Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 10 Operations Research Ex 10.2 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2

Question 1.
What is the Assignment problem?
Solution:
Suppose that we have ‘m1 jobs to be performed on ‘n’ machines. The cost of assigning each job to each machine is Cij (i = 1, 2, … n and j = 1, 2, … , n). Our objective is to assign the different jobs to the different machines (one job per machine) to minimize the overall cost. This is known as an assignment problem.

Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2

Question 2.
Give mathematical form of assignment problem.
Solution:
Consider the problem of assigning n jobs to n machines (one job to one machine). Let Cij be the cost of assigning ith job to the jth machine and xij represents the assignment of ith job to the jth machine.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 1
xij is missing in any cell means that no assignment is made between the pair of job and machine.(i.e) xij = 0.
xij is present in any cell means that an assignment is made their. In such cases xij = 1
The assignment model can written in LPP as follows
Minimize Z = \(\sum_{i=1}^{m}\) \(\sum_{j=1}^{n}\) Cij xij
Subject to the constrains
\(\sum_{i=1}^{n}\) xij = 1, j = 1, 2, …. n
\(\sum_{j=1}^{n}\) xij = 1, i = 1,2,….n and xij =0 (or) 1 for all i, j

Question 3.
What is the difference between Assignment Problem and Transportation Problem?
Solution:
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 2

Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2

Question 4.
Three jobs A, B, and C one to be assigned to three machines U, V, and W. The processing cost for each job machine combination is shown in the matrix given below. Determine the allocation that minimizes the overall processing cost.
(cost is in Rs per unit)
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 3
Solution:
Here the number of rows and columns are equal.
∴ The given assignment problem is balanced.
Step 1.
Select the smallest element in each row and subtract this from all the elements in its row.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 4
Look for atleast one zero in each row and each column.
Here each and every row and columns having exactly one zero No need for step 2 go to step 3.

Step 3.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 5
Mark the zero by □ Mark other zeros in its column by X.
Since each row and each column contains exactly one assignment, all three machines have been assigned a job.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 6
The Optimal assignment (minimum) cost = 46

Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2

Question 5.
A computer centre has got three expert programmers. The centre needs three application programmes to be developed. The head of the computer centre, after studying carefully the programmes to be developed, estimates the computer time in minutes required by the experts to the application programme as follows.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 7
Assign the programmers to the programme in such a way that the total computer time is the least.
Solution:
Here the number of rows and columns are equal.
∴ The given assignment problem is balanced.
Step 1.
Select the smallest element in each row and subtract this from all the elements in its row.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 8

Step 2.
Select the smallest element in each column and subtract this from all the elements in its column.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 9

Step 3. (Assignment)
Examine the rows with exactly one zero, mark the zero by □. Mark other zeros in its column by X.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 10

Step: 4
Now examine the columns with exactly one zero marks the zero by □. Mark other zeros in its row by X.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 11
Thus all three assignments have been made. The optimal assignment schedule and total cost is
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 12
The optimal assignment (minimum) cost = Rs 280

Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2

Question 6.
A departmental head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the time man would take to perform each task is given below.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 13
How should the tasks to allocated to subordinates so as to minimize the total man-hours?
Solution:
Here the number of rows and columns are equal.
∴ The given assignment problem is balanced.
Step 1.
Select the smallest element in each row and subtract this from all the elements in its row.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 14

Step 2.
Select the smallest element in each column and subtract this from all the elements in its column.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 15

Step 3. (Assignment)
Examine the rows with exactly one zero Mark the zero by □. Mark other zeros in its row by X.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 16

Step 4.
Now examine the columns with exactly one zero. Mark the zero by □ Mark other zeros in its row by X.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 17

Step 5.
Cover all the zeros of table 4 with three lines, since three assignments were made check (✓) row S since it has no assignment.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 18

Step 6.
Develop the new revised tableau. Examine those elements that are not covered by a line in table 5. Take the smallest element. This is 1 (one) our case. By subtracting 1 from the uncovered cells.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 19
[Adding 1 to elements (Q, S, R) that line at the intersection of two lines]

Step 7.
Go to step 3 and repeat the procedure until you arrive at an optimal assignment.

Step 8.
Determine an assignment.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 20
Thus all the four assignment have been made. The optimal assignment schedule and total time is
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 21
The optimum time (minimum) = 41 Hrs.

Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2

Question 7.
Find the optimal solution for the assignment problem with the following cost matrix.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 22
Solution:
Here the number of rows and columns are equal.
∴ The given assignment problem is balanced.
Step 1.
Select the smallest element in each row and subtract this from all the elements in its row.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 23

Step 2.
Select the smallest element in each column and sub tract this from all the elements in its column.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 24

Step 3. (Assignment)
Examine the rows with exactly one zero. Mark the zero by □ Mark other zeros in its column by X
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 25
Thus all the four assignments have been made. The optimal assignment schedule and total cost.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 26
The Optimum cost (minimum) = Rs 37

Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2

Question 8.
Assign four trucks 1, 2, 3 and 4 to vacant spaces A, B, C, D, E and F so that distance travelled is minimized. The matrix below shows the distance.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 27
Solution:
Since the number of columns is less than the number of rows, the given assignment problem is unbalanced one. To balance it, introduce two dummy columns with all the entries zeros.
The revised assignment problem is
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 28
Here only 4 tasks can be assigned to 4 vacant spaces.
Step 1.
It is not necessary, since each row contains zero entry. Go to step 2.

Step 2.
Select the smallest element in each column and subtract this from all the elements in its column.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 29

Step 3. (Assignment)
Since each row contains more than one zeros. Go to step 4.

Step 4.
Examine the columns with exactly one zero, mark the zero by □ Mark other zeros in its rows by X.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 30

Step 5.
Here all the four assignments have been made we can assign d1 for D then we will get d2 for E.
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 31
The optimal assignment schedule and total distance is
Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2 32
∴ The Optimum Distant (minimum) = 12 units

Samacheer Kalvi 12th Business Maths Guide Chapter 10 Operations Research Ex 10.2

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