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# Lecture 2

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### Lecture 2

1. 1. ___________________________________________________________________________ Operations Research Linear ProgrammingLinear Programming
2. 2. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Modeling ProcessModeling Process Real-WorldReal-World ProblemProblem Recognition andRecognition and Definition of theDefinition of the ProblemProblem Formulation andFormulation and Construction ofConstruction of the Mathematicalthe Mathematical ModelModel SolutionSolution of the Modelof the Model InterpretationInterpretation Validation andValidation and SensitivitySensitivity AnalysisAnalysis of the Modelof the Model ImplementationImplementation
3. 3. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research  linear objective functionlinear objective function  linear constraintslinear constraints  decision variablesdecision variables Mathematical ModelMathematical Model  maximizationmaximization  minimizationminimization  equationsequations ==  inequalitiesinequalities ≤≤ oror ≥≥  nonnegativity constraintsnonnegativity constraints
4. 4. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Example - PinocchioExample - Pinocchio  2 types of wooden toys:2 types of wooden toys: trucktruck traintrain  Inputs:Inputs: wood - unlimitedwood - unlimited carpentry labor – limitedcarpentry labor – limited finishing labor - limitedfinishing labor - limited  Objective:Objective: maximize total profit (revenue – cost)maximize total profit (revenue – cost)  Demand:Demand: trucks - limitedtrucks - limited trains - unlimitedtrains - unlimited
5. 5. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Example - PinocchioExample - Pinocchio TruckTruck TrainTrain PricePrice 550 CZK550 CZK 700 CZK700 CZK Wood costWood cost 50 CZK50 CZK 70 CZK70 CZK Carpentry laborCarpentry labor 1 hour1 hour 2 hours2 hours Finishing laborFinishing labor 1 hour1 hour 1 hour1 hour Monthly demand limitMonthly demand limit 2 000 pcs.2 000 pcs. ∞∞ Worth per hourWorth per hour Available per monthAvailable per month Carpentry laborCarpentry labor 30 CZK30 CZK 5 000 hours5 000 hours Finishing laborFinishing labor 20 CZK20 CZK 3 000 hours3 000 hours
6. 6. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Graphical Solution of LP ProblemsGraphical Solution of LP Problems Feasible areaFeasible area Objective functionObjective function Optimal solutionOptimal solution x1 x2 z
7. 7. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Graphical Solution of LP ProblemsGraphical Solution of LP Problems Feasible area - convex setFeasible area - convex set A set of pointsA set of points SS is ais a convex setconvex set if the line segment joiningif the line segment joining any pair of points inany pair of points in SS is wholly contained inis wholly contained in SS.. Convex polyhedronsConvex polyhedrons
8. 8. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Graphical Solution of LP ProblemsGraphical Solution of LP Problems Feasible area – corner pointFeasible area – corner point A pointA point PP in convex polyhedronin convex polyhedron SS is ais a corner pointcorner point if it doesif it does not lie on any line joining any pair of other (thannot lie on any line joining any pair of other (than PP) points in) points in SS..
9. 9. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research© Jan Fábry Graphical Solution of LP ProblemsGraphical Solution of LP Problems Basic Linear Programming TheoremBasic Linear Programming Theorem The optimal feasible solution, if it exists, will occurThe optimal feasible solution, if it exists, will occur at one or more of the corner points.at one or more of the corner points. Simplex methodSimplex method
10. 10. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Graphical Solution of LP ProblemsGraphical Solution of LP Problems 1000 3000 x1 x2 20000 A 2000 1000 B C D E Corner point x1 x2 z A 0 0 0 B 2000 0 900 000 C 2000 1000 1 450 000 D 1000 2000 1 550 000 E 0 2500 1 375 000
11. 11. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Interpretation of Optimal SolutionInterpretation of Optimal Solution  Decision variablesDecision variables  Binding / Nonbinding constraint (Binding / Nonbinding constraint (≤≤ oror ≥≥))  Objective valueObjective value = 0= 0 Slack/SurplusSlack/Surplus variablevariable > 0> 0 Slack/SurplusSlack/Surplus variablevariable
12. 12. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Special Cases of LP ModelsSpecial Cases of LP Models Unique Optimal SolutionUnique Optimal Solution z x1 x2 A
13. 13. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Special Cases of LP ModelsSpecial Cases of LP Models Multiple Optimal SolutionsMultiple Optimal Solutions z x1 x2 B C
14. 14. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Special Cases of LP ModelsSpecial Cases of LP Models No Optimal SolutionNo Optimal Solution z x1 x2
15. 15. Linear ProgrammingLinear Programming ___________________________________________________________________________ Operations Research Special Cases of LP ModelsSpecial Cases of LP Models NoNo FeasibleFeasible SolutionSolution x1 x2