Determine whether the following linear programming problem is infeasible, unboun

Question

Determine whether the following linear programming problem is infeasible, unbounded, or has multiple optimal solutions. Draw a graph and explain your conclusion.

Maximize 50x + 100y
Subject to:
2x + y > 15
2x + 2y < 10
y > 5
x, y > 0

Explanation / Answer

Since I can't show you a picture, I hope describing it will be enough. To start, let's look at the constraints working from the bottom up. With x,y > 0, we are in the first quadrant with x and y being positive, and the x- and y-axis as boundaries. Next, y < 5, we draw a horizontal line where y = 5 as a maximum value for y. If you are shading regions, we went from the entirety of the first quadrant to everything under y = 5. For x + y < 5, we draw a line from (0,5) to (5,0) to form a triangle at the origin with everything inside it being shaded. Finally we have 2x + y > 15. For this, draw a line from (0,15) to (7.5,0) and shade everything above this line. As you can see, this constraint does not fall inside the triangle made by the others, so the LP problem is infeasible as not all constraints can be met by an singular point or group of points.

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