# Pune University BE (Production Engineering) Operations Research Question Papers

B.E. (Production)

OPERATIONS RESEARCH (2008 Pattern) (Sem. – I)

Time :3 Hours]                                                                                              [Max. Marks :100

Instructions to the candidates:-

1)            Attempt question 1 or 2; 3 or 4 and 5 or 6 from section I.

2)            Attempt question 7 or 8; 9 or 10 and 11 or 12 from section II.

3)            Answers to section I and Section II be written in separate answer books.

4)            Use of calculators and Normal distribution probability tables is per permitted.

SECTION – I Unit – I

Q1) a) Solve by Simplex method :

 Maximize : Z = 6x1 + 8*2 5x1 + 1°*2 < 60 4*1 + * < 40 x1 , *2 > 0

b) The postal department is considering the purchase of vehicles to pick up and deliver mail from various offices. They are considering three types of vehicles. The costs of each of these are 5, 10 and 8 lakhs per vehicles. These require crew of 2, 4 and 4 persons per day considering multiple shifts respectively. They expect those to run for 60, 100 and 80 kilometers. Based on fuel economy, the operating cost per day for these vehicles are 200, 350 and 300 Rs/day respectively. They have budget restriction of Rs. 1.6 crore and have 80 people available for crew. How many vehicles of each type to be purchase so as to minimize the operating costs?

(Only formulate LPP. Do not solve it)                                                            

OR

Q2) a) Solve by Dual Simplex method :

 Ma*imize : Z = 2*i + 2 *2 4 2*1 + *2 > 4 *i + 2 *2 2 > 3 2*i + 2 *2 2 < 12 *i *2 > 0

b) Discuss Duality and its applications.

Unit – II

Q3) a) Following table represents cost of projects (lakhs) for particular bidder.

Find out optimal assignment of bidders to projects so as to minimize total cost. Find also total cost of the projects. One bidder is to be given only one project.    

 Projects Bidders I II III IV A 10 11 16 25 B 13 33 41 60 C 20 50 83 96 D 26 71 110 135

 

b) Discuss Travelling Salesman Problem.

OR

Q4) a) How unbalanced Assignment problem is solved? How maximization Assignment problem is solved?                                                                                                             

b) A company has three factories F1, F2, and F3 and goods are supplied to 4 different cities D1, D2, D3 and D4. The table shows per unit cost of transportation. The supply capacities and demand are as shown in the table.

 Factories Consumption centers Capacity D1 D2 D3 D4 F1 3 1 7 4 300 F2 2 6 5 9 400 F3 8 3 3 2 500 Demand 250 350 400 200

i)             Find BFS by VAM.                                                                                   

ii)           Test if VAM solution optimal?                                                               

Unit – III

Q5) a) Discuss Integer programming application.                                      

b)         How goals are set? How priority is set for different goals in Goal programming? Discuss.                                                                                                                               

c)        Discuss Bellman principle of optimality in view to Dynamic programming.



OR

Q6) a) What is 0-1 programming? How 0 – 1 IP problems are formulated? State applications.                                                                                                        

b) The following table represents arcs and the distances. A person wants to go from city 1 to city 10. The various distances are given in a table. Find the optimal path by Dynamic programming.                                                                                          

 Arc Distance Arc Distance Arc Distance Arc Distance 1-2 5 2-7 8 4-6 5 6-9 7 1-3 5 3-5 8 4-7 7 7-8 5 1-4 6 3-6 10 5-8 6 7-9 7 2-5 4 3-7 5 5-9 8 8-10 8 2-6 7 4-5 4 6-8 9 9-10 9

SECTION – II Unit – IV

Q7) a) Electronic equipment contains 500 resisters. When any resister fails it is replaced individually and the cost is Rs. 20 per resister. If all resisters are replaced at the same time the cost is Rs 5 per resister. The % surviving rate S(i) at the end of month ‘i’ is given in the table.                                                                                                             

 % Survival Rate Month (i) 0 1 2 3 4 5 S(i) 100 90 75 55 30 0

What is the optimum replacement plan?

b) Discuss dominance rule in Game theory with example.

OR

Q8) a) Discuss net present worth and future worth.

b) Solve the game by graphical method or sub – game method

 Strategies Player B b1 b2 b3 PlayerA al 3 1 4 a2 5 8 2
 Unit – V

Q9) a) Arrival rate of the customers at the banking counter follows Poisson distribution with mean 30 per hour. The service rate of the counter also follows Poisson distribution with mean of 60 per hour. Find 

i)             Probability of having zero customers in the system.

ii)           Probability of having 2 customers in the system.

iii)        Expected customers in the system

iv)         Mean customers in queue.

v)            Average waiting time in queue.

b) Derive equation for Economic Production Quantity. State your assumptions.    

OR

Q10) a) An automobile factory manufactures a particular type of gear in batches within the factory. The gear is used in the final assembly. The particulars of the gear are :

 3 00 00 000 per year

Demand rate                  10 000

Production rate              25 000

Set up cost                     180

Carrying cost                 2

Annual demand Find :

i)             Economic production Quantity.

ii)           Time between two setups.

iii)        Production period.

units/day

units/day

Rs/set-up

Rs per unit per year

iv)         Annual holding cost.                                                                                

v)            Annual set up cost and                                                                            

vi)         Annual total cost.                                                                                       b) Discuss minimum cost service rate.                                                                                      

Unit VI

Q11) a) Network is given below with three time estimates.

 Act 1-2 1-3 1-4 2-5 2-6 3-6 4-7 5-7 6-7 a* 5 18 26 16 15 6 7 7 3 b** 10 22 40 20 25 12 12 9 5 m*** 8 20 33 18 20 9 10 8 4

a* – Optimistic time estimate,                      b** – Pessimistic time estimate,

m*** – most likely time estimate

i)             Draw AOA network.                                                                                 

ii)           Find expected durations and variances.                                                

iii)        Find critical activities and critical path.                                               

iv)         How much is expected project duration?                                             

v)            Find probability that project takes more than 48 days.                      

vi)         Find probability that project is completed in 45 days.                       

vii)       Find the expected project duration for 95% confidence level? 

b) State different types of floats and discuss any one.                                        

OR

Q12) a) Network Immediate Predecessor (IP) table is given below.

 Activity A B C D E F G H I IP – – – A,B B D D,E C,F,G C,G Days 4 13 10 5 11 5 2 9 8
 i)             Draw AOA or AON network.                                                                  

ii)           Find critical activities and critical path.                                                

iii)        How long is the project duration?                                                          

iv)         Tabulate Early Start schedule and Late start schedule times.           

v)            Tabulate Total floats, Free floats, and Independent floats for any two non – critical activities.                                                                                                     

b) Compare CPM and PERT techniques.