The problem encountered in the stepping stone method of optimality test can be overcome try MODI method because we don’t have to evaluate the empty cells one by one, all of them can be evaluated simultaneously. This is considerably time saving. The method has the following steps.
Step – I
Set up the cost matrix of the problem only with the costs in those cells in which allocations have been made.

X 
Y 
A 
Rs.2000 

B 
Rs.2500 
Rs.2700 
C 

Rs.1700 
Step II.
Let there be set of number V_{j} (V_{1}, V_{2}) across the top of the matrix and a set of number U; (U_{1}, U_{2}, U_{3}) across the left side so that their sums equal the costs entered in the matrix shown above.


V_{1}=0 200 
V_{2}=200 
2500 
U_{1} 
Rs.2000 

2500 
U_{2} 
Rs.2500 
Rs.2700 
1500 
U_{3} 
Rs.1700 
U_{1}+V_{1} =2000 U_{2}+V_{2}=2700
U_{2}+V_{1} =2500 U_{2}+V_{2}= 1700
Let V_{1} = 0 then U_{1} = 2000, U_{2}= 2500
V_{2} =2700 – 2500 = 200
U_{3}=1500
Step III.
Leave the already filled cells vacant and fill the vacant cells with sums of U_{i} and V_{j}. This is shown in the matrix below:


0 V_{1} 
200 V_{2} 
2000 
U_{1} 
… 
2200 (V_{1}+V_{2}) 
2500 
U_{2} 
… 
… 
1500 
U_{3} 
1500 (U_{2}+V_{1}) 
… 
Step IV.
Subtract the vacant values now filled in step III from the original cost matrix. This will result in cell evaluation matrix and is shown below for the example in hand.
… 
53802200=3180 
… 
… 
25501500=1050 

Step V.
If any of the cell evaluation turns out to be negative, then the feasible solution is not optimal. If the values are positive the solution is optimal. In the present example, since both the cell evaluation values are positive, the feasible solution is optimal.
Let us take another example where some of the evolutions turns out to be negative to explain the entire procedure.
Let us assume the following transportation model for this purpose.
Distribution centres → (Press in the centre)


P 
Q 
R 
S 
Supply 
Plants 
A 
200 
300 
1100 
700 
6 (100) 

B 
100 
0 
600 
100 (1) 
1 (100) 10 (300) 

C 
500 
800 
1500 
900 
17 (Total) 

Demand 
7 
5 
3 
2 



(100) 
(300) 
(500) 
(600) 

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