Note:
this material was developed by Prof. Endre Boros, Director of the
Operations Research Center, RUTCOR, for a guest lecture in 2008.
An example for a linear programming problem
Description
Giapetto’s woodcarving shop manufactures two types of wooden toys:
sol
diers
and
trains
. A
soldier
sells for $27, and costs in materials and labor $24.
A
train
sells for $21 and costs in material and labor $19. The manufacturing
process consists of two main operations:
carpentry
and
finishing
.
A
soldier
requires 2 hours of
finishing
and 1 hour of
carpentry
, while a
train
requires
11 hours of each. Each week Giapetto has ample raw material, but has some
limitations in work force, namely he can only have 80 hours of
carpentry
and
100 hours of
finishing
done.
Furthermore, demand for
trains
is unlimited,
however only 40
soldiers
can be sold, weekly.
How many
soldiers
and
trains
should Giapetto planning on weekly, so
that he
meets these limitations
, and
gains the maximum profit
?
Formulation of a mathematical model
Decision variables
: In order to answer Giapetto’s quastion, we need to tell
him only
two
numbers: the number of
soldiers
and the number of
trains
he
has to produce weekly. Let us denote by
x
1
= the number of soldiers produced each week
x
2
= the number of trains produced each week
We do not know in advance these numbers, but will determine them via
solving a mathematical problem.
Objective function
: Giapetto has a clear objective, which is to increase
his profit as mush as possible. Let us denote his profit by
z
.
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 Fall '09
 Boros
 Operations Research, Linear Programming, Optimization, Simplex algorithm, George Dantzig, Giapetto

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