he RMC Corporation blends three raw materials to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of 1/2 ton of material 1 and 1/2 ton of material 3. A ton of solvent base is a mixture of 2/5 ton of material 1, 2/5 ton of material 2, and 1/5 ton of material 3. RMCs production is constrained by a limited availability of the three raw materials. For the current production period, RMC has the following quantities of each raw material: material 1, 20 tons; material 2, 16 tons; material 3, 15 tons. Management wants to achieve the following P₁ priority level goals:

Goal 1: Produce at least 35 tons of fuel additive.

Goal 2: Produce at least 25 tons of solvent base.

Assume there are no other goals.

1. Is it possible for management to achieve both P₁ level goals given the constraints on the amounts of each material available?

   	Amount needed to
   Raw Material	Achieve Both P1 Goals
   1
   2
   3

Choices - Yes, it is possible

No, it is impossible

b. Treating the amounts of each material available as constraints, formulate a goal programming model to determine the optimal product mix. Assume that both P₁ priority level goals are equally important to management. If you don't need the variable in the model, enter "0". If you need a negative number, enter minus sign with it. If necessary, enter the numbers as a common fraction.

Let
	x1 = the number of tons of fuel additive produced
	x2 = the number of tons of solvent base produced
	d1+ = the amount by which the number of tons of fuel additive produced exceeds the target value of 35 tons
	d1- = the amount by which the number of tons of fuel additive produced is less than the target of 35 tons
	d2+ = the amount by which the number of tons of solvent base produced exceeds the target value of 25 tons
	d2- = the amount by which the number of tons of solvent base is less than the target value of 25 tons

Min	d1-	+	d2-
s.t.
	x1	+	x2							Material 1
	x1	+	x2							Material 2
	x1	+	x2							Material 3
	x1	+	x2	+	d1+	+	d1-	=		Goal 1
	x1	+	x2	+	d2+	+	d2-	=		Goal 2
x1, x2, d1+, d1-, d2+, d2- 0

c. Choose the correct graph that solves the model formulated in part b.

(iii)		(iv)

d. If goal 1 is twice as important as goal 2, choose the correct graph for the optimal product mix?

(i)		(ii)
(iii)		(iv)

P.S. The graphs do not seem to work with the copy and paste function. A drawing/image of the correct graph will suffice. Please and thank you!