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IndE 310 Linear Programming

IndE 310 Linear Programming. UW Industrial Engineering Instructor: Prof. Zelda Zabinsky. Operations Research. “ The Science of Better”: The discipline of applying advanced analytical methods to help make better decisions

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IndE 310 Linear Programming

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  1. IndE 310Linear Programming UW Industrial Engineering Instructor: Prof. Zelda Zabinsky

  2. Operations Research • “The Science of Better”: The discipline of applying advanced analytical methods to help make better decisions • Operations – problems that concern how to conduct and coordinate the operations within an organization • Research – use of the scientific method to address these problems • Interdisciplinary – brings together: • mathematics • statistics • industrial engineering • management (“management science”) • systems engineering... • Highly application oriented

  3. Business Manufacturing Project management Scheduling Facility layout and location Many more Service Project management Transportation and logistics Marketing Queueing Many more Economics & Finance Auctions Portfolio selection Capital investment Many more Health Scheduling Queueing Biotechnology And more Military Applications of Operations Research

  4. Examples of O.R. Applications • China Select and schedule projects for future energy needs • New Haven Health Department Design an effective needle exchange program to combat HIV • Continental Airlines Optimize the reassignment of crews to flights • IBM Reengineer supply chain • Proctor and Gamble Redesign production and distribution system • Many more

  5. Operations Research Structuring the real-life situation into a mathematical model, abstracting the essential elements so that a solution relevant to the decision maker's objectives can be sought This involves looking at the problem in the context of the entire system (Understand and) Model (and verify) Exploring the structure of such solutions and developing systematic procedures for obtaining them Developing a solution, including the mathematical theory, if necessary, that yields an optimal value of the system measure of desirability (or possibly comparing alternative courses of action by evaluating their measure of desirability) (Strategize and) Solve (and make recommendations)

  6. Operations Research Modeling Toolset Queueing Theory Markov Chains PERT/ CPM Network Programming Dynamic Programming Simulation Markov Decision Processes Inventory Theory Linear Programming Stochastic Programming Forecasting Integer Programming Decision Analysis Nonlinear Programming Game Theory

  7. Operations Research Modeling Toolset 311 Queueing Theory 310 Markov Chains PERT/ CPM Network Programming Dynamic Programming Simulation Markov Decision Processes Inventory Theory Linear Programming 312 Stochastic Programming Forecasting Integer Programming Decision Analysis Nonlinear Programming Game Theory 312 312

  8. IndE 310 • Linear Programming • Modeling • Solving • Simplex method • Foundations of the simplex method • Duality and sensitivity • Transportation and Assignment Problems • Network Problems • Shortest path • Minimum spanning tree • Minimum cost, maximum flow • PERT/CPM

  9. Interesting Links • “Operation Everything”, The Boston Globe, June 27, 2004 http://www.boston.com/globe/search/stories/reprints/operationeverything062704.html • Links available on the course web

  10. O.R. Modeling Approach

  11. O.R. Modeling Approach • Define the problem and gather data • Formulate a (mathematical) model to represent the problem • Develop a computer-based procedure for deriving solutions • Test model and refine it • Prepare for the ongoing application of the model • Implement

  12. Defining the problem and gathering data • First order of business: • Study the relevant system • Develop a well-defined statement of the problem • Right answer for the wrong problem? • Defining objectives • What data are needed? • Collect them (easily said)

  13. Formulating a (mathematical) model • We need to create a model that can be used for • Abstracting the essence of the subject • Showing interrelationships • Facilitating analysis • A model: E=mc2 • A mathematical model usually consists of: • Decision variables • Parameters • Objective function • Constraints

  14. Deriving solutions • A common theme in O.R.: Search for an optimal solution • Develop a procedure for deriving solutions to the mathematical model • e.g. what is the best speed to obtain the highest possible energy? • What kind of a procedure? • Usually based on theoretical foundations • Exact vs. heuristic (slow vs. fast?) • Post-optimality and sensitivity analysis

  15. Testing, preparing and implementing • Need to validate the model • “Debugging” • Plausible results? • Prepare to apply as prescribed by management • Operating procedures, supporting systems, managerial reports… • Implementation • Coordination, detailed indoctrination, new courses of action

  16. Peeking into Chapter 3Wyndor Glass Co. Example • Wyndor Glass produces glass windows and doors • They have 3 plants: • Plant 1: makes aluminum frames and hardware • Plant 2: makes wood frames • Plant 3: produces glass and makes assembly • Two products proposed: • Product 1: 8’ glass door with aluminum siding • Product 2: 4’ x 6’ wood framed glass window • Some production capacity in the three plants is available to produce a combination of the two products • Problem is to determine the best product mix to maximize profits

  17. Peeking into Chapter 3Wyndor Glass Co. Data • The O.R. team has gathered the following data: • Number of hours of production time available per week in each plant • Number of hours of production time needed in each plant for each batch of new products • Estimated profit per batch of each new product

  18. Peeking into Chapter 3Formulate LP Model • Identify the parameters (activities/values you cannot control) • Identify the decision variables (activities/values that you can control and need to make a decision on) • Identify the objective function (function of the decision variables for minimization/maximization) • Identify the constraints (limitations you cannot control)

  19. Peeking into Chapter 3Graphical Solution Setup • The model:

  20. Prototype ExampleGraphical Solution • The model:

  21. LP Modeling • Wyndor Glass Co. is only one of a vast number of applications that LP can be used to address • In general, the most common type of an LP addresses: The allocation of limitedresourcesto competingactivitiesformaximizing the value of these activities

  22. LP Terminology The allocation of limitedresourcesto competingactivitiesformaximizing the value of these activities • Activities, n • Resources, m • Decision variables – or level of activities, x • Objective function – or value of activities, Z • Constraints • functional • non-negativity

  23. LP Terminology • Feasible region, feasible solution • Infeasible solution • Optimal solution • Extreme-point – or corner-point feasible solutions • Parameters

  24. Standard (Canonical) Form of an LP Model Maximize Z = c1x1 + c2x2 + … + cnxn subject to a11x1 + a12x2 + … + a1nxn ≤ b1 a21x1 + a22x2 + … + a2nxn ≤ b2 … am1x1 + am2x2 + … + amnxn ≤ bm x1≥ 0, x2≥ 0, …, xn≥ 0

  25. Matrix Form of an LP Model

  26. Other Forms that can be Converted into Standard Form • Objective function: Minimize Z=cx (instead of Maximize Z=cx) • Functional constraints: Ax ≥ b or Ax = b (instead of Ax ≤ b) • Non-negativity constraints: x unrestricted in sign (instead of x ≥ 0)

  27. Assumptions of Linear Programming • Proportionality • Additivity • Divisibility • Certainty

  28. LP Modeling Examples

  29. A Transportation Example • A company has 2 plants and 3 warehouses • Supply at plants 100 units in Plant 1, 200 units in Plant 2 • Sales potential at warehouses 150 units, 200 units, and 350 units at Warehouses 1, 2 and 3, respectively • Revenue 12 $/unit, 14 $/unit and 15 $/unit at Warehouses 1, 2 and 3, respectively • Cost of manufacturing one unit at plant i and shipping to w/h j: • How many units to ship from each plant to each w/h to maximize profits?

  30. Personnel Scheduling (p.56) • Union Airways is adding more flights and needs to hire additional customer service agents • Each agent works an eight-hour shift • The five possible shifts are • Shift 1: 6:00 am – 2:00 pm • Shift 2: 8:00 am – 4:00 pm • Shift 3: Noon – 8:00 pm • Shift 4: 4:00 pm – Midnight • Shift 5: 10:00 pm – 6:00 am

  31. Personnel Scheduling Minimum number of agents needed per two-hour time periods:

  32. Reclaiming Solid Wastes (p.52) • A recycling center takes four types of material • Material 1: Newsprint • Material 2: White paper • Material 3: Mixed paper • Material 4: Cardboard • Three products are reclaimed • Grades A, B, C • The SAVE-IT company wants to determine the amount of each grade to produce and the mix of materials in each grade to maximize profit

  33. Reclaiming Solid Wastes • Products • Solid waste materials • Use at least half of each material collected • Cannot use more than $30,000 per week for treatment of mat’ls

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