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In this paper, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is construed. The value of a linguistic matrix game is proven as a non-decreasing function of the linguistic values in the payoffs, and, hence, a pair of auxiliary linguistic linear programming (LLP) problems is formulated to obtain the linguistic lower bound and the linguistic upper bound of the interval-valued linguistic value of such class of games. The duality theorem of LLP is also adopted to establish the equality of values of the interval linguistic matrix game for players I and II. A flowchart to summarize the proposed algorithm is also given. The methodology is then illustrated via a hypothetical example to demonstrate the applicability of the proposed theory in the real world. The designed algorithm demonstrates acceptable results in the two-player constant-sum game problems with interval-valued 2-tuple linguistic payoffs.

The 2-tuple linguistic model initiated by Herrera and Martinez has become one of the most useful computing with words approaches. It provides an effective way of representing the linguistic evaluation in terms of 2-tuples to avoid information loss in decision-making process. This research aims at extending the combinative distance based assessment (CODAS) method by virtue of 2-tuple linguistic Fermatean fuzzy Hamacher aggregation operators. We put forth several different types of 2-tuple linguistic Fermatean fuzzy aggregation operators, including the 2-tuple linguistic Fermatean fuzzy Hamacher weighted averaging operator, the 2-tuple linguistic Fermatean fuzzy Hamacher weighted geometric operator, the 2-tuple linguistic Fermatean fuzzy Hamacher Hybrid averaging operator, and the 2-tuple linguistic Fermatean fuzzy Hamacher Hybrid geometric operator. Some fundamental properties of the proposed 2-tuple linguistic Fermatean fuzzy aggregation operators are investigated. Furthermore, we develop a 2-tuple linguistic Fermatean fuzzy CODAS method with the proposed aggregation operators for solving multi-attribute group decision-making problems. Numerical examples regarding the selection of construction companies and McDonald’s franchisees are used to show the effectiveness of the proposed extended CODAS approach. Finally, a comparative analysis is conducted to validate the proposed approach and show its advantages.

In this paper, we investigate the multiple attribute decision-making problems with picture 2-tuple linguistic information. Then, we utilize arithmetic and geometric operations to develop several picture 2-tuple linguistic aggregation operators. The prominent characteristic of these proposed operators is studied. Then, we have utilized these operators to develop some approaches to solving the picture 2-tuple linguistic multiple attribute decision-making problems. Finally, a practical example for enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.

This paper aims to put forth the theory of 2-tuple linguistic groups concerning the binary operation in the conventional sense. For this, a formal methodology has been introduced to prove that a predefined nonempty linguistic term set, LT , and the interval, [ - 1 2 , 1 2 ] , forms a group. Further, we have proved that a set of all 2-tuple linguistic information, LT ¯ ≡ L T × [ - 1 2 , 1 2 ] , and numerical interval, [ - n , n ] , where n is presumed to be a positive integer, also forms a group. Later on, we develop a one-to-one correspondence and homomorphic group relation between the set of all 2-tuple linguistic information and numerical interval, [ - n , n ] . Henceforth, a similarity relation between the two groups is obtained. Finally, a practical application is defined by proposing the notion of a 2-tuple linguistic bipolar graph to illustrate the usefulness and practicality of the group isomorphic relation.

In this contribution, a two-player constant-sum 2-tuple linguistic matrix game is described, and a linguistic linear programming (LLP) approach is proposed to solve this class of games. The proposed approach can be perceived as a unified mechanism in the sense that it can be adopted to solve linguistic matrix game problems, LLP problems, and linguistic multi-attribute decision-making (MADM) problems. The latter is exhibited by presenting examples of linguistic MADM problems modeled as two-player constant-sum linguistic matrix games with Nature as the second player.

Considering that interval 2-tuple linguistic model can solve the interval linguistic multi-criteria decision-making problems, and Choquet integral can deal with the multi-criteria group decision-making problems with correlated criteria, in the revelation of these two methods, this paper combined them together to cope with the interval linguistic multi-criteria group decision-making problems with correlated criteria. Because of the limitations of 2-tuple linguistic representation model, a modified composite scale based on 2-tuple linguistic model was proposed which can better improve the accuracy of decision-making. First, using Choquet integral and interval 2-tuple linguistic model, two aggregation operators with their properties were introduced to better reflect correlations among criteria, i.e., the interval 2-tuple correlated averaging operator and the interval 2-tuple correlated geometric operator. Then, a multi-criteria group decision-making method based on the above operators was proposed. Finally, the proposed method was compared with the existing ones based on the same illustrative example to confirm its feasibility and rationality.

The aim of this paper is to propose the 2-tuple spherical fuzzy linguistic aggregation operators and a decision-making approach to deal with uncertainties in the form of 2-tuple spherical fuzzy linguistic sets. 2-tuple spherical fuzzy linguistic operators have more flexibility than general fuzzy set. We proposed a numbers of aggregation operators, namely 2-tuple spherical fuzzy linguistic weighted average, 2-tuple spherical fuzzy linguistic ordered weighted average, 2-tuple spherical fuzzy linguistic hybrid average, 2-tuple spherical fuzzy linguistic weighted geometric, 2-tuple spherical fuzzy linguistic ordered geometric, and 2-tuple spherical fuzzy linguistic hybrid geometric operators. The distinguishing feature of these proposed operators is studied. At that point, we have used these operators to design a model to deal with multiple attribute decision-making issues under the 2-tuple spherical fuzzy linguistic information. Then, a practical application for best company selection for feeds is given to prove the introduced technique and to show its practicability and effectiveness. Besides this, a systematic comparison analysis with other existent methods is conducted to reveal the advantage of our method. Results indicate that the proposed method is suitable and effective for decision making problems.

This paper attempts to develop an outranking method based on intuitionistic 2-tuple fuzzy linguistic information and apply it to multi-criteria group decision-making problems. The theory of intuitionistic 2-tuple fuzzy linguistic elements (I2FLEs) is useful for modeling impressions and quantifying the ambiguous nature of subjective judgments in a convenient manner. The family of ELECTRE methods is well known and widely used outranking method, but it has not been investigated within the intuitionistic 2-tuple fuzzy linguistic environment. ELECTRE uses the concept of an outranking relationship. Therefore, the directional Hausdorff distance, which uses I2FLEs, is proposed and the outranking relations are subsequently defined using this distance. Furthermore, using the proposed score and accuracy functions of I2FLEs, this paper also develops an outranking method based on intuitionistic fuzzy support function, intuitionistic fuzzy risk function, intuitionistic fuzzy credibility function, and the net outranking flow index similar to the PROMETHEE method. An illustrative example is given to demonstrate the practicality and effectiveness of the developed approaches. Finally, a comparative discussion of two decision-making methods is conducted to demonstrate the advantages of the proposed method over the other.

The general mitigating and aggravating circumstances of criminal liability are elements attached to the crime that, when they occur, affect the punishment quantum. Cuban criminal legislation provides a catalog of such circumstances and some general conditions for their application. Such norms give judges broad discretion in assessing circumstances and adjusting punishment based on the intensity of those circumstances. In the interest of broad judicial discretion, the law does not establish specific ways for measuring circumstances’ intensity. This gives judges more freedom and autonomy, but it also imposes on them more social responsibility and challenges them to manage the uncertainty and subjectivity inherent in this complex activity. This paper proposes a model to aid the linguistic assessment of circumstances’ intensity and to provide linguistic and numerical recommendations to determine an appropriate punishment interval. M-LAMAC determines the collective evaluation of circumstances of the same type, determines the prevalence of a type of circumstance by means of a compensation function, recommends the required modification in the input interval, and finally recommends a numerical interval adjusted to the judges’ initially expressed preferences. The model’s applicability is demonstrated by means of several experiments on a fictitious case of bank document forgery.

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 non-membership fuzzy linguistic terms to represent uncertain assessments information of alternatives in LMCDMs, 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 TOPSIS 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.