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Operation Research (Elective I) (October 2010)

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Total No. of Questions : 12] [Total No. of Pages : 7 P1037 [3864]-139 B.E. (Mechanical) OPERATION RESEARCH (2003 Course) (Elective - I) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidates: 1) 2) 3) 4) 5) Answer three questions from section-I and three questions from section-II. Answers to the two sections should be written in separate books. Figures to the right indicate full marks. Use of logarithmic tables, slide rule, Mollier charts, electronic pocket calculator and steam tables is allowed. Assume suitable data, if necessary. SECTION - I Q1) a) Solve the following Linear Programming problem graphically. Maximize Subjected to [6] Z = 8000 X1 + 7000 X2 3X1 + X2 < 66 X1 + X2 < 45 3X1 < 20, X2 < 40 X1, X2 > 0 b) The ABC printing company is facing a tight financial squeeze and is attempting to cut costs wherever possible. At present it has only one printing contract and the book is selling well in both the hardcover and paperback edition. It has just received a request to print more copies if the book in either hardcover or paperback form. Printing cost of hardcover book is Rs.600 per 100 while for that paperback is only Rs.500 per 100. Although the company is attempting to economize, it does not wish to lay off any employee. Therefore it feels obliged to run its two printing press at least 80 and 60 hours per week respectively. Press I can produce 100 hardcover books in 2 hours or 100 paperback books in 1 hour. Press II can produce 100 hardcover books in 1 hour or 100 paperback books in 2 hours. Determine how many books of each type should be printed in order to minimize costs. [10] OR P.T.O. Q2) Solve the following Linear Programming problem graphically. [16] Z = 2X1 + 3X2 + 3X3 3X1 + X2 + 4X3 < 600 2X1 + 4X2 + 2X3 > 480 2X1 + 3X2 + 3X3 = 540 X1, X2, X3 > 0 Maximize Subjected to Q3) A company has four manufacturing plant and five warehouses. Each plat manufactures same product which is sold at different prices in each warehouse area. Cost of manufacturing and cost of raw material is different in each plant due to various factors. The capacities of the plant are also different. The data are given in following table. Item Plant 1 2 3 4 Manufacturing cost per unit (Rs.) 12 10 8 8 Raw material cost per unit (Rs.) 8 7 7 5 100 200 120 80 Capacity per unit time The company has five warehouses. The sales price, transportation cost and demands are given in the table. Transportation Cost per unit (Rs.) Sales Price per unit (Rs.) Demand (units) Warehouse Plant 1 Plant 2 Plant 3 Plant 4 A 4 7 4 3 30 80 B 8 9 7 8 32 120 C 2 7 6 10 28 150 D 10 7 5 8 34 70 E 2 5 8 9 30 90 a) Formulate the problem as Transportation Problem to maximize the profit. [4] b) Find the solution using VAM. [5] c) Test for optimality and find the optimal solution. [9] OR [3864]-139 2 Q4) a) A company is engaged in manufacturing 5 brands if packed snacks. It is having five manufacturing setups, each capable of producing any of its brands, one at a time. The cost to make a brand on these setups vary according to the following table. [10] S1 S2 S3 S4 S5 B1 4 6 7 5 11 B2 7 3 6 9 5 B3 8 5 4 6 9 B4 9 12 7 11 10 B5 7 5 9 8 11 Assume five setups are S1, S2, S3, S4, S5 and five brands are B1, B2, B3, B4, B5. Find the optimum assignment of products on these setups resulting minimum cost. b) Show that assignment model is a special case of transportation model.[4] c) Discuss the travelling salesman problem. [4] Q5) a) Derive formula for economic lot size model with constant demand. [6] b) A shopkeeper has a uniform demand of an item of 50 items per month. He buys it from a supplier at a cost of Rs.6.00 per item and the cost of ordering us Rs.10.00 each time. If the stock holding costs are 20% per year of the stock value, how frequently he should replenish his stock? Suppose the supplier offers a 5% discount on orders between 200 and 999 items and 10% discount on orders exceeding 1000. Can the shopkeeper reduce his cost by taking advantage of either of these discounts? [10] OR Q6) a) Explain necessity of maintaining inventory. b) Explain important characteristics of dynamic programming. [4] [4] c) Solve the following integer linear programming problem using cutting plane method. [8] Maximize Subjected to [3864]-139 Z = X1 + X2 3X1 + 2X2 < 5 X2 < 2 X1, X2 > 0 & are integers. 3 SECTION - II Q7) a) Solve the following game. [10] Player B I II III IV V 1 10 81 32 43 93 Player 2 59 63 39 69 73 A 3 71 20 5 27 84 4 34 14 44 44 69 b) Write a short note on group replacement policy for items that fails suddenly. Which of the policies group replacement or individual replacement is better? Why? [6] OR Q8) a) Find the cost per period of individual replacement policy of an installation of 300 bulbs given the data : i) Cost of replacing an individual bulb is Rs.2.00. ii) Conditional probability of failure is given below. Week No. 0 1 3 4 0.1 Conditional probability of failure 0 2 0.3 0.7 1.0 Also calculate number of bulbs that would fail during each of four weeks. [10] b) Solve the 2 5 game graphically. [6] Player B I II III IV V Player 5 5 0 1 8 A [3864]-139 1 2 8 4 1 6 5 4 Q9) a) Explain Kendall s notations for representing queuing model. [6] b) A bank has two tellers working on saving account. The first teller handles withdrawals only. The second teller handles deposits only. It has been found that the service time distribution for the deposits and withdrawal both are exponential with mean service time of 3 minutes per customer. Depositors are found to arrive in Poisson fashion throughout the day with mean arrival rate of 16 per hour. Withdrawers also arrive in Poisson fashion with mean arrival rate of 14 per hour. What would be the effect on the average waiting time of the depositors and withdrawers if each teller could handle both withdrawal and deposits? What would be the effect if this could only be accomplished by increasing the service time to 3.5 minutes? [10] OR Q10) a) There are seven jobs each of which has to go through machine A & B in order A-B. The processing time in hours is given below : [8] Job 1 2 3 4 5 6 7 Machine A (Hrs) 3 12 15 6 10 11 9 Machine B (Hrs) 8 10 10 6 12 1 3 Determine the sequence of the jobs that will minimize the total elapsed time. Also find the total completion time and idle time of each machine. b) A company manufactures around 200 mopeds depending upon the availability of raw material and other conditions. The daily production has been varying from 196 mopeds to 204 mopeds whose probability distribution is as follows. [8] Production/day 196 197 198 199 200 201 202 203 204 Probability 0.05 0.09 0.12 0.14 0.20 0.15 0.11 0.08 0.06 The finished mopeds are transported in a specially designed three storied lorry that can accommodate only 200 mopeds. Using the following 15 random numbers 82, 89, 78, 24, 53, 61, 18, 45, 04, 23, 50, 77, 27, 54 and 10. Simulate the process to find out i) ii) [3864]-139 Average number of mopeds waiting in factory. Number of empty spaces in lorry. 5 Q11) a) A small project consists of 13 activities. Their precedence relationship and duration in days is given in table. [14] Activity Predecessor Duration (Days) A --- 6 B A 4 C B 7 D A 2 E D 4 F E 10 G --- 2 H G 10 I J,H 6 J --- 13 K A 9 L C,K 3 M I,L 5 i) Construct the project network. ii) Find the Critical Path. iii) Find total completion time of the project. b) Differentiate between CPM and PERT. OR [3864]-139 6 [4] Q12) A project schedule has following characteristics. Activity To Tm Tp 1-2 1 2 3 2-3 1 2 3 2-4 1 3 5 3-5 3 4 5 4-5 2 3 4 4-6 3 5 7 5-7 4 5 6 6-7 6 7 8 7-8 2 4 6 7-9 4 6 8 8-10 1 2 3 9-10 3 5 [18] 7 a) Construct the project network. b) Find the expected duration and variance for each activity. c) Find the critical path and expected project completion time. d) What is the probability that the entire project will be completed in 30 days. Use the following data : Z 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 xxxx [3864]-139 7

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