A transportation problem with fuzzy-stochastic cost

In this paper, we consider some transportation problems (TPs) with different types of fuzzy-stochastic unit transportation costs and budget constraints. These fuzzy stochastic costs are reduced to corresponding crisp ones in two different ways. For the first method, using the definition of αα-cut of the fuzzy numbers, expectation is taken separately on both lower and upper αα-cuts and then mean expectation is calculated with the help of signed distance. In the second procedure, we realize fuzzy random events (ξ⩾r)(ξ⩾r) and (ξ⩽r)(ξ⩽r) for the fuzzy random variable (ξ)(ξ). Using credibility measure of these events, mean chances for the above fuzzy random events are calculated and then expectation is taken to get the crisp expressions. The reduced deterministic problems of the fuzzy stochastic TP are solved using a real coded genetic algorithm with Roulette wheel selection, arithmetic crossover and random mutation. Few numerical examples are demonstrated to find the optimal solutions of the proposed models.