This question paper contains 8 printed pages]
S. No. of Question Paper : 1677 Roll No.
Unique Paper Code : 2362301 F-3
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Name of the Paper : Introduction to Operational Research & Linear Programming
Name of the Course : B.Tech. (Computer Science) Allied Courses
Semester : III Maximum Marks: 75
Duration: 3 Hours
(Write your Roll No. on the top immediately on receipt of this question paper.)
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All questions carry equal marks.
There are three sections in the paper.
All sections are compulsory.
Attempt any five questions from each Section.
Use of simple calculator is allowed.
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Section A
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Define the term O.R. and write its applications in different areas of real life.
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Explain any two of the following:
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Slack variables
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Artificial variables
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Dual prices.
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What do you mean by linearly dependent and linearly independent vectors? Check whether
the given set is linearly dependent or linearly independent.
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What do you mean by extreme point in convex set? Find all extreme points from the system of equations given below :
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2x1 + x2 - x3 = 2
3x1 + 2x2 + x3 = 3
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Define convex set. Does the union of convex sets is a convex set ?
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Show that the following set is convex :
S = {(x1, x2) | x1 + x2 = 3, x1 = 0, x2 = 0}.
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Consider the following LPP with two variables :
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Maximize Z = 2x1 + 3x2
Subject to:
2x1 + x2 = 4
x1 + 2x2 = 5
x1, x2 = 0.
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Determine all the basic solutions of the problem, and classify them as feasible and infeasible.
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Verify graphically that the solution obtained in (a) is the optimum LP solution. Hence, conclude that the optimum solution can be determined algebraically by considering the basic feasible solutions only.
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Section B
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Day Trader wants to invest a sum of money that would generate an annual yield of at least $ 10000. Two stocks groups are available: blue chips and high tech, with average annual yields of 10% and 25%, respectively. Though high-tech stocks provide higher yield, they are more risky, and Trader wants to limit the amount invested in these stocks to no more than 60% of the total investment. Formulate the above problem as a LPP to maximize an annual yield.
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Solve the given LPP graphically :
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Maximize Z = x1 + 2x2
Subject to:
x1 - x2 = 1
x1 + 2x2 = 3
x1, x2 = 0.
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Consider the following LP :
Maximize Z = x1
Subject to:
5x1 + x2 = 4
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6x1 + x3 = 8
3x1 + x4 = 3
x1, x2, x3, x4 = 0.
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Solve the problem by inspection (do not use the Gauss-Jordan row operations), and justify the answer in terms of the basic solutions of the Simplex method.
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Repeat (a) assuming that the objective function calls for minimizing z = x1.
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Consider the two-dimensional solution space in figure given below.
Suppose that the objective function is given as
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Maximize z = 6x1 + 3x2.
If the simplex iterations start at point A, identify the path to the optimum point D.
Determine the entering variable, the corresponding ratios of the feasibility condition, and the change in the value of z..
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For the following LP, identify three alternative optimal basic solutions and then write a general expression for all the non-basic alternative optima :
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Maximize Z = 2x1 + 4x2
Subject to:
x1 + 2x2 = 5
x1 + x2 = 4
x1, x2 = 0
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Use Big-M method to solve :
Maximize Z = 12x1 + 20x2
Subject to:
6x1 + 8x2 = 100
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7x1 + 12x2= 120
x1, x2 = 0.
Section C
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Comment on the future of artificial variables at the optimal table of phase 1.
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Write the dual of the following LPP :
Minimize Z`= x1 + x2 + x3
Subject to:
x1 - 3x2 + 4x3 = 5
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x1 - 2x2 = 3
2x2 - x3 = 4,
x1, x2 = 0 and x3 is unrestricted.
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What do you understand by feasibility and optimality ranges of the variables in LPP ?
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Consider the following LP model:
Maximize Z = 4x1 + 10x2
Subject to:
2x1 + x2 = 50
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2x1 + 5x2= 100
2x1 + 3x2 = 90
x1, x2 = 0
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Check the optimality and feasibility of each of the following basic solution :
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Basic variables = (x1, x2, x5), Inverse =
5/8 -1/8 0 -1/4 1/4 0 -1/2 -1/2 1 -
Use dual simplex method to solve the given LPP :
Minimize Z = 3x1 + x2
Subject to:
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x1 + x2 = 1
2x1 + 3x2 = 2
x1, x2 = 0.
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Consider the following LP:
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Maximize Z = 5x1 + 12x2 + 4x3
Subject to:
x1 + 2x2 + x3 = 10
2x1-x2 + 3x3 = 8
x1, x2, x3 = 0.
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Using x4 and A as starting variables, the optimal tableau is given as :
| Basic | x1 | x2 | x3 | x4 | A | Solution |
|---|---|---|---|---|---|---|
| Z | 0 | 0 | 3/5 | 29/5 | -2/5+M | 274/5 |
| x2 | 0 | 1 | -1/5 | 2/5 | -1/5 | 12/5 |
| x1 | 1 | 0 | 7/5 | 1/5 | 2/5 | 26/5 |
Write the associated dual problem and determine its optimal solution.
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