Consider the convex set given by

Question

Consider the convex set given by

3 x1+6 x2 ≤16

5 x1+2 x2 ≥18

x1≥0,x2≥0

(a) Introduce a slack variable x3≥0 to convert the first inequality to an equation.

Answer: 3x[1] + 6x[2] + x[3]=16

(b) Introduce a slack variable x4≥0 to convert the second inequality to an equation.

Answer: 5x[1] + 2x[2] - x[4] = 18

(c) Enter the 2x5 augmented matrix that comes from your answers to (a) and (b), with pivots in the third and fourth columns

Answer: row 1: (3,6,1,0,16) row 2: (-5,-2,0,1,-18)

(d) Find a basic solution that is not feasible. It should have four components, x1,x2,x3,x4

(e) Find a feasible basic solution. It should have four components, x1,x2,x3,x4.

Answer: (19/6,13/12,0,0)

I did questions (a),(b), and (c) by myself and it is correct, and (e) is the answer from one of the tutors and it is correct also but for (d) the answer that the tutor provided was wrong. may I know I did (d)?

Answer 1

To find a basic solution that is not feasible, we need to set some of the variables to zero and solve the resulting system of equations. Let's set x3 and x4 to zero and solve for x1 and x2:

From equation (a):

3x1 + 6x2 + x3 = 16

Setting x3 = 0:

3x1 + 6x2 = 16

From equation (b):

5x1 + 2x2 - x4 = 18

Setting x4 = 0:

5x1 + 2x2 = 18

Now we have a system of two equations with two variables. Solving these equations, we get:

3x1 + 6x2 = 16  -->  x1 = (16 - 6x2) / 3

5x1 + 2x2 = 18  -->  x1 = (18 - 2x2) / 5

Setting the two expressions for x1 equal to each other:

(16 - 6x2) / 3 = (18 - 2x2) / 5

Cross-multiplying and simplifying:

5(16 - 6x2) = 3(18 - 2x2)

80 - 30x2 = 54 - 6x2

24 = 24x2

x2 = 1

Substituting x2 = 1 into one of the equations, we get:

3x1 + 6(1) = 16

3x1 + 6 = 16

3x1 = 10

x1 = 10/3 = 3.33

Therefore, a basic solution that is not feasible is (x1, x2, x3, x4) = (3.33, 1, 0, 0).