Learning Objectives – The Transportation Problems

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INTRODUCTION:

The transportation model is a linear program which is about the determination of the minimal cost of one commodity when being transferred from various places. Reaching up to the destinations also includes forwarding and clearing agents (C and F), facilities, and location. Planning and control. Thus the transportation model is a linear model though simplex algorithm can solve it. But the process becomes a lengthy one that will increase the chances of errors. Thus lies the significance of this for simple calculations.

Researches had been done since 1940 to solve the problem and complications of transportation. Because most of the time things get complex due to numerous locations and sources with their individual capacities and requirements. There was a common issue among every problem, and that was to minimise the transportation cost. Since then the significance of linear programming was determined. But before the discovery of linear programming simplex method was also applied but as stated above it made things more complicated and severe.

DEFINITION:

The transportation model can be defined as the determination of only one commodity that is being transported from one destination to various locations. Therefore for the determination must include the following set of information:

  1. The amount demanded in each destination
  2. The availability of the commodity in every source
  • The minimum transportation cost between the two places.

Therefore considering only one commodity transporting between the two sources, the minimum cost could only be determined if expressed in mathematical terms.

Like, let us consider:

m= number of sources (factory locations)

n= number of destinations

ai= number of commodity at the source (i = 1,2,3,….m)

bj= number of commodity at the destination ( j = 1,2,3,….n)

Cij= minimum transportation cost of the commodity between source location i and destination j

In the above, the determination of some units to be transported from one source i (ai) to destination j (bj) is clear enough to reduce the transportation cost.

Let us now consider,

Xij = the number of commodities being transported from source i to destination j.

Now we have to find Xij (non-negative value) which satisfies the following constraints:

i=1 ʃ n Xij= aifor i= 1, 2, 3…….m (availability at source i constraint)

j=1ʃ mXij= bjfor j = 1, 2, 3………..n (requirement at destination j constraint)

The total cost of transportation Z (the objective function)can be written as

Z = i=1 ʃnj=1 ʃ m XijXij

As in the above lines we see that Xijis a linear function hence it is a LPP.

Thus from the above things we conclude that the transportation cost of a particular route is proportional to the number of units transported. But things that must be kept in mind that are the source supply cannot exceed the availability of unit commodity at the source.

Hence,

j=1 ʃmxij≤aj = 1, 2, 3 ….m

Similarly, requirement of the commodity at destination must be equal to or more than the demand, i.e.,

i=1 ʃ m  xij≤ bj= 1,2,3 …n

But it is seen that in reality, the supply might not be equal or more than the demand. In such situations, the transportation model must be balanced.

 

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