___________________________________________________________________________ Operations Research  Jan Fábry Linear Programming

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

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry  linear objective function  linear constraints  decision variables Mathematical Model  maximization  minimization  equations =  inequalities  or   nonnegativity constraints

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

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Example - Pinocchio TruckTrain Price 550 CZK 700 CZK Wood cost 50 CZK 70 CZK Carpentry labor 1 hour 2 hours Finishing labor 1 hour Monthly demand limit pcs.  Worth per hour Available per month Carpentry labor 30 CZK hours Finishing labor 20 CZK hours

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Graphical Solution of LP Problems Feasible area Objective function Optimal solution x1x1 x2x2 z

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry 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

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry 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.

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry 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

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Graphical Solution of LP Problems x1x1 x2x A 1000 B C D E

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

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Special Cases of LP Models Unique Optimal Solution z x1x1 x2x2 A

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Special Cases of LP Models Multiple Optimal Solutions z x1x1 x2x2 B C

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Special Cases of LP Models No Optimal Solution z x1x1 x2x2

Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Special Cases of LP Models No Feasible Solution x1x1 x2x2