Hesitant Fuzzy Entropy and Cross-Entropy and Their Use in Multiattribute Decision-Making Zeshui Xu, 1,∗ Meimei Xia 2,† 1 Institute of Sciences, PLA University of Science and Technology, Nanjing 210007, PR China 2 School of Economics and Management, Tsinghua University, Beiing 100084, PR China We introduce the concepts of entropy and cross-entropy for hesitant fuzzy information, and discuss their desirable properties. Several measure formulas are further developed, and the relationships among the proposed entropy, cross-entropy, and similarity measures are analyzed, from which we can find that three measures are interchangeable under certain conditions. Then we develop two multiattribute decision-making methods in which the attribute values are given in the form of hesitant fuzzy sets reflecting humans’ hesitant thinking comprehensively. In one method, the weight vector is determined by the hesitant fuzzy entropy measure, and the optimal alternative is obtained by comparing the hesitant fuzzy cross-entropies between the alternatives and the ideal solutions; in another method, the weight vector is derived from the maximizing deviation method and the optimal alternative is obtained by using the TOPSIS method. An actual example is provided to compare our methods with the existing ones. C  2012 Wiley Periodicals, Inc. 1. INTRODUCTION Entropy, cross-entropy, and similarity measures are three important research topics in the fuzzy set theory, which have been widely used in practical applica- tions, such as pattern recognition, medical diagnosis, clustering analysis, image processing, and decision-making. Entropy is the measure of fuzziness. 1 Since its appearance, entropy has received great attentions. De Luca and Termini 2 put forward some axioms to describe the fuzziness degree of a fuzzy set, 3 and proposed several entropy formulas based on Shannon’s function. Kaufmann 4 introduced an entropy formula for the fuzzy set by the metric distance between its membership degree function and the membership function of its nearest crisp set. Another method pre- sented by Yager 5 is to view the fuzziness degree of the fuzzy set in terms of a lack of ∗ Author to whom all correspondence should be addressed: e-mail: xuzeshui@263.net. † e-mail: meimxia@163.com. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 27, 799–822 (2012) C  2012 Wiley Periodicals, Inc. View this article online at wileyonlinelibrary.com. • DOI 10.1002/int.21548