Given the linear programming problem below where X₁ represents the number of belts a company produces and X₂ represents the number of pairs of gloves produced. The first constraint is for the number of square yards leather available and the second for the number of hours of skilled labor available.

Max z = 4X₁ + 3X₂

s.t. X₁ + X₂ 40

2X₁ + X₂ 60

The solution is X₁ = X₂ = 20 with max Z = 140. The final tableau is shown below

Z	X1	X2	S1	S2	RHS	Basis
1	0	0	2	1	140	Z =140
0	0	1	2	-1	20	X2 = 20
0	1	0	-1	1	20	X1 = 20

Use this information to answer the following questions.

1. Show that if c1, the coefficient of x1 in the objective function, is between 1 and 5 the current basis remains optimal.

2. Suppose c1 = 5. Find the new maximum z value.

3. Show that if c2 is between 2 and 4 the current basis remains optimal.

4. Show that if the available leather is between 30 and 60 yards the current basis remains optimal.

5. Show that if the number of hours of skilled labor is between 40 and 80 the current basis remains optimal.

6. The company is considering manufacturing leather hats. Each hat would contribute $5 to profits and would use 2 yards of leather and 2 hours of skilled labor. Should the company manufacture hats?

7. Find the amount that hats should contribute to profits on order to make the company indifferent to manufacturing them