ASSIGNMENT MODEL IN QUANTITATIVE ANALYSIS FOR MANAGEMENT DECISIONS

 

There are three steps in the assignment method:

Step 1. Find the opportunity cost table by;

   a. Subtracting the smallest number in each row of the original cost table        or matrix from every number in the that row; and 

b. then subtracting the smallest number in each column of the table obtained in part (a) from every number in that column 

Step 2. Test the table resulting from step 1 to see whether an optimal assignment can be made.  The procedure is to draw the minimum number  of vertical and horizontal straight lines necessary to cover all zeros in the table.  If the number of lines equals either the number of rows or columns in the table, then an optimal assignment can be made.  If the number of lines is less than the number of rows or columns, proceed to step 3.

Step 3. revise the present opportunity cost table.  This is done by subtracting the smallest number not covered by line from every other uncovered number. This same smallest number is also added to any number(s) lying at the intersection of horizontal and vertical lines. We then return to step 2 and continue with cycle until an optimal assignment is possible.

 

STEP 1 NOW - FIND THE OPPORTUNITY COST TABLE

PERSON	PROJECT
	1	2	3
A	N11	N14	N6
B	8	10	11
C	9	12	7

 

SOLUTION: ROW OPPORTUNITY COST  TABLE 

PERSON	PROJECT
	1	2	3
A	5	8	0
B	0	2	3
C	2	5	0

The next step is simply take the cost in the above Table and subtract the smallest number in each column from the number in that column.  The resulting total opportunity costs are given in the Table below.  Please know what is row and column as students still confuse about the two.

 

TOTAL OPPORTUNITY COST TABLE 

PERSON	PROJECT
	1	2	3
A	5	6	0
B	0	0	3
C	2	3	0

STEP 2:  TEST FOR AN OPTIMAL ASSIGNMENT 

Looking for the last table, we see that there are four possible zero opportunity cost assignments.  In this step, a simple test is designed to help us determine whether an optimal assignment can be made.  The method consists of finding the minimum number of Straight lines (vertical and horizontal) necessary to cover all zeros in the costs table.  Each line is drawn so that it covers as many zeros as possible at one time. If the number of lines equals the number of rows or columns in the table, then an optimal assignment can be made.  If on the other hand, the number of lines is less than the number of rows or columns, an optimal assignment cannot be made.  Test for optimal solution is shown in table below:

TEST FOR OPTIMALITY 

PERSON	PROJECT
	1	2	3
A	5	6	0
B	0	0	3
C	2	3	0

Pls, I posted this in haste to help. So, I could note drawn the lines. But you can see that we can only have two lines where there are zeros. I.e where we have 0.3.0 there should be a line and where we have 0.0.3there is line. I have painted it green instead of lines. We are striking or drawing lines where we have 2 zeros and above.

STEP 3:  REVISE THE OPPORTUNITY COST TABLE 

Often, we need to revise the table in order to shift one (or more) of the zero cost from its present location (covered by lines) to a new uncovered location in the table.

This is accomplished by subtracting the smallest number not covered by a line from all numbers not covered by a straight line.  This same smallest number is then added to every number (including zeros) lying at the intersection of any two lines.

The smallest uncovered number in the last table, 2, is subtracted from each of the four uncovered numbers. Also, 2 is added to the number covered by the intersecting horizontal and vertical lines.  The results of step 3 are shown in Table below.

REVISE OPPORTUNITY COST TABLE 

PERSON	PROJECT
	1	2	3
A	3	4	0
B	0	0	5
C	0	1	0

To test for an optimal assignment, we return to step 2 and find the minimum number of lines necessary to cover the zeros in the revise opportunity cost table.  Since it requires three lines, as shown in the table below, to cover the zeros, an optimal assignment can be made.

  

THE FINAL TEST OF OPTIMALITY 

PERSON	PROJECT
	1	2	3
A	3	4	0
B	0	0	5
C	0	1	0

The optimal assignments are as follows:

Assignment.                     Cost

A to Project 3.                    6

B to Project 2.                   10

C to project 1.                     9

Total cost.                       N25

 Note: That Green areas should be lines not green. Strike a line through them all where you have 2 or more zeros.