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RMC, Inc. is a small firm that produces a variety of chemical products. In a particular production process, three raw materials are blended to produce two products: a fuel additive and a solvent base.

- Each ton of fuel additive is a mixture of [tex]\frac{2}{5}[/tex] ton of material 1 and [tex]\frac{3}{5}[/tex] ton of material 3.
- Each ton of solvent base is a mixture of [tex]\frac{1}{2}[/tex] ton of material 1, [tex]\frac{1}{5}[/tex] ton of material 2, and [tex]\frac{3}{10}[/tex] ton of material 3.
- The profit contribution is $40 for every ton of fuel additive produced and $30 for every ton of solvent base produced.

RMC's production is constrained by a limited availability of the three raw materials. For the current production period, RMC has available the following quantities of each raw material:

- Material 1: 20 tons
- Material 2: 5 tons
- Material 3: 21 tons

Assuming that RMC is interested in maximizing the total profit contribution, do the following:

A. **Decision Variables:**
- Let [tex]x[/tex] be the tons of fuel additive produced.
- Let [tex]y[/tex] be the tons of solvent base produced.

B. **Objective Function:**
- Maximize profit: [tex]40x + 30y[/tex]

C. **Model Constraints:**
- Material 1: [tex]\frac{2}{5}x + \frac{1}{2}y \leq 20[/tex]
- Material 2: [tex]\frac{1}{5}y \leq 5[/tex]
- Material 3: [tex]\frac{3}{5}x + \frac{3}{10}y \leq 21[/tex]

D. **Non-Negativity Statement:**
- [tex]x \geq 0[/tex]
- [tex]y \geq 0[/tex]

Answer :

Final answer:

A. Decision Variables: X1, X2

B. Objective Function: Maximize Profit = $40X1 + $30X2

C. Model Constraints: Material 1: (2/5)X1 + (1/2)X2 ≤ 20 tons, Material 2: (1/5)X2 ≤ 5 tons, Material 3: (3/5)X1 + (3/10)X2 ≤ 21 tons

D. X1 ≥ 0 and X2 ≥ 0.

Explanation:

To maximize the total profit contribution for RMC, Inc., we must first define the elements of the linear programming model:

A. DECISION VARIABLES:

X1: Tons of fuel additive produced.

X2: Tons of solvent base produced.

B. OBJECTIVE FUNCTION:

Maximize Profit = $40X1 + $30X2

C. MODEL CONSTRAINTS:

Material 1: (2/5)X1 + (1/2)X2 ≤ 20 tons

Material 2: (1/5)X2 ≤ 5 tons

Material 3: (3/5)X1 + (3/10)X2 ≤ 21 tons

D. NON-NEGATIVITY STATEMENT:

X1 ≥ 0 and X2 ≥ 0

This model includes variables representing the quantities of fuel additive and solvent base to produce, the goal to maximize profit, and constraints defined by the availability of the three raw materials. Balancing these elements will allow RMC, Inc. to achieve the maximum possible profit within their resource limits.

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Rewritten by : Jeany

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