In this paper, we propose an extension of hesitant fuzzy sets, i.e., proportional hesitant fuzzy sets (PHFSs), with the purpose of accommodating proportional hesitant fuzzy environments. The components of PHFSs, which are referred to as proportional hesitant fuzzy elements (PHFEs), contain two aspects of information provided by a decision-making team: the possible membership degrees in the hesitant fuzzy elements and their associated proportions. Based on the PHFSs, we provide a novel approach to addressing fuzzy multi-attribute group decision making (MAGDM) problems. Different from the traditional approach, this paper first converts fuzzy MAGDM (expressed by classical fuzzy numbers) into proportional hesitant fuzzy multi-attribute decision making (represented by PHFEs), and then solves the latter through the proposal of a proportional hesitant fuzzy TOPSIS approach. In this process, preferences of the decision-making team are calculated as the proportions of the associated membership degrees. Finally, a numerical example and a comparison are provided to illustrate the reliability and effectiveness of the proposed approach.

Metabolic syndrome is worldwide public health problem and is a serious threat to people's health and lives. Understanding the relationship between metabolic syndrome and the physical symptoms is a difficult and challenging task, and few studies have been performed in this field. It is important to classify adults who are at high risk of metabolic syndrome without having to use a biochemical index and, likewise, it is important to develop technology that has a high economic rate of return to simplify the complexity of this detection. In this paper, an artificial intelligence model was developed to identify adults at risk of metabolic syndrome based on physical signs; this artificial intelligence model achieved more powerful capacity for classification compared to the PCLR (principal component logistic regression) model. A case study was performed based on the physical signs data, without using a biochemical index, that was collected from the staff of Lanzhou Grid Company in Gansu province of China. The results show that the developed artificial intelligence model is an effective classification system for identifying individuals at high risk of metabolic syndrome.

With complexity and uncertainty having an increasing impact on the decision-making environment, much attention is being paid to the development and application of multiple criteria group decisionmaking (MCGDM) models owing to the potential for fully exploiting the diverse strengths and expertise of various members. In general, inevitable interactions among decision makers (DMs), when a number of DMs share similar knowledge and experiences, can have a significant impact on the management of decision information directly or indirectly related to DMs, and can easily lead to distorted and unconvincing decision outcomes. In order to model the MCGDM problem in which DMs share a similar background, a consolidated MCGDM model in the context of intuitionistic fuzzy sets (IFSs) is developed. First, we refine the constructive principles for intuitionistic fuzzy entropy (IFE) and use them as a basis to produce a novel IFE measure simultaneously factoring in the intuitionism and fuzziness of IFSs. With the aim of dealing with the impact on the specifications of the weights of DMs and criteria, an integrated method is then proposed based on the novel IFE measure, 2-additive fuzzy measure, and Choquet integral. Due to their capability of modeling effectively the interrelationships among arguments, the weighted intuitionistic fuzzy Bonferroni mean (WIFBM) and the weighted intuitionistic fuzzy geometric Bonferroni mean (WIFGBM) are introduced to fuse the individual evaluation values of alternatives on criteria. In addition, simple additive weighting based on the WIFBM or WIFGBM is applied to rank alternatives and select the best one. Finally, the feasibility and effectiveness of the proposed model are explored with a case study of an emergency plan decision-making problem accompanied with sensitivity and comparison analysis.INDEX TERMS Intuitionistic fuzzy sets, Bonferroni mean, intuitionistic fuzzy entropy, 2-additive fuzzy measure, choquet integral, multiple criteria group decision making.

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