To solve this problem, we need to create a linear programming (LP) model to minimize the cost of purchasing supplements while meeting the nutrient requirements. Here is the step-by-step formulation of the LP problem:

Decision Variables:

Let:

• $x_1$: grams of Memo+ to be purchased
• $x_2$: grams of Omega to be purchased

Objective Function:

The objective is to minimize the total cost of the supplements: $\text{Minimize: } Z = 20x_1 + 30x_2$ where:

• $20x_1$: cost in cents of Memo+ per gram
• $30x_2$: cost in cents of Omega per gram

Constraints:

Each nutrient has a minimum required amount, and the supplements provide specific amounts of each nutrient. We'll set up one constraint per nutrient.

1. SSS (at least 60 mg): $20x_1 + 30x_2 \geq 60$

2. BBB (at least 1000 mg): $500x_1 + 250x_2 \geq 1000$

3. GGG (at least 18 mg): $9x_1 + 2x_2 \geq 18$

4. HHH (at least 20 mg): $2x_1 + 10x_2 \geq 20$

5. LLL (at least 360 mg): $60x_1 + 90x_2 \geq 360$

Non-Negativity Constraints:

Since negative quantities of supplements are not possible: $x_1 \geq 0, \quad x_2 \geq 0$

Final LP Model:

$\text{Minimize: } Z = 20x_1 + 30x_2$ subject to:

20x_1 + 30x_2 &\geq 60 \quad \text{(SSS)} \\ 500x_1 + 250x_2 &\geq 1000 \quad \text{(BBB)} \\ 9x_1 + 2x_2 &\geq 18 \quad \text{(GGG)} \\ 2x_1 + 10x_2 &\geq 20 \quad \text{(HHH)} \\ 60x_1 + 90x_2 &\geq 360 \quad \text{(LLL)} \\ x_1, x_2 &\geq 0 \end{aligned}$$ This LP model minimizes costs while ensuring all nutrient requirements are met. --- Do you want me to solve this LP problem for optimal values of $$x_1$$ and $$x_2$$? ### Related Questions: 1. What is the graphical interpretation of this LP model? 2. How do the cost coefficients in the objective function affect the solution? 3. What happens if the nutrient requirements are increased? 4. Can this LP problem be solved using simplex method or software like Excel or Python? 5. What would the cost and nutrient balance be if $$x_1 = 0$$ (only Omega is used)? ### Tip: When solving LP problems, it's helpful to first check if the constraints are independent and feasible to ensure a valid solution exists.