Operations Research Answers

3(b) Three custom officers check the luggage of the passengers of an airport. The

passengers are found to arrive at an average rate of 30 per 8 hours a day. The amount

of time a custom officer spends with the passenger is found to have an exponential

distribution with mean service time 32 minutes. (5)

(i) Find the probability that all the custom officers are idle.

(ii) Find the expected number of passengers in the queues.

(iii) Find the expected waiting time of passenger in the system.

3. (a) A company has three factories 1 2 F ,F and F3

which supply goods to four warehouses

1 2 3 W ,W ,W and . W4

The daily factory capacities of 1 2 F ,F and F3

are, respectively, six

units, one unit and ten units. The demand of the warehouses 1 2 3 W ,W ,W and W4

are,

respectively, seven, five, three and two units. Unit transportation cost are as

follows: (5)

W1 W2 W3 W4

F1

2 3 11 7

F2

1 0 6 1

F3

5 8 15 9

Find an initial basic feasible solution by the Vogel’s approximation method.

2. (a) A firm makes two products A and B has a total production capacity of 9 tonnes per

day, with A and B utilizing the same production facilities. The firm has a

permanent contract to supply at least 2 tonnes of A per day to another company.

Each tone of A requires 20 machine hours of production time and each tone of B

requires 50 machine hours of production time. The daily maximum possible number

of machine hours is 360. All the firm’s output can be sold and the profit made is Rs.

80 per tonne of A and Rs. 120 per tonne of B. Formulate the problem of maximising

the profit as an LPP and solve it graphically.

8(b) Using graphical method, solve the game whose pay-off matrix is given as: (4) 

 Player B

 I II III IV 

 I 1 3 − 3 7 

Player A 

 II 2 5 4 − 6

6. (a) A company has 5 jobs to be processed by 5 mechanics. The following table gives the 

return in rupees when the th i job is assigned to the th j mechanic. i,( j = ,2,1 K ).5,

How should the jobs be assigned to the mechanics so as to maximize the overall 

return?

Jobs 

1 2 3 4 5 

1 22 28 30 18 30 

Mechanics 2 30 34 18 11 26 

3 31 17 23 20 27 

 4 12 28 31 26 26 

 5 19 23 30 25 29

5. A businessman needs five cabinets, 12 desks and 18 shelves cleaned out. He has two part 

time employees, Rashid and Ruby. Ruby can clean one cabinet, three desks and three 

shelves in a day while Rashid can clean one cabinet, two desks and 6 shelves in one 

day. Rashid is paid Rs. 22 per day and Ruby is paid Rs. 25 per day. In the order to 

minimize the cleaning costs, for how many days should Rashid and Ruby be 

employed? Formulate the problem as a linear programming problem and find its 

solution by the graphical method.

4. (a) For the transportation problem given below, check whether the given basic feasible 

solution is optimal. If not, modify the given solution and find an optimal solution and 

the optimal value for the problem. (5) 

6 1

25

9 3

45 70

11 5

5

2

50

8

55

10

85

12

5

4 7

90

85 35 50 45

3. Solve the following LPP by the two-phase simplex method. (10) 

 Max 1 2 3 Z = x + x − x

 Subject to 4x1 + x2 + x3 = 4

3 2 6 x1 + x2 − x4 =

 x1

, x2

, x3 ≥ 0

2. Solve the 4( × )3 game with pay off matrix. (10) 

6 5 6

7 4 5

8 6 5

8 5 8

A

 At each stage, clearly explain the steps involved.

1. Which of the following statements are true? Give a short proof or a counter example in 

support of your answer. (10) 

 (i) For any two square matrices A and B, AB = BA.

 (ii) If the following table is obtained in the intermediate stage while solving an LPP by 

the Simplex method, then the LPP has an unbounded solution: 

−1 − 2 0 0 0

1

1

x 1 2 −1 0 1

4

x 0 3 −1 1 2

0 4 −1 0 1

 (iii) The number of basic variables in a feasible solution of a transportation problem with 

m sources and n destinations is mn.

 (iv) An optimal assignment of the assignment problem with cost matrix C is also an 

optimal assignment of the assignment problem with cost matrix .

t C

 (v) )2,1( is an optimal solution to the following LPP: 

 Max 2 1 4 2 Z = x + x subject to 

 x1 + 2x2 ≤ 5

 4 x1 + x2 ≤

 x1

, x2 ≥ 0