N6U1/5N15 Reg. No

St. Joseph’s College of Arts & Science (Autonomous) St. Joseph’s College Road, Cuddalore – 607001 EMT618 - OPERATIONS RESEARCH - II

Time : 3 hrs

Max Marks :75 SECTION – A (5X2=10) Answer ALL Questions

1. Define mixed strategy. 2. Define pure I.P.P. 3. What is meant by inventory? 4. Write the Kendall’s notation for representing queuing model. 5. What is meant by PERT? SECTION – B (3X5=15) Answer any THREE Questions 6. For what value of  the game with the following payoff matrix is strictly determinable. Player B

Player A



6

2

-1



-7

-2

4



7. Explain about the branch and bounded technique. 8. Explain about the characteristics of queuing model.

~1~

N6U1/5N15

2Cs R . C1

9. Prove that EOQ 

10. Draw the network for the following project: Activity A B C D E F G H I J K L Immediate - A A B B C C F D G, E I Predecessor H Duration(Weeks) 10 9 7 6 12 6 8 8 4 11 5 7 Number the events by Fulkerson”s rules and find the critical path. Also find the time for completing the project. SECTION – C (5X10=50) Answer ALL Questions 11. a) Reduce the following game by dominance and find the game value. Player B 1 2 3 4 2 4 0 4 2 4 Player A 2 4 0 4 0 8 (or) b) Solve the following 2  4 game by graphical method. Player B 1 2 3 4 I II III IV

Player A

3 3 4 0

I II

3 5

3 4

4 3

0 7

12. a) Solve the following I.P.P Max. Z  x1  2 x2 Subject to the constraints x1  2 x2  12, 4 x1  3x2  14, x1 , x2 are non negative integers.

~2~

N6U1/5N15

(or) b) Solve the following mixed I.P.P by using branch and bound technique. Max. Z  x1  x2 Subject to the constraints 2 x1  5x2  16,6 x1  5x2  30, x1 , x2  0 and are integers. 13. a) Arrival rate of telephone calls at a telephone booth is according to Poisson distribution, with an average time of 9 minutes between two consecutive arrivals. The length of telephone call is assumed to be exponentially distributed, with mean 3 minutes. i) Determine the probability that a person arriving at the booth will have to wait. ii) Find the average queue length that forms from time to time. iii) The telephone company will install a second booth when convinced that an arrived would expect to have to wait at least four minutes for the phone. Find the increase in flow of arrivals which will justify a second booth. iv) What is the probability that an arrival will have to wait for more than 10 minutes before the phone is free? v) What is the probability that he will have to wait for more than 10 minutes before the phone is available and the call is also complete? (or) b) At a railway station only one train is handled at a time. The railway yard is sufficient only for two trains to wait while other is given signal to leave the station. Trains arrive at the station at an average rate of 6 per hour and the railway station can handle them on an average of 12 per hour. Assuming Poisson arrivals and exponential service distribution, find the steady state probabilities of the various number of trains in the system. Also find the average number of trains in the system. 14. a) A particular item has a demand of 9000 units/year. The cost of one procurement is Rs. 100 and the holding cost per unit is Rs. 2.40 per year. The replacement is instantaneous and no shortages are allowed . Determine

~3~

N6U1/5N15

(i) The economic lot size. (ii) The number of order per year. (iii) The time between orders. (or) b) A company has a demand of 12,000 units/year for an item and it can produce 200 such items per month. The cost of one setup is Rs. 400 and the holding cost/unit/month is Rs. 0.15. Find the optimum lot size and the total cost per year, assuming the cost of 1 unit as Rs. 4. Also find the maximum inventory, manufacturing time and total time. 15. a) The expected times and variances for the activities of a PERT network are given below. Determine the slack time for each event and the critical path. If the schedule completion time is 32 months, find the probability of completion on schedule. Activity Expected time(month) Duration

1-2 1-3 2-4 2-5 3-4 3-6 4-5 4-6 5-7 6-7 4 5 2 12 3 8 10 6 8 10 8 3 1 5 2 4 4 2 1 8 (or) b) A project schedule has the following characteristics: Activity 1-2 1-3 2-4 3-4 3-5 4-9 5-6 5-7 6-8 7-8 8-10 9-10 Time

4

1

1

1

6

5

4

(i) Construct the network. (ii) Compute TE and TL for each event. (iii)Find the critical path.

*************

~4~

8

1

2

5

7