Personal Meal Management - A Mathematical Model

Further to requests received to design a Mathematical Model to ensure one's satisfactory dining preferences vis a vis the cost of it and the variety, I have tried to make a Model that aims to capture the intent. While this has been written with a University Setting as a base it can be equally applied across all Dining facilities for everyone. I would be glad to receive any corrections on the model to make it more useful.

Creating a mathematical model to manage oneself and eating preferences in a university dining hall and cafes involves several factors. These include meal pricing, dietary preferences, nutritional needs, meal variety, and cost management. Here's a comprehensive model that incorporates these aspects:

Parameters:

1. ?????: Price of dining hall meals per day (paid in advance).
2. ??????: Average price per meal in cafes.
3. ??: Total monthly budget for meals.
4. ??: Number of days in a month.
5. ?????: Number of dining hall meals per day (usually fixed at 4 - Breakfast, Lunch, Snacks, Dinner).
6. ??????: Number of cafe meals per day (variable).
7. ?????: Number of distinct dishes in dining hall.
8. ??????: Number of distinct dishes in cafes.
9. ????: Nutritional requirements (e.g., calories, protein, carbs, fats).
10. ????: Price per unit of nutrient ?? in cafes.

Variables:

1. xdh: Number of days eating all meals in dining hall.
2. ??cf: Number of meals eaten in cafes per month.
3. ??????: Cost of cafe meals per month.
4. ?????: Cost of dining hall meals per month.

Constraints:

1. Budget Constraint: Cdh+Ccf≤B
2. Cost Calculation: 1. Cdh=xdh×Pdh 2. ??????=??????×??????
3. Meal Distribution Constraint: 4xdh+ycf=4N
4. Nutritional Requirements:

∑??(????×intake?from?dining?hall?meals)+∑??(????×intake?from?cafe?meals)≥∑?? ????

5. Variety Constraint:

Variety?in?dining?hall?meals ≥Vdh

Variety?in?cafe?meals≥??????

Objective Function:

The objective is to maximize satisfaction while adhering to budget and nutritional needs. Satisfaction can be modeled by a utility function U that considers the variety and preference for certain meals:

U=α(Variety?in?dining?hall?meals)+β(Variety?in?cafe?meals) +γ(Nutritional?adequacy)?δ(Cost)

where ??,??,??,?? are weights representing the importance of each factor.

Mathematical Formulation:

1. Budget Management: (xdh×Pdh)+(ycf×Pcf)≤B
2. Meal Distribution: 4xdh+ycf=4N
3. Nutritional Adequacy: k∑(Nk×intake?from?dining?hall?meals)+k∑(Nk×intake?from?cafe?meals)≥k∑Nk
4. Maximizing Satisfaction:

maximise U over xdh and ycf =α(Variety?in?dining?hall?meals)+β(Variety?in?cafe?meals)+ γ(Nutritional?adequacy)?δ(Cost)

Example Calculation:

Let's consider a specific example with the following values:

• ????h=?200 /day
• ??????=?150 /meal
• ??=?9000 /month
• ??=30 days
• ?????=4 (fixed)
• ?????=10 dishes
• ??????=20 dishes
• Nutritional needs: ????={2000?kcal,?50g?protein,?300g?carbs,?70g?fats}

1. Cost of Dining Hall Meals: ?????= ????? × 200
2. Cost of Cafe Meals: ??????=?????? ×150
3. Total Meals Constraint: 4?????+?????? =120
4. Budget Constraint: 200?????+150?????? ?≤ 9000

By solving these equations, we can determine the optimal number of days to eat at the dining hall and the number of cafe meals to maximize satisfaction while adhering to budget and nutritional needs. I was told that the optimization can be done using linear programming or other optimization techniques, but I am not aware of how to use them.

This model helps in effectively managing meal preferences, nutritional intake, and budget constraints in a university setting. Any help to further optimize the model is welcome including the use of Linear Programming/ Other Optimization techniques.