Motivated by intuitionistic fuzzy sets and fzzy linguistic approach, this article proposes the concept of linguistic intuitionistic fuzzy numbers (LIFNs) where membership and and nonmembership are represented as linguistic terms. In order to process the multiple attribute decision making (MADM) with LIFNs, we introduce the linguistic score index and linguistic accuracy index of the LIFN. Simultaneously, the operation laws for LIFNs are defined and the related properties of the operation laws are studied. Further, some aggregation operators are developed, involving the linguistic intuitionistic fuzzy weighted averaging (LIFWA) operator, linguistic intuitionistic fuzzy ordered weighted averaging (LIFOWA) operator and linguistic intuitionistic fuzzy hybrid averaging (LIFHA) operator, etc., which can be utilized to aggregate preference information taking the form of LIFNs. Based on the LIFWA and the LIFHA operators, we propose an approach to handle MADM under LIFNs environment. Finally, an illustrative example is given to verify the feasibility and effectiveness of the proposed method, which are then compared to other representative methods.

In the paper, we express uncertain assessments information in linguistic multi-criteria decision makings (LMCDMs) as linguistic intuitionistic fuzzy sets, i.e., the decision maker provides membership and nonmembership fuzzy linguistic terms to represent uncertain assessments information of alternatives in LM-CDMs, and present Hamming distance between two linguistic intuitionistic fuzzy sets. Then we propose the linguistic intuitionistic fuzzy set TOPSIS method for LMCDMs, compared with the traditional TOP-SIS method, we provide different the positive ideal solution, the negative ideal solution and the relative closeness degrees of alternatives, in addition, we design an algorithm to finish the linguistic intuitionistic fuzzy set TOPSIS method for LMCDMs. We utilize a LMCDM problem to illustrate the performance, usefulness and effectiveness of the linguistic intuitionistic fuzzy set TOPSIS method, and compare it with the hesitant fuzzy linguistic multi-criteria optimization and compromise solution (HFL-VIKOR) method, the symbolic aggregation-based method and the hesitant fuzzy linguistic TOPSIS (HFL-TOPSIS) method in the example, results show that the linguistic intuitionistic fuzzy set TOPSIS method is a useful and alternative method for LMCDMs.

In this paper, a decision model based on a fuzzy soft set and ideal solution approaches is proposed. This new decision-making method uses the divide-and-conquer algorithm, and it is different from the existing algorithm (the choice value based approach and the comparison score based approach). The ideal solution is generated according to each attribute (pros or cons of the attributes, with or without constraints) of the fuzzy soft sets. Finally, the weighted Hamming distance is used to compute all possible alternatives and get the final result. The core of the decision process is the design phase, the existing decision models based on soft sets mostly neglect the analysis of attributes and decision objectives. This algorithm emphasizes the correct expression of the purpose of the decision maker and the analysis of attributes, as well as the explicit decision function. Additionally, this paper shows the fact that the rank reversal phenomenon occurs in the comparison score algorithm, and an example is provided to illustrate the rank reversal phenomenon. Experiments indicate that the decision model proposed in this paper is efficient and will be useful for practical problems. In addition, as a general model, it can be extended to a wider range of fields, such as classifications, optimization problems, etc.