Modi Vam

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CD Tutorial 4 The MODI and VAM
Methods of Solving
Transportation Problems
Tutorial Outline
MODI METHOD
How to Use the MODI Method
Solving the Arizona Plumbing Problem with
MODI
VOGEL’S APPROXIMATION METHOD:
ANOTHER WAY TO FIND AN INITIAL
SOLUTION
DISCUSSION QUESTIONS
PROBLEMS
T4-2 CD TUTORIAL 4 THE MODI AND VAM METHODS OF SOLVING TRANSPORTATION PROBLEMS This tutorial deals with two techniques for solving transportation problems: the MODI method and Vogel’s Approximation Method (VAM).

MODI METHOD
The MODI (modified distribution) method allows us to compute improvement indices quickly for each unused square without drawing all of the closed paths. Because of this, it can often provide considerable time savings over other methods for solving transportation problems. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route. How to Use the MODI Method

In applying the MODI method, we begin with an initial solution obtained by using the northwest corner rule or any other rule. But now we must compute a value for each row (call the values R1, R2, R3 if there are three rows) and for each column (K1, K2, K3 ) in the transportation table. In general, we let The MODI method then requires five steps:

1. To compute the values for each row and column, set
Ri + Kj = Cij
but only for those squares that are currently used or occupied. For example, if the square at the intersection of row 2 and column 1 is occupied, we set R2 + K1 = C21. 2. After all equations have been written, set R1 = 0.

3. Solve the system of equations for all R and K values.
4. Compute the improvement index for each unused square by the formula improvement index (Iij) = Cij  Ri  Kj.
5. Select the largest negative index and proceed to solve the problem as you did using the stepping-stone method.
Solving the Arizona Plumbing Problem with MODI
Let us try out these rules on the Arizona Plumbing problem. The initial northwest corner solution is shown in Table T4.1. MODI will be used to compute an improvement index for each unused square. Note that the only change in the transportation table is the border labeling the Ri s (rows) and Kj s (columns).

We first set up an equation for each occupied square:
1. R1 + K1 = 5
2. R2 + K1 = 8
3. R2 + K2 = 4
4. R3 + K2 = 7
5. R3 + K3 = 5
Letting R1 = 0, we can easily solve, step by step, for K1, R2, K2, R3, and K3. 1. R1 + K1 = 5
0 + K1 = 5 K1 = 5
2. R2 + K1 = 8
R2 + 5 = 8 R2 = 3
3. R2 + K2 = 4
3 + K2 = 4 K2 = 1
R i
K j
C ij i j
i
j
ij
=
=
=
value assigned to row
value assigned to column
cost in square (cost of shipping from source to destination ) MODI METHOD T4-3
TABLE T4.1
Initial Solution to Arizona
Plumbing Problem in the
MODI Format
FROM
TO
ALBUQUERQUE BOSTON CLEVELAND
FACTORY
CAPACITY
DES MOINES
EVANSVILLE
FORT
LAUDERDALE
WAREHOUSE
REQUIREMENTS
5
8
4 3
100
Kj
Ri
R1
R2
R3
K1 K2 K3
200
200 300
100
100
300
100
4 3
9 7 5
300 200 200 700
4. R3 + K2 = 7
R2 + 1 = 7 R3 = 6
5. R3 + K3 = 5
6 + K3 = 5 K3 = 1
You can observe that these R and K values will not always be positive; it is common for zero and negative values to occur as well. After solving for the Rs and Ks in a few practice problems, you may become so proficient that the calculations can be done in your head instead of by writing the equations out. The next step is to compute the improvement index for each unused cell. That formula is improvement index = Iij = Cij  Ri  Kj

We have:
Because one of the indices is negative, the current solution is not optimal. Now it is necessary to trace only the one closed path, for Fort Lauderdale–Albuquerque, in order to proceed with the solution procedures.

The steps we follow to develop an improved solution after the improvement...
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