Worked Examples for Chapter 3
Example for Section 3.1
The Apex Television Company has to decide on the number of 27 and 20inch sets to be
produced at one of its factories. Market research indicates that at most 40 of the 27inch
sets and 10 of the 20inch sets can be sold per month. The maximum number of work
hours available is 500 per month. A 27inch set requires 20 workhours and a 20inch set
requires 10 workhours. Each 27inch set sold produces a profit of $120 and each 20inch
set produces a profit of $80. A wholesaler has agreed to purchase all the television sets
produced if the numbers do not exceed the maxima indicated by the market research.
(a) Formulate a linear programming model for this problem.
The decisions that need to be made are the number of 27inch and 20inch TV sets to be
produced per month by the Apex Television Company. Therefore, the decision variables
for the model are
x
1
= number of 27inch TV sets to be produced per month,
x
2
= number of 20inch TV sets to be produced per month.
Also let
Z = total profit per month.
The model now can be formulated in terms of these variables as follows.
The total profit per month is
Z = 120 x
1
+ 80 x
2
.
The resource constraints are:
(1) Number of 27inch sets sold per month:
x
1
≤
40
(2) Number of 20inch sets sold per month:
x
2
≤
10
(3) Workhours availability: 20 x
1
+ 10 x
2
≤
500.
Nonnegativity constraints on TV sets produced:
x
1
≥
0
x
2
≥
0
With the objective of maximizing the total profit per month, the LP model for this
problem is
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Z = 120 x
1
+
80 x
2
,
subject to
x
1
≤
40
x
2
≤
10
20 x
1
+
10
x
2
≤
500
and
x
1
≥
0,
x
2
≥
0.
(b) Use the graphical method to solve this model.
The constraint,
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 Spring '08
 Jon
 Linear Programming, Optimization, linear programming model, constraint boundary

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