Teaching a Traditional Optimization Course in a non-Traditional Manner James B. Orlin

1. Teaching a Traditional Optimization Course in a non-Traditional Manner James B. Orlin Mike Metzger MIT November 6, 2006

2. Part I: Introduction Quote of the day. If you treat individuals as they are they will remain as they are, but if you treat them as if they were what they ought to be and could be, they will become what they ought to be and could be. • Johann Wolfgang von Goethe.

3. Mike Trick Optimization Methods in Management Science I’ve scheduled league games for a major basketball conference. There are an incredible number of constraints, and it is very hard to solve. • Models, applications, and algorithms for optimization within Management Science. • finance, manufacturing, transportation, timetabling, games, operations, e-business, and more. • Linear programming, integer programming, network optimization, and decision analysis.

4. Syllabus

5. Our Goals Today • Lots of pedagogical advances over the years. • lot of description at INFORMS meetings • Focus on a range of techniques • visual learning, • active learning • interactive learning • use of technology • and more • End result: • students like the material better • more interested in follow on subjects • superior mastery of the material over students in previous years

6. Overview of the Structure of the Class • 2 Lectures per week per section (1.5 hours, required) • 1 small morning section, one large afternoon section • MIT students are used to 1 hour per lecture • Required recitation • weekly quizzes • Nearly weekly homework • 2 midterms and a final (and a project) • 1 professor, 3 TAs. 150 students

7. Structure of a Lecture • Intro • quotes • announcements • First half of class • Mental break • Second half of class • game to review the class material

8. Time line for 15.053 1995: Algorithm based 1996: taught 15.053 as an Excel-based class 1999: taught 15.053 as an algorithms based class 2001: introduced PowerPoint into classes; Mike Metzger was a student in the class

9. 15.053 class notes CLASS Time line continued 2004: taught both sections of 15.053 in the spring. Mike Metzger was one of my TAs. Hired undergrad graders. TAs helped develop educational materials 2005: introduced required recitations. 2007. dropped required text and replaced with our own notes

10. Part II: Pedagogy Imagination is more important than knowledge... • Albert Einstein You don't understand anything until you learn it more than one way. • Marvin Minsky

11. Who wants a piece of piece of candy? The Jetson’s diet problem. TV’s hit quiz show comes to 15.053 Overview of Pedagogy • Pedagogy has evolved from straight lecturing • Techniques adopted • Engaging Model Formulations • Algorithm animation using PowerPoint • Use of games in class • Active learning (talk to your neighbor)

12. A diet problem and its dual • Jane Jetson is trying to design a diet for George on her home food dispenser so as to satisfy nutritional requirements and minimize cost. Theme music

13. Spacely Sprockets has decided to make food pellets. Mr. Spacely wants to win over Jane Jetson by making, carbohydrate, fat, and protein pellets. How should he price them to win over Jane and maximize profit? Choose Prices

14. Discuss with partners • Usual: • left of the class solves Jane’s primal problem as best as possible by inspection • right half of the class solves Spacely’s dual problem by inspection • we collect answers from both, and observe that the primal opt is >= dual opt. • Do you have any other ways of getting students to understand the intuition behind weak duality? • (2 minutes)

15. Algorithm Animation • Powerpoint has lots of animation tools that are useful for illustrating concepts. Colors and transitions can simulate movement. Boxes can appear, and disappear. And one can use real movement. And pictures can help.

16. 5 4 3 2 1 1 2 3 4 5 6 2 Variable Simplex Illustrated Maximizez = 3 K + 5 S Start at any feasible corner point. Move to an adjacent corner point with better objective value. Continue until no adjacent corner point has a better objective value. S K

17. 5 4 3 2 1 1 2 3 4 5 6 An Alternative Representation Introducing: LPac Man S K

18. z z x1 x2 x3 x4 x1 x3 x4 1 1 1 3 -3 2 -2 0 0 0 0 0 = 3 0 0 0 0 0 -3 -3 3 3 1 1 0 0 6 = = -3 1 0 0 0 0 -4 -4 2 2 0 0 1 1 2 -4 0 1 Algorithm Animation: Simplex Pivoting Non-basic variable x2 becomes basic. Choose column 2. Basic variable x4 becomes non-basic. It was the basic variable in Constraint 2. 2 z = 0, x1 = 0, x2 = 0, x3 = 6, x4 = 2. Pivot on the 2. Choose constraint 2.

19. z 1 -1 0 0 1 2 0 3 0 1 -1.5 3 0 Pivoting to obtain a better solution New Solution: basic variables z, x2 and x3. Nonbasics: x1 and x4. -z x1 x2 x3 x4 1 3 -2 0 0 0 = 0 -3 3 1 0 6 = = 0 -2 -4 1 2 2 0 0 .5 1 2 1

20. What is the maximum number of diamonds that can be packed on a Chinese checkerboard. class exercise: how many can you pack? Each diamond covers 4 dots.

21. Here is a best possible

22. 10 9 11 5 7 6 8 12 2 3 4 1 For each circle, you can select at most one diamond incident to the circle. 5 slides later

23. An Excellent Integer Program Let S(i) be the diamonds that are incident with circle i of the Chinese Checkerboard. zLP = 27.5 And we found a solution by hand with zi = 27

24. Representing the Dual Problem Assign each circle (or node) a “weight.” The weight of a diamond d is w(d) = the sum of its node weights. The dual linear program: choose weights for each circle so that w(d) ≥ 1 for each diamond d. Minimize the weights of the circles. Class exercise: assigning ¼ to each circle gives a total weight of 30.25. Can you do better?

25. Node Weight 18 1 1 24 1/2 6 1/3 1/6 0 Each diamond has weight at least 1. The number of diamonds is at most 27.

26. Who wants a piece of CANDY? TV’s hit quiz show comes to 15.053

27. It helps students focuson what is important It keeps students engaged It is effective for students who are visual thinkers A, B, and C.

28. It is effective for students who are visual thinkers A, B, and C.

29. A, B, and C. Next question

30. A, B, and C. Next question

31. Which valid constraint should be added to the integer programming formulation of the TSP to eliminate this solution. 1 4 7 2 5 8 3 6 9 Both A and B Neither A nor B

32. Which valid constraint should be added to the integer programming formulation of the TSP to eliminate this solution. 1 4 7 2 5 8 3 6 9 Both A and B

33. Which valid constraint should be added to the integer programming formulation of the TSP to eliminate this solution. 1 4 7 2 5 8 3 6 9 Last Slide

34. Which valid constraint should be added to the integer programming formulation of the TSP to eliminate this solution. 1 4 7 2 5 8 3 6 9 Last Slide

35. Mental Break: Impossibilities Fooling around with alternating current is a waste of time. Nobody will use it, ever. Thomas Edison Rail travel at high speed is not possible because passengers, unable to breathe, would die of asphyxia. Dr. Dionysus Lardner, 1793-1859 Inventions have long since reached their limit, and I see no hope for future improvements. Julius Frontenus, 10 A. D.

36. Recitations, Tutorials, Homework, Exams, Etc.The role of the Head TA

37. Structure • Quiz • Statistical Break • Lesson • Summary

39. Quiz • Problem 1 (4 points total; 2 points each) • If exactly one of these points is optimal, which point(s) could it be? • If point B is optimal, must any other of these point(s) be optimal too? If so, which one(s)?

40. Statistical Break • Break Between Quiz and Lesson • It takes 5 minutes for them refocus • Interesting statistics! • Examples: • Favorite Airline • Exam Scores • Have You Heard of Welcome Back Kotter

42. Yes. 10 No. 90 Have you heard of Welcome Back Kotter

43. Recitation-”The Dual” • Consider the Following Statements • Try to Determine if they Are True or False • It is possible for an LP to have exactly 2 optimal solutions • Two different bases (and thus two different tableaus) can have the same corresponding basic feasible solution • We can have negative z-row coefficients and still be at an optimal solution.

45. Pivot- Geometry

46. Applications (Before) Ebel Mining Company owns two different mines that produce a given kind of ore. After crushing, ore is one of three grades: high, medium or low. Ebel must provide the parent company with at least 12 tons of high grade, 8 tons of medium grade, and 24 tons of low-grade ore per week. It costs $20,000 a day to run the first mine and $16,000 a day to run the second mine. In one day each mine produces the following tonnage:

47. Applications After. Dear 15.053 Class, My name is Jessica Simpson. I have been going through some tough times recently and am having a real problem with one of my cosmetic lines. The info for the line is on the next page. Recently though costs are changing based on market demand in addition to highly fluctuating resource costs. My problem is this we currently have an LP that we solve to find the optimal amount to produce of each product. However, every time a parameter changes, I am always forced to resolve the LP and this takes too long. I was hoping you guys could find a better way. Lately I have just been out of it. For example, Nick and I decided to split our Hummer in half, and now I need to buy a new one. Ohh yea, about the LP it seems to have been misplaced when I was moving out of my Malibu house. Please Help! - Jessica

48. Interaction • Video clip • More about video and educational technology later

49. Initial Concerns • Students bored in recitation • Rebellion to required recitation • Against MIT Grain

50. What We Found • Main Lesson • By putting out a good product, students didn’t complain. • If they are learning, they will be happy • The Fun Factor helped out • You can’t please everyone • But you can please 95%