Page 102 - DCOM303_DMGT504_OPERATION_RESEARCH

Unit 5: Transportation Problem

Introduction Notes

Transportation problem is a particular class of linear programming, which is associated with
day-to-day activities in our real life and mainly deals with logistics. It helps in solving problems
on distribution and transportation of resources from one place to another. The goods are
transported from a set of sources (e.g., factory) to a set of destinations (e.g., warehouse) to meet
the specific requirements. In other words, transportation problems deal with the transportation
of a product manufactured at different plants (supply origins) to a number of different warehouses
(demand destinations). The objective is to satisfy the demand at destinations from the supply
constraints at the minimum transportation cost possible. To achieve this objective, we must
know the quantity of available supplies and the quantities demanded. In addition, we must also
know the location, to find the cost of transporting one unit of commodity from the place of
origin to the destination. The model is useful for making strategic decisions involved in selecting
optimum transportation routes so as to allocate the production of various plants to several
warehouses or distribution centers.
The transportation model can also be used in making location decisions. The model helps in
locating a new facility, a manufacturing plant or an office when two or more number of locations
is under consideration. The total transportation cost, distribution cost or shipping cost and
production costs are to be minimized by applying the model.

5.1 Modeling of Transportation Problem

A transportation problem can be expressed in two ways.
1. Mathematical representation
2. Network representation

Obviously the method used for solving the problems is the formulation of transportation problem
through mathematical methods. But for understanding of the readers, network representation is
equally important.
Let us understand each of them one by one.

5.1.1 Mathematical Representation

The transportation problem applies to situations where a single commodity is to be transported
from various sources of supply (origins) to various demands (destinations).
Let there be m sources of supply S , S , .…..............S having a ( i = 1, 2,......m) units of supplies
1 2 m i
respectively to be transported among n destinations D , D ………D with b
1 2 n j
( j = 1,2…..n) units of requirements respectively. Let C be the cost for shipping one unit of the
ij
commodity from source i, to destination j for each route. If x represents the units shipped per
ij
route from source i, to destination j, then the problem is to determine the transportation schedule
which minimizes the total transportation cost of satisfying supply and demand conditions.
The transportation problem can be stated mathematically as a linear programming problem as
below:

Minimize Z =   c x
ij ij
 

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