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Question
A dietician wishes to mix two kinds ·of food X· and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B arid 8 units of vitamin C. The vitamin contents of one kg food is given below:
| Food | Vitamin A | Vitamin.B | Vitamin C |
| X | 1 unit | 2 unit | 3 unit |
| Y | 2 unit | 2 unit | 1 unit |
Orie kg of food X costs Rs 24 and one kg of food Y costs Rs 36. Using Linear Programming, find the least cost of the total mixture. which will contain the required vitamins.
Solution
Let one type of food be x and another type be y
| Food | Vitamin A | Vitamin B | Vitamin C |
| X | 1 units | 2 units | 3 units |
| Y | 2 units | 2 units | 1 units |
∴ According to the given condition
Minimize z = 24x + 36y
Subject to
`x + 2y >= 10`
`x + 2y >= 12 or x + y >= 6`
`x + y >= 8`
| X + 2y = 10 | x + y = 6 | 3x + 1y = 8 |
| `x/10 + y/5 = 1` | x + y = 6 | `x/(8/3) + y/8 = 1` |
| Corner points | Objective function z= 24x + 36y |
| A(10,0) | `z= 24xx 10 + 36 xx 0 = 240` |
| B(2,4) | `z = 24 xx 2 + 36 xx 4 = 192` |
| C(1,5) | `z = 24 xx 1 + 36 xx 5 = 204` |
| D(0,8) | `z = 0 xx 24 + 36 xx 8 = 288` |
Minimum value of z is 192 at B (2, 4)
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