Auxiliary Linear Programming Models Research Articles

A Mathematical Model for Fuzzyp-Median Problem with Fuzzy Weights and Variables

We investigate thep-median problem with fuzzy variables and weights of vertices. The fuzzy equalities and inequalities transform to crisp cases by using some technique used in fuzzy linear programming. We show that the fuzzy objective function also can be replaced by crisp functions. Therefore an auxiliary linear programming model is obtained for the fuzzyp-median problem. The results are compared with two previously proposed methods.

Linear programming technique for solving interval-valued constraint matrix games

The purpose of this paper is to propose an effective linear programming technique for solving matrix games in which the payoffs are expressed with intervals and the choice of strategies for players is constrained, i.e., interval-valued constraint matrix games. Because the payoffs of the interval-valued constraint matrix game are intervals, its value is an interval as well. In this methodology, the value of the interval-valued constraint matrix game is regarded as a function of values in the payoff intervals, which is proven to be monotonous and non-decreasing. By the duality theorem of linear programming, it is proven that both players always have the identical interval-type value and hereby the interval-valued constraint matrix game has an interval-type value. A pair of auxiliary linear programming models is derived to compute the upper bound and the lower bound of the value of the interval-valued constraint matrix game by using the upper bounds and the lower bounds of the payoff intervals, respectively. Validity and applicability of the linear programming technique proposed in this paper is demonstrated with a numerical example of the market share game problem.

Linear programming approach to matrix games with intuitionistic fuzzy goals

Abstract Adding an additional degree of non-membership, K. T. Atanassov introduced the concept of the intuitionistic fuzzy (IF) set (IF-set), which has rarely been applied to the game theory yet. The aim of this paper is to develop the concept and methodology of matrix games with IF-set goals in which goals of players are expressed with IF-sets and payoffs are expressed with real numbers rather than IF-sets. In this methodology, the concepts of IF-set goals and the solutions of matrix games with IF-set goals are proposed. It is proven that solutions of matrix games with IF-set goals can be obtained through solving the developed auxiliary linear programming models, which are the generalization of matrix games with fuzzy goals. The proposed methodology is illustrated with a numerical example. Furthermore, comparison analysis of the proposed methodology is conducted to show its advantages over matrix games with fuzzy goals.

Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers

The purpose of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of trapezoidal fuzzy numbers (TrFNs), which are a type of two-person non-cooperative games with payoffs expressed by TrFNs and players' strategies being constrained. In this methodology, it is proven that any Alfa-constrained matrix game has an interval-type value and hereby any constrained matrix game with payoffs of TrFNs has a TrFN-type value. The auxiliary linear programming models are derived to compute the interval-type value of any Alfa-constrained matrix game and players' optimal strategies. Thereby the TrFN-type value of any constrained matrix game with payoffs of TrFNs can be directly obtained through solving the derived four linear programming models with data taken from only 1-cut and 0-cut of TrFN-type payoffs. Validity and applicability of the models and method proposed in this paper are demonstrated with a numerical example of the market share game problem.

An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives

This article proposes a framework to handle multiattribute group decision making problems with incomplete pairwise comparison preference over decision alternatives where qualitative and quantitative attribute values are furnished as linguistic variables and crisp numbers, respectively. Attribute assessments are then converted to interval-valued intuitionistic fuzzy numbers (IVIFNs) to characterize fuzziness and uncertainty in the evaluation process. Group consistency and inconsistency indices are introduced for incomplete pairwise comparison preference relations on alternatives provided by the decision-makers (DMs). By minimizing the group inconsistency index under certain constraints, an auxiliary linear programming model is developed to obtain unified attribute weights and an interval-valued intuitionistic fuzzy positive ideal solution (IVIFPIS). Attribute weights are subsequently employed to calculate distances between alternatives and the IVIFPIS for ranking alternatives. An illustrative example is provided to demonstrate the applicability and effectiveness of this method.

Solving constrained matrix games with payoffs of triangular fuzzy numbers

Constrained matrix games with payoffs of triangular fuzzy numbers (TFNs) are a type of matrix games with payoffs expressed by TFNs and sets of players’ strategies which are constrained. So far as we know, no study have yet been attempted for constrained matrix games with payoffs of TFNs since there is no effective way to simultaneously incorporate the payoffs’ fuzziness and strategies’ constraints into classical and/or fuzzy matrix game methods. The aim of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of TFNs. In this methodology, we introduce the concepts of Alpha-constrained matrix games for constrained matrix games with payoffs of TFNs and the values. By the duality theorem of linear programming, it is proven that players’ gain-floor and loss-ceiling always have a common interval-type value and hereby any Alpha-constrained matrix game has an interval-type value. Moreover, using the representation theorem for the fuzzy set, it is proven that any constrained matrix game with payoffs of TFNs always has a TFN-type fuzzy value. The auxiliary linear programming models are derived to compute the lower and upper bounds of the interval-type value and optimal strategies of players for any Alpha-constrained matrix game. In particular, the mean and the lower and upper limits of the TFN-type fuzzy value of any constrained matrix game with payoffs of TFNs can be directly obtained through solving the derived three linear programming models with data taken from only 1-cut and 0-cut of payoffs. Hereby the TFN-type fuzzy value of any constrained matrix game with payoffs of TFNs are easily computed. The proposed method in this paper is compared with other methods and its validity and applicability are illustrated with a numerical example.

Linear programming approach to solve interval-valued matrix games

Matrix game theory is concerned with how two players make decisions when they are faced with known exact payoffs. The aim of this paper is to develop a simple and an effective linear programming method for solving matrix games in which the payoffs are expressed with intervals. Because the payoffs of the matrix game are intervals, the value of the matrix game is an interval as well. Based on the definition of the value for matrix games, the value of the matrix game may be regarded as a function of values in the payoff intervals, which is proven to be non-decreasing. A pair of auxiliary linear programming models is formulated to obtain the upper bound and the lower bound of the value of the interval-valued matrix game by using the upper bounds and the lower bounds of the payoff intervals, respectively. By the duality theorem of linear programming, it is proven that two players have the identical interval-type value of the interval-valued matrix game. Also it is proven that the linear programming models and method proposed in this paper extend those of the classical matrix games. The linear programming method proposed in this paper is demonstrated with a real investment decision example and compared with other similar methods to show the validity, applicability and superiority.

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

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.

Linear programming method for multiattribute group decision making using IF sets

The purpose of this paper is to develop a linear programming methodology for solving multiattribute group decision making problems using intuitionistic fuzzy (IF) sets. In this methodology, IF sets are constructed to capture fuzziness in decision information and decision making process. The group consistency and inconsistency indices are defined on the basis of pairwise comparison preference relations on alternatives given by the decision makers. An IF positive ideal solution (IFPIS) and weights which are unknown a priori are estimated using a new auxiliary linear programming model, which minimizes the group inconsistency index under some constraints. The distances of alternatives from the IFPIS are calculated to determine their ranking order. Moreover, some properties of the auxiliary linear programming model and other generalizations or specializations are discussed in detail. Validity and applicability of the proposed methodology are illustrated with the extended air-fighter selection problem and the doctoral student selection problem.

LEXICOGRAPHIC METHOD FOR MATRIX GAMES WITH PAYOFFS OF TRIANGULAR FUZZY NUMBERS

The purpose of the paper is to study how to solve a type of matrix games with payoffs of triangular fuzzy numbers. In this paper, the value of a matrix game with payoffs of triangular fuzzy numbers has been considered as a variable of the triangular fuzzy number. First, based on two auxiliary linear programming models of a classical matrix game and the operations of triangular fuzzy numbers, fuzzy optimization problems are established for two players. Then, based on the order relation of triangular fuzzy numbers the fuzzy optimization problems for players are decomposed into three-objective linear programming models. Finally, using the lexicographic method maximin and minimax strategies for players and the fuzzy value of the matrix game with payoffs of triangular fuzzy numbers can be obtained through solving two corresponding auxiliary linear programming problems, which are easily computed using the existing Simplex method for the linear programming problem. It has been shown that the models proposed in this paper extend the classical matrix game models. A numerical example is provided to illustrate the methodology.