A graph based group decision making approach with intuitionistic fuzzy preference relations Qiong Mou a,b , Zeshui Xu a,⇑ , Huchang Liao a a Business School, Sichuan University, Chengdu, Sichuan 610064, China b School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China article info Article history: Received 4 June 2016 Received in revised form 16 January 2017 Accepted 27 May 2017 Available online 30 May 2017 Keywords: Multi-criteria group decision making Intuitionistic fuzzy preference relation Best-worst method Consistency Healthcare appointment registration system abstract Intuitionistic fuzzy preference relation (IFPR) is an efficient tool in tackling comprehensive multi-criteria group decision making (MCGDM) problems via pairwise comparisons. Based on the intuitionistic fuzzy analytic hierarchy process (IFAHP) and the best-worst method (BWM), this paper aims to put forward a novel graph-based group decision making approach called the intuitionistic fuzzy best-worst method (IF-BWM) for MCGDM. To achieve this goal, we first aggregate the individual IFPRs provided by the deci- sion makers into a collective IFPR by the intuitionistic fuzzy weighted averaging (IFWA) operator. Afterwards, we draw the directed network according to the collective IFPR, and then design an algorithm to identify the best and worst criteria through computing the out-degrees and in-degrees of the directed network. Furthermore, to derive the weights of criteria, some mathematical models corresponding to the different definitions of consistent IFPR are developed. Finally, the procedure of the IF-BWM is proposed for practical applications and three numerical examples are given to illustrate the approach. Ó 2017 Published by Elsevier Ltd. 1. Introduction Analytic hierarchy process (AHP), as a classic theory of mea- surement, was originally introduced by Saaty (1980), and has become one of the most important decision making techniques. By decomposing a complex problem into a multi-level hierarchic structure of objectives, criteria, sub-criteria and alternatives, the AHP can assist the decision maker to describe the general decision operation when it was applied to decision making. The procedure of AHP can be divided into three steps: (1) providing a fundamental scale of relative magnitudes expressed in dominance units to rep- resent the judgments of pairwise comparisons; (2) deriving the ratio scales of relative magnitudes expressed in priority units from each set of comparisons; (3) synthesizing the ratio scales of prior- ities and then obtaining the ranking of alternatives (Saaty, 1990). The AHP has been applied comprehensively to solve various deci- sion making problems, such as the U.S.-OPEC Energy Conflict (Saaty, 1979), the marketing investment (Smyth & Lecoeuvre, 2015), the evaluation of information and communication technol- ogy (ICT) business alternatives (Angelou & Economides, 2009), and so on. In the classic AHP model, the relative magnitudes of pairwise comparisons over different criteria are represented by crisp numbers within the 1-9 scale. However, in some realistic situa- tions, people find that they encounter difficulties in assigning the crisp evaluation values to the comparison judgments due to some objective or subjective reasons such as knowledge limitation, indi- vidual interest and personal preferences, complexities and fuzzi- ness of the things, etc. Hence, even though the AHP has been popular and simple in handling multi-criteria decision making (MCDM) problems, it is often criticized for its inability to tackle the inherent uncertainty and vagueness effectively (Xu & Liao, 2014). In order to improve the ability of AHP, some innovative theo- ries, such as the fuzzy set theory (Zadeh, 1965) and the intuition- istic fuzzy set (IFS) theory (Atanassov, 2012), etc. have been applied to combine with the classical AHP. Thus, a succession of extended methods under uncertain circumstances have been developed, which include the fuzzy AHP (FAHP) (Ajami & Ketabi, 2012; Chena, Hsieha, & Do, 2015; Wang, Luo, & Hua, 2008) and the intuitionistic fuzzy analytic hierarchy process (IFAHP) (Liao & Xu, 2015; Xu, 2007; Xu & Liao, 2014), etc. Concerning the FAHP, the earliest study was initiated by Van Laarhoven and Pedrycz (1983). Through directly extending the classical AHP with, respec- tively, triangular fuzzy numbers and trapezoidal fuzzy numbers, Van Laarhoven and Pedrycz (1983) and Buckley (1985) derived fuzzy weights and fuzzy performance scores to rank alternatives. Boender, de Graan, and Lootsma (1989) proposed a more robust http://dx.doi.org/10.1016/j.cie.2017.05.033 0360-8352/Ó 2017 Published by Elsevier Ltd. ⇑ Corresponding author. E-mail addresses: mouqiong@126.com (Q. Mou), xuzeshui@263.net (Z. Xu), liaohuchang@163.com (H. Liao). Computers & Industrial Engineering 110 (2017) 138–150 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie