Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Students can download 12th Business Maths Chapter 10 Operations Research Additional Problems and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

I. One Mark Questions

Choose the correct Answer

Question 1.
Which of the following methods is used to verify the optimality of the current solution of the transportation problem?
(a) Least cost method
(b) Vogel’s method
(c) North-west comer rule
(d) None of these
Answer:
(a) Least cost method

Question 2.
The degeneracy’in the transportation problem indicates that _________
(a) Dummy allocations need to be added
(b) The problem has no feasible solution
(c) Multiple optimal solutions exist
(d) All of the above
Answer:
(c) Multiple optimal solutions exist

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 3.
The Hungarian method can also be used to solve ______
(a) Transportation problem
(b) Travelling salesman problem
(c) A linear programming problem
(d) All the above
Answer:
(b) Travelling salesman problem

Question 4.
An optimal solution of an assignment problem can be obtained only if, _________
(a) each row and column has only one zero element
(b) each row and column has at least one zero element
(c) The data is arranged in a square matrix
(d) None of the above
Answer:
(d) None of the above

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 5.
Say True or False.

  1. In a transportation problem, a single source may supply something to all destinations.
  2. A transportation model must have the same number of rows and columns.
  3. It is usually possible to find an optimal solution to a transportation problem that is degenerate.
  4. In a transportation problem, a dummy source is given a zero cost, while in an assignment problem, a dummy source is given a very high cost.
  5. The Hungarian method operates on the principle of matrix reduction, whereby the cost table is reduced to a set of opportunity costs.

Answer:

  1. True
  2. False
  3. True
  4. False
  5. True

Question 6.
Fill in the blanks.

  1. In a transportation problem, we must make the number of ________ and _______ equal.
  2. ______ or ______ are used to balance an assignment problem.
  3. The method of finding an initial solution based on opportunity costs is called _______
  4. ________ occurs when the number of occupied squares is less than the number of rows plus the number of columns minus one.
  5. Both transportation and assignment problems are members of a category of LP problems called ________
  6. In the case of an unbalanced problem, shipping cost coefficients of ______ are assigned to each dummy factory or warehouse.

Answer:

  1. units supplied, units demanded
  2. Dummy rows, dummy columns
  3. Vogel’s approximation method
  4. Degeneracy
  5. Network flow problems
  6. zero

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 7.
Match the following.

(a) Dummy column (i) Finding initial solution
(b) Northwest comer rule (ii) Rows ≠ Columns
(c) Hungarian method (iii) Supply ≠ Demand
(d) Feasible solution (iv) Assignment problem
(e) Unbalanced problem (v) All demand and supply constraints are met

Answer:
(a) – (iii)
(b) – (i)
(c) – (iv)
(d) – (v)
(e) – (ii)

Question 8.
The objective function of transportation problem is to ________
(a) Maximise total cost
(b) Minimise the total cost
(c) Total cost should be zero
(d) All the above
Answer:
(b) Minimise the total cost

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 9.
In transportation problem, optimal solution can be verified by using _______
(a) NWC
(b) LCM
(c) MODI method
(d) Matrix method
Answer:
(c) MODI method

Question 10.
The cells in the transportation problem can be classified as _______
(a) assigned cells and empty cells
(b) allocated cells and unallocated cells
(c) occupied and unoccupied cells
(d) assigned and unoccupied cells
Answer:
(c) occupied and unoccupied cells

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 11.
In transportation problem if total supply > total demand we add _________
(a) dummy row with cost 0
(b) dummy column with cost 0
(c) dummy row with cost 1
(d) dummy column with cost 1
Answer:
(b) dummy column with cost 0

Question 12.
In an LPP the objective function is to be ________
(a) Minimised
(b) Maximised
(c) (a) or (b)
(d) only (b)
Answer:
(c) (a) or (b)

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 13.
The method used for solving an assignment problem is called ________
(a) Reduced matrix method
(b) MODI method
(c) Hungarian method
(d) Graphical method
Answer:
(c) Hungarian method

II. 2 Mark Questions

Question 1.
Consider 3 jobs to be assigned to 3 machines. The cost for each combination is shown in the table below. Find the minimal job machine combinations.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 1
Solution:
Step 1:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 2
Step 2:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 3
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 4
Optimal assignment:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 5

Question 2.
Find an initial basic feasible solution by LCM.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 6
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 7
Total cost = (1 × 2) + (6 × 1) + (4 × 4) + (4 × 6)
= 2 + 6 + 16 + 24
= 48

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 3.
Find an initial basic feasible solution by the North West Corner Rule (NWC).
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 8
Solution:
Total demand = Total supply = 60
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 9
Total cost = (10 × 9) + (11 × 6) + (12 × 8) + (2 × 3) + (25 × 11)
= 90 + 66 + 96 + 6 + 275
= 533

Question 4.
Find an initial basic feasible solution using Least cost method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 10
Solution:
Total Demand = 5 + 8 + 7 + 14 = 34
Total Supply = 7 + 9 + 18 = 34
Since they are equal, problem is balanced.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 11
The minimum total transportation cost is = (7 × 10) + (2 × 70) + (7 × 40) + (3 × 40) + (8 × 8) + (7 × 20)
= 70 + 140 + 280 + 120 + 64 + 140
= Rs. 814

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 5.
Find the investment option using Maximin rule for the following:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 12
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 13
Max (5, -13, -5) = 5. Since the maximum payoff is 5, by maximin criteria, the decision is to invest in bonds.

III. 3 and 5 Marks Questions

Question 1.
Find an optimal solution to the following transportation problem by North West Corner Method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 14
Solution:
Total supply = 65 = Total demand. So the given problem is balanced.
First allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 15
Second allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 16
Third allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 17
Fourth allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 18
Total Transportation cost
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 19

Question 2.
Find an initial basic solution for the following transportation problem by Vogel’s Approximation method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 20
Solution:
Total demand = 72 + 102 + 41 = 215 and
Total supply = 76 + 82 + 77 = 235.
Total supply > Total demand. So we add a dummy constraint with 0 unit cost and with allocation 20 (235 – 215). The modified table is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 21
First allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 22
The maximum penalty is 16. Allot 20 units to cell (S2, Ddummy)
Second allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 23
Third allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 24
Fourth allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 25
The final allocation table is given below.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 26
The minimum total cost = (76 × 8) + (21 × 24) + (41 × 16) + (20 × 0) + (72 × 8) + (5 × 16) = 2424

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 3.
A company has 4 men available for 4 separate jobs. Only one man can work on anyone job. The cost of assigning each man to each job is given below. Find the optimal solution by the Hungarian method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 27
Solution:
The number of rows and columns are equal. So the given problem is a balanced assignment problem and we can get an optimal solution.
Step 1:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 28
Step 2:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 29
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 30
We are not able to assign job for person B. Proceed as follows. Draw a minimum number of vertical and horizontal lines to cover all the zeros.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 31
Subtract the smallest element 1 from all the uncovered elements and add it to the elements which lie at the intersection of two lines. Thus we obtain another reduced matrix for fresh assignment.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 32
Total cost is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 33

Question 4.
There are five machines and five jobs are to be assigned and the cost matrix is given below. Find the proper assignment.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 34
Solution:
Step 1: (Row-reduction)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 35
Step 2: (Column – reduction)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 36
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 37
We are not able to assign a machine to job D. We proceed as follows.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 38
The smallest uncovered element is 2. Subtract 2 from all those elements which are not covered. Add 2 all elements which are at the intersection of two lines. Then proceed with the new matrix.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 39
The assignment is as follows
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 40

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 5.
The cost of transportation from 3 sources to four destinations are given in the follow¬ing table. Obtain an initial basic feasible solution using
(i) North West Corner Rule (NWC)
(ii) Least Cost Method (LCM) and
(iii) Vogel’s Approximation Method (VAM)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 41
Solution:
(i) North West Corner Rule
We start by allotting the units to the North -West Comer cell. We show all the allocations in a single table.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 42
Total transportation cost is (200 × 4) + (50 × 2) + (350 × 7) + (100 × 5) + (200 × 3) + (1 × 300)
= 800 + 100 + 2450 + 500 + 600 + 300
= Rs. 4750

(ii) Least cost method (LCM)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 43
Transportation cost is = (250 × 2) + (200 × 3) + (150 × 7) + (100 × 5) + (200 × 3) + (300 × 1)
= 500 + 600+ 1050 + 500 + 600 + 300
= Rs. 3550

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

(iii) Vogel Approximation Method (VAM)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 44
There are five penalties which have the maximum value 2. The cell with the least cost is row 3 and hence select cell (3, D) for allocation.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 45
There are four penalties which have maximum value 2. Select cell (1, B) which has the least cost for allocation.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 46
The largest penalty is 6. Allot units to cell (2, A)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 47
The largest penalty is 3. Allot units to cell (3, B)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 48
We first allot 50 units to cell (3, C) which has less cost. Then the balance units we allot to cell (2, C). We get the final allocation table as follows.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 49
Transportation cost is = (250 × 2) + (200 × 3) + (250 × 5) + (150 × 4) + (50 × 3) + (300 × 1)
= 500 + 600 + 1250 + 600 + 150 + 300
= Rs. 3400

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