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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.

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.

This paper aims to present a novel approach for computing two-player constant-sum matrix games laid on the notion of a “symbolic proportional” linguistic term set. It is not always possible to lay assessments based on uniformly and symmetrically distributed linguistic term sets; hence, the defined concept motivates the decision-makers to represent their opinions using 2-tuples composed of two proportional linguistic terms. The proportional 2-tuple linguistic representation of payoffs concerns linguistic labels, which do not certainly have to be symmetrically distributed or do not have the conventional prerequisite of having uniform distance among them. This representation confers an opportunity to describe the linguistic payoffs of a matrix game using members of a continuous linguistic scale domain. In our work, we have defined a two-player constant-sum proportional linguistic matrix game and proposed an approach of Proportional Linguistic Linear Programming (PLLP) to evaluate these game problems. The framed PLLP problem is then transformed into a crisp LPP that can be easily solved, decreasing the computation complexities involved in solving the linguistic decision-making problems. This perspective of proportional 2-tuples provides decision-makers an approach to represent their opinions not by just using one label, rather by proportional linguistic labels of the form ( δ u i , γ u i + 1 ) , where u i and u i + 1 are two successive linguistic terms, with  0 ≤ δ , γ ≤ 1 and δ + γ = 1 . Besides, some test examples are also presented to show the consistency of our designed approach. Further, the PLLP formulation is utilized to solve a Multi-Criteria Decision-Making problem based on actual-time linguistic data.

A new type of bi-matrix game called the bi-matrix game with 2-tuple linguistic information is proposed and the linguistic linear programming model is used to obtain the mixed set. The problem of linguistic information is transformed into a mathematical problem that can be solved quickly, which reduces the uncertainty and complexity caused by linguistic information. This paper takes the media industry as an example to illustrate the superiority and effectiveness of the bi-matrix game with 2-tuple linguistic information.

In this paper, a multi-objective model for two-stage fixed charge transportation planning problem is studied. The transportation process is considered to occur from manufacturing plants to the distributers and then from distributers to the customers. The availabilities at the manufacturing plants, capacities of the distributers and demand of the customers, all are considered to be fuzzy numbers. The proposed model is formulated with three conflicting goals or objective functions. The first objective is to minimize the total transportation cost involved in the whole transportation process. The second objective is to maximize the total quantity of the products to be transported, whereas minimizing the total deterioration that occurred during the transportation process is considered to be the third objective function. Fuzzy linguistic relations or preferences among the three objective functions are studied. A linear membership function is used to represent the fuzzy relative preferences between the objective functions. For solving the multi-objective problem, fuzzy goal programming technique is adopted with some linear and nonlinear membership functions. Finally, the proposed model is illustrated and solved for some simulated numerical data and some sensitivity analysis for the problem is also discussed. The best results for the solved numerical problem are found when hyperbolic membership functions are considered to model the aspiration levels for objective functions, whereas comparatively less significant results are found when linear membership functions are used to model the aspiration levels for objective functions.

Personnel selection is a challenging problem for any organization. The success of a project is determined by the human resources that handle the project. To make better personnel selections, researchers have adopted multi-criteria decision-making (MCDM) approaches. Among these, fuzzy-based MCDM methods are most frequently used, as they handle vagueness and imprecision better. Intuitionistic fuzzy set (IFS) is a popular MCDM context which provides degree of membership and non-membership for preference elicitation. In this work, we propose a novel decision-making framework that consists of two stages. In the first stage, a new extension to the popular VIKOR method is presented under IFS context. The positive and negative ideal solutions are determined, and VIKOR parameters are calculated using transformation procedure. The proposed method combines the strength of both interval-valued fuzzy set and IFS that is more effective in handling vagueness with a simple formulation setup. In the second stage, a personnel selection problem is used to validate the proposed framework. Finally, the superiority and weakness of the proposed framework are discussed by comparison with other methods.

A Pythagorean fuzzy set is the superset of fuzzy and intuitionistic fuzzy sets, respectively. Yager proposed the concept of Pythagorean fuzzy sets in which he relaxed the condition that sum of square of both membership degree and nonmembership degree of an element of a set must not be greater than 1. This paper introduces two new techniques to solve LR-type fully Pythagorean fuzzy linear programming problems with mixed constraints having unrestricted LR-type Pythagorean fuzzy numbers as variables and parameters by introducing unknown variables and using a ranking function. Furthermore, we show the equivalence of both the proposed methods and compare the solutions obtained by the two techniques. Besides this, we solve an already existing practical model using proposed techniques and compare the result.

Game theory has found successful applications in different areas to handle competitive situations among different persons or organizations. Several extensions of ordinary game theory have been studied by the researchers to accommodate the uncertainty and vagueness in terms of payoffs and goals. Matrix games with payoffs represented by interval numbers, fuzzy numbers, and intuitionistic fuzzy numbers have considered only the quantitative aspects of the problems. But in many situations, qualitative information plays a crucial role in representing the payoffs of a game problem. This work presents a valuable study on matrix games with payoff represented by linguistic intuitionistic fuzzy numbers (LIFNs). First, the paper defines some new operational-laws for LIFNs based on linguistic scale function (LSF) and studies their properties in detail. Next, we define a new aggregation operator called ‘generalized linguistic intuitionistic fuzzy weighted average (GLIFWA)’operator for aggregating LIFNs. Several properties and special cases of GLIFWA operator are also discussed. The LSF provides an ability to consider the different semantic situations in a single formulation during the aggregation process. Further, the paper introduces some basic results of matrix games with payoffs represented by LIFNs. We develop solution methods using a pair of auxiliary linear/nonlinear-programming models derived from a pair of nonlinear bi-objective programming models. Finally, a real-life numerical example is considered to demonstrate the validity and applicability of the developed methods.

Decision-making is significant in economics, education, management, industries, and many real-life situations. Every decision-making problem does not have crisp parameters and restrictions, and may have uncertainties and qualitative information. Trapezoidal Hesitant fuzzy sets are highly beneficial for dealing with situations that are more uncertain and have some qualitative aspects. First, this paper introduces four score functions on the class of trapezoidal hesitant fuzzy elements, namely mean-position, right-spread, inference, and support position score function to define a complete ranking principle on trapezoidal hesitant fuzzy elements. Secondly, we propose a complete ranking principle on the class of trapezoidal Hesitant fuzzy sets by generalizing the four proposed score functions on the set of trapezoidal hesitant fuzzy elements. Thirdly, we extend our work to suggest another complete ranking principle on the new important sub-class of the trapezoidal Hesitant fuzzy set. Fourthly, we compare our proposed complete ranking principle with a few existing important ranking principles discussed in the trapezoidal Hesitant fuzzy sets (TrHFS) class. Further, we establish two new simplex algorithms on a special class of trapezoidal Hesitant fuzzy sets to solve fully Hesitant fuzzy linear programming problems (HFLPP). These new algorithms incorporate the proposed complete ranking principle on the trapezoidal Hesitant fuzzy set. Finally, we solve a few numerical examples of linear programming problems with fully Hesitant fuzzy information to show the applicability and potentiality of the proposed linear programming method. Complete ranking principle in the class of TrHFS and the use of complete ranking principle in solving a fully HFLPP is proposed for the first time in the literature.

The modern global economy is becoming more challenging and it is hardly possible to minimize the inventory cost for inventory practitioners in the coming days. Basically, most of the enterprises deal with deteriorating items having flexible demand rate and follow natural idle time in the entire inventory process. Moreover, traditionally most of the research articles have been made under non-stop time frame, but in reality, in a day–night scenario there exists a natural idle time and hence the time consumed for inventory run time may be viewed as single shift or periodic model. Here we formulate an economic order quantity (EOQ) inventory model considering natural idle time and deterioration under some constraints and minimize the average inventory cost. Then, the model is converted into an equivalent fuzzy model, taking the demand and all the cost parameters as linguistic polynomial fuzzy set (LPFS). To defuzzify the model, we have adopted indexing method as well as α -cut method. To validate the novelty, numerical experimentations have also been analyzed with the help of metaheuristic and evolutionary algorithms like goat search algorithm (GSA) and particle swarm optimization (PSO). Comparative analysis reveals that GSA approach can give finer optimum (− 10 % cost reduction) than other approaches. The main findings of this research give a new technique of (linguistic term) fuzzification–defuzzification of the proposed model and a new solution procedure to optimize the periodic deteriorating inventory model under GSA. To justify this model, sensitivity analysis and graphical illustration have been done. Scopes of future work have been discussed for further improvement of research on optimization problems using metaheuristic algorithms.