Please answer with detaield and complete solution. Thanks! 8. The Ministry of He
ID: 466368 • Letter: P
Question
Please answer with detaield and complete solution. Thanks!
8. The Ministry of Health have four medical teams available to allocated among four barrios. Each medical team will be sent to a single barrio. The Ministry of Health needs to determine how to assign these medical teams to these barrios in order to maximize the total effectiveness of the four teams. The measure of effectiveness being used is additional man- years of life. This measure equals the country's increased life expectancy in years times its population. The table below gives the estimated additional man-years of life in multiples of 1,000 if each of the four medical teams were assigned to each of the four barrios Barrio Medical Team a) What is the optimal assignment? b) Are other optimal assignments possible? If so, what are they?Explanation / Answer
The problem is assignment model problem, a special case of transportation model problem in Operations Management. The complete step-by-step solution is given below;
Step 1: The asssignment problem is a maximiation problem, which we have to convert into minimization problem to solve it. Hence to convert it into a minimization problem subtract all the values in the table from the maximum value in the table. In this case it is 19, hence subtract all values from 19 and rewrite the table;
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Step 2: Reduce the table row-wise
Sutract the smallest value in the row from all the values in that particular row. In case of row 1, the smallest value is 14 so subtract 14 from all the values in row 1. Repeat the same for all the rows you will get the following table.
Step 3: Reduce the table column-wise
Repeat step 2, but for column this time. So subtract the smallest value in a column from all the values in that particular column. In case of column 1, the smallest value is 2 hence subtract 2 from all the values of column. Following the same for all the columns the reduced table would look like followin
Step4: Assignment of Medical team to the respective Barrios with '0' in the row.
Now the assignment of all Medical teams is not poosible in this case since Team 2 and Team 4 both contains '0' in the same column i.e. column 2. Here we have to follow Hungarian method to solve it further. Follow the follwing steps.
Step 5: Draw the minimum number of lines in the table such it covers all the zeros in the table.
Since no. of lines < order of the matrix, this table does not give optimal solution.
Step 6 : Select the minimum value from those elements through which no line is passing. In this case it is 2.
Now subtract this minimum value from all the elements thorugh which no line is passing. And add this value to the elements on intersection of lines. The table should look like this; here minimum no. of lines crossing zeros = order of the matrix, this is a optimal solution.
Step 7 : Assign the Medical Teams to the Barrios with selecting '0' in the row/column such that there is no '0' in that row/column. In this case we select column 1 where there is only one '0' in row 1, hence we assign Team 1 to Barrios 1. Similarly, continue with the assignment with remaining rows and columns the solution is as given below;
There is only one solution to this problem since there are no two rows with zeros in the same column and hence there is only one optimal solution to this problem.
Barrio Medical Team 1 2 3 4 1 16 14 16 16 2 14 5 9 9 3 7 13 0 2 4 17 2 97