There are some underlying assumptions extracted from the transportation model:
Availability of the commodity: if the supply of the numerous sources is equal then it is called a balanced problem, and if it is more than the total demand of the destinations then it has to be balanced.
i.e. i=1ʃmai = j=1ʃn bi
A car manufacturing company has factories at places A, B and C. the cars have to reach destinations at cities X and Y. the capacity of the factories during the next quarter is 1000, 1500 and 1200 cars. The quarterly demand of the two cities X and Y 2300 and 1400 cars if transported by train then the per car transportation cost per km is Rs. 8.
Here is the chart showing the distance between the factory and destination.
Now calculate how many cars should be transported from which factory to which destination to minimise the cost?
Step I:The market distance and the cost of transportation per km are given to us, and that must be converted into costs.
A 2000 5380
B 2500 2700
C 2550 1770
Let us assume Xijas the number of cars that are transported from factories to the destination cities. Since the total supply (1000+1500+1200=3700) happens to equal the total demand (2300+1400=3700). Thus the transportation model is balanced.
X Y Supply
A X11 X12 1000
Source B X21 X22 1500
C X31 X32 1200
Demand 2300 1400 3700
Step II: minimize the transportation cost between the source and destination
I.e. Z = 2000 X11 + 5380 X12 + 2500 X21 + 2700 X22 + 2550 X31 + 1700 X32
In general, if C is the unit cost of transportation from ith source to jth destination, the objective is
Minimizei=1ʃ mj=1ʃn Cij Xij
Step III. Constraints
(a) Availability or supply of cars
X11 + X12 = 1000 (Source A)
X21 + X22 = 1500 (Source B)
X31 + X32 = 1200 (Source C)
As we have three factories hence there are three constraints.
(b) Demand of cars at the destination
X11 + X21 + X31 = 2300 (Destination center X)
X12 + X22 + X32 = 1400 (Destination center Y)
In this problem there are (m × n) variables i.e. (3 × 2= 6) variables and (m + n) constraints which is (3+2=6).
So from all the above steps it is evident enough that the objective function and constraint equations are linear as a result it can be solved by LPP or simplex method. But 6 variables or even more (in real life) will take a lot of time for the calculations. Therefore a computer aid is required. A point to remember is the simplex method best applicable for maximising the problems, but our motive is minimising the objective function.
SOLUTION OF THE TRANSPORTATION MODEL
Step I: Create a transportation model
X Y supply
Factories X11= 1000 Rs= 5380 1000
B Rs.2500 Rs. 2700
X21=1300 X22=1400 +1500
C Rs. 2550 Rs. 1700 +1200
Demand 2300 +1400 =3700
The above problem is thus balanced and when it is not so then a dummy source is createdto balance the demand and supply.
Step II: finding a basic feasible solution
There are many methods to determine the possible basicmethod. Out of the many methods, ‘north-west corner rule’ Dantzig is one.