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The Scientific World Journal
Volume 2014, Article ID 294183, 8 pages
Research Article

Analytical Properties of Credibilistic Expectation Functions

1Department of Industrial & Systems Engineering, National University of Singapore, Block E1A No. 05-19, 1 Engineering Drive 2, Singapore 117576
2Graduate School of Information, Production & Systems, Waseda University, Hibikino 2-7, Wakamatsu-ku, Kitakyshu, Fukuoka, Japan

Received 13 August 2013; Accepted 16 January 2014; Published 26 February 2014

Academic Editors: C. Park and D. von Rosen

Copyright © 2014 Shuming Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The expectation function of fuzzy variable is an important and widely used criterion in fuzzy optimization, and sound properties on the expectation function may help in model analysis and solution algorithm design for the fuzzy optimization problems. The present paper deals with some analytical properties of credibilistic expectation functions of fuzzy variables that lie in three aspects. First, some continuity theorems on the continuity and semicontinuity conditions are proved for the expectation functions. Second, a differentiation formula of the expectation function is derived which tells that, under certain conditions, the derivative of the fuzzy expectation function with respect to the parameter equals the expectation of the derivative of the fuzzy function with respect to the parameter. Finally, a law of large numbers for fuzzy variable sequences is obtained leveraging on the Chebyshev Inequality of fuzzy variables. Some examples are provided to verify the results obtained.