SPPU Mechanical Engineering (Semester 7)
Operation Research
May 2017
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 element Y and 8 grains of element Z. 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 three approach.
3 M

Solve any one question from Q.3 &Q.4
3 Solve the following transportation problem using Stepping Stone Method.
Plant Destination Supply
D1 D2 D3 D4
p1 20 30 50 17 7
p2 70 35 40 60 10
p3 40 12 60 25 8
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.
  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 shouls 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 firms's gain is another firm's loss. Firm A' s payoff matrix is given below:
Firm A 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 strategies for the two firms and the net outcome thereof.
6 M

6 A ball bearing manufacturing company is planning to install an additional plant which will require leasing new equipment for monthly payment of Rs. 60,000. Variable cost would be Rs. 20 per item and each item would retail for Rs.70.
a) How many ball bearing units must be sold in order to break even?
b) What would be profit or loss if 1,000 items are made &sold in a month?
c) 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 units
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 shortage 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 4 5 6 7 8
Maintenance Cost (Rs.) 100 250 400 600 900 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 product at Rs. 50 per unit from the manufacturer. The cost of ordering and transportation from the manufacturer is Rs. 75 per order. The cost of 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 th best interval between 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 a machine is estimated to be Rs. 20 per hour. The factory works 8 hours a day. The factory manager is considering 2 mechanics for repairing the machines. The first mechanic A take about 5 minutes, on an average, to repair a machine and demands wages at the rate of Rs. 10 per hour. The second mechanic B takes about 4 minutes in repairing a machine and demands wages at the rate of 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 should 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 known. Determine the order in which the jobs should be processed in order to minimize the total time required to turn out all the jobs. Also find the idle times for the three operations.

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 time 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)



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 network 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 patient for 30 minutes appointments. Some of the patients take more or less than 30 minutes depending on the types of dental work to be done. The following summary shows the various categoriesof 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 well the idleness of the doctor. Assume that all the patients show up at the clinic at exactly their scheduled arrived times, starting at 8 a.m. Use the following random numbers for handling the above problem: 40, 82, 11, 34, 25, 66, 17 and 19.
8 M

12(a) The time estimates (in weeks) for the activities of a PERT network are given below.
Activity t0 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 through it.
ii) Determine the expected project length.
iii) Calculate the standard deviation and variance of the project length.
iv) What is the 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 scheduled 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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