Search Results (9)

Decision-making involves selecting the best alternative based on evaluation criteria while considering environmental impacts. The translation of environmental factors into quantifiable mathematical expressions is challenging due to the inherent uncertainties. Decision-makers can address the subjective characteristics of alternatives by incorporating fuzzy set theory into decision-making processes where uncertainty and ambiguity exist. Game theory is introduced as another approach to enhance the robustness of decision-making models, leading to more informed and flexible decision outcomes. This approach promotes strategic thinking and aids decision-making by allowing individuals to visualize the potential consequences of different decisions under various conditions. This study proposes a fuzzy multi-criteria decision support system that provides a structured framework to address the complexities of agricultural land selection. The decision support system employs a two-person zero-sum game to identify the optimal land management option, considering the strategic interactions between players. The results from the payoff matrix reveal the equilibrium point, providing an ideal solution for more effective land use planning decisions. Full article

The choice of management strategy for companies operating in different sectors of the economy is of great importance for their sustainable development. In many cases, companies are in competition within the scope of the same activities, meaning that the profit of one company is at the expense of the other. The choice of strategies for each of the firms in this case can be optimized using game theory for a non-cooperative game case where the two players have antagonistic interests. The aim of this research is to develop a methodology which, in non-cooperative games, accounts for the benefits of different criteria for each of the strategies of the two participants. In this research a new integrated sequential interactive model for urban systems (SIMUS)–game theory technique for decision making in the case of non-cooperative games is proposed. The methodology includes three steps. The first step consists of a determination of the strategies of both players and the selection of criteria for their assessment. In the second step the SIMUS method for multi-criteria analysis is applied to identify the benefits of the strategies for both players according to the criteria. The model formation in game theory is drawn up in the third step. The payoff matrix of the game is formed based on the benefits obtained from the SIMUS method. The strategies of both players are solved by dual linear programming. Finally, to verify the results of the new approach we apply four criteria to make a decision—Laplace’s criterion, the minimax and maximin criteria, Savage’s criterion and Hurwitz’s criterion. The new integrated SIMUS–game theory approach is applied to a real example in the transport sector. The Bulgarian transport network is investigated regarding route and transport type selection for a carriage of containers between a starting point, Sofia, and a destination, Varna, in the case of competition between railway and road operators. Two strategies for a railway operator and three strategies for a road operator are examined. The benefits of the strategies for both operators are determined using the SIMUS method, based on seven criteria representing environmental, technological, infrastructural, economic, security and safety factors. The optimal strategies for both operators are determined using the game model and dual linear programming. It is discovered that the railway operator will apply their first strategy and that the road operator will also apply their first strategy. Both players will obtain a profit if they implement their optimal strategies. The new integrated SIMUS–game theory approach can be used in different areas of research, when the strategies for both players in non-cooperatives games need to be established. Full article

In this paper, we study the properties of Germeier’s scalarization applied for solving multicriteria games. The equilibria and the equilibrium values of such games, as a rule, make sets, and the problems of parametrizing and approximating these sets arise. Shapley proved that Nash equilibrium of multicriteria matrix game can be found by solving a two-parametric family of scalar games obtained with the help of linear scalarization of the criteria vector. We show that Germeier’s scalarization parametrizes the equilibria of the multicriteria game by using one-parametric family of scalar games. Germeier’s scalarization has certain advantages over the linear one, and we suggest it for approximating the multicriteria game equilibria with a finite set. For two-criteria games with $2\times 2$ matrices, we show by examples that there is no continuity of the values of scalar games in the scalarizing parameters. We prove one-sided (from the left or from the right) continuity for the game values. As a result, we come to convergence in Hausdorff metric for the set of equilibrium values obtained for $ϵ$-net on the simplex of scalarizing parameters to the value of the multicriteria game as $ϵ\to 0$. The constructed finite approximation may be helpful in practical applications, where players try to find a compromise in an iterative negotiating procedure under multiple criteria. Full article

This work provides a mathematical description of a process game for the safe driving of a ship that encounters other ships. State and control constraint variables, as well as a set of acceptable ship tactics, are taken into consideration. Multi-criteria optimization operations are developed as positional and matrix games, based on cooperative, non-cooperative, and classical (non-game) optimal steering control. Adequate algorithms for ship collision avoidance, relating to the above operations, are developed and verified through digital simulation of a real navigational situation using MATLAB/Simulink. Full article

The intuitionistic fuzzy set (IFS) is applied in various decision-making problems to express vagueness and showed great success in realizing the day-to-day problems. The principal aim of this article is to develop an approach for solving multi-criteria matrix game with intuitionistic fuzzy (I-fuzzy) goals. The proposed approach introduces the indeterminacy resolving functions of I-fuzzy numbers and discusses the I-fuzzy inequalities concept. Then, an effective algorithm based on the indeterminacy resolving algorithm is developed to obtain Pareto optimal security strategies for both players through solving a pair of multi-objective linear programming problems constructed from two auxiliary I-fuzzy programming problems. It is shown that this multi-criteria matrix game with I-fuzzy goals is an extension of the multi-criteria matrix game with fuzzy goals. Moreover, two numerical simulations are conducted to demonstrate the applicability and implementation process of the proposed algorithm. Finally, the achieved numerical results are compared with the existing algorithms to show the advantages of our algorithm. Full article

The classical matrix theory is deficient to express the vagueness of the real life. The fuzzy set theory has been successfully applied to bridge this gap. Much work has already been done on a two-person zero sum matrix game with fuzzy goals. In continuation, this paper is dedicated to define and study a multi-criteria two-person zero sum game with intuitionistic fuzzy goals. It is shown that solving such games is equivalent to solving two crisp multi object linear programming problems. Our work generalizes the previous study on a multi-criteria game with fuzzy goals by adopting the approach of linear programming with intuitionistic fuzzy sets. Finally, an illustrative numerical example is provided to elaborate the proposed approach. Full article

A multi-objective bi-matrix game model based on fuzzy goals is established in this paper. It is shown that the equilibrium solution of such a game model problem can be translated into the optimal solution of a multi-objective, non-linear programming problem. Finally, the results of this paper are demonstrated through a numerical example. Full article

In this article, we put forward the multi-objective matrix game model based on fuzzy payoffs. In order to solve the game model, we first discuss the relationship of two fuzzy numbers via the lower limit $-\frac{1}{2}$ of the possibility degree. Then, utilizing this relationship, we conclude that the equilibrium solution of this game model and the optimal solution of multicriteria linear optimization problems are of equal value. Finally, to illustrate the effectiveness and correctness of the obtained model, an example is provided. Full article