Linear Programming Operations Research Jan Fbry Linear Programming
Linear Programming ______________________________________ Operations Research Jan Fábry
Linear Programming Modeling Process Real-World Problem Implementation Recognition and Definition of the Problem Interpretation Validation and Sensitivity Analysis of the Model Formulation and Construction of the Mathematical Model Solution of the Model ______________________________________ Operations Research Jan Fábry
Linear Programming Mathematical Model Ø decision variables Ø linear objective function q maximization q minimization Ø linear constraints q equations = q inequalities or Ø nonnegativity constraints ______________________________________ Operations Research Jan Fábry
Linear Programming Example - Pinocchio Ø 2 types of wooden toys: truck train Ø Inputs: wood - unlimited carpentry labor – limited finishing labor - limited Ø Demand: trucks - limited trains - unlimited Ø Objective: maximize total profit (revenue – cost) ______________________________________ Operations Research Jan Fábry
Linear Programming Example - Pinocchio Truck Train Price 550 CZK 700 CZK Wood cost 50 CZK 70 CZK Carpentry labor 1 hour 2 hours Finishing labor 1 hour Monthly demand limit 2 000 pcs. Worth per hour Available per month Carpentry labor 30 CZK 5 000 hours Finishing labor 20 CZK 3 000 hours ______________________________________ Operations Research Jan Fábry
Linear Programming Graphical Solution of LP Problems Feasible area Objective function Optimal solution x 2 z x 1 ______________________________________ Operations Research Jan Fábry
Linear Programming Graphical Solution of LP Problems 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 ______________________________________ Operations Research Jan Fábry
Linear Programming Graphical Solution of LP Problems 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. ______________________________________ Operations Research Jan Fábry
Linear Programming Graphical Solution of LP Problems Basic Linear Programming Theorem The optimal feasible solution, if it exists, will occur at one or more of the corner points. Simplex method ______________________________________ Operations Research Jan Fábry
Linear Programming Graphical Solution of LP Problems x 2 3000 E 2000 D C 1000 B A 0 1000 2000 x 1 ______________________________________ Operations Research Jan Fábry
Linear Programming Interpretation of Optimal Solution Ø Decision variables Ø Objective value Ø Binding / Nonbinding constraint ( or ) Slack/Surplus variable = 0 Slack/Surplus variable > 0 ______________________________________ Operations Research Jan Fábry
Linear Programming Special Cases of LP Models Unique Optimal Solution x 2 A z x 1 ______________________________________ Operations Research Jan Fábry
Linear Programming Special Cases of LP Models Multiple Optimal Solutions x 2 B z C x 1 ______________________________________ Operations Research Jan Fábry
Linear Programming Special Cases of LP Models No Optimal Solution x 2 z x 1 ______________________________________ Operations Research Jan Fábry
Linear Programming Special Cases of LP Models No Feasible Solution x 2 x 1 ______________________________________ Operations Research Jan Fábry
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