The Transportation Problem is a special class of linear programming problems which deals with the transportation of a single homogeneous product from factories to warehouse. There are several well-known heuristics methods like Vogel’s Approximation Method (VAM), Northwest Corner Method (NWCM), Least Cost Method (LCM), TOC-VAM, RAM etc. are available to get the optimal solution but they can't gives the optimal solution directly for all transportation problem.
The word "directly" leads me to think that you want an optimal solution in one "step". This is not possible - that would imply that the transportation problem can be solved through some "greedy" procedure, but it can NOT. The heuristics you list only provide one FEASIBLE solution, but almost NEVER an optimal one - and not with any good guarantees on the quality either. There is a very simple way, however, to solve it. While there are specialized codes for solving the linear transportation problem, such as specialized simplex codes and even interior-point ones, you can transform it to a standard linear network flow problem with one source and one sink through the use of additional links from a dummy source to the actual sources with corresponding capacities equal to the capacity of each source - and cost zero, and similarly links with cost zero from the destinations to a dummy source whose capacities equal the demand for each source. (If these two sums do not add up to the same value, you do not have a balanced problem, whence you need a dummy node with a zero cost that balances the total.) Anyway, for this balanced problem you can use a simplex method or any of the good interior point methods out there, and solve it very fast.
I agree with Michael Patriksson's answer. In addition, I encourage exploration and understanding of the dual form of the problem. If you are new to this kind of thing, the primal dual relationship, and especially the interpretation of dual feasibility is very powerful. For an explanation in geographic terms see Taaffe, Gauthier and O'Kelly, Geography of Transportation.
There are several versions of the transportation problem ! Can you add some informations about your specific problem ?
I re-read your question, and notice that the methods you list tend to be suited to finding an initial basic feasible solution (BFS). That is, they are starting heuristics. There is absolutely NO reason that any of them should provide an optimal solution,. In any case, how would you know if it is or is not optimal? You need more theory. So, although it is not a single one step method, here is a process to find an optimal solution (in broad outline). Use the active variables (those in the basis from your initial guess/heuristic) to determine the prices associated with the origins and destinations respectively (the dual values). How? Well since X_ij > 0 implies V_j - U_i = C_ij it is possible to start with one value and walk through the active flows to determine the V and U numbers. They in turn can be used to determine the profit or loss of the inactive flows (those that are currently non basic). If there are any that have an opportunity to be profitable (price spread > C_ij) you need to move flows around, to exploit this source of profit ... thereby getting a new BFS, recomputing the prices, and so on ,....
There are lots of nuances -- some would prefer V_j + U_i ... the sign convention is immaterial.
At the end of this there should be a set of flows (X_ij) and dual variables (V U) that are provably optimal. How? Well all the inactive (non basic activities) are money losers, and the active ones (X_ij > 0) break even i.e. V_j - U_i = C_ij. Neat, eh?
Issues of degeneracy and so on will crop up, but try a simple case from a standard text book.
So far I know that there is no method which can directly give an optimal solution to a transportation problem. But, sometimes we may get directly an optimal solution once we are solving transportation problems. This will be considered an exception for a particular problem and for a particular method.
I agree that there is no one method that can provide the optimal solution at all times. This is simply the fact that some of these methods are "greedy search", in that they only look at the minimum cost for that iteration and this may not be the best in the next iteration.