Download DU (University of Delhi) B-Tech 3rd Semester Introduction of Operational Research and Linear Programming Question Paper

Sr. No. of Question Paper: 1509 F-7 Your Roll No..............

Unique Paper Code : 2362301

Name of the Paper : Introduction of Operational Research and Linear Programming

--- Content provided by‍ FirstRanker.com ---

Name of the Course : B.Tech Computer Science (Erstwhile FYUP) Allied Course

Semester : III

Duration : 3 Hours Maximum Marks: 75

Instructions for Candidates

1. Write your Roll No. on the top immediately on the receipt of this question paper.

--- Content provided by FirstRanker.com ---

2. Answer fifteen questions in all.
3. All questions carry equal marks.
4. Simple calculators are allowed.

1. Explain the importance of operational research in decision making. (5)
2. Find all possible basic solutions to the following set of linear equations.

   2x1 + 3x2 + x3 = 6

   --- Content provided by‌ FirstRanker.com ---

   x1 + 3x2 + 5x3 = 5

   Also check whether any of the above solutions is degenerate or not. (5)

3. Define the basis and dimension of a vector space. Test whether the set of vectors a1 = [1, 1, 0], a2 = [3, 0, 1], a3 = [5, 2, 1] form a basis of R³? (5)
4. Define a convex set. Examine the convexity of the following set :

   S = {(x1, x2): x1 + x2 = 1, ?x1, x2 ? R} (5)

   --- Content provided by‍ FirstRanker.com ---

5. A firm produces three products A, B and C. It uses two type of raw material I and II of which 5,000 and 7,500 units available. The raw material requirements per unit of the products are given below :

   Raw Material	Requirement per unit of product
   	A	B	C
   I	3	4	5
   II	5	3	5

   The labour time for each unit of product A is twice that of product B and three times that of product C. The entire labour force of the firm can produce the equivalent of 3,000 units. The minimum demand of the three products is 600, 650 and 500 units respectively. Formulate the problem as a linear programming problem (LPP) that will maximize the profit. (5)

6. Consider the following LPP :

   Maximize Z = 5x1 + 3x2

   Subject to: 3x1 + 5x2 <= 15

   --- Content provided by​ FirstRanker.com ---

   5x1 + 2x2 <= 10

   x1, x2 >= 0

   1. Determine all basic solutions of the problem and classify them as feasible and infeasible. (3)
   2. Show how the infeasible basic solutions are represented on the graphical solution space. (2)

--- Content provided by​ FirstRanker.com ---

7. Use graphical method to solve the following LPP :

   Minimise Z = 3x1 + 4x2

   Subject to 3x1 + 4x2 >= 240

   2x1 + x2 >= 100

   5x1 + 3x2 >= 120

   x1, x2 >= 0 (5)

   --- Content provided by‌ FirstRanker.com ---

8. Use Big M method to solve the following LPP

   Maximize Z = 10x1 + 20x2

   Subject to 2x1 + 4x2 >= 16

   x1 + 5x2 >= 15

   x1, x2 >= 0 (5)

   --- Content provided by‍ FirstRanker.com ---

9. How do you identify following in the optimal simplex table ?
   1. Alternate solution (2)
   2. Unbounded solution (2)
   3. Infeasible solution (1)

--- Content provided by FirstRanker.com ---

10. Solve the following LPP by dual simplex method

    Minimize Z = 3x1 - 2x2 + x3

    Subject to 3x1 + x2 + x3 >= 3

    -3x1 + 3x2 + x3 >= 6

    x1 + x2 + x3 <= 6

    x1, x2, x3 >= 0 (5)

    --- Content provided by‌ FirstRanker.com ---

11. Consider the following LPP

    Maximize Z = 3x1 + 4x2 + x3 + 7x4

    Subject to 8x1 + 3x2 + 4x3 + x4 <= 7

    2x1 + 6x2 + x3 + 5x4 <= 3

    x1 + 4x2 + 5x3 + 2x4 <= 8

    --- Content provided by‍ FirstRanker.com ---

    x1, x2, x3, x4 >= 0

    Its associated optimal simplex table is given as:

    Basic	x1	x2	x3	x4	x5	x6	x7	Solution
    Z	0	169/38	½	0	1/38	53/38	0	83/19
    x1	1	9/38	½	0	5/38	5/38	0	16/19
    x4	0	21/19	0	1	-1/19	-1/19	0	5/19
    x6	0	159/38	9/2	0	-1/38	-1/38	1	126/19

    Obtain the variations in cost coefficients which are permitted without changing the optimal solutions. (5)

12. Obtain the dual for the following primal problem :

    Minimize Z = 5x1 + 6x2 + x3

    --- Content provided by‍ FirstRanker.com ---

    Subject to x1 + 2x2 + x3 = 15

    -x1 + 5x2 <= 18

    4x1 + 7x2 <= 20

    x1, x2 >= 0, x3 unrestricted (5)

--- Content provided by FirstRanker.com ---

13. Consider the following LPP :

    Maximize Z = 4x1 + 14x2

    Subject to 2x1 + 7x2 + x3 = 21

    7x1 + 2x2 + x4 = 21

    x1, x2, x3, x4 >= 0

    Check the optimality and feasibility of the following basic solution :

    --- Content provided by⁠ FirstRanker.com ---

    Basic variables = (x2, x1), Inverse of the basis matrix = $\left(\begin{array}{cc}\frac{1}{7}& 0\\ \frac{-2}{7}& 1\end{array}\right)$ (5)

14. Find any three alternate optimal solution (if they exist) for the following LPP:

    Maximize Z = 2x1 + 4x2

    Subject to x1 + 2x2 <= 5

    x1 + x2 <= 4

    --- Content provided by‍ FirstRanker.com ---

    x1, x2 >= 0 (5)

15. Consider the LPP:

    Maximize Z = 2x1 + 4x2 + 4x3 - 3x4

    Subject to x1 + x2 + x3 = 4

    x1 + 4x2 + x4 = 8

    --- Content provided by‌ FirstRanker.com ---

    x1, x2, x3, x4>= 0

    By using x3 and x4 as the starting variables, the optimal table is given by

16. Basic	x1	x2	x3	x4	Solution
    Z	2	0	0	3	16
    x3	3/4	0	1	-1/4	2
    x2	1/4	1	0	1/4	2

    Write the associated dual problem, and determine its optimal solution in two ways. (5)

--- Content provided by‍ FirstRanker.com ---

17. Solve the following LPP by Two Phase method:

    Maximize Z = 3x1 + 2x2

    Subject to 2x1 + x2 <= 2

    3x1 + 4x2 >= 12

    x1, x2 >= 0 (5)

--- Content provided by⁠ FirstRanker.com ---

Visit FirstRanker.com for more information.