Prasun Kumar Nayak

LINEAR PROGRAMMING TECHNIQUE TO SOLVE TWO PERSON MATRIX GAMES WITH INTERVAL PAY-OFFS

A fuzzy two person interval game problem is proposed and treated in this paper which is not easily tackled by the conventional methods. First, with respect to this pay-off values, a necessary and sufficient condition for the existence of a saddle point is proved. Based on interval value model, we are to find the value of interval game without saddle point. Finally, example is given to illustrate the procedure and to indicate the performance of the proposed method.

Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages

This paper discusses a manufacturing inventory model with shortages where carrying cost, shortage cost, setup cost and demand quantity are considered as fuzzy numbers. The fuzzy parameters are transformed into corresponding interval numbers and then the interval objective function has been transformed into a classical multi-objective EPQ (economic production quantity) problem. To minimize the interval objective function, the order relation that represents the decision maker’s preference between interval objective functions has been defined by the right limit, left limit, center and half width of an interval. Finally, the transformed problem has been solved by intuitionistic fuzzy programming technique. The proposed method is illustrated with a numerical example and Pareto optimality test has been applied as well.

Matrix Games with Intuitionistic Fuzzy Pay-offs

Abstract In this paper two person zero sum matrix games are considered in which elements of payoff matrix are expressed with triangular intuitionistic fuzzy numbers (TIFNs). The purpose of this paper is to develop a solution methodology for solving such types of matrix games. For this, a pair of auxiliary intuitionistic fuzzy programming (IFP) models for each player are established. Then these two IFP problems are transformed into conventional crisp linear programming problems by defining a suitable defuzzification function to determine optimal strategies for each player and value of the game. The proposed method in this paper is illustrated with a numerical example to show the validity and applicability.

Matrix Game with Z-numbers

In this paper, a matrix game is considered in which the elements are represented as Z-numbers. The objective is to formalize the human capability for solving decision-making problems in uncertain situations. A ranking method of Z-numbers is proposed and used to define pure and mixed strategies. These strategies are then applied to find the optimal solution to the game problem with an induced pay off matrix using a min max, max min algorithm and the multi-section technique. Numerical examples are given in support of the proposed method.

Notes on triangular intuitionistic fuzzy numbers

The notion of triangular fuzzy number is put forward on the basis of intuitionistic fuzzy sense. The basic algebra and some non-linear operations of triangular intuitionistic fuzzy numbers (TIFNs) are devised. The average ranking index is introduced to find out order relations between two TIFNs. A method is described to approximate a TIFN to a nearly approximated interval number. Applying this result and using interval arithmetic, a bound unconstrained optimisation problem is solved whose coefficients are fixed TIFNs. Numerical examples are also given for illustration.

Solution of rectangular fuzzy games

A solution of m×n rectangular fuzzy game with payoff as imprecise numbers instead of crisp real numbers namely interval and triangular fuzzy numbers is considered here. Solution of such fuzzy games with pure strategies and the algebraic method to solve 2×2 fuzzy game without saddle point by using mixed strategies is also discussed. Here the m×n payoff matrix is reduced to 2×2 payoff matrix by dominance method. In this paper, we discuss a saddle point solution from an uncertain payoff matrix. Moreover solution method for fuzzy games has also been developed. Numerical example is provided to illustrate the method.

An alternative approach for solving fuzzy matrix games

In this paper, two-person matrix games is considered whose elements of pay-off matrix are triangular fuzzy numbers (TFNs). To solve such game a new methodology based on -cut of TFN is developed for each of the players. In this methodology, two auxiliary bi-objective linear programming (BOLP) models are derived. Then using average weighted approach these two BOLP models are decomposed into two auxiliary crisp linear programming (LP) problems. Finally, the value of the matrix game for each player is obtained by solving two corresponding auxiliary LP problems using the existing simplex method. Validity and applicability of this methodology are illustrated with practical example compared to existing methods.

Matrix games in intuitionistic fuzzy environment

This paper presents a new intuitionistic fuzzy optimisation (IFO) approach to solve matrix games under uncertainty. This approach is an application of the intuitionistic fuzzy set. First we have considered matrix game with intuitionistic fuzzy (IF) goals for each of a strategy in a payoff matrix in order to incorporate ambiguity of human judgements. The degree of membership is defined as the degree of satisfaction and the degree of nonmembership function have been taken for the degree of rejection. Then it converts the said problem into two conventional linear programming problems. Numerical example is provided to illustrate our approach.

An algorithm for solution of interval games

A new method of solution of a two person interval game without saddle point is proposed and treated in this paper. The solution is obtained by using multisection technique, whose concept is the multiple bi-section technique and the basis is the inequality relation of interval numbers. On the basis of multisection technique, max min and min max algorithm is given. An algorithm has been given and a numerical example is provided to illustrate the methodology.

Interval valued EOQ model with two types of defective items

Abstract In the classical inventory model, the issue of quality of the product is not considered. It is assumed that all the items procured are of perfect quality. However, in real life production environment it is observed that some of the items produced are defective. Here, we have considered an Economic Order Quantity (EOQ) model with imperfect quality items where the incoming lot has fractions of scrap and re-workable items. These fractions are considered to be known with a certain given percentage. The demand from customers end is met with the perfect and re-worked items and the scrap items are sold differently at a salvage cost in a secondary market. Also to represent the real life situations, the inventory parameters have been taken as interval numbers. The corresponding mathematical problem has been formulated as an un-constrained optimization problem and has been solved using Particle Swarm Optimization (PSO) technique. Finally, the model has been illustrated with a numerical example.