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15.053

15.053. February 5, 2002.

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15.053

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  1. 15.053 February 5, 2002 • 􀁺 Introduction to Optimization • Handouts: Lecture Notes

  2. Overview • Course Description • Course Administration and Logistics • What is Management Science? • Linear Programming Examples • MSR Marketing • GTC • Handouts: Syllabus and General Info. Lecture Notes Homework Set 1

  3. Slide #3 contains MIT specific information.

  4. Required Materials • Coursepack (Course Reader) • Includes copies of chapters from • Applied Mathematical Programming by Bradley,Hax & Magnanti • Class website: sloanspace.mit.edu • Other resources: • Operations Research by Winston • Introduction to Linear Optimization by Bertsimas & Tsitsiklis • Network Flows by Ahuja, Magnanti & Orlin

  5. Grading Policy • Homework assignments (27%) • Weekly ( 10 in total ) • Nonlinear grading scheme • Midterm Exams (25% each) • Two in class modterms • Final Exam (25%) • During Finals week • Last 1/3 of subject + modeling • Yes, it does add up to 102%.

  6. Course Policy (cont’d) • Webpage • Includes • class notes • Spreadsheets • Readings • assignments (assignment 1 is already there) • and more

  7. Active Learning • Occasionally I will introduce a break in the lecture for you to work on your own or with a partner • Please identify your “partner” now. • Those on aisle ends may be in a group of size 3.

  8. 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 Military Operations Research, later 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 found ORSA

  9. What is Management Science(Operations Research)? • Today: Operations Research and ManagementScience mean “the use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information, or in seeking further information if current knowledge is insufficient to reach a proper decision.” • c.f. Decision science, systems analysis,operational research, systems dynamics,operational analysis, engineering systems,systems engineering, and more.

  10. 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

  11. Operations Research Over the Years • 1947 • Project Scoop (Scientific Computation of Optimum Programs) with George Dantzig andothers. Developed the simplex method forlinear programs. • 1950's • Lots of excitement, mathematicaldevelopments, queuing theory, mathematicalprogramming.cf. A.I. in the 1960's • 1960's • More excitement, more development and grandplans. cf. A.I. in the 1980's.

  12. 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. Widespreadwillingness 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-ons to spreadsheets, modeling languages, large scaleoptimization. More intermixing of A.I. and O.R.

  13. Operations Research in the 00’s • LOTS of opportunities for OR as a field • Data, data, data • E-business data (click stream, purchases, othertransactional data, E-mail and more) • 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)

  14. Optimization As ageless as time

  15. Optimization in Nature Heron of Alexandria Angle of Incidence Angle of Incidence Angle of Reflection First Century AD

  16. FERM AT Angle α1 of Incidence Angle α2 of Refraction

  17. Calculus Maximum Minimum Fermat, Newton, Euler, Lagrange, G auss, and more

  18. Some of the themes of 15.053 • Optimization is everywhere • Models, Models, Models • The goal of models is “insight” notnumbers • paraphrase of Richard Hamming • Algorithms, Algorithms, Algorithms

  19. Optimization is Everywhere • The more you know about something, the more you see where optimization can be applied. • Some personal decision making • Finding the fastest route home (or to class) • Optimal allocation of time for homework • Optimal budgeting • Selecting a major

  20. Optimization is everywhere • Some MIT decision making • setting exam times to minimize overlap • assigning classes to classrooms and time slots while satisfying constraints • figuring out prices for parking and subsidies for public transportation so as to maximize fairness, and permit sufficient access • optimizing in fundraising

  21. Optimization is everywhere • Exercise: introduce yourself to your neighbor,and then mention a topic or two with which youare pretty familiar (summer job, or major, or yourparent’s occupation or whatever.) • Then brainstorm with your neighbor on placeswhere optimization occurs. • Then choose your favorite 2 or 3 applicationsfrom your list, and let’s share them.

  22. On 15.053 and Optimization Tools • Optimization is everywhere, but optimizationtools are not applied everywhere. • Goals in 15.053: present a variety of toolsfor optimization, and illustrate applicationsin manufacturing, finance, e-business,marketing and more. • When you see optimization problems arisein business (and you will), you will know thatthere are tools to help you out.

  23. Addressing managerial problems: Amanagement science framework • Determine the problem to be solved • Observe the system and gather data • Formulate a mathematical model of the problem and any important subproblems • Verify the model and use the model for predictionor analysis • Select a suitable alternative • Present the results to the organization • Implement and evaluate

  24. Linear Programming (our first tool,and probably the most important one.) • minimize or maximize a linear objective • subject to linear equalities andinequalities maximize 3x + 4y subject to 5x + 8y ≤ 24 x, y ≥ 0

  25. Here is the set of feasible solutions The optimal solution

  26. Terminology • Decision variables:e.g., x and y. • In general, there are quantities you can control to improve your objective which should completely describe the set of decisions to be made. • Constraints: e.g., 5x + 8y ≤ 24 , x ≥ 0 , y ≥ 0 • Limitations on the values of the decision variables. • Objective Function. e.g., 3x + 4y • Value measure used to rank alternatives • Seek to maximize or minimize this objective • examples: maximize NPV, minimize cost

  27. 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 20 15 10 15

  28. Formulating as a math model Work with your partner • What decisions need to be made? Expressthese as “decision variables.” • What is the objective? Express the objectivein terms of the decision variables. • What are the constraints? Express these interms of the decision variables. • If you have time, try to find the best solution.

  29. Gemstone Tool Company • Privately-held firm • Consumer and industrial market for construction tools • Headquartered in Seattle • Manufacturing plants in the US, Canada, and Mexico. • Simplifying assumptions, for purposes of illustration: • Winnipeg, Canada plant • Wrenches and pliers. • Made from steel • Injection molding machine • Assembly machine

  30. Data for the GTC Problem We want to determine the number of wrenches and pliers to produce given the available raw materials, machine hours and demand.

  31. To do with your partner • Work with your partner to formulate theGTC problem as a linear program. DONOT LOOK AHEAD IN THE NOTES. • Let P = number of pliers made • Let W = number of wrenches made

  32. Formulating the GTC Problem Step 1: Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured Step 2: Determine Objective Function Maximize Profit =

  33. The Formulation Continued Step 3: Determine Constraints Steel: Molding: Assembly: Plier Demand: Wrench Demand: We will show how to solve this in the next lecture.

  34. An Algebraic Formulation • J = set of items that are manufactured • e.g., S = {pliers, wrenches} • pj = unit profit from item j • dj = maximum demand for item j • xj = number of units of item j manufactured • M = set of manufacturing processes • e.g., M = {molding, and assembly} • bi = capacity of process i • aij = amount of capacity of process i used in making item j

  35. An Algebraic formulation • Maximize • subject to The same formulation works even if | J | = 10,000 and | M | = 100.

  36. An Algebraic formulation • Maximize • subject to

  37. Linear Programs • A linear function is a function of the form: • A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities. • Typically, an LP has non-negativity constraints.

  38. A non-linear program is permitted to havea non-linear objective and constraints. • maximize f(x,y) = xy • subject to x - y2/2 ≤ 10 3x – 4y ≥ 2 x ≥ 0, y ≥ 0

  39. An integer program is a linear programplus constraints that some or all of thevariables are integer valued. • Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ≥ 17 3x2 - x3 = 14 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 and x1 , x2, x3 are all integers

  40. An Algebraic formulation withequality constraints • Maximize • subject to

  41. Linear Programming Assumptions Proportionality Assumption Contribution from W is proportional to W Additivity Assumption Contribution to objective function from P is independent of W. 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). Maximize 4W + 3P 1.5W + P ≤ 15 ….

  42. 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.

  43. 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.

  44. 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.

  45. Summary • Answered the question: What is Operations Research & Management Science? and provided some historical perspective. • Introduced the terminology of linear programming • Two Examples: • MSR Marketing • Gemstone Tool Company • Small (2-dimensional) Linear Program, non-obvious solution • We will discuss this problem in detail in the next lecture

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