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Keywords = indeterminacy resolving approach

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1 pages, 126 KB  
Retraction
RETRACTED: Brikaa et al. Resolving Indeterminacy Approach to Solve Multi-Criteria Zero-Sum Matrix Games with Intuitionistic Fuzzy Goals. Mathematics 2020, 8, 305
by M. G. Brikaa, Zhoushun Zheng and El-Saeed Ammar
Mathematics 2025, 13(15), 2502; https://doi.org/10.3390/math13152502 - 4 Aug 2025
Viewed by 666
Abstract
The journal retracts the article “Resolving Indeterminacy Approach to Solve Multi-Criteria Zero-Sum Matrix Games with Intuitionistic Fuzzy Goals” [...] Full article
22 pages, 3682 KB  
Article
Shortest Path Solution of Trapezoidal Fuzzy Neutrosophic Graph Based on Circle-Breaking Algorithm
by Lehua Yang, Dongmei Li and Ruipu Tan
Symmetry 2020, 12(8), 1360; https://doi.org/10.3390/sym12081360 - 14 Aug 2020
Cited by 12 | Viewed by 4658
Abstract
The shortest path problem is a topic of increasing interest in various scientific fields. The damage to roads and bridges caused by disasters makes traffic routes that can be accurately expressed become indeterminate. A neutrosophic set is a collection of the truth membership, [...] Read more.
The shortest path problem is a topic of increasing interest in various scientific fields. The damage to roads and bridges caused by disasters makes traffic routes that can be accurately expressed become indeterminate. A neutrosophic set is a collection of the truth membership, indeterminacy membership, and falsity membership of the constituent elements. It has a symmetric form and indeterminacy membership is their axis of symmetry. In uncertain environments, the neutrosophic number can more effectively express the edge distance. The objectives in this study are to solve the shortest path problem of the neutrosophic graph with an edge distance expressed using trapezoidal fuzzy neutrosophic numbers (TrFNN) and resolve the edge distance according to the score and exact functions based on the TrFNN. Accordingly, the use of a circle-breaking algorithm is proposed to solve the shortest path problem and estimate the shortest distance. The feasibility of this method is verified based on two examples, and the rationality and effectiveness of the approach are evaluated by comparing it with the Dijkstra and Bellman algorithms. Full article
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30 pages, 2362 KB  
Article
RETRACTED: Resolving Indeterminacy Approach to Solve Multi-Criteria Zero-Sum Matrix Games with Intuitionistic Fuzzy Goals
by M. G. Brikaa, Zhoushun Zheng and El-Saeed Ammar
Mathematics 2020, 8(3), 305; https://doi.org/10.3390/math8030305 - 25 Feb 2020
Cited by 25 | Viewed by 3507 | Retraction
Abstract
The intuitionistic fuzzy set (IFS) is applied in various decision-making problems to express vagueness and showed great success in realizing the day-to-day problems. The principal aim of this article is to develop an approach for solving multi-criteria matrix game with intuitionistic fuzzy (I-fuzzy) [...] Read more.
The intuitionistic fuzzy set (IFS) is applied in various decision-making problems to express vagueness and showed great success in realizing the day-to-day problems. The principal aim of this article is to develop an approach for solving multi-criteria matrix game with intuitionistic fuzzy (I-fuzzy) goals. The proposed approach introduces the indeterminacy resolving functions of I-fuzzy numbers and discusses the I-fuzzy inequalities concept. Then, an effective algorithm based on the indeterminacy resolving algorithm is developed to obtain Pareto optimal security strategies for both players through solving a pair of multi-objective linear programming problems constructed from two auxiliary I-fuzzy programming problems. It is shown that this multi-criteria matrix game with I-fuzzy goals is an extension of the multi-criteria matrix game with fuzzy goals. Moreover, two numerical simulations are conducted to demonstrate the applicability and implementation process of the proposed algorithm. Finally, the achieved numerical results are compared with the existing algorithms to show the advantages of our algorithm. Full article
(This article belongs to the Special Issue Operations Research Using Fuzzy Sets Theory)
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