Sixth Semester B.E. Degree Examination, June/July 2013 Operations Research Question paper

USN 10CSIIS661 Sixth Semester B.E. Degree Examination, June/July 2013 Operations Research Time: 3 hrs. Max. Marks: 100 Note:Answer FWEfuli questions, selecting at least TWO questions from each part.

PART-A Â

1),a.Define operations research. List and explain the various phases of an operations research (08 Marks)

b. A,farmer has to plant two kinds of trees P and Q in a land of 400m2 area. Each P tree requires.at least 25m2 and Q tree requires 40m2 of land. The annual water requirement of P tree is 30Junits and of Q tree is 15 units per tree, while at month;(3,OOOunits of water is available It is also estimated that the ratio of the number of Q tree'\ to the number of P trees should not be less than 6119 and should not be more than 17/S. Then return per tree from P is expected to be one and half times as much as from Q tree.evaluvate the problem as an LPP model. (06 Marks)

C.2solve the following LPP@ Subject to the constraints ';/~ .X22 3, ACf,~" x, x~2 ,,\ And 5-0 '-J Xl, X2- . \:

2) a. Defme basic solution and obtain equations: 2Xl + 3X2+ 4X3= 10, ~ 3Xl + 4X2+ X3= 12 O~ , Also, classify the solutiQR~to

i) Basic feasible solution~n ~~~ Degenerate~ ~afc solution . . . 111) Non-degenerate feasible solution.

b. Solve the foll 10 \ ~ - , ~'\ X2220, ~' X3230 "I. ~ and xi, X2,X320. (06 Marks)

C. Solve the following LPP by two-phase simplex method: Maximize Z = 3Xl - X2 Subject to the constraints 2Xl + X22 2, x, + 3X2::;2, X2::;4 and xr, X22 0. (10 Marks) lof3 ~RT,-

3) a. Write any five key relationships be ' he primal and the dual problems.

b. Write the duals of the following L i) Maximize Z = 7xI + 4X2 Subject to the constraint 2xI - X3S; 10, 2 + X3S; 6 XI, X2,X3:2: O. ii) I + 2X2+ X3 constraints 2xI - 3X2+ X3S; 5, 4xI - 2X2 :2: 9, -Sx, + 4X2+ 3X3= 8 a XI, X2:2: 0 and X3is unrestricted.

c. :1 the following LPP by dual simplex method: inimize Z = 2xI + 2X2+ 4X3 Subject to the constraints 2xI + 3X2+ 5X3:2: 2, 3xI + X2+7X3 S; 3, XI + 4X2+ 6X3 S; 5 and XI, X2,X3:2: O. b. Solve the following LPP by Big-M method: Maximize Z = -Zx, - X2 Subject to the constraints 3xI + X2= 3, 4xI + 3X2 :2: 6, XI + 2X2 S; 4 and Xl, X2:2: O.

4) a. aoJve the following LPP by revised simplex method: M~ . ize Z = 2xI + X2 Sub the constraints 3xI + 4X2 S; 6, t X2S; 3 And 2:2: 0 b. Explain the fol i) Weak dualit perty ii) Strong duality p erty iii) Complementary solW' ns property iv) Complementary opti lutions prop

Part B

6) a. A company has 3 cement factories located in 3 cities X, Y and Z which supply cement to 4 project sites located in cities A, B, C and D. Each plant can supply 6, 1 and 10 truckloads of cement daily and the daily requirements of the projects are 7, 5, 3 and 2 truckloads respectively. The transportation cost (in thousands of rupees) per truck load of cement from each plant to each project site are shown below. 20f3 Projects ABC D 2 3 11 7 1 0 6 1 5 8 15 9 10CS/IS661 X Plants y Z _:":>_ .• ~~,'C\~~f '""-"''''i~ Determine the optimal distribution of the company so as to minimize the total transPQit'afi~n cost. Use V AM method to find the initial BFS. ;!r~2)\Iarks) Solve the following assignment problem: \¥:c~ Machines MJ M2 M3 M4 J1 11 17 8 16 h 9 7 12 6 Jobs h 13 16 15 12 h 21 24 17 28 J5 14 19 12 11 7 a. Define the following with i) Pay-off ii) Zero-sum game iii) Saddle point. b. Solve the following game by Dorni rinciple: II Player A AI 3 2 ' A2 3 4 A3 4 2 ~ 0 4 ing game by graphical method: Player B BI B2 B3 B4 Player A AI CIIII=II}] A2~ Write a short note on decision trees.;~. (08 Marks) (03 Marks) (06 Marks) 07 Marks) (9"J>:\arks)

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