SPPU Mechanical Engineering (Semester 7)
Operation Research
December 2016
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Solve any one question from Q.1 & Q.2(a,b)
1 A firm manufactures pain relieving pills in two sizes A and B. Size A contains 4 grains of element X, 7 grains of element Y and 2 grains of element Z. Size B contains 2 grains of element X, 10 grains of eleement Y and 8 grains of elementZ. It is found by users that it requires at least 12 grains of element X, 74 grains of element Y and 24 grains of element Z to provide immediate relief. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as standard L.P.P.
7 M

2(a) What is Decision Making? Explain and differentiate this under the condition of certainty and uncertainty.
4 M
2(b) State advantages and limitations of decision tree apporach.
3 M

Solve any one question from Q.3 & Q.4
3 Solve the following transportation problem using Steeping Stone Method.
Plants Destination Supply
D1 D2 D3 D4
P1 20 30 50 17 7
P2 70 35 40 60 10
P3 40 12 60 25 18
Requirement 5 8 7 15  
7 M

4 A team of 5 horses and 5 riders has entered a Jumping show contest. The number of penalty points to be expected when each rider rides any horse is shown below.
Horse Rider
  R1 R2 R3 R4 R5
H1 5 3 4 7 1
H2 2 3 7 6 5
H3 4 1 5 2 4
H4 6 8 1 2 3
H5 4 2 5 7 1

How should the horses be allotted to the riders so as to minimize the expected loss of the team?
7 M

Solve any one question from Q.5 & Q.6
5 Two firms are competing for business under the conditions such that one firm's gain is another firm's loss. Firm A' s payoff matrix is given below:
Firm B
  No Advertising Medium  Advertising Heavy Advertising
No Advertising 10 5 -2
Medium Advertising 13 12 15
Heavy  Advertising 16 14 10

Suggest optimal stratgies for the two firms and the net outcome thereof.
6 M

6 A ball bearing manufacturing company is planning to install ad additional plant which will require leasing new equipment for monthly payment of Rs.60,000. Variable cost would Rs.20 per item and each would retail for Rs.70.
i) How many ball bearing units must be sold in order to break even?
ii) What would be profit or loss if 1,000 items are made & sold in month?
iii) How many items must be sold to realize a profit of Rs. 40,000?
6 M

Solve any one question from Q.7(a,b) & Q.8(a,b)
7(a) The cost of parameters and other factors for a production inventory system of automobile pistons are given below. Demand per year = 6,000 unit Unit cost = Rs.40 Set-up Cost= Rs.500, Production Rate per year = 36,000 units, Holding cost per unit per year = Rs.8, Shortage cost per unit per year = Rs.20 Find
i) optimal lot size.
ii) number of shrotage and
iii) manufacturing time and time between set-ups
8 M
7(b) The cost of machine is Rs.6,100 and its scrap value is Rs. 100. The maintenance costs found from experience are as follows:
Year 1 2 3 5 6 7 8
Mainteance Cost (Rs.) 100 250 400 900 1,200 1,600 2,000

When should the machine be replaced?
8 M

8(a) A stockist has to supply 400 units of a product every Monday to his customers. He gets the product at Rs.50 per unit from the manufacturer is Rs. 75 per order The cost carrying inventory is 7.5% per year of the cost of product. Calculate:
i) Economic order quantity
ii) Total optimal cost
iii) Total weekly profit if the item is sold for Rs.55 per unit.
8 M
8(b) The following failure rates have been observed for a certain type of light bulbs:
End of week 1 2 3 4 5 6 7 8
Probability of  failure to date 0.05 0.13 0.25 0.43 0.68 0.88 0.96 1.00

The cost of replacing an individual failed bulb is Rs.1.25. The decision is made to replace all bulbs simultaneously at fixed intervals, and also to replace individual bulbs as they fail in service, If the cost of group replacement is 30 paise per bulb, what is the best interval beween group replacements? At what group replacement price per bulb would a policy of strictly individual replacement become preferable to the adopted policy?
8 M

Solve any one question from Q.9(a,b) & Q.10(a,b)
9(a) In a factory, the machines breakdown on an average rate of 10 machines per hour. The idle time cost of machine is estimated to be Rs. 20 per hour. The factory work 8 hours a day. The factory manager is coonsidering 2 mechanics for repairing the machines. The first mechanic A takes about 5 minutes, on an average, to repair a machine and demands wages at the rate of the Rs.10 per hour. The second mechanic B takes about 4 minutes in repairing a machine and deamands wages at the rate Rs. 15 per hour. Assuming that the rate of machine breakdown is Poisson distributed and the repair rate is exponentially distributed, which of the two mechanics shoud be engaged?
8 M
9(b) A machine operator has to perform three operations: turning, threading and knurling on a number of different jobs. The time required to perform these operations(in minutes) for each job is knowm. Determine the order in which the job should be processed in order minimize the total time required to turn out all the jobs. Also find the idle times for the three operations.
Job Time for turning(minutes) Time for threading(minutes) Time for knurling (minutes)
1 3 8 13
2 12 6 14
3 5 4 9
4 2 6 12
5 9 3 8
6 11 1 13
8 M

10(a) A super market has two sales girls at the sales counters. If the service time for each customer is exponential with a mean of 4 minutes. And if the people arrive in Poisson fashion at the rate of an 10 hour, then calculate:
i) Probability that a customer has to wait for being served?
ii) Expected percentage of idle time for each sales girl?
iii) If a customer has to wait, what is expected length of his waiting time?
8 M
10(b) A manufacturing company processes 6 different jobs on two machines A and B. Number of units of each job and its processing times on A &B are given in table. Find the optimal sequence. The total minimum elapsed time and idle time for either machine
Job No No. of Units of each job Processing time
Machine A(minutes) Machine B  (minutes)
1 3 5 8
2 4 16 7
3 2 6 11
4 5 3 5
5 2 9 7.5
6 3 6 14
8 M

Solve any one question from Q.11(a,b) & Q.12(a,b)
11(a) The utility data for a neatwork are given below. Determine the total, free, independent and interfering floats and identify critical path.
Activity 0-1 1-2 1-3 2-4 2-5 3-4 3-6 4-7 5-7 6-7
Duration 2 8 10 6 3 3 7 5 2 8
10 M
11(b) A dentist schedules all her patients for 30minutes appointments. Some of the patients take more or less than 30 minutes depending on the type of dental work to be done. The following summary shows the various categories of work, their probabilities and the time needed to complete work.
Category Time required(minutes)
Probability of category
Filling 45 0.40
Crown 60 0.15
Cleaning 15 0.15
Extraction 45 0.10
 Checkup 15 0.20

Simulate the dentist's clinic for four hours and determine the average waiting time for the patients as wll the idleness of the doctor. Assume that all the patients show up the clinic at exactly their scheduled arrived times, starting at 8 am. Use the following random numbers for handling the above problem: 40, 82, 11, 34, 25, 66, 17 and 79.
8 M

12(a) The time estimates ( in weeks) for the activities of a PERT network are given below.
Activity To Tm Tp
1-2 1 1 7
1-3 1 4 7
1-4 2 2 8
2-5 1 1 1
3-5 2 5 14
4-6 2 5 8
5-6 3 6 15

i) Draw the project network and identify all the paths throughit.
ii) Determine the expected project length.
iii) Calculate the standard deviation and variance of the project length .
iv) What is probability that the project will be completed?
1) At least 4 weeks earlier than expected time? 2) No more than 4 weeks later than expected time?
v) If the project due date is 19 weeks, what is the probability of not meeting the due date?
vi) The probability that the project will be completed on schedule if the scheduled completion time is 20 weeks. Vii) What should be the schedule completion time for the probability of completion to be 90%?
12 M
12(b) What do you mean by Goal programming? Where is it applicable?
6 M

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