• Revision:Transportation Problems

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Transportation Problem

The problem details trying to find the cheapest way in moving a product from a number of suppliers to a number of warehouses. The way these problems are normally represented are in a table:

\begin{tabular}{|c|c|c|c|c|}
\hline 
& \text{Warehouse 1} &  \text{Warehouse 2} & \text{Warehouse 3} & \text{Supply} \\ \hline 
\text{Bakery 1} & 3 & 6 & 7 & 2  \\ \hline 
\text{Bakery 2} & 4 & 3 & 5 & 8  \\ \hline 
\text{Bakery 3} & 6 & 7 & 9 & 5  \\ \hline 
\text{Supply}   & 7 & 5 & 3 & 15 \\ \hline
\end{tabular}

In this example, bakery two has 8 loaves of bread that it would like to sell. And, for example, it would cost £4 to send a loaf of bread from bakery 2 to warehouse 1.

Formation as a Linear Programming Problem

When you're forming problems in the form of a Linear Programming problem, you have to include two things, the objective function, that is, what you want to maximise or minimise; as well as the constraints. You also need to set your decision variables. This is done in the following way:

x_{ij} = The no. of loaves sent from bakery i to warehouse j

From here, you can set your constraints. Each of these formulae basically state that you can't send more bread out than a supermarket has, or that you can't send more bread to a warehouse that it wants.

x_{11} + x_{12} + x_{13} &=& 2

x_{21} + x_{22} + x_{23} &=& 8

x_{31} + x_{32} + x_{33} &=& 5

x_{11} + x_{21} + x_{31} &=& 7

x_{12} + x_{22} + x_{32} &=& 4

x_{13} + x_{23} + x_{33} &=& 3

The objective function is equal to the total cost of all of it; namely, the sum of each of the loaves of bread on each route, times the cost for that route.

P = 3x_{11} + 6x_{12} + 7x_{13} + 4x_{21} + 3x_{22} + 5x_{23} + 6x_{31} + 7x_{32} + 8x_{33}

This could be solved using the simplex method. Fortunately, you will never be asked to do it.

Finding an optimal solution

For an algorithm on how to find an optimal solution, go to Revision Notes: Transportation Algorithm

Unbalanced Problems

For information on unbalanced transportation problems, go to Revision Notes: Unbalanced Transportation Problems

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