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What Is Optimization?
1.1 Introduction
Optimization, or constrained optimization, or mathematical programming, is a mathematical procedure
for determining optimal allocation of scarce resources. Optimization, and its most popular special
form, Linear Programming (LP), has found practical application in almost all facets of business, from
advertising to production planning. Transportation and aggregate production planning problems are the
most typical objects of LP analysis. The petroleum industry was an early intensive user of LP for
solving fuel blending problems.
It is important for the reader to appreciate at the outset that the “programming” in Mathematical
Programming is of a different flavor than the “programming” in Computer Programming. In the
former case, it means to plan and organize (as in “Get with the program!”). In the latter case, it means
to write instructions for performing calculations. Although aptitude in one suggests aptitude in the
other, training in the one kind of programming has very little direct relevance to the other.
For most optimization problems, one can think of there being
two important classes of objects.
The first of these is
limited resources,
such as land, plant capacity, and sales force size. The second is
activities,
such as “produce low carbon steel,” “produce stainless steel,” and “produce high carbon
steel.”
Each activity consumes
or possibly
contributes
additional amounts of the
resources
. The
problem is to determine the best combination of activity levels that does not use more resources than
are actually available. We can best gain the flavor of LP by using a simple example.
1.2 A Simple Product Mix Problem
The Enginola Television Company produces two types of TV sets, the “Astro” and the “Cosmo”.
There are two production lines, one for each set. The Astro production line has a capacity of 60 sets
per day, whereas the capacity for the Cosmo production line is only 50 sets per day. The labor
requirements for the Astro set is 1 personhour, whereas the Cosmo requires a full 2 personhours of
labor. Presently, there is a maximum of 120 manhours of labor per day that can be assigned to
production of the two types of sets. If the profit contributions are $20 and $30 for each Astro and
Cosmo set, respectively, what should be the daily production?
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Chapter 1
What is Optimization?
A structured, but verbal, description of what we want to do is:
Maximize
Profit contribution
subject to
Astro production lessthanorequalto Astro capacity,
Cosmo production lessthanorequalto Cosmo capacity,
Labor used lessthanorequalto labor availability.
Until there is a significant improvement in artificial intelligence/expert system software, we will
need to be more precise if we wish to get some help in solving our problem. We can be more precise if
we define:
A
= units of Astros to be produced per day,
C
= units of Cosmos to be produced per day.
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 Spring '11
 AHFDKA
 Optimization, optimal solution, dual price, Astros, dual prices

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