construction industry's distinct context. Theoretical and practical contributions are expected through the FmCRITIC method, which will provide a refined tool for more strategic bidding in Taiwan's unique construction sector.

3 Methodology

3.1 Fuzzy sets and spherical fuzzy

To address the inherent uncertainty in human judgments, linguistic variables are regarded as more effective descriptors than precise numerical values. To quantify these linguistic terms, fuzzy theories have been introduced, developed, and widely adopted in the realm of decision-making. In 2019, Gündoğdu and Kahraman introduced spherical fuzzy sets (SFS) through the amalgamation of Pythagorean fuzzy sets and neutrosophic fuzzy sets [33]. The SFN is consequently defined using three key parameters encompassing membership, non-membership, and hesitancy. This formulation not only enables decision-makers to express their hesitancy but also provides them with a broader range of preferences for their judgments. The definition of SFN and its associated operators is presented as follows:Definition 1 Let's consider the universe of discourse denoted as T, and a spherical fuzzy number represented as A˜ in T. This spherical fuzzy number is defined by three essential parameters: the degree of hesitancy (δA˜), membership (αA˜), and non-membership (βA˜) as Eqs. (1), (2).

(1) A˜={⟨t,(αA˜(t),βA˜(t),δA˜(t))|t∈T}

whereαA˜:T→[0,1],βA˜:T→[0,1],δA˜:T→[0,1]

And(2) 0≤αA˜2(t)+βA˜2(t)+δA˜2(t)≤1∀t∈T

Definition 2 Let's consider two SFNs A˜=(αA˜,βA˜,δA˜) and B˜=(αB˜,βB˜,δB˜) in the universes of discourse T1 and T2. Their addition, multiplication, and power operators can be defined as Eqs. (3), (4), (5), (6), (7), (8), (9), (10), (11), (12):

(3) A˜⊕B˜={αA˜2+αB˜2−αA˜2αB˜2,βA˜βB˜,(1−αB˜2)δA˜2+(1−αA˜2)δB˜2−δA˜2δB˜2}

(4) A˜⊗B˜={αA˜αB˜,βA˜2+βB˜2−βA˜2βB˜2,(1−βB˜2)δA˜2+(1−βA˜2)δB˜2−δA˜2δB˜2}

(5) εA˜={1−(1−αA˜2)ε,βA˜ε,(1−αA˜2)ε−(1−αA˜2−δA˜2)ε},ε>0

(6) A˜ε={αA˜ε,1−(1−βA˜2)ε,(1−βA˜2)ε−(1−βA˜2−δA˜2)ε},ε>0

(7) A˜⊕B˜=B˜⊕A˜

(8) A˜⊗B˜=B˜⊗A˜

(9) ε(A˜⊕B˜)=εA˜⊕εB˜,ε>0

(10) ε1A˜⊕ε2A˜=(ε1+ε2)A˜,ε1>0,ε2>0

(11) (A˜⊗B˜)ε=A˜ε⊗B˜ε,ε>0

(12) A˜ε1⊗A˜ε2=A˜ε1+ε2,ε1>0,ε2>0

Definition 3 The score function and accuracy function of SFS are defined as follows for defuzzification and comparison according to Eq. (13), (14), (15).

(13) A˜<B˜ifandonlyifi.Score(A˜)<Score(B˜)orii.Score(A˜)=Score(B˜)andAccuracy(A˜)<Accuracy(B˜)

where(14) Score(A˜)=(αA˜−δA˜)2+(βA˜−δA˜)2

(15) Accuracy(A˜)=αA˜2+βA˜2+δA˜2

3.2 Spherical fuzzy modified CRITIC method

To harness the benefits of both the CRITIC method and Spherical Fuzzy Numbers, this study has undertaken an ambitious endeavor to not only integrate these two methodologies but also adapt and tailor them to better align with the specific requirements and objectives of the new application problem under consideration. The integration of the CRITIC method, known for its ability to assess the importance of criteria through intercriteria correlation, and Spherical Fuzzy Numbers, which offer a versatile framework for handling uncertainty and ambiguity in decision-making, represents a novel and innovative approach. By harmonizing these two techniques, the study aims to capitalize on their respective strengths and create a unified decision-making framework that can address the complexities and intricacies of the given problem more effectively. The proposed approach is delineated through a series of meticulously following steps.Step 1 Identification of Experts and Factors

In this initial phase, a careful selection process is undertaken to assemble a group of experts, denoted as k=1,2,…,n, who possess extensive experience and a high degree of expertise within the relevant research field. Subsequently, the factors influencing bidding decisions, represented as j=1,2,…,m, are delineated based on a synthesis of prior research findings and valuable input from these selected experts.Step 2 Expert Prioritization

Recognizing that experts may vary in terms of their experience, knowledge, and level of expertise, it becomes imperative to assign appropriate weights to each expert to account for these differences. To achieve this, the expertise of each expert, denoted as the k th expert, is expressed as an SFN, formulated as A˜k=(αk,βk,δk). These fuzzy representations are utilized to calculate the crisp weights for each expert, as detailed in Eq. (16), (17), (18) [34]. This weight assignment process ensures that the contributions of each expert are appropriately weighted, thus enhancing the accuracy and reliability of the decision-making process.(16) ϑk=1−((1−αk2)+βk2+δk2)/3∑k(1−((1−αk2)+βk2+γk2)/3)

where(17) ∑k=1Kϑk=1

and(18) 0≤αk2+βk2+δk2≤1

Step 3 Construction of Spherical Fuzzy Decision Matrix

In this pivotal step, the assembled group of experts offers their invaluable linguistic evaluations regarding the significance of the various factors in the decision-making process. These linguistic assessments are subsequently transformed into corresponding SFNs, as exemplified in Table 1. The spherical fuzzy decision matrix (X˜) is represented as Eq. (19), (20).(19) X˜=[x˜jk]m×n

Table 1 Linguistics term and corresponding SFN.

Table 1Linguistics term	Notation	Corresponding SFN	
No important	NI	(0.270, 0.730, 0.270)	
Less important	LI	(0.385, 0.615, 0.385)	
Medium important	MI	(0.500, 0.500, 0.500)	
Important	I	(0.615, 0.385, 0.385)	
Very Important	VI	(0.730, 0.270, 0.270)	

Where(20) x˜jk=(αjk,βjk,δjk)

Step 4 Identification of Crisp Decision Matrix

In this step, the transformation from Spherical Fuzzy Decision Matrix (as obtained in the previous step) to Crisp Decision Matrix, as presented in Eq. (21), (22), is achieved through a defuzzification process, as defined by Equation (14). This process serves to convert the fuzzy and uncertain information contained within the Spherical Fuzzy Numbers into crisp and deterministic values, facilitating the subsequent stages of analysis and decision-making.(21) Y=[yjk]m×n

where(22) yjk=(αx˜jk−δx˜jk)2+(βx˜jk−δx˜jk)2

Step 5 Construction of Weighted Decision Matrix

Moving forward, the influence of each expert's assigned weights, as determined in Step 2, is incorporated into the decision-making process. This integration is executed through the application of Eq. (23), (24), which serves to combine the individual assessments provided by the experts, considering their respective expertise levels and contributions.(23) Z=[zjk]m×n

where(24) zjk=yjkϑk,j=1…m

Step 6 Calculation of Factors' Mean

Then, the mean (average) value within each factor is determined as part of the decision-making process. This calculation is carried out following the guidelines outlined in Eq. (25), which offers a method for computing the central tendency or average value of the data within each factor.(25) b‾j=∑k=1nbjkn

Step 7 Calculation of Factors' Standard Deviation

In this step, the standard deviation of each factor is computed as part of the decision-making process. This calculation is performed in accordance with Eq. (26), which provides a method for assessing the variability or dispersion of values within each factor.(26) σj=∑k=1n|bjk−b‾j|2n

Step 8 Calculation of Factors' linear correlation coefficient

For each factor, the importance vector (vj) is defined in accordance with Eq. (27). For every combination of the j th factor and the l th factor, the linear correlation coefficient is computed. The outcome of these computations yields the correlation coefficient matrix (R), which is generated using Eq. (28).(27) vj=[z1j,z2j,…,zkj,…,znj]

(28) R=[rjl]m×m

where rjl represents the linear correlation coefficient between the vectors vj and vl.Step 9 Determination of Factors' Information Content

The amount of information content associated with the j th factor is estimated using Eq. (29). This equation provides a method for quantifying the information content or informational significance of the j th actor within the context of the decision-making problem.(29) Ij=σj∑l=1m(1−rjl),j=1…m

Step 10 Determination of Indicator Weights

In this crucial step, the weights assigned to each factor (wj) are established as per Eq. (30).(30) wj=Ijb‾j∑j=1mIjb‾j,j=1…m

3.3 The proposed spherical fuzzy objective weighting framework

The proposed spherical fuzzy objective weighting framework is a novel methodology tailored for improving bidding decision accuracy in the construction industry as shown in Fig. 1. Initially, a comprehensive literature review coupled with an expert survey informs the problem definition stage, pinpointing critical factors for bidding. The subsequent data collection and transformation stage leverages these identified factors, where linguistic importance evaluations are rendered into SFNs. These SFNs are then transformed, and a defuzzification process is applied, yielding crisp values that facilitate the computation of each factor's standard deviation, linear correlation coefficients, and mean importance. Following this, the SFmCRITIC analysis stage takes center stage, where the information content of each factor—indicative of their distinctiveness—is calculated from the standard deviation and correlation coefficients. This analysis culminates in the estimation of weights for each factor, integrating the calculated mean importance and information content. The framework's primary goal is to output a set of objective weights for the factors affecting bidding decisions.Fig. 1 The proposed Spherical Fuzzy Objective Weighting Framework.

Fig. 1

4 Case study

4.1 Factor weighting for bidding decisions in Taiwan construction industry

In a compelling case study within the dynamic landscape of the Taiwan construction industry, a comprehensive survey was undertaken to tap into the wealth of expertise held by 134 seasoned professionals. These experts, comprising project managers, engineering supervisors, and procurement managers, brought to the table over a decade of hands-on work experience in their respective domains. The research aimed to harness the collective wisdom and insights of these industry stalwarts, offering a deep dive into their perspectives on critical facets of construction. Fig. 2, a visual representation included in the study, outlines the composition structure of the participating experts, providing a glimpse into the diversity and richness of their backgrounds. Due to the substantial size of the expert pool participating in the survey, coupled with the observation of relatively consistent expertise levels among the participants, a pragmatic approach was adopted. In this approach, equal weighting was assigned to all experts, recognizing each expert's valuable contribution to the study as equally significant.Fig. 2 The structure of the experts.

Fig. 2

Drawing upon the collective insights and expertise of industry experts in Taiwan, a comprehensive survey has identified a total of twenty-three crucial factors, as shown in Table 2, that are widely acknowledged to exert a significant influence on the overall success of bidding decisions. The first factor, denoted as F1, pertains to the current location of the contractor company. This geographical aspect can significantly impact logistics, accessibility to the project site, and the overall cost framework. Factor F2, the identity and reputation of the client, is critical as it reflects the client's credibility and the likelihood of smooth project execution. Availability of resources constitutes the next three factors: F3, F4, and F5, representing the accessibility of required labor, materials, and equipment, respectively. These are foundational for project initiation and execution. The current workload, identified as F6, influences a contractor's capacity to take on new projects without compromising quality. The availability of required cash, noted as F7, is essential for ensuring adequate cash flow throughout the project lifecycle. F8 highlights the importance of experience in similar projects, which can greatly influence the bidding strategy and project delivery. Factor F9, the profit potential of previous similar projects, serves as a historical indicator of financial viability. The complexity of the project, F10, dictates the degree of planning and risk management required, while the duration of the project, F11, affects scheduling and resource allocation. Factor F12, the probable number and identity of competitors, shapes the competitive landscape and can influence bid pricing. The project type, F13, can determine the specialized skills and approaches needed. Factor F14 considers the identity of the consultant and designer, which can influence the project's technical demands and execution. Expected profitability, F15, directly relates to the financial desirability of the project. Payment terms, F16, can impact cash flow and financial planning, while regulations for adjusting prices based on the consumer price index, F17, offer a measure of financial protection against inflation. The project's location and accessibility, F18, influence logistics and operational efficiency, and the size of the project, F19, dictates the scale of management and labor required. The complexity of payment terms, F20, can affect the ease of financial dealings and the potential for disputes. The promptness of the client in the payment process, F21, is crucial for maintaining cash flow. Completeness of tender documents, F22, ensures clarity in expectations and requirements, while the time available for tendering preparation, F23, affects the quality and thoroughness of the bid.Table 2 Crucial factors in bidding decisions.

Table 2Notation	Factors	Classification	
F1	Current location of the contractor company	Logistical	
F2	Identify and Reputation of the client	Social	
F3	Availability of required labors	Technical	
F4	Availability of materials	Technical	
F5	Availability of equipment	Technical	
F6	Current workload	Operational	
F7	Availability of required cash	Economic	
F8	Experience in similar project	Technical	
F9	The profit potential of previous similar project	Economic	
F10	Complexity of the project	Technical	
F11	Duration of project	Operational	
F12	Probable number and identity of competitors	Market	
F13	Project type	Technical	
F14	Identity of consultant and designer	Social	
F15	Expected profitability	Economic	
F16	Payment terms	Economic	
F17	Regulations for adjusting prices based on the consumer price index	Economic	
F18	Location of project and accessibility	Logistical	
F19	Size of project	Operational	
F20	The complexity of payment terms	Economic	
F21	Promptness of client in the payment process	Economic	
F22	Completeness of documents	Legal/Procedural	
F23	Time available for tendering preparation	Operational	

In the subsequent phase of the study, the outcomes derived from the survey, specifically the linguistic assessments of the importance assigned to various factors, underwent a comprehensive synthesis and transformation process. This intricate procedure aimed to convert these linguistic evaluations into corresponding SFNs, as shown in Table 1. The transformation results are presented in Table A1 in the Appendix. To create the crisp decision matrix, a crucial defuzzification process was executed, following the principles outlined in Eq. (22). The results of this defuzzification procedure are thoughtfully compiled and presented in Table A2. For each factor under examination, two key statistical parameters, namely the mean and standard deviation, were calculated using Eq. (25) and Eq. (26), respectively. These calculations are instrumental in summarizing the central tendency and variability associated with each factor, offering valuable insights into their characteristics and distribution within the dataset.

As illustrated in Fig. 3, the availability of required labor, with an importance mean of 0.75, highlights it as the paramount concern, likely due to its direct impact on the feasibility and timeline of projects. Close behind, the availability of materials (F4) and the identity and reputation of the client (F2), with importance means of 0.67 and 0.65, respectively, underscore the reliance on quality supplies and reliable partnerships. The factors concerning financial aspects of projects, such as expected profitability (F15), payment terms (F16), the complexity of payment terms (F20), and the promptness of client payments (F21), all have means exceeding 0.58, reflecting a significant emphasis on economic considerations. These numbers indicate that financial terms and conditions play almost as pivotal a role as the more tangible resources like labor and materials. Moderately ranked factors, with means around the mid-point of the scale, include the availability of equipment (F5), current workload (F6), and available cash (F7), suggesting these are also necessary but perhaps more manageable concerns. Project complexity (F10) and regulations for price adjustments (F17) hover at a 0.54 mean, pointing to a moderate influence on decision-making, possibly due to their indirect effects on project execution and profitability. Interestingly, experience in similar projects (F8) and the probable number of competitors (F12) are deemed less influential, with the lowest importance means of 0.36 and 0.33. This could indicate a trend towards focusing on present capabilities and project specifics over historical performance and competition.Fig. 3 The factors' mean.

Fig. 3

In comparison with the average baseline, factors such as F3 (Availability of required labor), F4 (Availability of materials), F2 (Identity and Reputation of the client), F15 (Expected profitability), F16 (Payment terms), F20 (The complexity of payment terms), and F21 (Promptness of client in payments process) stand out with importance means above the average, indicating that they are considered more critical than other factors in the decision-making process for bidding. Particularly, F3, with the highest importance mean of 0.75, far exceeds the average, suggesting that the availability of labor is a highly influential factor, likely due to its direct impact on project execution and timelines. Conversely, factors like F8 (Experience in similar projects), F12 (Probable number and identity of competitors), and F13 (Project type), which have importance below the average, are deemed less influential in bidding decisions within the Taiwanese construction industry context. These lower values imply that while these factors are acknowledged, they may not be pivotal in the decision to pursue a bid. The factors that hover around the average mark, such as F5 (Availability of equipment), F6 (Current workload), and F19 (Size of the project), may indicate aspects of the bidding process that are important but not necessarily key differentiators in the decision to bid. They represent areas where firms may meet industry standards but may not necessarily leverage for a competitive advantage.

On the other hand, as shown in Fig. 4, the standard deviations of the importance ratings for the twenty-three factors offer a glimpse into the consensus or lack thereof among experts regarding the impact of each factor on bidding decisions. A lower standard deviation indicates greater agreement among experts, while a higher standard deviation suggests more varied opinions. Factors F8 (Experience in similar projects) and F12 (Probable number and identity of competitors), with the lowest standard deviations of 0.37 and 0.39, respectively, suggest that experts feel relatively similar about their impact—possibly considering these factors as either uniformly important or uniformly secondary. Likewise, F5 (Availability of equipment) and F6 (Current workload), both at 0.39, are areas where there seems to be a more common ground among experts' perceptions. Conversely, F2 (Identity and Reputation of the client) has the highest standard deviation at 0.45, indicating a broader range of opinions on how significantly this factor affects bidding outcomes. This could mean that some experts see client reputation as pivotal, whereas others may place less emphasis on it, perhaps due to experiences where other factors outweighed client identity in the decision-making process. Factors related to project logistics and execution, such as the availability of labor (F3) and materials (F4), and financial considerations, such as payment terms (F16), show standard deviations above 0.43, which reveals a moderately high variance in expert opinion. This variation may stem from differing market conditions, project types, or experiences that make these factors more or less critical, depending on the context. Factors with standard deviations clustering around 0.40 to 0.42, including F1 (Current location of the contractor company), F15 (Expected profitability), F21 (Promptness of the client in payments process), and F23 (Time available for tendering preparation), indicate a moderate dispersion in expert opinions. These figures suggest that while there is some consensus, there's also a significant breadth in how each expert values these factors based on individual or collective experiences.Fig. 4 The factors' standard deviation.

Fig. 4

In comparison with the average baseline, standard deviation values higher than the average (such as for F2, F3, F4, F16, F20, F21, and F23) indicate a wider divergence in expert opinions regarding the importance of these factors in the bidding process. This might suggest that experiences and perspectives on factors like the identity and reputation of the client (F2), the availability of required labor (F3), and payment terms (F16) can vary greatly, which may be due to their subjective nature or the varying circumstances under which different firms operate. On the other hand, factors like F8 (Experience in similar projects), F5 (Availability of equipment), and F12 (Probable number and identity of competitors), which have standard deviations below the average, reflect a tighter consensus among experts on their importance. This could be interpreted to mean that there is a common understanding in the industry regarding how much these factors should impact bidding decisions. Factors with standard deviation values equal to the average (F7, F11, F14, F15, F18) signify that the level of agreement about their importance is precisely what might be expected in general. The opinions on these factors do not differ substantially from the norm, indicating a uniformity in how these factors are generally perceived across the industry.

In the subsequent phase of the proposed framework, a pivotal step involves the calculation of the linear correlation coefficients for each pair of factors. This calculation is conducted following the procedures outlined in Step 8, as detailed in Section 3.2. The linear correlation coefficients provide a quantitative measure of the degree and direction of the relationships between these factor pairs. The linear correlation coefficient matrix of the factors is presented in Table 3. The coefficients range from −1 to +1, where a value of +1 indicates a perfect positive correlation, −1 is a perfect negative correlation, and 0 implies no correlation. Starting with the strongest correlations, which are highlighted, we can infer that some factors have a significantly strong positive relationship. For instance, F20 (Financial Stability of the contractor) has a very high correlation with F21 (Promptness of the client in the payments process), indicated by a coefficient of 0.655, suggesting that contractors who are financially stable may also experience timely payments from clients. This could reflect that more stable contractors are able to select projects with reliable clients or that they manage their cash flow more effectively, leading to prompt payments. The correlation between F7 (Company's previous experience with the client) and F8 (Experience in similar projects) is also strong (0.408), implying that companies with a history of working with a client are likely to have relevant project experience. This could be interpreted as a tendency for companies to build expertise and repeat business in particular types of projects or with specific clients. It's also interesting to note the moderate positive correlations between several factors that might be inherently linked, such as F15 (Expected profitability) with F16 (Payment terms by the client) and F17 (Financial resources of the contractor), which are 0.424 and 0.340, respectively. Expected profitability could be influenced by favorable payment terms, and both are naturally linked to a contractor's financial resources. On the other hand, many factors have near-zero or very low correlation coefficients with each other, indicating no substantial direct linear relationship in how they are rated for importance. For example, F1 (Current location of the contractor company) and F2 (Identity and Reputation of the client) have a coefficient of −0.004, suggesting no meaningful linear correlation between these factors. Some factors have a slight negative correlation, such as F23 (Time available for tendering preparation) with F15 (Expected profitability), albeit very weakly at −0.029, which might hint at the idea that rushed tendering could potentially impact profitability but not strongly enough to make a definitive assertion.Table 3 Factors’ linear correlation coefficient.

Table 3Factor	F1	F2	F3	F4	F5	F6	F7	F8	F9	F10	F11	F12	F13	F14	F15	F16	F17	F18	F19	F20	F21	F22	F23	
F1	1.000	−0.004	−0.037	0.033	−0.051	−0.013	0.071	0.052	0.048	−0.122	−0.003	0.032	−0.040	0.091	−0.057	−0.016	−0.073	−0.122	−0.024	0.027	−0.058	0.003	−0.052	
F2	−0.004	1.000	0.166	0.073	0.147	−0.029	−0.043	−0.036	0.126	−0.080	−0.133	−0.082	−0.028	0.105	−0.050	−0.025	−0.098	−0.096	0.147	0.086	−0.007	0.083	0.009	
F3	−0.037	0.166	1.000	0.472	0.144	0.024	0.134	0.174	0.180	0.154	0.124	0.122	0.126	0.076	0.169	0.272	0.141	0.271	0.165	0.222	0.291	0.226	0.067	
F4	0.033	0.073	0.472	1.000	0.137	0.069	0.168	0.130	0.092	0.189	0.351	0.095	0.131	−0.018	0.179	0.223	0.289	0.247	0.138	0.117	0.199	0.221	0.045	
F5	−0.051	0.147	0.144	0.137	1.000	0.186	0.118	0.113	0.256	0.152	0.094	0.381	0.136	0.052	0.139	0.066	0.155	0.101	0.239	0.130	0.137	0.231	0.068	
F6	−0.013	−0.029	0.024	0.069	0.186	1.000	0.320	0.293	0.271	0.313	0.218	0.187	0.176	0.279	0.283	0.179	0.172	0.184	0.250	0.258	0.286	0.131	0.079	
F7	0.071	−0.043	0.134	0.168	0.118	0.320	1.000	0.408	0.168	0.396	0.442	0.178	0.136	0.095	0.416	0.250	0.243	0.229	0.193	0.311	0.385	0.247	−0.103	
F8	0.052	−0.036	0.174	0.130	0.113	0.293	0.408	1.000	0.325	0.381	0.298	0.296	0.273	0.207	0.216	0.190	0.069	0.141	0.265	0.256	0.290	0.216	0.098	
F9	0.048	0.126	0.180	0.092	0.256	0.271	0.168	0.325	1.000	0.380	0.191	0.246	0.210	0.300	0.214	0.136	0.263	0.131	0.254	0.252	0.252	0.305	0.020	
F10	−0.122	−0.080	0.154	0.189	0.152	0.313	0.396	0.381	0.380	1.000	0.288	0.148	0.191	0.269	0.539	0.395	0.386	0.342	0.438	0.444	0.390	0.272	0.074	
F11	−0.003	−0.133	0.124	0.351	0.094	0.218	0.442	0.298	0.191	0.288	1.000	0.233	0.240	0.020	0.274	0.273	0.299	0.312	0.197	0.243	0.375	0.247	0.064	
F12	0.032	−0.082	0.122	0.095	0.381	0.187	0.178	0.296	0.246	0.148	0.233	1.000	0.479	0.210	0.286	0.319	0.293	0.347	0.396	0.265	0.400	0.286	0.261	
F13	−0.040	−0.028	0.126	0.131	0.136	0.176	0.136	0.273	0.210	0.191	0.240	0.479	1.000	0.298	0.134	0.190	0.262	0.307	0.476	0.348	0.422	0.269	0.203	
F14	0.091	0.105	0.076	−0.018	0.052	0.279	0.095	0.207	0.300	0.269	0.020	0.210	0.298	1.000	0.104	0.170	0.037	0.208	0.326	0.213	0.141	0.060	0.124	
F15	−0.057	−0.050	0.169	0.179	0.139	0.283	0.416	0.216	0.214	0.539	0.274	0.286	0.134	0.104	1.000	0.424	0.340	0.302	0.319	0.343	0.393	0.262	−0.029	
F16	−0.016	−0.025	0.272	0.223	0.066	0.179	0.250	0.190	0.136	0.395	0.273	0.319	0.190	0.170	0.424	1.000	0.333	0.326	0.363	0.445	0.462	0.323	0.168	
F17	−0.073	−0.098	0.141	0.289	0.155	0.172	0.243	0.069	0.263	0.386	0.299	0.293	0.262	0.037	0.340	0.333	1.000	0.505	0.297	0.385	0.384	0.347	0.052	
F18	−0.122	−0.096	0.271	0.247	0.101	0.184	0.229	0.141	0.131	0.342	0.312	0.347	0.307	0.208	0.302	0.326	0.505	1.000	0.348	0.362	0.422	0.264	0.249	
F19	−0.024	0.147	0.165	0.138	0.239	0.250	0.193	0.265	0.254	0.438	0.197	0.396	0.476	0.326	0.319	0.363	0.297	0.348	1.000	0.496	0.423	0.374	0.301	
F20	0.027	0.086	0.222	0.117	0.130	0.258	0.311	0.256	0.252	0.444	0.243	0.265	0.348	0.213	0.343	0.445	0.385	0.362	0.496	1.000	0.655	0.348	0.027	
F21	−0.058	−0.007	0.291	0.199	0.137	0.286	0.385	0.290	0.252	0.390	0.375	0.400	0.422	0.141	0.393	0.462	0.384	0.422	0.423	0.655	1.000	0.434	0.072	
F22	0.003	0.083	0.226	0.221	0.231	0.131	0.247	0.216	0.305	0.272	0.247	0.286	0.269	0.060	0.262	0.323	0.347	0.264	0.374	0.348	0.434	1.000	0.155	
F23	−0.052	0.009	0.067	0.045	0.068	0.079	−0.103	0.098	0.020	0.074	0.064	0.261	0.203	0.124	−0.029	0.168	0.052	0.249	0.301	0.027	0.072	0.155	1.000	

Leveraging both the standard deviation of the factors and the linear correlation coefficients between these factors, a crucial step involves the quantification of the information content contributed by each factor. This quantification process is conducted in accordance with Eq. (29), a methodical formula that considers both the variability and interrelationships among the factors. The results of this information content assessment are thoughtfully illustrated and presented in Fig. 5. It seems that the highest information content is associated with F1 (9.31) and the peak at F2 (9.77), suggesting these factors might be the most critical in the dataset's context. Such values could indicate that factors like the contractor's current location (F1) and the identity and reputation of the client (F2) hold substantial importance in the decision-making process. The values then appear to decrease and exhibit a downward trend from F3 to F6, where F6 reaches 6.90, indicating a relative decline in the information content and possibly their perceived importance. Subsequent factors from F7 through F14 fluctuate within a narrow band, suggesting a more consistent but moderate level of importance across these factors. A slight increase is observed from F15 onwards, reaching another high at F23 (8.47). This suggests that the factor represented by F23, possibly related to the time available for tendering preparation or another context-specific element, again becomes more critical in terms of the information it provides for decision-making.Fig. 5 Factors' information content.

Fig. 5

Ultimately, the weights assigned to each of the factors are established through a calculation detailed in Eq. (30). The resulting weights, which carry significant implications for the decision-making process, are visually depicted and thoughtfully presented in Fig. 6. From this treemap, it seems that the factors are not evenly weighted, signifying a variance in their influence on bidding decisions in Taiwan construction industry. The most substantial weights are assigned to F2 (7.57 %), F3 (7.30 %), and F4 (6.45 %), which suggests that these factors are deemed highly influential in the process. These could be factors like financial stability, the company's current workload, or market conditions, all of which are often crucial considerations in the bidding process. On the lower end of the scale, the factors F12 (2.52 %), F8 (2.70 %), and F13 (3.11 %) have the least weight, indicating they have a smaller impact on decision-making. These might be factors considered less critical, such as the geographic location of the project or the availability of subcontractors. The intermediate weights, which form the majority of the factors (like F6 at 3.70 %, F11 at 3.85 %, and F14 at 4.27 %), point to factors that have a moderate impact. These likely represent important but not overriding considerations, such as company experience, staff availability, or specific project risks.Fig. 6 Weight of critical factor for bidding decisions in Taiwan construction industry.

Fig. 6

As shown in Fig. 7, F2 (Identity and Reputation of the client) is ranked as the most critical factor with the top position. The significance placed on the client's identity and reputation suggests that who the client is can dramatically influence a firm's decision to bid, potentially due to the impact on project success and payment reliability. The following are F3 (Availability of required labor) and F4 (Availability of materials), ranked 2 and 3, respectively. These factors are pivotal, highlighting the fundamental importance of resource availability in the execution of construction projects [25]. At the other end of the spectrum, F12 (Probable number and identity of competitors), ranked at the bottom (23), indicates that this factor is considered the least influential in bidding decisions. This could reflect a market condition where competition is either not perceived as a significant threat or is overshadowed by other more pressing concerns. Intermediate rankings are occupied by factors like F15 (Expected profitability) and F16 (Payment terms), at ranks 4 and 5, which are still highly influential but slightly less so than the top three. These suggest that economic factors, while important, may not be as immediately impactful as the availability of resources when making a bid.Fig. 7 The rank of the critical factor for bidding decisions in Taiwan construction industry.

Fig. 7

Factors such as F5 (Availability of equipment), F6 (Current workload), and F19 (Size of project), with lower rankings of 18, 16, and 20, are seen as less critical. However, given their rankings are above the lowest, they still contribute to the decision-making process, albeit to a lesser extent. Interestingly, factors that could be assumed to be significant, such as F8 (Experience in similar projects) and F13 (Project type), are positioned towards the lower end of the ranking scale (22 and 21). This might indicate a shift in focus towards real-time, actionable factors over historical experience or the nature of the project.

4.2 Comparative analysis

In this section, the comparative analysis between rankings obtained from the SFmCRITIC method and the LOPCOW method offers a valuable perspective on the varying importance of factors in the bidding process. As shown in Fig. 8, the agreement between the two methods on factors like the probable number and identity of competitors (F12) and project type (F13) as least influential confirms a methodological consensus on their lower impact. Yet, discrepancies in other factors, such as the current location of the contractor (F1) and the completeness of documents (F22), with SFmCRITIC assigning higher importance, indicate method-specific sensitivities to logistical and procedural elements. The SFmCRITIC method's emphasis on factors that address uncertainty may explain why it ranks certain factors differently than LOPCOW. These disparities in factor rankings are not just academic; they have practical implications for firms in prioritizing resources and shaping bidding strategies. The methodology's verification comes from its consistency in ranking high-impact factors and its robustness in managing the intricacies of expert judgment under uncertainty. Such methodological validation is crucial, as it demonstrates the SFmCRITIC method's reliability and effectiveness in real-world scenarios. By providing a consistent and objective framework for evaluating critical bidding factors, this method stands as a verified approach for firms seeking to navigate the competitive landscape of Taiwan's construction industry.Fig. 8 The SFmCRITIC and LOPCOW ranking.

Fig. 8

4.3 Managerial implications

The implementation of the SFmCRITIC method revealed several key findings that bear significant implications for the construction industry. Chief among these is the clear identification of factors that hold the most weight in bidding decisions. Factors such as the identity and reputation of the client (F2), availability of required labor (F3), and availability of materials (F4) emerged as top influencers, reflecting the industry's prioritization of reliable partnerships, resource availability and supply chain robustness. Interestingly, the results indicate that while factors like the current workload (F6) and experience in similar projects (F8) are important, they are not as decisive in the bidding process as might be expected. This could suggest a strategic shift in focus within the industry toward real-time capabilities and the current financial landscape rather than historical performance. The application of the SFmCRITIC method also highlighted the lesser weight of factors such as the probable number and identity of competitors (F12) and project type (F13). This finding might reflect confidence within the industry that firms can compete effectively regardless of competitor actions or project nature, possibly due to a strong emphasis on internal strengths and capabilities.

The benefits of applying the SFmCRITIC method extend beyond just understanding the importance of factors. It provides a systematic approach to handling the uncertainties inherent in the bidding process. By utilizing a fuzzy logic system, firms can better navigate the often-subjective judgments involved in evaluating projects, leading to more confident and informed decision-making. The results of this study carry practical implications. They offer a roadmap for companies to reassess their bidding strategies, perhaps shifting the emphasis toward enhancing client relationships and ensuring resource availability. This shift could streamline bidding processes, reduce risk, and ultimately increase the success rate of bids. Furthermore, the results point to potential training and development areas, suggesting that companies should focus on building expertise in areas identified as most influential.

5 Conclusion

This research has made significant strides in advancing the field of decision-making within the Taiwan construction industry by introducing the spherical fuzzy extension of the modified CRITIC (SFmCRITIC) method. By meticulously combining the precision of the CRITIC method and the nuanced handling of uncertainty provided by Spherical Fuzzy Numbers, we have formulated a novel decision-support tool that enhances the bidding decision process. The findings underscore the importance of certain critical factors over others, such as F2, F3, and F4, which indicate aspects like financial robustness, market understanding, and specialized skills. These insights empower construction firms to allocate resources strategically, improving areas that are crucial for bidding success while avoiding excessive investment in less impactful factors.

The study provides a quantifiable and systematic approach to ascertain the relative importance of bidding factors, which was hitherto often left to subjective managerial interpretation. By providing a data-driven foundation, the SFmCRITIC method allows for a more nuanced and strategic approach to decision-making that aligns with the dynamics of the industry and the firm's specific strategic goals.

While the SFmCRITIC method offers significant advancements in the strategic bidding process within Taiwan's construction industry, it is important to acknowledge the limitations of this study. One key limitation is the potential variability in expert judgments, which may influence the weighting of the factors despite the use of spherical fuzzy sets to capture such nuances. The method's reliance on accurate and comprehensive expert input means that any bias or error in expert opinion could affect the results. Additionally, the dynamic nature of the construction industry means that the relative importance of factors may evolve over time, and the static nature of this study may not capture such shifts. Another limitation is the contextual focus on Taiwan's construction industry, which may limit the generalizability of the findings to other regions or sectors with different economic and regulatory environments. Future research should explore the applicability of the SFmCRITIC method across diverse geographical locations and in different industry settings to validate its effectiveness and adaptability. Finally, the integration of fuzzy theory with the CRITIC method, while innovative, requires a deep understanding of both concepts, which might present a learning curve for practitioners and may affect the widespread adoption of this approach. Addressing these limitations in future studies will help refine the method and expand its practical utility.

Ethical approval

The study was explained to participants through a questionnaire. They were informed that they would participate in the survey that all data would be de-identified and only reported in the aggregate. All participants acknowledged an informed consent statement in order to participate in the study.

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Informed consent

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability statement

Data included in article/supp. material/referenced in article.

CRediT authorship contribution statement

Yih-Tzoo Chen: Writing – review & editing, Writing – original draft, Project administration, Methodology, Investigation, Funding acquisition, Data curation, Conceptualization. Thi-Hien Dao: Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. I-Hsin Chiu: Writing – review & editing, Writing – original draft, Visualization, Validation, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix Table A1 The spherical fuzzy decision matrix

Table A1Factor	Expert 1	Expert 2	Expert 134	
F1	(0.385, 0.615, 0.385)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	
F2	(0.270, 0.730, 0.270)	(0.385, 0.615, 0.385)	(0.730, 0.270, 0.270)	
F3	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	
F4	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	
F5	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.385, 0.615, 0.385)	
F6	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	
F7	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	(0.615, 0.385, 0.385)	
F8	(0.615, 0.385, 0.385)	(0.615, 0.385, 0.385)	(0.615, 0.385, 0.385)	
F9	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	
F10	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	
F11	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	
F12	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	
F13	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	
F14	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	
F15	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	
F16	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	(0.615, 0.385, 0.385)	
F17	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.385, 0.615, 0.385)	
F18	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.385, 0.615, 0.385)	
F19	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	
F20	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	
F21	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	
F22	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	
F23	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.385, 0.615, 0.385)	

Table A2 The crisp decision matrix

Table A2Factor	Expert 1	Expert 2	Expert 134	
F1	0.0529	0	0.0529	
F2	0.2116	0.0529	0.2116	
F3	0.0529	0.2116	0.0529	
F4	0	0.2116	0.0529	
F5	0	0.2116	0.0529	
F6	0.0529	0.2116	0.2116	
F7	0.2116	0.0529	0.0529	
F8	0.0529	0.0529	0.0529	
F9	0.0529	0.2116	0	
F10	0.2116	0.2116	0.0529	
F11	0.2116	0.2116	0.0529	
F12	0	0.2116	0.0529	
F13	0.0529	0.2116	0.2116	
F14	0.0529	0.2116	0.2116	
F15	0.2116	0.2116	0.2116	
F16	0.2116	0.0529	0.0529	
F17	0.2116	0.2116	0.0529	
F18	0.0529	0.2116	0.0529	
F19	0	0.2116	0.2116	
F20	0.2116	0.2116	0.2116	
F21	0.2116	0.2116	0.2116	
F22	0.0529	0.2116	0	
F23	0	0.2116	0.0529	

Appendix A Supplementary data

The following are the Supplementary data to this article:Multimedia component 1

Multimedia component 1

Multimedia component 2

Multimedia component 2

Appendix A Supplementary data to this article can be found online at https://doi.org/10.1016/j.heliyon.2024.e36420.
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