Consider the region given by

Question

Consider the region given by

x≤1

−2 x+y ≥5

x+y−z ≤2

x,y,z ≥0

(a) Formulate the problem using slack variables s1≥0 ,s2≥0 and s3≥0.Write down the basic solution obtained by choosing the slack variables as pivots. Enter in the form (x,y,z,s1,s2,s3)

(b) Write down the basic solution obtained by choosing z,s1, and s2 as pivots. Enter in the form (x,y,z,s1,s2,s3)

 (c) Write down the basic solution obtained by choosing x,y,z as pivots. Enter in the form (x,y,z,s1,s2,s3)

Answer 1

(a) To formulate the problem using slack variables, we introduce s1, s2, and s3 as slack variables for the inequalities:

1) x ≤ 1 (no slack variable needed)

2) -2x + y + s1 = 5

3) x + y - z + s2 = 2

The basic solution obtained by choosing the slack variables as pivots is:

(x, y, z, s1, s2, s3) = (1, 0, 0, 0, 5, 2)

(b) To obtain the basic solution by choosing z, s1, and s2 as pivots, we set them as basic variables:

1) x = 1 - s3

2) y = 5 - 2x - s1

3) z = x + y - 2 - s2

The basic solution obtained is:

(x, y, z, s1, s2, s3) = (1, 3, 1, 0, 0, 0)

(c) To obtain the basic solution by choosing x, y, z as pivots, we set them as basic variables:

1) x = 1

2) y = -2 + 5 - s1 = 3 - s1

3) z = 1 + (3 - s1) - 2 - s2 = 2 - s1 - s2

The basic solution obtained is:

(x, y, z, s1, s2, s3) = (1, 3, 2 - s1 - s2, s1, s2, 0)