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=1}ʃ^{m}a_{i = j=1}ʃ^{n }b_{i}

- Transportation of commodity/terms: the model assumes that the commodities are being transported from the source to the destination very conveniently.
- The certainty of per unit transportation cost: obviously there is some transportation cost between the sources and destination.
- Independent per cost unit: per cost unit does not depend on the quantity being transported from source to destination.
- A number of units being transportedare proportional to the transportation cost of any route.
- The objective is to minimise the transportation cost for transporting commodities between two places.

**Example 5.1**

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.

X Y

1000 2690

1250 1350

1275 850

Now calculate how many cars should be transported from which factory to which destination to minimise the cost?

**Solution:**

**Step I:**The market distance and the cost of transportation per km are given to us, and that must be converted into costs.

X Y

A 2000 5380

B 2500 2700

C 2550 1770

Let us assume X_{ij}as 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 X_{11 }X_{12 }1000

Source B X_{21 }X_{22 }1500

C X_{31 }X_{32 }1200

Demand 2300 1400 3700

**Step II:** minimize the transportation cost between the source and destination

I.e. Z = 2000 X_{11 }+ 5380 X_{12 }+ 2500 X_{21 }+ 2700 X_{22 }+ 2550 X_{31 }+ 1700 X_{32}

In general, if C is the unit cost of transportation from ith source to jth destination, the objective is

Minimize_{i=1}ʃ ^{m}_{j=1}ʃ^{n} C_{ij }X_{ij}

**Step III**. Constraints

(a) Availability or supply of cars

X_{11} + X_{12} = 1000 (Source A)

X_{21} + X_{22} = 1500 (Source B)

X_{31} + X_{32} = 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

Distribution centers

X Y supply

A Rs.2000

Factories X_{11}= 1000 Rs= 5380 1000

B Rs.2500 Rs. 2700

X_{21}=1300 X_{22}=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.