Transportation Technique

by Mahesh P. Kumar

Introduction
Transportation is one of the various techniques of operations research ,and is based on fundamental assumption of linear programming.

Transportation is concerned with movement of goods from one place to another place.In transportation we do not study modes of transportation rather we study few transportation situations ,such as transporting goods from plant/ factory to a particular warehouse,from warehouse to a retailer , or from retailer to a consumer.The place from where the goods start moving is called source/origin and the place where these are supposed to reach is called destination.

The transportation problem is to transport the goods that are stored at various origins to different destinations in such a manner that total supply is exhausted to meet the total demand , and at the same time total transportation cost is minimized.

General Transportation Table(3 by3type)
. D-I D-II D-III Supply
O-I C11(x11) C12(x12) C13(x13) S1
O-II C21(x21) C22(x22) C23(x23) S2
O-III C31(x31) C32(x32) C33(x33) S3
Dem. D1 D2 D3 TD/TS

A Transportation Problem:
. D-I D-II D-III Supply
O-I 7 5 2 25
O-II 3 4 1 75
O-III 9 8 6 100
Dem. 50 60 90 200


SOLUTION:

Transportation Table-I
(North West Corner Method)
. D-I D-II D-III Supply
O-I 7(25) 5 2 25
O-II 3(25) 4(50) 1 75
O-III 9 8(10) 6(90) 100
Dem. 50 60 90 200

The above given transportation table is obtained through following procedure:

Step-1

Select the north west corner i.e. upper left corner of the transportation matrix.In the above given table cell(I,I) is towards the north west.Here corresponding supply is 25 units whereas corresponding demand is 50 units ,and we allocate 25 units i.e. the quantity which is minimum of the corresponding supply and corresponding demand.

Step-2

Adjust respective supply and demand numbers i.e. as we allocate 25 units in cell(I,I),we are left with demand balance of 25 units whereas whole of the supply is exhausted.

Step-3

(A) If the supply of the first row is exhausted we move downward the first cell in the second row and repeat the step-2 .

(B)If demand of the forst column is met we move horizontally to the next cell in the second column and repeat the step-2. 

 After allocating 25 units in cell(1,1) 
             we allocate 25 units in cell(2,1) ,
then        we allocate 50 units in cell(2,2) ,
then        we allocate 10 units in cell(3,2) ,
finally     we allocate 90 units in cell(3,3)

Step-4

Continue the procedure till total supply is used to meet the total demand.
Transportation Table-I
(Least Cost Entry Method)
. D-I D-II D-III Supply
O-I 7 5(10) 2(15) 25
O-II 3 4 1(75) 75
O-III 9(50) 8(50) 6 100
Dem. 50 60 90 200


Transportation Table-I
(Vogel's Approximation Method)
. D-I D-II D-III Supply
O-I 7 5 2(25) 25
O-II 3(50) 4 1(25) 75
O-III 9 8(60) 6(40) 100
Dem. 50 60 90 200


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(a)Cost Matrix of Basic Cells(Cij=Ui+Vj)
. D-I D-II D-III Ui
O-I - - 2 2
O-II 3 - 1 1
O-III - 8 6 6
Vj 2 2 0(let) .



(a)Cost Matrix of Non Basic Cells
(^ij=Ui+Vj-Cij)
. D-I D-II D-III Ui
O-I 7 5 - 2
O-II - 4 - 1
O-III 9 - - 6
Vj 2 2 0 .


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Mahesh P. Kumar
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