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Operations Research (Elective I) (October 2009)

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Total No. of Questions : 6] P971 [Total No. of Pages :5 [3664] - 130 B.E. (Mechanical) OPERATIONS RESEARCH (2003 Course) (Sem. - I) (Elective - I) Time : 3 Hours] Instructions to the candidates: 1) 2) 3) 4) 5) [Max. Marks:100 Answers to the two sections should be written in separate books. Neat diagrams must be drawn wherever necessary. Figures to the right indicate full marks. Use of logarithmic tables, slide rules, Mollier charts, electronic pocket calculator and steam tables in allowed. Assume suitable data, if necessary. SECTION - I Q1) A company makes two kinds of leather belts. Belt A is a high quality belt and belt B is of a lower quality. The respective profits are Rs. 4 and Rs. 3 per belt. Each of type A requires twice as much time as belt of type B, and if all the belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 belts per day (both A and B combined). Belt A requires a fancy buckle and only 400 days per day are available. There are only 70 buckles available for belt B. What should be daily production of each type of belt? Formulate the linear programming problem and solve it by simplex method. [16] OR Q1) Solve the following LP problem by using the two-phase simplex method: Minimize Z = xl + x2 [16] Subject to the constraints 2x1+ 4x2 > = 4 x1 + 7x2 > = 7 And x1, x2 > = 0. P.T.O. Q2) Solve Transportation Problem by all methods and check optimality by MODI method. [16] X Y Z REQUIREMENT A 2 1 5 7 B 3 0 8 5 C 11 6 15 3 D 7 1 9 2 SUPPLY 6 1 10 OR Q2) a) Solve the following assignment problem. [8] P A B C D E b) i) ii) Q R S T 10 3 10 7 7 5 9 7 11 9 13 18 2 9 10 15 13 2 7 4 16 6 2 12 12 Explain in detail the procedure for solving travelling Salesman problem. [4] Explain that assignment is a special case of transportation problem. [4] Q3) a) Derive the formula for Economic Lot Size model with constant demand.[8] b) The production department for a company requires 3,600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs. 36 and the cost of carrying inventory is 25% of the investment in the inventories. The price is Rs. 10 per kg. [10] Find i) Optimal lot size. ii) Optimal order cycle time. iii) Minimum yearly variable inventory cost. iv) Minimum yearly total cost. OR Q3) a) Explain the EOQ models with quantity discounts. [3664]-130 2 [8] b) The annual demand of a product is 10,000 units. Each unit costs Rs.100. if orders placed in quantities below 200 units but for orders of 200 or above, the price is Rs.95. The annual inventory holding costs is 10% of the value of item and the ordering cost is Rs. 5 per order. Find the economic lot size. [10] SECTION - II Q4) a) Solve the game whose payoff matrix is given below : [8] Player B B2 Player A b) B4 4 2 4 0 0 4 0 8 3 3 4 0 B3 2 4 2 4 B1 What is replacement? Describe some important replacement situations.[8] OR Q4) a) A firm has a machine whose purchase price is Rs.1,00,000. It s running cost and resale price (Rs.) at the end of different years are as follows: Year 1 Running cost 7,500 2 3 4 5 6 8,500 10,000 12,500 17,500 27,500 Resale price 85,000 76,500 70,000 60,000 40,000 15,000 i) Obtain the economic life of the machine and the minimum average cost. ii) The firm has obtained a contract to supply the goods produced by machine, for the period of five years from now. After this time period, the firm does not intend to use the machine. If the firm has a machine of this type which is one year old, what replacement policy should it adopt if it intends to replace the machine not more [8] than once? b) Write short notes (any 2) : i) Two person zero sum. ii) Principle of dominance. iii) Pure strategy in game theory. [3664]-130 3 [8] Q5) a) 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 (the time taken to jump a train) distribution is also exponential with an average of 36 minutes. Calculate i) Expected queue size (line length) ii) Probability that the queue size exceeds 10. If the input of trains increases to an average of 33 per day what will be the change in (i) and (ii)? [8] b) Find the sequence that minimizes the total time required in performing the following jobs on three machines in the order ABC. Processing times (in hours) are given in the following table: Job : 1 2 3 4 5 Machine A : 8 10 6 7 11 Machine B : 5 6 2 3 4 Machine C : 4 9 8 6 5 [8] OR Q5) a) Derive the difference equations for the queuing model {(M/M1): ( /FCFS)}. [8] b) Arrivals at a telephone booth are considered to be Poisson with an average time of 10 minutes between one arrival and the next. The length of phone call assumed to be distributed exponentially, with mean 3 minutes, i) What is the probability that a person arriving at the booth will have to wait? ii) The telephone department will install a second booth when convinced that an arrival would expect waiting for at least 3 minutes for a phone call. By how much should the flow of arrivals increase in order to justify a second booth? iii) What is the average length of the queue that form from time to time? iv) What is the probability that it will take him more than 10 minutes altogether to wait for the phone and complete his call? [8] Q6) A small project is composed of 7 activities whose time estimates are listed in the table below. Activities are identified by their beginning (i) and ending (j) node numbers. [3664]-130 4 a) Draw the project network. b) Find the expected duration and variance for each activity. What is the expected project length? c) Calculate the variance and standard deviation of the project length. What is the probability that the project will be completed? i) At least 4 weeks earlier than expected time? ii) No more than 4 weeks later than expected time? d) If the project due date is 19 weeks, what is the probability of not meeting the due date? [18] Given : Z Prob. 0.50 0.3085 0.67 0.2514 1.0 0.1587 1.33 0.0918 2.00 0.0228 OR Q6) Write short notes (any three) : [18] a) Updating of the network. b) Project crashing. c) Types of float. d) Role of statistical techniques in PERT. kbkb [3664]-130 5

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