Question Paper: Operations Research : Question Paper Dec 2014 - Mechanical Engg. (Semester 7) | Visveswaraya Technological University (VTU)
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## Operations Research - Dec 2014

### Mechanical Engg. (Semester 7)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
1 (a) Define operations research. Explain the phases of operations research.(6 marks) 1 (b) A firm manufactures two products A and B on which the profit earned per unit are 3 and 4 respectively. Each product is processed on two machines M1 and M2. Product A requires one minute of processing time on MI and two minutes on M2while B requires one minute on M1 and one minute on M2. Machine M1 is available for not more than 7 hrs. 30 mins while machine M2 is available for 10 hrs during any working day. Find the number of units of product A and B to be manufactured to get maximum profit(14 marks) 2 (a) Solve the following LPP using simplex method
Maximize Z-3x1+2x2
Subjected to constraints
x2+x2≤4
x1-x2≤2
x1, x2≥0
(10 marks)
2 (b) Solve the given problem by using Big-M method.
Maximum Z-- 2x1-x2
Subject to constraints:
3x1+x2-3
4x1+3x2,\ge 6
x1-2x2\le and
x1,x2\ge0
(10 marks)
3 (a) ABC limited has three production shops supplying a product to 5 warehouses. The cost of production varies from shop to shop, cost of transportation from shop to shop, cost of transportation from shop to warehouses also varies. Each shop has a specific production capacity of each warehouse has certain amount of requirement. The cost of transportation are as given below

 Shop Warehouse Capacity Cost of production I II III IV V A 6 4 4 7 5 100 14 B 5 6 7 4 8 125 16 C 3 4 6 3 4 175 15 Requirement 60 80 85 105 70

Find the optimum quantity to be supplied from each shop to different warehouse at minimum cost.(12 marks) 3 (b) A ABC company has 5 takes and 5 Persons to perform. Determine the optimal assignment that minimizes the total cost
 Jobs Machines A B C D E P 6 7 5 9 4 Q 7 5 10 9 6 R 5 4 3 6 5 S 8 3 5 6 4 T 4 7 5 6 6
(8 marks)
4 (a) Explain the importance of integer programming(5 marks) 4 (b) Solve the following linear programming by Gomory technique
Maximum Z=x1+x2
Subject to 2x1+x2\le6
4x+5x2\le 20
x2 x2\ge and intergers
(15 marks)
5 (a) Define the following
i) Normal time
ii) Crash time
iii) Free float
(6 marks)
5 (b) R and D activity has 7 activities for which the three time estimate are given below along with its preceding activity
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 Activity Preceding activity Optimistic time (a) Most likely time(m) Pessimistic time (b) A - 4 6 8 B A 6 10 12 C A 8 18 24 D B 9 9 9 E C 10 14 18 F A 5 5 5 G D,E,F 8 10 12

i) Draw PERT network
ii) find EST,Lst and slack for each node
iii)Find critical path expected project duration
(14 marks)
6 (a) Briefly explain queuing system and its characteristics(6 marks) 6 (b) Arrival rate of telephone call at a telephone booth are according to poisson distribution , with an average time of 9 minute between two consecutive arrivals. The length of the telephone call is assumed to be exponentially distributed, with mean 3 minute
i) Determine the probability that a person arriving at the booth will having to wait.
ii) Find the average queue length
iii) The telephone company will install a second booth when convinced that an arrival would expect to have to wait at least four minute for the phone. Find the increase in flow rate of arrival which justify a second booth.
iv) What is the probably that he will have to wait for more than 10 minute before the phone is free?
(14 marks)
7 (a) Explain clearly the following terms
i)Pay off matrix
iii) Fair game
(6 marks)
7 (b) Use dominance to find the optimum strategies for the both player
$$\begin{matrix} B_{1} & B_{2} &B_{3} &B_{4} &B_{5} &B_{6} \end{matrix}\\\begin{matrix} A_{1}\\A_{2} \\A_{3} \\A_{4} \\A_{5} \end{matrix}\begin{bmatrix} 4 &2 &0 &2 &1 &1 \\4 &2 &1 &3 &2 &2 \\4 &3 &7 &-5 &1 &2 \\4 &3 &4 &-1 &2 &2 \\4 &3 &3 &-2 &2 &2 \end{bmatrix}$$
(7 marks)
7 (c) Solve the game by graphical method
$$\begin{matrix} b_{1} &b_{2} \end{matrix}\\\begin{matrix} a_{1}\\a_{2} \\a_{3} \\a_{4} \end{matrix}\begin{bmatrix} 1 &-3 \\3 &5 \\-1 &6 \\4 &1 \end{bmatrix}$$
(7 marks)
8 (a) Define i) Total elapsed time ii) Idle time(4 marks) 8 (b) List the assumption made while dealing sequencing problem(4 marks) 8 (c) We have five jobs which must go through the machines A,B and C in the order ABC. Determine a sequence for job taht will minimize the total elapsed time and idle time for each machine.
 Job number Processing time in hours 1 2 3 4 5 Machine A 5 7 6 9 5 Machine B 2 1 4 5 3 Machine C 3 7 5 6 7
(12 marks)