# M-CFIS-R: Mamdani Complex Fuzzy Inference System with Rule Reduction Using Complex Fuzzy Measures in Granular Computing

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Vietnam Academy of Science and Technology, Graduate University of Science and Technology, Hanoi 010000, Vietnam

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Faculty of Computer Science and Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi 010000, Vietnam

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Institute of Information Technology, Vietnam Academy of Science and Technology, Hanoi 010000, Vietnam

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Department of Industrial Management, National Taiwan University of Science and Technology, No. 43, Section 4, Keelung Road, Taipei 10607, Taiwan

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Taiwan Building Technology Center, National Taiwan University of Science and Technology, No. 43, Section 4, Keelung Road, Taipei 10607, Taiwan

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VNU Information Technology Institute, Vietnam National University, Hanoi 010000, Vietnam

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School of Information Technology, Deakin University, 221 Burwood Highway, Burwood Victoria 3125, Australia

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Author to whom correspondence should be addressed.

Received: 10 April 2020 / Revised: 20 April 2020 / Accepted: 22 April 2020 / Published: 3 May 2020

(This article belongs to the Special Issue Applications of Fuzzy Optimization and Fuzzy Decision Making)

Complex fuzzy theory has strong practical background in many important applications, especially in decision-making support systems. Recently, the Mamdani Complex Fuzzy Inference System (M-CFIS) has been introduced as an effective tool for handling events that are not restricted to only values of a given time point but also include all values within certain time intervals (i.e., the phase term). In such decision-making problems, the complex fuzzy theory allows us to observe both the amplitude and phase values of an event, thus resulting in better performance. However, one of the limitations of the existing M-CFIS is the rule base that may be redundant to a specific dataset. In order to handle the problem, we propose a new Mamdani Complex Fuzzy Inference System with Rule Reduction Using Complex Fuzzy Measures in Granular Computing called M-CFIS-R. Several fuzzy similarity measures such as Complex Fuzzy Cosine Similarity Measure (CFCSM), Complex Fuzzy Dice Similarity Measure (CFDSM), and Complex Fuzzy Jaccard Similarity Measure (CFJSM) together with their weighted versions are proposed. Those measures are integrated into the M-CFIS-R system by the idea of granular computing such that only important and dominant rules are being kept in the system. The difference and advantage of M-CFIS-R against M-CFIS is the usage of the training process in which the rule base is repeatedly changed toward the original base set until the performance is better. By doing so, the new rule base in M-CFIS-R would improve the performance of the whole system. Experiments on various decision-making datasets demonstrate that the proposed M-CFIS-R performs better than M-CFIS.