Researchers have developed a multi-criteria decision-making framework that utilizes (p, q)-fractional fuzzy sets and Yager operations to evaluate and rank technological forecasting methods. Published in the journal Discover Analytics in January 2026, the study addresses the uncertainty and ambiguity inherent in selecting the most effective predictive mechanisms for manufacturing and industry. The authors, including Saifullah, Saleem Abdullah, Marya Nawaz, and Hameed Gul Ahmadzai, propose an extended CODAS method that operates within a novel fuzzy environment where the weights of criteria and the preferences of decision-makers are completely unknown. This approach aims to provide a more robust and rational basis for strategic decisions that can significantly impact financial outcomes and organizational stability.
Traditional fuzzy sets often struggle to model uncertainty when the sum of membership and non-membership degrees exceeds one or when specific parameters approach unity. The new framework introduces a condition where the weighted sum of the positive membership degree and the negative membership degree, scaled by parameters p and q, remains within the interval of zero to one. This mathematical structure provides a larger domain of flexibility for dealing with incoherent and ambiguous data in complex decision-making scenarios.
To facilitate the aggregation of data within this new framework, the researchers developed a series of Yager averaging aggregation operators. These include the (p, q)-fractional fuzzy Yager weighted averaging operator, the ordered weighted averaging operator, and the hybrid weighted averaging operator. The study establishes the operational laws for these sets using Yager t-norms and t-conorms, which are fundamental logical operations in fuzzy logic. The authors prove several key properties of these operators, including idempotency, boundedness, monotonicity, and reducibility, ensuring that the mathematical model is consistent and reliable for practical application.
The CODAS method is originally designed for multi-criteria decision-making in stochastic environments and uses both Euclidean and Taxicab distances to evaluate alternatives relative to a negative ideal solution. By integrating the proposed Yager aggregation operators, the extended method can handle uncertain input data more effectively. The algorithm involves constructing decision matrices, aggregating expert opinions, calculating score values, and determining the relative assessment of each alternative based on its distance from the negative ideal solution.
To demonstrate the applicability of the proposed model, the authors applied it to a real-world case study involving the selection of the most appropriate technological forecasting method. Five distinct forecasting strategies were evaluated against five criteria derived from a literature review and expert opinion in technology management. Three experts provided their assessments in the form of (p, q)-fractional fuzzy decision matrices, reflecting the inherent uncertainty in their judgments.
The computational results of the proposed aggregation operators showed that the patent analysis method was determined to be the best suitable technique for technological forecasting. The study compared the results of the extended (p, q)-fractional fuzzy CODAS method with existing methods, including the Pythagorean Fuzzy Grey Relational Analysis (PyF-GRA) and the Fractional Fuzzy TOPSIS methods. The comparison utilized the same decision matrices and expert weights to ensure a fair evaluation. The results indicated that the proposed model outperformed the traditional techniques, providing a more accurate and reliable ranking of the forecasting methods.
A sensitivity analysis was conducted to examine the influence of the parameters p and q on the final rankings. The study observed that while the overall score values fluctuated when different values were assigned to p and q, the relative ranking of the alternatives remained consistent. This stability suggests that the model is robust against variations in the optimism or pessimism of decision-makers, as the parameters can be interpreted as degrees of optimism. The consistent rankings indicate that the method provides balanced results that are not easily impacted by subjective biases in parameter selection.
The authors highlight that the precision of forecasting is essential for the high quality of outcomes and the future success of companies. Wrong decisions in this domain can lead to serious financial losses and social instability. By providing a structured strategy to cope with errors and ambiguities, the proposed framework helps decision-makers balance their perspectives, experiences, and data collection with suitable evaluation methods. The model is particularly advantageous in situations where decisions are complex and involve balancing conflicting objectives, offering a clear path to the optimal alternative.
The study also discusses the limitations of previous fuzzy set extensions, such as q-rung orthopair fuzzy sets, which do not encompass an analysis of fractional numbers. The introduction of the (p, q)-fractional fuzzy set addresses this gap by incorporating fractional parameters into the membership constraints. This allows for a more detailed handling of uncertain information compared to earlier models. The authors note that the proposed method can be extended to include heterogeneous data, which is more suitable for different criteria and more comparable to real-world scenarios where various types of information are present.
Future research directions outlined by the team include applying the framework to other decision-making problems, such as IoT cloud platform selection, building decisions, and site selection. They also plan to combine the (p, q)-fractional fuzzy environment with other aggregation operators, such as Einstein, Frank, and Bonferroni mean operators, to develop unique multi-criteria decision-making techniques. The study contributes to the growing field of fuzzy decision-making by offering a novel mathematical tool that enhances the ability of organizations to make informed choices in uncertain technological landscapes.
Within the mathematical architecture of the framework, the authors formally quantify uncertainty by defining the hesitancy degree alongside positive membership and negative non-membership functions. For any non-empty universal set, the hesitancy degree is expressed as one minus the quotient of the positive membership degree divided by p minus the quotient of the negative non-membership degree divided by q. To evaluate individual fuzzy numbers, the system relies on two diagnostic measures: a score function and an accuracy function. The score function subtracts the scaled negative membership from the scaled positive membership, producing values bounded between negative one and positive one. Conversely, the accuracy function computes the sum of these two scaled membership components, yielding values restricted to the interval between zero and one. In the ranking algorithm, options exhibiting higher score values are prioritized, while the accuracy function serves as a decisive tiebreaker whenever competing alternatives achieve identical score values.
The algebraic manipulation of these sets is governed by customized Yager operational laws constructed through specific t-norm and t-conorm functions controlled by a positive parameter theta. Under these rules, the researchers established exact formulations for multiplying two fuzzy numbers, adding them, multiplying a fuzzy number by a positive scalar, and raising a fuzzy number to an exponential power. Each operation integrates minimum functions alongside p-th and q-th root transformations. To demonstrate the computational feasibility of these operational laws, the authors presented a numerical test case evaluating two distinct fuzzy numbers with assigned parameters p equal to two, q equal to three, theta equal to two, and a scalar multiplier of four. Through step-by-step calculation, the operations generated verified numerical outputs for addition, multiplication, scalar scaling, and power elevation, confirming that the Yager formulations behave consistently within the defined boundaries of fractional fuzzy spaces.
The procedural implementation of the extended CODAS method introduces precise distance calculations to determine how alternatives deviate from an unfavorable baseline. After normalizing and aggregating the decision matrices using the weighted averaging operator, the algorithm calculates the negative ideal solution by identifying the minimum weighted score across each criterion. In the study’s experimental evaluation, this negative ideal benchmark yielded exact coordinates of minus 0.0101, minus 0.0132, minus 0.0104, minus 0.0201, and minus 0.0147. Each alternative is subsequently measured against this benchmark using both Euclidean distance, which captures geometric separation, and Taxicab distance, which sums absolute discrepancies. These metrics are combined within a relative assessment matrix regulated by a threshold parameter ranging from 0.01 to 0.05. The threshold function activates when absolute distance differences exceed a specified cut-off, ensuring that minor metric parity does not distort the evaluation score.
In contextualizing the technological forecasting case study, the authors distinguished between two foundational analytical paradigms identified across technology management literature: normative methods and explanatory approaches. Normative techniques operate by conceptualizing a preferred technological future and drafting the necessary action plans to realize that objective, whereas explanatory frameworks analyze empirical observations and prevailing environmental conditions to project future trajectories. To select the most suitable mechanism from among the five evaluated strategies, the panel of three experts assigned relative criterion weights calculated as 0.39 for the first criterion, 0.35 for the second, and 0.26 for the third. Furthermore, in the comparative validation phase contrasting the proposed framework against Pythagorean fuzzy Grey Relational Analysis and fractional fuzzy TOPSIS, the researchers implemented an objective entropy estimation technique to derive criterion weight vectors directly from the aggregated data, confirming the superior discriminatory power of their methodology.
Subject of Research: Development and application of a multi-criteria decision-making model using (p, q)-fractional fuzzy sets and Yager operations for technology forecasting.
Article Title: Multi criteria decision making approach using (p, q)-fractional fuzzy sets and Yager operations
Article References: Saifullah, Abdullah, S., Nawaz, M., & Ahmadzai, H. G. (2026). Multi criteria decision making approach using (p, q)-fractional fuzzy sets and Yager operations. Discover Analytics, 4(1), Article 4. https://doi.org/10.1007/s44257-025-00050-5
Image Credits: AI Generated
DOI: 10.1007/s44257-025-00050-5
Keywords: multi-criteria decision making, fractional fuzzy sets, Yager operations, technological forecasting, CODAS method, fuzzy logic, aggregation operators, uncertainty modeling, technology management, decision analysis, fuzzy mathematics, strategic planning, expert systems, ranking algorithms, industrial forecasting, mathematical modeling, risk assessment, optimization methods, fuzzy decision theory, innovation management
News Source: Reid Dalton. (November 14, 2026). New Fuzzy Logic Model Ranks Technology Forecasting Methods. Scienmag.
