Operations Research Exercise 10.2

Dear Class 12 Samacheer Kalvi students, here are the Text Book Solutions for Exercise 10.2 Business Maths Chapter 10 Operations Research for your reference and study.

You can find links to the other exercises below.

Solutions to Operations Research Exercise 10.1

Solutions to Operations Research Exercise 10.3

Solutions to Operations Research Exercise 10.4

Operations Research Exercise 10.2

$\text{1. What is the Assignment problem?} \\$
$\text{The assignment problem is a particular case of transportation problem }\\$
$\text{for which more efficient (less-time consuming) solution method }\\$
$\text{has been devised by KUHN and FLOOD. Suppose that we }\\$
$\text{have 'm' jobs to be performed on 'n' machines . The cost of }\\$
$\text{assigning each job to each machine is }\; C_{ij }\; \text{(i =1,2,…,n and}\\$
$\text{j = 1,2,…,n). Our objective is to assign the different jobs to } \\$
$\text{the different machines(one job per machine) to minimize }\\$
$\text{ the overall cost. This is known as assignment problem.}\\$

$\text{2. Give mathematical form of Assignment problem.}\\$
$\text{Minimize Z =}\sum_{i=1}^{m} \sum_{j=1}^{n}C_{ij}x_{ij}\\$
$\text{Subject to the constrains} \\$
$\sum_{i=1}^{n}x_{ij}=1\; j=1,2....n\\$
$\sum_{j=1}^{n}x_{ij}=1\; j=1,2....n\; \text{and}\; x_{ij}=0\\$
$\text{or 1 for all ij}\\$

$\text{3. What is the difference between Assignment Problem and Transportation Problem}\\$

Assignment Problem	Transportation Problem
One-to-one allocation	Many-to-many allocation
Square matrix (n × n)	Rectangular matrix (m × n)
No quantities involved	Quantities of goods are shipped
Cost or time per assignment	Cost per unit transported
Exactly one assignment per row and column	Supply and demand constraints allow multiple allocations
Objective: Minimize total assignment cost/time	Objective: Minimize total transportation cost

$\text{4. Three jobs A, B and C one to be assigned to three machines }\\$
$\text{U, V and W. The processing cost for each job machine combination }\\$
$\text{is shown in the matrix given below. Determine the allocation }\\$
$\text{that minimizes the overall processing cost. (cost is in ₹ per unit)}\\$

$\text{Number of rows=Number of columns}\\$
$\text{Therefore, the given assignment problem is balanced.}\\$
$\text{Step 1: Select the smallest element in each row and subtract it}\\$
$\text{from all the elements in its row.}\\$

$\text{Step 2: Select the smallest element in each column and subtract it}\\$
$\text{from all the elements in its column.}\\$

$\text{Step 3: From the elements without a line choose the least element i.e 5}\\$
$\text{Add this element(5) to the element where the lines interesct namely 1.}\\$
$\text{Subtract this element(5) from the other elements with no line over it.}\\$
$\text{Leave the remaining elements as is. }\\$

$\text{Step 4: Examine the rows with exactly one zero and circle }\\$
$\text{ them. Cross off any additional zeros in that column.}\\$

$\text{Step 5: Examine the columns with exactly one zero and }\\$
$\text{ circle them. Cross off any additional zeros in that row.}\\$

$\text{Repeat step 4 and 5}\\$

$\text{The optimal assignment schedule is:}\\$

$\text{The optimal assignment cost (minimum)is:Rs.46}\\$

$\text{5. A computer centre has got three expert programmers. The centre }\\$
$\text{needs three application programmes to be developed. The head of }\\$
$\text{the computer centre, after studying carefully the programmes }\\$
$\text{to be developed, estimates the computer time in minitues required}\\$
$\text{by the experts to the application programme as follows.}\\$

$\text{Assign the programmers to the programme in such a way } \\$
$\text{that the total computer time is least.} \\$

$\text{Step 2: Select the smallest element in each column and subtract it}\\$
$\text{from all the elements in its column.}\\$

$\text{Step 3: Examine the rows with exactly one zero and }\\$
$\text{ circle them. Cross off any additional zeros in that row.}\\$

$\text{Step 4: Examine the columns with exactly one zero and }\\$
$\text{ circle them. Cross off any additional zeros in that row.}\\$

$\text{The optimal assignment schedule is:}\\$

$\text{The optimal assignment cost (minimum)is:Rs.280}\\$

$\text{6. A departmental head has four subordinates and four tasks }\\$
$\text{to be performed. The subordinates differ in efficiency and }\\$
$\text{ the tasks differ in their intrinsic difficulty. His estimates }\\$
$\text{ of the time each man would take to perform each task is given below:}\\$

$\text{How should the tasks be allocated to subordinates so as to minimize the total manhours?} \\$

$\text{Number of rows=Number of columns}\\$
$\text{Therefore, the given assignment problem is balanced.}\\$
$\text{Step 1:Select the smallest element in each row and subtract it}\\$
$\text{from all the elements in its row.}\\$

$\text{Step 2:Select the smallest element in each column and subtract it}\\$
$\text{from all the elements in its column.}\\$

$\text{Step 3: From the elements without a line choose the least element i.e 1}\\$
$\text{ Add this element(1) to the element where the lines interesct namely 9 and 23.}\\$
$\text{Subtract this element(5) from the other elements with no line over it.}\\$
$\text{Leave the remaining elements(with line over it) as is. }\\$

$\text{Step 4: Examine the rows with exactly one zero and }\\$
$\text{ circle them. Cross off any additional zeros in that row.}\\$

$\text{Step 5: Examine the columns with exactly one zero and }\\$
$\text{ circle them. Cross off any additional zeros in that row.}\\$

$\text{The optimal assignment schedule is:}\\$

$\text{The optimal assignment cost (minimum)is: 41 hours}\\$

$\text{7. Find the optimal solution for the assignment problem with the following cost matrix.}\\$

$\text{Step 2: Select the smallest element in each column and subtract it}\\$
$\text{from all the elements in its column.}\\$

$\text{Step 3: Examine the rows with exactly one zero and circle }\\$
$\text{ them. Cross off any additional zeros in that row.}\\$

$\text{The optimal assignment schedule is:}\\$

$\text{The optimal assignment cost (minimum)is: Rs.37}\\$

$\text{8. Assign four trucks 1, 2, 3 and 4 to vacant spaces A, B, C, D, E and F}\\$
$\text{ so that distance travelled is minimized. The matrix below shows the distance.}\\$

$\text{Number of rows \# Number of columns}\\$
$\text{Therefore, the given assignment problem is unbalanced.}\\$
$\text{To balance it, introduce 2 dummy columns with all entries as 0.}\\$
$\text{The revised assignment problem is given below:} \\$

$\text{Step 1: Select the smallest element in each row and subtract it}\\$
$\text{from all the elements in its row.}\\$
$\text{Step 2: Select the smallest element in each column and subtract it}\\$
$\text{from all the elements in its column.}\\$

$\text{Step 3: Examine the rows with exactly one zero and circle }\\$
$\text{ them. Cross off any additional zeros in that row.}\\$

$\text{Step 4: Examine the columns with exactly one zero and }\\$
$\text{ circle them. Cross off any additional zeros in that row.}\\$

$\text{The optimal assignment schedule is:}\\$

$\text{The optimal assignment cost (minimum)is: Rs.12}\\$

Operations Research Exercise 10.2

Operations Research Exercise 10.2

Related Posts:

Leave a Reply Cancel reply

Operations Research Exercise 10.2

Related Posts:

Reader Interactions

Leave a Reply Cancel reply