2. 1 Teaching a Traditional Optimization Course in a non-Traditional Manner James B. Orlin Mike Metzger MIT November 6, 2006

3. 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.

4. 3 Optimization Methods in Management Science 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. Mike Trick I’ve scheduled league games for a major basketball conference. There are an incredible number of constraints, and it is very hard to solve.

5. 4 Topic Introduction Applications of Linear and Non-Linear Programming Geometry of LP Simplex Method 1 Simplex Method 2 Sensitivity Analysis LP Duality 1 LP Duality 2 Game Theory Introduction to Networks Maximum Flows. Min Cost Flows The Network Simplex Algorithm IP Models More IP Models Branch and Bound Cutting Planes Decision Trees 1 Decision Trees 2. Syllabus

6. 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

7. 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

8. 7 Structure of a Lecture Intro –quotes –announcements First half of class Mental break Second half of class game to review the class material

9. 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

10. 9 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 CLASS 15.053 class notes

11. 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

12. 11 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) The Jetson’s diet problem. Who wants a piece of piece of candy? TV’s hit quiz show comes to 15.053

13. 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. Space Burgers Saturn Shakes Venus Flies Requirements Carbs 1409 Fat 4208 Protein 41212 Cost (in $100s) 662 Theme music

14. 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? Upper bound CarbsFatProtein on Cost Space Burgers1446 Saturn Shakes4215.5 Venus Flies0022 Choose Prices

15. 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)

16. 15 Algorithm Animation Powerpoint has lots of animation tools that are useful for illustrating concepts. Boxes can appear, and disappear. Colors and transitions can simulate movement. And one can use real movement. And pictures can help.

17. 16 2 Variable Simplex Illustrated 123456 1 2 3 4 5 K 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. Maximize z = 3 K + 5 S

18. 17 An Alternative Representation 123456 1 2 3 4 5 K S Introducing: LPac Man

19. 18 Algorithm Animation: Simplex Pivoting 2001 310 201 0 0 310 201 0 0 -2001 ==== 2 6 x1x1 x2x2 x4x4 x3x3 z = 0 10 01 00 x4x4 x3x3 1 0 0 z -3 -4 -3 -4 3 x1x1 -3 -4 3 z = 0, x 1 = 0, x 2 = 0, x 3 = 6, x 4 = 2. Non-basic variable x 2 becomes basic. Basic variable x 4 becomes non-basic. It was the basic variable in Constraint 2. Choose column 2. Choose constraint 2. Pivot on the 2. 2

20. 19 2 x1x1 -3310 -401 Pivoting to obtain a better solution ==== 2 6 2 x2x2 x4x4 x3x3 3-200 -z 0 0 1 = 0 -210.51 0012301-1.53 New Solution: basic variables z, x 2 and x 3. Nonbasics: x 1 and x 4. 1 0 0 z

21. 20 What is the maximum number of diamonds that can be packed on a Chinese checkerboard. Each diamond covers 4 dots. class exercise: how many can you pack?

22. 21 Here is a best possible

23. 22 2 34 1 5 76 8 9 10 11 12 For each circle, you can select at most one diamond incident to the circle. 5 slides later

24. 23 An Excellent Integer Program Let S(i) be the diamonds that are incident with circle i of the Chinese Checkerboard. z LP = 27.5 And we found a solution by hand with z i = 27

25. 24 Representing the Dual Problem The dual linear program: choose weights for each circle so that w(d) ≥ 1 for each diamond d. Minimize the weights of the circles. Assign each circle (or node) a “weight.” The weight of a diamond d is w(d) = the sum of its node weights. Class exercise: assigning ¼ to each circle gives a total weight of 30.25. Can you do better?

26. 25 Node Weight 1 1/2 1/3 1/6 0 Each diamond has weight at least 1. 18 1 24 6 The number of diamonds is at most 27.

27. 26 Who wants a piece of CANDY? TV’s hit quiz show comes to 15.053

28. Who wants another piece of candy 50-50 15.053 AB CD What is the advantage of using algorithm animation? 27 It keeps students engaged It helps students focus on what is important It is effective for students who are visual thinkers A, B, and C.

29. Who wants another piece of candy 50-50 15.053 AB CD What is the advantage of using algorithm animation? 28 It is effective for students who are visual thinkers A, B, and C.

30. Who wants another piece of candy 50-50 15.053 AB CD What is the advantage of using algorithm animation? 29 Next question A, B, and C.

31. Who wants another piece of candy 50-50 15.053 AB CD What is the advantage of using algorithm animation? 30 Next question A, B, and C.

32. Who wants another piece of candy 50-50 15.053 AB CD What is the advantage of using algorithm animation? 31 Both A and BNeither A nor B Which valid constraint should be added to the integer programming formulation of the TSP to eliminate this solution. 1 23 4 56 7 89

33. Who wants another piece of candy 50-50 15.053 AB CD What is the advantage of using algorithm animation? 32 Which valid constraint should be added to the integer programming formulation of the TSP to eliminate this solution. 1 23 4 56 7 89 Both A and B

34. Who wants another piece of candy 50-50 15.053 AB CD What is the advantage of using algorithm animation? 33 Last Slide Which valid constraint should be added to the integer programming formulation of the TSP to eliminate this solution. 1 23 4 56 7 89

35. Who wants another piece of candy 50-50 15.053 AB CD What is the advantage of using algorithm animation? 34 Last Slide Which valid constraint should be added to the integer programming formulation of the TSP to eliminate this solution. 1 23 4 56 7 89

36. 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.

37. 36 Recitations, Tutorials, Homework, Exams, Etc. The role of the Head TA

38. 37 Structure Quiz Statistical Break Lesson Summary

39. 38

40. 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)?

41. 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. 41

43. 42 No. 90 Yes. 10 Have you heard of Welcome Back Kotter

44. 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. 44

46. 45 Pivot- Geometry

47. 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:

48. 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

49. 48 Interaction Video clip More about video and educational technology later

50. 49 Initial Concerns Students bored in recitation Rebellion to required recitation –Against MIT Grain

51. 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%

52. 51 Office Hours Structure –Twice per week for two hours each –Attendance can be as high as 60 Students –One TA leads each session Team teaching is a new approach –Small vs. large session –Goal: To help students understand the material on the homework sets –Management style

53. 52 Example According to recent reports in US Weekly, after 12 consecutive nights of partying, Jessica has decided to leave Nick. Currently they are trying to determine how to optimally divide some of their possessions. They hire “you” to try and help them accomplish this task. To start, you ask them to assign “value points” to all of the items under consideration such that they total 100. Their allocation is shown in the table below.

54. 53 The Use of Excel The main question –How to balance Excel Vs. Theory Students: –Have disliked Excel –However, after completing our course often say it is the most useful tool we taught.

55. 54 The Use of Excel Staff –Hard to grade Excel –Very important for Management students A new approach –Take Excel and separate from problem sets and Use as a case study –To be tried in spring 2007

56. 55 Our Idea The Diet Problem as an ongoing example and case study –Formulate your problem –Solve using simplex –Take dual and interpret as Grocer’s Problem –Study distribution costs using networks –Look into decision modeling with stochastic supply

57. 56 Modeling and Abstract modeling More emphasis on modeling abstract scenarios Building managerial skills while building on theory Framing the problem Min  c ij x ij s.t.  a ij x j <= b i for all i x >= 0

58. 57 Exams-Testing the “Why” not “How” JB was performing the simplex method; however, at exactly one intermediary step, he made the following error: instead of pivoting in a variable with a negative objective row coefficient, he instead pivoted in a variable a with a positive objective row coefficient. He chooses the exiting variable using the usual minimum ratio test. Assume after this initial error JB continues to solve the problem using the simplex method and does not make any additional errors. Also assume that no basis is degenerate.

59. 58 Testing the “why” not “how” It is possible that the pivot that JB took will result in a basic solution that is not feasible. It is possible that the pivot JB took will result in an optimal basis. It is possible that there are fewer simplex pivots with the error than if the error had not been made.

60. 59 Exam Management Testing depth of knowledge without penalizing students who understand the basics Our Solution: Short Answer Questions –Say we have an LP with only one variable. Which of the following could be the set of all feasible solutions? i) A single point, e.g. 2 ii) A line segment, e.g. [2, 4] iii) Two disjoint line segments, e.g, [2, 3]  [6, 7]. iv) A ray or infinite half line, e.g. [2, ∞)

61. 60 Suppose we have a primal-dual pair where each has a finite optimal solution. Say we remove the ith constraint for the primal. What can you say about the dual corresponding to this new problem? –i)The dual could be infeasible. –ii)The dual could still have an optimum solution. –iii)The dual could be unbounded.

62. 61 Use of Technology Three recent additions to the subject –Online PowerPoint reviews –Homework solution podcasts –Online videos of recitations and reviews

63. 62 Online PowerPoint Reviews An alternative to standard textbook readings –Solution: Online Tutorials Topics: –Abstraction of LP models –Solving systems of equations –Converting an LP to standard form –Degeneracy –Sensitivity analysis using Excel –Integer programming formulations

64. 63 With the Assistance of Some Friends

65. 64 Degeneracy in Linear Programming Tim, the turkey I heard that today’s tutorial is all about Ellen DeGeneres Ellen DeGeneres Sorry, Tim. But the topic is just as interesting. It’s about degeneracy in Linear Programming. Degeneracy? Students at MIT shouldn’t learn about degeneracy. And I heard that 15.053 students have already studied convicts’ sex. Reverend Jerry Falwell Actually, they studied convex sets.

66. 65 What is degeneracy? As you know, the simplex algorithm starts at a corner point and moves to an adjacent corner point by increasing the value of a non-basic variable x s with a negative value in the z-row (objective function). Typically, the entering variable x s does increase in value, and the objective value z improves. But it is possible that that x s does not increase at all. It will happen when one of the RHS coefficients is 0. In this case, the objective value and solution does not change, but there is an exiting variable. This situation is called degeneracy. x1x1 -3310 -4201 ==== 2 6 x2x2 x4x4 x3x3 3-200 z 0 0 1 = 2 1 00 A basic feasible solution is called degenerate if one of its RHS coefficients (excluding the objective value) is 0. This bfs is degenerate.

67. 66 Nooz, the most trusted name in fox. Great. I now know what degeneracy is. Now we can move on to other matters. Ollie, the computationally wise owl. No, Nooz**. This tutorial has many more slides. Degeneracy adds complications to the simplex algorithm. And if you understand what occurs under degeneracy, you really understand what is going on with the simplex algorithm. ** As you know, “No, Nooz” is good news.” Cleaver Professor Orlin apologizes for this bad pun, but feels that he could not resist the temptation.

68. 67 PodCast’s MIT Project Launched in Spring 2006 –15.053 selected as launch course for homework solutions via podcasts –Over 85% reported using videos on regular basis –Turned HW Solutions into tutorials –Review Session 2 Set the current MIT PodCast download record in a single night Crashed Server!

69. 68 PodCast’s Show Example