# Transportation Problems

What is a transportation problem?

In a business transaction products would have to be carried from the supplier to the buyer. This is possible by transportation only. However, any of the various means of transportation can be used. But the main thing is that the transportation would involve a certain cost. Transportation cost is a cost which cannot be eliminated. It is immaterial who bears the cost the buyer or the supplier.

Transportation problem is the problem of minimizing the transportation cost. This problem is not so acute where there is one source (supplier’s place) and one destination (buyer’s place). But the problem becomes acute when the supplier has various plants and the buyer has various warehouses. Here, the necessity of forming combinations of plants and warehouses would arise to minimize the transportation. The least expensive method of transportation should be found out. This is known as Optimization of transportation problem.

Transportation Matrix:

Matrix is nothing but combination of various mathematical numbers arranged in different rows and columns.

Now, to see how a transportation matrix is formed we would take a practical example. Further discussion on solving of transportation matrix would be done keeping this example in the center. There is one assumption made, while taking the example, that supply and demand are equal. However, in practice in many cases these two may be different.

Example:

TC Mallet Trucking Company has a contract to move 115 truck loads of sand per week among 3 sand washing plants, W, X and Y and 3 destinations A, B and C. Cost and volume information are give below:

Destination Requirements Plant Capacity

A 45 W 35

B 50 X 40

C 20 Y 40

115 115

Cost information>>

From To destination A To Destination B To destination C

Plant W 5 10 10

Plant X 20 30 20

Plant Y 5 8 12

Based on this data we want to form a transportation matrix.

In figure A represents different places from where the goods are to be transported i.e. the plants W, X, and Y in our example. B represents the capacities of each plant. C represents different destinations and D represents the requirements of each particular destination.

E consists of different squares, each of which represents a combination of a plant and a destination e.g. square WA represents the combination of plant W and destination A. In round circles, quantity to be transported from particular plant to a particular warehouse are to be written because until we don’t know the quantities to be transported from each warehouse to different destinations. In each of these squares, on right hand corner at the top cost figures would be written e.g. in case of square WA, 5 is written because transportation cost per truck load from plant W to destination A is Rs 5.

In this way the transportation matrix would be formed.

North West Corner rule

Now actual solving process of the transportation problem starts. For this purpose, North-West Corner Rule is applied. North West corner Rule states: Start with the North West corner of the matrix. Do not move till all the rim requirements of that row are exhausted. And while moving to the right to the next column exhaust all the rim requirements of that particular column

Following, this rule we would fill up the quantities in the squares of the transportation matrix. The matrix would now be as shown in Figure

The squares in which there is some quantity are known as used squares. The others are said to be unused squares. The number of used should be equal to total rim requirements. If it is not equal tot his then a condition known as degeneracy comes into existence. We would consider the problem of degeneracy at a later stage. We find that in our problem there is a degeneracy and so we proceed further now We would now form a total cost schedule to seen that each of the steps as we go further, brings us to some reduction in price. So, let us form a total cost schedule in accordance with the quantities we have allotted to different combination of plants and destinations.