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 2 tons of material 1 and 1 ton of material 3. A ton of solvent base is a mixture of 1 ton of material 1, 1 ton of material 2, and 1 ton of material 3. RMC's production is constrained by the 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: 5 tons
- Material 3: 21 tons

Management wants to achieve the following priority level goals:

- Goal 1: Produce at least 30 tons of fuel additive.
- Goal 2: Produce at least 15 tons of solvent base.

Assume there are no other goals.

(a) Is it possible for management to achieve both priority level goals given the constraints on the amounts of each material available? If not, which constraint is the limiting factor?

- Yes. It is possible to satisfy both priority level goals.
- No. There is an insufficient amount of Material 1.
- No. There is an insufficient amount of Material 2.
- No. There is an insufficient amount of Material 3.

(b) Treating the amounts of each material available as constraints, formulate a goal programming model to determine the optimal product mix. Assume that both priority level goals are equally important to management.

Let \( x_1 \) be the number of tons of fuel additive produced, \( x_2 \) be the number of tons of solvent base produced, \( d_1^- \) be the deviation variable which is less than the value of goal 1, \( d_2^- \) be the deviation variable which is less than the value of goal 2.

Minimize:
\[ Z = d_1^- + d_2^- \]

Subject to:
1. Material 1 constraint: \( 2x_1 + x_2 \leq 20 \)
2. Material 2 constraint: \( x_2 \leq 5 \)
3. Material 3 constraint: \( x_1 + x_2 \leq 21 \)
4. Goal 1: \( x_1 + d_1^- = 30 \)
5. Goal 2: \( x_2 + d_2^- = 15 \)

\( x_1, x_2, d_1^-, d_2^- \geq 0 \)