lecture 2
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___________________________________________________________________________ Operations Research
Linear ProgrammingLinear Programming
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Modeling ProcessModeling Process
Real-WorldReal-WorldProblemProblem
Recognition and Recognition and Definition of the Definition of the
ProblemProblem
Formulation and Formulation and Construction of Construction of
the Mathematical the Mathematical ModelModel
SolutionSolutionof the Modelof the Model
InterpretationInterpretationValidation and Validation and
Sensitivity Sensitivity AnalysisAnalysis
of the Modelof the Model
ImplementationImplementation
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
linear objective functionlinear objective function
linear constraintslinear constraints
decision variablesdecision variables
Mathematical ModelMathematical Model
maximizationmaximization minimizationminimization
equations equations == inequalities inequalities or or
nonnegativity constraintsnonnegativity constraints
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
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
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Graphical Solution of LP ProblemsGraphical Solution of LP Problems
Feasible areaFeasible area
Objective functionObjective function
Optimal solutionOptimal solutionx1
x2
z
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Graphical Solution of LP ProblemsGraphical Solution of LP Problems
Feasible area - convex setFeasible area - convex set
A set of points A set of points SS is a is a convex setconvex set if the line segment joining if the line segment joining any pair of points in any pair of points in SS is wholly contained in is wholly contained in SS..
Convex polyhedronsConvex polyhedrons
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Graphical Solution of LP ProblemsGraphical Solution of LP Problems
Feasible area – corner pointFeasible area – corner point
A point A point PP in convex polyhedron in convex polyhedron SS is a is a corner pointcorner point if it does if it does not lie on any line joining any pair of other (than not lie on any line joining any pair of other (than PP) points in ) points in
SS..
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 occur The 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
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Graphical Solution of LP ProblemsGraphical Solution of LP Problems
1000
3000
x1
x2
2000 0
A
2000
1000
B
C
D
E
Corner point x 1 x 2 zA 0 0 0B 2000 0 900 000C 2000 1000 1 450 000D 1000 2000 1 550 000E 0 2500 1 375 000
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Interpretation of Optimal SolutionInterpretation of Optimal Solution
Decision variablesDecision variables
Binding / Nonbinding constraint (Binding / Nonbinding constraint ( or or ))
Objective valueObjective value
= 0= 0Slack/SurplusSlack/Surplus
variablevariable
> 0> 0Slack/SurplusSlack/Surplus
variablevariable
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Special Cases of LP ModelsSpecial Cases of LP Models
Unique Optimal SolutionUnique Optimal Solution
z
x1
x2
A
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Special Cases of LP ModelsSpecial Cases of LP Models
Multiple Optimal SolutionsMultiple Optimal Solutions
z
x1
x2
B
C
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Special Cases of LP ModelsSpecial Cases of LP Models
No Optimal Solution No Optimal Solution
z
x1
x2
Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research
Special Cases of LP ModelsSpecial Cases of LP Models
No No FeasibleFeasible Solution Solution
x1
x2