Let's break down and approach the College meal selection problem as described in the image. Kathy Roniger has five nutritional requirements that need to be met for a meal plan, and there are seven food items available, each with a given cost and nutrient contribution.

Problem Statement (Restated):

We are tasked with formulating a Linear Programming (LP) problem to minimize the total cost of the meal while satisfying the given nutritional constraints:

1. Between 900 and 1,500 calories.
2. At least 4 milligrams of iron.
3. No more than 50 grams of fat.
4. At least 26 grams of protein.
5. No more than 50 grams of carbohydrates.

(a) Formulating the LP Problem:

Decision Variables:

Let $x_i$ represent the amount (in pounds) of food item $i$ (where $i = 1, 2, \dots, 7$) to be selected.

Objective Function:

Minimize the total cost of the meal: $\text{Minimize } Z = c_1 x_1 + c_2 x_2 + \dots + c_7 x_7$ where $c_i$ is the cost per pound of food item $i$.

Constraints:

The constraints will capture the nutritional requirements.

1. Calories (between 900 and 1,500): $900 \leq a_1 x_1 + a_2 x_2 + \dots + a_7 x_7 \leq 1500$ where $a_i$ represents the calorie contribution of food item $i$.

2. Iron (at least 4 mg): $b_1 x_1 + b_2 x_2 + \dots + b_7 x_7 \geq 4$ where $b_i$ is the iron contribution of food item $i$.

3. Fat (no more than 50 grams): $f_1 x_1 + f_2 x_2 + \dots + f_7 x_7 \leq 50$ where $f_i$ is the fat contribution of food item $i$.

4. Protein (at least 26 grams): $p_1 x_1 + p_2 x_2 + \dots + p_7 x_7 \geq 26$ where $p_i$ is the protein contribution of food item $i$.

5. Carbohydrates (no more than 50 grams): $carb_1 x_1 + carb_2 x_2 + \dots + carb_7 x_7 \leq 50$ where $carb_i$ is the carbohydrate contribution of food item $i$.

Additionally, all decision variables must be non-negative: $x_1, x_2, \dots, x_7 \geq 0$

(b) What is the Cost per Meal?

To find the actual cost per meal, we would need the full nutritional data and costs for each food item (which should be provided in a table in the textbook or document). Once the LP problem is solved (using simplex method or software like Excel Solver), the optimal values of $x_1, x_2, \dots, x_7$ can be found. Plugging these values into the objective function will give us the minimum total cost.

(c) Is this a Well-Balanced Diet?

To determine if this is a well-balanced diet, we would have to:

• Check whether the nutritional requirements (calories, iron, fat, protein, and carbohydrates) are satisfied within healthy limits.
• Compare the nutrients against recommended dietary guidelines.

Next Steps:

If you provide the table with the nutrient contributions and costs for each of the seven food items, I can help solve the LP problem in detail.

Would you like to provide more details from the table?

5 Relative Questions:

1. How can we set up the LP in Excel Solver or any LP solver tool?
2. What are typical methods used to solve LP problems like this?
3. What is the role of dual variables in linear programming?
4. How would adding more constraints affect the feasibility of the LP problem?
5. Can we introduce variability in costs or nutritional values and still keep the solution optimal?

Tip:

When formulating LP problems, always check if your constraints are both feasible and bounded. This ensures that the solution space has a well-defined optimal solution.