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Math 1b Practical, March 6, 2009
Intoduction to Linear Programming — Part 1
A
linear programming problem
(LP problem) is one that asks for the minimum or
maximum of some linear function (the ‘objective function’) of variables
x
1
,x
2
,...,x
n
over a domain deFned by linear inequalities and equations (linear ‘constraints’) on
x
.
Example A.
Maximize 2
x
1
+
x
2
+
x
4
subject to
3
x
1
+2
x
2
−
x
3
−
2
x
4
=1
1
+
x
3
x
4
=2
1
≥
0
2
≥
0
3
≥
0
4
≥
0
.
Example B.
Minimize
x
+
y
+
z
subject to
x
−
y
z
≥
5
+
y
−
4
z
≥
7
≥
0
,y
≥
0
,z
≥
0
.
Example C.
Minimize
x
+
y
+
z
subject to
x
+
y
≥
2
y
+3
z
≥
11
,
7
x
y
+
z
≥
3
,
2
x
+
y
+
z
≥
3
.
Often the constraints include the requirement that the variables be nonnegative, but
that is not required in general.
Practical examples arise in many areas. Suppose a factory makes products A, B,
and C. Each requires ‘cutting’, ‘folding’, and ‘packaging’. The times, in person-hours,
required for each product to be processed are listed below, along with the total number of
person-hours available in each department, and also the proFt per unit of the products.
ABC
available
cutting
10
5
2
2000
folding
3
9
4
1500
packaging
1
1
2
400
proFt
10
15
20
You are entrusted to maximize proFts. You decide to make
x
units of A,
y
units of B,
and
z
units of C. You might want
x, y, z
to be integers, but let’s ignore that. But you