AM 121: Intro to Optimization Models and Methods

1. AM 121: Intro to OptimizationModels and Methods Lecture 2: Intro to LP, Linear algebra review. Lecture 1: Course Overview Ozlem Ergun SEAS/Georgia Tech TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

2. Who is Ozlem?Visiting Professor at Applied MathIn normal life Associate Professor and Director of Center for Health and Humanitarian Logistics Industrial and Systems Engineering at Georgia Tech Methodological interests: Network Flows/Combinatorial Optimization/ Decentralized Decision Making/Mechanism Design/Collaborative Game Theory Applied to: Transportation & Logistics/Supply Chain Network Design/Humanitarian & Emergency Management Operations/Societal Impact

3. Lecture 1: Lesson Plan • Overview of topics (with motivating examples) • Linear Programming • Integer Programming • Stochastic Programming • Review syllabus, schedule

4. Mathematical Programming • Set of methods to model and solve optimization problems with continuous or integer variables • Programming ≠ computer programming • Programming = Planning

5. Mathematical Programming Real-world problems • We emphasize modeling • Methods with elegant math • Important applications … Model LP IP MDP Method Branch-and-Bound Simplex

6. Prominent Applications • Optimal crew scheduling saves American airlines 20 million dollars per year • Traffic control of Hanshin Expressway in Osaka, Japan saves 17 million driver hours per year • Google pageRank– a tool for indexing the Web –is defined by a Markov chain. • Optimal sourcing saves Procter & Gamble 100 million dollars per year. • Many advances in medical decision making and robot navigation rely on the Markov decision process framework. • Clearing Nationwide Kidney Exchanges, potentially saving hundreds of lives.

7. Campaign stops: The problem of the travelling politicianBy WILLIAM COOK (Georgia Tech) • What's the quickest way to hit every county in Iowa? • http://campaignstops.blogs.nytimes.com/2011/12/21/the-problem-of-the-traveling-politician/?emc=eta1

8. Solution

9. Yale Study Reports Clean Needle Project Helps Check AIDSBy MIREYA NAVARRO on ED KAPLAN’s work (Yale) • A program here to give intravenous drug users new needles for old ones is helping to curb the spread of the AIDS virus, a new study says. • The study, released today by Yale University researchers and the New Haven Health Department, estimates that the eight-month-old needle-exchange program has reduced new infections with HIV, the virus that causes AIDS, by 33 percent among participants. The sharing of contaminated needles is the major way that the virus is transmitted in New Haven, New York City and other urban areas. • http://www.nytimes.com/1991/08/01/nyregion/yale-study-reports-clean-needle-project-helps-check-aids.html?scp=1&sq=Kaplan%20%22needle%20exchange%22&st=cse

10. Got Toxic Milk?By LAWRENCE M. WEIN (Stanford) • In general, two threats are viewed as the most dangerous: anthrax, which is as durable as it is deadly, and smallpox, which is transmitted very easily and kills 30 percent of its victims. • But there is a third possibility that, while it seems far more mundane, could be just as deadly: terrorists spreading a toxin that causes botulism throughout the nation's milk supply. • http://www.nytimes.com/2005/05/30/opinion/30wein.html?ex=1162357200&en=6bbd4ad7194d987c&ei=5070

11. Haiti’s Eternal WeightBy REGINALD DESROCHES, OZLEM ERGUN and JULIE SWANN (Georgia Tech) • IT has been six months since the earthquake in Haiti left more than 300,000 people dead and destroyed 280,000 homes and businesses. Haiti still faces a long road to recovery, but one of the biggest things literally standing in its way is earthquake debris. • According to our research and conversations with aid groups in Haiti, less than 5 percent of this has been removed since January, and even less has been properly disposed of. A draft of the United States Army Corps of Engineers’ debris management plan says it would take a dump truck with a 20-cubic-yard bed 1,000 days to clear the debris, if it carried 1,000 loads a day — or about three years. But the current rate of removal is much lower. Based on our calculations, partially from the United States Agency for International Development’s reports on debris removal programs, we estimate that it could take 20 years or more. • http://www.nytimes.com/2010/07/08/opinion/08desroches.html

12. Rough Schedule • 6 weeks on linear programming • 4 weeks on integer programming • 2 weeks stochastic optimization • Several examples of applications • game theory, extreme optimization • Several guest lectures on cool applications!

13. Optimization • The problem of making decisions to maximize or minimize an objective, maybe in the presence of complicating constraints • Decision variables • Objective function • Constraints

14. A Very Simple Optimization Problem • minimize y = (x-2)2 +3 • minimize y = (x-2)2 +3 s.t. x ≥ 3 In this course, we focus on linear models.

15. A Linear Program x1 and x2 : variables

16. How would you establish the optimality of this algebraically?

17. Distributing Goods through a Distribution Network

18. Distributing Goods through a Distribution Network • Decision variables: Amount to ship on seven shipping lanes, xF1-F2, xF1-DC, xF1-w1, xF2-DC, xDC-W2, xW1-W2, xW2-W1. • Objective: Minimize the total shipping cost • Constraints • Amount shipped out – amount shipped in = required amount (flow balance) • Amount shipped on a shipping lane does not exceed its capacity (capacity) • Amount shipped on any shopping lane is nonnegative (non-negativity)

19. Produce Animal Feed Mix(UN WFP optimal aid food baskets) • Produce the feed mix that satisfy the nutrition requirements with minimal cost • The mixture must meet: • Calcium: at least 0.8% but not more than 1.2% • Protein: at least 22% • Fiber: at most 5% • Nutrient Contents and Costs of Ingredients

20. What we’ll cover for LP • Modeling: Given a problem, formulate the linear programming mathematical model. • Solve the formulated LPs • Simplex method • Sensitivity analysis • Duality

21. Integer Programming (IP) • Decision variables can only take integer values • Linear programming model + integrality constraints • Eg. max 5x1 + 8x2 s.t. x1 + x2 ≤ 6 5x1 + 9x2 ≤ 45 x1, x2 ≥ 0, integer

22. LP vs. IP • LP • Continuous decision variables • Potentially larger feasible space • IP • Discrete decision variables – have to take integer values • Potentially smaller feasible space • Generally harder to solve • The natural idea of rounding LP solution for IP does not work in general.

23. (376/193,950/193) LP solution 5 4 3 2 1 (5,0) IP solution 0 0 1 2 3 4 5

24. Sudoku

25. Sudoku • Decision variables • : equals 1 if cell at row i and column j has value v, otherwise 0. • Constraints • Each row must have one 1 … 9 • Each column must have one 1 … 9 • Each 3-by-3 section must have one 1 … 9 • Each cell must have a value from 1 … 9 • Objective • Anything

26. The Assignment Problem • There are n people available to carry out n jobs • Each person has to be assigned to exactly one job • Person i if assigned to job j incurs cost cij • Goal: find a minimum cost assignment

27. What we’ll cover for IP • Modeling: Given a problem, formulate the linear programming mathematical model. • Solve the formulated IPs • Branch-and-Bound • Cutting plane • Branch-and-cut • advanced solution techniques

28. Introducing Uncertainty • Both LP and IP are deterministic, with known model parameters • But many real world problems have uncertainty at the time a decision needs to be made • Stochastic programming is an approach for modeling optimization problems that involve uncertainty

29. What we’ll cover for stochastic programming • Markov chains (to model the underlying stochastic process) • Markov decision processes • Two-stage stochastic optimization

30. Markov Chains in One Slide • A stochastic process ( a sequence of random variables) that has Markov property • Markov property: state of the process at time t+1 only depends on the state at time t (history does not matter!) • Eg. Get up time

31. Markov Decision Processes (MDP) • Rather than passively observing the Markov chain, we can proactively make a decision about which action to take at a state to affect the transition probability • The decision process of choosing the optimal action is referred as a Markov Decision Process

32. A MDP Example: Machine Maintenance Goal: find a maintenance plan that minimizes cost

33. What to expect after the course • Be able to transform a real world problem into a mathematical programming model • Master the techniques to solve the formulated mathematical programming model • Be able to solve the problem using commercial software packages, CPLEX/AMPL • Feel empowered!

34. Please return the completed survey after today’s class! • Ozlem has office hours Wednesday at 10:00-11:00 in MD 111A.

35. Syllabus • Prerequisite: AM 21b or M21b will help with the linear algebra in the first 1/3 of the course • Recommended Text: “Operations Research: Models and Methods”, by Jensen and Bard. • Optional text: “AMPL”, by Fourer, Gay and Kernighan • TFs: Chris Coey, Xiao Cong, Jack Greenberg, Lily Hsiang, and David Zhang • Attend sections! • A lot going on at the beginning of the course! • Bulletin board, feedback form! • News on the course website (can subscribe RSS)

36. Syllabus • Weekly problem sets (mixture of modeling, computational exercises, and theory). Encourage use of Maple/ Matlab • Two “Extreme optimization” team assignments • what’s that? (Extreme values: communication, simplicity, feedback, courage, respect) • 2 out-of-class evening quizzes; No final. • Grade breakdown is (roughly) 50 % problem sets, 15% extreme optimization, 35% quizzes