To get the solution for an unbalanced transportation problem in operation research at first it is required to balance the problem by introducing some dummy sources or destinations. But in the real field how it is possible to consider the dummy sources? Opine please.
To be clear the dummy source cannot help make something from nothing. It is simply a way to account for differences in the total amount demanded (D) and supplied (S). If D > S, we have a shortage, and the places are going to have to get less than the ideal (in a sense they are rationed). We could use an imaginary extra amount to solve this (perhaps at an artificial high cost so we do not use it unless absolutely needed). If on the other hand S > D, we have plenty to go around, and the artificial sink is a place where we will send the surplus after we have taken care of the demand.
I have a feeling however that your real question is for the case D > S, and how do we make up the difference? In that case the best we can do is to decide where to send the scarce supplies and perhaps to learn something from the shadow prices about how valuable extra supplies would be.
Dummy sources have no practical use. It is only used to solve the problem. You need to consider the sources except dummy in your optimization problem.
One remark about the sentence:
"To get the solution for an unbalanced transportation problem [...] it is required to balance the problem by introducing some dummy sources or destinations."
If you use many existing optimization programs, you don't really need to balance the problem by yourself. You can enter unbalanced constraints - in this sense it is not required, the software will do it for you.
I agree that some packages will solve SOME of these problems, but they cannot solve infeasible problems. So, if D > S (the total demand is more than the supply) and the problem is set up with say equality constraints, then there is going to be an easy to detect infeasibility. Of course one way, as I think you are suggesting is that the problem could be formulated with inequality constraints. Even in this case, if the SUM_ j Xij = D_ j you are going to have infeasible problem if SUM_ j D_ j > SUM_i S_i. Right?
Right.
By formulating the D > S case with inequalities I meant inequalities in the demand constraints only, i.e. SUM_ j Xij = S_i and SUM_ i Xij
Unbalanced transportation problem can also be solved in Two stages. There are so many papers in the literature on two stage transportation problem.
Gourav, I guess that Mollah's question was about "standard" one-stage transportation problem.
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