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प्रश्न
A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours of work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.
उत्तर
Let x articles of model A and y articles of model B be made.
A number of articles cannot be negative.
Therefore, x, y ≥ 0
According to the question, the making of a model A requires 2 hrs. work by a skilled man and model B requires 1 hr by a skilled man
2x + y ≤ 40
The making of a model A requires 2 hrs. work by a semi-skilled man model B requires 3 hrs. work by a semi-skilled man.
2x + 3y ≤ 80
Total profit = Z = 15x + 10y which is to be maximised
Thus, the mathematical formulation of the given linear programming problem is Max Z = 15x + 10y
subject to
2x + y ≤ 40
2x + 3y ≤ 80
x ≥ 0
y ≥ 0
The feasible region determined by the system of constraints is
The corner points are `"A" (0, 80/3)`, B(10,20), C(20,0)
The values of Z at these corner points are as follows
| Corner point | Z= 15x+10y |
| A | `800/3` |
| B | 350 |
| C | 300 |
The maximum value of Z is 300 which is attained at C(20, 0)
Thus, the maximum profit is Rs 300 obtained when 10 units of a deluxe model and 20 unit of the ordinary model is produced.
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