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Friday, 1 May 2015

Operations Research- Model question paper for B.E/B.Tech Engineering, MSc Mathematics,Statistics



1(a) Prove that if a linear function assumes its minimum at two different points of a convex set, then it assumes this minimum on the entire line segment between the points. (6 Marks)
(b) Determine graphically whether [1,2]T is a convex combination of [1,1]T and [2,-1]T. (4Marks)
(c) Explain different steps of dual simplex method. (6 Marks)
(d) Fay Klein had developed two types of handcrafted, adult games that she sells to department stores throughout the country. Although the demand for these games exceeds her capacity to produce them, Ms. Klein continues to work alone and to limit her workweek to 50 Hours. Game I take 3.5 Hours to produce and bring a profit of $28, while game II requires 4 Hours to complete and brings a profit of $31. How many games of each type should Ms. Klein produce weekly if her objective is to maximize total profit?  (4 Marks)
2(a) Write a detailed note on duality and sensitivity analysis in linear programming. (8 Marks)
(b) What is Karmarkar’s algorithm? Explain. (4 Marks)
(c) Write a detailed note on Branching and Bounding algorithm in integer programming. (8 Marks)
3(a) Prove that if the costs in any row or any column of a transportation tableau are uniformly reduced by the same number (positive or negative), then the resultant problem has the same optimal solution as the original problem. (8 Marks)  
(b) Show by means of an example that an optimal itinerary for the traveling salesperson problem may not still be optimal when the constraint that each location be visited only once is dropped. (6 Marks)
(c) Use the Gomory algorithm to maximize: z=x1+9x2+x3 subject to: x1+2x2+3x3≤ 9   3x1+2x2+2x3 ≤ 15   with:   all variables nonnegative and integral. (6 Marks)
4(a) Write a note on sequential-search techniques. Distinguish between Fibonacci search and Golden-Mean search. (6 Marks)
(b) Approximate the location of the global maximum on [0,π] of f(x)=x2sinx, using a three-point search of the unrestricted interval with five functional evaluations. How good is this approximation? (6 Marks)
(c) Write a note on Hooke-Jeeves’ pattern search. (4 Marks)
(d) Derive Newton-Raphson formula. (4 Marks)
5(a) What is a shortest-route problem? How is it solved? (6 Marks)
(b) Write a note on critical path computations for PERT. (6 Marks)
(c) What is an inventory? How will you determine fixed order quantities? (4 Marks)
(d) The ABC retail company has the following data available for one of its items: D=10000 units; S=$ 20.00; C=25% of the acquisition cost $25.00. Find (a) economic order quantity; (b) number of orders per year; (c) total inventory cost. (4 Marks)
6(a) What is standard error of estimate? Explain. (4 Marks)
(b) Write a note on forecasting time series with multiplicative model. (6 Marks)
(c) Differentiate between stable games and unstable games. (6 Marks)
(d) Two ranchers have brought a dispute over a 6-yard-wide strip of land that separates their properties to a referee. Both claim the strip as entirely their own. Both ranchers are aware that that referee will ask each party to submit a confidential proposal for settling the dispute fairly and will then accept that proposal which gives the most. If both proposals give equally or not at all, the referee will split the difference, setting the boundary in the middle of the 6-yard width. Determine the rancher’s best proposals, if proposals are restricted to integral amounts. (6 Marks)
7(a) Explain Naïve decision criteria. (4 Marks)
(b) Explain the four-step procedure which is used to determine von Neumann utilities for a finite number of payoffs. (4 Marks)
(c) What is a multistage decision process? Explain. (6 Marks)
(d) A woman has ticket to a football game on a day for which the weather bureau predicts rain with likelihood of 40%. She can stay home and watch the game on television, the preferable choice under rainy conditions, or she can go to the stadium, the preferable choice under dry conditions. Which decision should she make? (6 Marks)
8(a) What is a stochastic matrix? Differentiate between ergodic matrices and regular matrices. (6 Marks)
(b) A new television set arrives for inspection every 3 min and is taken by a quality control engineer on a first-come, first-served basis. There is only one engineer on duty, and it takes exactly 4 min to inspect each new set. Determine the average number of sets waiting to be inspected over the first half-hour of a shift, if there are no sets awaiting inspection at the beginning of the shift. (4 Marks)
(c) What are balking function and reneging function? Explain. (6 Marks)
(d) A linear Markovian death process initialized at 10 members experiences an average weekly death rate µ = 0.6. Determine the probability of having a population of at least eight members after three days, and the expected size of the population at that time. (4 Marks)

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