Operations Research Answers

8. (a) Write the dual of the following LPP: (5)

Minimize 16 1 9 2 21 3 Z = x + x + x

Subject to the constraints

x1 + x2 + x3 = 16

2x1 + x2 + x3 ≥12

x1

, x2 ≥ 0

3

x -unrestricted.

7. Use dual simplex method to solve the following LPP. (10)

Min 1 2 2 3 3

z = x + x + x

Subject to

x1 − x2 + x3 ≥

x1 + x2 + 2x3 ≤ 8

x1 − x3 ≥ 2

x1

, x2

, x3 ≥ .0

6. (a) A contractor has to supply 10,000 bearings per day to an automobile manufacturer.

He finds that when he starts production run, he can produce 25,000 bearings per day.

The cost of holding a bearing in stock for one year is Rs. 2 and the set up cost of a

production run is Rs. 180. Find the EOQ. How frequently should the production run

he made? (5)

5(b) A department has five employees with five jobs to be performed. The time (in hours)

each men will take to perform each job is given in the table below: (5)

Employees

I II III IV V

A 10 5 13 15 16

B 3 9 18 13 6

C 10 7 2 2 2

D 7 11 9 7 12

E 7 9 10 4 12

How should the jobs be assigned, one job per employee, so as to minimize the total

man-hours?

5. (a) Use the simplex method to solve the following L.P.P. (5)

Max 4 1 3 2

z = x + x

Subject to

2x1 + x2 ≤1000

x1 + x2 ≤ 800

400 x1 ≤

x2 ≤ 700

.0 , x1

x2 ≥

4. (a) A television repairman finds that the time spent on his jobs has an exponential

distribution with a mean of 30 minutes. If he repairs sets in the order in which they

come in, and if arrival of sets follows a Poission distribution approximately with an

average rate of 10 per 8 hours day, what is the repairman’s expected idle time each

day, How many jobs are ahead of the average set just brought in?

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(b) Find the sequence of jobs that minimizes the total elapsed time required to complete

the following task on two machines.

Task A B C D E F G

I 2 5 4 9 8 5 4

II 6 8 7 4 9 8 11

Also, find the optimal elapsed time.

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.