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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 \(\frac{2}{5}\) ton of material 1 and \(\frac{3}{5}\) ton of material 3. A ton of solvent base is a mixture of \(\frac{1}{2}\) ton of material 1, \(\frac{1}{5}\) ton of material 2, and \(\frac{3}{10}\) ton of material 3. RMC's 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: 5 tons
- Material 3: 21 tons

Management wants to achieve the following P1 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 P1 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 P1 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 P1 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, \(d_{pi}\) be the deviation variable which exceeds the value of goal \(i\), and \(d_{ni}\) be the deviation variable which is less than the value of goal \(i\), for \(i = 1, 2\).

Minimize: \(d_{n1} + d_{n2}\)

Subject to:
\[
\begin{aligned}
&\frac{2}{5}x_1 + \frac{1}{2}x_2 + d_{n1} - d_{p1} \leq 20 \quad \text{(Material 1)} \\
&\frac{1}{5}x_2 + d_{n2} - d_{p2} \leq 5 \quad \text{(Material 2)} \\
&\frac{3}{5}x_1 + \frac{3}{10}x_2 + d_{n3} - d_{p3} \leq 21 \quad \text{(Material 3)} \\
&x_1 \geq 30 \quad \text{(Goal 1)} \\
&x_2 \geq 15 \quad \text{(Goal 2)} \\
&x_1, x_2, d_{ni}, d_{pi} \geq 0 \quad \text{for } i = 1, 2, 3
\end{aligned}
\]

(c) Use the graphical goal programming procedure to find the optimal solution for the model formulated in part (b).

\((x_1, x_2) =\)

(d) If goal 1 is twice as important as goal 2, what is the optimal product mix?

\((x_1, x_2) =\)

Answer :

a. No. There is an insufficient amount of Material 1.

a. The constraint on Material 1 availability is the limiting factor preventing the achievement of both P1 level goals. The required amount of Material 1 for producing both the fuel additive and solvent base exceeds the available quantity.

This constraint restricts the production capacity and makes it impossible to meet the desired production goals for both products simultaneously.

b. The goal programming model formulated ensures that the deviation variables, which represent the extent to which the goals are not met, are minimized. By setting up the constraints based on the availability of each raw material and the production goals, the model aims to find the optimal product mix that satisfies both goals to the greatest extent possible.

c. The graphical goal programming procedure involves plotting the feasible region and identifying the point that minimizes the deviations from the goals. By graphically analyzing the constraints and objective functions, the optimal solution, indicating the quantities of fuel additive and solvent base to produce, can be determined.

d. If goal 1 is twice as important as goal 2, the weights assigned to the deviation variables corresponding to each goal are adjusted accordingly. Then, the goal programming model is solved to find the optimal product mix that minimizes the weighted sum of deviations from both goals.

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Rewritten by : Jeany

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