SOME ALGORITHMS FOR FUZZY TRANSPORTATION PROBLEMS

Abstract

At the turn of the nineteenth century, reducing complex real world systems to precise mathematical models was the main trend in science and engineering. In the middle of the century, Operations Research began to be applied to the real life decision making problems and became an integral part of the problem solving tools. Among various optimization tools, the most extensively used is linear programming. It serves as a decision making aid and critical tool in engineering, industry, management, economics, service organizations and many other areas. The most successful and important application of linear programming is found in physical distribution of goods, commonly known as transportation problem. Transportation plays a vital role in the nancial development of a country. Sometimes it is among the most signi cant factors which decide the survival and growth of an enterprise. The precise mathematical models were only approximation to the practical problems as the real life data is not always deterministic,. Therefore to deal with this imprecision/ uncertainty, Zadeh proposed the concept of fuzzy set theory. It has enlarged the applications of linear programming and hence transportation models to the real life situations as it allows to consider the tolerances for decision variables in a natural and direct way. The present thesis is devoted to the study of transportation problems with parameters represented by fuzzy numbers, called the fuzzy or the fully fuzzy transportation problems. We have studied the single objective fully fuzzy transportation problem, fuzzy and fully fuzzy multi-objective multi-commodity solid transportation problem, time minimization fuzzy transportation problem and fuzzy non-linear i ii programming problem. We have discussed both the cases of single objective unbalanced fully fuzzy transportation problem. Unlike the existing methods of solving fully fuzzy unbalanced transportation problems, we have proposed methods to nd the facility/facilities at which the de cit in supply should be increased to meet the demand at a minimum cost or the client(s) to whom the excess supply be transported for future demand. We have studied the fuzzy multi-objective multi-commodity solid transportation problem with parameters as trapezoidal fuzzy numbers and suggested methods to nd its crisp as well as fuzzy optimal compromise solution without any condition on the total supply, the total demand or on the conveyance capacity. We have also showed that after balancing the problem, the expected value model always gives a feasible and hence an optimal compromise solution for these problems. Further, we have worked on the multi-commodity solid transportation problem considering the safety factor during transportation. An additional constraint for the budget of each client is also considered. After that, we have discussed a time minimizing fuzzy transportation problem and proposed a straightforward and easy technique to solve it. Finally the multi-objective non-linear programming problemshave been studied in the optimistic and pessimistic decision maker's view point.