February 1, 2005

Transcription

1 February 1, 2005 Introduction to Optimization Handouts: Lecture Notes Info Sheet 1-page subject description Note: no laptops in class. Note: is not a prerequisite for this semester. However, we will assume that you are comfortable in working with matrices and in solving systems of equations (the level of matrix operations assumed in 18.02) 1

2 Introductions James Orlin Professor of Operations Research at Sloan Co-director of the MIT Operations Research Center 2

3 Course Administration Goals for this Lecture Background on Operations Research (Management Science) and Optimization Course Themes and Goals Linear Programming Examples MSR Marketing DTC Some time to choose a homework partner and fill in a info sheet 3

4 Required Materials Introduction to Operations Research (4 th edition) by Wayne Winston Class Website: - - It contains subject information (which is accessible from web.mit.edu/jorlin/www/) Assignment 1 will be due next Thursday 4

5 Grading Policy Homework assignments (25%) Midterm Exams (40%: 20% each) March 10 th, 7:30 to 9:30 PM April 21 st, 7:30 to 9:30 PM Final Exam (20%) During Finals week Recitation quizzes (16%) Extra credit mini-project (5%) 5

6 Active Learning Occasionally I will introduce a break in the lecture for you to work on your own or with an in-class partner. Cognitive balancing. Please identify your partner now. Those on aisle ends may be in a group of size 3. 6

7 What is Operations Research? What is Management Science? World War II : British military leaders asked scientists and engineers to analyze several military problems Deployment of radar Management of convoy, bombing, antisubmarine, and mining operations. The result was called Operations Research MIT was one of the birthplaces of OR Professor Morse at MIT was a pioneer in the US Founded MIT OR Center and helped to found ORSA 7

8 What is Management Science (Operations Research)? Operations Research (O.R.) is the discipline of applying advanced analytical methods to help make better decisions. It is the Science of Better 8

9 Voices from the past Waste neither time nor money, but make the best use of both. -- Benjamin Franklin Obviously, the highest type of efficiency is that which can utilize existing material to the best advantage. -- Jawaharlal Nehru It is more probable that the average man could, with no injury to his health, increase his efficiency fifty percent. -- Walter Scott 9

10 Operations Research Over the Years 1947 Project Scoop (Scientific Computation of Optimum Programs) with George Dantzig and others. Developed the simplex method for linear programs. 1950's Lots of excitement, mathematical developments, queuing theory, mathematical programming. 1960's More excitement, more development and grand plans. 10

11 Operations Research Over the Years 1970's Disappointment, and a settling down. NPcompleteness. More realistic expectations. 1980's Widespread availability of personal computers. Increasingly easy access to data. Widespread willingness of managers to use models. 1990's Improved use of O.R. systems. Further inroads of O.R. technology, e.g., optimization and simulation add-ins to spreadsheets, modeling languages, large scale optimization. More intermixing of A.I. and O.R. 11

12 Operations Research in the 2000 s LOTS of opportunities for OR as a field Data, data, data E-business data (click stream, purchases, other transactional data, and more) Sensor data The human genome project and its outgrowth Need for more automated decision making Need for increased coordination for efficient use of resources (Supply chain management) 12

13 Some Skills for Operations Researchers Modeling Skills Take a real world situation, and model it using mathematics Methodological Toolkit Optimization Probabilistic Models 13

14 Optimization As ageless as time Thanks to Tom Magnanti for these slides

15 Optimization in Nature Heron of Alexandria b Angle of Incidence Angle of Incidence b Angle of Reflection a First Century A.D

16 FERMAT Angle α 2 of Refraction Angle α 1 of Incidence

17 Calculus Maximum Minimum Fermat, Newton, Euler, Lagrange, Gauss, and more 17

18 Some of the themes of Optimization is everywhere Models, Models, Models The goal of models is insight not numbers paraphrase of Richard Hamming Algorithms, Algorithms, Algorithms 18

19 Optimization is Everywhere It is embedded in language, and part of the way we think. firms want to maximize value to shareholders people want to make the best choices We want the highest quality at the lowest price When playing games, we want the best strategy When we have too much to do, we want to optimize the use of our time etc. Take 3 minutes with your partner to brainstorm on where optimization might be used. (business, or sports, or personal uses, or politics, or ) Optimize the. or optimal. American Airlines Optimal crew scheduling saves $20 million/year Optimal yield management contributes $500 million per year Revenue management saves National Car rental $56 million in one year Better truck dispatching at Reynolds Metals improves on-time delivery and reduces annual freight costs by $7 million Improved shipment routing saves Yellow freight over $17.3 million/year GTE local access capacity expansion saves $30 million per year Optimizing its global supply chains saves Digital Equipment over $300 million Restructuring North America operations, Proctor and Gamble reduces plants by 20%, saving $200 million/year Taha Steel (India) maximizes production in response to power shortages contributing $73 million Improved production planning at Sadia (Brazil) saves $50 million over three years Production optimization at Harris Corporation improves on-time deliveries from 75% to 90% Optimal traffic control of the Hanshin Expressway in Osaka saves 17 million driver hours with benefits of $320 million Better scheduling of hydro and thermal generating units saves Southern Company $140 million 19

20 From Google Search Search phrase number of hits optimize the supply chain 12,000 optimize (the) return 16,490 optimal experience 39,800 optimal investment 57,000 optimal system 63,800 optimal decision 66,000 optimize your PC 81,000 optimal choice 163,000 optimal design 285,000 optimal health 382,000 20

21 On and Optimization Tools Some goals in : Present a variety of tools for optimization Illustrate applications in manufacturing, finance, e- business, marketing and more. Prepare students to recognize opportunities for mathematical optimization as they arise 21

22 Linear Programming minimize or maximize a linear objective subject to linear equalities and inequalities Example. Max is in a pie eating contest that lasts 1 hour. Each torte that he eats takes 2 minutes. Each apple pie that he eats takes 3 minutes. He receives 4 points for each torte and 5 points for each pie. What should Max eat so as to get the most points? Step 1. Determine the decision variables Let x be the number of tortes eaten by Max. Let y be the number of pies eaten by Max. 22

23 Max s linear program Step 2. Determine the objective function Step 3. Determine the constraints Maximize z = 4x + 5y (objective function) subject to 2x + 3y 60 (constraint) x 0 ; y 0 (non-negativity constraints) A feasible solution satisfies all of the constraints. x = 10, y = 10 is feasible; x = 10, y = 15 is infeasible. An optimal solution is the best feasible solution. The optimal solution is x = 30, y = 0. 23

24 Terminology Decision variables: e.g., x and y. In general, these are quantities you can control to improve your objective which should completely describe the set of decisions to be made. Constraints: e.g., 2x + 3y 24, x 0, y 0 Limitations on the values of the decision variables. Objective Function. e.g., 4x + 5y Value measure used to rank alternatives Seek to maximize or minimize this objective examples: maximize NPV, minimize cost 24

25 MSR Marketing Inc. adapted from Frontline Systems Need to choose ads to reach at least 1.5 million people Minimize Cost Upper bound on number of ads of each type TV Radio Mail Newspaper Audience Size 50,000 25,000 20,000 15,000 Cost/Impression $500 $200 $250 $125 Max # of ads

26 Formulating as a math model Work with your partner 2 minutes 1. The decisions are how many ads of each type to choose. Let x 1 be the number of TV ads selected. Let x 2, x 3, x 4 denote the number of radio, mail, and newspaper ads. These are the decision variables. 2. What is the objective? Express the objective in terms of the decision variables. 3. What are the constraints? Express these in terms of the decision variables. 4. If you have time, try to find the best solution. write on the board the decision variables the objective function and the constraints 26

27 David s Tool Corporation (DTC) Motto: We may be no Goliath, but we think big. Manufacturer of slingshots kits and stone shields. 27

28 Data for the DTC Problem Stone Gathering time Stone Smoothing Delivery time Slingshot Kits Stone Shields Resources 2 hours 3 hours 100 hours 1 hour 2 hours 60 hours 1 hour 1 hour 50 hours Demand Profit 3 shekels 5 shekels 28

29 To do with your partner Take two minutes to work with your partner to formulate the DTC problem as a linear program. Let K = number of slingshot kits made Let S = number of stone shields made 29

30 Formulating the DTC Problem as an LP Step 1: Determine Decision Variables K = number of slingshot kits manufactured S = number of stone shields manufactured Step 2: Write the Objective Function as a linear function of the decision variables Maximize Profit = Step 3: Write the constraints as linear functions of the decision variables subject to 30

31 The Formulation Continued Step 3: Determine Constraints Stone gathering: Smoothing: Delivery: Slingshot demand: Shield demand: We will show how to solve this in Lecture 3. 2K + 3S <= 100 K + 2S <= 60 K + S <= 50 K <= 40 S <= 50 K >= 0 and S >= 0 We ignore integrality constraints. 31

32 Addressing managerial problems: A management science framework 1. Determine the problem to be solved 2. Observe the system and gather data 3. Formulate a mathematical model of the problem and any important subproblems 4. Verify the model and use the model for prediction or analysis 5. Select a suitable alternative 6. Present the results to the organization 7. Implement and evaluate 32

33 How Problems get large, and what to do. Suppose that there are 10,000 products and 100 raw materials and processes that lead to constraints. New technique used: write an algebraic version of the model 33

34 An Algebraic Formulation n = number of items that are manufactured e.g., in the previous example, n = 2; m = number of resource constraints e.g., m = 3, {gathering, smoothing, and delivery.} x j = number of units of item j manufactured p j = unit profit from item j, e.g., p 1 = 3; Decision Variables Data for objective function d j = maximum demand for item j; e.g., b i = amount of resource i available a ij = amount of resource i used in making item j Data for constraints 34

35 An Algebraic formulation Maximize subject to n px j j j=1 n j= 1 ax bfor i= 1to m ij j i x d for j = 1to n j j x 0 for j = 1 to n j 35

36 Linear Programs A linear function is a function of the form: f(x 1, x 2,..., x n ) = c 1 x 1 + c 2 x c n x n = i=1 to n c i x i e.g., 3x 1 + 4x 2-3x 4. A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities. e.g., 3x 1 + 4x 2-3x 4 7 x 1-2x 5 = 7 Typically, an LP has non-negativity constraints. 36

37 A non-linear program is permitted to have a non-linear objective and constraints. maximize f(x,y) = xy subject to x - y 2 /2 10 3x 4y 2 x 0, y 0 37

38 An integer program is a linear program plus constraints that some or all of the variables are integer valued. Maximize 3x 1 + 4x 2-3x 3 3x 1 + 2x 2 - x x 2 - x 3 = 14 x 1 0, x 2 0, x 3 0 and x 1, x 2, x 3 are all integers 38

39 Linear Programming Assumptions Maximize 4 K + 3 S 2 K + 3 S 10. Proportionality Assumption Contribution from K is proportional to K Additivity Assumption Contribution to objective function from K is independent of S. Divisibility Assumption Each variable is allowed to assume fractional values. Certainty Assumption. Each linear coefficient of the objective function and constraints is known (and is not a random variable). 39

40 Some Success Stories Optimal crew scheduling saves American Airlines $20 million/yr. Improved shipment routing saves Yellow Freight over $17.3 million/yr. Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 million/yr. GTE local capacity expansion saves $30 million/yr. Mention Cindy Barnhart s success at United Airlines: her fleet routing models are estimated to save United $100,000,000 per year. Our solutions have an estimated savings of $25 million per year. Mention our work with CSX Mention Interfaces. 40

41 Other Success Stories (cont.) Optimizing global supply chains saves Digital Equipment over $300 million. Restructuring North America Operations, Proctor and Gamble reduces plants by 20%, saving $200 million/yr. Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hours/yr. Better scheduling of hydro and thermal generating units saves southern company $140 million. 41

42 Success Stories (cont.) Improved production planning at Sadia (Brazil) saves $50 million over three years. Production Optimization at Harris Corporation improves on-time deliveries from 75% to 90%. Tata Steel (India) optimizes response to power shortage contributing $73 million. Optimizing police patrol officer scheduling saves police department $11 million/yr. Gasoline blending at Texaco results in saving of over $30 million/yr. 42

43 Summary Answered the question: What is Operations Research & Management Science? and provided some historical perspective. Introduced the terminology of linear programming Two Examples: 1. MSR Marketing 2. David s Tool Company Lecture check problems. Section 3.1 Problems