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...