Fuzzy linear programming problems: models and solutions

Dynamic programming approach for fuzzy linear programming problems FLPs and its application to optimal resource allocation problems in education system

This research is about the development of a dynamic programming model for solving fuzzy linear programming problems. Initially, fuzzy dynamic linear programming model FDLP is developed. This research revises the established dynamic programming model for solving linear programming problems in a crisp environment. The mentioned approach is upgraded to address the problem in an uncertain environment. Dynamic programming model can either be passing forward or backward. In the proposed approach backward dynamic programming approach is adopted to address the problem. It is then followed by implementing the proposed method on the education system of Pakistan. The education system of Pakistan comprises of the Primary, Middle, Secondary, and Tertiary education stages. The problem is to maximize the efficiency of the education system while achieving the targets with minimum usage of the constrained resources. Likewise the model tries to maximize the enrollment in the Primary, Middle, Secondary and Tertiary educational categories, subject to the total available resources in a fuzzy uncertain environment. The solution proposes that the enrollment can be increased by an amount 9997130, by increasing the enrollment in the Middle and Tertiary educational categories. Thus the proposed method contributes to increase the objective function value by 30%. Moreover, the proposed solutions violate none of the constraints. In other words, the problem of resources allocation in education system is efficiently managed to increase efficiency while remaining in the available constrained resources. The motivation behind using the dynamic programming methodology is that it always possesses a numerical solution, unlike the other approaches having no solution at certain times. The proposed fuzzy model takes into account uncertainty in the linear programming modeling process and is more robust, flexible and practicable.

Finding the Solution of Balanced and Unbalanced Intuitionistic Fuzzy Transportation Problems by Using Different Methods With Some Software Packages

In this chapter, two different methodologies are proposed to find out the optimal solution to the balanced and unbalanced intuitionistic fuzzy transportation problems (UBIFTPs). In addition, the parameter of both the balanced and UBIFTPs are considered to be triangular intuitionistic fuzzy numbers (TIFNs). Two new methodologies, respectively method-1 and method-2, are presented in this chapter. Proposed method-1 is based on linear programming technique, and proposed method-2 is based on modified distribution method. Both the methodologies are used to solve the balanced and UBIFTPs. The ideas of the proposed methodologies are illustrated with the help of real-life numerical examples. The solutions obtained by the proposed methodologies are checked with some software (e.g., MATLAB, LINGO) and the computer code related to the proposed problems is also given. The unique results, comparative study, discussions, and the merits of the proposed methodologies are all given. At the end of the chapter, future work is mentioned.