Question

For the given graph of a Linear Programming Problem, write all the constraints satisfying the given feasible region.

Hint:

To write the constraints for a Linear Programming Problem from a graph, find the equations of the lines forming the boundaries of the feasible region and translate them into inequalities.

Answer 1

The feasible region is determined by the constraints derived from the given graph. The vertices of this region are identified as $ A(0, 200) $, $ B(50, 250) $, $ C(150, 150) $, and $ D(200, 0) $. The inequalities corresponding to these constraints are established by calculating the equations of the lines passing through pairs of these points. 1. Line segment AB:
The slope \( m \) between points $ A(0, 200) $ and $ B(50, 250) $ is calculated as: \[ m = \frac{250 - 200}{50 - 0} = \frac{50}{50} = 1 \] Using the point-slope form with point $ A(0, 200) $: \[ y - 200 = 1(x - 0) \quad \Rightarrow \quad y = x + 200 \] The corresponding constraint is: \[ y \leq x + 200 \] 2. Line segment BC:
The slope \( m \) between points $ B(50, 250) $ and $ C(150, 150) $ is: \[ m = \frac{150 - 250}{150 - 50} = \frac{-100}{100} = -1 \] Using the point-slope form with point $ B(50, 250) $: \[ y - 250 = -1(x - 50) \quad \Rightarrow \quad y = -x + 300 \] The corresponding constraint is: \[ y \leq -x + 300 \] 3. Line segment CD:
The slope \( m \) between points $ C(150, 150) $ and $ D(200, 0) $ is: \[ m = \frac{0 - 150}{200 - 150} = \frac{-150}{50} = -3 \] Using the point-slope form with point $ C(150, 150) $: \[ y - 150 = -3(x - 150) \quad \Rightarrow \quad y = -3x + 750 \] The corresponding constraint is: \[ y \leq -3x + 750 \] 4. Line segment DO (origin):
The segment connecting point $ D(200, 0) $ to the origin $ O(0, 0) $ lies on the x-axis. The slope \( m \) is: \[ m = \frac{0 - 0}{200 - 0} = 0 \] The equation of this line is: \[ y = 0 \] Since the feasible region is above or on the x-axis, the constraint is: \[ y \geq 0 \] Summary of Constraints:
The complete set of constraints defining the feasible region is: \[ y \leq x + 200 \] \[ y \leq -x + 300 \] \[ y \leq -3x + 750 \] \[ y \geq 0 \] These inequalities collectively define the feasible region for the given Linear Programming Problem.