Dealing with uncertainty is always a challenging problem, and different tools have been proposed to deal with it. Recently, a new model that is based on hesitant fuzzy sets has been presented to manage situations in which experts hesitate between several values to assess an indicator, alternative, variable, etc. Hesitant fuzzy sets suit the modeling of quantitative settings; however, similar situations may occur in qualitative settings so that experts think of several possible linguistic values or richer expressions than a single term for an indicator, alternative, variable, etc. In this paper, the concept of a hesitant fuzzy linguistic term set is introduced to provide a linguistic and computational basis to increase the richness of linguistic elicitation based on the fuzzy linguistic approach and the use of context-free grammars by using comparative terms. Then, a multicriteria linguistic decision-making model is presented in which experts provide their assessments by eliciting linguistic expressions. This decision model manages such linguistic expressions by means of its representation using hesitant fuzzy linguistic term sets.

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Hesitant Fuzzy Linguistic Term Sets for Decision Making

Multi-Criteria Decision Aid (MCDA) or Multi-Criteria Decision Making (MCDM) methods have received much attention from researchers and practitioners in evaluating, assessing and ranking alternatives across diverse industries. Among numerous MCDA/MCDM methods developed to solve real-world decision problems, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) continues to work satisfactorily across different application areas. In this paper, we conduct a state-of-the-art literature survey to taxonomize the research on TOPSIS applications and methodologies. The classification scheme for this review contains 266 scholarly papers from 103 journals since the year 2000, separated into nine application areas: (1) Supply Chain Management and Logistics, (2) Design, Engineering and Manufacturing Systems, (3) Business and Marketing Management, (4) Health, Safety and Environment Management, (5) Human Resources Management, (6) Energy Management, (7) Chemical Engineering, (8) Water Resources Management and (9) Other topics. Scholarly papers in the TOPSIS discipline are further interpreted based on (1) publication year, (2) publication journal, (3) authors' nationality and (4) other methods combined or compared with TOPSIS. We end our review paper with recommendations for future research in TOPSIS decision-making that is both forward-looking and practically oriented. This paper provides useful insights into the TOPSIS method and suggests a framework for future attempts in this area for academic researchers and practitioners.

Review: A state-of the-art survey of TOPSIS applications

With respect to multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers, some new group decision making analysis methods are developed. Firstly, some operational laws, score function and accuracy function of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers are introduced. Then two new aggregation operators: induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator are proposed, and some desirable properties of the I-IFOWG and I-IIFOWG operators are studied, such as commutativity, idempotency and monotonicity. An I-IFOWG and IFWG (intuitionistic fuzzy weighted geometric) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers. Further, we extend the developed models and procedures based on I-IIFOWG and IIFWG (interval-valued intuitionistic fuzzy weighted geometric) operators to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of interval-valued intuitionistic fuzzy numbers. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness.

Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making

In this paper, we investigate the hesitant fuzzy multiple attribute decision making (MADM) problems in which the attributes are in different priority level. Motivated by the ideal of prioritized aggregation operators [R.R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008) 263-274], we develop some prioritized aggregation operators for aggregating hesitant fuzzy information, and then apply them to develop some models for hesitant fuzzy multiple attribute decision making (MADM) problems in which the attributes are in different priority level. Finally, a practical example about talent introduction is given to verify the developed approaches and to demonstrate its practicality and effectiveness.

Hesitant fuzzy prioritized operators and their application to multiple attribute decision making

Computing with Words (CW) methodology has been used in several different environments to narrow the differences between human reasoning and computing. As Decision Making is a typical human mental process, it seems natural to apply the CW methodology in order to create and enrich decision models in which the information that is provided and manipulated has a qualitative nature. In this paper we make a review of the developments of CW in decision making. We begin with an overview of the CW methodology and we explore different linguistic computational models that have been applied to the decision making field. Then we present an historical perspective of CW in decision making by examining the pioneer papers in the field along with its most recent applications. Finally, some current trends, open questions and prospects in the topic are pointed out.

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Computing with words in decision making: foundations, trends and prospects

Interval-valued intuitionistic fuzzy (IVIF) sets are useful to deal with fuzziness inherent in decision data and decision-making processes. The aim of this paper is to develop a nonlinear-programming methodology that is based on the technique for order preference by similarity to ideal solution to solve multiattribute decision-making (MADM) problems with both ratings of alternatives on attributes and weights of attributes expressed with IVIF sets. In this methodology, nonlinear-programming models are constructed on the basis of the concepts of the relative-closeness coefficient and the weighted-Euclidean distance. Simpler auxiliary nonlinear-programming models are further deduced to calculate relative-closeness of IF sets of alternatives to the IVIF-positive ideal solution, which can be used to generate the ranking order of alternatives. The proposed methodology is validated and compared with other similar methods. A real example is examined to demonstrate the applicability and validity of the methodology proposed in this paper.

Corrections to “TOPSIS-Based Nonlinear-Programming Methodology for Multi-attribute Decision Making With Interval-Valued Intuitionistic Fuzzy Sets”

The concept of an intuitionistic fuzzy number (IFN) is of importance for quantifying an ill-known quantity, and the ranking of IFNs is a very difficult problem. The aim of this paper is to introduce the concept of a triangular IFN (TIFN) as a special case of the IFN and develop a new methodology for ranking TIFNs. Firstly the concepts of TIFNs and cut sets as well as arithmetical operations are introduced. Then the values and ambiguities of the membership function and the non-membership function for a TIFN are defined. A new ranking method is developed on the basis of the concept of a ratio of the value index to the ambiguity index and applied to multiattribute decision making problems in which the ratings of alternatives on attributes are expressed with TIFNs. The validity and applicability of the proposed method, as well as analysis of the comparison with other methods, are illustrated with a real example.

A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems

Linear programming method for multiattribute group decision making using IF sets

Multiattribute decision making (MADM) with multiple formats of information, which is called heterogeneous MADM for short, is very complex and interesting in applications. The purpose of this paper is to extend the Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) for solving heterogeneous MADM problems which involve intuitionistic fuzzy (IF) sets (IFSs), trapezoidal fuzzy numbers (TrFNs), intervals and real numbers. In this method, DM's preference is given through pair-wise comparisons of alternatives with hesitation degrees which are represented as IFSs. The IF consistency and inconsistency indices are defined on the basis of pair-wise comparisons of alternatives. Each alternative is assessed on the basis of its distance to a fuzzy ideal solution (FIS) unknown a priori . Based on the defined IF consistency and inconsistency indices, we construct a new fuzzy mathematical programming model, which is solved by the developed method of fuzzy mathematical programming with IFSs. Once the FIS and the attribute weights are obtained, we can calculate the distances of all alternatives to the FIS, which are used to determine the ranking order of the alternatives. A supplier selection example is presented to demonstrate the validity and applicability of the proposed method.

Fuzzy LINMAP approach to heterogeneous MADM considering comparisons of alternatives with hesitation degrees

Fuzziness is inherent in decision data and decision making process. In this paper, interval-valued intuitionistic fuzzy (IVIF) sets are used to capture fuzziness in multiattribute decision making (MADM) problems. The purpose of this paper is to develop a methodology for solving MADM problems with both ratings of alternatives on attributes and weights being expressed with IVIF sets. In this methodology, a weighted absolute distance between IF sets is defined using weights of IF sets. Based on the concept of the relative closeness coefficients, we construct a pair of nonlinear fractional programming models which can be transformed into two simpler auxiliary linear programming models being used to calculate the relative closeness coefficient intervals of alternatives to the IVIF positive ideal solution, which can be employed to generate ranking order of alternatives based on the concept of likelihood of interval numbers. The proposed method is illustrated with a real example.

Deng-Feng Li

Papers

Corrections to “TOPSIS-Based Nonlinear-Programming Methodology for Multi-attribute Decision Making With Interval-Valued Intuitionistic Fuzzy Sets”

A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems

Linear programming method for multiattribute group decision making using IF sets

Fuzzy LINMAP approach to heterogeneous MADM considering comparisons of alternatives with hesitation degrees

Linear programming method for MADM with interval-valued intuitionistic fuzzy sets