PNS-WINGS-AHP–EWM集成框架:数字农业多准则决策分析方法

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Journals Symmetry Volume 18 Issue 9 10.3390/sym18091435
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Table of Contents
Abstract
Introduction
Literature Review
Fundamental Concepts and Procedures
Pythagorean Neutrosophic WINGS-AHP–EWM Combination Method
Results
Discussion
Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix B
References
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Open AccessArticle
PNS-WINGS-AHP–EWM Integrated Framework for Multi-Criteria Decision Analysis: An Application-Oriented Decision Framework in Digital Agriculture
by Yawen Wang 1 andKecheng Zhang 2,*
1
College of Economics and Management, Shandong Agricultural University, Tai’an 271018, China
2
School of Business Administration, Shandong Women’s University, Jinan 250300, China
*
Author to whom correspondence should be addressed.
Symmetry 2026, 18(9), 1435; https://doi.org/10.3390/sym18091435
Submission received: 7 July 2026 / Revised: 21 August 2026 / Accepted: 25 August 2026 / Published: 27 August 2026
(This article belongs to the Special Issue Symmetry/Asymmetry in Fuzzy Multi-Criteria Decision-Making: Application in Management and Engineering)

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Abstract
Digital-agriculture development increasingly requires multi-criteria decision analysis (MCDA) approaches to evaluate complex and interrelated influencing factors. However, existing methods often face difficulties in representing uncertain expert judgments, capturing inter-factor influence structures, and integrating complementary weighting information. To address these challenges, this study develops an integrated Pythagorean Neutrosophic Sets (PNS)–Weighted Influence Non-linear Gauge System (WINGS)–Analytic Hierarchy Process (AHP)–Entropy Weight Method (EWM) framework for digital-agriculture factor evaluation. Specifically, PNS is employed to represent uncertain linguistic assessments through truth, indeterminacy, and falsity information; WINGS is applied to identify the direction and intensity of expert-perceived influence relationships among factors; and a coordinated weighting mechanism integrates AHP-based preference information with entropy-based cross-expert dispersion information. The proposed framework is applied to a digital-agriculture case to identify key influencing factors and determine development priorities. The results indicate that Economic Development Level (W1) ranks first in the final composite priority assessment, followed by Digital Technology Level (W6) and Digital Infrastructure (W5). Furthermore, the WINGS analysis reveals the perceived influence structure among factors and identifies Digital Infrastructure (W5) as the strongest net-influencing factor within the network. Comparative analyses suggest that the proposed framework provides a systematic representation of uncertainty, influence relationships, and complementary weighting information for digital-agriculture factor evaluation. This study highlights the application potential of integrating established MCDM techniques to support complex agricultural decision-making problems.
Keywords: multi-criteria decision analysis; Pythagorean neutrosophic sets; weighted influence nonlinear gauge system; Analytic Hierarchy Process; Entropy Weight Method; digital agriculture

1. Introduction
   Digital agriculture, emerging from the convergence of information technology and agricultural production, has become an important driver of agricultural transformation worldwide. Meanwhile, population growth and accelerated urbanization continue to increase the demand for agricultural products. However, agricultural systems face multiple challenges, including limited land availability, labor constraints, climate variability, and changing socioeconomic conditions. These challenges have accelerated the transition toward digitalization, intelligent management, and sustainable development, positioning digital agriculture as an important topic in both scientific research and policy discussions [1,2].
   To support this transition, extensive studies have investigated the factors influencing digital-agriculture development. Existing research has identified several important determinants, including economic development level, social capital participation, policy support, human capital quality, digital infrastructure, digital technology level, operational scale, profitability, and openness to global markets. Understanding the relative importance of these factors is essential for designing effective development strategies. However, existing analytical approaches still face several limitations that restrict their ability to provide comprehensive decision support.
   First, many studies analyze influencing factors separately and treat them as independent criteria within conventional multi-criteria decision-making (MCDM) frameworks [3,4]. Such approaches may overlook the complex interactions, feedback effects, and interdependencies among factors in digital-agriculture systems. Consequently, the resulting rankings often provide a static evaluation of factor importance but may not reveal the relative influence positions of different factors during the digital transformation process [5]. Second, many existing methods rely on precise numerical evaluations and provide limited support for representing uncertainty, hesitation, and inconsistency in expert judgments [6]. As expert assessments of digital-agriculture development often involve subjective perceptions and incomplete information, directly converting linguistic judgments into crisp values may reduce the richness of the original evaluation information [7].
   Despite considerable methodological progress, existing studies remain fragmented across four interconnected analytical dimensions. At the information representation level, expert judgments are frequently transformed into numerical scores, which may fail to preserve linguistic uncertainty and indeterminacy. At the structural analysis level, conventional weighting and ranking approaches generally assume criterion independence and therefore have limited capability to capture direct, indirect, and self-influence relationships among digital-agriculture factors [3,4,5]. At the weighting level, the Analytic Hierarchy Process (AHP) incorporates expert preferences but may be affected by subjective judgment, whereas the Entropy Weight Method (EWM) reflects data variation but does not consider expert preference information or network relationships [8,9]. At the integration level, uncertainty representation, influence analysis, and weight determination are often conducted separately. Therefore, existing approaches may identify factor importance or influence structures individually, but they rarely establish a continuous analytical process connecting uncertain expert assessments, inter-factor relationships, and final priority determination.
   Motivated by these limitations, this study integrates Pythagorean Neutrosophic Sets (PNS), the Weighted Influence Non-linear Gauge System (WINGS), AHP, and EWM into a sequential and functionally differentiated analytical framework. Rather than simply combining multiple techniques, the proposed framework assigns each method a specific analytical role. PNS is used at the input stage to represent linguistic evaluations through truth, indeterminacy, and falsity components. After aggregation and score transformation, the resulting influence matrix is used as the input for WINGS, which analyzes direct, indirect, and self-influence relationships and identifies network prominence and net-influence positions. Subsequently, AHP and EWM provide complementary perspectives on factor importance: AHP reflects expert preference information, whereas EWM captures cross-expert dispersion information derived from the individual AHP priority distributions. Their results are integrated through a combination-weighting procedure, and the resulting weights are further linked with WINGS-based network prominence to generate composite priorities.
   The contribution of this study lies in establishing and applying this connected analytical workflow for digital-agriculture factor evaluation rather than developing entirely new individual methods. The framework provides both an expert-perceived influence structure and a composite priority ranking. Moreover, the potential effects of multi-stage processing, including possible information compression, are examined through subsequent comparative and sensitivity analyses.
   The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 introduces the fundamental concepts of PNS. Section 4 presents the integrated methodology. Section 5 reports the empirical results and analysis. Section 6 discusses the comparative analysis, theoretical implications, policy implications, limitations, and future research directions. Finally, Section 7 summarizes the main findings and conclusions of this study.
2. Literature Review
   2.1. Evolution of Digital Agriculture and Its Influencing Factors
   Certain stages in agricultural development have involved profound transformations and have therefore been described as agricultural revolutions, such as the Green Revolution [10,11]. The emergence of digital technologies is expected to promote a new stage of agricultural transformation by improving the efficiency, sustainability, and management capability of both crop production and livestock systems. This transition has attracted increasing attention from researchers, policymakers, and practitioners [12].
   The concept of digital agriculture was introduced in 1997 as a data-driven agricultural development approach that integrates geospatial technologies and information-based systems [13]. By leveraging digital tools and resources, digital agriculture emphasizes the application of digital technologies throughout the agricultural production process, with the objectives of improving productivity, increasing farmers’ income, and enhancing the competitiveness of agricultural products [14]. In recent years, digital agriculture has also been associated with concepts such as precision agriculture and smart farming, which originated from technology-enabled crop management practices in the 1990s [15,16]. Accordingly, researchers have proposed various definitions of digital agriculture from different perspectives, reflecting its multidisciplinary characteristics [17,18,19].
   Recent studies have increasingly focused on identifying the factors that influence digital-agriculture development. Existing research has highlighted the importance of technological progress, economic conditions, digital infrastructure, policy support, operational scale, profitability, social capital participation, human capital quality, and market openness. Specifically, Shepherd emphasized the role of technological advancement in transforming traditional agricultural management models [20]. Shen demonstrated that regional economic conditions significantly influence the development level of digital agriculture [21]. Luo highlighted the importance of digital infrastructure in supporting digital transformation [22], while Tian examined the role of governmental policy support in facilitating agricultural digitalization [23]. Chen further suggested that the adoption of digital technologies is closely associated with the expansion of agricultural operation scale [24]. In addition, other studies have identified profitability, global market openness, social capital participation, and human capital quality as important determinants of digital-agriculture development [25,26,27,28].
   Although previous studies have identified numerous influencing factors, an important research question remains: how can the relative importance of these factors and their interrelationships be systematically evaluated? Existing studies often examine individual factors separately, providing limited insights into the interactions among multiple dimensions of digital-agriculture development. Therefore, a comprehensive quantitative framework is required to identify key factors and reveal their structural relationships. Furthermore, because evaluations of digital-agriculture development frequently involve expert knowledge, incomplete information, and subjective judgment, decision-making approaches capable of handling uncertainty are needed.
   In this study, the decision problem is defined as a regional-level prioritization problem within the Chinese digital-agriculture context. The evaluation object is not an individual farm, enterprise, or specific technology project, but rather the regional enabling environment that supports digital-agriculture development. The primary decision stakeholders are regional agricultural authorities and public development agencies, which need to determine the relative priorities of policy attention, infrastructure investment, capability development, and implementation resources. The resulting ranking aims to support medium-term planning over an approximately three-to-five-year horizon by identifying structural development priorities rather than predicting short-term agricultural performance or determining specific budget allocations.
   2.2. Methodological Foundation: From Fuzzy Sets to Pythagorean Neutrosophic Sets
   To address uncertainty in complex decision-making problems, fuzzy set theory has been widely adopted as an effective approach for representing imprecise information. Zadeh introduced fuzzy sets in 1965, using membership functions to describe uncertain and vague information [29]. However, because fuzzy sets rely on a single membership degree, they may have limited capability in representing situations involving hesitation or conflicting information [30].
   To overcome this limitation, Atanassov proposed intuitionistic fuzzy sets (IFS), which extend fuzzy sets by incorporating membership and non-membership degrees as well as hesitation information [31]. Nevertheless, IFS requires that the sum of membership and non-membership degrees does not exceed one, which may restrict its applicability in certain complex decision environments [32,33]. Subsequently, Yager introduced Pythagorean fuzzy sets (PFS), allowing the squared sum of membership and non-membership degrees to satisfy a less restrictive condition and thereby expanding the representation space of uncertain information [34,35,36,37]. Meanwhile, Smarandache proposed neutrosophic set theory, which explicitly introduces an indeterminacy dimension in addition to truth and falsity memberships, enabling a more comprehensive description of uncertain and incomplete information [38,39,40,41].
   Building on these developments, Jansi et al. introduced the concept of Pythagorean neutrosophic sets (PNS) in 2019 [7]. PNS integrates truth, falsity, and indeterminacy components, providing a flexible representation of information that involves uncertainty, ambiguity, and incomplete knowledge [42,43]. This characteristic is particularly relevant for digital-agriculture factor evaluation, where expert judgments regarding factor relationships may contain both supportive and uncertain perceptions. Compared with conventional fuzzy sets and intuitionistic fuzzy sets, PNS provides a broader representation space by explicitly incorporating indeterminacy information [44]. Therefore, PNS has been applied in various uncertain decision-making contexts, including multi-criteria decision analysis, artificial intelligence, and decision-support systems [45,46,47].
   Existing PNS research has gradually expanded from theoretical development toward practical applications. Early studies mainly focused on mathematical properties and operational rules [48,49], whereas recent studies have explored its application in complex decision-support problems [50]. By retaining multiple dimensions of uncertain information, PNS provides a useful foundation for decision frameworks that require the representation of expert uncertainty.
   Recent MCDM studies have also explored the integration of uncertainty modeling and structural analysis. Muyulema-Allaica et al. combined FAHP with DEMATEL to identify causal and effect factors in industrial symbiosis resource optimization [51]. Petrović et al. incorporated triangular fuzzy numbers into AHP to address uncertainty in weight determination under expert subjectivity [52]. Turgay et al. developed a decision model based on the T-spherical hesitant fuzzy rough set to represent incomplete and conflicting information using truth, falsity, and indeterminacy dimensions [53]. These studies demonstrate the importance of uncertainty-aware decision models. However, existing approaches generally focus on specific analytical aspects, such as uncertainty representation or influence analysis, rather than integrating uncertainty modeling, influence-network analysis, and hybrid weighting within a unified decision process.
   To clarify the methodological positioning of the proposed framework, Table 1 compares representative MCDM approaches from the perspectives of uncertainty representation, influence analysis, and weight determination.
   Table 1. Analytical comparison of representative MCDM methods related to uncertainty quantification, influence-structure analysis, and hybrid weighting.
   2.3. Analytical Methods: WINGS, AHP, and EWM
   The Decision-Making Trial and Evaluation Laboratory (DEMATEL) is a widely used MCDM method for analyzing complex systems through influence relationship modeling [54]. Building upon DEMATEL, Michnik proposed the Weighted Influence Non-linear Gauge System (WINGS), which extends the analysis by incorporating self-influence and considering broader network characteristics beyond pairwise relationships [55]. Previous studies have applied WINGS to analyze complex inter-factor relationships and identify influential elements in different decision contexts. For example, Wang et al. demonstrated its capability in analyzing interconnected factors and their internal relationships [56]. Kannan et al. applied WINGS to prioritize barriers in decision systems [57], while Kumar et al. used it to investigate factors affecting blockchain technology adoption in supply chains [58]. However, WINGS generally requires numerical input information and therefore has limited capability in directly handling linguistic uncertainty [59].
   The Analytic Hierarchy Process (AHP) is a classical MCDM approach that combines qualitative judgments with quantitative analysis and has been widely applied in complex decision-making problems since the 1970s [60,61]. By incorporating expert preferences, AHP provides an effective approach for subjective weight determination. However, its results may be influenced by expert judgment and consistency issues [8].
   In contrast, the Entropy Weight Method (EWM) determines indicator weights according to information entropy and the dispersion of observations [9]. Although EWM is commonly classified as an objective weighting method because its weights follow predefined mathematical rules once the input matrix is fixed, the informational meaning of the resulting weights depends on the nature of that input matrix. In the present study, the EWM input consists of expert-specific AHP priority distributions; therefore, EWM is interpreted as extracting cross-expert dispersion information rather than as introducing an independent objective data source. EWM has been applied in various evaluation and optimization problems, including industrial efficiency assessment, risk evaluation,