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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

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Chapter 2 Linear Programming Models: Graphical and Computer Methods

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  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

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