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CD Tutorial
3
The Simplex Method
of Linear Programming
Tutorial
Outline
CONVERTING THE CONSTRAINTS TO
EQUATIONS
SETTING UP THE FIRST SIMPLEX TABLEAU
SIMPLEX SOLUTION PROCEDURES
SUMMARY OF SIMPLEX STEPS FOR
MAXIMIZATION PROBLEMS
ARTIFICIAL AND SURPLUS VARIABLES
SOLVING MINIMIZATION PROBLEMS
S
UMMARY
K
EY
T
ERMS
S
OLVED
P
ROBLEM
D
ISCUSSION
Q
UESTIONS
P
ROBLEMS
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CD T
UTORIAL
3T
HE
S
IMPLEX
M
ETHOD OF
L
INEAR
P
ROGRAMMING
Most realworld linear programming problems have more than two variables and thus are too com
plex for graphical solution. A procedure called the
simplex method
may be used to find the optimal
solution to multivariable problems. The simplex method is actually an algorithm (or a set of instruc
tions) with which we examine corner points in a methodical fashion until we arrive at the best solu
tion—highest profit or lowest cost. Computer programs and spreadsheets are available to handle the
simplex calculations for you. But you need to know what is involved behind the scenes in order to
best understand their valuable outputs.
CONVERTING THE CONSTRAINTS TO EQUATIONS
The first step of the simplex method requires that we convert each inequality constraint in an LP for
mulation into an equation. Lessthanorequalto constraints (
≤
) can be converted to equations by
adding
slack variables
, which represent the amount of an unused resource.
We formulate the Shader Electronics Company’s product mix problem as follows, using linear
programming:
Maximize profit = $7
X
1
+ $5
X
2
subject to LP constraints:
where
X
1
equals the number of Walkmans produced and
X
2
equals the number of WatchTVs produced.
To convert these inequality constraints to equalities, we add slack variables
S
1
and
S
2
to the left
side of the inequality. The first constraint becomes
2
X
1
+ 1
X
2
+
S
1
= 100
and the second becomes
4
X
1
+ 3
X
2
+
S
2
= 240
To include all variables in each equation (a requirement of the next simplex step), we add slack vari
ables not appearing in each equation with a coefficient of zero. The equations then appear as
Because slack variables represent unused resources (such as time on a machine or laborhours avail
able), they yield no profit, but we must add them to the objective function with zero profit coeffi
cients. Thus, the objective function becomes
Maximize profit = $7
X
1
+ $5
X
2
+ $0
S
1
+ $0
S
2
SETTING UP THE FIRST SIMPLEX TABLEAU
To simplify handling the equations and objective function in an LP problem, we place all of the
coefficients into a tabular form. We can express the preceding two constraint equations as
S
OLUTION
M
IX
X
1
X
2
S
1
S
2
Q
UANTITY
(RHS)
S
1
2
1
1
0
100
S
2
4
3
0
1
240
The numbers (2, 1, 1, 0) and (4, 3, 0, 1) represent the coefficients of the first equation and second
equation, respectively.
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This note was uploaded on 08/10/2010 for the course MGMT 400 taught by Professor Frankenstein during the Spring '10 term at DeVry Chicago O'Hare.
 Spring '10
 Frankenstein

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