In the following problem, x₁, x₂ and x₃ are respectively the number of kilograms of produced paint from Type 1, 2 and 3. For producing each type of paint, three kinds of raw materials (M₁, M₂ and M₃) should be mixed together. There is an availability limit on the number of kilograms of each kind of raw material, as shown in the right-hand side of each of the constraints. The objective function of this formulated problem is to maximize the profit at the end.

Max z= 10x₁ + 12x₂ + 8x₃

s.t.

2x₁ + 3x₂ + 3x₃ 50 (Availability of M₁ in kilograms)

x₁ + 2x₂ + x₃ 20 (Availability of M₂ in kilograms)

4x₁ + 2x₂ + 2x₃ 40 (Availability of M₃ in kilograms)

x₁, x₂, x₃ 0

1. Assume that based on new market estimates, the profit per unit of paint Type 2 has reduced to $9. How will this change impact the optimal solution for this problem? i.e., what will be the updated optimal values for decision variables and the objective function based on this change?
2. If the company has the option to increase the available kilograms of M₁ or M₃, and each kilogram of increase costs the company $1, would you recommend an increase to any of the resources (raw materials) availability? If yes, in which one? Why?
3. The producing company can increase the available kilograms of M₂ from 20 to 30. If this increase costs the company $48, would you recommend such an increase? Why?

Using Solver, the optimal solution for this problem is x1=2.5,x2=2.5,x3=12.5,z=$155. Also, the following is the sensitivity analysis report for this problem: Variahle relle