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MA3264 Mathematical Modelling Lecture 8

MA3264 Mathematical Modelling Lecture 8. Chapter 7 Discrete Optimization Modelling. Example page 238.

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MA3264 Mathematical Modelling Lecture 8

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  1. MA3264 Mathematical ModellingLecture 8 Chapter 7 Discrete Optimization Modelling

  2. Example page 238 How many* tables and how many* bookcases should a carpenter make each week to maximize profit? He realizes a profit of $25 per table and $30 per bookcase. He has 600 feet of lumber per week and 40 hours of labor per week. Each table requires 20 feet of lumber and 5 hours of labor. Each bookcase requires 30 feet of lumber and 4 hours of labor. He has signed contracts to deliver 4 tables and 2 bookcases every week. How many* : need not be integers since part of a table or bookcase can be made in a week

  3. Mathematical Formulation decision variables Maximize objective function Subject to constraints

  4. Mathematical Formulation decision variables Maximize objective function Subject to constraints Question what real world entity does each decision variable represent ?

  5. Mathematical Formulation decision variables Maximize objective function Subject to constraints Question what real world entity does the objective function represent ?

  6. Mathematical Formulation decision variables Maximize objective function Subject to constraints Question what real world entity does each constraint represent ?

  7. Mathematical Formulation decision variables Maximize objective function Subject to constraints Question what real world entities have been abstracted out of this formulation ?

  8. Solution decision variables Maximize objective function Subject to constraints Question can we set the derivatives = 0 to solve this maximization problem ?

  9. Constraints constraints Question what are the regions consisting of all points that satisfy the 1st , 2nd ,3rd, 4th constraint ?

  10. Constraint Line Question What equation describes the doted line ?

  11. Constraint Line Answer The equation above describes the dotted line.

  12. Constraint Region Question What region is described by the inequality above ?

  13. Constraint Region THIS RED REGION

  14. Feasible Region constraints Question what region consisting of all points that satisfy all fourconstraints ?

  15. Feasible Region is the red region

  16. Feasible Region is both closed and bounded

  17. Objective Function is continuous on the feasible region

  18. Objective Function not necessarily unique must have a maximum at some point p in the feasible region

  19. Objective Function A similar argument applied to edges (to be shown using visualizer) shows that f has a maximum at a vertex of the feasible region the simplex method

  20. Suggested Reading Introduction and Section 7.1 overview of discrete optimization modelling p. 236- 249 Linear Programming 1: Geometric Solutions p. 250-259

  21. Tutorial 8 Due Week 20–24 Oct Problem 1. Page 245 Problem 1 Problem 2. Page 245 Problem 2 Problem 3. Page 258 Problem 3 Problem 4. Page 259 Problem 4, a Problem 5. Page 259 Problem 4, b Problem 6. Page 259 Problem 4, c

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