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4
O N L I N E T U T O R I A L
The MODI and VAM Methods
of Solving Transportation Problems
DISCUSSION QUESTIONS
1. Vogel’s approximation method gives a good initial solution
because it makes each allocation on the basis of the opportunity
2. The major difference between the MODI and stepping-stone
methods is in the procedure used to test for optimality. In the
solution.
END-OF-TUTORIAL PROBLEMS
T4.1 (a) Hardrock’s initial solution using the VAM method is
T4.2 Hardrock’s problem now requires the addition of a dummy
project (destination) because supply exceeds demand. The VAM
initial solution is as follows:
The optimal solution, whose total cost is $980, is shown
in the following table. An alternate optimal solution also ex-
ists:
T4.3 (a) Using the northwest corner rule for the Saussy Lumber
Company data, the following initial solution is reached:
ONLINE TUTORIAL 4 THE MODI A ND VAM METH O D S OF SO LVING TR A N S PO R T A T I O N PRO B L E M S 373
(b) Solving the Saussy Lumber Company problem with MODI,
12 12 1 2
13 13 1 3
31 31 3 1
32 32 3 2
Pineville customer 2 3 0 1 2
Pineville customer 3 = 2 0 2 0
best improvement index Mapletown customer 1 = 3 1 3 1
Mapletown customer 2 = 2 1 1 0
I C R K
I C R K
I C R K
I C R K
− = = − − = − − = +
− = − − = − − =
− = − − = − − = −
− = − − = − − =
The final solution is also evaluated using MODI below.
Calculations of the Ri’s, Kj’s and improvement indices are:
1 1 11 1 1
0 3 or 3
R K C K K
+ = + = =
374 ONLINE TUTORIAL 4 THE MODI AND VAM METHODS OF SOLVING TRANSPORTATION PROBLEMS
T4.4 Krampf Lines Railway Company’s initial northwest corner
solution is shown below:
To test for improvement with MODI, we set up an equation
for each occupied square:
1
1 1 1 1
0
50 0 50 or 50
R
R K K K
=
+ = + = =
= − − = − − − = +
= − − = − − − = +
13 13 1 3
index 60 0 ( 40) 100
index 70 0 ( 90) 160
C R K
C R K
Second Krampf solution—cost = 5,500 miles:
1 2 2
30 30
R K K
+ = =
ONLINE TUTORIAL 4 THE MODI A ND VAM METH O D S OF SO LVING TR A N S PO R T A T I O N PRO B L E M S 375
Third and optimal Krampf solution = 3,100 miles:
T4.5 Jessie Cohen Clothing Group’s first VAM assignment table:
In the initial assignment table above, we see that the Z row has
This second VAM table (above) indicates that the greatest differ-
ence is now in the B column (4). We may assign up to 15 units to
the W − B square without exceeding the demand at store B.
376 ONLINE TUTORIAL 4 THE MODI AND VAM METHODS OF SOLVING TRANSPORTATION PROBLEMS
Third VAM assignment with W’s requirement satisfied:
The third VAM table involves assigning 20 units to the W − C route.
This is done because column C has the highest difference and square
ONLINE TUTORIAL 4 THE MODI A ND VAM METH O D S OF SO LVING TR A N S PO R T A T I O N PRO B L E M S 377
(b) MODI technique to test for optimality:
1
0
99
R
R K K
=
+ = =
34 34 3 4
_ _ _ _ _
index = = 7 ( 10) 5 = +
C R K 12
Because all improvement indices are zero or pos-
