To transform this LP to a standard form, we need to the transform the inequality
constraints into equalities by adding slack variables. Thus, the standard LP is
maximize
10
x
11
+ 10
x
12
+ 15
x
21
+ 15
x
22
subject to
x
11
+
x
21
+
s
1
= 200
x
12
+
x
22
+
s
2
= 250
x
11
+
x
21

x
12

x
22
+
s
3
= 5

x
11

x
21
+
x
12
+
x
22
+
s
4
= 5
x
11
,x
12
,x
21
,x
22
,s
1
,s
2
,s
3
,s
4
≥
0
.
Solution to the modiﬁcation of Exercise 5, page 83:
The given problem is not LP, so at ﬁrst, we provide its LP formulation. The ﬁrst
step is to eliminate the objective function given by maximization by introducing a
new variable
y
. The equivalent formulation is
maximize
y
subject to
y
≤ 
x
1

x
2
+ 3
x
3

y
≤  
x
1
+ 3
x
2

x
3

x
1
,x
2
,x
3
≥
0
,
y
is unrestricted
.
Note that
y
is a variable in the preceding problem.
The next step toward LP formulation is to eliminate nonlinear constraints involv
ing absolute value. By doing so, we obtain the following LP formulation:
maximize
y
subject to
x
1

x
2
+ 3
x
3

y
≥
0

x
1
+
x
2

3
x
3

y
≥
0

x
1
+ 3
x
2

x
3

y
≥
0
x
1

3
x
2
+
x
3

y
≥
0
x
1
,x
2
,x
3
≥
0
,
y
is unrestricted
.
Now, we provide the standard form of the preceding LP. To do so, we need to
introduce a slack variable for each of the inequality constraints, and we need to write
y
as
y
=
x
+
4

x

4
with
x
+
4
≥
0 and
x

4
≥
0 (since
y
is unrestricted). Thus, the
standard form of the preceding LP is:
maximize