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Explore the recognition, formulation, and implementation of mathematical models for real-world problems in linear programming. Understand decision variables, objective functions, constraints, and solutions. Dive into examples like the Pinocchio toy manufacturing scenario and graphical solutions for LP problems. Learn about feasible areas, convex sets, corner points, optimization theorems, simplex methods, slack variables, and special cases of LP models.

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  1. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry

  2. Recognition and Definition of the Problem Formulation and Construction of the Mathematical Model Real-WorldProblem Interpretation Validation and Sensitivity Analysis of the Model Solution of the Model Implementation Linear Programming Modeling Process ___________________________________________________________________________ Operations Research  Jan Fábry

  3. Linear Programming Mathematical Model • decision variables • linear objective function • maximization • minimization • linear constraints • equations = • inequalities  or  • nonnegativityconstraints ___________________________________________________________________________ Operations Research  Jan Fábry

  4. 2 types of wooden toys: truck train wood - unlimited carpentry labor – limited finishing labor - limited • Inputs: • Demand: trucks - limited trains - unlimited • Objective: maximize total profit (revenue – cost) Linear Programming Example - Pinocchio ___________________________________________________________________________ Operations Research  Jan Fábry

  5. Linear Programming Example - Pinocchio ___________________________________________________________________________ Operations Research  Jan Fábry

  6. x2 Objective function z Optimal solution x1 Linear Programming Graphical Solution of LP Problems Feasible area ___________________________________________________________________________ Operations Research  Jan Fábry

  7. Feasible area - convex set A set of points S is a convex set if the line segment joining any pair of points in S is wholly contained in S. Convex polyhedrons Linear Programming Graphical Solution of LP Problems ___________________________________________________________________________ Operations Research  Jan Fábry

  8. Feasible area – corner point A point P in convex polyhedron S is a corner point if it does not lie on any line joining any pair of other (than P) points in S. Linear Programming Graphical Solution of LP Problems ___________________________________________________________________________ Operations Research  Jan Fábry

  9. Basic Linear Programming Theorem The optimal feasible solution, if it exists, will occur at one or more of the corner points. Simplex method Linear Programming Graphical Solution of LP Problems ___________________________________________________________________________ Operations Research  Jan Fábry

  10. x2 3000 E D 2000 C 1000 B A x1 1000 2000 0 Linear Programming Graphical Solution of LP Problems ___________________________________________________________________________ Operations Research  Jan Fábry

  11. Slack/Surplusvariable Slack/Surplusvariable = 0 > 0 Linear Programming Interpretation of Optimal Solution • Decision variables • Objective value • Binding / Nonbinding constraint (or ) ___________________________________________________________________________ Operations Research  Jan Fábry

  12. x2 A z x1 Linear Programming Special Cases of LP Models Unique Optimal Solution ___________________________________________________________________________ Operations Research  Jan Fábry

  13. x2 B z C x1 Linear Programming Special Cases of LP Models Multiple Optimal Solutions ___________________________________________________________________________ Operations Research  Jan Fábry

  14. x2 z x1 Linear Programming Special Cases of LP Models No Optimal Solution ___________________________________________________________________________ Operations Research  Jan Fábry

  15. x2 x1 Linear Programming Special Cases of LP Models No Feasible Solution ___________________________________________________________________________ Operations Research  Jan Fábry

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