The 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 1/5 ton of material 1, 3/5 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, 16 tons; material 2, 18 tons; material 3, 12 tons. Management wants to achieve the following P₁ priority level goals:

Goal 1: Produce at least 20 tons of fuel additive.

Goal 2: Produce at least 15 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	fill in the blank 1
   2	fill in the blank 2
   3	fill in the blank 3

   Yes, it is possibleNo, it is impossible

2. 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 20 tons
   	d1- = the amount by which the number of tons of fuel additive produced is less than the target of 20 tons
   	d2+ = the amount by which the number of tons of solvent base produced exceeds the target value of 15 tons
   	d2- = the amount by which the number of tons of solvent base is less than the target value of 15 tons

   Min	fill in the blank 5d1-	+	fill in the blank 6d2-
   s.t.
   	x1	+	x2						fill in the blank 9	Material 1
   	fill in the blank 10x1	+	x2						fill in the blank 12	Material 2
   	x1	+	x2						fill in the blank 15	Material 3
   	fill in the blank 16x1	+	fill in the blank 17x2	+	fill in the blank 18d1+	+	fill in the blank 19d1-	=	fill in the blank 20	Goal 1
   	fill in the blank 21x1	+	fill in the blank 22x2	+	fill in the blank 23d2+	+	fill in the blank 24d2-	=	fill in the blank 25	Goal 2
   x1, x2, d1+, d1-, d2+, d2- 0

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

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

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

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

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