An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives
Highlights
► A framework is proposed to handle multiattribute group decision making problems. ► Incomplete preference relations over alternatives are furnished by decision-makers. ► Assessments are given as linguistic variables and numerical values. ► Raw data are converted to interval-valued intuitionistic fuzzy numbers. ► A linear program is established to derive attribute weights and the positive ideal solution.
Introduction
When facing a decision situation, a decision-maker (DM) often has to evaluate a finite set of alternatives against multiple attributes. This process can be conveniently modeled as a multiattribute decision making (MADM) problem. Several formal procedures have been proposed to deal with MADM such as the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) (Hwang & Yoon, 1981) and the Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) (Srinivasan & Shocker, 1973). The LINMAP proves to be a practical and useful technique for determining attribute weights and a positive-ideal solution based on a DM’s pairwise comparisons of alternatives. In the traditional LINMAP, performance ratings are known precisely and given as crisp values. Under many practical decision situations, it is hard, if not impossible, to obtain exact assessment values due to inherent vagueness and uncertainty in human judgment. As such, Zadeh (1965) puts forward a powerful paradigm, fuzzy set theory, to handle ambiguity information that often arises in human decision processes. The LINMAP has subsequently been extended to handle MADM with fuzzy judgment data (Li & Yang, 2004).
In Zadeh’s fuzzy set, an element’s membership to a particular set is defined as a real value μ between 0 and 1 and its nonmembership is implied to be 1 − μ. This extension of traditional binary logic provides a powerful framework to characterize vagueness and uncertainty. The treatment of nonmembership as a complement of membership essentially omits a DM’s hesitation in the decision making process. To facilitate further characterization of uncertainty and vagueness, Atanassov (1986) introduces intuitionistic fuzzy sets (IFSs), depicted by real-valued membership, nonmembership, and hesitancy functions. Due to its capability of accommodating hesitation in human decision processes, IFSs have been widely recognized as flexible and practical tools for tackling imprecise and uncertain decision information (Xu & Cai, 2010) and have been widely applied to the field of decision modeling. For instance, Li (2005) proposes a linear programming method to handle MADM using IFSs; Wei (2010) develops an intuitionistic fuzzy weighted geometric operator-based approach to solve multi-attribute group decision making (MAGDM) problems; Li, Chen, and Huang (2010) extend the LINMAP method to solve MAGDM with intuitionistic fuzzy information.
An IFS is characterized by real-valued membership and nonmembership functions defined on [0, 1], and the hesitancy function can be easily derived based on the aforesaid two functions. In some decision situations with highly uncertain and imprecise judgment, it could pose a significant challenge to require that membership and nonmembership be identified as exact values. To address this issue, IFSs are further extended to interval-valued intuitionistic fuzzy sets (IVIFSs) (Atanassov & Gargov, 1989) where membership and nonmembership are represented as interval-valued functions. Since its inception, significant research has been conducted to develop and enrich the IVIFS theory, such as investigations on the correlation and correlation coefficients of IVIFSs (Bustince and Burillo, 1995, Hong, 1998, Hung and Wu, 2002), fuzzy cross entropy of IVIFSs (Ye, 2011), relationships between IFSs, L-fuzzy sets, interval-valued fuzzy sets and IVIFSs (Deschrijver, 2007, Deschrijver, 2008, Deschrijver and Kerre, 2007), similarity measures of IVIFSs (Wei et al., 2011, Xu and Chen, 2008), and comparison of the interval-valued intuitionistic numbers (IVIFNs) (Li and Wang, 2010, Wang et al., 2009, Xu, 2007). Thanks to their advantage in coping with uncertain decision data, IVIFSs have been widely applied to decision models with multiple attributes (Li, 2010a, Li, 2010b, Wang et al., 2009, Park et al., 2011, Li, 2011, Wang et al., 2011, Wei, 2010, Wei, 2011, Xu, 2007, Xu and Yager, 2008, Xu et al., 2011). Recently, researchers started to address MAGDM problems involving IVIFS decision data. For instance, Park, Kwun, Park, and Park (2009) investigate group decision problems based on correlation coefficients of IVIFSs. Xu (2010) introduces certain IVIFN relations and operations and proposes a distance-based method for group decisions. Ye (2010) develops a MAGDM method with IVIFNs to solve the partner selection problem of a virtual enterprise under incomplete information. Yue (2011) puts forward an approach to aggregate interval numbers into IVIFNs for group decisions. Chen, Wang, and Lu (2011) propose a framework to tackle MAGDM problem based on interval-valued intuitionistic fuzzy preference relations and interval-valued intuitionistic fuzzy decision matrices.
To the authors’ knowledge, little research has been carried out to handle MAGDM problems in which attribute values are converted to IVIFNs with unknown attribute weights and incomplete pairwise comparison preference relations on alternatives. In this research, the focus is to further extend the LINMAP method and develop a new approach to MAGDM problems with IVIFN decision data. In this paradigm, it is assumed that raw decision data are furnished as linguistic variables (for qualitative attributes) and numerical values (for quantitative attributes), then IVIFNs are constructed to reflect fuzziness and uncertainty contained in attribute assessment values and DMs’ subjective judgment. Group consistency and inconsistency indices are defined for pairwise comparison preference relations on alternatives. A linear program is proposed for deriving the interval-valued intuitionistic fuzzy positive ideal solution (IVIFPIS) and attribute weights. The distances of alternatives to the IVIFPIS are calculated to determine their ranking orders for individual DMs. Finally, a group ranking order can be generated using the Borda function (Hwang & Yoon, 1981). An earlier version of this paper was presented at a conference and published in the proceedings (Wang, Wang, & Li, 2011). This manuscript has significantly expanded the research reported therein by refining the modeling process, addressing certain technical deficiency, and furnishing two theorems to reveal useful properties of the proposed framework.
The remainder of the paper is organized as follows. Section 2 provides preliminaries on IVIFSs and Euclidean distance between IVIFNs. Section 3 formulates the MAGDM problem with IVIFNs and defines group consistency and inconsistency indices. Section 4 proposes an approach to handle MAGDM problems with IVIFNs, and a linear program is established to estimate the IVIFPIS and attribute weights. Section 5 presents a numerical example to demonstrate how to apply the proposed approach, followed by some concluding remarks in Section 6.
Section snippets
Preliminaries
Let Z be a fixed nonempty universe set, an IFS A in Z is an object in the following form (Atanassov, 1986):where and , satisfying , .
μA(z) and νA(z) denote, respectively, the degree of membership and nonmembership of element z to set A. In addition, for each IFS A in Z, πA(z) = 1 − μA(z) − νA(z) is often referred to as its intuitionistic fuzzy index, representing the degree of indeterminacy of z to A. Obviously, 0 ⩽ πA(z) ⩽ 1 for every z ∈ Z.
An MAGDM problem and group consistency measurement
This section presents an MAGDM problem with IVIFNs and defines group consistency and inconsistency indices.
A linear programming approach to the MAGDM problem
As the group inconsistency index B reflects the overall inconsistency between the derived Euclidean distance and the DMs’ judgment, the smaller the B, the better the model characterizes the DMs’ decision rationales. Therefore, a sensible attribute weight vector and IVIFPIS x∗ is to minimize the group inconsistency index B (Li et al. (2010) apply the similar treatment to handle multiattribute group decision making with intuitionistic fuzzy sets). Based on this consideration, the
An illustrative example
This section presents an MAGDM problem about recommending undergraduate students for graduate admission to demonstrate how to apply the proposed approach.
Without loss of generality, assume that there are three committee members (i.e., DMs) d1, d2, and d3, and four students x1, x2, x3, and x4 as the finalists after preliminary screening. All DMs agree to evaluate these candidates against four attributes, academic records (a1), college English test Band 6 score (a2), teamwork skills (a3), and
Conclusions
In a typical MAGDM problem, both quantitative and qualitative attributes are often involved and assessed with imprecise data and subjective judgment. This article first proposes mechanisms for converting numerical quantitative assessments and linguistic qualitative values into IVIFN decision data. Based on incomplete pairwise comparison preference relations furnished by the DMs, group consistency and inconsistency indices are introduced. The converted IVIFN decision data and group consistency
Acknowledgments
Kevin W. Li would like to acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) under its Discovery Grant program.
References (37)
Intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1986)- et al.
Interval-valued intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1989) - et al.
Correlation of interval-valued intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1995) - et al.
A multicriteria group decision-making approach based on interval-valued intuitionistic fuzzy sets: a comparative perspective
Expert Systems with Applications
(2011) Arithmetic operators in interval-valued fuzzy set theory
Information Sciences
(2007)A representation of t-norms in interval-valued L-fuzzy set theory
Fuzzy Sets and Systems
(2008)- et al.
On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision
Information Sciences
(2007) A note on correlation of interval-valued intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1998)- et al.
Correlation of intuitionistic fuzzy sets by centroid method
Information Sciences
(2002) Multiattribute decision making models and methods using intuitionistic fuzzy sets
Journal of Computer and System Sciences
(2005)