Linear Programming
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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
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Operations Research
 Jan Fábry
Linear Programming
Mathematical Model
 decision variables
 linear objective function
 maximization
 minimization
 linear constraints
 equations
=
 inequalities  or 
 nonnegativity constraints
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Example - Pinocchio
 2 types of wooden toys:
truck
train
 Inputs:
wood - unlimited
carpentry labor – limited
finishing labor - limited
 Demand:
trucks - limited
trains - unlimited
 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
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
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Operations Research
 Jan Fábry
Linear Programming
Graphical Solution of LP Problems
Feasible area
Objective function
Optimal solution
x2
z
x1
___________________________________________________________________________
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
x2
Corner point
3000
A
B
C
D
E
E
2000
D
x2
z
0
2000
2000
1000
0
0
0
1000
2000
2500
0
900 000
1 450 000
1 550 000
1 375 000
C
1000
B
A
0
x1
1000
2000
x1
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Interpretation of Optimal Solution
 Decision variables
 Objective value
 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
x2
A
z
x1
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Operations Research
 Jan Fábry
Linear Programming
Special Cases of LP Models
Multiple Optimal Solutions
x2
B
z
C
x1
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Special Cases of LP Models
No Optimal Solution
x2
z
x1
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Special Cases of LP Models
No Feasible Solution
x2
x1
___________________________________________________________________________
Operations Research
 Jan Fábry