1063-6706 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2019.2945251, IEEE Transactions on Fuzzy Systems 1 Information measures in Atanassov’s intuitionistic fuzzy environment and their application in decision-making Vishnu Singh, Shiv Prasad Yadav and Radko Mesiar Abstract—This paper considers some information measures such as normalized divergence measure, similarity measure, dissimilarity measure and normalized distance measure in an intuitionistic fuzzy environment (IFE), which measure the uncer- tainty and hesitancy, and which can be applied to the selection of alternatives in group decision problems. We introduce and study the continuity of considered measures. Next, we prove some results that can be used to generate measures for fuzzy sets as well as for Atanassov’s intuitionistic fuzzy sets (AIFSs) and we also prove some approaches to construct point measures from set measures in IFE. We define the weight set for one and many preference orders of alternatives. Next, we investigate the properties and results related to the weight set. Based on the weight set, we develop a model for finding the uncertain weights corresponding to attributes. Also, we develop a model to finding positive certain weights corresponding to each attribute by using uncertain weights. Finally, an algorithm for choosing the best alternative according to the preference orders of alternatives in decision-making problems is proposed and its validity is shown with the help of a numerical example. Index Terms—Fuzzy set, intuitionistic fuzzy set, normalized divergence measure, dissimilarity measure, inclusion measure, normalized distance measure. I. I NTRODUCTION Most of the information present in the real-life is uncertain in nature. Generally, the decision maker (DM) can not handle completely such complex information. Zadeh [42] introduced the concept of fuzzy set (FS) theory to model uncertainty by assigning the degree of an association called the membership degree. Several authors have given different types of measures to deal the uncertain information and have studied their theoretical properties and also established the interrelationship between them ([5], [10], [13], [17], [21], [26], [44]). There are several kinds of information which contain uncertainty as well as vagueness. Such information can not be modeled by FS theory. For such situation, Atannasov [1] gave the concept of the intuitionistic fuzzy set which handles the uncertainty as well as hesitation by assigning degrees known as membership Vishnu Singh and Shiv Prasad Yadav are associated with the Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667,India (e-mail: vsingh1@ma.iitr.ac.in; spyorfma@gmail.com). The first author is thankful to the Ministry of Human Resource and Development(MHRD), Govt. of India, with grant number OH-31-23-200-428. Radko Mesiar is associ- ated with the Faculty of Civil Engineering, Slovak University of Technology, Radlinskeho 11, Bratislava Sk-81005, Slovakia (e-mail: mesiar@math.sk), and with the Faculty of Science, Department of Algebra and Geometry, Palack´ y University Olomouc, 17. listopadu 12, Olomouc, 771 46, Czech Republic. He acknowledges the support of grants VEGA 1/0006/19 and of the project of Grant Agency of the Czech Republic (GA ˇ CR) no. 18-06915S. degree and non-membership degree. If the sum of membership and non-membership degrees at each point of the universe is one, then the Atannasov’s intuitionistic fuzzy set (AIFS) becomes FS. If the sum of membership and non-membership degrees at each point of the universe lies in the open interval (0, 1), then the AIFS is called pure AIFS. Atanassov [2] has also proposed some operations on AIFSs. During last decades, AIFS theory played an important role in modeling uncertain and vague systems, received much attention from the researchers and meaningful results were obtained in the field of decision-making problems [31], pattern recognition ([14], [34]) to name a few. Decision making is one of the popular branches of Operations Research in which the problem to choose the best alternative from a given set of feasible alternatives is considered. There exist several processes in literature but there are mainly four stages required to choose the best alternative: (i) Evaluate the set of feasible alternatives from given information; (ii) Determine the weight vector corresponding to alternatives or attributes which depend on DM; (iii) Aggregate alternatives by taking weight vector given by DM; (iv) Rank the alternatives in order of preferences and select the best one. A. Overview of the literature There are several information measures in IFE, such as divergence measures, similarity measures, dissimilarity mea- sures and distance measures. They model uncertain and vague information. The inclusion between two AIFSs can be mea- sured by the concept of inclusion measure [20] and the commonality between two AIFSs can be measured by the concept of similarity measure ([22], [23]). Also, new similarity measures are constructed and used in pattern recognition [14]. Moreover, several authors established the relationship between point similarity measures and similarity measures ([6], [24], [28]). The concept of distance measure in IFE and different types of distance measures are given in [37]. The concept of H-max distance measure of AIFSs is given in [31] and it is used in decision-making problems. Bustince [4] gave the indicator of interval inclusion grade for interval- valued FSs and their application to approximate reasoning. Grzegorzewski [19] gave distances and orderings in a family of IF numbers. The comparative analysis between similarity measures and distance measures in IFE are discussed from a pattern recognition point of view [34] and theoretical point of view [36]. The concept of divergence measures, local