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Question
Diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1500 calories Two foods F1 and F2 cost Rs. 50 and Rs. 75 per unit respectively. Each unit of food F1 contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas each unit of food F2 contains 100 units of vitamins, 2 units of minerals and 30 calories. Formulate the above problem as L.P.P. to satisfy the sick person's requirements at minimum cost.
Solution
Let x1 : Number of units of Food F1
x2 : Number of units of food F2
Since the number of unit of foods cannot be negative.
∴ x1 ≥ 0 and x2 ≥ 0
| Product | Food F1 | Food F2 | Minimum Requirement |
| Vitamins | 200 | 100 | 4000 |
| Minerals | 1 | 2 | 50 |
| Calories | 40 | 30 | 1500 |
| Cost per unit | Rs. 50 | Rs. 75 | - |
From the table
For vitamins 200x1 + 100x2 ≥ 4000
For Minerals x1 + 2x2 ≥ 50
For Calories 40x1 + 30x2 ≥ 1500
Total cost z = 50x1 + 75x2
∴ The L.P.P. is :
Minimize Z = 50x1 + 75x2
Subject to the constraints
200x1 + 100x2 ≥ 4000
x1 + 2x2 ≥ 50
40x1 + 30x2 ≥ 1500 and x1 , x2 ≥ 0
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