Let's say we have:
- Two supply points - Houston and Dalas, which produce 160 and 200 products.
- Two transshipment points - Chicago and Los Angeles
- Two demand points - San Francisco and New York, which require 140 products each.
- Transportation costs, given in the following table:
$\begin{array} \centering & Chicago & LA & S.F. & New York & Supply\\ Dallas & 9 & 14 & 26 & 29 & 200 \\ Houston & 16 & 13 & 27 & 26 & 160 \\ Chicago & 0 & 7 & 17 & 18 & B \\ LA & 7 & 0 & 15 & 17 & B \\ Demand & ? & ? & 140 & 140 & \end{array}$
I've solved Transshipment problems with balanced demand and supply by converting the task into a transportation problem by making a table (its rows are the supply and transshipment points and its columns are the transhipment and demand points) and putting $M$ as transportation costs for non-existent paths. However, in this problem we produce 360 products but demand only 280. Therefore, we need to add an additional fake point, which demands 80 products. Thus, a few questions arise:
- Is this fake point a pure demand point or it is a transshipment point?
- What are the costs between the fake ending point and all other points in the transport table? This should depend on whether we make a path between the fake point and every other point, or only between the fake point and all transshipment and ending points (thus excluding the starting points). Thus, are all transportation costs zero, as in a regular transport problem, or there are Ms as well (M is a very large number)?
- In order to convert the transshipment problem into a regular transportation problem, we also calculate the supply and demand of the transshipment points. The supply is equal to $B$ and the demand is equal to $the original demand + B$. We define $$B= max\{\text{total supply}, \text{total demand}\}$$ In ordinary transshipment problems, where a balance was present, the total supply and the total demand were equal. For this particular problem, if I calculate B by the old demand and supply ($max\{360, 280\}$) or by the new ones($max\{360, 360\}$), it is the same and it will be in every task. But what is the correct way to calculate it? From the old or from the new values?
So far, I assume the table should look like this:
$
\begin{array}
\centering
& Chicago & LA & S.F. & New York & Fake point & \\
Dallas & 9 & 14 & 26 & 29 & 0 & 160 \\
Houston & 16 & 13 & 27 & 26 & 0 & 200 \\
Chicago & 0 & 7 & 17 & 18 & 0 & B \\
LA & 7 & 0 & 15 & 17 & 0 & B \\
& B & B & 140 & 140 & 80 &
\end{array}$