Linear Programming Models: Graphical Methods

# Linear Programming Models: Graphical Methods

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## Linear Programming Models: Graphical Methods

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1. Linear Programming Models:Graphical Methods noha hussein elkhidir

2. Requirements of a Linear Programming Problem • Formulating Linear Programming Problems. • Graphical Solution to a Linear Programming Problem • Graphical Representation of Constraints • Corner-Point Solution Method noha hussein elkhidir

3. Linear Programming Applications • Production. • Diet Problem Example • Labor Scheduling Example. noha hussein elkhidir

4. When you complete this module you should be able to: - Formulate linear programming models, including an objective function and constraints • Graphically solve an LP problem with the corner-point method. - Construct and solve a minimization and maximization problem noha hussein elkhidir

5. - Formulate production-mix, diet, and labor scheduling problems noha hussein elkhidir

6. What about Linear Programming? - A mathematical technique to help plan and make decisions relative to the trade-offs necessary to allocate resources - Will find the minimum or maximum value of the objective - Guarantees the optimal solution to the model formulated noha hussein elkhidir

7. Requirements of an LP Problem • LP problems seek to maximize or minimize some quantity (usually profit or cost) expressed as an objective function • The presence of restrictions, or constraints, limits the degree to which we can pursue our objective noha hussein elkhidir

8. There must be alternative courses of action to choose from • The objective and constraints in linear programming problems must be expressed in terms of linear equations or inequalities noha hussein elkhidir

9. Steps in Developing a Linear Programming (LP) Model • Formulation • Solution noha hussein elkhidir

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

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

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

13. Flair Furniture Co. Data • Other Limitations: • Make no more than 450 chairs • Make at least 100 tables noha hussein elkhidir

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

15. 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) noha hussein elkhidir

16. 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 noha hussein elkhidir

17. 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) noha hussein elkhidir

18. 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. noha hussein elkhidir

19. 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 noha hussein elkhidir

20. 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 noha hussein elkhidir

21. C 1000 600 450 0 Max Chair Line C = 450 Min Table Line T = 100 Feasible Region 0 100 500 800 T noha hussein elkhidir

22. 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 noha hussein elkhidir 0 100 200 300 400 500 T