A novel mathematical programming method for interval-valued Pythagorean fuzzy MCGDM with incomplete weight information

Zhen Jin; Xiaofang Deng; Gaili Xu; Zhen Jin; Xiaofang Deng; Gaili Xu

doi:10.3934/math.2026088

AIMS Mathematics

2026, Volume 11, Issue 1: 2131-2187. doi: 10.3934/math.2026088

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A novel mathematical programming method for interval-valued Pythagorean fuzzy MCGDM with incomplete weight information

1.
School of Arts and Sciences, Guangzhou Maritime University, Guangzhou 510725, China
2.
School of Artificial Intelligence, Guangzhou Maritime University, Guangzhou 510725, China
3.
School of Mathematics and Statistics, Guilin University of Technology, Guilin 541002, China

Received: 12 November 2025 Revised: 29 December 2025 Accepted: 07 January 2026 Published: 23 January 2026
MSC : 03E72, 90B50

The interval-valued Pythagorean fuzzy set possesses a strong capability to characterize uncertainty and fuzziness, and has been widely applied to multi-criteria group decision-making. However, extant studies seldom consider the fuzzy truth degrees in pairwise comparisons of alternatives and often overlook incomplete information regarding criteria weights. Therefore, this paper investigated interval-valued Pythagorean fuzzy multi-criteria group decision-making, incorporating both interval-valued Pythagorean fuzzy truth degrees for pairwise comparisons and incomplete information on criterion weights. First, recognizing that decision-makers may have different weights under different criteria, their weights with respect to each criterion were determined based on the relative closeness of each alternative to the positive ideal solution and the negative ideal solution under that criterion. To derive the criteria weights, this paper defined the interval-valued Pythagorean fuzzy positive ideal solution and the interval-valued Pythagorean fuzzy negative ideal solution, and established the interval-valued Pythagorean fuzzy group consistency index and inconsistency index. By minimizing the group inconsistency index, a bi-objective interval-valued Pythagorean fuzzy programming model was constructed and skillfully transformed into a linear programming model to compute the criteria weights. Subsequently, the relative closeness degree of each alternative for each decision-maker was calculated and used to generate individual rankings of the alternatives. To obtain a collective ranking, a multi-objective allocation model was established and then converted into a single-objective programming model for the solution. Finally, a wireless network selection example was provided to demonstrate the effectiveness of the proposed method.
- interval-valued Pythagorean fuzzy set,
- multi-criteria group decision-making,
- incomplete information
Citation: Zhen Jin, Xiaofang Deng, Gaili Xu. A novel mathematical programming method for interval-valued Pythagorean fuzzy MCGDM with incomplete weight information[J]. AIMS Mathematics, 2026, 11(1): 2131-2187. doi: 10.3934/math.2026088

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Abstract

The interval-valued Pythagorean fuzzy set possesses a strong capability to characterize uncertainty and fuzziness, and has been widely applied to multi-criteria group decision-making. However, extant studies seldom consider the fuzzy truth degrees in pairwise comparisons of alternatives and often overlook incomplete information regarding criteria weights. Therefore, this paper investigated interval-valued Pythagorean fuzzy multi-criteria group decision-making, incorporating both interval-valued Pythagorean fuzzy truth degrees for pairwise comparisons and incomplete information on criterion weights. First, recognizing that decision-makers may have different weights under different criteria, their weights with respect to each criterion were determined based on the relative closeness of each alternative to the positive ideal solution and the negative ideal solution under that criterion. To derive the criteria weights, this paper defined the interval-valued Pythagorean fuzzy positive ideal solution and the interval-valued Pythagorean fuzzy negative ideal solution, and established the interval-valued Pythagorean fuzzy group consistency index and inconsistency index. By minimizing the group inconsistency index, a bi-objective interval-valued Pythagorean fuzzy programming model was constructed and skillfully transformed into a linear programming model to compute the criteria weights. Subsequently, the relative closeness degree of each alternative for each decision-maker was calculated and used to generate individual rankings of the alternatives. To obtain a collective ranking, a multi-objective allocation model was established and then converted into a single-objective programming model for the solution. Finally, a wireless network selection example was provided to demonstrate the effectiveness of the proposed method.

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