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Systems Approach in Civil Engineering (Elective I) (October 2009)

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Total No. of Questions : 12] P1149 [Total No. of Pages : 5 [3664]-106 B.E. (Civil) SYSTEMS APPROACH IN CIVIL ENGINEERING (2003 Course & 1997 Course) (Elective - I) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidates: 1) 2) 3) 4) 5) 6) Answer three questions from Section I and three questions from Section II. 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 rule, Mollier charts, electronic pocket calculator and steam tables is allowed. Assume suitable data, if necessary. SECTION - I Q1) a) Minimize Z = x1 + 2x2 + 3x3 Subject to x1 x2 + x3 4 x1 + x2 + 2x3 8 x2 x3 2 x1, x2, x3 0 Use Big M Method. [12] b) Explain the principle of Big M Method. How does it affect the value of the objective function? [4] OR Q2) a) Solve the problem in Q1 (a) above by Two Phase Method. b) Explain with suitable sketches. i) Unbounded solution in L.P. ii) Infeasible solution in L.P. [12] [4] Q3) In a regional rural water supply scheme, water is to be supplied from three sources to four different villages. The quantity of water available at each source and that required by each village and the unit cost of transport are given in the following table [18] a) Find the Initial Feasible solution by i) N-W Corner Method. ii) VAM. P.T.O. b) Using the solution obtained by VAM, find the distribution policy which will minimize the cost of transportation. Sources A B C Demand 1 5 11 15 5 Villages 2 5 8 18 15 Supply 3 9 13 16 35 4 12 13 21 45 40 30 30 OR Q4) a) Five civil contractors have submitted their bids for 4 projects. One project will be awarded to one contractor only, who has submitted the minimum bid. Find the optimal assignment if the cost bid by each contractor for the projects is given below (unit in lakh rupees), so that the total cost of the assignment is minimum. Contractor D is unable to execute project [10] No.3. Which one of the contractors does not get a project? Contractors A B C D E 1 18 25 22 26 25 Projects 2 3 26 28 29 31 26 30 27 24 31 4 26 28 27 26 29 b) How will you solve an Assignment Model for maximization? [2] c) Write the dual of the problem in Q1(a). [6] Q5) a) Use Fibonacci Method to maximize Z = 75x x3 in the range 0 to 15 to an accuracy of 1%. Carry out the first four iterations only. [10] b) Explain the Dichotomous Search Technique. [6] OR Q6) a) Use Newton s Method to minimize [12] 2 f ( x ) = 2 x12 + 2 x2 + 2 x1 x2 3 x1 2 x2 Take the starting point as (0, 0) b) What is a Hessian Matrix? Explain its significance. [3664]-106 2 [4] SECTION - II Q7) a) Use Lagrange Multiplier Technique to optimise [10] 2 Z = 3 x12 + 4 x1 x2 + 6 x2 Subject to 4x1 + 5x2 = 26. b) What is Bellman s principle of optimality? Give any 4 applications of Dynamic programming to Civil Engineering. [6] OR Q8) Six units of power are to be distributed among three regions designated as A, B & C. The expected returns from each region depending upon the number of units of power supplied are given below. Use Dynamic Programming to determine an optimum power allocation so as to maximize the returns. Write the recursive equation for each stage. [16] Power supplied (in units) 0 1 2 3 4 5 6 Returns from Regions (in money units) A B C 0 0 0 10 30 24 20 40 48 60 44 50 62 46 52 64 48 54 66 50 56 Q9) a) There are 10 jobs which are to be processed on two machines A and B, first on machine A and then on machine B. The processing times in minutes, of these jobs are given below. Determine the sequence for the 10 jobs so as to minimize the total elapsed time. Also find the idle times [9] of the machines. Processing Times in minutes Job No. Machine A Machine B 1 40 8 2 20 24 3 6 10 4 20 16 5 10 12 6 4 24 7 16 8 8 14 20 b) What are the various components of a queueing system? [3664]-106 3 9 6 12 10 8 2 [3] c) Customers arrive at an airport ticket booking counter according to Poisson distribution at the rate of 2 every 5 minutes. It takes 1.5 minutes on an average to serve a customer and the service time follows exponential distribution. [6] Find : i) The proportion of time the facility remains idle. ii) The expected number of customers waiting but currently not being served at a particular point of time. iii) The expected time a customer spends in the system. OR Q10) The interarrival time and the service time in a waiting line problem have the following frequency distribution based on 100 such arrivals. [18] Interarrival Time in minutes Frequency 3 6 9 12 15 18 6 9 25 37 16 7 Service Time in minutes Frequency 4 6 8 10 12 4 10 18 44 24 Calculate : a) The average waiting time and percentage waiting time of an arrival. b) The average idle time and percentage idle time of the server. c) Mean queue length. Simulate 10 arrivals and use the following random numbers. Arrivals Service 15 9 19 11 61 90 49 64 54 37 73 29 85 43 96 78 31 87 22 56 Q11) a) Explain the assumptions in a Two-Person-Zero Sum Game. b) Define saddle point and Fair Game. [4] [4] c) The pay off matrix in a game between A and B is as follows. Determine the strategies of each player and the value of the game. The payoffs are for player A. [8] B1 B2 B 3 B4 A1 6 9 10 1 A2 7 8 9 2 A3 9 8 16 2 A4 4 5 0 4 [3664]-106 4 OR Q12) a) What is an Annuity? What are the different types of Annuities? [8] b) A company is interested in investing in any one of the following two projects. The relevant data pertaining to the two projects is given below. Rank the projects according to the B/C ratio and state which option would be the best. [8] Particulars Project A Project B 1. Initial Investment (lakh Rupees) 37 45 2. Net Annual Benefits (lakh Rupees) 5 7 3. Useful Life (Years) 15 10 4. Salvage Value (in lakh Rupees) 1 2 10% 10% 5. Discount Rate [3664]-106 5

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Additional Info : 2003 & 1997 Course
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