Transportation Model

Transportation methods of solving linear programming problem aim at minimizing cost of (material handling) sending goods form dispatch stations to reviving ends. Generally, specific  quantities of goods are sipped form several dispatch stations to a number of receiving centers. Linear programming approach in such situations tends to find, how many goods may be transported from which dispatch station to which receiving end in order to make the shipment most economical (least costly). Transportation technique in such problems gives a feasible solution which is improved in a number of subsequent steps till the optimum (best) feasible solution is arrived at.

Assumptions

1.    Availability of quantity.

Quantity available for distribution at different sources or depots is equal to total requirement of different consumption centers.

                               m                n
                               ∑ ai   =       ∑ bj
                               i-1              j=1

quantity available = quantity required.

2.    Transportation of items.

Items can be conveniently transported form every production centre to every consumption centre.

3.    Cost per unit.

The per unit transportation cost of items form one production centre to another consumption contain certain.

4.    Independent cost.

      The per unit cost of transportation is independent of the quantity

5.    Objective.

The objective of such an arrangement is to minimize the total cost of transportation for the organization as a whole.

Few basic terms with reference to the transportation problem

1. Feasible Solution (FS).

A set of non-negative individual allocations (x _{ij  ³} 0 ) which simultaneously removes deficiencies is called a feasible solution.

2. Basic feasible solution (BFS).

A feasible solution to a m-origin, n-destination problem is said to be basic if the number of positive allocations are m + n-1i.e., one less than the sum of rows and columns.

3. Optimal solution.

A feasible solution is said to be optimal if it minimizes the total transportation cost. The optimal solution itself may or may not be a basic solution. This is done through successive improvements to the initial basic feasible solution until no further decrease in transportation cost is possible.

There are two methods for solving transportation problems.

(a)    Vogel’s approximate method

It was developed by W.R. Vogel and gives good approximation to the solution. It provides optimum solution in simple problems.In complex transportation problems also, the first solution is obtained using Vogel’s method which is further worked upon and tested for optimality by stepping stone method or modified distribution method.

(b)North west corner method.

It derives its name from the fact that initial allocation of resources is started from the north-west corner of the matrix. Ignoring other things (i.e. cost) and simply considering the factory capacity and dealer’s requirements, the minimum of the two placed in north-west corner.

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