1 Diet Problem We wish to solve the diet problem. The Diet Problem can be formulated mathematically as a linear programming problem as shown below. Sets F =set of foods N =set of nutrients Parameters aij =amount of nutrient (j) in food (i), i F, j N ci =cost per serving of food (i), i F Fmini =minimum number of required servings of food i, i F Fmaxi =maximum allowable number of servings of food i, i F Nminj =minimum required level of nutrient j, j N Nmaxj =maximum allowable level of nutrient j, j N Variables xi =number of servings of food (i) to purchase/consume, i F Objective Function: Minimize the total cost of the food Minimize iF ci xi Constraint Set 1: For each nutrient j N, at least meet the minimum required level. iF aij xi Nminj, j N 1

Constraint Set 2:For each nutrient j N, do not exceed the maximum allowable level. iF aij xi Nmaxj, j N Constraint Set 3:For each food i F, select at least the minimum required number of servings. xi Fmini, i F Constraint Set 4:For each food i F, do not exceed the maximum allowable number of servings. xi Fmaxi, i F

2 Problem Data The problem dataset can be found by the path code/datasets/diet_problem.xlsx 3 Questions: 1. What is the optimal solution to minimizing cost? 2. Write objective 2 (in the code as obj2) to minimize the total number of carobhydrates. What is the optimal solution to this problem? 3. Use the multi-objective code from the in class exercise to plot the pareto frontier. Include that plot in your solution. 4. Setup a new model where the number of carbohydrates is at most to 250. What is the new optimal solution? What is the cost value for this solution? 5. In addition to carbohydrates being at most 250, we want to add the following conditions: (a) At most 2 types of meats (including eggs) are included in the diet. (b) If broccocli is included, then carrots must also be included with at least 1 serving. (c) If soup is included, then we must also have tortilla chips with at least 1 serving. Write constraints to model these conditions. Add them to your homework solutions. Then code them and determine the optimal solution (to minimize cost) using these constraints.