R5311304-operations Research

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Code No: R5311304

III B.Tech I Semester(R05) Supplementary Examinations, May 2009 OPERATIONS RESEARCH (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) What is a model ? Discuss various classification schemes of models. (b) Discuss in brief the role of OR models in decision making process.

[8+8]

2. (a) Give a mathematical formulation of the transportation problem and the simplex problem, what are the differences in the nature of problems that can be solved by these methods. (b) Find the optimum solution to the transportation problem for which the cost, origin-availabilities and destination-requirements are given in table 1. [6+10]

Table 1: 3. (a) What are the three strategies of replacement of items which follow sudden failure mechanisms? Explain each of them with examples. (b) A firm is considering replacement of a machine, whose cost price is Rs. 12,200 and the scrap value Rs. 200. The running (maintenance and operating) costs are found from experience to be as follows: Year : 1 2 3 4 5 6 7 8 Running Cost (Rs) : 200 500 800 1,200 1,800 2,500 3,200 4,000 When should the machine be replaced? [6+10] 4. (a) Let A = { ajj } be an m × n payoff matrix for a zero-sum two-person game. Define a saddle point for matrix A and show that the value of the game is equal to the saddle value. (b) Show how a ‘game’ can be formulated as a liner programming problem. [8+8] 5. 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 the service time distribution is also exponential with an average 36 minutes, calculate the following: (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 be the change in (i) and (ii)? [16] 6. (a) Define the terms i. Lead time. ii. Reorder point. iii. Buffer stock. (b) A company uses 24,000 units of a raw material which costs Rs.12.50 per unit. Placing each order costs Rs.22.50 and the carrying cost is 5.4% per year of the average inventory. Find the economic order quantity and the total inventory cost (Including the cost of the material). [6+10] 7. Solve the following LPP by dynamic programming: Maximum Z = 3x1 +8x2 , Subject to x1 +4x2 ≤ 8, x2 ≤ 2, x1 , x2 ≥ 0.

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8. What is simulation? Discuss about application of simulation.

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Code No: R5311304

III B.Tech I Semester(R05) Supplementary Examinations, May 2009 OPERATIONS RESEARCH (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) What is a model ? Discuss various classification schemes of models. (b) Discuss in brief the role of OR models in decision making process.

[8+8]

2. (a) Give the mathematical formulation of an assignment problem. (b) Solve the assignment problem given in table 1.

[6+10]

Table 1: 3. (a) Explain the difference between age replacement and preventive maintenance. (b) The following failure rates have been observed for a certain type of light bulbs. End of week : Probability of failure to date :

1 0.05

2 0.13

3 0.25

4 0.43

5 0.68

6 0.88

7 0.96

8 1.00

The cost of replacing an individual bulb is Rs. 2.25, 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 60 paise per bulb and the total number of bulbs is 1000, what is the best interval between group replacement? [6+10] 4. (a) Differentiate between strictly determinable games and non-determinable games. (b) With the help of an appropriate example establish the relationship between ‘Game theory’ and ‘Linear Programming’. [8+8] 5. (a) Discuss about Kendal’s Notation for queuing models. (b) A person repairing radios finds that the time spent on the radio sets has an exponential distribution with mean 20 minutes. If the radios are repaired in the order in which they come in and their arrival is approximately Poisson with an average rate of 15 for 8-hour day, what is the repairman’s expected idle time in each day? How many jobs are ahead of the average set just brought in? [6+10] 6. (a) Classify inventory. (b) Find the economic lot size, that associates with total cost and the length of time between two orders, given that the set-up cost is Rs.100, daily holding cost per unit of inventory is 5 paise and daily demand is approximately 30 units. [6+10] 7. Solve the following problem: Maximize Z = u21 + u22 + u23 , Subject to u1 2 . u2 2 . u3 2 = 6, u1 , u2 , u3 all positive integers.

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8. By taking an example of simple inventory problem, explain phases of simulation. [16] ?????

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Code No: R5311304

III B.Tech I Semester(R05) Supplementary Examinations, May 2009 OPERATIONS RESEARCH (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Explain the meaning of basic feasible solution and degenerate solution to a LPP. (b) Solve the following LPP using Big - M method Maximize Z = -2x1 - x2 subject to the constraints 3x1 + x2 = 3 4x1 + 3x2 ≥ 6 x1 + 2x2 ≤ 4 x1 , x2 ≥ 0.

[6+10]

2. A manufacturer has distribution located at Agra. Allahabad and Kolkata. These center have available 40, 20 and 40 units of his product. His retail outlets at A, B, C, D and E requires 25,10,20,30 and 15 units of the product, respectively. The shipping cost per unit (in rupees) between each center and outlet is given in the following table. Distribution Centre Agra Allahabad Kolkata

A 55 35 40

Retail Outlets B C D 30 40 50 30 100 45 60 95 35

E 40 60 30

Determine the optimal shipping cost.

[16]

3. (a) What are the situations which makes the replacement of items necessary? (b) Madras Cola Inc. uses a bottling machine that costs Rs. 50,000 when new. Table below gives the expected operating costs per year in the annual expected production per year and the salvage value of the machine. The wholesale price for a bottle of drink is Re. 1.00. Data Associated with Age of Bottling Machine Age 1 2 3 4 5 Operating costs (Rs.) : 7,000 8,000 10,000 14,000 20,000 Production (Bottles) : 2,08,000 2,08,000 2,00,000 1,90,000 1,75,000 Salvage value (Rs.) : 30,000 19,000 15,000 12,000 10,000 When should machine be replaced?

[6+10]

4. (a) State the rules for detecting a saddle point. (b) Find the saddle point (or points) and hence solve the following game :  A

I II III

I  -5 5 4

B II 2 5 -2

III 1 4 0

[8+8]

 IV 20  6 -5

5. A barber with one man takes exactly 25 minutes to complete one hair cut. If customers arrive in a Poisson fashion at an average rate of one every 40 minutes, how long on an average must a customer wait for service? [16] 6. A newspaper boy buys a paper for Rs.2.60 each and sells them for Rs.3.60 each. He cannot return unsold newspapers. Daily demand has the following distribution. No. of Customers Probability

23 0.01

24 0.03

25 0.06

26 0.1

27 0.2

28 0.25

29 0.15

30 0.1

31 0.05

32 0.05

If each day’s demand is independent of the previous days, how many papers should he order each day?

[16]

7. Solve the following LPP by dynamic programming: Maximum Z = 3x1 +8x2 , Subject to x1 +4x2 ≤ 8, x2 ≤ 2, x1 , x2 ≥ 0.

[16]

8. Discuss about various types of simulation models.

[16]

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Code No: R5311304

III B.Tech I Semester(R05) Supplementary Examinations, May 2009 OPERATIONS RESEARCH (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Describe briefly the different phases of OR. (b) What are the essential characteristics of OR? Explain the role of computers in this field.

[8+8]

2. (a) What is a transportation problem? Explain. (b) Determine an initial basic feasible solution to the transportation problem given in table 1. Also find an optimal solution. [6+10]

Table 1: 3. A company is considering to replace grinder X presently of worth Rs 10,000 by a new grinder Y of Rs 20,000 but will be economic in running expenditures. The expected life of grinder X is 5 years with running expenditures of Rs 4,000 in fist year and then additional increase of Rs 400 per year for next four years. For the new grinder, the annual running cost is Rs 1,000 per year and scrap value of Rs 2,000. As an advisor to the company, find (a) The present value of the cost of old and new grinders considering 12 per cent normal rate interest. (b) Suggest whether the old grinder be replaced by the new grinder, assuming the life of new grinder to be 5 years. [16] 4. (a) Explain the “best strategy” on the basis of minimax criterion of optimalities. (b) Describe the maximin principle of game theory. What do you understand by pure strategies and saddle point? [8+8] 5. 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 the service time distribution is also exponential with an average 36 minutes, calculate the following: (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 be the change in (i) and (ii)? [16] 6. (a) What are the advantages and Disadvantages of inventory? (b) A certain item costs Rs. 235 per ton. The monthly requirement is 5 tons and each time the stock is replenished, there is a set-up cost of Rs. 1,000. The costs of carrying inventory has been estimated at 10% of the value of the stock per year. What is the optimum order quantity? [6+10] 7. Minimize Z = y 21 + y 22 + y 23 Subject to y1 + y2 + y3 = 10, When (a) y1 , y2 , y3 are non-negative, (b) y1 , y2 , y3 are non-negative integers.

[16]

8. Discuss about ‘Simulation-applications’.

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