b.tech me - or i semester (r07) regular examinations,november 2

8/6/2019 B.tech ME - Or I Semester (R07) Regular Examinations,NOVEMBER 2

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Code No: 07A7EC03 R07 Set No. 2

IV B.Tech I Semester Regular Examinations,NOVEMBER 2010

OPERATIONS RESEARCH

Common to Mechanical Engineering, Mechatronics, Automobile Engineering

Time: 3 hours Max Marks: 80

Answer any FIVE Questions

All Questions carry equal marks

1. Solve the following L.P.P by dynamic programming method:Maximize Z = 2x1+5x2Subject to 2x1+x2 430,

2x2 460x1, x2 0. [16]

2. (a) What are the applications of OR techniques?

(b) Discuss the significance and scope of OR in modern management. [8+8]

3. (a) Discuss about significance of inventory.

(b) A stockist purchases an item at the rate of Rs. 40 per piece from a manufac-turer. 2,000 units of the item are required per year. What should be the orderquantity per order if the cost per order is Rs.15 and the inventory charges peryear are 20 per cent? [6+10]

4. Discuss about various types of simulation models. [16]

5. A game has the payoff matrix

A =

0 21 1

Show that E(x,y)=1-2x(y-1/2) and deduce that in solution of the game the firstplayer follows a pure strategy while the second has infinite number of mixed strate-gies. [8+8]

6. A T.V. repairman finds that the time spent on his jobs have an exponential distri-bution with mean of 30 minutes. If he repairs sets in the order in which they comein, and if the arrival of sets is approximately Poisson distribution with an averagerate of 10 per 8 hour day, what is repairmens expected idle time each day? Howmany jobs are ahead of the average set just brought in? [16]

7. (a) Explain Vogels Approximation method for obtaining an initial basic feasiblesolution of a transportation problem.

(b) Goods have to be transported from sources S1, S2 and S3 to destinations D1,

D2 and D3. The transportation cost per unit, capacities of the sources andrequirements of the destinations are given in table 7b.

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table 7b

Determine a transportation schedule so that the total cost is minimized. [6+10]8. (a) In the theory of replacement models, construct an equation for the cost of

maintaining a system as a function of the control variable t (the number ofperiods between group replacement).

(b) It has been suggested by a data processing firm that they adopt a policy ofperiodically replacing all the 1000 tubes in a certain piece of equipment. Atube is known to have the mortality distribution ( probability of failure) shownin the table below :

Tube failures/week : 1 2 3 4 5Probability of failure : 0.3 0.1 0.1 0.2 0.3

The cost of replacing the tubes on an individual basis is estimated to be Re. 1.00per tube and the cost of a group replacement is Re. 0.30 per tube. Compare thecost of preventive replacement with that of remedial replacement.

[6+10]

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OPERATIONS RESEARCH





1. (a) What is duality principle? Explain.

(b) Use the duality principle to solve the following L.P.P and find the solution ofprimal from the dual solution.

Minimize z = 4x1 + 2x2 + 3x3 subjected to: x1 + 4x3 52x1 + 3x2 + x3 4and x1, x2, x3 0. [4+12]

2. The inter arrival time of customer follows a discrete distribution as shown in thefollowing table:

Inter arrival time Probability(in Min)

1 0.202 0.253 0.354 0.105 0.086 0.02

Write the GPSS block. [16]

3. Trains arrive at the yard every 15 minutes and the service time is 33 minutes. Ifthe line capacity of the yard is limited to 4 trains, find

(a) The probability that the yard is empty.

(b) The average number of trains in the system. [16]

4. Find number of each of three items to be included in a package so that value ofpackage will be maximum. Total weight of package must not exceed 5 Kgs. [16]

Item Weight in kgs Value in Rs.1 1 302 3 803 2 65

5. (a) Write a short note on Assignment problem.

(b) Five operators have to be assigned to five machines. The assignment costs are

given in table 5b.

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table 5bOperator A cannot operate machine III and operator C cannot operate ma-chine IV. Find the optimum assignment schedule. [6+10]

6. (a) What is descount rate? How is it computed?

(b) A truck owner finds from his past records that the maintenance costs per yearof a truck whose purchase price is Rs. 8,000 are as give below:

Year : 1 2 3 4 5 6 7 8Maintenance cost (Rs) 1,000 1,300 1,700 2,000 2,900 3,800 4,800 6,000

Resale price (Rs) 4,000 2,000 1,200 600 500 400 400 400

Determine at which time it is profitable to replace the truck. [6+10]

7. (a) Discuss equivalence of matrix game and the problem of linear programming.Explain the method of solving a zero-sum two persons game as a linear pro-gramming problem.

(b) Find the minimax and maxmin value of the following game : [8+8]

3 7 1 34 8 0 6

6 9 2 4

8. A baking company sells cake by weight. It makes a profit of 50 paise a kg on every

pound sold on the day it is baked. It disposes of all cakes not sold on the day theyare baked at a loss of 12 paise a kg. If the demand is known to be rectangularbetween 2,000 and 3,000 kgs, determine the optimum daily amount baked. [16]

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OPERATIONS RESEARCH





1. A barber takes exactly 25 minutes to complete one hair cut. If customers arrivein a Poisson fashion at an average rate of one every 40 minutes, how long on anaverage must a customer wait for service? What is the probability that a customer

need not wait for service? [16]2. (a) Explain the replacement strategy for items of low cost which fail suddenly.

(b) The following failure rates have been observed for a certain type of light bulb.

End of Week : 1 2 3 4 5 6 7 8Probability of failure : 0.05 0.13 0.25 0.43 0.68 0.88 0.96 1.00

The cost of replacing an individual bulb is Rs. 2.25. The decision is madeto replace all bulbs simultaneously at fixed intervals, and also to replace in-dividual bulbs as they fail in service. If the cost of group replacement is 60

paise per bulb and the total number of bulbs is 1,000, what is the best intervalbetween group replacements? [6+10]

3. (a) Is OR a discipline or a profession, or set of techniques, or a philosophy, or anew name of an old thing? Justify your answer.

(b) How can OR models be classified? Explain. [8+8]

4. (a) Solve the game given in table 4a by reducing to 2 2 game by graphicalmethod.

table 4a

(b) Which competitive situation is called a game? [10+6]

5. Find the most economic batch quantity of a product on a machine of the productionrate of the item on the machine with 200 pieces/day and the demand is uniformat the rate of 100 pieces/day. The set-up cost is Rs.200 per batch and the cost ofholding one item in inventory is Rs.0.81 per day. How will the batch quantity varyif the machine production rate was infinite? [16]

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6. Use dynamic programming to find the maximum value of the product y1, y2, y3...... y

n.

When y1 + y2 + y3 + ...... +yn = Q. [16]

7. What is the importance of simulation and modeling? Explain utility of simulationto solve inventory problems. [16]

8. (a) How will you solve an Assignment problem, where a particular assignmentprohibited?

(b) A machine operator processes four types of items on his machine and he mustchoose a sequence for them. The set-up cost per change depends on the itemcurrently on machine and the item to be made as given in table 8b. If he

processes each of the item once and only once each week, then how should hesequence the items on his machine? Use the method of traveling salesman forthe problem to solve this. [6+10]

table 8b

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OPERATIONS RESEARCH





1. In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day.Assuming that the inter-arrival time follows an exponential distribution and theservice time distribution is also exponential with an average 36 minutes, calculate

the following: (Assuming 24-hour working day).(a) The mean queue size.

(b) The probability that the queue size exceeds 10.

(c) If the input of goods trains increases to an average 33 per day, what will bethe change in (a) and (b)? [16]

2. Maximize Z = 3x1+5x2Subject to x1 4

x2 6,3x1+2x2 18x1, x2 0.

Solve the problem using dynamic programming method. [16]

3. (a) Classify inventory.

(b) Find the economic lot size, that associates with total cost and the length oftime between two orders, given that the set-up cost is Rs.100, daily holdingcost per unit of inventory is 5 paise and daily demand is approximately 30units. [6+10]

4. (a) What is the principle of duality in LPP? Explain its advantages.

(b) Use duality to solve the LPPMinimize z = 8x1 - 2x2 - 4x3Subjected to

x1 - 4x2 - 2x3 2x1 +x2 - 3x3 - 1-3x1 - x2 + x3 -1x1, x2, x3 0. [6+10]

5. (a) Define:

i. Competitive game;

ii. Pure strategies;iii. Mixed strategies

iv. Two-person zero-sum (or rectangular) game,

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v. Payoff matrix.

(b) Player A and B, each take out one or two matches and guess how manymatches the opponent has taken. If one of the players guesses correctly, thenthe loser to pay him as many rupees as the sum of the number held by bothplayers. Otherwise, the pay out is zero. Write down the payoff matrix andobtain the optimal strategies of both players. [8+8]

6. What is simulation? Discuss about application of simulation. [16]

7. A manufacturing company has three factories F1, F2 and F3 with monthly manu-facturing capacities of 7,000, 4,000 and 10,000 units of a product. The product isto be supplied to seven stores. The factory capacities, store equipments and units

cost (in rupees) of shipping from each factory to each store are given in table 7.

table 7Find the optimal transportation plan so as to minimize the total transportationcost.

[16]

8. (a) Explain how the theory of replacement is used in the following problems:

i. Replacement of items whose maintenance cost varies with time.

ii. Replacement of items that fail completely.

(b) The following mortality rates have been observed for a certain type of fuse:

Week : 1 2 3 4 5Percent failing by the end 5 15 35 57 100

There are 1,000 fuses in use and it costs Rs. 5 to replace an individual fuse.If all fuses were replaced simultaneously it would cost Rs 1.25 per fuse. It isproposed to replace all fuses at fixed intervals of time, whether or not they

have burnt out, and to continue replacing burnt out fuses as they fail. Atwhat intervals the group replacement should be made? Also prove that thisoptimal policy is superior to the straight forward policy of replacing each fuseonly when it fails. [6+10]

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b.tech me - or i semester (r07) regular examinations,november 2

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