3
O N L I N E T U T O R I A L
The Simplex Method of Linear Programming
1. The fundamental purpose of the simplex procedure is to ena-
ble solutions to be found for sets of simultaneous equations in
which the number of variables exceeds the number of equations.
The simplex procedure is:
◼ Identify the pivot column by finding the minimum cj – zj.
2. Differences between graphical and simplex methods:
◼ Graphical can be used only when two variables are in the
model.
◼ Graphical must evaluate all corner points (if corner point
method is used); simplex checks a lesser number of
356 ONLINE TUTORIAL 3 THE SIM PLE X METHOD OF LINEAR PR OGRAM MING
Copyright ©2014 Pearson Education, Inc.
Solution Mix
5
Quantity
5
x2
1
3
x1
0
5
−
0
Solution Mix
5
Quantity
5
x2
1
3
x1
0
5
−
0
12
Profit 3 5 3 2 5 6 6 30 $36
xx
= + = + = + =
To solve this problem using the simplex method, the con-
straint equations must first be rewritten as equalities with the
appropriate slack variables:
x2 + S1 = 6
3x1 + 2x2 + S2 = 18
and then the system of equations entered into the simplex tab-
leau as shown below:
Initial simplex tableau:
cj
Solution Mix
3
5
0
0
Quantity
x1
x2
S1
S2
0
0
1
0
6
0
3
2
1
zj
0
0
0
0
0
0
1
0
6
0
3
2
1
zj
0
0
0
0
Second simplex tableau:
cj
Solution Mix
3
5
0
0
Quantity
x1
x2
S1
S2
5
x2
0
1
1
0
6
0
S2
3
0
−
2
1
6
0
5
5
0
0
5
5
0
cj
−
zj
3
0
−
5
0
3x1 + x2 75
The optimal solution is found at the intersection of the two
constraints:
x1 + 2x2 80
3x1 + x2 75
To solve these equations simultaneously, begin by writing
them in the form shown below:
x1 + 2x2 = 80
3x1 + x2 = 75
Multiply the second equation by
−
2 and add it to the first:
+ = → + =
1 2 1 2
2 80 2 80
x x x x
or:
2
66 33
2
x==
358 ONLINE TUTORIAL 3 THE SIM PLE X METHOD OF LINEAR PR OGRAM MING
x1 + 2x2
−
S1 + A1 = 80
3x1 + x2
−
S2 + A2 = 75
and then the system of equations entered into the simplex
tableau as shown on the next page.
2x1 + x2 + 3x3 + S1 = 120
2x1 + 6x2 + 4x3 + A1 = 240
and then the system of equations entered into the simplex tableau
as shown on the next page.
Copyright ©2014 Pearson Education, Inc.
x1 = 0, x2 = 17.143, x3 = 34.286
and:
Profit = 582 6/7 = $582.86
T3.6 The original equations are:
Objective: 4x1 + 1x2 (minimize)
Subject to: 3x1 + x2 = 3
4x2 + 3x2 6
x1 + 2x2 3
x1, x2 0
3x1 + x2 + A1 = 3
4x1 + 3x2 – S1 + A2 = 6
x1 + 2x2 + S2 = 3
360 ONLINE TUTORIAL 3 THE SIM PLE X METHOD OF LINEAR PR OGRAM MING
and then the system of equations entered into the simplex
tableau as shown below:
356 ONLINE TUTORIAL 3 THE SIM PLE X METHOD OF LINEAR PR OGRAM MING
Copyright ©2014 Pearson Education, Inc.
12
Profit 3 5 3 2 5 6 6 30 $36
xx
= + = + = + =
To solve this problem using the simplex method, the con-
straint equations must first be rewritten as equalities with the
appropriate slack variables:
x2 + S1 = 6
3x1 + 2x2 + S2 = 18
and then the system of equations entered into the simplex tab-
leau as shown below:
Initial simplex tableau:
cj
Solution Mix
3
5
0
0
Quantity
x1
x2
S1
S2
Second simplex tableau:
cj
Solution Mix
3
5
0
0
Quantity
x1
x2
S1
S2
5
x2
0
1
1
0
6
0
S2
3
0
−
2
1
6
cj
−
zj
3
0
−
5
0
3x1 + x2 75
The optimal solution is found at the intersection of the two
constraints:
x1 + 2x2 80
3x1 + x2 75
To solve these equations simultaneously, begin by writing
them in the form shown below:
x1 + 2x2 = 80
3x1 + x2 = 75
Multiply the second equation by
−
2 and add it to the first:
+ = → + =
1 2 1 2
2 80 2 80
x x x x
or:
2
66 33
2
x==
358 ONLINE TUTORIAL 3 THE SIM PLE X METHOD OF LINEAR PR OGRAM MING
x1 + 2x2
−
S1 + A1 = 80
3x1 + x2
−
S2 + A2 = 75
and then the system of equations entered into the simplex
tableau as shown on the next page.
2x1 + x2 + 3x3 + S1 = 120
2x1 + 6x2 + 4x3 + A1 = 240
and then the system of equations entered into the simplex tableau
as shown on the next page.
Copyright ©2014 Pearson Education, Inc.
x1 = 0, x2 = 17.143, x3 = 34.286
and:
Profit = 582 6/7 = $582.86
T3.6 The original equations are:
Objective: 4x1 + 1x2 (minimize)
Subject to: 3x1 + x2 = 3
4x2 + 3x2 6
x1 + 2x2 3
x1, x2 0
3x1 + x2 + A1 = 3
4x1 + 3x2 – S1 + A2 = 6
x1 + 2x2 + S2 = 3
360 ONLINE TUTORIAL 3 THE SIM PLE X METHOD OF LINEAR PR OGRAM MING
and then the system of equations entered into the simplex
tableau as shown below: