9.4
Some Characteristics Of Integer Programs—A Sample Problem
287
subject to:
D

J
X
j
=
1
d
i j
x
i j
≥
0
(
i
=
1
,
2
, . . . ,
I
),
J
X
j
=
1
x
i j
=
1
(
i
=
1
,
2
, . . . ,
I
),
I
X
i
=
1
x
i j
≤
y
j
I
(
j
=
1
,
2
, . . . ,
J
),
S
j

I
X
i
=
1
p
i
x
i j
=
0
(
j
=
1
,
2
, . . . ,
J
),
J
X
j
=
1
f
j
(
s
j
)
≤
B
,
y
1
+
y
2

2
y
≥
0
,
y
3
+
y
4
+
2
y
≥
2
,
x
i j
,
y
j
,
y
binary
(
i
=
1
,
2
, . . . ,
I
;
j
=
1
,
2
, . . . ,
J
).
Atthispointwemightreplaceeachfunction
f
j
(
s
j
)
byanintegerprogrammingapproximationtocomplete
the model. Details are left to the reader. Note that if
f
j
(
s
j
)
contains a fixed cost, then new fixedcost variables
need not be introduced—the variable
y
j
serves this purpose.
The last comment, and the way in which the conditional constraint ‘‘
y
j
=
0 implies
x
i j
=
0
(
i
=
1
,
2
, . . . ,
I
)
’’ has been modeled above, indicate that the formulation techniques of Section 9.2 should not
be applied without thought. Rather, they provide a common framework for modeling and should be used in
conjunction with good modeling ‘‘common sense.’’ In general, it is best to introduce as few integer variables
as possible.
9.4
SOME CHARACTERISTICS OF INTEGER PROGRAMS—A SAMPLE PROBLEM
Whereas the simplex method is effective for solving linear programs, there is no single technique for solving
integer programs. Instead, a number of procedures have been developed, and the performance of any particular
technique appears to be highly problemdependent. Methods to date can be classified broadly as following
one of three approaches:
i) enumeration techniques, including the branchandbound procedure;
ii) cuttingplane techniques; and
iii) grouptheoretic techniques.
In addition, several composite procedures have been proposed, which combine techniques using several of
these approaches. In fact, there is a trend in computer systems for integer programming to include a number
of approaches and possibly utilize them all when analyzing a given problem. In the sections to follow, we
shall consider the first two approaches in some detail. At this point, we shall introduce a specific problem
and indicate some features of integer programs. Later we will use this example to illustrate and motivate the
solution procedures. Many characteristics of this example are shared by the integer version of the custom
molder problem presented in Chapter 1.
The problem is to determine
z
*
where:
z
*
=
max
z
=
5
x
1
+
8
x
2
,
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
288
Integer Programming
9.5
subject to:
x
1
+
x
2
≤
6
,
5
x
1
+
9
x
2
≤
45
,
x
1
,
x
2
≥
0
and
integer
.
The feasible region is sketched in Fig. 9.8. Dots in the shaded region are feasible integer points.
Figure 9.8
An integer programming example.
If the integrality restrictions on variables are dropped, the resulting problem is a linear program. We will
call it the
associated linear program
. We may easily determine its optimal solution graphically. Table 9.1
depicts some of the features of the problem.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 SETHURAMAN
 Linear Programming, Optimization, 1 j, 10 percent, 2.5 percent, L j

Click to edit the document details