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प्रश्न
A firm manufactures two products A and B on which the profits earned per unit are ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M1 and M2. Product A requires one minute of processing time on M1 and two minutes on M2, While B requires one minute on M1 and one minute on M2. Machine M1 is available for not more than 7 hrs 30 minutes while M2 is available for 10 hrs during any working day. Formulate this problem as a linear programming problem to maximize the profit.
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उत्तर
(i) Variables: Let x1 represents the product A and x2 represents the product B.
(ii) Objective function:
Profit earned from Product A = 3x1
Profit earned from Product B = 4x2
Let Z = 3x1 + 4x2
Since the profit is to be maximized, we have maximize Z = 3x1 + 4x2
(iii) Constraints:
| M1 | M2 | |
| Requirement for A | 1 min | 2 min |
| Requirement for B | 1 min | 1 min |
M1 is available for 7 hrs 30 min = 7 × 60 + 30 = 450 min
M2 is available for 10 hrs = 10 × 60 = 600 min
∴ x1 + x2 ≤ 450 .....[for M1]
2x1 + x2 ≤ 600 ......[for M2]
(iv) Non-negative restrictions:
Since the number of products of type A and B cannot be negative, x1, x2 ≥ 0.
Hence, the mathematical formulation of the LLP is maximize
Z = 3x1 + 4x2
Subject to the constraints
x1 + x2 ≤ 450
2x1 + x2 ≤ 600
x1, x2 ≥ 0
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