r05410301 operations research

Code No: R05410301 Set No. 1

IV B.Tech I Semester Regular Examinations, January 2005OPERATIONS RESEARCH

(Electrical & Electronic Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

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1. (a) Give a general form of LPP.

(b) Solve the following LPP by penalty MethodMaximize Z = x1 + 2x2 + 3x3 - x4

subject to the constraintsx1 + 2x2 + 3x3 = 152x1 + x2 +5 x3 ≥ 20x1 + 2x2 + x3 + x4 = 10x1, x2, x3, x4 ≥ 0. [6+10]

2. ABC Limited has three production shops which supply a product to five warehouse.The cost of production varies from shop to shop and cost of transportation from oneshop to a warehouse also varies. Each shop has a specific production capacity andeach warehouse has certain amount of requirement. The costs of transportation aregiven below:

I II III IV V SupplyA 6 4 4 7 5 100B 5 3 7 4 8 125C 3 4 6 3 4 175

Demand 60 80 85 105 70 400

The cost of manufacturing the product at different production shops is

Shop Variable Fixed CostA 14 7,000B 16 4,000C 15 5,000

Find the optimum quantity to be supplied from each shop to different warehousesat minimum total cost. [16]

3. (a) Write a note on replacement of capital items when money value is considered.

(b) The following table gives the running costs per year and resale price of acertain equipment whose purchase price is Rs. 5000.

Year : 1 2 3 4 5 6 7 8Running costs (Rs.) : 1500 1600 1800 2100 2500 2900 3400 4000Resale value (Rs.) : 3500 2500 1700 1200 800 500 500 500

At what year is the replacement due? [6+10]

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4. (a) Let A = {aij} 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 isequal to the saddle value.

(b) Show how a ‘game’ can be formulated as a liner programming problem. [8+8]

5. 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 repairmen’s expected idle time each day? Howmany jobs are ahead of the average set just brought in? [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 andeach time the stock is replenished, there is a set-up cost of Rs. 1,000. Thecosts of carrying inventory has been estimated at 10% of the value of the stockper year. What is the optimum order quantity? [6+10]

7. Solve the following LPP by dynamic programming:Maximize Z = 3x1+8x2,Subject to

x1+4x2 ≤ 8,x2 ≤ 2,x1, x2 ≥ 0. [16]

8. (a) Explain how ‘Simulation’ is useful in optimization.

(b) By taking an example of simple inventory problem, explain phases of simula-tion. [6+10]

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

(b) Use duality to solve the LPPMinimize z = 8x1 - 2x2 - 4x3

Subjected tox1 - 4x2 - 2x3 ≥ 2x1 +x2 - 3x3 ≥ - 1-3x1 - x2 + x3 ≥ -1x1, x2, x3 ≥ 0. [6+10]

2. (a) Explain Least Cost method for obtaining an initial basic feasible solution of atransportation problem.

(b) A company has three factories I, II, III and four warehouses 1, 2, 3 and 4. Thetransportation cost (in Rs.) per unit from each factory to each warehouse isgiven in table 2b. The requirements of each warehouse and the capacity ofeach factory are given below:

table 2bFind the minimum cost transportation schedule. [6+10]

3. (a) Discuss in brief the replacement policy for the items that deteriorate withtime.

(b) A plant manager is considering replacement policy for a new machine. Heestimates the following costs (all costs in rupees).

Year : 1 2 3 4 5 6Replacement cost at the beginning of year 100 110 125 140 160 190Resale value at the end of year: 60 50 40 25 10 0Operating costs: 25 30 40 50 65 80

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Find an optimal replacement policy and corresponding minimum cost. [6+10]

4. (a) Solve the game, given in figure 4a by reducing to 2 × 2 game by graphicalmethod.

table 4a(b) Explain the best strategy on the basis of minimax criterion of optimalities.

[10+6]

5. The arrivals at a telephone booth are considered to be following Poisson processwith an average time of 10 minutes between one arrival and the next. Length of thephone call is assumed to be distributed exponentially with a mean of 3 minutes.

(a) What is the probability that a person arriving at the booth will have to wait?

(b) What is the average length of queue that forms from time to time? [16]

6. Find the optimum order quantity for a product for which the price breaks are asfollows:

Quantity Unit Cost (Rs.)0 ≤ Q1 < 800 Re. 1.00800 ≤ Q2 Re. 0.98

The yearly demand for the product is 1,600 units per year, cost of placing an orderis Rs. 6. The cost of shortage is 10% per year. [16]

7. Solve the following LPP by dynamic programming:Maximize Z = 3x1+8x2,Subject to

x1+4x2 ≤ 8,x2 ≤ 2,x1, x2 ≥ 0. [16]

8. (a) By taking an example of simple queuing problem, explain phases of simulation.

(b) How can simulation help to optimize the inventory? [6+10]

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1. (a) What are the advantages of using linear programming.

(b) A farmer has 100 acre farm. He can sell all tomatoes, lettuce, or radishes hecan raise. The price he can obtain is Rs 1.00 per kg for tomatoes, Rs 0.75 ahead for lettuce and Rs 2.00 per kg for radishes. The average yield per acre is2000 kg of tomatoes, 3000 heads of lettuce and 1000 kgs of radishes. Fertilizeris available at Rs 0.50 per kg and the amount required per acre is 100 kgseach for tomatoes and lettuce, and 50 kgs for radishes. Labour required forsowing and harvesting per acre is 5 man-days for tomatoes and radishes, and6 man-days for lettuce. A total of 400 man-days of labour are available atRs 20.00 per man-day. Formulate this as a Linear-Programming model tomaximize the farmers total profit. [6+10]

2. Find the optimum solution to the transportation problem given in table 2 for whichthe cost, origin-availabilities, and destination-requirements are given. [16]

table 2

3. A firm has a machine whose purchase price Rs. 1,00,000. Its running cost andresale price at the end of different years are as follows:

Year : 1 2 3 4 5 6Running Cost (Rs) : 7,500 8,500 10,000 12,500 17,500 27,500Resale price (Rs) : 85,000 76,500 70,000 60,000 40,000 15,000

Obtain the economic life of the machine and the minimum average cost. [16]

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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 repairman is to be hired to repair machines which break down at an average rateof 3 per hour. The breakdowns follow Poisson distribution; Non-productive time ofa machine is to cost Rs.16 per hour. Two repairmen have been interviewed: one isslow but cheap while the other is fast but expensive. The slow repairman chargesRs. 8 per hour and he services broken-down machines at the rate of 4 per hour.The fast repairman demands Rs.10 per hour and he services at an average rate of6 per hour. Which repairman should be hired? [16]

6. Find the optimalorder quantity for a product when the annual demand for theproducts is 500 units, the cost of storage per unit per year is 10% of the unit costand ordering cost per order is Rs.180. The unit costs are given below: [16]

Quantity Unit Cost (Rs.)0 ≤ q1< 500 Rs.25.00

500 ≤ q2< 1, 500 Rs.24.801, 500 ≤ Q3 < 3, 000 Rs. 24.60

3, 000 ≤ q4 Rs.24.40

7. Solve the following LPP by dynamic programming:Maximize Z = 8x1+7x2

Subject to2x1+x2 ≤ 8,5x1+2x2 ≤ 5,x1, x2

≥ 0. [16]

8. A company trading in motor vehicle spares wishes to determine the stock it shouldcarry for the items in its range. Demand is not certain and there is a lead time forstock replenishment for one item X. The following information is obtainded:Demand (units/day) : 3 4 5 6 7

Probability : 0.10 0.20 0.30 0.30 0.10[16]

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1. Solve the following problem by simplex methodMaximize Z = x1 - x2 + 3x3

subject to the constraintsx1 + x2 + x3 ≤ 102x1 - x3 ≤ 22x1 - 2x2 +3x3 ≤ 0 x1, x2, x3 ≥ 0. [16]

2. Use graphical method to minimize the time needed to process the following jobs onthe machines. Also calculate the total time elapsed to complete both the jobs.Job 1 Sequence A B C D E

Time (hrs) 2 6 5 4 7Job 2 Time C B D A E

(hrs)5 6 5 7 4 8

[16]

3. (a) Write a note on replacement of capital items when money value is considered.

(b) The following table gives the running costs per year and resale price of acertain equipment whose purchase price is Rs. 5000.

Year : 1 2 3 4 5 6 7 8Running costs (Rs.) : 1500 1600 1800 2100 2500 2900 3400 4000Resale value (Rs.) : 3500 2500 1700 1200 800 500 500 500

At what year is the replacement due? [6+10]

4. (a) State the major limitations of the game theory.

(b) Player A is paid Rs. 8 if two coins turn both heads and Rs. 10 if two coins turnboth tails. Player B is paid Rs. 3 when the two coins do not match. Giventhe choice of being A or B, which one would you choose and what would beyour strategy? [4+12]

5. A product manufacturing plant at a city distributes its products by trucks, loaded atthe factory warehouse. It has its own fleet of trucks of a private transport company.This transport company has complained that sometimes its trucks have to wait inline and thus the company loses money paid for a truck and driver of waiting truck.The company has asked the plant manager either to go in for a second warehouseor discount prices equivalent to the waiting time. The data available is:Average arrival rate of all trucks = 3 per hour.Average service rate is = 4 per hour.The transport company has provided 40% of the total number of trucks. Assumingthat these rates are random according to Poisson distribution, determine:

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(a) The probability that a truck has to wait?

(b) The waiting time of a truck that has to wait,

(c) The expected waiting time of company trucks per day. [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 andeach time the stock is replenished, there is a set-up cost of Rs. 1,000. Thecosts of carrying inventory has been estimated at 10% of the value of the stockper year. What is the optimum order quantity? [6+10]

7. An oil company has 8 units of money available for exploration of three sites. If oilis present at a site, the probability of finding it depends upon the amount allocatedfor exploiting the site, as given below:

Units of money allocated

0 1 2 3 4 5 6 7 8Site 1 0.0 0.0 0.1 0.2 0.3 0.5 0.7 0.9 1.0Site 2 0.0 0.1 0.2 0.3 0.4 0.6 0.7 0.8 1.0Site 3 0.0 0.1 0.1 0.2 0.3 0.5 0.8 0.9 1.0

The probability that oil exists at the sites 1, 2 and 3 is 0.4, 0.3 and 0.2 respectively.Find the optimal allocation of money. [16]

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

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r05410301 operations research

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