Linear Programming

1
Linear Programming

• Graphical Solution Procedure

2
Two Variable Linear Programs

• When a linear programming model consists of only
  two variables, a graphical approach can be
  employed to solve the model.
• There are few, if any, real life linear
  programming models with two variables.
• But the graphical illustration of this case helps
  develop the terminology and approaches for
  solving larger models.

3
5-Step Graphical Solution Procedure

• Graph the constraints.
• The resulting set of possible or feasible points
  is called the feasible region.
• Set the objective function value equal to any
  number and graph the resulting objective function
  line.
• Move the objective function line parallel to
  itself until it touches the last point(s) of the
  feasible region.
• This is the optimal solution.
• Solve for the values of the variables of the
  optimal solution.
• This involves solving 2 equations in 2 unknowns.
• Solve for the optimal value of the objective
  function.
• This is done by substituting the variable values
  into the objective function formula.

4
Graphing the Feasible Region
2X1 1X2 10003X1 4X2 24001X1 1X2
7001X1 - 1X2 350 X1, X2 0
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Graphing the Feasible Region
2X1 1X2 10003X1 4X2 24001X1 1X2
7001X1 - 1X2 350 X1, X2 0
6
Graphing the Feasible Region
2X1 1X2 10003X1 4X2 24001X1 1X2
7001X1 - 1X2 350 X1, X2 0
2X1 1X2 1000
3X1 4X2 2400
7
Graphing the Feasible Region
2X1 1X2 10003X1 4X2 24001X1 1X2
7001X1 - 1X2 350 X1, X2 0
2X1 1X2 1000
1X1 1X2 700
Redundant Constraint
3X1 4X2 2400
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Graphing the Feasible Region
2X1 1X2 10003X1 4X2 24001X1 1X2
7001X1 - 1X2 350 X1, X2 0
2X1 1X2 1000
1X1 1X2 700
1X1 - 1X2 350
FeasibleRegion
3X1 4X2 2400
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Characterization of Points
3X1 4X2 2400
2X1 1X2 1000
1X1 - 1X2 350
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Graphing the Objective Function
Set the objective function value equal to any
number and graph it.
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Identifying the Optimal Point
Move the objective function line parallel to
itself until it touches the last point of the
feasible region.
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Determining the Optimal Point
Solve the 2 equations in 2 unknowns that
determines the optimal point.
Multiply top equation by 4
8X1 4X2 40003X1 4X2 2400
Subtract second equation from first
5X1 1600 or X1 320
Substituting in first equation
2(320) 1X2 1000 or X2 360
(320, 360)
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Determining the Optimal Objective Function Value
Substitute x-values into objective function
MAX 8 X1 5 X2
4360
(320, 360)
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Review

• 5 Step Graphical Solution Procedure
• Graph constraints
• Graph objective function line
• Move the objective function line parallel until
  it touches the last point of the feasible region
• Solve for optimal point
• Solve for the optimal objective function value
• Redundant Constraints
• Identification of points
• Infeasible
• Interior
• Boundary
• Extreme