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Question
Minimum and maximum z = 5x + 2y subject to the following constraints:
x-2y ≤ 2
3x+2y ≤ 12
-3x+2y ≤ 3
x ≥ 0,y ≥ 0
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Solution
x-2y ≤ 2
3x+2y ≤ 12
-3x+2y ≤ 3
x ≥ 0,y ≥ 0
Converting the inequations into equations, we obtain the lines
x-2y = 2 .....(i)
3x+2y = 12......(ii)
-3x+2y = 3.......(iii)
x = 0,y =0
From the graph, we get the corner points as
A(0, 5), B(3.5, 0.75), C(2, 0), D(1.5, 3.75), O(0, 0)
The values of the objective function are:
| Point (x, y) | Values of the objective function Z = 5x + 2y |
| A(0, 5) | 5 × 0 + 2 × 5 = 10 |
| B(3.5, 0.75) | 5 × 3.5 + 2 × 0.75 = 19 (Maximum) |
| C(2, 0) | 5 × 2 + 2 × 0= 10 |
| D(1.5, 3.75) | 5 × 1.5 + 2 × 3.75 = 15 |
| O(0, 0) | 5 × 0 + 2 × 0 = 0 (Minimum) |
The maximum value of Z is 19 and its minimum value is 0.
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