Sathi Mukherjee

Solution of a class of Intuitionistic Fuzzy Assignment Problem by using similarity measures

In this paper, we set mathematical models of the assignment problem with restriction on person cost depending on efficiency/qualification and restriction on job cost where both the costs are considered as intuitionistic fuzzy numbers. Restriction of qualification is in the form of the maximum intuitionistic fuzzy cost which can be offered to a person depending on his/her efficiency/qualification. Further restrictions on the intuitionistic fuzzy cost which can be spent for doing a particular job makes the problem of Intuitionistic Fuzzy Assignment Problem with Restrictions (IFAPR) more realistic than the problems found in the literature so far. Consideration of Intuitionistic Fuzzy Numbers (IFNs) for representing the costs makes the problem more general in the sense that it considers both the degree of acceptance and the degree of rejection. A heuristic method has been constructed for showing the existence of the solution so that both the constraints are satisfied. The methodology for solving the problem consists of two algorithms. The concept of relative degree of similarity measures to the Positive Ideal Intuitionistic Fuzzy Solution (PIIFS) has been applied under Atanassovs intuitionistic fuzzy environment. A well established intuitionistic fuzzy ranking method has been used here for comparing the IFNs using their score functions and the accuracy degrees. Mathematical model of the problem has been established. Numerical examples show the effectiveness of this method.

Dijkstra’s Algorithm for Solving the Shortest Path Problem on Networks Under Intuitionistic Fuzzy Environment

In this paper, a well known problem called the Shortest Path Problem (SPP) has been considered in an uncertain environment. The cost parameters for traveling each arc have been considered as Intuitionistic Fuzzy Numbers (IFNs) which are the more generalized form of fuzzy numbers involving a degree of acceptance and a degree of rejection. A heuristic methodology for solving the SPP has been developed, which aim to exploit tolerance for imprecision, uncertainty and partial truth to achieve tractability, robustness and low cost solution corresponding to the minimum-cost path or the shortest path. The Modified Intuitionistic Fuzzy Dijkstra’s Algorithm (MIFDA) has been proposed in this paper for solving Intuitionistic Fuzzy Shortest Path Problem (IFSPP) using the Intuitionistic Fuzzy Hybrid Geometric (IFHG) operator. A numerical example illustrates the effectiveness of the proposed method.

Solution of a Multi-Objective and Multi-Index Real-Life Transportation Problem Using Different Fuzzy Membership Functions

The paper presents the application of an exponential membership function to fuzzy programming technique for solving a multi-objective and multi-index real-life transportation problem. It focusses on the minimization of the transportation cost, deterioration rate and underused capacity of the transported raw materials like coal, iron ore, etc. from different sources to different destination sites at Durgapur Steel Plant, Durgapur, West Bengal, India, by different transportation modes like train, trucks, etc. A special type of non-linear (exponential) membership function is assigned to each objective function. Thus, the paper presents fuzzy programming approach with an exponential function to a real-life transportation problem and develops a non-dominated compromise solution. In addition, the interval-valued numbers used in supply and demand parameters represent the uncertainties in a real-life problems. The optimization models have a wide use in real-life multi-objective transportation problems. The use of fuzzy programming technique has been successfully concluded for the focussed transportation problem taking a linear and a hyperbolic membership function. But the numerical illustration shows that exponential membership function gives a better and comparable result for the problem.

Solution of a class of generalized assignment problem

The paper presents the solution of a multi objective multi-index generalized assignment problem using Fuzzy Programming Technique which is a new concept. It emphasizes on minimization of the job cost and time at different sites where the jobs are being performed using different machines (medium). Multi objective assignment problem or multi objective generalized assignment problem have been solved in different ways by different mathematicians. But they have considered only two indices (jobs and alternatives). Here we have considered more than two indices and so it becomes a multi-index generalized assignment problem which makes it superior to the previous methods. This type of problem has not been solved earlier by our proposed method. Moreover in this paper Fuzzy Programming Technique has been used considering linear, exponential and hyperbolic (non-linear) membership functions for getting optimal solutions. Finally a real life application has been presented.

A new fuzzy programming technique approach to solve fuzzy transportation problem

The paper presents an application of a new method - Modified Fuzzy Programming Technique(MFTP) for the Fuzzy optimal solution to the Single Objective FuzzyTransportation Problem(SOFTP) with Fuzzy parameters in terms of Triangular Fuzzy Numbers without defuzzifying the problem. Numerical illustrations are solved to show the better results and comparisons are also provided to show the same. The problem is formulated where parameters of cost objective functions, supply and demand are taken as Triangular Fuzzy Numbers. The proposed method is applied for the first time to solve the SOFTP. First, the SOFTP is converted into a Multi-Objective Transportation Problem and then solved by the MFTP.

Decision Making In Assignment Problem With Multiple Attributes Under Intuitionistic Fuzzy Environment

In this paper we develop a methodology to solve the multiple attribute assignment problems where the attributes are considered to be Intuitionistic Fuzzy Sets (IFS). We apply the concept of similarity measures of IFS to solve the Intuitionistic Fuzzy Multi‐Attribute Assignment Problem (IFMAAP). The weights of the attributes are determined from expert opinion. An illustrative example is solved to verify the developed approach and to demonstrate its practicality.