Optimal Healthy Menu for Cost-Efficient Nutrition

Finding the most cost-effective yet nutritious food choices for a healthy diet is essential. In this scenario, we aim to meet specific nutritional goals by determining the optimal amount of stir-fry and boba to purchase while considering factors like calories, sugar, calcium, and food costs. By analyzing the nutritional values and costs per ounce or fluid ounce of each item, we can optimize our food purchasing decisions to stay within budget while meeting our health targets.

• Nutrition
• Healthy eating
• Food cost
• Meal optimization
• Nutritional goals

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1. Warm-up: What to eat? We are trying healthy by finding the optimal amount of food to purchase. We can choose the amount of stir-fry (ounce) and boba (fluid ounces). Healthy Squad Goals 2000 Calories 2500 Sugar 100 g Calcium 700 mg Food Cost Calories Sugar Calcium Stir-fry (per oz) 1 100 3 20 Boba (per fl oz) 0.5 50 4 70 What is the cheapest way to stay healthy with this menu? How much stir-fry (ounce) and boba (fluid ounces) should we buy?

3. Announcements Assignments: HW3 (online) Due Tonight, 10 pm HW4 (online) Due 2/14, 10 pm P1: Search and Games due yesterday!! P2: Linear/Integer Programming Due 2/23, 10pm (1 week after the exam) Exam 1 Feb 16!

4. AI: Representation and Problem Solving Linear Programming Instructor: Stephanie Rosenthal Slide credits: CMU AI with drawings from http://ai.berkeley.edu

5. Warm-up: What to eat? We are trying healthy by finding the optimal amount of food to purchase. We can choose the amount of stir-fry (ounce) and boba (fluid ounces). Healthy Squad Goals 2000 Calories 2500 Sugar 100 g Calcium 700 mg Food Cost Calories Sugar Calcium Stir-fry (per oz) 1 100 3 20 Boba (per fl oz) 0.5 50 4 70 What is the cheapest way to stay healthy with this menu? How much stir-fry (ounce) and boba (fluid ounces) should we buy?

6. Optimization Graphical Representation Problem Description Optimization Representation ??? min ? s.t. ?? ?

7. Constraint Satisfaction Problems Map coloring ??? ? s.t. ? ????????? ??????????? Notation Alert!

8. Optimization Formulation Diet Problem ??? ? s.t. ? ????????? ??????????? Healthy Squad Goals 2000 Calories 2500 Sugar 100 g Calcium 700 mg Cost Calories Calciu m Food Sugar Stir-fry (per oz) 1 100 3 20 Notation Alert! Boba (per fl oz) 0.5 50 4 70

9. Optimization Formulation Diet Problem min ? ????(?) Objective s.t. ? ????????? ??????????? Healthy Squad Goals 2000 Calories 2500 Sugar 100 g Calcium 700 mg Cost Calories Calciu m Food Sugar Stir-fry (per oz) 1 100 3 20 Notation Alert! Boba (per fl oz) 0.5 50 4 70

10. Optimization Formulation Diet Problem min ? s.t. ???????? ? ????????? ????? ? ????? calcium ? ????? ????(?) Healthy Squad Goals 2000 Calories 2500 Sugar 100 g Calcium 700 mg Cost Calories Food Sugar Calcium Stir-fry (per oz) 1 100 3 20 Boba (per fl oz) 0.5 50 4 70

11. Optimization Formulation Diet Problem min ?1, ?2 s.t. 100 ?1+ 50 ?2 2000 100 ?1+ 50 ?2 2500 3 ?1+ 4 ?2 100 20 ?1+ 70 ?2 700 1 ?1+ 0.5 ?2 Healthy Squad Goals 2000 Calories 2500 Sugar 100 g Calcium 700 mg Cost Calories Food Sugar Calcium Stir-fry (per oz) 1 100 3 20 Boba (per fl oz) 0.5 50 4 70 Notation Alert!

12. Optimization Formulation Diet Problem min ?1, ?2 s.t. ?1,1 ?1+ ?1,2 ?2 ?1 ?2,1 ?1+ ?2,2 ?2 ?2 ?3,1 ?1+ ?3,2 ?2 ?3 ?4,1 ?1+ ?4,2 ?2 ?4 ?1 ?1+ ?2 ?2 Cost 1 ? = 0.5 Limit 2000 2500 100 700 Calorie min Calorie max Sugar Calcium 100 100 3 20 50 50 4 70 ? = ? = Notation Alert!

13. Optimization Formulation Diet Problem ??? min ? s.t. ?1,1 ?1+ ?1,2 ?2 ?1 ?2,1 ?1+ ?2,2 ?2 ?2 ?3,1 ?1+ ?3,2 ?2 ?3 ?4,1 ?1+ ?4,2 ?2 ?4 Cost 1 ? = 0.5 Limit 2000 2500 100 700 Calorie min Calorie max Sugar Calcium 100 100 3 20 50 50 4 70 ? = ? = Notation Alert!

14. Optimization Formulation Diet Problem ??? min ? s.t. ?1,1 ?1 ?1,2 ?2 ?1 ?2,1 ?1+ ?2,2 ?2 ?2 ?3,1 ?1+ ?3,2 ?2 ?3 ?4,1 ?1 ?4,2 ?2 ?4 Cost 1 ? = 0.5 Limit 2000 2500 100 700 Calorie min Calorie max Sugar Calcium 100 100 3 20 50 50 4 70 ? = ? =

15. Optimization Formulation Diet Problem ??? min ? s.t. ?1,1 ?1+ ?1,2 ?2 ?1 ?2,1 ?1+ ?2,2 ?2 ?2 ?3,1 ?1+ ?3,2 ?2 ?3 ?4,1 ?1+ ?4,2 ?2 ?4 Cost 1 ? = 0.5 Limit 2000 2500 100 700 Calorie min Calorie max Sugar Calcium 100 100 3 20 50 50 4 70 ? = ? =

16. Optimization Formulation Diet Problem ??? min ? s.t. ?? ? Cost 1 ? = 0.5 Limit 2000 2500 100 700 Calorie min Calorie max Sugar Calcium 100 100 3 20 50 50 4 70 ? = ? = Notation Alert!

17. Poll 1 What has to increase to add more nutrition constraints? ??? min ? s.t. ?? ? Select all that apply A) length ? B) length ? C) height ? D) width ? E) length ?

18. Poll 1 What has to increase to add more nutrition constraints? ??? min ? s.t. ?? ? 2000 2500 100 700 100 100 3 20 50 50 4 70 ?1 ?2 1 ? = ? = ? = ? = 0.5

19. Poll 2 What has to increase to add more menu items? ??? min ? s.t. ?? ? Select all that apply A) length ? B) length ? C) height ? D) width ? E) length ?

20. Poll 2 What has to increase to add more nutrition constraints? ??? min ? s.t. ?? ? 2000 2500 100 700 100 100 3 20 50 50 4 70 ?1 ?2 1 ? = ? = ? = ? = 0.5

21. Poll 3 If ? ? ?, which of the following also equals ?? ??? min ? s.t. ?? ? Select all that apply A) length ? B) length ? C) length ? Notation Alert!

22. Linear Programming Linear objective with linear constraints ??? min ?. s.t. ?? ? As opposed to general optimization min ?. s.t. ??? 0, ? = 1 ? ?? ?0? ?? = ??, ? = 1 ?

23. Linear Programming Different formulations Inequality form General form Standard form ??? ??? + ? ??? min ?. s.t. ?? ? min. ? s.t. ?? ? ?? = ? min. ? s.t. A? = ? ? 0 Important to pay attention to form!

24. Linear Programming Different formulations Inequality form General form Standard form ??? ??? + ? ??? min ?. s.t. ?? ? min. ? s.t. ?? ? ?? = ? min. ? s.t. A? = ? ? 0 Can switch between formulations!

25. Optimization Graphical Representation Problem Description Optimization Representation ??? min ? s.t. ?? ?

26. Graphics Representation Geometry / Algebra I Quiz What shape does this inequality represent? ?1 ?1+ ?2 ?2 ?1

27. Graphics Representation Geometry / Algebra I Quiz What shape does this inequality represent? ?1 ?1+ ?2 ?2= ?1 ?1 ?1+ ?2 ?2 ?1 ?1,1 ?1+ ?1,2 ?2 ?1 ?2,1 ?1+ ?2,2 ?2 ?2 ?3,1 ?1+ ?3,2 ?2 ?3 ?4,1 ?1+ ?4,2 ?2 ?4

28. Poll 4 What is the relationship between the half plane: ?1 ?1+ ?2 ?2 ?1 and the vector: ? ?1,?2 ?2 Infeasible II I Feasible III IV ?1

29. Poll 4 ? and initial point ?(0), Given the cost vector ?1,?2 Which unit vector step ?will cause ?(1)= ?(0)+ ? to have the lowest cost ???(1)? II I ? III IV Notation Alert!

30. Cost Contours ? where will Given the cost vector ?1,?2 ??? = 0 ? ??? = 1 ? ??? = 2 ? ??? = -1 ? ??? = -2 ?

31. Poll 5 As the magnitude of ? increases, the distance between the contours lines of the objective ???: A) Increases B) Decreases