1 / 28

Chapter 2 Linear Programming Models: Graphical and Computer Methods

Chapter 2 Linear Programming Models: Graphical and Computer Methods. Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA 99223 chen@jepson.gonzaga.edu. Steps in Developing a Linear Programming (LP) Model. Formulation Solution

zaide
Download Presentation

Chapter 2 Linear Programming Models: Graphical and Computer Methods

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 2Linear Programming Models:Graphical and Computer Methods Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA 99223 chen@jepson.gonzaga.edu

  2. Steps in Developing a Linear Programming (LP) Model • Formulation • Solution • Interpretation and Sensitivity Analysis

  3. Properties of LP Models • Seek to minimize or maximize • Include “constraints” or limitations • There must be alternatives available • All equations are linear

  4. Example LP Model Formulation:The Product Mix Problem Decision: How much to make of > 2 products? Objective: Maximize profit Constraints: Limited resources

  5. Example: Flair Furniture Co. Two products: Chairs and Tables Decision: How many of each to make this month? Objective: Maximize profit

  6. Flair Furniture Co. Data • Other Limitations: • Make no more than 450 chairs • Make at least 100 tables

  7. Decision Variables: T = Num. of tables to make C = Num. of chairs to make Objective Function: Maximize Profit Maximize $7 T + $5 C

  8. Constraints: • Have 2400 hours of carpentry time available 3 T + 4 C < 2400 (hours) • Have 1000 hours of painting time available 2 T + 1 C < 1000(hours)

  9. More Constraints: • Make no more than 450 chairs C < 450 (num. chairs) • Make at least 100 tables T > 100 (num. tables) Nonnegativity: Cannot make a negative number of chairs or tables T > 0 C > 0

  10. Model Summary Max 7T + 5C (profit) Subject to the constraints: 3T + 4C < 2400 (carpentry hrs) 2T + 1C < 1000 (painting hrs) C < 450 (max # chairs) T > 100 (min # tables) T, C > 0 (nonnegativity)

  11. Using Excel’s Solver for LP Recall the Flair Furniture Example: Max 7T + 5C (profit) Subject to the constraints: 3T + 4C < 2400 (carpentry hrs) 2T + 1C < 1000 (painting hrs) C < 450 (max # chairs) T > 100 (min # tables) T, C > 0 (nonnegativity) Go to file 2-1.xls

  12. Add a new constraint • A new constraint specified by the marketing department. • Specifically, they needed to ensure theat the number of chairs made this month is at least 75 more than the number of tables made. The constraint is expressed as: C - T > 75

  13. Revised Model for Flair Furniture Max 7T + 5C (profit) Subject to the constraints: 3T + 4C < 2400 (carpentry hrs) 2T + 1C < 1000 (painting hrs) C < 450 (max # chairs) T > 100 (min # tables) - 1T + 1C > 75 T, C > 0 (nonnegativity) Go to file 2-2.xls

  14. End of Chapter 2 • No Graphical Solution will be discussed

  15. Graphical Solution • Graphing an LP model helps provide insight into LP models and their solutions. • While this can only be done in two dimensions, the same properties apply to all LP models and solutions.

  16. C 600 0 Carpentry Constraint Line 3T + 4C = 2400 Intercepts (T = 0, C = 600) (T = 800, C = 0) Infeasible > 2400 hrs 3T + 4C = 2400 Feasible < 2400 hrs 0 800 T

  17. C 1000 600 0 Painting Constraint Line 2T + 1C = 1000 Intercepts (T = 0, C = 1000) (T = 500, C = 0) 2T + 1C = 1000 0 500 800 T

  18. C 1000 600 450 0 Max Chair Line C = 450 Min Table Line T = 100 Feasible Region 0 100 500 800 T

  19. C 500 400 300 200 100 0 Objective Function Line 7T + 5C = Profit 7T + 5C = $4,040 Optimal Point (T = 320, C = 360) 7T + 5C = $2,800 7T + 5C = $2,100 0 100 200 300 400 500 T

  20. C 500 400 300 200 100 0 Additional Constraint Need at least 75 more chairs than tables C > T + 75 Or C – T > 75 New optimal point T = 300, C = 375 T = 320 C = 360 No longer feasible 0 100 200 300 400 500 T

  21. LP Characteristics • Feasible Region: The set of points that satisfies all constraints • Corner Point Property: An optimal solution must lie at one or more corner points • Optimal Solution: The corner point with the best objective function value is optimal

  22. Special Situation in LP • Redundant Constraints - do not affect the feasible region Example: x < 10 x < 12 The second constraint is redundant because it is less restrictive.

  23. Special Situation in LP • Infeasibility – when no feasible solution exists (there is no feasible region) Example: x < 10 x > 15

  24. Special Situation in LP • Alternate Optimal Solutions – when there is more than one optimal solution C 10 6 0 Max 2T + 2C Subject to: T + C < 10 T < 5 C < 6 T, C > 0 All points on Red segment are optimal 2T + 2C = 20 0 5 10 T

  25. Special Situation in LP • Unbounded Solutions – when nothing prevents the solution from becoming infinitely large C 2 1 0 Direction of solution Max 2T + 2C Subject to: 2T + 3C > 6 T, C > 0 0 1 2 3 T

More Related