The aim of the research article is not only to propose a solution procedure to solve multi-objective fuzzy stochastic programming problem by using genetic-algorithm-based fuzzy programming method, but also to apply the computational techniques for transportation of the hazardous waste materials. In this article, routing and siting problems for nuclear hazardous waste material are studied and solved. The amount of waste materials generated in the nuclear reactors follows normal distribution. The two considered objective functions are about route selection which includes minimum travel time and minimum number of houses along the way, taking the safety measures into consideration. A multi-objective fuzzy stochastic mathematical model is formulated with the above mentioned objective functions and the route selection as the constraints. The proposed solution procedure is illustrated by a numerical example and a case study.