Public Policy Modeling
Simplex Method
Wednesday, March 16, 2016
Hun Myoung Park, Ph.D.
Public Management & Policy Analysis Program
Graduate School of International Relations
Simplex Method
• Algorithm to solving LP problems.
• Developed by George Dantzig in 1947
• Search for optimal solution by examining corner
point feasible solutions (corner solutions)
• Iteration algorithm keeps searching until finding
the optimal solution.
• Efficient algorithm to reach the optimal solution
by focusing on corner solutions
• Informative to provide valuable information such
as shadow price (sensitivity analysis)
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Augmented Form 1
• Convert the functional inequality constraints to
equivalent equality constraints by introducing
slack variables
Original form
Max Z=3x1+5x2,
Subject to
x1 <= 4
2x2 <= 12
3x1 + 2x2 <= 18
Augmented form
Z-3x1-5x2=0
Subject to
x1 + s 1 = 4
2x2 + s2 = 12
3x1 + 2x2 + s3 = 18
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Augmented Form 2
• The original form has been augmented by some
supplementary variables (slack variables)
• Augmented solution is a solution for the original
variables that has been augmented by the
corresponding values of the slack variables.
• Basic solution is an augmented corner-point
solution and
• Basic feasible (BF) solution is an augmented
corner point feasible (CPF) solution
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Simplex Method 1
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Tabular form is often used
Use augmented form with equality constraints
Construct a simplex tableau
A simplex tableau summarizes coefficients in
the middle and result values on the right side
Objective function Z always appears on the top.
The Z value appear on the right side column
Right side contains the (initial) basic feasible
solution.
See Excel Worksheet for an example.
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Simplex Method 2
• An example of an Initial Simplex Tableau
Basic
Coefficients
RHS
Z
x1
x2
s1
s2
s3
Z
1
-3
-5
0
0
0
0
S1
0
1
0
1
0
0
4
S2
0
0
2
0
1
0
12
S3
0
3
2
0
0
1
18
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Simplex Method 4
• InitializationOptimality TestIteration
Optimality test… Initialization: construct the
initial simplex tableau
– Decision variables remain nonbasic variables (set
equal to zero)
– Slack variables become basic variables that appear
on the left.
• Optimality test: if and only if every coefficient in
row for Z is nonnegative (>=0) current BF
solution is optimal. Otherwise, go to an iteration.
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Simplex Method 5
• Iteration 1: determine the entering basic variable
(to be basic variables in the next iteration) with
the negative coefficient having the largest
absolute value. The column for such variable is
called pivot column.
• Iteration 2: determine the leaving basic variable
(to be nonbasic variables in the next iteration)
using minimum ratio test.
• See pp.103-107 of Hillier & Lieberman (2010)
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Simplex Method 6
• Minimum ratio test:
– Pick out each positive (>0) coefficient in the pivot
column
– Divide right side values by positive coefficients
– Identify the row having the smallest ratio
– The row (basic variable) is leaving basic variable and
is called pivot row.
– The value in pivot column and row is pivot number.
• Minimum ratio test determine which basic
variable drops to zero first as the entering basic
variable is increased.
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Simplex Method 7
• Iteration 3: solve for the new basic feasible
solution by Gauss-Jordan method of elimination
– Divide the pivot row by the pivot number
– Identify a row other than pivot row that has nonzero
coefficient in the pivot column
– Multiply the pivot row by the coefficient and then
subtract the pivot row from the row of interest.
– Repeat it to other row having nonzero coefficient in
the pivot column
• Optimality Test to see if every coefficients in row
Z are nonnegative.
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Further Directions
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Duality Theory: primal versus dual problem
Transportation and Assignment
Integer (Binary) Programming
Goal Programming
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