Consider the constraints 2x + y ≤ 6 , y − x ≤ 2 , y + 1 ≥ x , x, y ≥ 0 .

Question

Consider the constraints 2x + y ≤ 6, y − x ≤ 2, y + 1 ≥ x, x, y ≥ 0 .

(a) Plot the constraints and shade the feasible region. In your plot, label each line which forms the boundary of the feasible region.

(b) Write the feasible region in matrix-vector form Ax ≤ b, x ≥ 0.

(c) Find all vertices of the feasible region

Answer 1

Here is a graph of the boundaries of the solution set:

https://www.desmos.com/calculator/npuqldnvud

The simplex method locates the boundaries, then the vertices of the boundaries, and finally moves from one vertex to the next until the optimum solution is found.

One of the vertices has the best solution.

You can do the same thing by locating the boundaries, then checking out each vertex to find which solution is the best. The simplex method takes a little practice to use it properly, so I am not going to describe it here.

Answer 2

Let's start by drawing the region.

Part (a)

The last two constraints are x > = 0 and y > = 0. This region is the first quadrant including the axes.

So now we can concentrate only on the first quadrant when drawing the other lines.

These are easiest if we first put them in slope-intercept form.

So solve 2x + y ≤ 6 for y to get y <= -2x + 6.

Plot the y intercept at (0,6). Count using the slope. Down 2, right 1. Repeat several times.

connect the dots using a solid line. (Solid because of the = sign.)

Lightly shade below the line because of the < sign.

Put the next equation into slope-intercept form.

So solve y - x <= 2. for y to get. y <= x + 2

Plot the y intercept at (0,2) and use the slope to count up 1, right 1. Repeat several times connecting the dots using a solid line. Lightly shade below the line because of the < sign.

Put the next equation into slope-intercept form.

So solve. y + 1 >= x for y to get. y >= x - 1

Plot the y intercept at (0,-1) and use the slope to count up 1, right 1. Repeat several times connecting the dots using a solid line. Lightly shade above this line because of the > sign.

Part (b)

To write the feasible region in matrix-vector form, first arrange each equation in Ax + By = C format.

So. 2x + 1y = 6 So the A matrix is the coefficients of x and y. 2 1

-1x + 1y = 2 -1 1

-1x + 1y = -2 -1 1

1x + 0y = 0 0 1

0x + 1y = 0 1 0

The x matrix is simply the unknown values in vertical format.

x

y

The b matrix is the constants after the equal signs.

So. 6

2

1

0

0

Part c

The feasible region is a 5 sided area where all your shadings overlap. The vertices are at

(0,0), (0,2), (1,0), (1.5, 3.5), and (2.5, 1.5)