A departmental head has four subordinates, and four tasks to be performed. The subordinates differ in efficiency, and the tasks differ in their intrinsic difficulty. His estimate, of the time each subordinate would take to perform each task is given in the matrix below:
| Tasks | Subordinnate | |||
| E | F | G | H | |
| A | 18 | 26 | 17 | 11 |
| B | 13 | 28 | 14 | 26 |
| C | 38 | 19 | 18 | 15 |
| D | 19 | 26 | 24 | 10 |
How should the task be allocated, one to a subordinate, so as to minimize the total time?
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Write the dual of the following LPP:
Maximize z = 2x1 + x2
Subject to x1 + 5x2 < 10
x1 + 3x2 > 6
2x1 + 2x2 < 8
x2 > 0, x1 is unrestricted in sign.
Which of the following statements are true or false? Give reasons for your answer.
i) The number of basic variables in a feasible solution of a transportation problem with m sources and n destinations is mn.
ii) An optimal assignment of the assignment problem with cost matrix C is also an optimal assignment of the assignment problem with cost matrix Ct .
iii) (1, 2) is an optimal solution to the following LPP:
Max Z = 2x + 4x
subject to x1 + 2x2 < 5
x1 + x2 < 4
x1 ,x2 > 0
iv) The intersection of a finite number of convex sets is not a convex set.
v) If value of the 2X2 matrix game is 4, then p > 4.
A businessman needs 5 cabinets, 12 desks and 18 shelves cleaned out. He has two part time employees, Rashid and Ruby. Ruby can clean 1 cabinet, 3 desks and 3 shelves in a day while Rashid can clean 1 cabinet, 2 desks and 6 shelves in one day. Rashid is paid Rs.22 per day and Ruby is paid Rs.25 per day. In order to minimize the cleaning costs, for how many days should Rashid and Ruby be employed? Formulate the problem as a linear programming problem and find its solution by simplex method.
Determine all the basic feasible solutions to the equations
x1 + x2 + 2x3 = 4
2x1 - x2 + x3 = 2
Identify the degenerate basic feasible solutions.
If and C = [1,2] compute A'B, AC, B'A + CB, whenever defined. If you think any of these is not defined, give reason for saying so.
Test for convexity the following sets:
S = {(x, y) | x2 + y2 >1, y > x, y > - x}
S = {(x, y) | x2 + y2 <16, x < 2, y > 2}
Is the set of vectors{(1,2,3),(3,4,1),(2,3,2)} linearly independent? Give reason for the answer.