The Graphical Method

The Graphical Method
for Solving LP’s
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Key Terms in Graphing
•
•
•
•
•
Optimal solution
Feasible solution space
Corner point
Redundant constraint
Slack
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Formulating LP Models
• Formulating linear programming models involves
the following steps:
1. Define the decision variables.
2. Determine the objective function.
3. Identify the constraints.
4. Determine appropriate values for parameters and
determine whether an upper limit, lower limit, or
equality is called for.
5. Use this information to build a model.
6. Validate the model.
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Solving Linear Programming
Problems
• Graphical Technique
¾ First graph the constraints:
the solution set of the system is that region
(or set of ordered pairs), which satisfies
ALL the constraints. This region is called
the feasible set
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Solving Linear Programming Problems:
Graphical Technique continued
￭ Locate all the corner points of the graph:
the coordinates of the corners will be
determined algebraically
It is important to note that the optima is
obtained at the boundary of the solution set
and furthermore at the corner points.
For linear programs, it can be shown that the
optima will always be obtained at corner
points.
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Solving Linear Programming Problems:
Graphical Technique continued
￭ Determine the optimal value:
test all the corner points to see which yields the
optimum value for the objective function
Objective function
Feasible
set
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Optimum
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Example
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Example
x1 = quantity of server model
1 to produce
x2 = quantity of server model
2 to produce
maximize Z = 60x1+50x2
Subject to:
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Feasible Region Based on a Plot of the First Constraint
(assembly time) and the Nonnegativity Constraint
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Completed Graph of the Server Problem Showing All of the
Constraints and the Feasible Solution Space
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A Completed Graph of the Server Problem Showing the
Assembly and Inspection Constraints and the Feasible
Solution Space
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Finding the Optimal Solution
• The extreme point approach
– Involves finding the coordinates of each corner
point that borders the feasible solution space
and then determining which corner point
provides the best value of the objective
function.
– The extreme point theorem
– If a problem has an optimal solution at least one
optimal solution will occur at a corner point of
the feasible solution space.
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The
The Extreme
Extreme Point
Point Approach
Approach
1. Graph the problem and identify the feasible solution
space.
2. Determine the values of the decision variables at each
corner point of the feasible solution space.
3. Substitute the values of the decision variables at each
corner point into the objective function to obtain its value
at each corner point.
4. After all corner points have been evaluated in a similar
fashion, select the one with the highest value of the
objective function (for a maximization problem) or lowest
value (for a minimization problem) as the optimal
solution.
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Graph of Server Problem with Extreme Points of the Feasible
Solution Space Indicated
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Extreme Point Solutions for the Server
Problem
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The
The Objective
Objective Function
Function
(Iso-Profit
(Iso-Profit Line)
Line) Approach
Approach
• This approach directly identifies the optimal
corner point, so only the coordinates of the
optimal point need to be determined.
– Accomplishes this by adding the objective
function to the graph and then using it to
determine which point is optimal.
– Avoids the need to determine the coordinates of
all of the corner points of the feasible solution
space.
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The Server Problem with Profit Lines of $300,
$600, and $900
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Finding the Optimal Solution to the
Server Problem
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Graphing—Objective
Graphing—Objective Function
Function Approach
Approach
1. Graph the constraints.
2. Identify the feasible solution space.
3. Set the objective function equal to some amount
that is divisible by each of the objective function
coefficients.
4. After identifying the optimal point, determine
which two constraints intersect there.
5. Substitute the values obtained in the previous step
into the objective function to determine the value
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of
the
at
the
Operaciones
A Comparison of Maximization and
Minimization Problems
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Example
Minimization
Determine the values of decision variables x1
and x2 that will yield the minimum cost in the
following problem. Solve using the objective
function approach.
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Graphing the Feasible Region and Using the Objective
Function to Find the Optimum for Example
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Example
Example
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Summary of Extreme Point Analysis for
Example
1
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Computing the Amount of Slack for the Optimal Solution to
the Server Problem
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Special Conditions in LP Models
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Special Conditions in LP Models
• A number of special conditions may occur
in LP problems:
–
–
–
–
Alternate Optimal Solutions
Redundant Constraints
Unbounded Solutions
Infeasibility
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Example of Alternate Optimal Solutions
X2
What if the price of Aqua-Spas generates a profit
of $450 instead of $350?
250
objective function level curve
450X1 + 300X2 = 78300
200
MAX: 350X1 + 300X2
S.T.:
1X1 + 1X2 <= 200
150
9X1 + 6X2 <= 1566
12X1 + 16X2 <= 2880
100
X1 >= 0
X2 >= 0
alternate optimal solutions
50
0
0
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Example of a Redundant Constraint
X2
What if 225 pumps are available instead of 200?
1X1 + 1X2 <= 225
250
Pumps
boundary line of tubing constraint
200
boundary line of pump constraint
Redundant constraint
150
boundary line of labor constraint
100
Feasible Region
50
0
0
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X1
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Example of an Unbounded Solution
Consider the following problem:
MAX: Z= X1 + X2
S.T.:1X1 + 1X2 >= 400
-1X1 + 2X2 <= 400
X1 >= 0
X2 >= 0
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Example of an Unbounded Solution
X2
1000
objective function
X1 + X2 = 600
800
-X1 + 2X2 = 400
objective function
X1 + X2 = 800
600
400
200
X1 + X2 = 400
0
0
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X1
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Example of Infeasibility
Consider the following problem:
MAX: Z= X1 + X2
S.T.:1X1 + 1X2 <= 150
1X1 + 1X2 >= 200
X1 >= 0
X2 >= 0
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Example of Infeasibility
X2
250
200
X1 + X2 = 200
feasible region for
second constraint
150
100
feasible region
for first
constraint
50
X1 + X2 = 150
0
0
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PROBLEMS.
Hillier, Frederick S. & Hillier, Mark S.
“Introduction to Management Science”.
2nd. Ed. USA, Mc Graw Hill - Irwin, 2003.
870 pp.
M. En C. Eduardo Bustos Farías
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