International Journal of Computer Applications (0975 – 8887) Volume 106 – No.8, November 2014 22 Linear Programming Problem with Intuitionistic Fuzzy Numbers Anil Kumar Nishad Department of Mathematics, Banaras Hindu University, Varanasi-221005, India S.R. Singh Department of Mathematics Banaras Hindu University, Varanasi-221005, India ABSTRACT: In many real life optimization problems, the parameters are often imprecise and are difficult to be represented in discrete quantity. One of the approaches to model such situation is considering these imprecise parameters as intuitionistic fuzzy numbers and then approximating these by its expected interval value. Further in process of solution, membership function for each objective function are constructed by computing best and worst acceptable solutions and deal the constraints of the problem with ranking of intuitionistic fuzzy number with a concept of feasibility degree. The paper presents a computational algorithm for solution of objective functions at different feasibility degree. The developed algorithm has been illustrated by implementing on a linear programming problem as well as on a multi objective linear programming problem (MOLPP) in intuitionistic fuzzy environment. Keywords Intuitionistic Fuzzy Set, Trapezoidal Intuitionistic Fuzzy Number (TIFN), Expected Interval of Fuzzy Number 1. INTRODUCTION Modeling a financial or a production planning problem needs some prior information about its feasibility and possible outcome. In many situations, it also needs financial analysis about resource utilization and optimal profit or gain. Such analysis needs the complete information about various parameters such as profit coefficients, resource limitations, constraints as well as its objectives and other goals. As a matter of fact in real life production planning problems, it is often difficult to get discrete and exact information for various parameters affecting the process. Even in many situations the information available are imprecise or vague. Under such situations it is difficult to have the mathematical formulation to solve the mathematical programming problem using a linear programming technique. For such situations, fuzzy set developed by Zadeh[22] played a vital role in modeling the optimization problem having imprecision in parameters and was initiated by Zimmermann[23,24,25] as fuzzy linear programming problem. One of the major difficulties to study such fuzzy linear programming problems with fuzzy coefficients is how to compare these fuzzy numbers. Thus an important issue of ranking of fuzzy numbers and its approximation method took considerable interest amongst the researchers. Some of the authors who made significant contributions in the area are Dubois and Prade[10], Heliperrn[15], Adrian[1,2]. This growing discipline attracted many authors to extend the theory of fuzzy sets to various application areas of industrial planning, production planning, agricultural production planning, economics etc. Atanossov[4, 5] extended the fuzzy set theory to intuitionistic fuzzy sets. This extended new set, named as intuitionistic fuzzy set, has a feature to accommodate hesitation factor of including an element in a fuzzy set apart from the feature of degree of belonging and non belonging. This extension of fuzzy set to intuitionistic fuzzy set attracted research workers as well as planners to apply this new set in the field of decision sciences. Thus an extension of deterministic optimization to intuitionistic fuzzy optimization was initiated by Angelov[3].The Angelov study was motivated by Zimmermann visualization of a fuzzy set to explain the degree of satisfaction of respective condition and was expressed by their membership function. Angelov[3] in his study extended the Bellman and Zadeh[6] approach of maximizing the degree of (membership function)acceptance of the objective functions and constraints to maximizing the degree of acceptance and minimizing the degree of rejection of objective functions and constraints. In view of its suitability of intuitionistic fuzzy set in modeling systems having imprecise parameters, a considerable research work has been carried out in the direction of ranking of intuitionistic fuzzy numbers. Further development of approximation methods are needed for development of intuitionistic optimization techniques (please see Hassan[14], Grzegorzewski[13], Parvathi and Malathi[21]. Nishad et al. [7,20] have also worked on developing the ranking method for intuitionistic fuzzy numbers and have applied it on intuitionistic fuzzy optimization. There are many more authors, who worked on the ranking methods and approximation of intuitionistic fuzzy number (please see Inuiguchi and Tanaka[16]). Recently Dubey et al [11,12] have studied fuzzy linear programming with intuitionistic fuzzy numbers. The present work is a motivation towards the application of intuitionistic fuzzy numbers to optimization problem and develops a computational method for solution of such optimization problems. The study is presented in the following sections: Section 2 is preliminaries to intuitionistic fuzzy set and intuitionistic fuzzy numbers needed for consequent sections. Section 3 comprise of modeling of an intuitionistic fuzzy optimization problem and its solution algorithm. Section 4 illustrates the implementation of the theory developed in section 3 to a linear programming problem as well as to a multi objective linear programming problem. Last section presents the results of the undertaken problem and provides a brief discussion on the developed method. 2. PRELIMINARIES Definition 1. Fuzzy Set Let is a collection of objects denoted by then a fuzzy set in is a set of ordered pairs: , where is called the membership function or grade of membership of in that maps to the membership space . Definition 2. Intuitionistic Fuzzy Set Let is a collection of objects then an intuitionistic fuzzy set in is a defined as : where