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Abstract
Multi index transportation problem (MITP) is the transportation problem which have more than two indices,, such as source, destination, commodity, type of conveyance, and so on.In multi-index transportation problem there are different cases which affect in the solution methods. 4-index transportation problem will be discuss in case every source own his enough vehicles capacity to transport his existing commodity or in another meaning the source and the destination are independent while the commodity and capacity are dependent.The mathematical model for 4-index transportation problem proposed and the problem solved in deterministic (by using graph solution) and fuzzy environment.The problem will divided into sub problems at every available capacity let (k)to every type of commodity let (l). The shape of the sub problem is a classic transportation problem (the number of sub problems=[(k)*(l)]. the problems can be solved by a graph method or by any package, and the solution for the multi-index problem will be equal the summation of the solutions of the sub problems. N {u2013}Vehicle cost varying transportation problem expresses the situation when there is no ability to provide enough vehicles to transport the available commodity.The transportation problem in N-Vehicle type has three indexes of sources, destinations and conveyances