Introduction and Mathematical Formulation – Transportation Problems

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There are multiple facets associated with a business and while dealing with these facets one needs to be very careful. A single mistake could cost quite an amount and it is to be averted at the earliest.

Amongst the problems that is to be faced by the concerned people, this whole aspect of Transportation problem is one that is to be dealt with extreme care. This problem generally deals with distribution of a specific product from its place of origin to various sources through a single mode of transportation with a minimal cost associated with it.

Example: Let us take that there are X factories which produce a particular commodity that is required by Y number of markets. In such a scenario, it is taken as supply to be S1, S2…SM, with the demands of the market,is taken as D1, D2……DM.

Associated issues of transportation problem:

There is a very important issue that is associated with Transportation problem that is Assignment Problem. Here a columnar format is taken into consideration where supply is measured in rows in regards to availability of resources as availability of men, vehicle, and capital. In the same manner, columns represent the various job that has to be done within a span of time.

One of the most important factors of this consideration is that for each of the machinery, there is only a singular person who is associated, and the work proceeds in that manner.

Theorems associated with Transportation problems:

Given that it is one of the most important factors that needs to be checked in regards to a particular production process, there are certain important theorems that needs understanding.

  • Theorem 1: The transportation problem has to be balanced, where a specific condition has to be satisfied to get an acceptable condition.
  • Theorem 2: It is always taken that basic variables within a basic feasible solution are m x n, where transportation problem is takenm + n-1

There are certain points to note apart from merely understanding the theorems. It is these points that ensure that theorems that have been given stand their ground.

Points to note:

  • If it is seen that basic variable varies between positive and zero value, it can be taken that corresponding cell that is present is either occupied cell or basic cell.
  • If a basic variable takes zero as its value, it is taken that basic feasible solution is degenerate.
  • However, in case of non-basic variables, point taken is said to be zero.
  • Also, in case of non-basic variables, the corresponding cells is known as non-occupied cell/non-basic cell/non-allocated cell.

There is another important factor that needs to be taken into consideration in this case. It is known as Loop.

This loop is a closed circuit connection in a transportation set that helps in connecting the occupied cells by fulfilling certain conditions.

Conditions kept for loop completion:

  • Each line present can only connect 2 occupied cells.
  • The number of connected cells has to be even in every case.
  • It has a set of vertical and horizontal lines that connect the cells.
  • It is taken that lines skip the cell that is kept in the middle of the 3 adjacent lines to ensure that the condition mentioned above of each line connecting 2 cells is

Thus, with these aspects taken care of and other priorities fulfilled, a transportation problem can be placed properly and get its ideal solution.

That transportation problem that does not have any loop, it is considered to be a basic transportation problem.


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