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प्रश्न
The postmaster of a local post office wishes to hire extra helpers during the Deepawali season, because of a large increase in the volume of mail handling and delivery. Because of the limited office space and the budgetary conditions, the number of temporary helpers must not exceed 10. According to past experience, a man can handle 300 letters and 80 packages per day, on the average, and a woman can handle 400 letters and 50 packets per day. The postmaster believes that the daily volume of extra mail and packages will be no less than 3400 and 680 respectively. A man receives Rs 225 a day and a woman receives Rs 200 a day. How many men and women helpers should be hired to keep the pay-roll at a minimum ? Formulate an LPP and solve it graphically.
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उत्तर
Let the postmaster hire x men and y women.
Clearly, x≥0, y≥0
Also, it is given that the number of temporary helpers must not exceed 10.
∴ x+y≤10
The given information can be represented in the tabular forms as
| Men (x) |
Women (y) |
Minimum Volume of Mails | |
| Letters | 300 | 400 | 3400 |
| Packages | 80 | 50 | 680 |
| Payroll (Rs) | 225 | 200 |
Thus, the given LPP can be stated mathematically as follows:
Minimise Z = 225x + 200y
Subject to the constraints:
x+y≤10 .....(1)
300x+400y≥3400 (constraint on letters)
⇒3x+4y≥34 .....(2)
80x+50y≥680 (constraint on packages)
8x+5y≥68 .....(3)
and x, y≥0 .....(4)
Converting the inequations into equations, we obtain the lines x + y = 10, 3x + 4y = 34, 8x + 5y = 68, x = 0 and y = 0.
These lines are drawn and the feasible region of the LPP is shaded. It is observed that the feasible region is a point (6, 4).
The value of the objective function at this point is given as Z = 2,150.
So, the minimum value of Z is 2150 at the point (6, 4).
Hence, the postmaster should hire 6 men and 4 women to keep the pay-roll at a minimum of Rs 2,150.
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| A | (12, 0) | 4(12) + 5(0) | 48 | |
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| C | ( ___, ___ ) | 4( ___) + 5(___ ) | ______ | |
| D | (0, 10) | 4(0) + 5(10) | 50 |
∴ Z is minimum at ___ ( ___, ___ ) with the value ___
