Soft Computing

, Volume 21, Issue 19, pp 5787–5804 | Cite as

Interval-valued intuitionistic fuzzy programming technique for multicriteria group decision making based on Shapley values and incomplete preference information

  • Wenkai Zhang
  • Yanbing Ju
  • Xiaoyue Liu
Methodologies and Application


In the classical LINear programming technique for Multidimensional Analysis of Preference (LINMAP), the criteria are assumed to be independent, which may cause distorted results. The aim of this paper is to employ the LINMAP to develop an interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for multicriteria group decision making considering not only the importance of criteria and that of their ordered positions but also the interactions among criteria and those among their ordered positions. In the spirit of LINMAP, new group consistency and group inconsistency indices based on cross-entropy and Shapley values are firstly defined by considering the interactions among criteria and among their ordered positions. To simultaneously determine the fuzzy measures on criteria set and on ordered set, a bi-objective IVIF programming model is then constructed by minimizing the inconsistency index and maximizing the consistency index. The proposed IVIF programming model is subsequently solved by transforming it into a multiobjective programming model. Furthermore, some generalizations of the proposed programming model are investigated. Finally, two examples are given to illustrate the application of the proposed method, and its superiority is demonstrated by comparing the performance of the proposed method with that of the existing methods.


Multicriteria group decision making (MCGDM) Interval-valued intuitionistic fuzzy sets (IVIFSs) Interval-valued intuitionistic fuzzy programming Shapley values Incomplete preference information structure 



The authors thank the editor and the anonymous reviewers for their valuable comments which helped us to improve the quality of this paper significantly. This research is supported by Program for New Century Excellent Talents in University (NCET-13-0037), Humanities and Social Sciences Foundation of Ministry of Education of China (No. 14YJA630019), and China Scholarship Council (No. 201506030031).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Management and EconomicsBeijing Institute of TechnologyBeijingChina
  2. 2.School of Information Technology and ManagementUniversity of International Business and EconomicsBeijingChina

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