Find an initial basic feasible solution to the following transportation problem by the North-West corner method. Check whether your solution is optimal.
| 6 | 4 | 1 | 5 | 14 |
| 8 | 9 | 2 | 7 | 16 |
| 4 | 3 | 6 | 2 | 5 |
| 6 | 10 | 15 | 4 |
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Prove that the dual of dual of a LPP is primal LPP.
A company has 5 jobs to be processed by 5 mechanics. The following table gives the return in rupees when the ith job is assigned to the jth mechanic ( i, j = 1,2,....5) . How should the jobs be assigned to the mechanics so as to maximize the overall return?
| 1 | 2 | 3 | 4 | 5 | |
|---|---|---|---|---|---|
| 1 | 22 | 28 | 30 | 18 | 30 |
| 2 | 30 | 34 | 18 | 11 | 26 |
| 3 | 31 | 17 | 23 | 20 | 27 |
| 4 | 12 | 28 | 31 | 26 | 26 |
| 5 | 19 | 23 | 30 | 25 | 29 |
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}
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 |
A firm manufactures pills in two sizes A and B. A contains 2 grains of aspirin, 5 grains of bicarbonate and 1 grains of codeine. B contains 1 grain of aspirin, 8 grains of bicarbonate and 6 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providing immediate effect. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a standard LPP and solve it graphically.
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.