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Yager has proposed the decision making under measure-based granular uncertainty, which can make decision with the aid of Choquet integral, measure and representative payoffs. The decision making under measure-based granular uncertainty is an effective tool to deal with uncertain issues. The intuitionistic fuzzy environment is the more real environment. Since the decision making under measure-based granular uncertainty is not based on intuitionistic fuzzy environment, it cannot effectively solve the decision issues in the intuitionistic fuzzy environment. Then, when the issues of decision making are under intuitionistic fuzzy environment, what is the decision making under measure-based granular uncertainty with intuitionistic fuzzy sets is still an open issue. To deal with this kind of issues, this paper proposes the decision making under measure-based granular uncertainty with intuitionistic fuzzy sets. The decision making under measure-based granular uncertainty with intuitionistic fuzzy sets can effectively solve the decision making issues in the intuitionistic fuzzy environment, in other words, it can extend the decision making under measure-based granular uncertainty to the intuitionistic fuzzy environment. Numerical examples are applied to verify the validity of the decision making under measure-based granular uncertainty with intuitionistic fuzzy sets. The experimental results demonstrate that the decision making under measure-based granular uncertainty with intuitionistic fuzzy sets can represent the objects successfully and make decision effectively. In addition, a practical application of applied intelligence is used to compare the performance between the proposed model and the decision making under measure-based granular uncertainty. The experimental results show that the proposed model can solve some decision problems that the decision making under measure-based granular uncertainty cannot solve.

Entailment for measure-based belief structures can extend the possible probability value range of variables on a space and obtain more information from variables. However, if the variable space comes from intuitionistic fuzzy sets, the classical entailment for measure-based belief structures will not work in this issue. To deal with this situation, we propose the entailment for intuitionistic fuzzy sets based on generalized belief structures in this paper to apply the entailment for measure based belief structures on space, which is made up of non-membership degree, membership degree and hesitancy degree of a given intuitionistic fuzzy sets. Numerical examples are mentioned to prove the effectively and flexibility of this proposed entailment model. The experimental results indicate that the proposed algorithm can extend the possible probability value range of variables of space efficiently and obtain more information from intuitionistic fuzzy sets.belief function, Dempster-Shafer evidence theory, entailment, generalized belief structures, intuitionistic fuzzy sets | INTRODUCTIONIn real life, there are a lot of uncertainties in the real world. To address the uncertainties, many mathematical theories are proposed, such as probability theory, 1 fuzzy sets (FS), 2 Dempster-Shafer evidence theory, 3,4 intuitionistic fuzzy sets (IFS), 5,6 information quality, 7-11 Z-number, 12 fuzzy reasoning, 13-15 D-number, 16-18 entropy function, [19][20][21][22] and belief structure. 23,24 Among these theories and models, the IFS 25 considering the degree of hesitancy, membership, and nonmembership of objects, can deal with the uncertainties more flexibly and accurately, which has been used in many fields widely, 26 such as the uncertainty decision making, 27,28 pattern recognition 29 and so on.Recently, Yager proposed the entailment for measure based belief structures with a space, which has the promising aspect. 30 However, what the entailment for measure-based belief structures for a given IFS is still an open issue to be addressed. This paper proposes entailment for intuitionistic fuzzy sets based on generalized belief structures, which treats hesitancy degree, non-membership degree and membership degree of a given intuitionistic fuzzy set as three variables in a space X . Assume V takes its value in the X and is an uncertain variable. If g 1 and g 2 be two generalized belief structures on X and g 2 contains g 1 , then the belief of the statement that V is true based on g 2 will be smaller than that of the statement that V is true based on g 1 and the plausibility of the statement that V is true based on g 2 will be bigger than that of the statement that V is true based on g 1 .The remain of this paper is structured as follows. Section 2 introduces the preliminary. Section 3 presents the entailment for intuitionistic fuzzy sets based on generalized belief structures. Section 4 illustrates the flexibility and accuracy of the entailment for intuitionistic fuzzy sets based on generalized belief structures. Section 5 summarizes the...

In practical application problems, the uncertainty of an unknown object is often very difficult to accurately determine, so Yager proposed the interval-valued entropies for Dempster-Shafer structures, which is based on Dempster-Shafer structures and classic Shannon entropy and is an interval entropy model. Based on Dempster-Shafer structures and classic Shannon entropy, the interval uncertainty of an unknown object is determined, which provides reference for theoretical research and provides help for industrial applications. Although the interval-valued entropies for Dempster-Shafer structures can solve the uncertainty interval of an object very efficiently, its application scope is only a traditional probability space. How to extend it to the evidential environment is still an open issue. This paper proposes interval-valued belief entropies for Dempster-Shafer structures, which is an extension of the interval-valued entropies for Dempster-Shafer structures. When the belief entropy degenerates to the classic Shannon entropy, the interval-valued belief entropies for Dempster-Shafer structures will degenerate into the interval-valued entropies for Dempster-Shafer structures. Numerical examples are applied to verify the validity of the interval-valued belief entropies for Dempster-Shafer structures. The experimental results demonstrate that the proposed entropy can obtain the interval uncertainty value of a given uncertain object successfully and make decision effectively.

Refined expected value decision rules can refine the calculation of the expected value and make decisions by estimating the expected values of different alternatives, which use many theories, such as Choquet integral, PM function, measure and so on. However, the refined expected value decision rules have not been applied to the orthopair fuzzy environment yet. To address this issue, in this paper we propose the refined expected value decision rules under the orthopair fuzzy environment, which can apply the refined expected value decision rules on the issues of decision making that is described in the orthopair fuzzy environment. Numerical examples were applied to verify the availability and flexibility of the new refined expected value decision rules model. The experimental results demonstrate that the proposed model can apply refined expected value decision rules in the orthopair fuzzy environment and solve the decision making issues with the orthopair fuzzy environment successfully.